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--- abstract: | I review several central open questions concerning GRB Jets. -   I discuss a new estimate of the beaming correction for the rate of GRBs $\sim 75\pm 20$. -   I discuss the universal structured jet (USJ) model and conclude that while jets might be structured they are less likely to be universal. -   I discuss recent observations of a sideways expansion of a GRB afterglow and compare these with current hydrodynamics simulations of jet evolution. -   I discuss the implications of resent outliers to the energy-angle relation. author: - Tsvi Piran title: The Beaming Factor and Other Open Issues in GRB Jets --- \[1999/12/01 v1.4c Il Nuovo Cimento\] Introduction ============ The realization that the relativistic outflow in GRBs is in the form of jets has direct implication to their energy budget and their rates. This has, in turn, further implications on the nature of the inner engines. For example prior to the understanding that GRBs are beamed, events such as GRB 990123 with an isotropic equivalent energy of more than $10^{54}$ erg were hard to explain. With beaming the energy output of this event is a“mere" $10^{51}$ erg and it is compatible with a simple stellar mass progenitor. Evidence of jetted GRBs arises from observations [@Harrisonetal99; @Staneketal99] of the predicted achromatic breaks in the afterglow light curves [@Rhoads99; @SPH99]. Further support is given by the comparison of long term radio calorimetry with the energy of the prompt emission [@Waxmanetal98]. The time of the jet break provides an estimate of the jet angle [@SPH99]: $$\theta =0.16 {\rm rad }(n/E_{k,iso,52})^{1/8} t_{b,days}^{3/8} = 0.07 {\rm rad } (n/E_{k,\theta,52})^{1/6} t_{b,days}^{1/2}, \label{eq:time}$$ where $t_{b,days}$ is the break time in days and $E_{k,iso,52}$ is the “isotropic equivalent" kinetic energy in units of $10^{52}$ergs, while $E_{k,\theta,52}$ is the real kinetic energy in the jet i.e: $E_{k,\theta,52}=(\theta^2/2) E_{k,iso,52}$. Frail et al. [@Frail01] and Panaitescu and Kumar [@PanaitescuK01] have estimated the opening angles $\theta$ for several GRBs with known redshifts. They find that the total gamma-ray energy release, when corrected using a beaming factor, $f_b$ $$E_\gamma = f_b E_{\gamma,iso} \equiv {\theta^2 \over 2 } E_{\gamma,iso} ,$$ is clustered. This energy-angle relation is commonly called the Frail relation. A precursor of this discovery was found already in 1999 by Sari, Piran & Halprn [@SPH99] who found that the two brightest bursts known at that time were beamed. Bloom, Frail & Kulkarni [@BloomFrailKulkarni03] confirmed this clustering around $\sim 1.3\times 10^{51}$ erg on a larger sample. This result is remarkable as it involves two seemingly unrelated quantities, $\theta$ that is determined from the break in the afterglow light curve and $E_{\gamma,iso}$ which is a property of the prompt emission. The fact that the product of these two unrelated quantities is a constant is, in my mind, an indication that our overall model is correct. There are two leading models for the jet structure and for the interpretation of the jet break and the beaming angle. According to the original uniform jet model (UJ) the energy per solid angle is roughly constant within some finite opening angle, $\theta$, and it sharply drops outside $\theta$. Within the UJ model, the observed break corresponds to the jet opening angle, $\theta$ (see Eq. \[eq:time\]). The Frail relation implies, here, that the total energy released in GRBs is constant and that the differences in the isotropic equivalent energies arise from variations in the opening angles. According to the alternative universal structured jet (USJ) model [@Lipunov_Postnov_Pro01; @Rossi02; @Zhang02] all GRB jets are intrinsically identical. The energy per solid angle varies as a function of the angle from the jet axis. The jet break corresponds to the viewing angle of the observer and the Frail relation imposes a specific distribution of energy per unit angle in the jet: $$\label{eq:usj} {\cal E}(\theta)=\left\{% \begin{array}{ll} E_0/ (\pi \theta^2) , & \hbox{for $\theta>\theta_c$;} \\ E_0/(\pi \theta_c^2), & \hbox{for $\theta<\theta_c$,} \\ \end{array}% \right.$$ where $E_0$ is a constant and $\theta_{c}$ is the core angle of the jet [@Rossi02]. While the UJ model contains a free function, the luminosity function or the corresponding opening angle distribution, the USJ model is complectly determined by the Frail relation. Its apparent luminosity function has the form $\Phi(L)\propto L^{-2}$ [@Pernaetal03]. We can use this lack of freedom within the USJ model to test it. The question which model is the correct one is still an open one. There are many other open questions concerning the structure of the jets and their evolution. In this talk I review some of these open questions. I refer the reader to a recent review [@p04] for a more extended overview on GRBs in general and on GRB jets in particular. The Beaming and the Rate of GRBs ================================ The overall GRB rate depends clearly on the amount that GRBs are beamed. Within the UJ model this has been measured traditionally in terms of the beaming correction factor, $f_b^{-1}$, which is defined as the ratio of total number of bursts to the observed ones. To estimate the overall GRB rate we need the average beaming correction $\langle f_b^{-1} \rangle $ such that $n_{true}= \langle f_b^{-1} \rangle n_{obs}$. The average is performed over the observed distribution. Taking into account the fact that for every observed burst there are $f_b^{-1} $ that are not observed, Frail et al., [@Frail01] estimated the average beaming correction[^1] $$\langle f_b^{-1} \rangle_{F01} = {1\over N} \sum_i {2 \over \theta_i^{2}} \simeq 520\pm 85 ,$$ where the sum is over the observed distribution. However, this calculation overestimates the actual beaming correction. In the [*intrinsic*]{} luminosity distribution there are many low luminosity bursts that have large opening angles. These bursts dominate the rate estimate. However, they are rather weak and can be observed only to small distances. Hence they are under-represented in the observed distribution [@gpw05]. This effect can be taken into account in the following way. For a given burst with a luminosity $L$ we define the volume from which such a burst can be detected: $$V_L \equiv \int_0^{z(L)} dz\,(dV/dz)R_{GRB}(z)/[R_{GRB}(0)(1+z)],$$ where $R_{GRB}(z)$ is the comoving rate of GRBs and $z_m(L)$ is the maximal redshift from which a burst with a luminosity L can be detected. Similarly $V_\infty=V_{L=\infty}$ is the whole effective volume of the observable universe. The [*intrinsic*]{} beaming correction can be written as: $$\langle f_b^{-1} \rangle\equiv {\sum_i (2 \theta_i^{-2}) (V_\infty/V_{L_i}) \over \sum_i (V_\infty/V_{L_i})} .$$ This estimate is, of course, somewhat model dependent as it requires an assumption on $R_{GRB}$. Guetta, Piran & Waxman [@gpw05] find that for several models in which GRBs follow the SFR $\langle f_b^{-1} \rangle_{int} =75 \pm 25$, about a factor of 8 smaller than the previous estimate of $520\pm 85$ that did not take this effect into account. Following [@gpw05] we can also estimate the beaming correction for the rate of GRBs within the USJ model in the following way. The total flux of GRBs per year (or per any other unit of time) is an observed quantity obtained by summing over the observed distribution. A comparison of this total flux with the total energy emitted by a single burst can tell us directly the total number of bursts. Integrating over the energy distribution (Eq. \[eq:usj\]) we obtain the total energy that a burst with a USJ emits: $$E_{\rm USJ} = 2 [\int_0^{\theta_c}(E_0/\theta_c^2)^2 \theta d\theta + \int_{\theta_{c}}^{\theta_{\rm max}} E_0 \theta^{-1} d\theta] = E_0 [1+2 \log ( \theta_{\rm max}/\theta_c) ],$$ where $\theta_{max}$ is the maximal angle to which the jet extends. This immediately implies that the ratio of UJs (emitting each $E_0$) to USJs (emitting each $E_0 [1+2 \log ( \theta_{\rm max}/\theta_c) ]$) required to explain the observed flux is $$N_{\rm UJ} /N_{\rm USJ} = [1+2 \log ( \theta_{\rm max}/\theta_c) ].$$ The upper and lower limits of this integral are uncertain but the logarithmic dependance implies that the factor cannot be smaller than 2 or much larger than 5. This implies that the number of USJs required to produce the observed flux is about factor of 4 below the corresponding number of UJs. Hence, the average beaming correction for USJs is $\sim 20\pm 10$. Universal Structured Jets ========================= One of the intriguing open questions concerning GRB jets is their angular distribution. The two leading models are the UJ and USJ discussed earlier. The differences between USJ and UJ have crucial implications to the question of the nature of GRBs’ inner engines and their progenitors. First, the universality of the USJ requires that more or less the same process operates with the same parameters within different GRBs. Second, a USJ carries roughly five times more energy than a UJ. This implies, for a USJ, an energy budget of $\sim 10^{52}$ erg. The rate of USJs is, correspondingly, smaller by a factor of five. It is therefore, important to ask whether there are observations that can distinguish between the two models. ![[**(a)**]{}: The 2D distribution density, $dn(z,\ln\theta)/dz d\ln\theta$, of the GRB rate as a function of $z$ and $\ln\theta$ in the USJ model. The white contour lines confine the minimal area that contains $1\;\sigma$ of the total probability. The circles denote $16$ bursts with known $z$ and $\theta$ [@BloomFrailKulkarni03]. [**(b)**]{}: A limited redshift range, $0.8<z<1.7$ (containing $10$ out of the $16$ data points) in which both redshift selections effects and the sensitivity to the unknown GRB rate are minimized. From [@ngg04].](fig1){width="10cm" height="10cm"} \[fig:1\] Perna, Sari & Frail [@Pernaetal03] calculated the expected distribution of the observed opening angles within the USJ model assuming that GRBs follow the SFR. Following them we define the number of bursts in the interval $(\theta, \theta+d\theta)$ and $(z, z+dz)$ as: $${dn\over d\theta dz} d\theta dz = sin \theta { R_{GRB}(z) \over (1+z)} { dV(z) \over dz} {\cal T}(\theta,z) \label{eq:dndz}$$ where, $V(z)$ is the comoving volume element and ${\cal T}(\theta,z)$ depends on the distribution of GRB duration (see [@Pernaetal03; @ngg04] for details). Perna et al., [@Pernaetal03] integrated over the redshift distribution and found, with reasonable assumptions on ${\cal T}$ a remarkable agreement between the expected angular distribution $dn(\theta)/d\theta \equiv \int(dn/d \theta dz) dz$ and the observed angular distribution, lending a strong support to the USJ model. However, Nakar, Granot & Guetta [@ngg04] compared of the two dimensional distribution $dn/d\theta dz$ with the observed one. They found (see Fig. \[fig:1\]) that the two distribution disagree strongly. The observed points are very far from the location of the peak of the expected distributions. Some are even in a non-allowed region. This implies that the agreement between the observed angle distribution and the one predicted by the USJ model was just a coincidence and should not be taken as supporting this model. A similar discrepancy arose when Guetta et al., [@gpw05] compared the observed count ($C_{max}/C_{min}$) distribution for the BATSE long bursts sample with the one expected from the USJ model. Guetta et al., [@gpw05] found that the USJ model predicts a significant excess of weak bursts as compared with the observed distribution. Selection effects and uncertainties mean that these discrepancies are insufficient to rule out the USJ model. Still these discrepancies certainly imply that the agreement found for the angular distribution alone cannot be used, as hoped, to demonstrate the validity of this model. If it will turn out that these conclusions hold with additional data, we will have to reconsider the validity of the USJ. Both discrepancies could be removed if we consider Structured Jets that are not universal. That is if we allow an angular dependence of the flow parameters (such as $\Gamma$ and/or $E$) but we do not require that all jets are similar. Namely we could replace USJ with SJ. While such a solution is viable it clearly takes away some of the simplicity and the elegance of the USJ mode. Jet Evolution ============= ![[*upper panel*]{}: $R_\bot$ as a function of $T_d$ for different sideways expansion models in ISM. The energy to external density ratio $E/n=0.6\cdot 10^{51}$ erg  cm$^3$ and $\theta_0=0.06 rad$. [*Lower panel*]{}: $R_\bot$ as a function of $T_d$ for different opening angles in ISM, with a constant $E/n=0.6\cdot 10^{51}$ erg cm$^3$. $T_j$ is in days, $\beta_s=\beta_{thermal}$. From [@onp04] []{data-label="fig:3"}](fig3.eps){width="10cm" height="10cm"} An important question that determines the observed light curve of the afterglow is the sideways expansion of the jet after $\Gamma \sim \theta^{-1}$, where $\Gamma$ is the Lorentz factor. The observations of the radio afterglow of GRB 030329 provided a unique opportunity to test this issue. Taylor et al., [@tfbk04; @tmp+04] have measured the size of the radio afterglow of GRB 030329 between 20 and 300 days after the burst. Oren, Nakar & Piran [@onp04] (see also [@grl05]) compared the observed sizes with several schematic models of spherical and jetted propagation. The remarkable results is that the apparent size of the afterglow is rather insensitive to the details of the model. This robustness is a good indication for the validity of the model, as it is difficult to force it to have different values and the values obtained fit the observations. On the other hand it is a drawback when one wishes to use the size to determine the parameters of the outflow. It is insensitive to these parameters. Oren et al., [@onp04] find that the image of a spherically expanding fireball is largest with expansion as $t^{0.6}$, while the size of a sideways expanding jet (at $v\approx c$) increases as $t^{0.5}$. As can be seen in Fig. \[fig:3\] there is practically no difference between an expansion at the speed of light or at the sound speed, $c\sqrt{3}$. On the other hand the size of a non-expanding jet increases only as $t^{0.25}$. A comparison of these models to the observations (see Fig. \[fig:3\]) shows that the non-expanding jet model is inconsistent with the observations. While an addition of the faster expansion during the Newtonian phase [@grl05] can somewhat alleviate the problem, it is not clear that this can be done with reasonable parameters [@onp04]. Based on simple analytic model Sari Piran & Halpern [@SPH99] estimated that the relativistic jet will expand sideways almost at the speed of light. On the other hand Panaitescu & Meszaros [@PanaitescuMeszaros99] estimated that the jet will expand only at the sound speed, $c/\sqrt{3}$. Both are consistent with the observations. More recently Kumar & Granot [@KumarGranot03] integrated the hydrodynamic equations over the radial direction and obtained one dimensional simplified hydrodynamics equations. Solving these equations they find no or little sideways expansion. Similar results were obtained in a two dimensional numerical integration of the full hydrodynamic equations [@Granot01; @Cannizzoetal04]. Remarkably this does not influence much the observed light curve (see [@Granot01]). We are left with a puzzle why do the numerical simulation indicate little or no expansion while the observations suggest a rather rapid sideways expansion. One may guess that the current computations are not refined enough to follow the jet evolution (see [@pg01] for a detailed discussion of the problematics of these computations). We probably have to wait for a higher resolution codes to resolve this problem. Another possibility is that the size of the radio afterglow does not trace well the size of the expanding jet. Detailed emission computations have to be carried out to determine this possibility. The Implication of Outliers to the Frail Relation and a Speculation =================================================================== We begun stressing the importance of the Frail (energy-angle) relation. However, Berger et al., [@Bergeretal03_030329] pointed out that in addition to the well known weak GRB 980425 there are three other outliers to this relation GRBs 980326, 980519 and 030329 . Intriguingly enough all outliers are weaker relative to the common value of $\sim 10^{51}$ erg. I would like to suggest a simple explanation to this phenomenon. It seems that the common value indicates indeed the available energy reservoir. Under optimal conditions a significant fraction of this energy is released as $\gamma$-rays. An intriguing question, incidentally, is how is this conversion so efficient? These efficient cases produce the brightest and easiest to detect bursts. The less efficient case are less powerful (in $\gamma$-rays) and hence are easily missed. GRB 980425 would not have been discovered if it was not so close. Another outlier, GRB 030329, is also nearby at z=0.168. The redshift of the other two outliers is unknown. Thus, I conclude with a speculation that as time progresses and more energies, redshifts and jet angles will become available it will turn out that the Frail relationship is satisfied as an inequality with $\sim 10^{51}$ erg being the upper limit to the $\gamma$-ray energy. This work was supported by EU-RTN “GRBs Enigma and a Tool” and by US-Israel BSF. I thank J. Granot, D. Guetta, E. Nakar and E. Waxman for helpful discussions. [10]{} . . . . . . . . . . . . . . . in , (2001) pp.  312–+. . . , [12483]{}[2004]{}. . . in , (2001) pp. 300+. . [^1]: Frail et al., [@Frail01] estimate the observed beaming factor distribution as $p_{obs}(f_b) = (f_b/f_0)^{\alpha+1}$ and $(f_b/f_0)^{\beta+1}$ for $f_b\lessgtr f_0$ respectively. They find that $\alpha$ is poorly constraint by the data while $\beta =-2.77_{-0.3}^{+0.24}$ and $log(f_0)= -2.91_{-0.06}^{+0.07}$ and obtain $\langle f_b^{-1} \rangle_{F01}$ by integrating over this distribution.
--- abstract: 'In order to reduce backgrounds from radon-daughter plate-out onto detector surfaces, an ultra-low-radon cleanroom is being commissioned at the South Dakota School of Mines and Technology. An improved vacuum-swing-adsorption radon mitigation system and cleanroom build upon a previous design implemented at Syracuse University that achieved radon levels of $\sim$0.2Bqm$^{-3}$. This improved system will employ a better pump and larger carbon beds feeding a redesigned cleanroom with an internal HVAC unit and aged water for humidification. With the rebuilt (original) radon mitigation system, the new low-radon cleanroom has already achieved a $>$300$\times$ reduction from an input activity of $58.6\pm0.7$Bqm$^{-3}$ to a cleanroom activity of $0.13\pm0.06$Bqm$^{-3}$.' author: - 'J. Street' - 'R. Bunker' - 'C. Dunagan' - 'X. Loose' - 'R.W. Schnee' - 'M. Stark' - 'K. Sundarnath' - 'D. Tronstad' title: 'Construction and Measurements of an Improved Vacuum-Swing-Adsorption Radon-Mitigation System' --- INTRODUCTION TO RADON MITIGATION ================================ A potential source of dominant backgrounds for many rare-event searches or screening detectors is from radon daughter plate-out [@LRT2013simgen]. Backgrounds from $^{210}$Pb and the recoiling $^{206}$Pb nucleus from the $\alpha$ decay of $^{210}$Po are the dominant low-energy backgrounds for XMASS [@LRT2015xmass], SuperCDMS Soudan [@supercdms_lowmass2014] and EDELWEISS [@LRT2013edelweiss], and are expected to be dominant for SuperCDMS SNOLAB and the BetaCage screener [@LRT2013bunker; @LRT2015schnee] without improvements. Radon daughters on surfaces may dominate for SuperNEMO [@LRT2013SuperNEMO] and CUORE [@LRT2015cuore]. Both neutrons from ($\alpha,n$) reactions and $^{206}$Pb recoils are important for LZ [@lux2014backgrounds], XENON1T, and DArKSIDE. Storing and assembling detector components may be possible using vacuum glove boxes or by cleaning components upon commissioning ( [@LRT2013schneeEP]). However, when the components are large or require delicate assembly, a vacuum glove box can be impractical. Cleaning can also be impractical when the components are delicate or have complex geometries. To create a radon-mitigated, breathable-air environment one may consider two system classes: continuous flow through a filtration column and swing flow through two or more filtration columns. The filtration columns are usually filled with activated carbon. The continuous flow system ( [@nemoLRT2006]) operates on the basis that some considerable fraction of radon decays before exiting the column. For an ideal column, the final radon concentration $C_\text{final}=C_\text{initial}\exp{\left(-t/t_\text{Rn}\right)}$, where $C_\text{initial}$ is the radon concentration of the input air, $t$ is the characteristic break-through time of the filter, and $t_\text{Rn}=5.38$days is the mean lifetime of radon. To increase the break-through time, and therefore make a continuous flow system practical, one must cool the carbon to reduce desorption of radon. Continuous flow systems are relatively simple and robust, are commercially available, and typically achieve reduction factors of $\sim$1000$\times$, to $\sim$10–30mBqm$^{-3}$. In a swing flow system, two or more filtration columns are used. While filtering through one column the other is regenerated using either low pressure or high temperatures to allow radon to desorb efficiently and be exhausted. For a vacuum-swing-adsorption (VSA) system ( [@LRT2004Pocar; @PocarThesis; @LRT2013schneeVSA]), high-radon input air is filtered through the first column while the second column is pumped down to $\sim$1Torr. Well before the break-through time, the path of the high-radon input air is switched so that it flows through the second column instead, allowing the first to regenerate. For an ideal column, no radon reaches the output. Swing flow systems are more complicated both in operation and analysis. A VSA system ($e.g.$ Fig. \[VSAdiagram\]) can potentially outperform a continuous flow system at a lower cost. Temperature-swing systems ( [@LRT2010HallinRadon]) should provide best performance but at the highest cost and complexity. We use the VSA technique because of its lower cost and potential for excellent radon reduction. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- (a)![Schematic diagram $(a)$ and photo $(b)$ of the SDSM&T VSA system. High-radon air is brought in either from outdoors ($\sim$10Bq m$^{-3}$) or from the building ($\sim$100Bq m$^{-3}$). The air is dried and passed through one of the two carbon columns (open circles denote open valves, while $\otimes$ denotes a closed valve). The radon in the air adsorbs preferentially to the carbon (compared to N$_2$ or O$_2$). The radon slowly migrates through the column, adsorbing and desorbing, while low-radon air flows to the cleanroom. A small amount of this low-radon air flows back into the other carbon column, while it is pumped to a few Torr, helping radon to desorb from the carbon and be exhausted. Before any radon breaks through the left column, the system swings and the cycle repeats. []{data-label="VSAdiagram"}](VSA_diagram2015e.png "fig:"){width="0.49\linewidth"} (b)![Schematic diagram $(a)$ and photo $(b)$ of the SDSM&T VSA system. High-radon air is brought in either from outdoors ($\sim$10Bq m$^{-3}$) or from the building ($\sim$100Bq m$^{-3}$). The air is dried and passed through one of the two carbon columns (open circles denote open valves, while $\otimes$ denotes a closed valve). The radon in the air adsorbs preferentially to the carbon (compared to N$_2$ or O$_2$). The radon slowly migrates through the column, adsorbing and desorbing, while low-radon air flows to the cleanroom. A small amount of this low-radon air flows back into the other carbon column, while it is pumped to a few Torr, helping radon to desorb from the carbon and be exhausted. Before any radon breaks through the left column, the system swings and the cycle repeats. []{data-label="VSAdiagram"}](VSA_image2015.png "fig:"){width="45.00000%"} ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- THE SDSM&T RADON-MITIGATION SYSTEM AND CLEANROOM ================================================ The system built at the South Dakota School of Mines & Technology (SDSM&T) consists of two parts: a VSA radon filter and a low-radon cleanroom. Figure \[VSAdiagram\] shows the VSA rebuilt as it was at Syracuse University [@LRT2013schneeVSA] (based on the design in [@LRT2004Pocar; @PocarThesis]) with a few modifications due to the geometry of its new living space. The system has a blower that takes in air from either outdoors or inside the lab at a 100cfm capacity. The input air is dried with a dehumidifier and passes through chillers to maintain air temperature. Each column is filled with 125kg activated carbon. With the use of a booster to accelerate pumping speed at low pressure, a column being purged is evacuated to $<$2.5Torr in 7min. High-radon air is filtered through one column for 40min while the other column is regenerated, giving a full swing-cycle period of 80min. The low-radon cleanroom is an evolution of the Syracuse University cleanroom design. The cleanroom was built with materials having sufficiently low emanation and permeation of radon. The area of the cleanroom is 21$\times$9.25ft$^2$ with an additional 5$\times$9.25ft$^2$ of internal anteroom space and a ceiling height of 8ft. Recent measurements indicate that the cleanroom is class 100 when empty. The cleanroom was constructed primarily of aluminum. All seams were sealed with butyl rubber and aluminum tape making the cleanroom very leak-tight and easily over-pressured. Although the VSA system can provide low-radon air at 100cfm, the cleanroom can be over-pressured effectively (to $\sim$0.25 inches of water) with only 15cfm of make-up air. The windows are made of $\geq$ 1/8 inch polycarbonate, contributing $\leq8$mBqm$^{-3}$ of $^{222}$Rn activity to the cleanroom. If diffusion through the windows becomes the dominant source of radon (following the upgrades described below), they will be covered with metal and/or thickened. Low-radon air is delivered to the cleanroom from the VSA system via galvanized-steel, 22-gauge spiral ducting. A notable feature of this cleanroom is the internal placement of the HVAC. During operation, the HVAC circulates low-radon air through HEPA filters and then back into the cleanroom at $\sim$1000cfm. A negative pressure region forms behind the blower inside the HVAC. Placing the HVAC outside the cleanroom presents the difficulty of sealing it sufficiently to prevent high-radon air from leaking into the circulation path. Leaks of radon into the HVAC limited the system at Syracuse. Situating the HVAC inside the cleanroom mitigates this issue. The VSA system and cleanroom, commissioned at SDSM&T, recently achieved a radon reduction of $>$300$\times$ to $0.13\pm0.06$Bqm$^{-3}$, as shown in Fig. \[VSAresults\]a. About 100cfm of air was drawn from the lab and conditioned to a temperature of $\sim$16C and dew point of $-$17C. A fraction of this air was used as input to the VSA system, resulting in 20cfm of low-radon make-air for the cleanroom. This result is consistent with the performance of the system while at Syracuse but has the added benefit (because the input air has higher activity than at Syracuse) of demonstrating the greater activity reduction of which this system is capable. The SDSM&T system indicates that the VSA technique is a viable low-cost alternative to continuous flow systems. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- (a)![$(a)$ Radon activity of the input air (higher, orange error bars with 7-day average) compared to that of the air in the cleanroom (lower, blue error bars with 5-day average), showing a $>$300$\times$ reduction in activity from $58.6\pm0.7$Bqm$^{-3}$ to $0.13\pm0.06$Bqm$^{-3}$. $(b)$ Expected contributions from the initial water (blue dashes), from diffusion (green $+$’s), and from emanation (purple dots) to the total radon activity in the water aging tank (black solid). Radon emanation is negligible compared to radon diffusion, which limits this system for aging times longer than about 40 days. The corresponding contribution to the radon activity in the cleanroom (red circles) is diluted by over 5 orders of magnitude. []{data-label="VSAresults"}](radon_reduction.png "fig:"){width="48.00000%"} (b)![$(a)$ Radon activity of the input air (higher, orange error bars with 7-day average) compared to that of the air in the cleanroom (lower, blue error bars with 5-day average), showing a $>$300$\times$ reduction in activity from $58.6\pm0.7$Bqm$^{-3}$ to $0.13\pm0.06$Bqm$^{-3}$. $(b)$ Expected contributions from the initial water (blue dashes), from diffusion (green $+$’s), and from emanation (purple dots) to the total radon activity in the water aging tank (black solid). Radon emanation is negligible compared to radon diffusion, which limits this system for aging times longer than about 40 days. The corresponding contribution to the radon activity in the cleanroom (red circles) is diluted by over 5 orders of magnitude. []{data-label="VSAresults"}](Humidification_activityd.png "fig:"){width="48.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Low-Radon Humidification System and Other Planned Upgrades ----------------------------------------------------------- The VSA radon mitigation system tends to reduce relative humidity of the cleanroom air to $\sim$5%. Using water for humidification without reducing its activity would introduce radon into the cleanroom. To reduce water activity, we will age the water until a sufficient fraction of radon has decayed away. The water to be used is expected to have an activity of $\sim$1000Bqm$^{-3}$. If no aging is done and the water is used for increasing the relative humidity in the cleanroom, the activity contribution to the cleanroom would be $\sim$6mBqm$^{-3}$ because the vapor density of water at 35% relative humidity is $\sim$$6\times10^{-6}$gcm$^{-3}$. As shown in Fig. \[VSAresults\]b, after aging, the activity will be reduced by 100$\times$ and will contribute only 100’s of $\upmu$Bqm$^{-3}$. The low-radon humidification system consists of two large (110gal) tanks made of low-density polyethylene (LDPE). Water from a local reverse osmosis system fills the first tank. The water is aged until its activity is sufficiently low (about 10Bqm$^{-3}$) and then transferred to the second tank. Upon transferring, the first tank is again filled and the aging process repeats. The second tank supplies low-radon water to the cleanroom humidification system at a rate of about 3galday$^{-1}$. The system is designed such that the aged water in the second tank lasts longer than the aging time. In addition, an in-line filter will prevent any particulates from entering the cleanroom. Sufficient aging time is determined by the intrinsic activity of the water, with consideration of radon emanation from the LDPE tank, and radon diffusion through the tank walls, as shown in Fig. \[VSAresults\]b. Aging for 35days is sufficient to contribute only 100’s of $\upmu$Bqm$^{-3}$ to the cleanroom. Further aging makes little difference as the total activity asymptotically approaches the contribution from diffusion, which limits this system. If a lower activity is needed in the future, switching to metal tanks would reduce the contribution from humidification to the order of 10nBqm$^{-3}$, because LDPE has a diffusion coefficient on order of 10$^{-11}$m$^2$s$^{-1}$ while metal is $\sim$10$^{-15}$m$^2$s$^{-1}$ [@RadonDiffusionJiranek]. Other radon-mitigation upgrades that have yet to be installed are taller carbon columns (providing an additional 175kg activated carbon each) and a more powerful roughing pump, which should allow easier maintenance as well as providing a small improvement in performance. In principle, the increased height of the carbon columns should provide additional reduction in radon concentration of another factor $\sim$100–1000$\times$, potentially providing world-leading radon reduction at a modest cost. ACKNOWLEDGMENTS =============== This work was supported in part by the National Science Foundation (Grants No. PHY-1205898, PHY-1506633, and PHY-1546843), the Department of Energy (Grant No. DE-AC02-05CH1123), and the state of South Dakota. [15]{} natexlab\#1[\#1]{}\[1\][“\#1”]{} url \#1[`#1`]{}urlprefix\[2\]\[\][[\#2](#2)]{} H. [Simgen]{}, “[Radon assay and purification techniques]{},” in *Topical Workshop on Low Radioactivity Techniques: LRT 2013*, edited by L. [Miramonti]{}, and L. [Pandola]{}, 2013, vol. 1549 of *American Institute of Physics Conference Series*, pp. 102–107. K. [Kobayashi]{}, “[Surface purity control during XMASS detector refurbishment]{},” in *Topical Workshop on Low Radioactivity Techniques: LRT 2015*, edited by J. L. [Orrell]{}, American Institute of Physics Conference Series, 2015, [https://www.npl.washington.edu/indico/contributionDisplay.py?contribId=24&confId=5]{}. R. [Agnese]{} [*et al.*]{} (SuperCDMS Collaboration), *Physical Review Letters* **112**, 241302 (2014), . X.-F. [Navick]{}, and [EDELWEISS Collaboration]{}, “[Background suppression in the Edelweiss-III experiment]{},” in *Topical Workshop on Low Radioactivity Techniques: LRT 2013*, edited by L. [Miramonti]{}, and L. [Pandola]{}, 2013, vol. 1549 of *American Institute of Physics Conference Series*, pp. 148–151. R. [Bunker]{}, Z. [Ahmed]{}, M. A. [Bowles]{}, S. R. [Golwala]{}, D. R. [Grant]{}, M. [Kos]{}, R. H. [Nelson]{}, R. W. [Schnee]{}, A. [Rider]{}, B. [Wang]{}, and A. [Zahn]{}, “[The BetaCage, an ultra-sensitive screener for surface contamination]{},” in *Topical Workshop on Low Radioactivity Techniques: LRT 2013*, edited by L. [Miramonti]{}, and L. [Pandola]{}, 2013, vol. 1549 of *American Institute of Physics Conference Series*, pp. 132–135. R. W. [Schnee]{}, [The BetaCage, an Ultra-sensitive Screener for Surface Contamination]{}, [Talk at the Topical Workshop on Low Radioactivity Techniques: LRT 2015]{} (2015), [https://www.npl.washington.edu/indico/contributionDisplay.py?contribId=48&confId=5]{}. J. [Mott]{}, and [SuperNEMO Collaboration]{}, “[Low-background tracker development for SuperNEMO]{},” in *Topical Workshop on Low Radioactivity Techniques: LRT 2013*, edited by L. [Miramonti]{}, and L. [Pandola]{}, 2013, vol. 1549 of *American Institute of Physics Conference Series*, pp. 152–155. B. [Wang]{}, “[Background analysis techniques in the CUORE experiment]{},” in *Topical Workshop on Low Radioactivity Techniques: LRT 2015*, edited by J. L. [Orrell]{}, American Institute of Physics Conference Series, 2015, [https://www.npl.washington.edu/indico/contributionDisplay.py?contribId=82&confId=5]{}. D. S. [Akerib]{} [*et al.*]{} (LUX Collaboration), *Astroparticle Physics* **62**, 33–46 (2014), . R. W. [Schnee]{}, M. A. [Bowles]{}, R. [Bunker]{}, K. [McCabe]{}, J. [White]{}, P. [Cushman]{}, M. [Pepin]{}, and V. E. [Guiseppe]{}, “[Removal of long-lived $^{222}$Rn daughters by electropolishing thin layers of stainless steel]{},” in *Topical Workshop on Low Radioactivity Techniques: LRT 201*, edited by L. [Miramonti]{}, and L. [Pandola]{}, 2013, vol. 1549 of *American Institute of Physics Conference Series*, pp. 128–131. A. [Nachab]{}, “[Radon reduction and radon monitoring in the NEMO experiment]{},” in *AIP Conf. Proc. 897: Topical Workshop on Low Radioactivity Techniques: LRT 2006*, edited by P. [Loaiza]{}, American Institute of Physics, Melville, NY, 2007, pp. 35–39. A. [Pocar]{}, “[Low background techniques for the Borexino nylon vessels]{},” in *Topical Workshop on Low Radioactivity Techniques: LRT 2004.*, edited by B. [Cleveland]{}, R. [Ford]{}, and M. [Chen]{}, 2005, vol. 785 of *American Institute of Physics Conference Series*, pp. 153–162, . A. [Pocar]{}, *[Low Background Techniques and Experimental Challenges for Borexino and its Nylon Vessels]{}*, Ph.D. thesis, Princeton University (2003). R. W. [Schnee]{}, R. [Bunker]{}, G. [Ghulam]{}, D. [Jardin]{}, M. [Kos]{}, and A. S. [Tenney]{}, “[Construction and measurements of a vacuum-swing-adsorption radon-mitigation system]{},” in *Topical Workshop on Low Radioactivity Techniques: LRT 2013*, edited by L. [Miramonti]{}, and L. [Pandola]{}, 2013, vol. 1549 of *American Institute of Physics Conference Series*, pp. 116–119. D. [Grant]{}, A. [Hallin]{}, S. [Hanchurak]{}, C. [Krauss]{}, S. [Liu]{}, and R. [Soluk]{}, “[Low Radon Cleanroom at the University of Alberta]{},” in *Topical Workshop on Low Radioactivity Techniques: LRT 2010*, edited by R. [Ford]{}, 2011, vol. 1338 of *American Institute of Physics Conference Series*, pp. 161–163. M. [Jiranek]{}, and M. [Kotrbata]{} *Radiation Protection Dosimetry* **145**, 178–183 (2011).
--- abstract: 'The article presents an improved theoretical description of dissociative recombination of HCO$^+$ and DCO$^+$ ions with a low energy electron. In the previous theoretical study (Phys. Rev. A [**74**]{}, 032707) on HCO$^+$, the vibrational motion along the CO coordinate was neglected. Here, all vibrational degrees of freedom, including the CO stretch coordinate, are taken into account. The theoretical dissociative recombination cross-section obtained is similar to the previous theoretical result at low collision energies ($<$0.1 eV) but somewhat larger at higher ($>$0.1 eV) energies. Therefore, the present study suggests that motion along the CO coordinate does not play a significant role in the process at low collision energies. The theoretical cross-section is still approximately 2-3 times lower than the data from a recent merged-beam experiment.' author: - 'Nicolas Douguet, Viatcheslav Kokoouline' - 'Chris H. Greene' title: 'Theoretical rate of dissociative recombination of HCO$^{+}$ and DCO$^{+}$ ions' --- The HCO$^+$ ion has been known for more than a century and was the first ion discovered in interstellar space via microwave spectroscopy [@Buhl]. It was classified as unidentified element at the time of discovery. Later Klemperer [@Klemperer] suggested the linear HCO$^+$ ion as a candidate, which was later confirmed by experiments (see, for example Ref. [@Woods]). These last two decades, HCO$^+$ and other small polyatomic ions have extensively been studied theoretically: their electronic structure, potential energy surfaces and equilibrium geometry have been systematically investigated as, for example, in Refs. [@Yamaguchi; @Palmieri]. The spectroscopy of neutral HCO in the energy range relevant to these DR studies has been extensively mapped out by Grant and coworkers [@GrantV]. Dissociative recombination (DR) of molecular ions like HCO$^+$ plays an important role in chemistry of interstellar clouds and therefore allows astronomers to probe remotely various characteristics of these clouds. In space, the HCO$^+$ ion can be formed by several possible associations such as H$_2$ + CO$^+$ or CH + O and also H$_3^+$ + CO [@rowe92; @Mcgregor] and destroyed by DR. Different types of laboratory experiments have been performed in order to study DR of HCO$^+$: afterglow plasma, merged-beam and storage ring experiments [@rowe92; @Gougousi; @adams84; @lepadellec97; @amano90; @poterya05]. From the laboratory experiments it is now known that DR in HCO$^+$ proceeds mainly into the H+CO channel: HCO$^+$ +e$^-$ $\rightarrow$ H + CO. On the other hand, at present, there is no consensus among different experimental measurements of the actual DR rate coefficient: as they differ by up to a factor of ten [@lepadellec97; @amano90]. The theory of DR in diatomic ions has been reasonably well developed in recent decades. For triatomic ions, only recently has theory been able to provide meaningful results for the simplest triatomic ion, H$_3^+$ [@orel93; @kokoouline01; @kokoouline03a]. The theoretical description of DR in triatomic molecular ions is a difficult problem in part because several different (electronic and vibrational) degrees of freedom have to be taken into account. Several approximations have also been made in a recent theoretical study of DR in HCO$^+$ [@Ivan]. The obtained DR cross-section was about factor 2.5 smaller than the lowest experimental cross-section [@lepadellec97]. One of the possible reasons why the theoretical cross-section was smaller in Ref. [@Ivan] than the experimental one is the approximation of the frozen CO coordinate fixed at its equilibrium value. Although the main dissociation pathway does not involve this coordinate, it was argued [@Ivan] that the CO vibration could increase the probability to capture the electron and increase the overall DR cross-section. In the present study, we improve the previously developed DR treatment in HCO$^+$ and investigate the explicit role of the CO vibration. The theoretical treatment presented here resembles in many respects the approach applied previously to the H$_3^+$ [@kokoouline01] and HCO$^+$ [@Ivan] target ions. Below, we describe the new elements of the theoretical approach. We represent the Hamiltonian of the ion+electron system as $H=H_{ion}+H_{el}$, where $H_{ion}$ is the ionic Hamiltonian and $H_{el}$ describes the electron-ion interaction. Consider first the ionic Hamiltonian $H_{ion}$ written in the center-of-mass reference frame. We use Jacobi coordinates to represent all vibrational degrees of freedom: Introducing [*G*]{} as the center of mass of C-O, the set of internuclear coordinates is represented by the quartet $\mathcal{Q}$=$\{ R_{\rm{CO}},R_{G\rm{H}},\theta,\varphi \}$. Here $R_{\rm{CO}}$ and $R_{G\rm{H}}$ represent respectively the distances C-O and [*G*]{}-H, $\theta$ is the bending angle between $\overrightarrow{\rm{OC}}$ and $ \overrightarrow{G\rm{H}}$, $\varphi$ is the azimuthal orientation of the bending. Here, we consider $R_{G\rm{H}}$ as the adiabatic coordinate representing the dissociation path. In the previous study [@Ivan], the inter-nuclear distance $R_{\rm{CO}}$ was fixed at its equilibrium value ($R_{\rm{CO}}$=2.088 a.u) and the $R_{\rm{CH}}$ coordinate was treated as the dissociative coordinate. Note that even though we use an adiabatic representation, our inclusion of nonadiabatic coupling effects means that we are not utilizing an adiabatic approximation since the adiabatic eigenstates form a complete basis set. The vibrational Hamiltonian in the Jacobi coordinates is: $$\begin{aligned} \label{eq:hamiltonian} H_{ion}={-\frac{\hbar^2}{2\mu_{\rm{CO}}}\frac{\partial^{2}}{\partial R_{\rm{CO}}^{2}}} -{\frac{\hbar^2}{2\mu_{\rm{H-CO}}}\frac{\partial^{2}}{\partial R_{\rm{H-CO}}^{2}}}+\nonumber\\ +{\frac{\hat L^{2}(\theta,\varphi)}{2\mu_{\rm{H-CO}} R_{\rm{H-CO}}^{2}}}+{V(R_{\rm{CO}},R_{G\rm{H}},\theta)}\,,\end{aligned}$$ where $\mu_{\rm{CO}}$ and $\mu_{\rm{H-CO}}$ are respectively the reduced masses of the C–O and H–CO pairs; $\hat L^{2}(\theta,\varphi)$ is the familiar operator of angular momentum corresponding to relative rotation of H and the CO axis. Representing the ion by the above Hamiltonian depending on $\{ R_{\rm{CO}},R_{G\rm{H}},\theta,\varphi \}$ only, we neglected by rotational motion of the CO bond in space, but included relative rotation of H and CO. This approximation is justified by a large CO/H mass ratio. As a result of the approximation, the projection $m_{\varphi}$ of the angular momentum $\hat L$ on the CO axis is conserved. We solve the Schrödinger equation with the Hamiltonian (\[eq:hamiltonian\]) keeping the $R_{G\rm{H}}$ coordinate fixed; this determines vibrational wave functions $\Phi_{ m_{\varphi},l}(R_{G\rm{H}};R_{CO},\theta,\varphi)$ and corresponding adiabatic energies $U_{m_{\varphi},l}(R_{G\rm{H}})$ that depend parametrically on $R_{G\rm{H}}$. Several of these curves are shown in Fig. \[fig:ionic\_potential\]. The lowest adiabatic curves can approximately be characterized by quantum numbers {$v_{1},v_{2}^{m_\varphi},v_{3}$} of the four normal modes of the HCO$^+$ ion. ![(Color online) Adiabatic curves $U_{m_{\varphi},l}(R_{G\rm{H}})$ versus the adiabatic coordinate $R_{G\rm{H}}$. The curves are labeled with quantum numbers $\{v_1v_2^{m_\varphi}v_3\}$ of normal modes of HCO$^+$. The $v_1$ quantum in our model corresponds approximately to motion along $R_{G\rm{H}}$ and therefore is not defined for the adiabatic curves because $R_{G\rm{H}}$ is not quantized. The curves $\{v_12^{\pm2}0\}$ and $\{v_12^00\}$ are almost degenerate.[]{data-label="fig:ionic_potential"}](fig1.eps){width="8cm"} A reasonable measure of accuracy of the adiabatic approximation is provided by comparing the energy splitting between our adiabatic potentials $U_{m_{\varphi},l}(R_{G\rm{H}})$ with exact calculations of the corresponding vibrational splittings [@Palmieri] (Table \[tab:vibr\_ene\]). The obtained values are practically the same as in the previous study [@Ivan] also shown in the table, since a similar (but not identical) adiabatic approximation was used in that study. $\{v_1v_2^{l},v_3\}$ Present calculation Previous calculation[@Ivan] Puzzarini [*et al.*]{}[@Palmieri] ---------------------- --------------------- ----------------------------- ----------------------------------- $10^00$ 343 363 383.1 $01^10$ 91 92 103.0 $02^00$ 182 181 203.5 $03^10$ 275 273 304.9 $04^00$ 369 362 403.8 $00^01$ 298 no value 270.6 : Comparison of vibrational energies obtained in the adiabatic approximation with the exact calculation from Ref. [@Palmieri]. The result of the previous study [@Ivan], where a different adiabatic approximation was used is also shown. In that study CO was not quantized and thus, $00^01$ was not calculated. The overall error is about $12 \%$, which translates into about 25 $\%$ for vibrational wave functions. The energies are given in meV.[]{data-label="tab:vibr_ene"} The structure of the electronic part $H_{el}$ of the total Hamiltonian $H$ is the same as in the previous study [@Ivan]: It includes $ns\sigma $, $np\pi ^{-1}$, $np\sigma $, $np\pi ^{+1}$, and $nd\sigma $ electronic states only. In the basis of the five electronic states, the Hamiltonian has the following block-diagonal form for each principal quantum number $n$: $$H_{int}(\mathcal{Q})=\left( \begin{array}{ccccc} E_{s\sigma } &0 & 0 & 0 &0\\ 0&E_{p\pi } & \delta e^{i\varphi } & \gamma e^{2i\varphi } &0\\ 0& \delta e^{-i\varphi } & E_{p\sigma }& \delta e^{i\varphi } &0\\ 0&\gamma e^{-2i\varphi } & \delta e^{-i\varphi } & E_{p\pi }&0\\ 0&0 & 0 & 0&E_{d\sigma } \end{array} \right) \,, \label{eq:Hint}$$ where $E_{s\sigma }$, $E_{p\sigma }$, $E_{p\pi }$, and $E_{d\sigma }$ are the energies of the corresponding electronic states at the linear ionic configuration; $\delta $ and $\gamma $ are the real non-Born-Oppenheimer coupling parameters. The couplings $\delta $ and $\gamma $ depend on $R_{G\rm{H}},\ R_{\rm{CO}}$, and $\theta $ and responsible for the Renner-Teller interaction. They are zero for linear geometry of the ion. The parameters in the above Hamiltonian are obtained from [*ab initio*]{} calculations of Ref. [@larson05] as discussed in Ref. [@Ivan]. In the present method, the electron-ion interaction Hamiltonian $H_{int}(\mathcal{Q})$ is now used to construct the $5\times 5$ reaction matrix $K_{i,i'}(\mathcal{Q})$ written in the same basis of electronic states as $H_{int}(\mathcal{Q})$. Once the adiabatic states $\Phi_{ m_{\varphi},l}(R_{G\rm{H}};R_{\rm{CO}},\theta,\varphi)$, energies $U_{m_{\varphi},l}(R_{G\rm{H}})$, and the reaction matrix $K_{i,i'}(\mathcal{Q})$ are obtained, we take $R_{G\rm{H}}$ as the adiabatic coordinate and apply the quantum-defect approach that has been already used in a number of DR studies of triatomic and diatomic ions [@kokoouline01; @moshbach05]. We construct the reaction matrix $\mathcal{K}_{j,j^{\prime }}(R_{G\rm{H}})$ $$\mathcal{K}_{\{m_\varphi,l,i\},\{m_\varphi,l,i\}^{\prime }}(R_{G\rm{H}})=\langle \Phi _{m_\varphi,l}|K_{i,i'}(\mathcal{Q})|\Phi _{m' _\varphi,l'}\rangle\,,$$ where the integral is taken over the three internuclear coordinates, $R_{\rm{CO}}$, $\varphi$, and $\theta$. The reaction matrix $\mathcal{K}_{j,j^{\prime }}$ thus obtained has many channels and parametrically depends on $R_{G\rm{H}}$. For each $R_{G\rm{H}}$ value, we then obtain a number of resonances with energies $U_a(R_{G\rm{H}})$ and widths $\Gamma_a(R_{G\rm{H}})$. The resonances correspond to the autoionizing electronic states of the neutral molecule at frozen $R_{G\rm{H}}$. The fixed-$R_{G\rm{H}}$ width of the resonances is the reciprocal of the fixed-$R_{G\rm{H}}$ resonance autoionization lifetime, which of course is not an experimentally-observable resonance width since $R_{G\rm{H}}$ has not been quantized. The energies and widths of the resonances are then used to calculate the cross-section for electron capture by the ion. Depending on whether or not a particular neutral potential curve $U_a(R_{G\rm{H}})$ is energetically open for direct dissociation, two different formulas are appropriate to use for the cross-section calculation. For the neutral states energetically open for direct dissociation, we have $$\label{eq:DA} \sigma= \frac{2\pi^2}{k_o^2} \sum_a\frac{\Gamma_{a}(R_{G\rm{H}})}{|U'_a(R_{G\rm{H}})|}|\chi^+_o(R_{G\rm{H}})|^2\,.$$ However, the following equation should be used $$\label{eq:sum_over_channels} \sigma=\frac{2\pi^2}{k_o^2}\sum_c|\langle\chi^{\rm res}(R_{G\rm{H}}) | \sqrt{\Gamma_{a}(R_{G\rm{H}})}|\chi^+_o(R_{G\rm{H}})\rangle|^2 n_c^3\,$$ for the $U_a(R_{G\rm{H}})$ curves that are energetically closed to direct dissociation [@Ivan]. In the above equations, $k_o$ is the asymptotic wave number of the incident electron, which depends on the initial state $o$ of the target molecular ion; $\chi^+_o(R_{G\rm{H}})$ is the initial vibrational wave function of the ion; $\chi^{\rm res}(R_{G\rm{H}})$ is the quantized radial wave function of the $U_a(R_{G\rm{H}})$ curve. The sum in the first equation includes all neutral states open for direct dissociation. The sum in the above equation is over all closed channels $c$ that produce potential curves $U_a(R_{G\rm{H}})$ closed to direct dissociation [@Ivan]. The total DR cross-section for HCO$^+$ is mainly determined by the second sum because the electron is most likely captured into one of the lowest closed channels that cannot dissociate directly. As mentioned previously, the formulas above describe the cross-section for capture of the electron. It is equal to the DR cross-section only if the probability of subsequent autoionization is negligible compared to the dissociation probability, after the electron has been captured by the ion. The projection $M=m_\varphi+\lambda$ of the total angular momentum on the CO molecular axis, where $\lambda$ is the projection of the electronic angular momentum on the CO axis, is a conserved quantity in our model. Therefore, the resonances and the cross-section are calculated separately for each value of $|M|$. Since $\lambda$ can only be 0 or $\pm$1 in our model ($\sigma$ and $\pi$ states) and the initial vibrational state of the ion has $m_{\varphi}=0$, the possible values of $|M|$ are 0 and 1. The total cross-section for electron capture by the ion in the ground vibrational level is given [@Ivan] by $\langle\sigma^{total}\rangle=\langle\sigma^{M=0}\rangle + 2\langle\sigma^{M=1}\rangle$. ![(Color online) The figure shows the calculated DR cross-section for HCO$^+$ (solid line) as a function of the incident electron energy. The experimental [@lepadellec97] (cross symbols) and previous theoretical [@Ivan] (dashed line) cross-sections are also shown for comparison. The theoretical curves include a convolution over the experimental electron energy distribution according to the procedure described in Ref. [@moshbach05] with $\Delta E_{\perp}=\Delta E_{||}=3$ meV. []{data-label="fig:merged-beam"}](fig2.eps){width="8cm"} Figures \[fig:merged-beam\] and \[fig:thermal-rate\] summarize the results of the present calculation. Fig. \[fig:merged-beam\] compares the present results with the experimental data from a merged-beam experiment [@lepadellec97] and with the previous theoretical study [@Ivan]. The theoretical results are almost identical (about 10% different) for electron energies below 0.1 eV. However they differ significantly at higher energies, where the present calculation gives a higher cross-section. Both curves are smaller than the experimental data by a factor of 2-3. Therefore, the approximation of the frozen C-O bond employed in Ref. [@Ivan] is apparently justified for low electron energies but appears to deteriorate at higher energies. This result can be rationalized as follows. For small electron energies, the CO vibration plays a negligible role because only a few resonances are associated with excited CO vibrational modes. In addition, normally, widths of these resonances are relatively small due to small relevant coupling in the corresponding reaction matrix elements: The largest coupling elements in the matrix are associated with the Renner-Teller coupling, which is active when $m_\varphi$ is changed. However, when the total energy of the system becomes close to (but below) the first CO-excited level $\{00^01\}$ of the ion (0.3 eV above the ground vibrational level), the Rydberg series of resonances associated with the $\{00^01\}$ level becomes more dense and, more importantly, they become mixed with the Rydberg series of the resonances associated with $\{03^10\}$. The latter are coupled relatively strongly to the ground vibrational level $\{00^00\}$ of the ion by the Renner-Teller coupling. ![(Color online) Theoretical (dashed lines) and experimental DR thermal rates for HCO$^+$ and DCO$^+$. The only available experimental data point for DCO$^+$ is shown as a diamond symbol. The other symbols and the solid line represent data from experiments with HCO$^+$. []{data-label="fig:thermal-rate"}](fig3.eps){width="8cm"} Figure \[fig:thermal-rate\] shows the thermal rate coefficients obtained in the present study for HCO$^+$ and DCO$^+$ and compares them with available experimental data. Somewhat analogous to Fig. \[fig:merged-beam\], the theoretical DR rate coefficient for HCO$^+$ is smaller than the rate coefficient obtained from the merged-beam experiment [@lepadellec97] by about a factor of 3. The majority of the other experimental thermal rate coefficients shown in the figure are obtained in plasma experiments. These rates are significantly higher than the merged-beam experimental data [@lepadellec97]. In the previous treatment [@Ivan], the DR rate coefficient obtained for DCO$^+$ was approximately 30% smaller than for HCO$^+$. In the present study, the DCO$^+$ coefficient is smaller than the one in HCO$^+$ by only 10%. The only experimental data available for DCO$^+$ is the rate coefficient $2.6\times10^{-7}$cm$^{3}/s$ obtained at T=95 K [@adams84]. The same study provides the rate coefficient for HCO$^+$, which is by 10% larger. [*Conclusion.*]{} The theoretical DR cross-section obtained in the previous study [@Ivan] was smaller by a factor 2-3 than the lowest measured experimental cross-section. However, the previous theory did not account for vibration along the CO coordinate, and the main purpose of the present study was to assess the validity of the frozen CO approximation employed there. It suggests that the CO vibration does not play a significant role in the DR process at energies below 0.1 eV, but starts to be important at higher energies, when the total energy of the ion+electron system approaches that of the first excited CO vibrational mode. This study suggests also that reduced dimensionality can be used in DR studies of small polyatomic ions as long as one includes (1) the dissociative coordinate and (2) the vibrational coordinates responsible for the highest probability of electron capturing. For HCO$^+$ the dominant dissociative coordinate is the CH bond (or $R_{G\rm{H}}$), whereas the vibrational coordinates responsible for the electron [*capture*]{} are $\theta$ and $\varphi$. Comparing with our previous study [@Ivan], the effect of the CO vibration seems to be larger for DCO$^+$, which can be explained by a larger D/CO mass ratio. In the present and previous theoretical studies, it was assumed that $s$- and $p$-wave-dominated eigenchannels are not mixed. This is justified to some extent by the fact that the [*ab initio*]{} energies used here account for the mixing, at least, at static geometries of the ion. 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--- abstract: 'Nikos Voglis had many astronomical interests, among them was the question of the origin of galactic angular momentum. In this short tribute we review how this subject has changed since the 1970’s and how it has now become evident that gravitational tidal forces have not only caused galaxies to rotate, but have also acted to shape the very cosmic structure in which those galaxies are found. We present recent evidence for this based on data analysis techniques that provide objective catalogues of clusters, filaments and voids.' author: - 'Bernard Jones & Rien van de Weygaert' title: | Cosmic Order out of Primordial Chaos:\ a tribute to Nikos Voglis --- Some early history ================== [r]{}[1.5in]{} -0.25in ![image](joneswey.fig1.eps){width="1.5in"} It was in the 1970’s that Nikos Voglis first came to visit Cambridge, England, to attend a conference and to discuss a problem that was to remain a key area of personal interest for many years to come: the origin of galaxy angular momentum. It was during this period that Nikos teamed up with Phil Palmer to create a long lasting and productive collaboration. The fundamental notion that angular momentum is conserved leads one to wonder how galaxies could acquire their angular momentum if they started out with none. This puzzle was perhaps one of the main driving forces behind the idea that cosmic structure was born out of some primordial turbulence. However, by the early 1970’s the cosmic turbulence theory was falling into disfavour owing to a number of inherent problems (see @Jones76 for a detailed review of this issue). The alternative, and now well entrenched, theory was the gravitational instability theory in which structure grew through the driving force of gravitation acting on primordial density perturbations. The question of the origin of angular momentum had to be addressed and would be central to the success or failure of that theory. [@Peebles69] provided the seminal paper on this, proposing that tidal torques would be adequate to provide the solution. However, this was for many years mired in controversy. Tidal torques had been suggested as a source for the origin of angular momentum since the late 1940’s when @Hoyle49 invoked the tidal stresses exerted by a cluster on a galaxy as the driving force of galaxy rotation. Although the idea as expounded was not specific to any cosmology, there can be little doubt that Hoyle had his Steady State cosmology in mind. The @Peebles69 version of this process specifically invoked the tidal stresses between two neighbouring protogalaxies, but it was not without controversy. There were perhaps three sources for the ensuing debate: - Is the tidal force sufficient to generate the required angular momentum&gt; - Are tidal torques between proto-galaxies alone responsible for the origin of galactic angular momentum? - Tidal torques produce shear fields – what is the origin of the observed circular rotation? [@oort1970] and [@harrison1971] had both argued that the interaction between low-amplitude primordial perturbations would be inadequate to drive the rotation: they saw the positive density fluctuations as being “shielded” by a surrounding negative density region which would diminish the tidal forces. This doubt was a major driving force behind “alternative” scenarios for galaxy formation. The last of these was a more subtle problem since, to some, even if tidal forces managed to generate adequate shear flows, the production of rotational motion would nonetheless require some violation of the Kelvin circulation theorem. Although the situation was clarified by @Jones76 it was not until the exploitation of N-Body cosmological simulations that the issue was considered to have been resolved. It was into this controversy that Nikos stepped, asking precisely these questions. A considerable body of his later work (much of it with Phil Palmer, see for example @PalVog83) was devoted to addressing these issues at various levels. Since these days our understanding of the tidal generation of galaxy rotation has expanded impressively, mostly as a result of ever more sophisticated and large N-body simulation [e.g. @efstathjones79; @jonesefstath79; @Barnes87; @Porciani02; @Bosch02; @Bett07]. What remains is Nikos’ urge for a deeper insight, beyond simulation, into the physical intricacies of the problem. Angular Momentum generation: the tidal mechanism {#sec:tidaltorque} ================================================ -0.25truecm ![image](joneswey.fig2.eps){width="3.6in"} -0.75truecm \[fig:torques\] In order to appreciate these problems it is helpful to look at a simplified version of the tidal model as proposed by Peebles. Consider two neighbouring, similar sized, protogalaxies [**A**]{} and [**B**]{} (figure  \[fig:torques\]). We can view the tidal forces exerted on [**B**]{} by [**A**]{} from either the reference frame of the mass center of [**A**]{} or from the reference frame of the mass center of [**B**]{} itself. These forces are depicted by arrows in the figure: note that relative to the mass center of [**B**]{} the tidal forces act so as to stretch [**B**]{} out in the direction of [**A**]{}. To a first approximation, the force gradient acting on [**B**]{} can be expressed in terms of the potential field $\phi(\bf x)$ in which [**B**]{} is situated: $$T_{ij} = \displaystyle {\partial {F_i} \over \partial {x_j}} = \displaystyle {{{\partial^2 \phi} \over {\partial x_i \partial x_j}}} - {1 \over 3} \delta_{ij} \nabla^2 \phi \label{eq:tidetensor}$$ where the potential field is determined from the fluctuating component of the density field via the Poisson equation [^1]. The flow of material is thus a shear flow determined by the principal directions and magnitudes of inertia tensor of the blob [**B**]{}. Viewed as a fluid flow this is undeniably a shear flow with zero vorticity as demanded by the Kelvin circulation theorem [^2]. So how does the vorticity that is evident in galaxy rotation arise? The answer is twofold. Shocks will develop in the gas flow and stars will form: the Kelvin Theorem holds only for nondissipative flows. Then, a “gas” of stars does not obey the Kelvin Theorem since it is not a fluid (though there is a six-dimensional phase space analogue for a stellar “gas”). The magnitude and direction of angular momentum vector is related to the inertia tensor. $I_{ij}$, of the torqued object and the driving tidal forces described by the tensor $T_{ij}$ of equation (\[eq:tidetensor\]). In 1984, based on simple low-order perturbation theory, @White84 wrote an intuitively appealing expression for the angular momentum vector $L_i$ of a protogalaxy having inertia tensor $I_{mk}$: $$L_i \propto \epsilon_{ijk} T_{jm} I_{mk}, \label{eq:angmom}$$ where summation is implied over the repeated indices. This was later taken up by @CatTheuns99 in a high-order perturbation theory discussion of the problem. However, there is in these treatments an underlying assumption, discussed but dismissed by @CatTheuns99, that the tensors $T_{ij}$ and $I_{ij}$ are statistically independent. Subsequent numerical work by @LeePen00 showed that this assumption is not correct and that ignoring it results in an incorrect estimator for the magnitude of the spin. The approach taken by @LeePen00 [@LeePen01] is interesting: they write down an equation for the autocorrelation tensor of the angular momentum vector in a given tidal field, averaging over all orientations and magnitudes of the inertia tensor. On the basis of equation (\[eq:angmom\]) one would expect this tensor autocorrelation function to be given by $$\langle L_i L_j | {\bf T} \rangle \propto \epsilon_{ipq} \epsilon_{jrs} T_{pm}T_{rn} \langle I_{mq} I_{ns}\rangle$$ where the notation $\langle L_i L_j | {\bf T} \rangle$ is used to emphasise that $T_{ij}$ is regarded as a given value and is not a random variable. The argument then goes that the isotropy of underlying density distribution allows us to replace the statistical quantity $\langle I_{mq} I_{ns}\rangle$ by a sum of Kronecker deltas leaving only $$\langle L_i L_j | {\bf T} \rangle \propto {1 \over 3} \delta_{ij} + ({1 \over 3} \delta_{ij} - T_{ik}T_{kj} ) \label{eq:indepIT}$$ It is then asserted that if the moment of inertia and tidal shear tensors were uncorrelated, we would have only the first term on the right hand side, ${1 \over 3} \delta_{ij}$: the angular momentum vector would be isotropically distributed relative to the tidal tensor. In fact, in the primordial density field and the early linear phase of structure formation there is a significant correlation between the shape of density fluctuations and the tidal force field [@bond1987; @weyedb1996]. Part of the correlation is due to the anisotropic shape of density peaks and the internal tidal gravitational force field that goes along with it [@icke1973]. The most significant factor is that of intrinsic spatial correlations in the primordial density field. It is these intrinsic correlations between shape and tidal field that are at the heart of our understanding of the Cosmic Web, as has been recognized by the Cosmic Web theory of [@bondweb96]. The subsequent nonlinear evolution may strongly augment these correlations (see e.g. fig. \[fig:cosmicwebtide\]), although small-scale highly nonlinear interactions also lead to a substantial loss of the alignments: clusters are still strongly aligned, while galaxies seem less so. Recognizing that the inertia and tidal tensors may not be mutually independent, [@LeePen00; @LeePen01] write $$\langle L_i L_j | {\bf T} \rangle \propto {1 \over 3} \delta_{ij} + c ({1 \over 3} \delta_{ij} - T_{ik}T_{kj} )$$ where $c = 0$ for randomly distributed angular momentum vectors. The case of mutually independent tidal and inertia tensors is described by $c = 1$ (see equation \[eq:indepIT\]). They finally introduce a different parameter $a = 3c/5$ and write $$\langle L_i L_j | {\bf T} \rangle \propto {{1+a} \over 3} \delta_{ij} - a T_{ik}T_{kj}$$ which forms the basis of much current research in this field. The value derived from recent study of the Millenium simulations by @LeePen07 is $a \approx 0.1$. Gravitational Instability ========================= In the gravitational instability scenario, [e.g. @peebles1980], cosmic structure grows from an intial random field of primordial density and velocity perturbations. The formation and molding of structure is fully described by three equations, the [*continuity equation*]{}, expressing mass conservation, the [*Euler equation*]{} for accelerations driven by the gravitational force for dark matter and gas, and pressure forces for the gas, and the [*Poisson-Newton equation*]{} relating the gravitational potential to the density. A general density fluctuation field for a component of the universe with respect to its cosmic background mass density $\rho_{\rm u}$ is defined by $$\delta({\bf r},t)\,=\,\frac{\rho({\bf r})-\rho_{\rm u}}{\rho_{\rm u}}\,.$$ Here ${\bf r}$ is comoving position, with the average expansion factor $a(t)$ of the universe taken out. Although there are fluctuations in photons, neutrinos, dark energy, etc., we focus here on only those contributions to the mass which can cluster once the relativistic particle contribution has become small, valid for redshifts below 100 or so. A non-zero $\delta({\bf r},t)$ generates a corresponding total peculiar gravitational acceleration ${\bf g}({\bf r})$ which at any cosmic position ${\bf r}$ can be written as the integrated effect of the peculiar gravitational attraction exerted by all matter fluctuations throughout the Universe: $${\bf g}({\bf r},t)\,=\,- 4\pi G \bar{\rho}_m (t)a(t) \int {\rm d}{\bf r}'\,\delta({\bf r}^\prime,t)\,{\displaystyle ({\bf r}-{\bf r}^\prime) \over \displaystyle |{\bf r}-{\bf r}^\prime|^3}\ . \label{eq:gravstab}$$ Here $\bar{\rho}_{m}(t)$ is the mean density of the mass in the universe that can cluster (dark matter and baryons). The cosmological density parameter $\Omega_m (t)$ is defined by $\rho_{\rm u}$, via the relation $\Omega_m H^2 = (8\pi G/3) \bar{\rho_{m}} $ in terms of the Hubble parameter $H$. The relation between the density field and gravitational potential $\Phi$ is established through the Poisson-Newton equation: $$\nabla^2 \Phi = 4 \pi G \bar{\rho}_{m}(t) a(t)^2 \ \delta({\bf r},t) .$$ The peculiar gravitational acceleration is related to $\Phi({\bf r},t)$ through ${\bf g}=-\nabla \Phi/a$ and drives peculiar motions. -0.2truecm 0.25cm \[fig:millenniumhier\] -0.75truecm In slightly overdense regions around density excesses, the excess gravitational attraction slows down the expansion relative to the mean, while underdense regions expand more rapidly. The underdense regions around density minima expand relative to the background, forming deep voids. Once the gravitational clustering process has progressed beyond the initial linear growth phase we see the emergence of complex patterns and structures in the density field. Large N-body simulations all reveal a few “universal” characteristics of the (mildly) nonlinear cosmic matter distribution: its hierarchical nature, the anisotropic and weblike spatial geometry of the spatial mass distribution and the presence of huge underdense voids. These basic elements of the Cosmic Web [@bondweb96; @weybondgh2008] exist at all redshifts, but differ in scale. Fig. \[fig:millenniumhier\], from the state-of-the-art “Millennium simulation”, illustrates this complexity in great detail over a substantial range of scales. The figure zooms in on the dark matter distribution at five levels of spatial resolution and shows the formation of a filamentary network connecting to a central cluster. This network establishes transport channels along which matter will flow into the cluster. The hierarchical nature of the structure is clearly visible. The dark matter distribution is far from homogeneous: a myriad of tiny dense clumps indicate the presence of dark halos in which galaxies, or groups of galaxies, will have formed. Within the context of gravitational instability, it is the gravitational tidal forces that establish the relationship between some of the most prominent manifestations of the structure formation process. It is this intimate link between the Cosmic Web, the mutual alignment between cosmic structures and the rotation of galaxies to which we wish to draw attention in this short contribution. Tidal Shear =========== When describing the dynamical evolution of a region in the density field it is useful to distinguish between large scale “background” fluctuations $\delta_{\rm b}$ and small-scale fluctuations $\delta_{\rm f}$. Here, we are primarily interested in the influence of the smooth large-scale field. Its scale $R_b$ should be chosen such that it remains (largely) linear, i.e. the r.m.s. density fluctuation amplitude $\sigma_{\rho}(R_b,t) \lesssim 1$. To a good approximation the smoother background gravitational force ${\bf g}_{\rm b}({\bf x})$ (eq. \[eq:gravstab\]) in and around the mass element includes three components (apart from rotational aspects). The [*bulk force*]{} ${\bf g}_{\rm b}({\bf x}_{pk})$ is responsible for the acceleration of the mass element as a whole. Its divergence ($\nabla \cdot {\bf g}_{\rm b}$) encapsulates the collapse of the overdensity while the tidal tensor $T_{ij}$ quantifies its deformation, $$g_{\rm b,i}({\bf x})\,=\,g_{\rm b,i}({\bf x}_{pk})\,+\,a\,\sum_{j=1}^3\,\left\{{\displaystyle 1 \over \displaystyle 3 a}(\nabla \cdot {\bf g}_{\rm b})({\bf x}_{pk})\,\delta_{\rm ij}\,-\,T_{ij}\right\} (x_j - x_{pk, {\rm j}})\,.$$ The tidal shear force acting over the mass element is represented by the (traceless) tidal tensor $T_{ij}$, $$\begin{aligned} T_{\rm ij}&\,\equiv\,&-{\displaystyle 1 \over \displaystyle 2 a}\, \left\{ {\displaystyle \partial g_{{\rm b},i} \over \displaystyle \partial x_i} + {\displaystyle \partial g_{{\rm b},j} \over \displaystyle \partial x_j}\right\}\,+\, {\displaystyle 1 \over \displaystyle 3 a} (\nabla \cdot {\bf g}_{\rm b}) \,\delta_{\rm ij}\end{aligned}$$ in which the trace of the collapsing mass element, proportional to its overdensity $\delta$, dictates its contraction (or expansion). For a cosmological matter distribution the close connection between local force field and global matter distribution follows from the expression of the tidal tensor in terms of the generating cosmic matter density fluctuation distribution $\delta({\bf r})$ [@weyedb1996]: $$\begin{aligned} && T_{ij}({\bf r})={\displaystyle 3 \Omega H^2 \over \displaystyle 8\pi}\, \int {\rm d}{\bf r}'\,\delta({\bf r}')\ \left\{{\displaystyle 3 (r_i'-r_i)(r_j'-r_j) - |{\bf r}'-{\bf r}|^2\ \delta_{ij} \over \displaystyle |{\bf r}'-{\bf r}|^5}\right\} - {\frac{1}{2}}\Omega H^2\ \delta({\bf r},t)\ \delta_{ij} . \nonumber \label{eq:quadtide}\end{aligned}$$ The tidal shear tensor has been the source of intense study by the gravitational lensing community since it is now possible to map the distribution of large scale cosmic shear using weak lensing data. See for example @HirSel04 [@massey2007]. 0.25truecm -5.0truecm The Cosmic Web ============== Perhaps the most prominent manifestation of the tidal shear forces is that of the distinct [*weblike geometry*]{} of the cosmic matter distribution, marked by highly elongated filamentary, flattened planar structures and dense compact clusters surrounding large near-empty void regions (see fig. \[fig:millenniumhier\]). The recognition of the [*Cosmic Web*]{} as a key aspect in the emergence of structure in the Universe came with early analytical studies and approximations concerning the emergence of structure out of a nearly featureless primordial Universe. In this respect the Zel’dovich formalism [@zeldovich1970] played a seminal role. It led to the view of structure formation in which planar pancakes form first, draining into filaments which in turn drain into clusters, with the entirety forming a cellular network of sheets. The Megaparsec scale tidal shear forces are the main agent for the contraction of matter into the sheets and filaments which trace out the cosmic web. The anisotropic contraction of patches of matter depends sensitively on the signature of the tidal shear tensor eigenvalues. With two positive eigenvalues and one negative, $(-++)$, we will see strong collapse along two directions. Dependent on the overall overdensity, along the third axis collapse will be slow or not take place at all. Likewise, a sheetlike membrane will be the product of a $(--+)$ signature, while a $(+++)$ signature inescapably leads to the full collapse of a density peak into a dense cluster. For a proper understanding of the Cosmic Web we need to invoke two important observations stemming from intrinsic correlations in the primordial stochastic cosmic density field. When restricting ourselves to overdense regions in a Gaussian density field we find that mildly overdense regions do mostly correspond to filamentary $(-++)$ tidal signatures [@pogosyan1998]. This explains the prominence of filamentary structures in the cosmic Megaparsec matter distribution, as opposed to a more sheetlike appearance predicted by the Zeld’ovich theory. The same considerations lead to the finding that the highest density regions are mainly confined to density peaks and their immediate surroundings. The second, most crucial, observation [@bondweb96] is the intrinsic link between filaments and cluster peaks. Compact highly dense massive cluster peaks are the main source of the Megaparsec tidal force field: filaments should be seen as tidal bridges between cluster peaks. This may be directly understood by realizing that a $(-++)$ tidal shear configuration implies a quadrupolar density distribution (eqn. \[eq:quadtide\]). This means that an evolving filament tends to be accompanied by two massive cluster patches at its tip. These overdense protoclusters are the source of the specified shear, explaining the canonical [*cluster-filament-cluster*]{} configuration so prominently recognizable in the observed Cosmic Web. the Cosmic Web and Galaxy Rotation: MMF analysis ================================================ With the cosmic web as a direct manifestation of the large scale tidal field we may wonder whether we can detect a connection with the angular momentum of galaxies or galaxy halos. In section \[sec:tidaltorque\] we have discussed how tidal torques generate the rotation of galaxies. Given the common tidal origin we would expect a significant correlation between the angular momentum of halos and the filaments or sheets in which they are embedded. It was [@LeePen00] who pointed out that this link should be visible in alignment of the spin axis of the halos with the inducing tidal tensor, and by implication the large scale environment in which they lie. In order to investigate this relationship it is necessary to isolate filamentary features in the cosmic matter distribution. A systematic morphological analysis of the cosmic web has proven to be a far from trivial problem, though there have recently been some significant advances. Perhaps the most rigorous program, with a particular emphasis on the description and analysis of filaments, is that of the [*skeleton*]{} analysis of density fields by [@novikov2006; @sousbie2007]. Another strategy has been followed by [@hahn2007] who identify clusters, filaments, walls and voids in the matter distribution on the basis of the tidal field tensor $\partial^2 \phi/\partial x_i \partial x_j$, determined from the density distribution filtered on a scale of $\approx 5h^{-1}\hbox{\rm Mpc}$. ![image](joneswey.fig5.eps){width="12.0cm"} The one method that explicitly takes into account the hierarchical nature of the mass distribution when analyzing the weblike geometries is the Multiscale Morphology Filter (MMF), introduced by [@aragonmmf2007]. The MMF dissects the cosmic web on the basis of the multiscale analysis of the Hessian of the density field. It starts by translating an N-body particle distribution or a spatial galaxy distribution into a DTFE density field [see @weyschaap2007]. This guarantees a morphologically unbiased and optimized density field retaining all features visible in a discrete galaxy or particle distribution. The DTFE field is filtered over a range of scales. By means of morphology filter operations defined on the basis of the Hessian of the filtered density fields the MMF successively selects the regions which have a bloblike (cluster) morphology, a filamentary morphology and a planar morphology, at the scale at which the morphological signal is optimal. By means of a percolation criterion the physically significant filaments are selected. Following a sequence of blob, filament and wall filtering finally produces a map of the different morphological features in the particle distribution. With the help of the MMF we have managed to find the relationship of shape (inertia tensor) and spin-axis of halos in filaments and walls and their environment. On average, the long axis of filament halos is directed along the axis of the filament; wall halos tend to have their longest axis in the plane of the wall. At the present cosmic epoch the effect is stronger for massive halos. Interestingly, the trend appears to change in time: low mass halos tended to be more strongly aligned but as time proceeds local nonlinear interactions affect the low mass halos to such an extent that the situation has reversed. The orientation of the rotation axis provides a more puzzling picture (fig. \[fig:wallspin\]). The rotation axis of low mass halos tends to be directed along the filament’s axis while that of massive halos appears to align in the perpendicular direction. In walls there does not seem to exist such a bias: the rotation-axis of both massive and light haloes tends to lie in the plane of the wall. At earlier cosmic epochs the trend in filaments was entirely different: low mass halo spins were more strongly aligned as large scale tidal fields were more effective in directing them. During the subsequent evolution in high-density areas, marked by strongly local nonlinear interactions with neighbouring galaxies, the alignment of the low mass objects weakens and ultimately disappears. Tidal Fields and Void alignment =============================== A major manifestation of large scale tidal influences is that of the alignment of shape and angular momentum of objects [see @bondweb96; @desjacques2007]. The alignment of the orientations of galaxy haloes, galaxy spins and clusters with larger scale structures such as clusters, filaments and superclusters has been the subject of numerous studies [see e.g. @binggeli1982; @bond1987; @rhee1991; @plionis2002; @basilakos2006; @trujillo2006; @aragon2007; @leevrard2007; @leespringel2007]. Voids are a dominant component of the Cosmic Web [see e.g. @tully2007; @weyrom2007], occupying most of the volume of space. Recent analytical and numerical work [@parklee2007; @leepark2007; @platen2008] discussed the magnitude of the tidal contribution to the shape and alignment of voids. @leepark2007 found that the ellipticity distribution of voids is a sensitive function of various cosmological parameters and remarked that the shape evolution of voids provides a remarkably robust constraint on the dark energy equation of state. [@platen2008] presented evidence for significant alignments between neigbouring voids, and established the intimate dynamic link between voids and the cosmic tidal force field. Voids were identified with the help of the Watershed Void Finder (WVF) procedure [@platen2007]. The WVF technique is based on the topological characteristics of the spatial density field and thereby provides objectively defined measures for the size, shape and orientation of void patches. Void-Tidal Field alignments: formalism -------------------------------------- In order to trace the contributions of the various scales to the void correlations [@platen2008] investigated the alignment between the void shape and the tidal field smoothed over a range of scales $R$. The alignment function ${\mathcal A}_{TS}(R_1)$ between the local tidal field tensor $T_{ij}(R_1)$, Gaussian filtered on a scale $R_1$ at the void centers, and the void shape ellipsoid is determined as follows. For each individual void region the shape-tensor $\mathcal{S}_{ij}$ is calculated by summing over the $N$ volume elements $k$ located within the void, $$\begin{aligned} \qquad \mathcal{S}_{ij}& = - &\sum_{k} x_{ki} x_{kj} \qquad \ \ \ \ \ \ \ \ \textrm{(offdiagonal)} \\ \qquad \mathcal{S}_{ii}& = &\sum_{k} \left({\bf x}_k^2-x_{ki}^2\right) \qquad\ \ \textrm{(diagonal)}\,,\nonumber \end{aligned}$$ where ${\bf x}_k$ is the position of the $k$-th volume element within the void, with respect to the (volume-weighted) void center ${\bf {\overline{r}}}_v$, i.e. ${\bf x}_k = {\bf r}_k - {\bf {\overline{r}}}_v$. The shape tensor $\mathcal{S}_{ij}$ is related to the inertia tensor $\mathcal{I}_{ij}$. However, it differs in assigning equal weight to each volume element within the void region. Instead of biasing the measure towards the mass concentrations near the edge of voids, the shape tensor $\mathcal{S}_{ij}$ yields a truer reflection of the void’s interior shape. The smoothing of the tidal field is done in Fourier space using a Gaussian window function ${\hat W}^*({\bf k};R)$: $$\begin{aligned} T_{ij}({\bf r};R)\,=\, {\displaystyle 3 \over \displaystyle 2} \Omega H^2 \int \d3k\left({\displaystyle k_i k_j \over \displaystyle k^2}-{\displaystyle 1 \over \displaystyle 3} \delta_{ij}\right)\,{\hat W}^*({\bf k};R)\,{\hat \delta}({\bf k})\, {\rm e}^{-{\rm i}{\bf k}\cdot{\bf r}}&&\, \nonumber\end{aligned}$$ Here, ${\hat \delta}({\bf k})$ is the Fourier amplitude of the relative density fluctuation field at wavenember $\bf k$. Given the void shape ${\mathcal S}_{ij}$ and the tidal tensor $T_{ij}$, for every void the function ${\Gamma}_{TS}(m,R_1)$ at the void centers is determined: $$\begin{aligned} {\Gamma}_{TS}(m;R_1)&\,=\,&\ -\ {\displaystyle \sum\nolimits_{i,j} {\tilde {\mathcal S}}_{m,ij}\,T_{ij}({\bf r}_m;R_1) \over \displaystyle {\tilde {\mathcal S}}_m\,\,T({\bf r}_m;R_1)}\end{aligned}$$ where $T({\bf r}_m;R_1)$ is the norm of the tidal tensor $T_{ij}({\bf r_m})$ filtered on a scale $R_1$ and The void-tidal alignment ${\mathcal A}_{TS}(R_1)$ at a scale $R$ is then the ensemble average $$\begin{aligned} {\mathcal A}_{TS}(R_1)&\,=\,&\langle\,\Gamma_{TS}(R_1)\,\rangle\,. \label{eq:ats}\end{aligned}$$ which we determine simply by averaging $\Gamma_{TS}(m,R_1)$ over the complete sample of voids. Void-Tidal Field alignments: results ------------------------------------ A visual impression of the strong relation between the void’s shape and orientation and the tidal field is presented in the lefthand panel of fig \[fig:cor2\] (from [@platen2008]). The tidal field configuration is depicted by means of (red-coloured) tidal bars. These bars represent the compressional component of the tidal force field in the slice plane, and have a size proportional to its strength and are directed along the corresponding tidal axis. The bars are superimposed on the pattern of black solid watershed void boundaries, whose orientation is emphasized by means of a bar directed along the projection of their main axis. The compressional tidal forces tend to be directed perpendicular to the main axis of the void. This is most clearly in regions where the forces are strongest and most coherent. In the vicinity of great clusters the voids point towards these mass concentrations, stretched by the cluster tides. The voids that line up along filamentary structures, marked by coherent tidal forces along their ridge, are mostly oriented along the filament axis and perpendicular to the local tidal compression in these region. The alignment of small voids along the diagonal running from the upper left to the bottom right is particularly striking. A direct quantitative impression of the alignment between the void shape and tidal field, may be obtained from the righthand panel of fig. \[fig:cor2\]. The figure shows $\mathcal{C}_{TS}$ (dotted line), the alignment between the compressional direction of the tidal field and the shortest shape axis. It indicates that the tidal field is instrumental in aligning the voids. To further quantify and trace the tidal origin of the alignment one can investigate the local shape-tide alignment function ${\mathcal A}_{TS}$ (eqn. \[eq:ats\]) versus the smoothing radius $R_1$. This analysis reveals that the alignment remains strong over the whole range of smoothing radii out to $R_1 \approx 20-30h^{-1}\hbox{\rm Mpc}$ and peaks at a scale of $R_1 \approx 6h^{-1} \hbox{\rm Mpc}$. This scale is very close to the average void size, and also close to the scale of nonlinearity. This is not a coincidence: the identifiable voids probe the linear-nonlinear transition scale. The remarkably strong alignment signal at large radii than $R_1>20h^{-1}\hbox{\rm Mpc}$ (where ${\mathcal A}_{TS}\approx 0.3$), can only be understood if large scale tidal forces play a substantial role in aligning the voids. Final remarks ============= The last word on the origin of galactic angular momentum has not been said yet. It is now a part of our cosmological paradigm that the global tidal fields from the irregular matter distribution on all scales is the driving force, but the details of how this works have yet to be explored. That is neither particularly demanding nor particularly difficult, it is simply not trendy: there are other problems of more pressing interest. The transition from shear dominated to rotation dominated motion is hardly explored and will undoubtedly be one of the principal by-products of cosmological simulations with gas dynamics and star formation. The role of tidal fields has been found to be more profound than the mere transfer of angular momentum to proto-objects. The cosmic tidal fields evidently shape the entire distribution and dynamics of galaxies: they shape what has become known as the “cosmic web”. Although we see angular momentum generation in cosmological N-Body simulations it is not clear that the simulations do much more than tell us what happened: galaxy haloes in N-Body models have acquired spin by virtue of tidal interaction. We draw comfort from the fact that the models give the desired result. Nikos Voglis’ approach was somewhat deeper: he wanted to understand things at a mechanistic level rather than simply to simulate them and observe the result. In that he stands in the finest tradition of the last of the great Hellenistic scientists, Hipparchus of Nicaea, who studied motion of bodies under gravity. Perhaps we should continue in the spirit of Nikos’ work by trying to understand things rather than simply simulate them. Nikos was a good friend, a fine scientist and certainly one of the kindest people one could ever meet. It was less than one year ago when we met for the last time at the [*bernard60*]{} conference in Valencia. We were of course delighted to see him and we shall cherish that brief time together. Acknowledgments =============== We thank Panos Patsis and George Contopoulos for the opportunity of delivering this tribute to our late friend. We are grateful to Volker Springel for allowing us to use figure 3. We particularly wish to acknowledge Miguel Aragón-Calvo and Erwin Platen for allowing us to use their scientific results: their contributions and discussions have been essential for our understanding of the subject. [99]{} M.A., [Jones]{} B.J.T., [van de Weygaert]{} R., [van der Hulst]{} J.M.: Astrophys. J. **655**, L5 (2007) M.A., [Jones]{} B.J.T., [van de Weygaert]{} R., [van der Hulst]{} J.M.: Astron. Astrophys. **474**, 315 (2007) J., [Efstathiou]{} G.: Astrophys. J. **319**, 575 (1987) S., [Plionis]{} M., [Yepes]{} G., [Gottlöber]{} S., [Turchaninov]{} V.: Mon. Not. Roy. Astron. Soc. **365**, 539 (2006) P., [Eke]{} V., [Frenk]{} C.S., [Jenkins]{} A., [Helly]{} J., [Navarro]{} J.: Mon. 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--- abstract: 'We rederive a popular nonsemisimple fusion algebra in the braided context, from a Nichols algebra. Together with the decomposition that we find for the product of simple Yetter–Drinfeld modules, this strongly suggests that the relevant Nichols algebra furnishes an equivalence with the triplet $W$-algebra in the $(p,1)$ logarithmic models of conformal field theory. For this, the category of Yetter–Drinfeld modules is to be regarded as an *entwined* category (the one with monodromy, but not with braiding).' address: 'Lebedev Physics Institute, Russian Academy of Sciences' author: - 'A.M. Semikhatov' title: 'Fusion in the entwined category of Yetter–Drinfeld modules of a rank-1 Nichols algebra' --- Introduction ============ The idea to construct “purely algebraic” counterparts of vertex-operator algebras (conformal field theories) has a relatively long history [@[KLx]; @[Fink]; @[T]; @[MS]; @[FRS]; @[FFRS]]. In [@[FGST]; @[FGST2]; @[FGST3]; @[FGSTq]; @[NT]; @[GT]; @[BFGT]; @[BGT]], this idea was developed for nonsemisimple—logarithmic—CFT models, which have been intensively studied recently (see [@[AA]; @[CR]; @[HY]; @[AN]; @[VJS]; @[AM-2p]; @[HLZ]; @[GRW]; @[FSS-11]; @[FSS-12]; @[VGJS]; @[GV]; @[RGW-12]] and the references therein). In [@[STbr]], further, a braided and arguably “more fundamental” algebraic counterpart of logarithmic CFT was proposed. It is given by Nichols algebras [@[Nich]; @[AG]; @[AS-onthe]; @[AS-pointed]; @[Andr-remarks]]; the impressive recent progress in their theory (see [@[Heck-class]; @[Heck-Weyl]; @[AHS]; @[ARS]; @[GHV]; @[GH-lyndon]; @[AFGV]; @[AAY]; @[Ag-0804-standard]; @[Ag-1008-presentation]; @[Ag-1104-diagonal]] and the references therein) is a remarkable “spin-off” of Andruskiewitsch and Schneider’s program of classification of pointed Hopf algebras. Associating Nichols algebras with CFT models implies that certain CFT-related structures must be reproducible from (some) Nichols algebras. Here, we take the simplest, rank-1 Nichols algebra ${\mathfrak{B}}\!_p$ of dimension $p\geq2$ and, from the category of its Yetter–Drinfeld modules, extract a commutative associative $2p$-dimensional algebra on the ${\,\boldsymbol{\mathsf{x}}}(r)_{\nu}$, $1\leq r\leq p$, $\nu\in{\mathbb{Z}}_2$: $$\label{the-algebra} {\,\boldsymbol{\mathsf{x}}}(r_1)_{\nu_1}{\,\boldsymbol{\mathsf{x}}}(r_2)_{\nu_2} = \sum_{\substack{s=|r_1-r_2|+1\\ \text{step}=2}}^{p-1-|r_1+r_2-p|} {\,\boldsymbol{\mathsf{x}}}(s)_{\nu_1+\nu_2} \,\,+\,\,\sum_{\substack{s = 2p-r_1-r_2+1\\ \text{step}=2}}^{p} {\,\boldsymbol{\mathsf{p}}}(s)_{\nu_1+\nu_2},$$ with $$ {\,\boldsymbol{\mathsf{p}}}(r)_{\nu}= \begin{cases} 2{\,\boldsymbol{\mathsf{x}}}(r)_{\nu} + 2{\,\boldsymbol{\mathsf{x}}}(p-r)_{\nu+1},& r<p,\\ {\,\boldsymbol{\mathsf{x}}}(p)_\nu,& r=p. \end{cases}$$ This is the FHST fusion algebra [@[FHST]] (also see [@[GT]]), which makes part of what we know from [@[NT]] (also see [@[AM-3]]) to be an equivalence of representation categories—of the *triplet algebra* $W(p)$ in the $(p,1)$ logarithmic conformal models [@[Kausch]; @[Gaberdiel-K]; @[Gaberdiel-K-2]; @[Gaberdiel-K-3]; @[FHST]] and of a small quantum $s\ell_2$ at the $2p$th root of unity, proposed in this capacity in [@[FGST]; @[FGST2]] and then used and studied, in particular, in [@[MN]; @[FHT]; @[Ar]; @[KoSa]; @[S-yd]] (this quantum group had appeared before in [@[AGL]; @[Su]; @[X]]). The reoccurrence of the fusion algebra in the braided approach advocated in [@[STbr]], together with some other observations, supports the idea that Nichols algebras are *at least as good as* the quantum groups proposed previously [@[FGST]; @[FGST2]; @[FGST3]; @[FGSTq]; @[S-q]] for the logarithmic version of the Kazhdan–Lusztig correspondence (the correspondence between vertex-operator algebras and quantum groups).[^1] Algebra  arises here as an algebra in the center of the category of Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules; the ${\,\boldsymbol{\mathsf{x}}}(r)_{\nu}$ are certain images of the simple Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules ${{\mathscr{X}}}(r)_{\nu}$.[^2] More is actually true: from the study of the representation theory of ${\mathfrak{B}}\!_p$, we obtain that the tensor product of simple Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules decomposes as $$\label{the-fusion} {{\mathscr{X}}}(r_1)_{\nu_1}{\otimes}{{\mathscr{X}}}(r_2)_{\nu_2} = \bigoplus_{\substack{s=|r_1-r_2|+1\\ \text{step}=2}}^{p-1-|r_1+r_2-p|} {{\mathscr{X}}}(s)_{\nu_1+\nu_2} \,\,\oplus\,\,\bigoplus_{\substack{s = 2p-r_1-r_2+1\\ \text{step}=2}}^{p} {{\mathscr{P}}}[s]_{\nu_1+\nu_2},$$ where ${{\mathscr{P}}}[p]_{\nu}={{\mathscr{X}}}(p)_{\nu}$ and ${{\mathscr{P}}}[r]_{\nu}$ for $1\leq r\leq p-1$ is a reducible Yetter–Drinfeld ${\mathfrak{B}}\!_p$-module with the structure of subquotients $$\label{P-mod-ind} \raisebox{-2\baselineskip}{${{\mathscr{P}}}[r]_{\nu} \ =\ \ $} \xymatrix@=12pt{ &{\mathscr{X}}(p-r)_{\nu+1} \ar@/_12pt/[dl] \ar[dr] &\\ {\mathscr{X}}(r)_{\nu}\ar[dr] &&{\mathscr{X}}(r)_{\nu+2},\ar@/_12pt/[dl] \\ &{\mathscr{X}}(p-r)_{\nu+1}& }$$ Decompositions  were conjectured in [@[STbr]] and are proved here. The ${{\mathscr{X}}}(r)_\nu$ and ${{\mathscr{P}}}[r]_\nu$ do not exhaust all the category of Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules, but make up “the most significant part of it,” and relations , together with the structure of ${{\mathscr{P}}}[r]_\nu$, already seem to imply that the category of Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules is equivalent to the $W(p)$ representation category. This requires an important clarification, however. In the braided category of Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules, the simple objects are the ${{\mathscr{X}}}(r)_{\nu}$ labeled by $1\leq r\leq p$ and $\nu\in{\mathbb{Z}}_4$ (and, accordingly, $\nu\in{\mathbb{Z}}_4$ in ${{\mathscr{P}}}[r]_\nu$, and so on). There are twice as many objects as in the category of $W(p)$ representations [@[FHST]; @[AM-3]; @[NT]]. But the presumed equivalence is maintained for *entwined* categories [@[Brug]]—those endowed with only “double braiding” $D_{{{\mathscr{Y}}},{{\mathscr{Z}}}}=c_{{{\mathscr{Z}}},{{\mathscr{Y}}}}{\mathbin{\raisebox{1pt}{\,$\scriptscriptstyle\circ$\,}}}c_{{{\mathscr{Y}}},{{\mathscr{Z}}}}$ (the *monodromy* on the $W(p)$ side). The properties of double braiding can be axiomatizedwithout having to resort to the braiding itself [@[Brug]]. This defines a *twine structure* and, accordingly, an entwined category. Remarkably, it was noted in [@[Brug]] that > “many significant notions apparently related to $c$ actually depend only on $D$ or \[the twist\] $\theta$. The $S$-matrix, and the subcategory of transparent objects, which play an important role in the construction of invariants of 3-manifolds, are defined purely in terms of the double braiding. More surprisingly, the invariants of ribbon links … do not depend on the actual braiding, but only on $D$.” In the entwined category of Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules, the objects with $\nu$ and $\nu+2$ in their labels are isomorphic, which sets $\nu\in{\mathbb{Z}}_2$ and resolves the “representation doubling problem”; everything else on the algebraic side appears to be already “fine-tuned” to ensure the equivalence. (We do not go as far as modular transformations in this paper, but the above quotation suggests that dealing with entwined categories is not an impediment to rederiving the $W(p)$ modular properties at the Nichols algebra level, in a “braided version” of what was done in [@[FGST]].) It may also be worth noting that we derive  and  independently (of course, from the same structural results on Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules, but not from one another). In particular, is obtained by directly composing the *action* of ${\,\boldsymbol{\mathsf{x}}}(r_1)_{\nu_1}$ and ${\,\boldsymbol{\mathsf{x}}}(r_2)_{\nu_2}$ on Yetter–Drinfeld modules, with ${\,\boldsymbol{\mathsf{x}}}(r)_{\nu}:{{\mathscr{Y}}}\to{{\mathscr{Y}}}$ given by “running ${{\mathscr{X}}}(r)_{\nu}$ along the loop” in the diagram (with the notation to be detailed in what follows) $$\label{the-loop} \begin{tangles}{l} {\mathrm{id}}\step[2.5]\coev\\ \vstr{50}\dh\step[1.5]\ddh\step[2]{\mathrm{id}}\\[-2pt] \step[.25]\obox{2}{\mathsf{B}^2}\step[2.25]{\mathrm{id}}\\[-2pt] \vstr{50}\hdd\step[1.5]\hd\step[2]{\mathrm{id}}\\ {\mathrm{id}}\step[2.5]\O{{\boldsymbol{\pmb{\vartheta}}}}\step[1.5]\hdd\\[-2pt] {\mathrm{id}}\step[2.25]\obox{2}{\mathsf{B}}\\[-2pt] \vstr{33}{\mathrm{id}}\step[2.5]{\mathrm{id}}\step[1.5]{\mathrm{id}}\\ \hh{\mathrm{id}}\step[2.75]{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{15}b\makeatother} \end{tangles}$$ As such, the ${\,\boldsymbol{\mathsf{x}}}(r)_{\nu}$ depend only on $\nu\in{\mathbb{Z}}_2$—there is no “${\mathbb{Z}}_4$ option” for them.[^3] This paper is organized as follows. For the convenience of the reader, we summarize the relevant points from [@[STbr]] in Sec. \[sec:Nich\]; a very brief summary is that for a Nichols algebra ${\mathfrak{B}}(X)$,a category of its Yetter–Drinfeld modules can be constructed using another braided vector space $Y$ (whose elements are here called “vertices,” and the Yetter–Drinfeld modules the “multivertex” modules). In Sec. \[sec:duality\], we introduce duality and the related assumptions that make it possible to write diagrams . In Sec. \[sec:p\], everything is specialized to a rank-1 Nichols algebra ${\mathfrak{B}}\!_p$ (depending on an integer $p\geq2$). First and foremost, “everything” includes multivertex Yetter–Drinfeld modules. We actually construct important classes of these modules quite explicitly (Appendix [**\[app:modules\]**]{}), which allows proving  and also establishing duality relations among the modules. We also study their braiding, find the ribbon structure, and finally use all this to derive  from  for ${\mathfrak{B}}\!_p$. Basic properties of Yetter–Drinfeld modules over a braided Hopf algebra are recalled in Appendix [**\[app:YD-axiom\]**]{}. The Nichols algebra of screenings {#sec:Nich} ================================= We summarize the relevant points of [@[STbr]] in this section. Screenings and ${\mathfrak{B}}(X)$ {#screenings-and-mathfrakbx .unnumbered} ---------------------------------- The underlying idea is that the nonlocalities associated with screening operators—multiple-integration contours, such as $$\label{crosses} \xymatrix@R=4pt@C=80pt{ \ar@{--}|(.2){{\textstyle\!\!{\times}\!\!}}|(.55){{\textstyle\!\!{\times}\!\!}}|(.8){{\textstyle\!\!{\times}\!\!}}[0,2]&& } = \iiint_{-\infty<z_1<z_2<z_3<\infty} s_{i_1}(z_1) s_{i_2}(z_2) s_{i_3}(z_3),$$ where $s_j(z)$ are the “screening currents”— allow introducing a *coproduct* by contour cutting, called “deconcatenation” in what follows: $$\begin{aligned} \label{cutting} \Delta:\xymatrix@R=4pt@C=26pt{ \ar@{--}|(.25){{\textstyle\!\!{\times}\!\!}}|(.5){{\textstyle\!\!{\times}\!\!}}|(.75){{\textstyle\!\!{\times}\!\!}}[0,2]&& }\mapsto{}& \xymatrix@R=4pt@C=9pt{ \ar@{--}|(.25){{\textstyle\!\!{\times}\!\!}}|(.5){{\textstyle\!\!{\times}\!\!}}|(.75){{\textstyle\!\!{\times}\!\!}}[0,3] &&&{\ \raisebox{-6pt}{\rotatebox{90}{\mbox{\ding{34}}}}\ }\ar@{--}[0,3]&&& } + \xymatrix@R=4pt@C=9pt{ \ar@{--}|(.33){{\textstyle\!\!{\times}\!\!}}|(.66){{\textstyle\!\!{\times}\!\!}}[0,3] &&&{\ \raisebox{-6pt}{\rotatebox{90}{\mbox{\ding{34}}}}\ }\ar@{--}|(.5){{\textstyle\!\!{\times}\!\!}}[0,3]&&& } \\ \notag &+ \xymatrix@R=4pt@C=9pt{ \ar@{--}|(.5){{\textstyle\!\!{\times}\!\!}}[0,3] &&&{\ \raisebox{-6pt}{\rotatebox{90}{\mbox{\ding{34}}}}\ }\ar@{--}|(.33){{\textstyle\!\!{\times}\!\!}}|(.66){{\textstyle\!\!{\times}\!\!}}[0,3]&&& } + \xymatrix@R=4pt@C=9pt{ \ar@{--}[0,3] &&&{\ \raisebox{-6pt}{\rotatebox{90}{\mbox{\ding{34}}}}\ }\ar@{--}|(.25){{\textstyle\!\!{\times}\!\!}}|(.5){{\textstyle\!\!{\times}\!\!}}|(.75){{\textstyle\!\!{\times}\!\!}}[0,3] &&& } $$ (with the line cutting symbol subsequently understood as ${\otimes}$). A *product* of “lines populated with crosses” is also defined, as the “quantum” shuffle product [@[Rosso-CR]], which involves a *braiding* between any two screenings. It is well known that these three structures—coproduct, product, and braiding—satisfy the braided bialgebra axioms [@[Rosso-CR]]. The *antipode* is in addition given by contour reversal. The braided Hopf algebra axioms are then satisfied for quite a general braiding (by far more general than may be needed in CFT); it is rather amusing to see how the braided Hopf algebra axioms are satisfied by merging and cutting contour integrals [@[STbr]]. The algebra *generated by* single crosses—individual screenings—is the Nichols algebra ${\mathfrak{B}}(X)$ of the braided vector space $X$ spanned by the different screening species (whose number is called the rank of the Nichols algebra). Nichols algebras {#nichols-algebras .unnumbered} ---------------- The Nichols algebras—“bialgebras of type one” in [@[Nich]]—are a crucial element in a classification program of *ordinary* Hopf algebras of a certain type (see [@[AG]; @[AS-pointed]; @[AS-onthe]; @[ARS]] and the references therein). Nichols algebras have several definitions, whose equivalence is due to [@[Sch-borel]] and [@[AG]]. The Nichols algebra ${\mathfrak{B}}(X)$ of a braided linear space $X$ can be characterized as a graded braided Hopf algebra ${\mathfrak{B}}(X)=\bigoplus_{n\geq0}{\mathfrak{B}}(X)^{(n)}$ such that ${\mathfrak{B}}(X)^{(1)}=X$ and this last space coincides with the space of *all primitive elements* $P(X)=\{x\in{\mathfrak{B}}(X) \mid\Delta x=x{\otimes}1 + 1{\otimes}x\}$ and it *generates all of ${\mathfrak{B}}(X)$* as an algebra.[^4] Nichols algebras occurred independently in [@[Wor]], in constructing a quantum differential calculus, as “fully braided generalizations” of symmetric algebras, $${\mathfrak{B}}(X) = k\oplus X\oplus\bigoplus_{r\ge 2} X^{\otimes r}/\ker{\mathfrak{S}_{r}},$$ where ${\mathfrak{S}_{r}}$ is the total braided symmetrizer (“braided factorial”). The space of vertices $Y$ {#the-space-of-vertices-y .unnumbered} ------------------------- In addition to the braided linear space $X$ spanned by the different screening species, we introduce the space of vertex operators taken at a fixed point, $$Y=\text{Span}(V_\alpha(0)),$$ where $\alpha$ ranges over the different primary fields in a given CFT model. CFT also yields the braiding $\Psi:X{\otimes}X\to X{\otimes}X$ of any two screenings (which is always applied to two screenings on the same line, as in ), as well as the braiding $\Psi:X{\otimes}Y\to Y{\otimes}X$ and $\Psi:Y{\otimes}X\to X{\otimes}Y$ of a screening and vertex (also on the same line, as in  below), and eventually the braiding $\Psi:Y{\otimes}Y\to Y{\otimes}Y$ of any two vertices, but a large part of our construction can be formulated without this last. The two braided vector spaces $X$ and $Y$ are all that we need in this section; the braiding $\Psi$ can be entirely general. Dressed vertex operators as ${\mathfrak{B}}(X)$-modules {#dressed-vertex-operators-as-mathfrakbx-modules .unnumbered} ------------------------------------------------------- We use the space $Y$ to construct ${\mathfrak{B}}(X)$-modules. Their elements are sometimes referred to in CFT as “dressed$/$screened vertex operators,” for example, $$\label{1-punctured} \xymatrix@R=4pt@C=70pt{ \ar@*{[|(1.6)]}@{-}|(.1){{\textstyle\!\!{\times}\!\!}}|(.35){{\textstyle\!\!{\times}\!\!}}|(.55){{\textstyle{\circ}}}|(.7){{\textstyle\!\!{\times}\!\!}}[0,2]&&\\ } = \iint_{-\infty<x_1<x_2<0}\!\!\!\!\!\!\! s_{i_1}(x_1) s_{i_2}(x_2)\; V_\alpha(0)\!\!\! \int\limits_{0<x_3<\infty} s_{i_3}(x_3).$$ It is understood that the $\times$ and $\circ$ are decorated with the appropriate indices read off from the right-hand side; but it is in fact quite useful to suppress the indices altogether and let $\times$ and $\circ$ respectively denote the entire spaces $X$ and $Y$, and we assume this in what follows. Because the integrations can be taken both on the left and on the right of the vertex position, the resulting modules are actually ${\mathfrak{B}}(X)$ bimodules. The left and right actions of ${\mathfrak{B}}(X)$ are by pushing the “new” crosses into the different positions using braiding; the left action, for example, can be visualized as $$\begin{gathered} \xymatrix@R=6pt@C=30pt{ \ar@{--}|(.5){{\textstyle\!\!{\times}\!\!}}[0,2]&& } {\mathbin{\pmb{.}}}\xymatrix@R=6pt@C=30pt{ \ar@*{[|(1.6)]}@{-}|(.3){{\textstyle{\circ}}}|(.65){{\textstyle\!\!{\times}\!\!}}[0,2]&& } = \\[6pt] \xymatrix@R=6pt@C=30pt{ \ar@*{[|(1.6)]}@{-}|(.1){{\textstyle\!\!{\times}\!\!}}|(.3){{\textstyle{\circ}}}|(.70){{\textstyle\!\!{\times}\!\!}}[0,2]&& } + \xymatrix@R=6pt@C=30pt{ \ar@*{[|(1.6)]}@{-}|(.3){{\textstyle{\circ}}}|(.5){{\textstyle\!\!{\times}\!\!}}|(.75){{\textstyle\!\!{\times}\!\!}}[0,2] \ar@/^10pt/[r]&& } + \xymatrix@R=6pt@C=30pt{ \ar@*{[|(1.6)]}@{-}|(.3){{\textstyle{\circ}}}|(.6){{\textstyle\!\!{\times}\!\!}}|(.95){{\textstyle\!\!{\times}\!\!}}[0,2] \ar@/^12pt/[rr]&& }\end{gathered}$$ where the arrows, somewhat conventionally, represent the braiding $\Psi$. Once again by deconcatenation, e.g., $$\begin{aligned} \delta_{\text{L}}:\xymatrix@R=4pt@C=28pt{ \ar@*{[|(1.6)]}@{-}|(.25){{\textstyle\!\!{\times}\!\!}}|(.5){{\textstyle\!\!{\times}\!\!}}|(.65){{\textstyle{\circ}}}| (.85){{\textstyle\!\!{\times}\!\!}}[0,2]&&} \mapsto{}& \xymatrix@R=4pt@C=12pt{ \ar@{--}[0,3] &&&{\ \raisebox{-6pt}{\rotatebox{90}{\mbox{\ding{34}}}}\ }\ar@*{[|(1.6)]}@{-}|(.25){{\textstyle\!\!{\times}\!\!}}|(.5){{\textstyle\!\!{\times}\!\!}}|(.65){{\textstyle{\circ}}}|(.85){{\textstyle\!\!{\times}\!\!}}[0,3] &&&} + \xymatrix@R=4pt@C=12pt{ \ar@{--}|(.5){{\textstyle\!\!{\times}\!\!}}[0,3] &&&{\ \raisebox{-6pt}{\rotatebox{90}{\mbox{\ding{34}}}}\ }\ar@*{[|(1.6)]}@{-}|(.3){{\textstyle\!\!{\times}\!\!}}|(.65){{\textstyle{\circ}}}|(.85){{\textstyle\!\!{\times}\!\!}}[0,3] &&&} \\ &{}+\xymatrix@R=4pt@C=12pt{\ar@{--}|(.3){{\textstyle\!\!{\times}\!\!}}|(.6){{\textstyle\!\!{\times}\!\!}}[0,3] &&&{\ \raisebox{-6pt}{\rotatebox{90}{\mbox{\ding{34}}}}\ }\ar@*{[|(1.6)]}@{-}|(.65){{\textstyle{\circ}}}|(.85){{\textstyle\!\!{\times}\!\!}}[0,3] &&&},\end{aligned}$$ these bimodules are also bicomodules and, in fact, Hopf bimodules over ${\mathfrak{B}}(X)$ (see [@[Besp-TMF]; @[Besp-next]; @[Besp-Dr-(Bi)]; @[Majid-book]] for the general definitions). Braid group diagrams and quantum shuffles {#braid-group-diagrams-and-quantum-shuffles .unnumbered} ----------------------------------------- A standard graphical representation for the multiplication in ${\mathfrak{B}}(X)$ and its action on its modules is in terms of braid group diagrams. For example, the above left action is represented as (to be read from top down) $$\label{Sh12} \begin{tangles}{l} {\object{\scriptstyle {\times}}}\step{\object{\scriptstyle \otimes}}\step {\object{\scriptstyle {\circ}}}\step{\object{\scriptstyle {\times}}}\\ \vstr{200}{\mathrm{id}}\step[2]{\mathrm{id}}\step{\mathrm{id}}\end{tangles}\ \to \begin{tangles}{l} {\object{\scriptstyle {\times}}}\step {\object{\scriptstyle {\circ}}}\step{\object{\scriptstyle {\times}}}\\ \vstr{200}{\mathrm{id}}\step[1]{\mathrm{id}}\step{\mathrm{id}}\end{tangles}\ + \ \begin{tangles}{l} {\object{\scriptstyle {\times}}}\step {\object{\scriptstyle {\circ}}}\step[1]{\object{\scriptstyle {\times}}}\\ \vstr{200}\hx\step{\mathrm{id}}\end{tangles}\ + \ \ \begin{tangles}{l} {\object{\scriptstyle {\times}}}\step {\object{\scriptstyle {\circ}}}\step[1]{\object{\scriptstyle {\times}}}\\ \hx\step{\mathrm{id}}\\ {\mathrm{id}}\step\hx \end{tangles} \ \ = ({\mathrm{id}}+ \Psi_{1} + \Psi_{2}\Psi_{1})(X{\otimes}Y{\otimes}X),$$ where we use the “leg notation,” in the right-hand side, letting $\Psi_i$ denote the braiding of the $i$th and $(i+1)$th factors in a tensor product (our notation and conventions are the same as in [@[STbr]]). The braid group algebra element ${{{\mathop{\text{{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont}\sf Sh}}\nolimits}^{}_{1,2}}}\equiv{\mathrm{id}}+ \Psi_{1} + \Psi_{2}\Psi_{1}$ occurring here is an example of quantum shuffles. The product in ${\mathfrak{B}}(X)$ is in fact the shuffle product $$\label{Sh-prod} {{{\mathop{\text{{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont}\sf Sh}}\nolimits}^{}_{r,s}}}:X^{\otimes r}{\otimes}X^{\otimes s}\to X^{\otimes(r+s)}$$ on each graded subspace. The antipode restricted to each $X^{\otimes r}$ is up to a sign given by the “half-twist”—the braid group element obtained via the Matsumoto section from the longest element in the symmetric group: $$\label{antipode} {\raisebox{.5pt}{\large$\mathpzc{S}\kern-1pt$}}_r= (-1)^r\, \Psi_1 (\Psi_2\Psi_1)(\Psi_3\Psi_2\Psi_1)\dots (\Psi_{r-1}\Psi_{r-2}\dots\Psi_1):X^{\otimes r} \to X^{\otimes r}$$ (with the brackets inserted to highlight the structure, and the sign inherited from reversing the integrations); for example, $${\raisebox{.5pt}{\large$\mathpzc{S}\kern-1pt$}}_5=- \ \begin{tangles}{l} \hstr{70}\vstr{50}\hx\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \hstr{70}\vstr{50}{\mathrm{id}}\step[1]\hx\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \hstr{70}\vstr{50}{\mathrm{id}}\step[1]{\mathrm{id}}\step[1]\hx\step[1]{\mathrm{id}}\\ \hstr{70}\vstr{50}\hx\step[1]{\mathrm{id}}\step[1]\hx\\ \hstr{70}\vstr{50}{\mathrm{id}}\step[1]\hx\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \hstr{70}\vstr{50}\hx\step[1]\hx\step[1]{\mathrm{id}}\\ \hstr{70}\vstr{50}{\mathrm{id}}\step[1]\hx\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \hstr{70}\vstr{50}\hx\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \end{tangles}$$ The Hopf bimodules alluded to above are (some subspaces in) $\bigoplus\limits_{r,s\geq 0}X^{\otimes r}{\otimes}Y{\otimes}X^{\otimes s}$, with the left and right ${\mathfrak{B}}(X)$ actions on these also expressed in terms of quantum shuffles as $$\begin{aligned} {{{\mathop{\text{{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont}\sf Sh}}\nolimits}^{}_{r,s+1+t}}}&:X^{\otimes r}{\otimes}\bigl(X^{\otimes s}{\otimes}Y{\otimes}X^{\otimes t}\bigr)\to \bigoplus_{i=0}^r X^{\otimes(s+r-i)}{\otimes}Y{\otimes}X^{\otimes(t+i)}\pagebreak[3]\\[-6pt] \intertext{and} {{{\mathop{\text{{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont}\sf Sh}}\nolimits}^{}_{s+1+t,r}}}&: \bigl(X^{\otimes s}{\otimes}Y{\otimes}X^{\otimes t}\bigr) {\otimes}X^{\otimes r}\to \bigoplus_{i=0}^r X^{\otimes(s+r-i)}{\otimes}Y{\otimes}X^{\otimes(t+i)}.\end{aligned}$$ Hopf-algebra diagrams {#hopf-algebra-diagrams .unnumbered} --------------------- The four operations on bi(co)modules of a braided Hopf algebra ${\mathfrak{B}}$ are standardly expressed as $$\begin{tangles}{ccccccc} \lu&\qquad\qquad&\ld&\qquad\qquad&\ru&\qquad\qquad&\rd\\ \end{tangles}$$ which are respectively the left module structure ${\mathfrak{B}}{\otimes}{{\mathscr{Z}}}\to{{\mathscr{Z}}}$, the left comodule ${{\mathscr{Z}}}\to{\mathfrak{B}}{\otimes}{{\mathscr{Z}}}$, the right module structure ${{\mathscr{Z}}}{\otimes}{\mathfrak{B}}\to{{\mathscr{Z}}}$, and the right comodule structures ${{\mathscr{Z}}}\to{{\mathscr{Z}}}{\otimes}{\mathfrak{B}}$. The product and coproduct in the braided Hopf algebra itself are denoted as $\begin{tangles}{l} \hcu \end{tangles}$  and $\begin{tangles}{l} \hcd \end{tangles}$. The braiding is still denoted as $\begin{tangles}{l} \vstr{67}\hx \end{tangles}\ $, but in contrast to the braid-group diagrams, each line now represents a copy of ${\mathfrak{B}}$ or a ${\mathfrak{B}}$ (co)module. Adjoint action and Yetter–Drinfeld modules {#adjoint-action-and-yetterdrinfeld-modules .unnumbered} ------------------------------------------ The left and right actions of a braided Hopf algebra ${\mathfrak{B}}$ on its Hopf bimodule ${{\mathscr{Z}}}$ give rise to the *left adjoint action* ${\mathfrak{B}}{\otimes}{{\mathscr{Z}}}\to{{\mathscr{Z}}}$: $$\label{adja} \begin{tangles}{l} \vstr{240}\lu\object{\raisebox{22pt}{\kern-4pt\tiny$\blacktriangleright$}} \end{tangles} \ \ = \ \ \begin{tangles}{l} \hcd\step{\mathrm{id}}\\ \vstr{80}{\mathrm{id}}\step\hx\\ \lu\step\O{{\mathpzc{S}}}\\ \vstr{75}\step\ru \end{tangles}$$ A fundamental fact is that *the space of right coinvariants in a Hopf bimodule is invariant under the left adjoint action*; this actually leads to an equivalence of categories, the category of Hopf bimodules and the category of Yetter–Drinfeld modules [@[Besp-TMF]; @[Besp-next]; @[Sch-H-YD]; @[Wor]]. We recall some relevant facts about Yetter–Drinfeld modules in Appendix \[app:YD-axiom\]. In our case of modules spanned by dressed vertex operators, the right coinvariants—all those $y$ that map as $y\mapsto y{\otimes}1$ under the right coaction—are simply the vertex operators dressed by screenings only from the left, i.e., elements of $X^{\otimes r}{\otimes}Y$, for example, $\xymatrix@C=40pt@1{ \ar@*{[|(1.6)]}@{-}|(.3){{\textstyle\!\!{\times}\!\!}}|(.6){{\textstyle\!\!{\times}\!\!}}|(.75){{\textstyle{\circ}}}[0,2]&& }$. In terms of *braid group* diagrams (with the lines representing the $X$ and $Y$ spaces), an example of the left adjoint action on such spaces is given by $$\label{adja.1.2} \begin{tangles}{l} \hstr{80}{\object{\scriptstyle {\times}}}\step{\object{\scriptstyle \otimes}}\step {\object{\scriptstyle {\times}}}\step{\object{\scriptstyle {\times}}}\step{\object{\scriptstyle {\circ}}}\\ \hstr{80}\vstr{280}{\mathrm{id}}\step[2]{\mathrm{id}}\step{\mathrm{id}}\step{\mathrm{id}}\end{tangles}\ \to\ \begin{tangles}{l} \\ \hstr{80}\vstr{320}{\mathrm{id}}\step{\mathrm{id}}\step{\mathrm{id}}\step{\mathrm{id}}\end{tangles} \;+ \ \begin{tangles}{l} \hstr{80}\vstr{50}{\mathrm{id}}\step{\mathrm{id}}\step{\mathrm{id}}\step{\mathrm{id}}\\ \hstr{80}\vstr{220}\hx\step{\mathrm{id}}\step{\mathrm{id}}\\ \hstr{80}\vstr{50}{\mathrm{id}}\step{\mathrm{id}}\step{\mathrm{id}}\step{\mathrm{id}}\\ \end{tangles} \;+ \; \begin{tangles}{l} \hstr{80}\vstr{160}\hx\step{\mathrm{id}}\step{\mathrm{id}}\\ \hstr{80}\vstr{160}{\mathrm{id}}\step\hx\step{\mathrm{id}}\end{tangles} \;- \; \begin{tangles}{l}\vstr{90} \hstr{90}\vstr{80}\hx\step{\mathrm{id}}\step{\mathrm{id}}\\ \hstr{90}\vstr{80}{\mathrm{id}}\step\hx\step{\mathrm{id}}\\ \hstr{90}\vstr{80}{\mathrm{id}}\step{\mathrm{id}}\step\hx\\ \hstr{90}\vstr{80}{\mathrm{id}}\step{\mathrm{id}}\step\hx \end{tangles} \;- \; \begin{tangles}{l} \hstr{90}\vstr{66}\hx\step{\mathrm{id}}\step{\mathrm{id}}\\ \hstr{90}\vstr{66}{\mathrm{id}}\step\hx\step{\mathrm{id}}\\ \hstr{90}\vstr{66}{\mathrm{id}}\step{\mathrm{id}}\step\hx\\ \hstr{90}\vstr{66}{\mathrm{id}}\step{\mathrm{id}}\step\hx\\ \hstr{90}\vstr{66}{\mathrm{id}}\step\hx\step{\mathrm{id}}\end{tangles} \;- \; \begin{tangles}{l} \hstr{90}\vstr{55}\hx\step{\mathrm{id}}\step{\mathrm{id}}\\ \hstr{90}\vstr{55}{\mathrm{id}}\step\hx\step{\mathrm{id}}\\ \hstr{90}\vstr{55}{\mathrm{id}}\step{\mathrm{id}}\step\hx\\ \hstr{90}\vstr{55}{\mathrm{id}}\step{\mathrm{id}}\step\hx\\ \hstr{90}\vstr{55}{\mathrm{id}}\step\hx\step{\mathrm{id}}\\ \hstr{90}\vstr{55}\hx\step{\mathrm{id}}\step{\mathrm{id}}\end{tangles}$$ where a single “new” cross arrives to each of the three possible positions in two ways, one with the plus and the other with the minus sign in front (which is something expected of an “adjoint” action). That the cross never stays to the right of $\circ$ is precisely a manifestation of the above invariance statement for the space of right coinvariants. This means that a number of termsthat follow when expressing  in terms of braid group diagrams cancel. The left adjoint action  can in fact be expressed more economically as follows. We define a modified left action $\ \begin{tangles}{l}\vstr{70}\lu \object{\raisebox{5.2pt}{\tiny$\bullet$}} \end{tangles}\ $ of ${\mathfrak{B}}(X)$ on its Hopf bimodules spanned by dressed vertex operators by allowing the “new” crosses to arrive only to the left of $\circ$, for example, $$\label{left-dot-example} \begin{tangles}{l} \hstr{80}{\object{\scriptstyle {\times}}}\step{\object{\scriptstyle \otimes}}\step {\object{\scriptstyle {\times}}}\step{\object{\scriptstyle {\times}}}\step{\object{\scriptstyle {\circ}}}\\ \hstr{80}\vstr{200}{\mathrm{id}}\step[2]{\mathrm{id}}\step{\mathrm{id}}\step{\mathrm{id}}\end{tangles}\ \to\ \begin{tangles}{l} \\ \hstr{80}\vstr{240}{\mathrm{id}}\step{\mathrm{id}}\step{\mathrm{id}}\step{\mathrm{id}}\end{tangles} \;+ \ \begin{tangles}{l} \hstr{80}\vstr{240}\hx\step{\mathrm{id}}\step{\mathrm{id}}\\ \end{tangles} \;+ \; \begin{tangles}{l} \hstr{80}\vstr{120}\hx\step{\mathrm{id}}\step{\mathrm{id}}\\ \hstr{80}\vstr{120}{\mathrm{id}}\step\hx\step{\mathrm{id}}\end{tangles}$$ (more crosses might be initially placed to the right of the vertex $\circ$; the action does not see them). In general, $\ \begin{tangles}{l}\vstr{70}\lu \object{\raisebox{5.2pt}{\tiny$\bullet$}} \end{tangles}\ $ is the map $$\label{FromLeftii} \ \begin{tangles}{l}\vstr{70}\lu \object{\raisebox{5.2pt}{\tiny$\bullet$}} \end{tangles}\ = {{{\mathop{\text{{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont}\sf Sh}}\nolimits}^{}_{r,s}}}:X^{\otimes r}{\otimes}\bigl(X^{\otimes s}{\otimes}Y\bigr) \to X^{\otimes(r+s)}{\otimes}Y.$$ Similarly, a modified right action $\ \begin{tangles}{l} \vstr{70}\object{\raisebox{5.1pt}{\tiny$\bullet$}}\ru \end{tangles}\ $ on the space of right coinvariants is defined by first letting the new cross to be braided with the vertex and then shuffling into all possible positions relative to the “old” crosses: $$\begin{tangles}{l} \hstr{80}{\object{\scriptstyle {\times}}}\step{\object{\scriptstyle {\times}}}\step {\object{\scriptstyle {\circ}}}\step{\object{\scriptstyle \otimes}}\step{\object{\scriptstyle {\times}}}\\ \hstr{80}\vstr{200}{\mathrm{id}}\step{\mathrm{id}}\step{\mathrm{id}}\step[2]{\mathrm{id}}\end{tangles}\ \to\ \begin{tangles}{l} \hstr{80}\vstr{240}{\mathrm{id}}\step{\mathrm{id}}\step\hx\\ \end{tangles}\ \ + \ \ \begin{tangles}{l} \hstr{80}\vstr{120}{\mathrm{id}}\step{\mathrm{id}}\step\hx\\ \hstr{80}\vstr{120}{\mathrm{id}}\step\hx\step{\mathrm{id}}\\ \end{tangles}\ \ + \ \ \begin{tangles}{l} \hstr{80}\vstr{80}{\mathrm{id}}\step{\mathrm{id}}\step\hx\\ \hstr{80}\vstr{80}{\mathrm{id}}\step\hx\step{\mathrm{id}}\\ \hstr{80}\vstr{80}\hx\step{\mathrm{id}}\step{\mathrm{id}}\end{tangles}$$ which in general is $$\label{FromRightii} \ \begin{tangles}{l} \vstr{70}\object{\raisebox{5.1pt}{\tiny$\bullet$}}\ru \end{tangles}\ = {{{\mathop{\text{{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont}\sf Sh}}\nolimits}^{}_{s,r}}}{\mathbin{\raisebox{1pt}{\,$\scriptscriptstyle\circ$\,}}}({\mathrm{id}}^{\otimes s}{\otimes}{\pmb{\Psi}}_{1,r}) :\bigl(X^{\otimes s}{\otimes}Y\bigr){\otimes}X^{\otimes r} \to X^{\otimes(s+r)}{\otimes}Y,$$ where ${\pmb{\Psi}}_{s,r}$ is the braiding of an $s$-fold tensor product with an $r$-fold tensor product. The $\ \begin{tangles}{l}\vstr{70}\lu \object{\raisebox{5.2pt}{\tiny$\bullet$}} \end{tangles}\ $ and $\ \begin{tangles}{l} \vstr{70}\object{\raisebox{5.1pt}{\tiny$\bullet$}}\ru \end{tangles}\ $ actions preserve the spaces of right coinvariants and commute with each other. The “economic” expression for adjoint action  is [@[STbr]] $$\label{dot-adja} \begin{tangles}{l} \vstr{200}\lu \object{\raisebox{18pt}{\kern-4pt\tiny$\blacktriangleright$}} \end{tangles} \ \ = \ \ \begin{tangles}{l} \hcd\step{\mathrm{id}}\\ \vstr{80}{\mathrm{id}}\step\hx\\ \vstr{100}\lu\object{\raisebox{8.4pt}{\tiny$\bullet$}} \step\O{{\mathpzc{S}}}\\ \vstr{75}\step\object{\raisebox{5.8pt}{\tiny$\bullet$}}\ru \end{tangles}$$ This diagram is the map $$\label{adja-formula} {{ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}}_{r,s}\equiv \sum_{i=0}^{r} {{{\mathop{\text{{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont}\sf Sh}}\nolimits}^{}_{r - i, s + i}}} \bigl({{{\mathop{\text{{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont}\sf Sh}}\nolimits}^{}_{s, i}}} {{\pmb{\Psi}}_{1,i}^{\uparrows}} \,{{\raisebox{.5pt}{\large$\mathpzc{S}\kern-1pt$}}_i^{\uparrow(s+1)}}\, {\pmb{\Psi}}_{i,s+1} \bigr)^{{\uparrow}(r-i)}: X^{\otimes r}{\otimes}(X^{\otimes s}{\otimes}Y)\to X^{\otimes(s+r)}{\otimes}Y.$$ Multivertex Yetter–Drinfeld modules {#multivertex-yetterdrinfeld-modules .unnumbered} ----------------------------------- More general, *multivertex*, Yetter–Drinfeld ${\mathfrak{B}}(X)$-modules can be constructed by letting two or more vertices (the $Y$ spaces) sit on the same line, e.g., $$\label{ex1} \xymatrix@R=6pt@C=50pt{ \ar@*{[|(1.6)]}@{-}|(.15){{\textstyle\!\!{\times}\!\!}} |(.3){{\textstyle{\circ}}}|(.45){{\textstyle\!\!{\times}\!\!}}|(.6){{\textstyle\!\!{\times}\!\!}}|(.75){{\textstyle\!\!{\times}\!\!}}|(.90){{\textstyle{\circ}}}[0,2]&& }\quad\text{or}\quad \xymatrix@R=6pt@C=50pt{ \ar@*{[|(1.6)]}@{-}|(.15){{\textstyle\!\!{\times}\!\!}} |(.25){{\textstyle{\circ}}}|(.4){{\textstyle\!\!{\times}\!\!}}|(.5){{\textstyle\!\!{\times}\!\!}}|(.65){{\textstyle{\circ}}} |(.8){{\textstyle\!\!{\times}\!\!}} |(.90){{\textstyle{\circ}}}[0,2]&& }$$ These diagrams respectively represent $X{\otimes}Y{\otimes}X^{\otimes 3}{\otimes}Y$ and $X{\otimes}Y{\otimes}X^{\otimes 2}{\otimes}Y{\otimes}X{\otimes}Y$ (in general, different spaces could be taken instead of copies of the same $Y$, but in our setting they are all the same). By definition, the ${\mathfrak{B}}(X)$ action and coaction on these are 1. \[2items\] the “cumulative” left adjoint action, and 2. deconcatenation up to the first $\circ$. The “cumulative” adjoint means that all the $\circ$ except the rightmost one are viewed on equal footing with the $\times$ under this action: the adjoint action of $X^{\otimes r}$ on the space $X^{\otimes s}{\otimes}Y{\otimes}X^{\otimes t}{\otimes}Y$ in a two-vertex module is given by ${{ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}}_{r,s+1+t}$. For example, the left adjoint action $\xymatrix@1@C=30pt{ \ar@{--} |(.5){{\textstyle\!\!{\times}\!\!}}[0,2]&& } \ { \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}\ \xymatrix@1@C=40pt{ \ar@*{[|(1.6)]}@{-} |(.3){{\textstyle{\circ}}}|(.65){{\textstyle\!\!{\times}\!\!}}|(.90){{\textstyle{\circ}}}[0,2]&& }$ is given by the braid group diagrams that are exactly those in the right-hand side of , with the corresponding strand representing not ${\times}{}=X$ but ${\circ}{}=Y$. The ${\mathfrak{B}}(X)$ coaction by deconcatenation up to the first vertex means, for example, that at most one $\times$ can be deconcatenated in each diagram in . For multivertex Yetter–Drinfeld modules, the form  of the adjoint action is valid if $\ \begin{tangles}{l}\vstr{70}\lu \object{\raisebox{5.2pt}{\tiny$\bullet$}} \end{tangles}\ $ is understood as the “cumulative” action preserving right coinvariants; for example, $$\xymatrix@1@C=30pt{ \ar@{--} |(.5){{\textstyle\!\!{\times}\!\!}}[0,2]&& } \ \mbox{\small$\bullet$} \ \xymatrix@1@C=40pt{ \ar@*{[|(1.6)]}@{-} |(.3){{\textstyle{\circ}}}|(.65){{\textstyle\!\!{\times}\!\!}}|(.90){{\textstyle{\circ}}}[0,2]&& }$$ is given just by the braid group diagrams in the right-hand side of  with the second strand representing not ${\times}{}=X$ but ${\circ}{}=Y$. Fusion product {#fusion-product .unnumbered} -------------- The multivertex Yetter–Drinfeld modules are not exactly tensor products of single-vertex ones—they carry a different action, which is *not* $(\mu_{{{\mathscr{Y}}}}{\otimes}\mu_{{{\mathscr{Z}}}}){\mathbin{\raisebox{1pt}{\,$\scriptscriptstyle\circ$\,}}}\Delta$, and the coaction is not diagonal either. They actually follow via a *fusion product* [@[STbr]], which is defined on two single-vertex Yetter–Drinfeld modules (each of which is the space of right coinvariants in a Hopf bimodule) as $$\label{fusion-def} \begin{tangles}{l} {\mathrm{id}}\step[1]\ld\\ \object{\raisebox{8pt}{\tiny$\bullet$}}\ru\step[1]{\mathrm{id}}\end{tangles}$$ which is the map $$\sum_{j=0}^{t} {{{\mathop{\text{{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont}\sf Sh}}\nolimits}^{}_{s, j}}} {{\pmb{\Psi}}_{1, j}^{\uparrows}}: (X^{\otimes s}{\otimes}Y){\otimes}(X^{\otimes t}{\otimes}Y)\to X^{\otimes s}{\otimes}Y{\otimes}X^{\otimes t}{\otimes}Y$$ on each $(s,t)$ component. For example, if $s=2$ and $t=3$, the top of the above diagram can be represented as $$\xymatrix@1@C=50pt{ \ar@*{[|(1.6)]}@{-}|(.15){{\textstyle\!\!{\times}\!\!}} |(.55){{\textstyle\!\!{\times}\!\!}}|(.80){{\textstyle{\circ}}}[0,2]&& }\ \ {\otimes}\ \ \xymatrix@1@C=50pt{ \ar@*{[|(1.6)]}@{-} |(.25){{\textstyle\!\!{\times}\!\!}}|(.5){{\textstyle\!\!{\times}\!\!}}|(.7){{\textstyle\!\!{\times}\!\!}} |(.85){{\textstyle{\circ}}}[0,2]&& }$$ and then in view of the definition of  $\ \begin{tangles}{l} \vstr{70}\object{\raisebox{5.1pt}{\tiny$\bullet$}}\ru \end{tangles}\ $, the meaning of  is that $j\geq 0$ crosses from the right factor are detached from their “native” module and sent to mix with the left crosses (the sum over $j$ is taken in accordance with the definition of the coaction). The construction extends by taking the fusion product of multivertex modules: the coaction in  is then the one just described, by deconcatenation up to the first vertex, and the $\ \begin{tangles}{l} \vstr{70}\object{\raisebox{5.1pt}{\tiny$\bullet$}}\ru \end{tangles}\ $ action on a multivertex module is “cumulative,” i.e., each cross acting from the right, e.g., on $\xymatrix@1@C=50pt{ \ar@*{[|(1.6)]}@{-}|(.15){{\textstyle\!\!{\times}\!\!}} |(.3){{\textstyle{\circ}}}|(.5){{\textstyle\!\!{\times}\!\!}}|(.7){{\textstyle\!\!{\times}\!\!}} |(.90){{\textstyle{\circ}}}[0,2]&& }$, arrives at each of the *five* possible positions. Duality in the category of Yetter–Drinfeld modules {#sec:duality} ================================================== We now consider duality in a braided category of representations of a braided Hopf algebra ${\mathfrak{B}}$. We briefly recall the standard definitions and basic properties, and then assume that duality exists in the setting of the preceding section; this then allows us to construct endomorphisms of the identity functor in Sec. \[sec:p\]. For a ${\mathfrak{B}}$-module ${{\mathscr{Z}}}$, we let ${{}^{\vee}\!{{\mathscr{Z}}}}$ denote the left dual module in the same (rigid) braided category. The duality means that there are coevaluation and evaluation maps $$\begin{tangles}{l} \vphantom{x}\\ \Coev\\ {\object{\scriptstyle {{\mathscr{Z}}}}}{\object{\scriptstyle \kern60pt{{}^{\vee}\!{{\mathscr{Z}}}}}} \end{tangles} \qquad\qquad\text{and}\qquad \begin{tangles}{l} {\object{\scriptstyle {{}^{\vee}\!{{\mathscr{Z}}}}}}{\object{\scriptstyle \kern60pt {{\mathscr{Z}}}}}\\ \Ev \end{tangles}$$ which are morphisms in the category and satisfy the axioms $$\begin{tangles}{l} \vstr{15}{\mathrm{id}}\\ \vstr{67}\hstr{50}{\mathrm{id}}\step[2]\coev\\ \vstr{67}\hstr{50}\ev\step[2]{\mathrm{id}}\\ \vstr{67}\hstr{50}\vstr{33}\step[4]{\mathrm{id}}\end{tangles} \quad=\ \ \ \begin{tangles}{l} \vstr{175}{\mathrm{id}}\end{tangles} \qquad\text{and}\qquad \begin{tangles}{l} \vstr{67}\hstr{50}\vstr{33}\step[4]{\mathrm{id}}\\ \vstr{67}\hstr{50}\coev\step[2]{\mathrm{id}}\\ \vstr{67}\hstr{50}{\mathrm{id}}\step[2]\ev\\ \vstr{15}{\mathrm{id}}\end{tangles} \quad=\ \ \ \begin{tangles}{l} \vstr{175}{\mathrm{id}}\end{tangles}$$ where the two straight lines are ${\mathrm{id}}_{{{}^{\vee}\!{{\mathscr{Z}}}}}$ and ${\mathrm{id}}_{{{\mathscr{Z}}}}$.  It follows that $$\begin{tangles}{l} \hxx\step[1]{\mathrm{id}}\\ \hh{\mathrm{id}}\step\ev \end{tangles} \ \ = \ \ \begin{tangles}{l} {\mathrm{id}}\step[1]\hx\\ \hh\ev\step[1]{\mathrm{id}}\end{tangles}$$ and similarly for the coevaluation. The dual ${{}^{\vee}\!{{\mathscr{Z}}}}$ to a left–left Yetter–Drinfeld ${\mathfrak{B}}$-module ${{\mathscr{Z}}}$ is a left–left Yetter–Drinfeld ${\mathfrak{B}}$-module with the action and coaction, *temporarily* denoted by $\ \begin{tangles}{l}\vstr{80}\lu \object{\raisebox{5.2pt}{$\oleft$}} \end{tangles}\ $  and $\ \begin{tangles}{l}\vstr{80} \ld\object{\raisebox{5.2pt}{$\oleft$}} \end{tangles}\ $,  defined as [@[Besp-next]] $$\label{on-dual} \begin{tangles}{l} {\mathrm{id}}\step[1]{\mathrm{id}}\\ \vstr{120}\lu[1] \object{\raisebox{9.3pt}{$\oleft$}}\\ \step[1]{\mathrm{id}}\end{tangles}\ \ = \ \ \begin{tangles}{l} \vphantom{x}\\ \O{{\mathpzc{S}}}\step{\mathrm{id}}\step\coev\\ \vstr{50}\hx\step[1]{\mathrm{id}}\step[2]{\mathrm{id}}\\ {\mathrm{id}}\step[1] \lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[2]{\mathrm{id}}\\ \ev\step[2]{\mathrm{id}}\end{tangles} \qquad\text{and}\qquad \begin{tangles}{l} \step[1]{\mathrm{id}}\\ \vstr{120}\ld[1] \object{\raisebox{9.3pt}{$\oleft$}}\\ {\mathrm{id}}\step[1]{\mathrm{id}}\end{tangles}\ \ = \ \ \begin{tangles}{l} {\mathrm{id}}\step[3]\coev\\ {\mathrm{id}}\step[2]\ld\step[2]{\mathrm{id}}\\ \vstr{50}{\mathrm{id}}\step[2]\hxx\step[2]{\mathrm{id}}\\ \ev\step\O{{\mathpzc{S}}^{{\scriptscriptstyle-1}}}\step[2]{\mathrm{id}}\end{tangles}$$ The definitions are equivalent to the properties (which, inter alia, imply that the evaluation is a ${\mathfrak{B}}$ module comodule morphism) $$\label{equiv-to} \begin{tangles}{l} {\mathrm{id}}\step[1]{\mathrm{id}}\step[2]{\mathrm{id}}\\ \vstr{120}\lu[1] \object{\raisebox{9.3pt}{$\oleft$}}\step[2]{\mathrm{id}}\\ \step[1]\ev \end{tangles}\ \ = \ \ \begin{tangles}{l} \O{{\mathpzc{S}}}\step{\mathrm{id}}\step[1]{\mathrm{id}}\\ \vstr{50}\hx\step[1]{\mathrm{id}}\\ {\mathrm{id}}\step\lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\\ \ev \end{tangles} \qquad\text{and}\qquad \begin{tangles}{l} \vstr{50}\step[1]{\mathrm{id}}\step[2]{\mathrm{id}}\\ \vstr{120}\ld[1]\object{\raisebox{9.3pt}{$\oleft$}}\step[2]{\mathrm{id}}\\[-4pt] {\mathrm{id}}\step[1]\ev \end{tangles}\ \ = \ \ \begin{tangles}{l} {\mathrm{id}}\step[2]\ld\\ \vstr{50}{\mathrm{id}}\step[2]\hxx\\ \ev\step\O{{\mathpzc{S}}^{{\scriptscriptstyle-1}}} \end{tangles}$$ We prove the Yetter–Drinfeld property for for $\ \begin{tangles}{l}\vstr{80}\lu \object{\raisebox{5.2pt}{$\oleft$}} \end{tangles}\ $  and $\ \begin{tangles}{l}\vstr{80} \ld\object{\raisebox{5.2pt}{$\oleft$}} \end{tangles}\ $  for completeness. In view of , it is easiest to verify the Yetter–Drinfeld axiom by establishing that $$\label{to-prove} \begin{tangles}{l} \vstr{90}\cd\step{\mathrm{id}}\step[2]{\mathrm{id}}\\ \vstr{50}{\mathrm{id}}\step[2]\hx\step[2]{\mathrm{id}}\\ \vstr{90}\lu[2] \object{\raisebox{6.2pt}{$\oleft$}} \step\hd\step[1.5]{\mathrm{id}}\\ \vstr{90}\ld[2]\object{\raisebox{6.2pt}{$\oleft$}}\step\ddh \step[1.5]{\mathrm{id}}\\ \vstr{50}{\mathrm{id}}\step[2]\hx\step[2]{\mathrm{id}}\\ \vstr{90}\cu\step\ev \end{tangles}\ \ = \ \ \begin{tangles}{l} \cd\step\ld\object{\raisebox{6.9pt}{$\oleft$}}\step[2]{\mathrm{id}}\\[-4pt] {\mathrm{id}}\step[2]\hx\step{\mathrm{id}}\step[2]{\mathrm{id}}\\ \cu\step\lu \object{\raisebox{7.2pt}{$\oleft$}}\step[2]{\mathrm{id}}\\ \step{\mathrm{id}}\step[3]\ev \end{tangles}$$ Pushing the new action and then the coaction “to the other side,” we see that the left-hand side of , by the above properties, is equal to $$\begin{tangles}{l} \hcd{\object{\scriptstyle \quad\checkmark}}\step[2]{\mathrm{id}}\step[1]{\mathrm{id}}\\[-4pt] \O{{\mathpzc{S}}}\step[1]\x\step[1]{\mathrm{id}}\\ \vstr{50}\hx\step[2]\hxx\\ {\mathrm{id}}\step[1]\hd\step[1]\hld\step{\mathrm{id}}\\ \dh\step[1]\hx\step[.5]{\mathrm{id}}\step{\mathrm{id}}\\[-.8pt] \step[.5]\hx\step \hlu\object{\raisebox{13.2pt}{\kern-4pt\tiny$\blacktriangleright$}} \step[.5]\ddh\\ \step[.5]\O{{\mathpzc{S}}^{{\scriptscriptstyle-1}}}\step[.75] {\makeatletter\@ev{0,\hm@de}{10,\hm@detens}{15}b\makeatother} \step[1.25]\dd{\object{\scriptstyle \checkmark}}\\ \step[.5]\cu \end{tangles} \quad=\quad \begin{tangles}{l} \O{{\mathpzc{S}}}\step[1]{\mathrm{id}}\step[2.5]{\mathrm{id}}\\ \hx\step[2.5]{\mathrm{id}}\\ {\mathrm{id}}\step[.5]\hcd\step[1]\ld\\ {\mathrm{id}}\step[.5]{\mathrm{id}}\step[1]\hx\step{\mathrm{id}}\\ {\mathrm{id}}\step[.5]\hcu\step \lu\object{\raisebox{8.1pt}{\kern-4pt\tiny$\blacktriangleright$}}\\ \hx\step[2]\ddh\\ \O{{\mathpzc{S}}^{{\scriptscriptstyle-1}}}\step[1]\ev \end{tangles} \quad=\quad \begin{tangles}{l} \O{{\mathpzc{S}}}\step[1]{\mathrm{id}}\step[1.5]{\mathrm{id}}\\ \vstr{50}\hx\step[1.5]{\mathrm{id}}\\ {\mathrm{id}}\step[.5]\hcd\step{\mathrm{id}}\\ \vstr{50}{\mathrm{id}}\step[.5]\hd\step[.5]\hx\\ {\mathrm{id}}\step[1]\hlu[1]\object{\raisebox{12.9pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[-.5]\hld \step[1]{\mathrm{id}}\\ \vstr{50}{\mathrm{id}}\step[.5]\ddh\step[.5]\hx\\ {\mathrm{id}}\step[.5]\hcu\step{\mathrm{id}}\\ \vstr{50}\hx\step[1.5]{\mathrm{id}}\\ \O{{\mathpzc{S}}^{{\scriptscriptstyle-1}}} \step[.75]{\makeatletter\@ev{0,\hm@de}{10,\hm@detens}{15}b\makeatother} \end{tangles} \quad=\quad \begin{tangles}{l} \hcd\step{\mathrm{id}}\step[1.5]{\mathrm{id}}\\ \vstr{50}{\mathrm{id}}\step\hx\step[1.5]{\mathrm{id}}\\ {\mathrm{id}}\step{\mathrm{id}}\step\O{{\mathpzc{S}}}\step[.5]\dd\\ {\mathrm{id}}\step{\mathrm{id}}\step\hlu\object{\raisebox{13pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[-.5]\hld\\ {\mathrm{id}}\step{\mathrm{id}}\step\O{{\mathpzc{S}}^{{\scriptscriptstyle-1}}}\step[.5]\d\\ \vstr{50}{\mathrm{id}}\step\hx\step[1.5]{\mathrm{id}}\\ \hcu\step[1.75]{\makeatletter\@ev{0,\hm@de}{0,\hm@detens}{15}b\makeatother} \end{tangles}$$ In the first diagram, we insert ${\raisebox{.5pt}{\large$\mathpzc{S}\kern-1pt$}}$ at the position of the upper checkmark and ${\raisebox{.5pt}{\large$\mathpzc{S}\kern-1pt$}}^{-1}$ into the same line, at the lower checkmark, and use the properties of the antipode, $$\begin{tangles}{l} \cd\\ \O{{\mathpzc{S}}}\step[2]\O{{\mathpzc{S}}} \end{tangles}\ = \ \begin{tangles}{l} \step\O{{\mathpzc{S}}}\\ \cd\\ \xx\\ \end{tangles} \qquad \text{and} \quad \begin{tangles}{l} \O{{\mathpzc{S}}^{{\scriptscriptstyle-1}}}\step[2]\O{{\mathpzc{S}}^{{\scriptscriptstyle-1}}}\\ \cu \end{tangles}\ = \ \begin{tangles}{l} \x\\ \cu\\ \step\O{{\mathpzc{S}}^{{\scriptscriptstyle-1}}} \end{tangles}$$ This readily gives the second diagram above, where we further recognize the right-hand side of the Yetter–Drinfeld axiom assumed for the module. After using it (the third diagram), and after another application of the properties of ${\raisebox{.5pt}{\large$\mathpzc{S}\kern-1pt$}}$ and ${\raisebox{.5pt}{\large$\mathpzc{S}\kern-1pt$}}^{-1}$, we obtain the fourth diagram, and it is immediate to see that it coincides with the right-hand side of  also rewritten by pushing $\ \begin{tangles}{l}\vstr{80}\lu \object{\raisebox{5.2pt}{$\oleft$}} \end{tangles}\ $  and $\ \begin{tangles}{l}\vstr{80} \ld\object{\raisebox{5.2pt}{$\oleft$}} \end{tangles}\ $ “to the other side.” Assuming a rigid category ------------------------- We further assume that the category of $n$-vertex Yetter–Drinfeld ${\mathfrak{B}}$-modules is rigid; this means that the dual modules are modules in the same category—in our case, multivertex Yetter–Drinfeld ${\mathfrak{B}}$-modules, and the action and coaction defined in  are just those in —and hence the evaluation map satisfies the properties $$\label{equiv-to-2} \begin{tangles}{l} {\mathrm{id}}\step[1]{\mathrm{id}}\step[2]{\mathrm{id}}\\ \vstr{120}\lu[1] \object{\raisebox{10pt}{\kern-4pt\tiny$\blacktriangleright$}} \step[2]{\mathrm{id}}\\ \step[1]\ev \end{tangles}\ \ = \ \ \begin{tangles}{l} \O{{\mathpzc{S}}}\step{\mathrm{id}}\step[1]{\mathrm{id}}\\ \vstr{50}\hx\step[1]{\mathrm{id}}\\ {\mathrm{id}}\step\lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\\ \ev \end{tangles} \qquad\text{and}\qquad \begin{tangles}{l} \vstr{50}\step[1]{\mathrm{id}}\step[2]{\mathrm{id}}\\ \vstr{120}\ld[1]\step[2]{\mathrm{id}}\\ {\mathrm{id}}\step[1]\ev \end{tangles}\ \ = \ \ \begin{tangles}{l} {\mathrm{id}}\step[2]\ld\\ {\mathrm{id}}\step[2]\hxx\\ \ev\step\O{{\mathpzc{S}}^{{\scriptscriptstyle-1}}} \end{tangles}$$ for any pair of Yetter–Drinfeld ${\mathfrak{B}}$-modules. Evidently, we then also have $$\label{equiv-to-3} \begin{tangles}{l} {\mathrm{id}}\step\coev\\ \vstr{67}\hx\step[1]\dd\\ {\mathrm{id}}\step\lu[1]\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}} \end{tangles} \ \ =\quad \begin{tangles}{l} \O{{\mathpzc{S}}}\step\coev\\ \lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[2]{\mathrm{id}}\\ \vstr{50}\step{\mathrm{id}}\step[2]{\mathrm{id}}\end{tangles} \qquad\text{and}\qquad \begin{tangles}{l} \coev\\ {\mathrm{id}}\step\ld\\ \hx\step[1]{\mathrm{id}}\end{tangles} \ \ = \ \ \begin{tangles}{l} \step[1]\coev\\ \ld\step[2]{\mathrm{id}}\\ \O{{\mathpzc{S}}^{{\scriptscriptstyle-1}}}\step{\mathrm{id}}\step[2]{\mathrm{id}}\end{tangles}$$ If the category ${{}\mbox{\small${}^{{\mathfrak{B}}}_{{\mathfrak{B}}}$}{\mathcal{Y\kern-3ptD}}}$ of Yetter–Drinfeld ${\mathfrak{B}}$-modules is rigid, then for each ${{\mathscr{Z}}}\in{{}\mbox{\small${}^{{\mathfrak{B}}}_{{\mathfrak{B}}}$}{\mathcal{Y\kern-3ptD}}}$, there is a morphism $\chi_{{{\mathscr{Z}}}}:{{\mathscr{Y}}}\to{{\mathscr{Y}}}$ for any ${{\mathscr{Y}}}\in{{}\mbox{\small${}^{{\mathfrak{B}}}_{{\mathfrak{B}}}$}{\mathcal{Y\kern-3ptD}}}$, defined as $$\label{YD-loop} \begin{tangles}{l} {\mathrm{id}}{\object{\scriptstyle \kern10pt{{\mathscr{Y}}}}}\step[3]{\object{\scriptstyle \kern-12pt{{\mathscr{Z}}}}}\coev {\object{\scriptstyle \kern12pt{{}^{\vee}\!{{\mathscr{Z}}}}}}\\[-4pt] \vstr{50}\dh\step[1.5]\dd\step[2]{\mathrm{id}}\\[-2pt] \step[.25]\obox{2}{\mathsf{B}^2}\step[2.75]{\mathrm{id}}\\[-2pt] \vstr{50}\hdd\step[1.5]\d\step[2]{\mathrm{id}}\\ {\mathrm{id}}\step[3]\O{\vartheta}\step[1]\dd\\ {\mathrm{id}}\step[2.5]\obox{2}{\mathsf{B}}\\ \vstr{33}{\mathrm{id}}\step[3]{\mathrm{id}}\step{\mathrm{id}}\\ \hh{\mathrm{id}}\step[3]\ev \end{tangles} \qquad = \qquad \begin{tangles}{l} {\mathrm{id}}{\object{\scriptstyle \kern10pt{{\mathscr{Y}}}}}\step[3]{\object{\scriptstyle \kern-12pt{{\mathscr{Z}}}}}\coev {\object{\scriptstyle \kern12pt{{}^{\vee}\!{{\mathscr{Z}}}}}}\\[-4pt] \vstr{50}\dh\step[1.5]\dd\step[2]{\mathrm{id}}\\[-2pt] \step[.25]\obox{2}{\mathsf{B}^2}\step[2.75]{\mathrm{id}}\\[-2pt] \vstr{50}\hdd\step[1.5]\d\step[2]{\mathrm{id}}\\ {\mathrm{id}}\step[3]\O{\vartheta}\step[2]{\mathrm{id}}\\ {\mathrm{id}}\step[3]\O{{\mbox{\LARGE${\times}$}}}\step[1]\dd\\ \vstr{50}{\mathrm{id}}\step[3]\hx\\ {\mathrm{id}}\step[3]{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother} \end{tangles}$$ where $\mathsf{B}$ is defined in  and $\vartheta$ is any ${\mathfrak{B}}$ module comodule morphism. In the second diagram, Bespalov’s “squared relative antipode” [@[Besp-TMF]] $$\label{eq:sigma2} \sigma_2\equiv\ \ \ \begin{tangles}{l} \vstr{50}{\mathrm{id}}\\ \O{{\mbox{\LARGE${\times}$}}}\\ \vstr{50}{\mathrm{id}}\end{tangles} \ \ \ = \ \ \ \begin{tangles}{l} \ld[2]\\ \O{{\mathpzc{S}}^2}\step[2]{\O{{\Pi_{_{\bullet}}}}}\\ \x\\ \object{\raisebox{8pt}{\tiny$\bullet$}}\ru[2] \end{tangles} \ \ \ = \ \ \ \begin{tangles}{l} \ld\\ \O{{\mathpzc{S}}}\step{\mathrm{id}}\\ \lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\\ \end{tangles} \qquad\text{such that}\qquad \begin{tangles}{l} \vstr{150}\lu\object{\raisebox{12.9pt}{\kern-4pt\tiny$\blacktriangleright$}}\\[-4pt] \step[1]\O{{\mbox{\LARGE${\times}$}}}\\ \step[1]{\mathrm{id}}\end{tangles} \quad =\quad \begin{tangles}{l} \x\\ \O{{\mbox{\LARGE${\times}$}}}\step[2]\O{{\mathpzc{S}}^2}\\ \x\\ \lu[2]\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}} \end{tangles}$$ (see [@[Besp-next]; @[BKLT]] for its further properties and use) occurs in view of . That the map defined by  is a ${\mathfrak{B}}$ module comodule morphism follows from the general argument that so are $\mathsf{B}$, evaluation, and coevaluation (and $\theta$). It is also instructive to see this by diagram manipulation (temporarily writing$\begin{tangles}{l} \O{\theta} \end{tangles}$for $\begin{tangles}{l} \O{\vartheta}\\ \O{{\mbox{\LARGE${\times}$}}} \end{tangles}$for brevity): $$\begin{tangles}{l} \step[1]\lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\\[-4pt] \step[1]\obox{2}{\eqref{YD-loop}}\\ \step[2]{\mathrm{id}}\end{tangles} \ \ = \ \ \begin{tangles}{l} \step[.5]\cd\step{\mathrm{id}}\\ \vstr{100}\hcd\step[1.5]\hx\\ {\mathrm{id}}\step[1]{\mathrm{id}}\step[1]\hld\step[1]\O{{\mathpzc{S}}}\\ \vstr{50}{\mathrm{id}}\step[1]\hx\step[.5]{\mathrm{id}}\step{\mathrm{id}}\\[-.5pt] \hcu\step\hlu\object{\raisebox{13pt}{\kern-4pt\tiny$\blacktriangleright$}}\step{\mathrm{id}}\\[-10pt] \vstr{50}\step[.5]{\mathrm{id}}\step[2]\hx\step[2]\vstr{100}\coev\\ \step[.5]\cu\step\x\step[2]{\mathrm{id}}\\ \step[1.5]\lu[2]\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[-.5]\hld\step[2]{\mathrm{id}}\step[2]{\mathrm{id}}\\ \step[3]{\mathrm{id}}\step[.5]\x\step[2]{\mathrm{id}}\\[-.5pt] \step[3]\hlu\object{\raisebox{13pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[2] \O{\theta}\step[2]{\mathrm{id}}\\ \step[3.5]{\mathrm{id}}\step[2]\x\\ \step[3.5]{\mathrm{id}}\step[2]\ev \end{tangles} \quad=\ \ \begin{tangles}{l} \step[.5]\cd\step{\mathrm{id}}\\ \vstr{50}\step[.5]{\mathrm{id}}\step[2]\hx\\ \step[.5]{\mathrm{id}}\step[1.5]\hld\step[1]\O{{\mathpzc{S}}}\step[1]\coev\\ \hcd\step[1]\dh\d\lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[2]{\mathrm{id}}\\ \vstr{50}{\mathrm{id}}\step[1]{\mathrm{id}}\step[1.5]{\mathrm{id}}\step[1]\hx\step[2]{\mathrm{id}}\\ \dh\step[.5]\dh\step[1]\lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[1]{\mathrm{id}}\step[2]{\mathrm{id}}\\ \step[.5]{\mathrm{id}}\step[1]\x\step[1]{\mathrm{id}}\step[2]{\mathrm{id}}\\ \step[.5]\lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[-.5]\hld\step[2]{\mathrm{id}}\step[1]{\mathrm{id}}\step[2]{\mathrm{id}}\\ \step[.5]\hdd\step[.5]\x\step[1]{\mathrm{id}}\step[2]{\mathrm{id}}\\ \step[.5]{\mathrm{id}}\step[1]{\mathrm{id}}\step[2]\hx\step[2]{\mathrm{id}}\\ \step[.5]\d\lu[2]\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\step\O{\theta}\step[2]{\mathrm{id}}\\ \step[1.5]\lu[2]\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[1]\x\\ \step[3.5]{\mathrm{id}}\step[1]\ev \end{tangles} \ \ =\ \ \begin{tangles}{l} \step[.5]\cd\step{\mathrm{id}}\\[-10pt] \vstr{50}\step[.5]{\mathrm{id}}\step[2]\hx\step[1]\vstr{100}\coev\\ \step[.5]{\mathrm{id}}\step[1]\ld\step[1]\hx\step[1]\dd\\ \hcd\step[.5]{\mathrm{id}}\step[1]\hx\step[1]\lu[1]\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\\ {\mathrm{id}}\step[1]{\mathrm{id}}\step[.5]\lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[-.5]\hld\step[1]{\mathrm{id}}\step[2]{\mathrm{id}}\\ \vstr{50}{\mathrm{id}}\step[1]\hx\step[.5]{\mathrm{id}}\step{\mathrm{id}}\step[2]{\mathrm{id}}\\ \hcu\step[1]\hlu\object{\raisebox{13pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[1]{\mathrm{id}}\step[2]{\mathrm{id}}\\ \vstr{50}\step[.5]{\mathrm{id}}\step[2]\hx\step[2]{\mathrm{id}}\\ \step[.5]\lu[2]\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[1]\O{\theta}\step[2]{\mathrm{id}}\\ \step[2.5]{\mathrm{id}}\step[1]\x\\ \step[2.5]{\mathrm{id}}\step[1]\ev \end{tangles} \ = \ \ \begin{tangles}{l} \step[.5]\cd\step{\mathrm{id}}\\[-10pt] \vstr{50}\step[.5]{\mathrm{id}}\step[2]\hx\step[1]\vstr{100}\coev\\ \step[.5]{\mathrm{id}}\step[1]\ld\step[1]\hx\step[1]\dd\\ \hcd\step[.5]{\mathrm{id}}\step[1]\hx\step[1]\lu[1]\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\\ {\mathrm{id}}\step[1]{\mathrm{id}}\step[.5]\lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[-.5]\hld\step[1]\hd\step[1.5]{\mathrm{id}}\\ {\mathrm{id}}\step[1]\hx\step[.5]\hd\step[1]{\mathrm{id}}\step[1.5]{\mathrm{id}}\\ \vstr{67}{\mathrm{id}}\step[1]{\mathrm{id}}\step[1]\hx\step[1]{\mathrm{id}}\step[1.5]{\mathrm{id}}\\ \vstr{67}{\mathrm{id}}\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\step[1]\hx\step[1.5]{\mathrm{id}}\\ {\mathrm{id}}\step[1]{\mathrm{id}}\step[1]\hx\step[1]\O{{\mathpzc{S}}}\step[1]\hdd\\ {\mathrm{id}}\step\lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[1]\O{\theta}\step[1]\lu[1]\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\\ \lu[2]\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}} \step[1]\x\\ \step[2]{\mathrm{id}}\step\ev \end{tangles}$$ In the first equality, we use only the Yetter–Drinfeld axiom, with $\mathsf{B}^2$ represented by the *first* diagram for $\mathsf{B}^2$ in ; the associativity of action was used in the second equality above; another use of the associativity in the lower part of the third diagram allows recognizing the left-hand side of ; the Yetter–Drinfeld property is then applied in the third equality together with the first property in , yielding the fourth diagram; there, we use that the property of $\sigma_2$ in  and the first property in  to obtain the last, fifth diagram, where an “antipode bubble” is annihilated, showing that, indeed, $$\begin{tangles}{l} \step[1]\lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\\[-4pt] \step[1]\obox{2}{\eqref{YD-loop}}\\ \step[2]{\mathrm{id}}\end{tangles} \ \ = \ \ \begin{tangles}{l} {\mathrm{id}}\step[2]{\mathrm{id}}\\ {\mathrm{id}}\step[1]\obox{2}{\eqref{YD-loop}}\\ \lu[2]\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}} \end{tangles}$$ The commutativity of  with coaction can be verified similarly. Ribbon structure ---------------- A ribbon structure is a morphism ${\boldsymbol{\pmb{\vartheta}}}:{{\mathscr{Y}}}\to{{\mathscr{Y}}}$ for every object ${{\mathscr{Y}}}$ such that $$\label{Rib-def} \begin{tangles}{l} \step[.25]\O{{\boldsymbol{\pmb{\vartheta}}}}\step[1.5]\O{{\boldsymbol{\pmb{\vartheta}}}}\\ \vstr{10}\step[.25]{\mathrm{id}}\step[1.5]{\mathrm{id}}\\[-2pt] \obox{2}{\mathsf{B}^2}\\ \vstr{33}\step[.5]{\mathrm{id}}\step[1]{\mathrm{id}}\end{tangles}\ \ = \begin{tangles}{l} \vstr{50}\step[.5]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \obox{2}{{\boldsymbol{\pmb{\vartheta}}}}\\ \vstr{50}\step[.5]{\mathrm{id}}\step[1]{\mathrm{id}}\end{tangles}$$ Whenever it exists, choosing $\vartheta={\boldsymbol{\pmb{\vartheta}}}$ in  makes $\chi_{{{\mathscr{Z}}}}$ “multiplicative” in ${{\mathscr{Z}}}$. To show this, we calculate $\chi_{{{\mathscr{W}}}}(\chi_{{{\mathscr{Z}}}}({{\mathscr{Y}}}))$ by sliding one of the diagrams along the ${{\mathscr{Y}}}$ line into the middle of the other and then expanding: $$\label{B--B} \kern-6pt \begin{tangles}{l} {\mathrm{id}}{\object{\scriptstyle \kern10pt{{\mathscr{Y}}}}}\step[3]{\object{\scriptstyle \kern-12pt{{\mathscr{Z}}}}}\coev {\object{\scriptstyle \kern12pt{{}^{\vee}\!{{\mathscr{Z}}}}}}\\[-3.9pt] \vstr{50}\dh\step[1.5]\dd\step[2]{\mathrm{id}}\\[-2pt] \step[.25]\obox{2}{\mathsf{B}^2}\step[2.75]{\mathrm{id}}\\[-2pt] \vstr{50}\hdd\step[1.5]\d\step[2]{\mathrm{id}}\\ {\mathrm{id}}\step[3]\O{\theta}\step[1]\dd\\ \vstr{50}{\mathrm{id}}\step[3]\hx\\ {\mathrm{id}}\step[3]{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother}\\ {\mathrm{id}}{\object{\scriptstyle \kern10pt{{\mathscr{Y}}}}}\step[3]{\object{\scriptstyle \kern-14pt{{\mathscr{W}}}}}\coev {\object{\scriptstyle \kern14pt{{}^{\vee}\!{{\mathscr{W}}}}}}\\[-.1pt] \vstr{50}\dh\step[1.5]\dd\step[2]{\mathrm{id}}\\[-2pt] \step[.25]\obox{2}{\mathsf{B}^2}\step[2.75]{\mathrm{id}}\\[-2pt] \vstr{50}\hdd\step[1.5]\d\step[2]{\mathrm{id}}\\ {\mathrm{id}}\step[3]\O{\theta}\step[1]\dd\\ \vstr{50}{\mathrm{id}}\step[3]\hx\\ {\mathrm{id}}\step[3]{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother} \end{tangles} \qquad = \ \ \begin{tangles}{l} \step[.5]\ld[2]\step[2] \\ \hcd\step[1.5]\x {\makeatletter\@ev{0,\hm@de}{25,\hm@detens}{50}t\makeatother} \step[5]{\mathrm{id}}\\ {\mathrm{id}}\step{\mathrm{id}}\step[1]\hld\step[1]\ld \step[5]{\mathrm{id}}\step[-2]\hh\coev\\ {\mathrm{id}}\step{\mathrm{id}}\step[1]{\mathrm{id}}\step[.5]{\mathrm{id}}\step[1]{\mathrm{id}}\step\d \step[1]\dd\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\ {\mathrm{id}}\step\hx\step[.5]{\mathrm{id}}\step[1]\O{{\mathpzc{S}}} \step[1.5]\obox{2}{\mathsf{B}^2}\step[1.5]{\mathrm{id}}\step[1]{\mathrm{id}}\\[-.8pt] {\mathrm{id}}\step{\mathrm{id}}\step[1]{\mathrm{id}}\step[.5]{\mathrm{id}}\step[1]{\mathrm{id}}\step\dd \step[1]\hd\step[1.5]{\mathrm{id}}\step[1]{\mathrm{id}}\\[-.9pt] \hcu\step[1]\hlu\object{\raisebox{13.1pt}{\kern-4pt\tiny$\blacktriangleright$}}\step\lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}} \step[2.5]\O{\theta}\step[1]\ddh\step[1]{\mathrm{id}}\\ \step[.5]{\mathrm{id}}\step[2]\x \step[2.5]\hx\step[1.5]{\mathrm{id}}\\ \step[.5]\lu[2]\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[2]\O{\theta} \step[2.5]{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother} \step[1.5]\ne{3}\\ \step[2.5]{\mathrm{id}}\step[2]\x\\ \step[2.5]{\mathrm{id}}\step[2]\ev \end{tangles} \ \ \ = \quad \begin{tangles}{l} \vstr{90}\step[1]\ld[2]\step[2]\Coev\\ \vstr{90}\step[.5]\hcd\step[1.5]\x \step[3]\d\\ \vstr{90}\ddh\step[1]{\mathrm{id}}\step[1]\hld{\object{\scriptstyle \checkmark}}\step[2]\d\step[3]\d \step[-2] \hh\coev\\ \vstr{90}{\mathrm{id}}\step[1.5]\hx\step[.5]\dh\step[2.5]\x\step[1]{\mathrm{id}}\step{\mathrm{id}}\\ {\mathrm{id}}\step[1.5]{\mathrm{id}}\step[.5]\hcd\step[.5]\hd\step[2]{\mathrm{id}}\step\ld\step[1]{\mathrm{id}}\step{\mathrm{id}}\\ \vstr{90}{\mathrm{id}}\step[1.5]{\mathrm{id}}\step[.5]{\mathrm{id}}\step[1]\hx\step\ld{\object{\scriptstyle \kern-6pt\checkmark}} \step\O{{\mathpzc{S}}}\step{\mathrm{id}}\step[1]{\mathrm{id}}\step{\mathrm{id}}\\ \vstr{90}{\mathrm{id}}\step[1.5]{\mathrm{id}}\step[.5]\lu\object{\raisebox{7pt}{\kern-4pt\tiny$\blacktriangleright$}}\step\hx\step{\mathrm{id}}\step\lu\object{\raisebox{7pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[1]{\mathrm{id}}\step{\mathrm{id}}\\ \vstr{90}\dh\step[1]\hd\step[1]\hx\step[1]\lu\object{\raisebox{7pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[2]{\mathrm{id}}\step[1]{\mathrm{id}}\step{\mathrm{id}}\\ \vstr{90}\step[.5]\d\step[.5]\hcu\step{\mathrm{id}}\step[2]\x\step[1]{\mathrm{id}}\step{\mathrm{id}}\\ \vstr{90}\step[1.5]\hcu\step[1.5]\x\step[2]\O{\theta}\step{\mathrm{id}}\step{\mathrm{id}}\\ \vstr{90}\step[2]\lu[2]\object{\raisebox{7pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[2]\O{\theta}\step[2]\hx\step[1]{\mathrm{id}}\\ \vstr{90}\step[4]{\mathrm{id}}\step[2]\d\step{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother}\step\dd\\ \vstr{90}\step[4]{\mathrm{id}}\step[3]\x\\ \vstr{90}\step[4]\step[3]\ev \end{tangles}$$ In the last diagram, we recognize the diagonal coaction (the two $\checkmark$) and action (two $\blacktriangleright$ just below the respective checkmarks) on a tensor product of two Yetter–Drinfeld modules, as in . In the bottom right part of the diagram, we recall that $\begin{tangles}{l} \O{\theta} \end{tangles}\ \ = \ \ \begin{tangles}{l} \O{{\boldsymbol{\pmb{\vartheta}}}}\\ \O{{\mbox{\LARGE${\times}$}}} \end{tangles}$and calculate $$\begin{tangles}{l} \step[-.5]\O{{\mbox{\LARGE${\times}$}}}\step[1.5]\O{{\mbox{\LARGE${\times}$}}}\step[1.5]{\mathrm{id}}\step{\mathrm{id}}\\ \step[-.5]\hd\step{\mathrm{id}}\step[1.5]{\mathrm{id}}\step{\mathrm{id}}\\ {\mathrm{id}}\step{\mathrm{id}}\step\ddh\step[1]{\mathrm{id}}\\ {\mathrm{id}}\step\hx\step\ddh\\ \d{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother} \step[1]\dd\\ \step\hx\\ \step{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother} \end{tangles} \ \ = \ \ \begin{tangles}{l} \step[-.5]\O{{\mbox{\LARGE${\times}$}}}\step[1.5]\O{{\mbox{\LARGE${\times}$}}}\step[1.5]{\mathrm{id}}\step{\mathrm{id}}\\ \step[-.5]\hd\step{\mathrm{id}}\step[1]\hdd\step[.5]\ddh\\ \hx\step[1]\hxx\\ {\mathrm{id}}\step\hx\step{\mathrm{id}}\\ \d{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother} \step[1]\dd\\ \step[1]\hx\\ \step{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother} \end{tangles} \ \ = \ \ \begin{tangles}{l} \step[-.5]\O{{\mbox{\LARGE${\times}$}}}\step[1.5]\O{{\mbox{\LARGE${\times}$}}}\step[1.5]{\mathrm{id}}\step{\mathrm{id}}\\ \step[-.5]\hd\step{\mathrm{id}}\step[1]\hdd\step[.5]\ddh\\ \hx\step[1]\hxx\\ \hx\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\ {\mathrm{id}}\step[1]\hx\step[1]{\mathrm{id}}\\ \hx\step[1]\hx\\ {\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother} \step[2]{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother} \end{tangles} \ \ = \ \ \begin{tangles}{l} \step[-.5]\O{{\mbox{\LARGE${\times}$}}}\step[1.5]\O{{\mbox{\LARGE${\times}$}}}\step[1.5]{\mathrm{id}}\step{\mathrm{id}}\\ \step[-.5]\hd\step{\mathrm{id}}\step[1]\hdd\step[.5]\ddh\\ \hx\step{\mathrm{id}}\step{\mathrm{id}}\\ \hx\step{\mathrm{id}}\step{\mathrm{id}}\\ {\mathrm{id}}\step\hx\step{\mathrm{id}}\\ \hx\step\hx\\ {\mathrm{id}}\step\hx\step{\mathrm{id}}\\ \Ev{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother} \end{tangles} \ \ = \ \ \begin{tangles}{l} \vstr{67}\step[.5]{\mathrm{id}}\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \vstr{67}\step[.5]\hx\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\[-1pt] \hld\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \vstr{67}{\mathrm{id}}\step[.5]\hx\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \vstr{67}{\mathrm{id}}\step[.5]{\mathrm{id}}\step[1]\hx\step[1]{\mathrm{id}}\\ \vstr{67}{\mathrm{id}}\step[.5]\hx\step[.5]\step[.5]\hx\\[-1pt] \hlu\object{\raisebox{13.1pt}{\kern-4pt\tiny$\blacktriangleright$}} \step[.5]\hld\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \vstr{67}\step[.5]{\mathrm{id}}\step[.5]{\mathrm{id}}\step[.5]\hx\step[1]{\mathrm{id}}\\[-1pt] \step[.5]{\mathrm{id}}\step[.5]\hlu\object{\raisebox{13.1pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \step[.5]\Ev{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother} \end{tangles} \ \ \stackrel{\checkmark}{=} \ \ \begin{tangles}{l} \vstr{90}\hld\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \vstr{90}\vstr{50}{\mathrm{id}}\step[.5]\hx\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\[-1pt] \vstr{90}\hlu\object{\raisebox{11.5pt}{\kern-4pt\tiny$\blacktriangleright$}} \step[-.5]\hld\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \vstr{50}{\mathrm{id}}\step[.5]\hx\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\[-1pt] \vstr{90}\hlu\object{\raisebox{11.5pt}{\kern-4pt\tiny$\blacktriangleright$}} \step[.5]\hld\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \vstr{50}\step[.5]{\mathrm{id}}\step[.5]{\mathrm{id}}\step[.5]\hx\step[1]{\mathrm{id}}\\[-1pt] \hld\step[.5]\hlu\object{\raisebox{13.1pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[.5]\hld\step[1]{\mathrm{id}}\\ \vstr{50}{\mathrm{id}}\step[.5]\hx\step[.5]{\mathrm{id}}\step[.5]\hx\\[-1pt] \vstr{90}\hlu\object{\raisebox{11.5pt}{\kern-4pt\tiny$\blacktriangleright$}} \step[.5]\hld\step[.5]\hlu\object{\raisebox{11.5pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[1]{\mathrm{id}}\\ \vstr{50}\step[.5]{\mathrm{id}}\step[.5]{\mathrm{id}}\step[.5]\hx\step[1]{\mathrm{id}}\\[-1pt] \vstr{90}\step[.5]{\mathrm{id}}\step[.5]\hlu\object{\raisebox{11.5pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \vstr{90}\step[.5]\Ev{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother} \end{tangles} \ \ = \ \ \begin{tangles}{l} \step[.5]{\mathrm{id}}\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \obox{2}{\mathsf{B}^2}\step[.5]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \step[.5]{\mathrm{id}}\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \obox{2}{{\mbox{\LARGE${\bigtimes}$}}}\step[.5]{\mathrm{id}}\step[1]{\mathrm{id}}\\ \step[.5]{\mathrm{id}}\step\hx\step{\mathrm{id}}\\ \step[.5]\hx\step\hx\\ \step[.5]{\mathrm{id}}\step\hx\step{\mathrm{id}}\\ \step[.5]\Ev{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother} \end{tangles}$$ where the first three equalities are elementary (and well-known) rearrangements, the fourth involves , and the checked equality is verified by repeatedly applying the Yetter–Drinfeld axiom in its right-hand side. The sixth diagram involves $\mathsf{B}^2$ in the upper part and the diagonal action and coaction  in the lower part, which gives the last equality. We therefore conclude that if  holds, then $$\begin{tangles}{l} \step[-.5]\O{{\boldsymbol{\pmb{\vartheta}}}}\step[1.5]\O{{\boldsymbol{\pmb{\vartheta}}}}\step[1.5]{\mathrm{id}}\step{\mathrm{id}}\\ \step[-.5]\O{{\mbox{\LARGE${\times}$}}}\step[1.5]\O{{\mbox{\LARGE${\times}$}}}\step[1.5]{\mathrm{id}}\step{\mathrm{id}}\\ \step[-.5]\hd\step{\mathrm{id}}\step[1]\ddh\step{\mathrm{id}}\\ {\mathrm{id}}\step\hx\step\ddh\\ \d{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother} \step[1]\dd\\ \step\hx\\ \step{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother} \end{tangles} \ \ = \ \ \begin{tangles}{l} \step[.25]\O{{\boldsymbol{\pmb{\vartheta}}}}\step[1.5]\O{{\boldsymbol{\pmb{\vartheta}}}}\step[1.75]{\mathrm{id}}\step[1.5]{\mathrm{id}}\\[-2pt] \obox{2}{\mathsf{B}^2}\step[1.5]{\mathrm{id}}\step[1.5]{\mathrm{id}}\\ \vstr{33}\step[.5]{\mathrm{id}}\step[1]{\mathrm{id}}\step[2]{\mathrm{id}}\step[1.5]{\mathrm{id}}\\ \obox{2}{{\mbox{\LARGE${\bigtimes}$}}}\step[.5]\dd\step[1]\ddh\\ \step[.5]{\mathrm{id}}\step\hx\step\dd\\ \step[.5]\hx\step\hx\\ \step[.5]{\mathrm{id}}\step\hx\step{\mathrm{id}}\\ \step[.5]\Ev{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother} \end{tangles} \ \ = \ \ \begin{tangles}{l} \vstr{50}\step[.5]{\mathrm{id}}\step[1]{\mathrm{id}}\step[2]{\mathrm{id}}\step[1.5]{\mathrm{id}}\\ \obox{2}{{\boldsymbol{\pmb{\vartheta}}}}\step[1.5]{\mathrm{id}}\step[1.5]{\mathrm{id}}\\ \vstr{50}\step[.5]{\mathrm{id}}\step[1]{\mathrm{id}}\step[2]{\mathrm{id}}\step[1.5]{\mathrm{id}}\\ \obox{2}{{\mbox{\LARGE${\bigtimes}$}}}\step[.5]\dd\step[1]\ddh\\ \step[.5]{\mathrm{id}}\step\hx\step\dd\\ \step[.5]\hx\step\hx\\ \step[.5]{\mathrm{id}}\step\hx\step{\mathrm{id}}\\ \step[.5]\Ev{\makeatletter\@ev{0,\hm@de}{5,\hm@detens}{10}b\makeatother} \end{tangles}$$ Substituting this in  shows that $\chi$ is indeed “multiplicative”: $\chi_{{{\mathscr{W}}}}(\chi_{{{\mathscr{Z}}}}({{\mathscr{Y}}}))=\chi_{{{\mathscr{W}}}{\otimes}{{\mathscr{Z}}}}({{\mathscr{Y}}})$. Rank-one Nichols algebra {#sec:p} ======================== We specialize the preceding sections to the case of a rank-one Nichols algebra ${\mathfrak{B}}\!_p$, whose relation to the $(p,1)$ logarithmic CFT models was emphasized in [@[STbr]]. An integer $p\geq 2$ is fixed throughout. Notation {#notation .unnumbered} -------- We fix the primitive $2p$th root of unity $$ {\mathfrak{q}}=e^{\frac{i\pi}{p}}$$ and introduce the $q$-binomial coefficients $$ {\mathchoice {{\mbox{\footnotesize$\displaystyle \genfrac{\langle}{\rangle}{0pt}{}{r}{s}$}}}{{\genfrac{\langle}{\rangle}{0pt}{}{r}{s}}} {{\genfrac{\langle}{\rangle}{0pt}{}{r}{s}}}{{\genfrac{\langle}{\rangle}{0pt}{}{r}{s}}}}={\raisebox{.5pt}{\mbox{\footnotesize$\displaystyle\frac{{\langler\rangle!\,}}{{\langles\rangle!\,}{\langler-s\rangle!\,}}$}}}, \quad{\langler\rangle!\,}={\langle1\rangle}\dots{\langler\rangle}, \quad{\langler\rangle}={\raisebox{.5pt}{\mbox{\footnotesize$\displaystyle\frac{q^{2r}-1}{q^2-1}$}}},$$ which are assumed to be specialized to $q={\mathfrak{q}}$. We sometimes use the notation $(a)_N= a\;\mathrm{mod}\;N\in\{0,1,\dots,N-1\}$. The braided Hopf algebra ${\mathfrak{B}}\!_p$ {#sec:screenings} --------------------------------------------- The rank-$1$ Nichols algebra ${\mathfrak{B}}\!_p$ is ${\mathfrak{B}}(X)$ for a one-dimensional braided linear space $X$. We fix an element $F$ (a single screening in the CFT language) as a basis in $X$. The braiding, taken from CFT, is $$\label{Psi-F-F} \Psi(F(r){\otimes}F(s))={\mathfrak{q}}^{2 r s} F(s){\otimes}F(r).$$ Shuffle product  then becomes $$F(r)\,F(s)={\mathchoice {{\mbox{\footnotesize$\displaystyle \genfrac{\langle}{\rangle}{0pt}{}{r+s}{r}$}}}{{\genfrac{\langle}{\rangle}{0pt}{}{r+s}{r}}} {{\genfrac{\langle}{\rangle}{0pt}{}{r+s}{r}}}{{\genfrac{\langle}{\rangle}{0pt}{}{r+s}{r}}}}F(r+s)$$ and coproduct  is $\Delta:F(r)\mapsto\sum\limits_{s=0}^r F(s){\otimes}F(r-s)$. The antipode defined in  acts as ${\raisebox{.5pt}{\large$\mathpzc{S}\kern-1pt$}}(F(r))=(-1)^r {\mathfrak{q}}^{r(r-1)}F(r)$. The algebra ${\mathfrak{B}}\!_p$ is the linear span of $F(r)$ with $0\leq r\leq p-1$. It can also be viewed as generated by a single element $F$, such that $F(r)={\raisebox{.5pt}{\mbox{\footnotesize$\displaystyle\frac{1}{{\langler\rangle!\,}}$}}}F^r$, $r\leq p-1$, with $F^p=0$. We write $F=F(1)$. Because $X$ is now one-dimensional, we can think of $\xymatrix@1@C=20pt{ \ar@{--}|(.55){{\textstyle\!\!{\times}\!\!}}[0,2]&& }$ as just $F$, and write $$F(r)={}\ \xymatrix@1@C=40pt{ \ar@{--}|(.2){{\textstyle\!\!{\times}\!\!}}|(.55){{\textstyle\!\!{\times}\!\!}}|(.8){{\textstyle\!\!{\times}\!\!}}[0,2]&& } \qquad\text{($r$ crosses)}.$$ Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules -------------------------------------------- We specialize the construction of Yetter–Drinfeld ${\mathfrak{B}}(X)$-modules in Sec. \[sec:Nich\] to ${\mathfrak{B}}\!_p$. The construction involves another braided vector space $Y$, a linear span of vertex operators present in the relevant CFT model. \[sec:dressedVO\] ### The vertices For the $(p,1)$ model corresponding to ${\mathfrak{B}}\!_p$ (see [@[FHST]]), $Y$ is a $2p$-dimensional space $$Y= \text{span}(V^a\mid a\in{\mathbb{Z}}_{4p})$$ with the diagonal braiding $$\label{Psi-V-V} \Psi(V^a{\otimes}V^{b}) = {\mathfrak{q}}^{\frac{a b}{2}} V^{b}{\otimes}V^a.$$ and with $$\label{Psi-F-V} \Psi(V^a{\otimes}{F(r)})={\mathfrak{q}}^{-a r}{F(r)}{\otimes}V^a, \quad \Psi({F(r)}{\otimes}V^a)={\mathfrak{q}}^{-a r}\,V^a{\otimes}{F(r)}.$$ This suffices for calculating the “cumulative adjoint” ${\mathfrak{B}}\!_p$ action on multivertex Yetter–Drinfeld modules, as we describe next. In what follows, the integers $a$, $b$, … are tacitly considered modulo $4p$. ### Multivertex Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules We saw in Sec. \[sec:Nich\] that multivertex Yetter–Drinfeld modules (see  and ) can be represented as an essentially “combinatorial” construction for the crosses to populate, in accordance with the braiding rules, line segments that are separated from one another by vertex operators, e.g., $\xymatrix@R=6pt@C=50pt@1{ \ar@*{[|(1.6)]}@{-}|(.15){{\textstyle\!\!{\times}\!\!}} |(.25){{\textstyle{\circ}}}|(.4){{\textstyle\!\!{\times}\!\!}}|(.5){{\textstyle\!\!{\times}\!\!}}|(.65){{\textstyle{\circ}}} |(.8){{\textstyle\!\!{\times}\!\!}} |(.90){{\textstyle{\circ}}}[0,2]&& }$, where ${\times}=X$ and ${\circ}=Y$ (for a finite-dimensional Nichols algebra, each “segment” can carry only finitely many crosses). In the rank-$1$ case, each cross can be considered to represent the $F$ element, and each segment is fully described just by the number of the $F$s sitting there. For example, each two-vertex Yetter–Drinfeld module is a linear span of $$\label{Vsatb} {V^{a,\,b}_{s,\,t}} =\xymatrix@R=4pt@C=70pt{ \ar@*{[|(1.6)]}@{-}|(.1){{\textstyle\!\!{\times}\!\!}}|(.25){{\textstyle\!\!{\times}\!\!}}|(.5){{\textstyle{\circ}}}^(.5){{}^{\scriptstyle a}}|(.65){{\textstyle\!\!{\times}\!\!}}|(.8){{\textstyle{\circ}}}^(.8){{}^{\scriptstyle b}}[0,2]&&} \!\!\!\!,$$ where $s$ and $t$ must not exceed $p-1$ ($s=2$ and $t=1$ in the picture) and $a$ and $b$ indicate $V^a$ and $V^b$. Because the braiding is diagonal, there is a ${\mathfrak{B}}\!_p$ module comodule for each fixed $a$ and $b$ (and $c$, … for multivertex modules). The simplest, one-vertex Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules are spanned by $$\label{Vsa} {V^{a}_{s}}=\xymatrix@R=4pt@C=70pt{ \ar@*{[|(1.6)]}@{-}|(.1){{\textstyle\!\!{\times}\!\!}}|(.35){{\textstyle\!\!{\times}\!\!}}|(.55){{\textstyle\!\!{\times}\!\!}}|(.7){{\textstyle{\circ}}}^(.7){{}^{\scriptstyle a}}[0,2]&&\text{\qquad\small($s$ crosses),}}$$ where $s$ ranges over a subset of $[0,\dots,p-1]$. The ${\mathfrak{B}}\!_p$ coaction is by “deconcatenation up to the first vertex” in all cases, i.e., $$\begin{aligned} \delta\,{V^{a}_{s}} &= \sum_{r=0}^s F(r) {\otimes}{V^{a}_{s-r}}, \\ \delta {V^{a,\,b}_{s,\,t}} &=\sum_{r=0}^s F(r) {\otimes}{V^{a,\,b}_{s-r,\,t}},\end{aligned}$$ and similarly for ${V^{a,\,b,\,c}_{s,\,t,\,u}}$, and so on. The ${\mathfrak{B}}\!_p$ action (which is the left adjoint action ) is then calculated as $$\begin{aligned} F{ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a}_{s}} &= \xi {\langles - a\rangle} {\langles + 1\rangle} {V^{a}_{s + 1}},\qquad \xi = 1-{\mathfrak{q}}^2, \\ \intertext{and the cumulative adjoint evaluates on multivertex spaces as} F{ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a,\,b}_{s,\,t}} &= \xi {\langles + 2 t - a - b\rangle} {\langles + 1\rangle} {V^{a,\,b}_{s + 1,\,t}} + \xi {\mathfrak{q}}^{2 s - a} {\langlet - b\rangle} {\langlet + 1\rangle} {V^{a,\,b}_{s,\,t + 1}}, \\ F{ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a,\,b,\,c}_{s,\,t,\,u}} &= \xi {\langles + 2 t + 2 u - a - b - c\rangle} {\langles + 1\rangle} {V^{a,\,b,\,c}_{s + 1,\,t,\,u}} \\ &{}+ {\mathfrak{q}}^{2 s - a} \xi {\langlet + 2 u - b - c\rangle} {\langlet + 1\rangle} {V^{a,\,b,\,c}_{s,\,t + 1,\,u}} + {\mathfrak{q}}^{2 s + 2 t - a - b} \xi {\langleu - c\rangle} {\langleu + 1\rangle} {V^{a,\,b,\,c}_{s,\,t,\,u + 1}},\end{aligned}$$ and so on. The braiding follows from , , and , for example, $$\label{braiding11} \Psi({V^{a}_{s}}{\otimes}{V^{b}_{t}}) = {\mathfrak{q}}^{{ \mathchoice{{\raisebox{.5pt}{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(a - 2 s) (b - 2 t)} {V^{b}_{t}}{\otimes}{V^{a}_{s}}.$$ Module types and decomposition ------------------------------ We now study the category of Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules in some detail: we find how the one-vertex and two-vertex spaces decompose into indecomposable Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules. We first forget about braiding and study only *the module comodule structure*; the action and coaction *are* related by the Yetter–Drinfeld axiom, but we try to avoid speaking of Yetter–Drinfeld modules before we come to the braiding. ### The relevant module comodules, which we construct explicitly in Appendix [**\[app:modules\]**]{}, are as follows: - simple $r$-dimensional module comodules ${{\mathscr{X}}}(r)$, $1\leq r\leq p$; for $r=p$, we sometimes use the special notation ${{\mathscr{S}}}(p)={{\mathscr{X}}}(p)$; - the $p$-dimensional extensions $$\begin{aligned} \label{V-def} \raisebox{-.7\baselineskip}{${{\mathscr{V}}}[r]\;={}$} &\xymatrix@=15pt{ &{{\mathscr{X}}}(p-r)\ar@/_12pt/[dl]\\ {{\mathscr{X}}}(r)& } \qquad\qquad\raisebox{-.7\baselineskip}{$1\leq r\leq p-1,$} \end{aligned}$$ where the arrow means that $\delta{{\mathscr{X}}}(p-r)\subset($the “trivial” piece ${\mathfrak{B}}\!_p{\otimes}{{\mathscr{X}}}(p-r))+{\mathfrak{B}}\!_p{\otimes}{{\mathscr{X}}}(r)$. - $2p$-dimensional indecomposable module comodules ${{\mathscr{P}}}[r]$ with the structure of subquotients $$\begin{aligned} \label{P-def} \raisebox{-1.55\baselineskip}{${{\mathscr{P}}}[r]\;={}$} &\xymatrix@=12pt{ &{\mathscr{X}}(p-r) \ar@/_12pt/[dl] \ar[dr] \\ {\mathscr{X}}(r)\ar[dr] &&{\mathscr{X}}(r)\ar@/_12pt/[dl] \\ &{\mathscr{X}}(p-r)& }\qquad\qquad\raisebox{-1.55\baselineskip}{$1\leq r\leq p-1.$} \end{aligned}$$ ### We also show in Appendix [**\[app:modules\]**]{} that the $p^2$-dimensional *one-vertex space* $$\mathbb{V}_p(1)\equiv \text{Span}({V^{a}_{s}} \mid 0\leq a,s\leq p-1)$$ decomposes into ${\mathfrak{B}}\!_p$ module comodules as $$\label{V1-decomp} \mathbb{V}_p(1) = {{\mathscr{S}}}(p)\;\oplus\bigoplus_{1\leq r\leq p-1}{{\mathscr{V}}}[r]$$ and the $p^4$-dimensional *two-vertex space* $$\mathbb{V}_p(2)\equiv \text{Span}({V^{a,\,b}_{s,\,t}} \mid 0\leq a,b,s,t\leq p-1)$$ decomposes as $$\label{V2-decomp} \mathbb{V}_p(2) = p^2 {{\mathscr{S}}}(p)\;\oplus \bigoplus_{1\leq r\leq p-1} 2 r (p - r) {{\mathscr{V}}}[r]\;\oplus \bigoplus_{1\leq r\leq p-1}(p - r)^2 {{\mathscr{P}}}[r].$$ *Multivertex spaces* give rise to “zigzag” Yetter–Drinfeld modules, which we do not consider here. ### Notation Compared with representation theory of Lie algebras, the role of highest-weight vectors is here played by *left coinvariants* ${V^{a}_{0}}$ and ${V^{a,\,b}_{0,\,t}}$. When a module comodule of one of the above types ${{\mathscr{A}}}={{\mathscr{X}}}$, ${{\mathscr{V}}}$, or ${{\mathscr{P}}}$ is constructed starting with a left coinvariant, we use the notation ${{\mathscr{A}}}_{0}^{\{a\}}$ or ${{\mathscr{A}}}_{0,\,t}^{\{a,\,b\}}$ to indicate the coinvariant, and sometimes also use the notation such as ${{\mathscr{X}}^{\{a,\,b\}}_{0,\,t}}(r)$ to indicate the dimension (although it is uniquely defined by $a$, $t$, $b$, and the module type). ### {#1-vertex} The module comodules that can be constructed starting with one-vertex coinvariants ${V^{a}_{0}}$ are classified immediately, as we show in [**\[app:1-vertex\]**]{}.The module comodule *generated* from ${V^{a}_{0}}$ under the ${\mathfrak{B}}\!_p$ action is isomorphic to ${{\mathscr{X}}}(r)$ whenever $(a)_p = r - 1$ ($1\leq r\leq p$). If $r\leq p-1$, then extension  follows immediately. ### {#sec:all-cases} The strategy to classify two-vertex ${\mathfrak{B}}\!_p$ module comodules according to their characteristic left coinvariant ${V^{a,\,b}_{0,\,t}}$ is to consider the following cases that can occur under the action of $F(s)$ on the left coinvariant. 1. $F(s){ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a,\,b}_{0,\,t}}$ is nonvanishing and not a coinvariant for all $s$, $1\leq s\leq p-1$. In this case, there are the possibilities that 1. \[caseL\] $F(s){ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a,\,b}_{0,\,t}}$ is a coinvariant, i.e., $F(s){ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a,\,b}_{0,\,t}}=\mathrm{const}\;{V^{a,\,b}_{0,\,t+s}}$, for some $s\leq p-1$, and 2. \[caseS\] $F(s){ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a,\,b}_{0,\,t}}$ is not a coinvariant for any $s\leq p-1$. 2. $F(s){ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a,\,b}_{0,\,t}}=0$ for some $s\leq p-1$. In this case, further possibilities are 1. \[casenone\] For some $s'<s$,  $F(s'){ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a,\,b}_{0,\,t}}$ is a coinvariant, and 2. $F(s'){ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a,\,b}_{0,\,t}}$ is not a coinvariant for any $s'<s$. We then distinguish the cases where 1. \[caseB\] ${V^{a,\,b}_{0,\,t}}$ is in the image of $F$, and 2. \[caseX\] ${V^{a,\,b}_{0,\,t}}$ is not in the image of $F$. We show in Appendix [**\[app:modules\]**]{} that these cases are resolved as follows in terms of the parameters $a$, $t$, and $r=(a + b - 2 t)_p + 1$: \[caseL\] : $1\leq r\leq p-1$ and either $t \leq (a)_p - r$  or  $(a)_p + 1 \leq t \leq p - r - 1$.  Then the left coinvariant is the leftmost coinvariant in , and the Yetter–Drinfeld module generated from it is the “left–bottom half” ${{\mathscr{L}}}(r)$ of ${{\mathscr{P}}}[r]$ (see [**\[sec:LB\]**]{}). \[caseS\] : $r=p$.  Then ${{\mathscr{X}}}(p)\equiv{{\mathscr{S}}}(p)$ is generated from the left coinvariant. \[casenone\] : is not realized. \[caseB\] : $1\leq r\leq p-1$ and either $t \geq p - r + (a)_p + 1$  or  $p - r \leq t \leq (a)_p$.  Then the bottom Yetter–Drinfeld submodule ${{\mathscr{B}}}(r)$ in ${{\mathscr{P}}}[p-r]$ is generated from the left coinvariant. \[caseX\] : $1\leq r\leq p-1$ and either $(t\leq (a)_p$ and $(a)_p - r + 1\leq t\leq p - 1 - r)$  or  $(t\geq (a)_p + 1$ and $p - r\leq t\leq p - r + (a)_p)$.  Then ${{\mathscr{X}}}(r)$ is generated from the left coinvariant. Braiding sectors {#sec:braiding-sectors} ---------------- The ${{\mathscr{X}}}(r)$ and the other module comodules appearing above satisfy the Yetter–Drinfeld axiom. Considering them as Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules means that isomorphic module comodules may be distinguished by the braiding. This is indeed the case: for example, shifting $a\to a+p$ in ${{\mathscr{X}}^{\{a\}}_0}$ or ${{\mathscr{X}}^{\{a,\,b\}}_{0,\,t}}$ does not affect the module comodule structure described in Appendix [**\[app:modules\]**]{}, but changes the braiding with elements of ${\mathfrak{B}}\!_p$ by a sign in accordance with . We thus have *pairs* $({{\mathscr{A}}}_\nu,{{\mathscr{A}}}_{\nu+1})$, $\nu\in{\mathbb{Z}}_2$, of isomorphic module comodules distinguished by a sign occurring in their braiding. In particular, there are $2p$ nonisomorphic simple Yetter–Drinfeld modules. Further, these Yetter–Drinfeld modules can be viewed as elements of a braided category, whose braiding (see ) involves . The dependence on $a$ in  is modulo $4p$, and hence we have not pairs but quadruples $({{\mathscr{A}}}_\nu)_{\nu\in{\mathbb{Z}}_4}$, with the different ${{\mathscr{A}}}_\nu$ distinguished by their braiding with other such modules. In particular, there are $4p$ nonisomorphic simple objects in this braided category of Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules [@[STbr]]. It is convenient to write $a=(a)_p-\nu p,$ $\nu\in{\mathbb{Z}}_4$ [@[STbr]], and introduce the notation ${{\mathscr{X}}}(r)_{\nu}$ for simple modules, with $${{\mathscr{X}}^{\{a\}}_0}\cong{{\mathscr{X}}}(r)_{\nu} \quad\text{whenever}\quad a=r-1-\nu p.$$ As before, $r$ is the dimension, and we sometimes refer to $\nu$ as the braiding sector or braiding index. For $\nu\in{\mathbb{Z}}_4$, the isomorphisms are in the braided category of Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules. The “quadruple structure” occurs totally similarly for other modules, including those realized in multivertex spaces; for example, for any $a,b\in{\mathbb{Z}}$, we have the isomorphisms among the simple Yetter–Drinfeld modules realized in the two-vertex space (cf. !): $${{\mathscr{X}}^{\{a,\,b\}}_{0,\,t}}\cong{{\mathscr{X}}}(r)_\nu \quad\text{whenever \ $a + b - 2 t = r-1 - \nu p$ \ and \ \eqref{cond:Xi}${}\lor{}$\eqref{cond:Xii} \ holds.}$$ For the reducible extensions as in , the two subquotients have adjacent braiding indices, and we conventionally use one of them in the notation for the reducible module: $$\label{X-X-nu} \raisebox{-.8\baselineskip}{\mbox{${{\mathscr{V}}}[r]_{\nu}\ =\ $}} \xymatrix@=15pt{ &{{\mathscr{X}}}(p-r)_{\nu + 1}\ar@/_12pt/[dl]\\ {{\mathscr{X}}}(r)_{\nu}& },$$ and ${{\mathscr{V}}^{\{a,\,b\}}_{0,\,t}}[r]_{\nu}\cong {{\mathscr{V}}}[r]_{\nu}$ whenever $a+b-2t = r-1 -\nu p$ and ${}\lor{}$ holds. In , the relevant braiding indices range an interval of three values, and we use the leftmost value in the notation for the entire reducible Yetter–Drinfeld module, which yields , with ${{\mathscr{P}}^{\{a,\,b\}}_{0,\,t}}[r]_{\nu} \cong{}{{\mathscr{P}}}[r]_{\nu}$ whenever $a+b-2t = r-1 -\nu p$ and holds. In the above formulas and diagrams, $\nu\in{\mathbb{Z}}_4$ if the modules are viewed as objects of the braided category of Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules. But *if the Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules are considered as an entwined category, then the braiding sectors $\nu$ and $\nu+2$ become indistinguishable, and hence $\nu\in{\mathbb{Z}}_2$*. In particular, there are $2p$ nonisomorphic simple objects in the entwined category of Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules. Proof of decomposition  ----------------------- Decomposition  can be derived from the list in [**\[sec:all-cases\]**]{} as follows. The fusion product  of two one-vertex modules is the map (assuming that $a,b\leq p-1$ to avoid writing $(a)_p$ and $(b)_p$) $$\label{VV-fusion} V^a_s{\otimes}V^b_t\mapsto \sum_{i=0}^bq^{-a i}{\mathchoice {{\mbox{\footnotesize$\displaystyle \genfrac{\langle}{\rangle}{0pt}{}{s+i}{s}$}}}{{\genfrac{\langle}{\rangle}{0pt}{}{s+i}{s}}} {{\genfrac{\langle}{\rangle}{0pt}{}{s+i}{s}}}{{\genfrac{\langle}{\rangle}{0pt}{}{s+i}{s}}}}{V^{a,\,b}_{s+i,\,t-i}}.$$ In evaluating ${{\mathscr{X}}^{\{a\}}_0}(s){\otimes}{{\mathscr{X}}^{\{b\}}_0}(t)$, this formula is applied for $0\leq s\leq a$ and $0\leq t\leq b$. Then the left coinvariants produced in the right-hand side are ${V^{a,\,b}_{0,\,u}}$, where $0\leq u\leq b$ *and* $u\leq a$. But the conditions defining the different items in the list in [**\[sec:all-cases\]**]{} have the remarkable property that the module ${{\mathscr{A}}}^{\{a,b\}}_{0,u}$ generated from each such coinvariant is as follows: $$\label{the-property} {{\mathscr{A}}}^{\{a,b\}}_{0,u}= \begin{cases} {{\mathscr{X}}^{\{a,\,b\}}_{0,\,u}},&a + b \leq p-1,\\ {{\mathscr{X}}^{\{a,\,b\}}_{0,\,u}},&a + b \geq p\ \text{ and } \ u \geq a + b - p + 2,\\ {{\mathscr{L}}^{\{a,\,b\}}_{0,\,u}},&a + b \geq p\ \text{ and } \ a + b - 2 u - p \geq 0,\\ {{\mathscr{S}}^{\{a,\,b\}}_{0,\,u}},&a + b \geq p\ \text{ and } \ a + b - 2 u - p = -1,\\ {{\mathscr{B}}^{\{a,\,b\}}_{0,\,u}},&a + b \geq p\ \text{ and } \ a + b - 2 u - p \leq -2. \end{cases}$$ This is established (*only* for $0\leq u\leq a,b\leq p-1$) by direct inspection of each case in the list at the end of [**\[sec:all-cases\]**]{}.  The module ${{\mathscr{L}}^{\{a,\,b\}}_{0,\,u}}$ is the “left–bottom half” of ${{\mathscr{P}}^{\{a,\,b\}}_{0,\,u}}$, and ${{\mathscr{B}}^{\{a,\,b\}}_{0,\,u}}$ is the bottom sub(co)module in another ${{\mathscr{P}}}$ module; the details are given in [**\[sec:LB\]**]{}. The crucial point is that each ${{\mathscr{L}}^{\{a,\,b\}}_{0,\,u}}$ can be extended to ${{\mathscr{P}}^{\{a,\,b\}}_{0,\,u}}$ (while the ${{\mathscr{B}}}$, on the other hand, are not interesting in that they are sub(co)modules in the ${{\mathscr{L}}}$ that are already present). We next claim that each of the ${{\mathscr{L}}}$s occurring in ${{\mathscr{X}}^{\{a\}}_0}(s){\otimes}{{\mathscr{X}}^{\{b\}}_0}(t)$ indeed occurs there together with the entire ${{\mathscr{P}}}$ module; this follows from counting the dimensions and from the fact that there are no more left coinvariants among the ${V^{a,\,b}_{v,\,w}}$ appearing in the right-hand side of  (and, of course, from the structure of the modules described in Appendix [**\[app:modules\]**]{}). Once it is established that each ${{\mathscr{L}}}$ occurs in  as a sub(co)module of the corresponding ${{\mathscr{P}}}$, it is immediate to see that  is equivalent to . Duality ------- We now recall Sec. \[sec:duality\]. The structures postulated there are indeed realized for the $n$-vertex Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules. ### One-vertex modules: $\mathrm{coev}$ and $\mathrm{ev}$ maps {#coev-1} For the irreducible Yetter–Drinfeld module ${{\mathscr{X}}^{\{a\}}_0}\cong{{\mathscr{X}}}(r)$ as in [**\[sec:Xi\]**]{}, the coevaluation map $\mathrm{coev}:k\to{{\mathscr{X}}^{\{a\}}_0}{\otimes}{{}^{^{\vee}}\kern-2pt{\mathscr{X}}^{\{a\}}_0}$ is given in terms of dual bases as $$\begin{tangles}{l} \vphantom{x}\\ \Coev\\ {\object{\scriptstyle {{\mathscr{X}}^{\{a\}}_0}}}{\object{\scriptstyle \kern60pt{{}^{^{\vee}}\kern-2pt{\mathscr{X}}^{\{a\}}_0}}} \end{tangles}\kern30pt \ = \sum_{s=0}^{r-1}{V^{a}_{s}}{\otimes}{U^{-a}_{s}}, \qquad r = (a)_p+1,$$ and the evaluation map $\mathrm{ev}: {{}^{^{\vee}}\kern-2pt{\mathscr{X}}^{\{a\}}_0}{\otimes}{{\mathscr{X}}^{\{a\}}_0}\to k$, accordingly, as $$\begin{tangles}{l} \vphantom{x}\\ {\object{\scriptstyle {{}^{^{\vee}}\kern-2pt{\mathscr{X}}^{\{a\}}_0}}}{\object{\scriptstyle \kern60pt{{\mathscr{X}}^{\{b\}}_0}}}\\[4pt] \Ev \end{tangles}\kern30pt \ : \ {U^{-a}_{s}}{\otimes}{V^{b}_{t}}\mapsto 1 \delta_{s,t}\delta_{a,b}.$$ We then use  to find the ${\mathfrak{B}}\!_p$ module comodule structure on the $U_s^a$. Simple calculation shows that $$\begin{aligned} F(r) U_s^a &= {\mathfrak{q}}^{r(r-1) - r a - 2 r s} {\mathchoice {{\mbox{\footnotesize$\displaystyle \genfrac{\langle}{\rangle}{0pt}{}{s}{r}$}}}{{\genfrac{\langle}{\rangle}{0pt}{}{s}{r}}} {{\genfrac{\langle}{\rangle}{0pt}{}{s}{r}}}{{\genfrac{\langle}{\rangle}{0pt}{}{s}{r}}}} (-\xi)^{r} \prod_{t=s-r}^{s-1}{\langlet+a\rangle}\;U^a_{s-r}, \\ \delta U^a_s &= \sum_{r=0}^{p-1-s} (-1)^r{\mathfrak{q}}^{-r a - 2 s r - r(r - 1)} F(r){\otimes}U_{s+r}^a.\end{aligned}$$ It follows that we can identify $U_{s}^{a} = (-1)^{a+s}{\mathfrak{q}}^{(s+1)(s+a-2)} V_{p - 1 - s}^{a - 2 + 2p}$ (the action and coaction—and in fact the braiding—are identical for both sides). The coevaluation and evaluation maps can therefore be expressed as $$\begin{tangles}{l} \vphantom{x}\\ \Coev\\ {\object{\scriptstyle {{\mathscr{X}}^{\{a\}}_0}}}\step[3]{\object{\scriptstyle {}^\vee\!{{\mathscr{X}}^{\{a\}}_0}}} \end{tangles}\kern15pt = \sum_{s=0}^{r-1}{V^{a}_{s}}{\otimes}{V^{2p-a-2}_{p-1-s}} (-1)^{a + s} {\mathfrak{q}}^{(s+1) (s - a - 2)},\quad r=(a)_p+1,$$ and $$\begin{tangles}{l}\Ev \end{tangles}\kern25pt :\ {V^{a}_{s}}{\otimes}{V^{b}_{t}}\mapsto {\bigl\langle{V^{a}_{s}},\,{V^{b}_{t}}\bigr\rangle\,}= (-1)^s{\mathfrak{q}}^{-s^2+s(a-1)} \delta_{s+t,p-1}\delta_{a+b,2p-2}.$$ For $a\not\equiv p-1\;\text{mod}\;p$, evidently, $a = r-1 - \nu p$ implies that $2 p - a - 2= p - r - 1 + (\nu + 1) p$, and therefore the module left dual to ${{\mathscr{V}}}[r]$ in , with $r=(a)_p+1$, can be identified as $$\raisebox{-\baselineskip}{\mbox{${}^{\vee}({{\mathscr{V}}}[r]_{\nu})={}$}} \xymatrix@=15pt{ .&{{\mathscr{X}}}(r)_{-\nu}\ar@/_12pt/[dl]\\ {{\mathscr{X}}}(p-r)_{-\nu-1}&}$$ where ${{\mathscr{X}}}(r)_{-\nu}$ is dual to ${{\mathscr{X}}}(r)_{\nu}$ in . *The properties expressed in  and  now hold*, as is immediate to verify. ### Two-vertex modules {#coev-2} Similarly to [**\[coev-1\]**]{}, for the ${U^{a,\,b}_{s,\,t}}$ that are dual to the two-vertex basis, $${\bigl\langle{U^{a,\,b}_{s,\,t}},\,{V^{c,\,d}_{u,\,v}}\bigr\rangle\,} =\delta_{a+c,0}\delta_{b+d,0}\delta_{s,u}\delta_{t,v},$$ it follows from  that $$F(r){U^{a,\,b}_{s,\,t}} =\sum_{u=0}^r(-1)^r {\mathfrak{q}}^{r(r-1)-r(b+2s+2t)} {c^{-a\kern-1pt,\,-b}_{s-r+u\kern-1pt,\,t-u}(r\kern-1pt,u)}{U^{a,\,b}_{s-r+u,\,t-u}}.$$ Replacing here $s\to p-1-s$ and $t\to p-1-t$ and noting that the coefficients ${c^{a\kern-1pt,\, b}_{ s\kern-1pt,\, t}( r\kern-1pt, u)}$ in  have the symmetry $${c^{a\kern-1pt,\, b}_{ s\kern-1pt,\, t}( r\kern-1pt, u)} = {\mathfrak{q}}^{2 r (r + 2 t + 2 s - a - b)} {c^{-a - 2\kern-1pt,\, -b - 2}_{ p - 1 - s - r + u\kern-1pt,\, p - 1 - t - u}( r\kern-1pt, u)}, \quad r\geq u,$$ we arrive at the identification $${U^{a,\,b}_{s,\,t}} = (-1)^{t + s} {\mathfrak{q}}^{(t + s + 2) (2 a + b + t + s - 3)} {V^{a - 2,\,b - 2}_{p - 1 - s,\,p - 1 - t}}.$$ Hence, under the pairing $${\bigl\langle{V^{a,\,b}_{s,\,t}},\,{V^{c,\,d}_{u,\,v}}\bigr\rangle\,} =(-1)^{s + t} {\mathfrak{q}}^{(s + t) (2 a + b + 1 - s - t)} \delta_{a+c,-2}\delta_{b+d,-2}\delta_{s+u,p-1}\delta_{t+v,p-1},$$ the module left dual to ${{\mathscr{P}}^{\{a,\,b\}}_{0,\,t}}$ can be identified with ${{\mathscr{P}}^{\{-a-2,\,-b-2\}}_{0,\,p-r-t-1}}$ (as before, $a+b-2t=r-1-\nu p$, $1\leq r\leq p-1$). The module dual to  has the structure $$\raisebox{-2\baselineskip}{\mbox{ ${}^{\vee}\bigl({{\mathscr{P}}^{\{a,\,b\}}_{0,\,t}}[r]_{\nu}\bigr) = {{\mathscr{P}}^{\{-a-2,\,-b-2\}}_{0,\,p-r-t-1}}[r]_{-2-\nu}\ \ =$}} \xymatrix@=12pt{ &&{\mathscr{X}}(p-r)_{-\nu-1} \ar@/_12pt/[dl] \ar[dr] &\\ &{\mathscr{X}}(r)_{-2-\nu}\ar[dr] &&{\mathscr{X}}(r)_{-\nu}\ar@/_12pt/[dl] \\ &&{\mathscr{X}}(p-r)_{-\nu-1}& }$$ Ribbon structure ---------------- We set $${\boldsymbol{\pmb{\vartheta}}}\, V^{a}_s = {\mathfrak{q}}^{{ \mathchoice{{\raisebox{.5pt}{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}((a + 1)^2 - 1)}V^{a}_s,$$ which obviously commutes with the ${\mathfrak{B}}\!_p$ action and coaction, and $$\label{Ribbonii} {\boldsymbol{\pmb{\vartheta}}}{V^{a,\,b}_{s,\,t}}= {\mathfrak{q}}^{{ \mathchoice{{\raisebox{.5pt}{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}((a + b - 2 t + 1)^2 - 1)} \sum_{i=0}^{s} {\mathfrak{q}}^{-i a} \xi^i {\mathchoice {{\mbox{\footnotesize$\displaystyle \genfrac{\langle}{\rangle}{0pt}{}{t + i}{i}$}}}{{\genfrac{\langle}{\rangle}{0pt}{}{t + i}{i}}} {{\genfrac{\langle}{\rangle}{0pt}{}{t + i}{i}}}{{\genfrac{\langle}{\rangle}{0pt}{}{t + i}{i}}}} \prod_{j=0}^{i-1}{\langlet + j - b\rangle}\, {V^{a,\,b}_{s - i,\,t + i}}$$ (we recall that $\xi=1-{\mathfrak{q}}^2$). Algebra  from  -------------- With the above ribbon structure, we now calculate diagram  in some cases. To maintain association with the diagram, we write $\chi_{{{\mathscr{Z}}}}({{\mathscr{Y}}})$ as ${{\mathscr{Y}}}{\mathbin{\mbox{\large$\looparrowdownleft$}}}{{\mathscr{Z}}}$ (the reasons for choosing the right action are purely notational$/$graphical). The calculations in what follows are based on a formula for the double braiding: for two one-vertex modules, the last diagram in  evaluates as $$\begin{gathered} \label{B2-VV} \mathsf{B}^2\left(V^{a}_s{\otimes}V^{b}_t\right) =\sum_{n=0}^{s + t} \sum_{i=n}^{s + t}\sum_{j=0}^{\min(i, t)} {\mathfrak{q}}^{a b + 2 j (j - 1) + (i - n - 1) (i - n) - 2 b j + a (n - 2 i - t)} \\[-6pt] \times{}\xi^{i - j} {\mathchoice {{\mbox{\footnotesize$\displaystyle \genfrac{\langle}{\rangle}{0pt}{}{i}{j}$}}}{{\genfrac{\langle}{\rangle}{0pt}{}{i}{j}}} {{\genfrac{\langle}{\rangle}{0pt}{}{i}{j}}}{{\genfrac{\langle}{\rangle}{0pt}{}{i}{j}}}} {\mathchoice {{\mbox{\footnotesize$\displaystyle \genfrac{\langle}{\rangle}{0pt}{}{s + t - j}{s}$}}}{{\genfrac{\langle}{\rangle}{0pt}{}{s + t - j}{s}}} {{\genfrac{\langle}{\rangle}{0pt}{}{s + t - j}{s}}}{{\genfrac{\langle}{\rangle}{0pt}{}{s + t - j}{s}}}} {\mathchoice {{\mbox{\footnotesize$\displaystyle \genfrac{\langle}{\rangle}{0pt}{}{s + t - n}{i - n}$}}}{{\genfrac{\langle}{\rangle}{0pt}{}{s + t - n}{i - n}}} {{\genfrac{\langle}{\rangle}{0pt}{}{s + t - n}{i - n}}}{{\genfrac{\langle}{\rangle}{0pt}{}{s + t - n}{i - n}}}} \prod_{\ell=0}^{i - j - 1}\!{\langle\ell + j - b\rangle} \;{V^{a,\,b}_{s + t - n,\,n}}.\end{gathered}$$ ### {#section-4} If ${{\mathscr{Y}}}$ is irreducible, ${{\mathscr{Y}}}\cong{{\mathscr{X}}}(r)_{\nu}$, then ${{\mathscr{X}}}(r)_{\nu}{\mathbin{\mbox{\large$\looparrowdownleft$}}}{{\mathscr{Z}}}$ can only amount to multiplication by a number; indeed, we find that $$\text{for all $x\in{{\mathscr{X}}^{\{a\}}_0}$ with $(a)_p\neq p-1$,}\quad x{\mathbin{\mbox{\large$\looparrowdownleft$}}}{{\mathscr{X}}^{\{b\}}_0} = {\lambda(a,b)}x,$$ where $${\lambda(a,b)} = \frac{{\mathfrak{q}}^{(a + 1)(b + 1)} - {\mathfrak{q}}^{-(a + 1)(b + 1)}}{{\mathfrak{q}}^{a + 1} - {\mathfrak{q}}^{-a - 1}}.$$ It is instructive to reexpress this eigenvalue by indicating the representation labels rather than the relevant coinvariants: for $a = r' - 1 - \nu' p$ and $b = r - 1 - \nu p$,we find that ${{\mathscr{X}}}(r')_{\nu'}{\mathbin{\mbox{\large$\looparrowdownleft$}}}{{\mathscr{X}}}(r)_{\nu}$ amounts to multiplication by $$\begin{aligned} \lambda(r',\nu';r,\nu) &= (-1)^{\nu'(r+1) + \nu r' + p \nu \nu'}\, {\raisebox{.5pt}{\mbox{\footnotesize$\displaystyle\frac{{\mathfrak{q}}^{r' r} - {\mathfrak{q}}^{-r' r}}{{\mathfrak{q}}^{r'} - {\mathfrak{q}}^{-r'}}$}}}\\ &=(-1)^{\nu'(r+1) + \nu r' + p \nu \nu'} \sum_{i=1}^{r} {\mathfrak{q}}^{r' (r + 1 - 2 i)}. \\ \intertext{The last form is also applicable in the case where $r'=p$, and ${{\mathscr{S}}}(p)_{\nu'}{\mathbin{\mbox{\large$\looparrowdownleft$}}}{{\mathscr{X}}}(r)_{\nu}$ amounts to multiplication by} \lambda(p,\nu';r,\nu) &= (-1)^{(\nu' + 1)(r-1-\nu p)} r.\end{aligned}$$ For ${{\mathscr{Y}}}={{\mathscr{V}}}[r]_{\nu}$ in , it may be worth noting that the identity $\lambda(r',\nu';r,\nu)=\lambda(p-r',\nu'+1;r,\nu)$, $1\leq r'\leq p-1$, explicitly shows that the action is the same on both subquotients. ### {#section-5} Next, the action ${{\mathscr{P}}}[r']_{\nu'}{\mathbin{\mbox{\large$\looparrowdownleft$}}}{{\mathscr{X}}}(r)_{\nu}$ has a diagonal piece, given again by multiplication by $\lambda(r',\nu';r,\nu)$, and a nondiagonal piece, mapping the top subquotient in $$\raisebox{-1.95\baselineskip}{${{\mathscr{P}}}[r']_{\nu'}\ {}={}\ $} \xymatrix@=12pt{ &{\mathscr{X}}(p-r')_{\nu'+1} \ar@/_12pt/[dl] \ar[dr] &\\ {\mathscr{X}}(r')_{\nu'}\ar[dr] &&{\mathscr{X}}(r')_{\nu'+2}\ar@/_12pt/[dl] \\ &{\mathscr{X}}(p-r')_{\nu'+1}& }$$ into the bottom subquotient. Specifically, in terms of the “top” and “bottom” elements defined in  and , we have $$u^{a, b}_t (1){\mathbin{\mbox{\large$\looparrowdownleft$}}}{{\mathscr{X}}}(r) = {\lambda(r', \nu'; r, \nu)} u^{a, b}_t (1) + \mu(r', \nu'; r, \nu) v^{a, b}_t (r + 1),$$ where $$\begin{gathered} \mu(r',\nu';r,\nu) = (-1)^{1 + \nu' r + \nu r' + p \nu' \nu} {\raisebox{.5pt}{\mbox{\footnotesize$\displaystyle\frac{{\mathfrak{q}}- {\mathfrak{q}}^{-1}}{({\mathfrak{q}}^{r'} - {\mathfrak{q}}^{-r'})^3}$}}} \\ {}\times\Bigl(({\mathfrak{q}}^{r' r} - {\mathfrak{q}}^{-r' r})({\mathfrak{q}}^{r'} + {\mathfrak{q}}^{-r'}) - r({\mathfrak{q}}^{r' r} + {\mathfrak{q}}^{-r' r})({\mathfrak{q}}^{r'} - {\mathfrak{q}}^{-r'})\Bigr).\end{gathered}$$ Because ${}{\mathbin{\mbox{\large$\looparrowdownleft$}}}{{\mathscr{X}}}(r)$ commutes with the ${\mathfrak{B}}\!_p$ action and coaction, and because ${{\mathscr{P}}^{\{a,\,b\}}_{0,\,t}}$ is generated by the ${\mathfrak{B}}\!_p$ action and coaction from $u^{a, b}_t(1)$, the action of ${{\mathscr{X}}^{\{c\}}_0}$ is thus defined on all of ${{\mathscr{P}}^{\{a,\,b\}}_{0,\,t}}$. ### {#section-6} Let ${\,\boldsymbol{\mathsf{x}}}(r)_{\nu}$ and ${\,\boldsymbol{\mathsf{p}}}(r)_{\nu}$ be the respective operations ${\mathbin{\mbox{\large$\looparrowdownleft$}}}{{\mathscr{X}}}(r)_{\nu}$ and ${\mathbin{\mbox{\large$\looparrowdownleft$}}}{{\mathscr{P}}}(r)_{\nu}$. We then have relations , which are the fusion algebra in [@[FHST]]. We see explicitly from the above formulas that ${{\mathscr{A}}}_{\nu'}{\mathbin{\mbox{\large$\looparrowdownleft$}}}{{\mathscr{X}}}(r)_{\nu}$ depends on both $\nu'$ and $\nu$ only modulo 2. Conclusion ========== The construction of multivertex Yetter–Drinfeld ${\mathfrak{B}}(X)$-modules has a nice combinatorial flavor: elements of the braided space $X$ populate line intervals separated by “vertex operators”—elements of another braided space $Y$, as $\xymatrix@R=6pt@C=70pt{ \ar@*{[|(1.6)]}@{-}|(.15){{\textstyle\!\!{\times}\!\!}} |(.25){{\textstyle{\circ}}}|(.4){{\textstyle\!\!{\times}\!\!}}|(.5){{\textstyle\!\!{\times}\!\!}}|(.65){{\textstyle{\circ}}} |(.8){{\textstyle\!\!{\times}\!\!}} |(.90){{\textstyle{\circ}}}[0,2]&&}\!\!\!\!$. This construction and the ${\mathfrak{B}}(X)$ action on such objects are “universal” in that they are formulated at the level of the braid group algebra and work for any braiding. However, even for diagonal braiding, extracting information such as fusion from Nichols algebras by direct calculation is problematic, except for rank $1$ (and maybe $2$). Much greater promise is held by the program of finding the modular group representation and then extracting the fusion from a generalized Verlinde formula like the one in [@[GT]]. Importantly, those Nichols algebras that *are* related to CFT (and some certainly are, cf. [@[2-boson]]) presumably carry an $SL(2,{\mathbb{Z}})$ representation on the center of their Yetter–Drinfeld category. Going beyond Nichols algebras ${\mathfrak{B}}(X)$ may also be interesting, and is meaningful from the CFT standpoint: adding the divided powers such as $F(p)$ in our ${\mathfrak{B}}\!_p$ case, which are not in ${\mathfrak{B}}(X)$ but do act on ${\mathfrak{B}}\!_p$-modules, would yield a braided (and, in a sense, “one-sided”) analogue of the infinite-dimensional quantum group that is Kazhdan–Lusztig-dual to logarithmic CFT models viewed as Virasoro-symmetric theories [@[BFGT]; @[BGT]]. ### Acknowledgments {#acknowledgments .unnumbered} The content of Sec. \[sec:Nich\] and Secs. [**\[sec:screenings\]**]{}–[**\[sec:dressedVO\]**]{} is the joint work with I. Tipunin [@[STbr]]. I am grateful to N. Andruskiewitsch, T. Creutzig, J. Fjelstad, J. Fuchs, A. Gainutdinov, I. Heckenberger, D. Ridout, I. Runkel, C. Schweigert, I. Tipunin, A. Virelizier, and S. Wood for the very useful discussions. A. Virelizier also brought paper [@[Brug]] to my attention. This paper was supported in part by the RFBR grant 10-01-00408 and the RFBR–CNRS grant 09-01-93105. Yetter–Drinfeld modules {#app:YD-axiom} ======================= In the category of left–left module comodules over a braided Hopf algebra ${\mathfrak{B}}$, a Yetter–Drinfeld (also called “crossed”) module [@[Sch-H-YD]; @[Besp-TMF]; @[Besp-next]] is a left module under an action  $\begin{tangle} \vstr{80}\lu \object{\raisebox{5.9pt}{\kern-4pt\tiny$\blacktriangleright$}} \end{tangle}\ :{\mathfrak{B}}{\otimes}{\mathscr{Y}}\to{\mathscr{Y}}$ and left comodule under a coaction  $\begin{tangle}\vstr{80}\ld\end{tangle}\ :{\mathscr{Y}}\to{\mathfrak{B}}{\otimes}{\mathscr{Y}}$ such that the axiom $$\label{yd-axiom} \begin{tangles}{l} \step{\object{\scriptstyle {\mathfrak{B}}}}\step[2]{\object{\scriptstyle {\mathscr{Y}}}}\\ \vstr{90}\cd\step{\mathrm{id}}\\ \vstr{50}{\mathrm{id}}\step[2]\hx\\ \vstr{90}\lu[2] \object{\raisebox{7pt}{\kern-4pt\tiny$\blacktriangleright$}} \step\hd\\ \vstr{90}\ld[2]\step\ddh\\ \vstr{50}{\mathrm{id}}\step[2]\hx\\ \vstr{90}\cu\step{\mathrm{id}}\end{tangles}\ \ = \ \ \begin{tangles}{l} \step{\object{\scriptstyle {\mathfrak{B}}}}\step[3]{\object{\scriptstyle {\mathscr{Y}}}}\\ \cd\step\ld\\ {\mathrm{id}}\step[2]\hx\step{\mathrm{id}}\\ \cu\step\lu\object{\raisebox{8.5pt}{\kern-4pt\tiny$\blacktriangleright$}} \end{tangles}$$ holds. The category ${{}\mbox{\small${}^{{\mathfrak{B}}}_{{\mathfrak{B}}}$}{\mathcal{Y\kern-3ptD}}}$ of Yetter–Drinfeld ${\mathfrak{B}}$-modules is monoidal and braided. The action and coaction on a tensor product of Yetter–Drinfeld modules are diagonal, respectively given by $$\label{act-coact-YD} \begin{tangles}{l} \hcd\step{\mathrm{id}}\step{\mathrm{id}}\\ {\mathrm{id}}\step\hx\step{\mathrm{id}}\\ \lu\object{\raisebox{8.5pt}{\kern-4pt\tiny$\blacktriangleright$}} \step \lu\object{\raisebox{8.5pt}{\kern-4pt\tiny$\blacktriangleright$}} \end{tangles} \qquad\text{and}\qquad \begin{tangles}{l} \ld\step\ld\\ {\mathrm{id}}\step\hx\step{\mathrm{id}}\\ \hcu\step{\mathrm{id}}\step{\mathrm{id}}\end{tangles}$$ For two Yetter–Drinfeld modules, their braiding and its inverse and square are given by $$\label{B+B2} \begin{tangles}{l} \vstr{67}\step[.5]{\mathrm{id}}\step{\mathrm{id}}\\ \obox{2}{\mathsf{B}}\\ \vstr{67}\step[.5]{\mathrm{id}}\step{\mathrm{id}}\end{tangles} \ \ = \ \ \begin{tangles}{l} \ld\step[1]{\mathrm{id}}\\ \vstr{50}{\mathrm{id}}\step\hx\\ \lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[1]{\mathrm{id}}\end{tangles} \ , \quad \begin{tangles}{l} \step[.5]{\mathrm{id}}\step{\mathrm{id}}\\ \obox{2}{\mathsf{B}^{-1}}\\ \step[.5]{\mathrm{id}}\step{\mathrm{id}}\end{tangles} \ \ = \ \ \begin{tangles}{l} \vstr{50}\step[1]\hx\\ \ld\step[1]{\mathrm{id}}\\ \vstr{67}\hxx\step[1]{\mathrm{id}}\\ {\mathrm{id}}\step\O{{\mathpzc{S}}^{\scriptscriptstyle-1}}\step[1]{\mathrm{id}}\\ {\mathrm{id}}\step\lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}} \end{tangles}\ , \quad\text{and}\quad \begin{tangles}{l} \step[.5]{\mathrm{id}}\step{\mathrm{id}}\\ \obox{2}{\mathsf{B}^{2}}\\ \step[.5]{\mathrm{id}}\step{\mathrm{id}}\end{tangles} \ \ = \ \ \begin{tangles}{l} \ld\step[2]{\mathrm{id}}\\ {\mathrm{id}}\step\x\\ \lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[2]{\mathrm{id}}\\ \ld\step[2]{\mathrm{id}}\\ {\mathrm{id}}\step\x\\ \lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[2]{\mathrm{id}}\end{tangles} \ \ = \ \ \begin{tangles}{l} \step[.5]\ld[2]\step[2]{\mathrm{id}}\\ \hcd\step[1.5]\x\\ {\mathrm{id}}\step{\mathrm{id}}\step[1]\hld\step[1]\ld\\ {\mathrm{id}}\step\hx\step[.5]{\mathrm{id}}\step[1]\O{{\mathpzc{S}}}\step[1]{\mathrm{id}}\\[-.8pt] \hcu\step[1]\hlu\object{\raisebox{13.1pt}{\kern-4pt\tiny$\blacktriangleright$}}\step\lu\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\\ \step[.5]{\mathrm{id}}\step[2]\x\\ \step[.5]\lu[2]\object{\raisebox{8pt}{\kern-4pt\tiny$\blacktriangleright$}}\step[2]{\mathrm{id}}\end{tangles}$$ Construction of Yetter–Drinfeld ${\mathfrak{B}}\!_p$ modules {#app:modules} ============================================================ One-vertex modules {#app:1-vertex} ------------------ One-vertex Yetter–Drinfeld ${\mathfrak{B}}\!_p$-modules [@[STbr]] are spanned by ${V^{a}_{s}}$ (see ) for a fixed $a\in{\mathbb{Z}}$ with $s$ ranging over a subset of $[0,\dots,p-1]$, under the ${\mathfrak{B}}\!_p$ action and coaction given in [**\[sec:dressedVO\]**]{}. ### Simple modules ${{\mathscr{X}}^{\{a\}}_0}$ {#sec:Xi} From each left coinvariant ${V^{a}_{0}}$, the action of ${\mathfrak{B}}\!_p$ generates a simple module comodule of dimension $(a)_p+1$: $${{\mathscr{X}}^{\{a\}}_0}=\text{Span}({V^{a}_{s}}\mid 0\leq s\leq (a)_p)$$ (simply because $F{ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a}_{(a)_p}}=0$ in accordance with the above formulas). The module comodule structure (in particular, the matrix of $F(r){ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}$ in the basis of ${V^{a}_{s}}$, Eq. ) depends on $a$ only modulo $p$, and hence there are just $p$ nonisomorphic simple one-vertex module comodules, for which we choose the notation ${{\mathscr{X}}}(r)$ indicating the dimension $1\leq r\leq p$; then there are the ${\mathfrak{B}}\!_p$ module comodule isomorphisms $$\begin{gathered} {{\mathscr{X}}^{\{a\}}_0}\cong{{\mathscr{X}}}(r)\quad \text{whenever} \quad (a)_p = r - 1.\end{gathered}$$ ### {#sec:S(p)} As noted above, we sometimes use a special notation ${{\mathscr{S}}}(p)={{\mathscr{X}}}(p)$. ### {#sec:Xext} For each $1\leq r\leq p-1$, ${{\mathscr{X}}^{\{a\}}_0}$ extends to a reducible module comodule ${{\mathscr{V}}}[r]$ with ${{\mathscr{X}}^{\{a\}}_0}\cong{{\mathscr{X}}}(r)$ as a sub(co)module and with the quotient isomorphic to ${{\mathscr{X}}}(p-r)$,as shown in . In terms of basis, this is $$\label{X-X} \xymatrix@=15pt{ &&&v^{a}(r+1)\ar@/_12pt/[dl]\ar^F[r]&\dots\ar^F[r]&v^{a}(p)\\ v^{a}(1)\ar^F[r]&\dots\ar^F[r]& v^{a}(r)}$$ where $$v^a(i)= F^{i - 1}{ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a}_{0}},\quad i \leq r,$$ is a basis in ${{\mathscr{X}}^{\{a\}}_0}(r)$, with the last, $r$th element given by $v^a(r)=C(r){V^{a}_{r - 1}}$ with a nonzero $C(r)$, and hence the upper floor starts with the element $v^a(r+1)=C(r){V^{a}_{r}}$. The downward arrow in  can be understood to mean $X\mapsto x_1$ whenever $\delta X=\sum_s F(s){\otimes}x_s$; this convention is a reasonable alternative to representing the same diagram as $$\xymatrix@=15pt{ &&&v^{a}(r+1)\ar^F[r] \ar@{{.}{--}{>}}|{F(r)}@/_18pt/[dlll] \ar@{{.}{--}{>}}@/_12pt/[dll] \ar@{{.}{--}{>}}|{F(1)}@/_8pt/[dl] &\dots\ar^F[r]&v^{a}(p)\\ v^{a}(1)\ar^F[r]&\dots\ar^F[r]& v^{a}(r)}$$ to express the idea that $\delta v^a(r+1)\in {\mathfrak{B}}\!_p{\otimes}\text{Span}(v^{a}(j)\mid 1\leq j\leq r + 1)$. The general form of the adjoint action on the one-vertex space is $$\begin{aligned} \label{Ad1-F(r)} F(r){ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a}_{s}} &= {\mathchoice {{\mbox{\footnotesize$\displaystyle \genfrac{\langle}{\rangle}{0pt}{}{r + s}{r}$}}}{{\genfrac{\langle}{\rangle}{0pt}{}{r + s}{r}}} {{\genfrac{\langle}{\rangle}{0pt}{}{r + s}{r}}}{{\genfrac{\langle}{\rangle}{0pt}{}{r + s}{r}}}}\,\xi^r \prod_{i=s}^{s + r - 1} {\langlei - a\rangle} {V^{a}_{s + r}},\end{aligned}$$ ### {#section-7} We verify that  holds by counting the total dimension of the modules just constructed: $$\dim{{\mathscr{S}}}(p)+\sum_{r=1}^{p-1}\dim{{\mathscr{V}}}[r]=p + (p - 1) p = p^2.$$ With the braiding , each of the above module comodules satisfies the Yetter–Drinfeld axiom. Two-vertex modules {#two-vertex-modules} ------------------ A two-vertex Yetter–Drinfeld module is a linear span of some ${V^{a,\,b}_{s,\,t}}$, $0\leq s,t \leq p-1$, for fixed integers $a$ and $b$ (see ).The left adjoint action of ${\mathfrak{B}}\!_p$ on these is given by $$\begin{aligned} \label{Ad2-F(r)} F(r){ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a,\,b}_{s,\,t}} &= \sum_{u=0}^{r} {c^{a\kern-1pt,\,b}_{s\kern-1pt,\,t}(r\kern-1pt,u)} {V^{a,\,b}_{s + r - u,\,t + u}},\end{aligned}$$ where $$\label{c-st} {c^{a\kern-1pt,\,b}_{s\kern-1pt,\,t}(r\kern-1pt,u)}= \xi^r {\mathfrak{q}}^{u (2 s - a)} {\mathchoice {{\mbox{\footnotesize$\displaystyle \genfrac{\langle}{\rangle}{0pt}{}{s + r - u}{r - u}$}}}{{\genfrac{\langle}{\rangle}{0pt}{}{s + r - u}{r - u}}} {{\genfrac{\langle}{\rangle}{0pt}{}{s + r - u}{r - u}}}{{\genfrac{\langle}{\rangle}{0pt}{}{s + r - u}{r - u}}}}\, {\mathchoice {{\mbox{\footnotesize$\displaystyle \genfrac{\langle}{\rangle}{0pt}{}{t + u}{u}$}}}{{\genfrac{\langle}{\rangle}{0pt}{}{t + u}{u}}} {{\genfrac{\langle}{\rangle}{0pt}{}{t + u}{u}}}{{\genfrac{\langle}{\rangle}{0pt}{}{t + u}{u}}}}\, \prod_{i=u}^{r - 1}{\langles + i + 2 t - a - b\rangle}\, \prod_{j=0}^{u - 1}{\langlet + j - b\rangle}.$$ The dependence on $b$ in  is modulo $p$, and on $a$, modulo $2p$. However, ${c^{a+p\kern-1pt,\,b}_{s\kern-1pt,\,t}(r\kern-1pt,u)}=(-1)^u {c^{a\kern-1pt,\,b}_{s\kern-1pt,\,t}(r\kern-1pt,u)}$ and the matrix of $F(r){ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}$ in the basis $({V^{a+p,\,b}_{s,\,t}})_{0\leq s,t\leq p-1}$ is the same as in the basis $((-1)^t{V^{a,\,b}_{s,\,t}})_{0\leq s,t\leq p-1}$; moreover, the coaction is unaffected by this extra sign. Hence, the *module comodule structure depends on both $a$ and $b$ modulo $p$*. We arrive at decomposition  by first listing all module comodules generated from *left coinvariants* $${V^{a,\,b}_{0,\,t}}= \xymatrix@1@C=70pt{ \ar@*{[|(1.6)]}@{-}|(.20){{\textstyle{\circ}}}^(.20){{}^{\scriptstyle a}}|(.3){{\textstyle\!\!{\times}\!\!}}|(.50){{\textstyle\!\!{\times}\!\!}}|(.65){{\textstyle\!\!{\times}\!\!}}|(.85){{\textstyle{\circ}}}^(.85){{}^{\scriptstyle b}}[0,2]&&} \qquad\text{($t$ crosses),}$$ and then studying their extensions. In accordance with , the algebra acts on left coinvariants as $$F(r){ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a,\,b}_{0,\,t}} = \sum_{s=0}^{r} c^{a,b}_t(r,s) {V^{a,\,b}_{r - s,\,t + s}},$$ with the coefficients $$\label{c-t} c^{a,b}_t(r,s)= c^{a,b}_{0,t}(r,s)= \xi^r {\mathfrak{q}}^{-s a} {\mathchoice {{\mbox{\footnotesize$\displaystyle \genfrac{\langle}{\rangle}{0pt}{}{t + s}{s}$}}}{{\genfrac{\langle}{\rangle}{0pt}{}{t + s}{s}}} {{\genfrac{\langle}{\rangle}{0pt}{}{t + s}{s}}}{{\genfrac{\langle}{\rangle}{0pt}{}{t + s}{s}}}} \prod_{i=s}^{r - 1}{\langlei + 2 t - a - b\rangle}\, \prod_{j=0}^{s - 1}{\langlet + j - b\rangle}.$$ In practical terms, the cases in [**\[sec:all-cases\]**]{} can be conveniently studies as follows. 1. \[case:Steinberg\] $F(r){ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a,\,b}_{0,\,t}}$ is nonvanishing and not a coinvariant for all $r$, $1\leq r\leq p-1$. 2. \[case:short-vanishing2\]${V^{a,\,b}_{0,\,t}}$ is not in the image of $F$ and $F(r){ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a,\,b}_{0,\,t}}$ vanishes for some $r\leq p-1$, i.e., $$c^{a,b}_t(r,s)=0, \quad 1\leq s\leq r.$$ 3. \[case:coinvariant\] $F(r){ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a,\,b}_{0,\,t}}$ is a coinvariant, i.e., $F(r){ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a,\,b}_{0,\,t}}=\mathrm{const}\;{V^{a,\,b}_{0,\,t+r}}$, for some $r\leq p-1$, which is equivalent to $$\begin{cases} c^{a,b}_t(r,s)=0,&0\leq s\leq r-1,\\ c^{a,b}_t(r,r)\neq 0. \end{cases}$$ Let $\beta=(a+b-2t)_p + 1$. For $0\leq b\leq p-1$, this is equivalent to $b=(2t + \beta - 1 - a)_p$. In fact, every triple $(a,b,t)$, $0\leq a,b,t\leq p-1$, can be uniquely represented as $$\label{abt-param} (a,b,t)=(a,\; (2 t + \beta - 1 - a)_p,\; t), \quad 1\leq \beta\leq p.$$ In this parameterization, coefficients  become $$\begin{gathered} c^{a,b}_t(r,s) = \xi^r {\mathfrak{q}}^{-s a} {\mathchoice {{\mbox{\footnotesize$\displaystyle \genfrac{\langle}{\rangle}{0pt}{}{t + s}{s}$}}}{{\genfrac{\langle}{\rangle}{0pt}{}{t + s}{s}}} {{\genfrac{\langle}{\rangle}{0pt}{}{t + s}{s}}}{{\genfrac{\langle}{\rangle}{0pt}{}{t + s}{s}}}} \prod_{i=s+1}^{r}{\langlei - \beta\rangle}\, \prod_{j=1}^{s}{\langlej - \beta + a - t\rangle}.\end{gathered}$$ and the analysis of the above cases becomes relatively straightforward. The results are as follows. ### Irreducible dimension-$p$ modules ${{\mathscr{S}}^{\{a,\,b\}}_{0,\,t}}(p)$ (case \[case:Steinberg\]) {#sec:S} A simple module comodule of dimension $p$, isomorphic to ${{\mathscr{S}}}(p)$ in [**\[sec:S(p)\]**]{}, is generated under the ${\mathfrak{B}}\!_p$ action from a coinvariant ${V^{a,\,b}_{0,\,t}}$ if and only if $$\label{cond:S} (a + b - 2 t)_p + 1=p.\pagebreak[3]$$When this condition is satisfied, we write ${{\mathscr{S}}^{\{a,\,b\}}_{0,\,t}}$, or even ${{\mathscr{S}}^{\{a,\,b\}}_{0,\,t}}(p)$, for this module comodule isomorphic to ${{\mathscr{S}}}(p)$.[^5] ### Reducible dimension-$p$ modules ${{\mathscr{V}}^{\{a,\,b\}}_{0,\,t}}[r]$ (case \[case:short-vanishing2\]) {#sec:X} A simple module comodule isomorphic to ${{\mathscr{X}}}(r)$ for some $1\leq r\leq p-1$ is generated under the action of ${\mathfrak{B}}\!_p$ from a coinvariant ${V^{a,\,b}_{0,\,t}}$ that is not itself in the image of $F$ if and only if $r=(a + b - 2 t)_p + 1$ and either of the two conditions holds:[^6] $$\begin{aligned} \label{cond:Xi} &t\leq (a)_p\quad\text{and} \quad (a)_p - r + 1\leq t\leq p - 1 - r,\\ \label{cond:Xii} &t\geq (a)_p + 1\quad\text{and} \quad p - r\leq t\leq p - r + (a)_p.\end{aligned}$$ In this case, we write ${{\mathscr{X}}^{\{a,\,b\}}_{0,\,t}}$ or ${{\mathscr{X}}^{\{a,\,b\}}_{0,\,t}}(r)$ for the corresponding module comodule: $$\label{2X-mod} {{\mathscr{X}}^{\{a,\,b\}}_{0,\,t}}\cong{{\mathscr{X}}}(r)\quad\text{whenever \ $r=(a + b - 2 t)_p + 1$ and~\eqref{cond:Xi} or~\eqref{cond:Xii} holds.}$$ Every ${{\mathscr{X}}^{\{a,\,b\}}_{0,\,t}}$ is further extended as in , which in terms of basis is now realized as $$\xymatrix@=15pt@C=12pt{ &&&*{\sum\limits_{s=0}^{r-1} {\langler\!-\!1\rangle!\,}c^{a,b}_t(r\!-\!1,s)\!{V^{a,\,b}_{r - s,\,t + s}}} \ar@/_20pt/[dl]\ar^(.8)F[r] &\dots \\ {V^{a,\,b}_{0,\,t}}\!\ar^(.65)F[r]&\dots\ar^(.17)F[r]& *{\sum\limits_{s=0}^{r-1} {\langler\!-\!1\rangle!\,}c^{a,b}_t(r\!-\!1,s) {V^{a,\,b}_{r -1 - s,\,t + s}}} }$$ with the south-west arrow meaning the same as in [**\[sec:Xext\]**]{}; the quotient is isomorphic to ${{\mathscr{X}}}(p-r)$. The notation${{\mathscr{V}}^{\{a,\,b\}}_{0,\,t}}[r]$ for this dimension-$p$ module comodule explicitly indicates the relevant left coinvariant and the dimension of the sub(co)module; the module comodule structure depends only on $r$: ${{\mathscr{V}}^{\{a,\,b\}}_{0,\,t}}[r]\cong{{\mathscr{V}}}[r]$. ### Three-floor modules ${{\mathscr{P}}^{\{a,\,b\}}_{0,\,t}}[r]$ (case \[case:coinvariant\]) {#sec:LB} We next assume that none of the above conditions , , is satisfied. An exemplary exercise shows that the negation of $\eqref{cond:S}\lor\eqref{cond:Xi}\lor\eqref{cond:Xii}$ is the “or” of the four conditions $$\begin{aligned} \label{P-cond-3} &t \geq p - r + (a)_p + 1,\\ \label{P-cond-4} &p - r \leq t \leq (a)_p,\\ \label{P-cond-1} &t \leq (a)_p - r,\\ \label{P-cond-2} &(a)_p + 1 \leq t \leq p - r - 1,\end{aligned}$$ where again $r=(a + b - 2 t)_p + 1$, $1\leq r\leq p-1$. The module generated from the coinvariant ${V^{a,\,b}_{0,\,t}}$ is then a sub(co)module in an indecomposable module comodule with the structure of subquotients $$ \xymatrix@=12pt{ &&{\mathscr{X}}(p-r') \ar@/_12pt/[dl] \ar[dr] &\\ &{\mathscr{X}}(r')\ar[dr] & &{\mathscr{X}}(r')\ar@/_12pt/[dl] \\ &&{\mathscr{X}}(p-r')& }$$ where $r'$ is either $r$ or $p-r$, as we now describe. 1. \[case:bot\]If $t+r\geq p$ (which means that either or holds), then the submodule generated from ${V^{a,\,b}_{0,\,t}}$ is isomorphic to ${{\mathscr{X}}}(r)$. We let it be denoted by ${{\mathscr{B}}^{\{a,\,b\}}_{0,\,t}}(r)$. (${{\mathscr{B}}}$ is for “bottom,” and ${{\mathscr{L}}}$ is for “left.”) 2. \[case:left\]If $t+r\leq p-1$ (which means that either or holds), then the submodule generated from ${V^{a,\,b}_{0,\,t}}$, denoted by ${{\mathscr{L}}^{\{a,\,b\}}_{0,\,t}}(r)$, is a $p$-dimensional reducible module comodule with ${{\mathscr{B}}^{\{a,\,b\}}_{0,\,t+r}}(p-r)\cong{{\mathscr{X}}}(p-r)$ as a sub(co)module and with the quotient isomorphic to ${{\mathscr{X}}}(r)$: $$\raisebox{-\baselineskip}{${{\mathscr{L}}^{\{a,\,b\}}_{0,\,t}}[r]={}$\ \ \ } \xymatrix@=12pt{{{\mathscr{X}}}(r)\ar[dr]\\ & {{\mathscr{X}}}(p-r). }$$ In terms of basis, this diagram is $$\xymatrix@C=15pt@R=12pt{ {V^{a,\,b}_{0,\,t}}\ar^F[r]&\dots\ar^(.15)F[r]& \sum\limits_{s=0}^{r-1} {\langler-1\rangle!\,}c^{a,b}_t(r-1,s) {V^{a,\,b}_{r -1 - s,\,t + s}}\ar[dr]^(.6)F\\ &&&{\langler\rangle!\,}{V^{a,\,b}_{0,\,t+r}}\ar^(.55)F[r]&\dots }$$ The set of all diagrams of this type actually describes both cases \[case:bot\] and \[case:left\]: according to whether a given coinvariant ${V^{a,\,b}_{0,\,u}}$ is or is not in the image of $F$, it occurs either in the bottom line (case \[case:bot\]) or in the upper line (case \[case:left\]) of the last diagram. Every such diagram is extended further, again simply because of the “cofree” nature of the coaction: $$\begin{gathered} \label{Xtended} {}\kern-20pt\xymatrix@C=15pt@R=12pt{ &&& *{\ T^{a,b}_{t+r}(r)} \ar@/_12pt/[ld] \\ {V^{a,\,b}_{0,\,t}}\ar^F[r]&\dots\ar^(.15)F[r]& *{\sum\limits_{s=0}^{r-1}{\langler-1\rangle!\,} c^{a,b}_t(r-1,s) {V^{a,\,b}_{r - s - 1,\,t + s}}} \ar[dr]^(.6)F\\ &&&*{{\langler\rangle!\,}c^{a,b}_t(r,r){V^{a,\,b}_{0,\,t+r}}\ } \ar^(.75)F[r]&\dots }\kern-20pt\end{gathered}$$ where, evidently, $$T^{a,b}_{t+r}(r)= \sum_{s=0}^{r - 1} {\langler-1\rangle!\,}c^{a, b}_t(r - 1, s) {V^{a,\,b}_{r - s,\,t + s}}.$$ Setting $$\begin{aligned} \label{uab} u^{a, b}_t(i)&=F^{i-1}{ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}T^{a,b}_{t+r}(r),\\ \label{vab} v^{a, b}_t(i)&=F^{i-1}{ \mathchoice{\mathbin{\blacktriangleright}} {\mathbin{\mbox{\small${\blacktriangleright}$}}} {\mathbin{{\blacktriangleright}}} {\mathbin{{\blacktriangleright}}}}{V^{a,\,b}_{0,\,t}},\end{aligned}$$ we have the full picture extending  as (omitting the ${}^{a,b}_{t}$ labels for brevity) $$\xymatrix@C=12pt@R=12pt{ &&& u(1) \ar@/_12pt/[dl]\ar^(.55)F[r]&\dots\ar^(.45)F[r]& u(p-r)\ar^F[dr] \\ v(1)\ar^F[r]&\dots\ar^(.45)F[r]& v(r)\ar[dr]^(.55)F&&&&u(p-r+1)\ar@/_12pt/[ld] \ar^(.65)F[r]&\dots\ar^(.45)F[r]&u(p)\\ &&&v(r+1) \ar^(.55)F[r]&\dots\ar^(.55)F[r]&v(p) }$$ Here,[^7] $$\delta u(1) = 1{\otimes}u(1) + {\raisebox{.5pt}{\mbox{\footnotesize$\displaystyle\frac{1}{{\langler-1\rangle!\,}}$}}}\,F(1){\otimes}v(r)+\dots$$ and, similarly, $$\delta u(p+r-1) = 1{\otimes}u(p+r-1) + {\raisebox{.5pt}{\mbox{\footnotesize$\displaystyle\frac{{\mathfrak{q}}^{-2 r}}{{\langler - 1\rangle!\,}}$}}}\,F(1){\otimes}v(p) + \dots.$$ To label such modules by the leftmost coinvariant ${V^{a,\,b}_{0,\,t}}$ (even though the entire module is not *generated from* this element), we write ${{\mathscr{P}}^{\{a,\,b\}}_{0,\,t}}$ to indicate both the module type and the characteristic coinvariant. An even more redundant notation is ${{\mathscr{P}}^{\{a,\,b\}}_{0,\,t}}[r]$, indicating the length $r$ of the left wing (which of course is $r=(a+b-2t)_p+1$). The module comodule structure depends only on $r$: $$\label{P-iso} {{\mathscr{P}}^{\{a,\,b\}}_{0,\,t}}[r] \cong {{\mathscr{P}}}[r].$$ To summarize, given a coinvariant ${V^{a,\,b}_{0,\,t}}$, holds if and only if (for $r=(a + b - 2 t)_p + 1$) $$\label{ProjectiveQ} \begin{aligned} &1\leq r\leq p-1 \text{ \ and}\\ &\text{$t \leq (a)_p - r$ \ or \ $(a)_p + 1 \leq t \leq p - r - 1$.} \end{aligned}$$ ### Completeness We verify  by counting the total dimension of the modules constructed. This gives $p^4$, the dimension of $\mathbb{V}_p(2)$, as follows.There are $p^2$ modules ${{\mathscr{S}}}(p)$ constructed in [**\[sec:S\]**]{}, $2 r (p - r)$ modules ${{\mathscr{X}}}(r)$ in [**\[sec:X\]**]{} for each $1\leq r\leq p-1$, making the total of $\frac{1}{3} p (p^2 - 1)$, and, finally, $(p - r)^2$ modules ${{\mathscr{L}}}(r)$ in [**\[sec:LB\]**]{} for each $1\leq r\leq p-1$, making the total of $\frac{1}{6} p (p - 1) (2 p - 1)$. Each ${{\mathscr{S}}}(p)$ is $p$-dimensional, each ${{\mathscr{X}}}(r)$ *extends to* a $p$-dimensional module, and each ${{\mathscr{L}}^{\{a,\,b\}}_{0,\,t}}[r]$ extends to a $2p$-dimensional module. The total dimension is $$p^2\cdot p + {\raisebox{.5pt}{\mbox{\footnotesize$\displaystyle\frac{1}{3}$}}}\, p (p^2 - 1) \cdot p + {\raisebox{.5pt}{\mbox{\footnotesize$\displaystyle\frac{1}{6}$}}}\,p (p - 1)(2 p - 1)\cdot 2 p = p^4.$$ ### Example Decomposition  is illustrated in Fig. \[fig:5-modules\] for $p=5$. $$\begin{matrix} \begin{pmatrix} {{\mathscr{X}}^{\{0,\,0\}}_{0,\,0}}\left(1\right)_0 \\ {{\mathscr{X}}^{\{0,\,0\}}_{0,\,1}}\left(4\right)_1 \\ {{\mathscr{L}}^{\{0,\,0\}}_{0,\,2}}\left[2\right]_1 \\ {{\mathscr{S}}^{\{0,\,0\}}_{0,\,3}}\left(5\right)_2 \\ {{\mathscr{B}}^{\{0,\,0\}}_{0,\,4}}\left(3\right)_2 \end{pmatrix} &\begin{pmatrix} {{\mathscr{X}}^{\{0,\,1\}}_{0,\,0}}\left(2\right)_0 \\ {{\mathscr{S}}^{\{0,\,1\}}_{0,\,1}}\left(5\right)_1 \\ {{\mathscr{X}}^{\{0,\,1\}}_{0,\,2}}\left(3\right)_1 \\ {{\mathscr{L}}^{\{0,\,1\}}_{0,\,3}}\left[1\right]_1 \\ {{\mathscr{B}}^{\{0,\,1\}}_{0,\,4}}\left(4\right)_2 \end{pmatrix} &\begin{pmatrix} {{\mathscr{X}}^{\{0,\,2\}}_{0,\,0}}\left(3\right)_0 \\ {{\mathscr{L}}^{\{0,\,2\}}_{0,\,1}}\left[1\right]_0 \\ {{\mathscr{B}}^{\{0,\,2\}}_{0,\,2}}\left(4\right)_1 \\ {{\mathscr{X}}^{\{0,\,2\}}_{0,\,3}}\left(2\right)_1 \\ {{\mathscr{S}}^{\{0,\,2\}}_{0,\,4}}\left(5\right)_2 \end{pmatrix} &\begin{pmatrix} {{\mathscr{X}}^{\{0,\,3\}}_{0,\,0}}\left(4\right)_0 \\ {{\mathscr{L}}^{\{0,\,3\}}_{0,\,1}}\left[2\right]_0 \\ {{\mathscr{S}}^{\{0,\,3\}}_{0,\,2}}\left(5\right)_1 \\ {{\mathscr{B}}^{\{0,\,3\}}_{0,\,3}}\left(3\right)_1 \\ {{\mathscr{X}}^{\{0,\,3\}}_{0,\,4}}\left(1\right)_1 \end{pmatrix} &\begin{pmatrix} {{\mathscr{S}}^{\{0,\,4\}}_{0,\,0}}\left(5\right)_0 \\ {{\mathscr{L}}^{\{0,\,4\}}_{0,\,1}}\left[3\right]_0 \\ {{\mathscr{L}}^{\{0,\,4\}}_{0,\,2}}\left[1\right]_0 \\ {{\mathscr{B}}^{\{0,\,4\}}_{0,\,3}}\left(4\right)_1 \\ {{\mathscr{B}}^{\{0,\,4\}}_{0,\,4}}\left(2\right)_1 \end{pmatrix} \\[8pt] \begin{pmatrix} {{\mathscr{X}}^{\{1,\,0\}}_{0,\,0}}\left(2\right)_0 \\ {{\mathscr{S}}^{\{1,\,0\}}_{0,\,1}}\left(5\right)_1 \\ {{\mathscr{X}}^{\{1,\,0\}}_{0,\,2}}\left(3\right)_1 \\ {{\mathscr{L}}^{\{1,\,0\}}_{0,\,3}}\left[1\right]_1 \\ {{\mathscr{B}}^{\{1,\,0\}}_{0,\,4}}\left(4\right)_2 \end{pmatrix} &\begin{pmatrix} {{\mathscr{X}}^{\{1,\,1\}}_{0,\,0}}\left(3\right)_0 \\ {{\mathscr{X}}^{\{1,\,1\}}_{0,\,1}}\left(1\right)_0 \\ {{\mathscr{X}}^{\{1,\,1\}}_{0,\,2}}\left(4\right)_1 \\ {{\mathscr{X}}^{\{1,\,1\}}_{0,\,3}}\left(2\right)_1 \\ {{\mathscr{S}}^{\{1,\,1\}}_{0,\,4}}\left(5\right)_2 \end{pmatrix} &\begin{pmatrix} {{\mathscr{X}}^{\{1,\,2\}}_{0,\,0}}\left(4\right)_0 \\ {{\mathscr{X}}^{\{1,\,2\}}_{0,\,1}}\left(2\right)_0 \\ {{\mathscr{S}}^{\{1,\,2\}}_{0,\,2}}\left(5\right)_1 \\ {{\mathscr{X}}^{\{1,\,2\}}_{0,\,3}}\left(3\right)_1 \\ {{\mathscr{X}}^{\{1,\,2\}}_{0,\,4}}\left(1\right)_1 \end{pmatrix} &\begin{pmatrix} {{\mathscr{S}}^{\{1,\,3\}}_{0,\,0}}\left(5\right)_0 \\ {{\mathscr{X}}^{\{1,\,3\}}_{0,\,1}}\left(3\right)_0 \\ {{\mathscr{L}}^{\{1,\,3\}}_{0,\,2}}\left[1\right]_0 \\ {{\mathscr{B}}^{\{1,\,3\}}_{0,\,3}}\left(4\right)_1 \\ {{\mathscr{X}}^{\{1,\,3\}}_{0,\,4}}\left(2\right)_1 \end{pmatrix} &\begin{pmatrix} {{\mathscr{L}}^{\{1,\,4\}}_{0,\,0}}\left[1\right]_{-1} \\ {{\mathscr{B}}^{\{1,\,4\}}_{0,\,1}}\left(4\right)_0 \\ {{\mathscr{L}}^{\{1,\,4\}}_{0,\,2}}\left[2\right]_0 \\ {{\mathscr{S}}^{\{1,\,4\}}_{0,\,3}}\left(5\right)_1 \\ {{\mathscr{B}}^{\{1,\,4\}}_{0,\,4}}\left(3\right)_1 \end{pmatrix} \\ \hdotsfor{5} \\ \begin{pmatrix} {{\mathscr{S}}^{\{4,\,0\}}_{0,\,0}}\left(5\right)_0 \\ {{\mathscr{L}}^{\{4,\,0\}}_{0,\,1}}\left[3\right]_0 \\ {{\mathscr{L}}^{\{4,\,0\}}_{0,\,2}}\left[1\right]_0 \\ {{\mathscr{B}}^{\{4,\,0\}}_{0,\,3}}\left(4\right)_1 \\ {{\mathscr{B}}^{\{4,\,0\}}_{0,\,4}}\left(2\right)_1 \end{pmatrix} &\begin{pmatrix} {{\mathscr{L}}^{\{4,\,1\}}_{0,\,0}}\left[1\right]_{-1} \\ {{\mathscr{B}}^{\{4,\,1\}}_{0,\,1}}\left(4\right)_0 \\ {{\mathscr{L}}^{\{4,\,1\}}_{0,\,2}}\left[2\right]_0 \\ {{\mathscr{S}}^{\{4,\,1\}}_{0,\,3}}\left(5\right)_1 \\ {{\mathscr{B}}^{\{4,\,1\}}_{0,\,4}}\left(3\right)_1 \end{pmatrix} &\begin{pmatrix} {{\mathscr{L}}^{\{4,\,2\}}_{0,\,0}}\left[2\right]_{-1} \\ {{\mathscr{S}}^{\{4,\,2\}}_{0,\,1}}\left(5\right)_0 \\ {{\mathscr{B}}^{\{4,\,2\}}_{0,\,2}}\left(3\right)_0 \\ {{\mathscr{L}}^{\{4,\,2\}}_{0,\,3}}\left[1\right]_0 \\ {{\mathscr{B}}^{\{4,\,2\}}_{0,\,4}}\left(4\right)_1 \end{pmatrix} &\begin{pmatrix} {{\mathscr{L}}^{\{4,\,3\}}_{0,\,0}}\left[3\right]_{-1} \\ {{\mathscr{L}}^{\{4,\,3\}}_{0,\,1}}\left[1\right]_{-1} \\ {{\mathscr{B}}^{\{4,\,3\}}_{0,\,2}}\left(4\right)_0 \\ {{\mathscr{B}}^{\{4,\,3\}}_{0,\,3}}\left(2\right)_0 \\ {{\mathscr{S}}^{\{4,\,3\}}_{0,\,4}}\left(5\right)_1 \end{pmatrix} &\begin{pmatrix} {{\mathscr{L}}^{\{4,\,4\}}_{0,\,0}}\left[4\right]_{-1} \\ {{\mathscr{L}}^{\{4,\,4\}}_{0,\,1}}\left[2\right]_{-1} \\ {{\mathscr{S}}^{\{4,\,4\}}_{0,\,2}}\left(5\right)_0 \\ {{\mathscr{B}}^{\{4,\,4\}}_{0,\,3}}\left(3\right)_0 \\ 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Yu.Bespalov and B.Drabant, *Hopf (bi-)modules and crossed modules in braided monoidal categories*, J. Pure and Applied Algebra 123 (1998) 105–129. S.Majid, *Foundations of Quantum Group Theory*, Cambridge University Press, 1995. P.Schauenburg, *Hopf modules and Yetter–Drinfel’d modules*, J. Algebra 169 (1994) 874–890. A.M.Semikhatov, *Virasoro central charges for Nichols algebras*, arXiv:1109.1767. Yu.Bespalov, T.Kerler, V.Lyubashenko, and V.Turaev, *Integrals for braided Hopf algebras*, q-alg/9709020. [^1]: But the actual motivation in [@[STbr]], which is yet to be tested on more advanced examples, was that Nichols algebras can actually do better than the “old” quantum groups. [^2]: The notation is fully explained below, but here we note that the module comodule structure, e.g., of ${{\mathscr{X}}}(r)_\nu$ depends only on $r$, whereas $\nu$ serves to distinguish isomorphic module comodules that nevertheless have different braiding. [^3]: Diagram  involves not only the squared braiding $\mathsf{B}^2$ of Yetter–Drinfeld modules but also, “in the loop,” the braiding itself (and the ribbon map ${\boldsymbol{\pmb{\vartheta}}}$). This does not affect the statement of the equivalence of entwined categories, but rather suggests exploring a further possibility, elaborating on the fact that the braiding of a Yetter–Drinfeld ${\mathfrak{B}}\!_p$-module *with itself* and *with its dual* also depends on $\nu\in{\mathbb{Z}}_2$, not $\nu\in{\mathbb{Z}}_4$ (and the same for the ribbon map). An entwined$'$ category might allow these braidings in addition to twines. This is similar to the idea of *twist equivalence* in the theory of Nichols algebras [@[AS-pointed]] (the similarity is not necessarily superficial if we recall that the braiding of “bare vertex operators” is diagonal for ${\mathfrak{B}}\!_p$). [^4]: An important technicality, noted in [@[A-about]; @[G-free]], is a distinction between quantum symmetric algebras [@[Rosso-inv]] and Nichols algebras proper; the latter are selected by the condition that the braiding be *rigid*, which in particular guarantees that the duals $X^*$ are objects in the same braided category with the $X$. [^5]: Condition  is actually worked out as follows: For odd $p$, it holds if and only if either $t \equiv {{ \mathchoice{{\raisebox{.5pt}{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(a + b + 1 + p)\;\text{mod}\;p}$ with $a + b$ even, or $t = { \mathchoice{{\raisebox{.5pt}{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(a + b + 1)$ with $a + b$ odd. For even $p$, it holds if and only if either $t \equiv{{ \mathchoice{{\raisebox{.5pt}{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(p + 1 + a + b)\;\text{mod}\;p}$ or $t = { \mathchoice{{\raisebox{.5pt}{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(a + b + 1)$ (which selects only odd $a+b$). [^6]: The logic of the presentation is that we assume that $0\leq a,b\leq p-1$, and hence $(a)_p=a$; but we do not omit the operator of taking the residue modulo $p$ because we refer to formulas given here also in the case where $a\in{\mathbb{Z}}$. [^7]: “The closure of the rhombus” in the above diagram is a good illustration of the use of the Yetter–Drinfeld axiom, which is also used in several other derivations without special notice. The “relative factor” ${\mathfrak{q}}^{-2r}$ in the next two formulas, in particular, is an immediate consequence of the Yetter–Drinfeld condition.
ELECTRONIC STRUCTURES OF QUANTUM DOTS AND THE ULTIMATE RESOLUTION OF INTEGERS C.G. Bao, Y.Z. He, and G.M. Huang Department of Physics, Zhongshan University, Guangzhou, 510275, China ABSTRACT: The orbital angular momentum L as an integer can be ultimately factorized as a product of prime numbers. We show here a close relation between the resolution of L and the classification of quantum states of an N-electron 2-dimensional system. In this scheme, the states are in essence classified into different types according to the $m(k)$-accessibility, namely the ability to get access to symmetric geometric configurations. The $% m(k)$-accessibility is an universal concept underlying all kinds of 2-dimensional systems with a center. Numerical calculations have been performed to reveal the electronic structures of the states of different types. This paper supports the Laughlin wave function and the composite fermion model from the aspect of symmetry. PACS: 73.20.Dx, 03.75.Fi, 02.10.Lh [**Introduction, the resolution of integers and the accessibility of the configurations with a m-fold axis**]{} The resolution of integers is a basic and important theorem in the primary theory of number. Each integer I can be ultimately factorized as a product of prime numbers as I= $2^{n_2}3^{n_3}5^{n_5}7^{n_7}\cdot \cdot \cdot $. Thus, like the elementary particles in physics, the prime numbers serve as elementary elements of integers. Let the set of the idempotent indexes be denoted as $\{n_i\}$ . Evidently, the character of an integer is determined by this set. It is possible that the speciality of the set $\{n_i\}$ of an integer would affect the role of the integer in nature. In quantum mechanics, integers play a very important role. A number of physical quantities are integers (if specific units are used), e.g., the number of particles in a system, the orbital angular momenta L and their components, the spin S, the charge Z , etc.. Therefore, there might be a connection between the theory of number and quantum mechanics. However, such a connection is not clear until now. In particular, how the idempotent series $% \{n_i\}$ play their role directly in physical world is not clear. For three dimensional systems there are magic integers ( e.g., the numbers 2, 8, 20, 28, 50, 82, $\cdot \cdot \cdot $ for the shell structures of nuclei; the numbers 2,10, 18, 36, 54, $\cdot \cdot \cdot $ for the periodic table of atoms, the numbers 13, 19, 25, 55, 71 $\cdot \cdot \cdot $ for the abundance of atomic clusters, etc.). It is not clear whether these magic integers have something special in their $\{n_i\}$ . Nonetheless, for two-dimensional quantum dots, we show here a close relation between the set $\{n_i\}$ of L as an integer and the electronic structures. Let us consider a two-dimensional system of N electrons confined in a quantum dot $^{1-3}$. N may be large or small, but is finite. It is assumed that the potential of confinement is isotropic so that the orbital angular momentum L together with the total spin S are conserved in an eigenstate $% \Psi _{LS}$ . It is well known that the spin-states are the basis-states of the two-row representation $\stackrel{\sim }{\lambda }$ of the permutation group, $\stackrel{\sim }{\lambda }$ =\[$\frac N2$+S,$\frac N2-$S\] . Let the $% i $-th spin-state of the $\stackrel{\sim }{\lambda }$ representation be denoted as $\chi _i^{\stackrel{\sim }{\lambda }}$ , $i$ = 1 to $d$ ( the dimension of $\stackrel{\sim }{\lambda }$ ). $\Psi _{LS}$ can be expanded as $\Psi _{LS}=\sum_iF_i^\lambda \chi _i^{\stackrel{\sim }{\lambda }}$(1) where $F_i^\lambda $ is a function of spatial coordinates and is a basis-state of the $\lambda $ representation, the conjugate representation of $\stackrel{\sim }{\lambda }$. . In (1) the antisymmetrization is assured. On the other hand, when the number of electrons N or N-1 can be factorized as a product of integers N=N$_A$N$_B$ or N$-$1=N$_C$N$_D$, the electrons may surround the center of confinement and form a geometric configuration with a m-fold axis, m=N$_A$, N$_B$, N$_C$, or N$_D$ . Such a configuration is called a $m(k)$ configuration, where k=N/m or (N-1)/m is the number of homocentric circles, each contains m electrons (some circles might have the same radius). When k=(N-1)/m, the $m(k)$ configuration would have an electron at the center. Some of the $m(k)$ configurations are in the domain of low total potential energy, these $m(k)$ are important to the electronic structures as we shall see. Since a rotation of a $m(k)$ about the center by $\frac{2\pi }m$ is equivalent to $k$ cyclic permutations of particles, we have$^4$ at a $m(k)$ $e^{i2\pi L/m}F_i^\lambda (12\cdot \cdot \cdot \cdot )=F_i^\lambda (23\cdot \cdot \cdot \cdot )=\sum_jG_{ji}^\lambda (p_c)F_j^\lambda (12\cdot \cdot \cdot \cdot )$(2) where $G_{ji}^\lambda (p_c)$ is the matrix element of the $\lambda $-representation associated with the $k$ cyclic permutations $p_c$ . There are totally $d$ such equations, they form a set of homogeneous linear equations. From the set we define a determinant $D(L,\lambda ,m)=|G_{ji}^\lambda (p_c)-\delta _{ij}e^{i2\pi L/m}|$(3) Evidently, if $D(L,\lambda ,m)$ is nonzero, the set of linear equations will have only zero solutions, and thereby the $F_i^\lambda $ must all be zero at the $m(k)$. In this case, an inherent nodal surface is imposed by symmetry and the $m(k)$ is therefore inaccessible$^5$. If a wave function is distributed in a domain containing an inaccessible configuration, the inherent nodal surface would cause an excited oscillation resulting in a great increase in energy. Hence, for low-lying states, the wave function would be far away from the inaccessible configuration. Anyway, whether $% D(L,\lambda ,m)$ is nonzero or zero would affect strongly the electronic structure of the state. Since $D(L,\lambda ,m)$ depends on L and S, evidently the electronic structures depend strongly on L and S. The calculation of $D(L,\lambda ,m)$ is not difficult if N is small. However, a general discussion is not easy due to the complexity in the general representation of [*S*]{}$_N$ group. Nonetheless, for polarized systems, the discussion becomes much simpler as follows. Let the group of states having the same $L$ be called a $L-series$. For polarized systems we have S=N/2 and $\lambda $ is totally-antisymmetric. In this case the discriminant reads $D(L,\lambda ,m)=(-1)^{(m-1)k}-e^{i2\pi L/m}=0$(4) If a couple of $m$ and $k$ fulfil (4), then the corresponding $m(k)$ is accessible to the $L-series$. Evidently, all the $m(k)$ are accessible to the L=0 series except the case of m even and k odd. Furthermore, all the $% m(k)$ are inaccessible to the L=1 series except the case of m=2 and k odd. When $L\geq 2$, from the resolution of integers, we have $L=2^{n_2}3^{n_3}5^{n_5}7^{n_7}(11)^{n_{11}}\cdot \cdot \cdot \cdot $(5.1) $m=2^{m_2}3^{m_3}5^{m_5}7^{m_7}(11)^{m_{11}}\cdot \cdot \cdot \cdot $(5.2) Inserting (5) into (4), we have $\exp [i\pi (2^{n_2+1-m_2}3^{n_3-m_3}5^{n_5-m_5}\cdot \cdot \cdot )]=(-1)^{(m-1)k}$(6) From (6) and by using a little primary knowledge of the theory of number we arrive at the following rules: RULE 1, If $m$ is odd, or if $m$ and $k$ are both even, then the $m(k)$ is accessible to the $L-series$ with $n_i\geq m_i$ (here $i=2,3,5,7,\cdot \cdot \cdot $) . RULE 2, If $m$ is even and $k$ is odd, then the $m(k)$ is accessible to the $% L-series$ with $n_2=m_2-1$ and $n_i\geq m_i$ . RULE 3, Let $mk=m^{\prime }k^{\prime }=I$ . If the integers $m$ and $% m^{\prime }$ do not have a common factor, and if both the $m(k)$ and $% m^{\prime }(k^{\prime })$ are accessible to a $L-series$ , then the product-configuration $mm^{\prime }(\frac I{mm^{\prime }})$ is also accessible to the $L-series$ . RULE 4, Let $mk=m^{\prime }k^{\prime }=I$ . If the configuration $mm^{\prime }(\frac I{mm^{\prime }})$ is accessible to a $L-series$, then both the $m(k)$ and $m^{\prime }(k^{\prime })$ are accessible to the $L-series$ . Alternatively, if $m(k)$ (or $m^{\prime }(k^{\prime })$ ) is inaccessible to a $L-series$ , the $mm^{\prime }(\frac I{mm^{\prime }})$ is also inaccessible to the $L-series$ . RULE 5, If $_{}L=mk(mk\pm j_o)/2\geq 0$, where $j_o$ is an odd integer , then the $m(k)$ is accessible to the $L-series$. For examples, (i) The 8(3) configuration of the N=24 or 25 system has $m_2=3$ and $m_i=0$ ($i\geq $3). According to RULE 2, this configuration is accessible to the L=4$j_o$ series, here $j_o$ is an arbitrary positive odd integer. (ii) According to RULE 1 the 3(6) is accessible to the L=3$j$ series, here $j$ is an arbitrary positive integer. According to RULE 2 the 2(9) is accessible to the L=$j_o$ series. Therefore both the 3(6) and 2(9) are accessible to the L=3$j_o$ series. Since 3 and 2 do not have a common factor, according to RULE 3, the 6(3) is also accessible to the L=3$j_o$ series. (iii) Since the 8(1) is accessible to the L=4$j_o$ series, according to RULE 4 both the 4(2) and 2(4) are also accessible to the series. Alternatively, since the 2(4) is inaccessible to the L=$j_o$ series, according to RULE 4 both the 4(2) and 8(1) are also inaccessible to the series. It is straight forward to know from the RULE 5 that all the $m(k)$ with $% mk= $N are accessible to the $L=$N(N-1)/2 $\pm j$ N states, and all the $% m(k) $ with $mk=$N-1 are accessible to the $L=$N(N-1)/2 $\pm $ $j$ (N-1) states . Therefore, all the $m(k)$ disregarding $mk=$N or N-1 are accessible to the $L=j_o$N(N-1)/2 states, i.e., these special states do not contain inherent nodal surfaces at any $m(k)$, therefore they are specially stable. When a magnetic field is applied, they are the strongest candidates of ground states. Here the reciprocal of $j_o$ is associated with the filling factor $\nu $ of the Hall effect$^{6,7}$, and the $L=j_o$N(N-1)/2 states are associated with $\nu =1,\frac 13,\frac 15,\cdot \cdot \cdot $ It is recalled that the famous Laughlin wave function$^{8,9}$. $\psi _\nu =[\Pi _{i<j}(z_i-z_j)^{j_o^{\prime }}]\exp (-\sum_jz_j^{*}z_j)$(7) has $L=j_o^{\prime }$N(N-1)/2 , here $j_o^{\prime }$ is also an odd integer. Thus, we have proved that this state does not contain inherent nodal surfaces at any $m(k)$ configuration, therefore the wave function can be smoothly distributed without nodes in the domain of low potential energy. This might be a reason that they are close to the exact solutions. [**Classification of quantum states in a 9-electron dot**]{} We shall see that the accessibility of the $m(k)$-configurations affects the electronic structures greatly. Let us investigate in detail a 9-electron dot. Although such a system has already been more or less concerned in the literatures$^{10-13}$, a precise calculation beyond the lowest Landau level approximation and a detailed analysis of the wave functions have not yet been performed. In the view of geometry there are the 9(1), 3(3), 8(1), 4(2), and 2(4) configurations. However, due to the RULE 1 and 2, only some of them are accessible to a specific $L-series$ . Furthermore, the 9(1) can be neglected due to having a much higher potential energy ( In 9(1) only nine bonds can be optimized, while in 8(1) sixteen bonds can be. Incidentally, the 9(1) would become more important in a 10-electron system, in that case eighteen bonds can be optimized). Thus the important $m(k)$ are the other four, they lie in the domain of lower potential energy. Based on their accessibility, the $L-series$ can be classified into eight types as shown in TABLE 1. Due to the RULE 1 and 2, the scheme depends straight on $\{n_i\}$ . For an example, the $L-series$ having $n_2=2$ and $n_3\geq 1$ are both 8(1)- and 3(3)-accessible. According to the RULE 4, the 4(2), and 2(4) are also accessible to this series. Thus they are inherently nodeless in all the important $m(k)$, and are grouped to type 1. Consequently, they are superior in stability and therefore particularly important. They have $L=j_o$N(N-1)/6, thus the above mentioned Laughlin states ($L=j_o^{\prime }$N(N-1)/2) are members of this type. The other members of this type are also candidates of the ground state and are associated with the filling factor $% \nu =3/j_o$ . For another example, the type 4 is 3(3)-accessible but 2(4)-, 4(2)- and 8(1)-inaccessible. Incidentally, the $n_i$ with $i\geq 5$ are irrelevant to the classification of the 9-electron system and therefore can be arbitrary. TABLE 1, Classification of states of a polarized 9-electron dot according to the $\{n_i\}$ of L. The $m(k)$ configurations accessible to a specific type are listed. ------------------------- $% \{n_i\}$ -- -- ------------------- $n_2=2,n_3\geq 1$ $n_2\geq 3,n_3\geq 1$ $n_2=1,n_3\geq 1$ $n_2=0,n_3\geq 1$ $n_2=2,n_3=0$ $n_2\geq 3,n_3=0$ $n_2=1,n_3=0$ $n_2=0,n_3=0$ ------------------------- The classification according to $\{n_i\}$ is in essence a classification according to the accessibility of the $m(k)$ , or in other words according to the inherent nodal surfaces. Thus the classification is model-independent and based simply on the fundamental principle of symmetry. To show the reasonableness of the classification, numerical results are given in the follows. The Hamiltonian reads $H=\sum_i[\frac{p_i^2}{2m*}+\frac 12m^{*}\omega _o^2r_i^2]+\frac{e^2}{4\pi \varepsilon }\sum_{i>j}\frac 1{r_{ij}}$(8) where $m^{*}$ is the effective mass, $\varepsilon $ the dielectric constant. It is assumed that $m^{*}=0.067m_e$ , $\varepsilon =12.4$ (for GaAs dots), and $\hbar \omega _o$ = 3 meV. In what follows meV and $\sqrt{\hbar /(m^{*}\omega _o)}$ will be used as units for energy and length, respectively. Let the single-electron harmonic oscillator states be denoted as $% |ll^{\prime },\omega \rangle ,$ they have energies $(l+l^{\prime }+1)\hbar \omega $ and angular momenta $l-l^{\prime }$ . With them antisymmetrized harmonic oscillator product states $\Phi _J=$ $|l_1l_1^{\prime },l_2l_2^{\prime },\cdot \cdot \cdot l_Nl_N^{\prime },\omega \rangle $ with $% \sum_i(l_i-l_i^{\prime })=L$ are constituted and are used as basis functions of eigenstates. Here $\omega $ is in general not equal to $\omega _o$ , but is adjustable to minimize the eigenenergies. For all the following calculations we have $l_i$ $\leq $ $25$ , $\sum_il_i^{\prime }\leq $ $3$ , i.e., higher Landau levels are included. In order to depress the number of basis functions, $\Phi _J$ are arranged in such a way so that $<\Phi _J|H|\Phi _J>\leq <\Phi _{J+1}|H|\Phi _{J+1}>$ . In such a sequence the one with a very large index $J$ is not important to the low-lying states. Then, $% H$ is diagonalized first in a space with $J$ starting from 1 to a given smaller number, and again to a larger number, and repeatedly, until a satisfied convergency is achieved, i.e., the eigenenergies have at least four effective figures and the correlated densities extracted from the related wave functions are nearly unchanged. It was found that, even in the case of N=19, $J\leq 8000$ is enough for our purpose if the variational parameter $\omega $ has been properly adopted and if L is not much larger than N(N-1)/2 (e.g., L $\leq$ 100 if N=9). Once an eigenstate $\Psi _{LS}$ is obtained, the associated 1-body, 2-body, and 3-body density functions $\rho _1$, $\rho _2$ and $\rho _3$ would be extracted (the results of $\rho _1$ will not be given here)$.$ For example, the 3-body density function is defined as $\rho _3(\stackrel{\rightarrow }{r}_1,\stackrel{\rightarrow }{r}_2,\stackrel{% \rightarrow }{r}_3)=\int d\stackrel{\rightarrow }{r}_4\cdot \cdot \cdot d% \stackrel{\rightarrow }{r}_9|\Psi _{LS}|^2$(9) It has been previously suggested$^{10-12}$ that some of the electrons ( N$% _{out}$ ) might be located in a ring outside, and form a N$_{out}$-ring-structure. For a 9-electron dot, it turns out that the total potential energy of a ring-structure with N$_{out}\leq 4$ or N$_{out}$=9 is much higher. The domain in coordinate space containing the 5- to 8-ring-structures are broad, where the total potential energy is low and flat . If symmetry is not taken into account, these ring-structures might be equally preferred by low-lying states. However, if the domain of a ring-structure contains an inaccessible $m(k)$ (e.g., the domain of a 6-ring structure contains the 3(3) which is inaccessible to the type 5 to 8), the ring-structure would be unfavorable because the existing inherent nodal surface would cause a great increase in energy. Thus, which ring-structures would be the better choice depends on the type of states. Let the $n$-th state and its energy of a $L-series$ be denoted as (L)$_n$ and E$_n$(L) . The $n$=1 state, namely the lowest of the series, is called the first-state. Let us define E$_{cusp}(L)=$E$_1$(L)-(E$_1$(L-1)+E$_1$(L+1))/2(10) Evidently, if E$_{cusp}(L)$ is negative, the $L-series$ would have a downward cusp. Now, let us first inspect the L=60 series with $\nu $=N(N-1)/2L=3/5 as an example of type 1. We have E$_1$(60)=272.86 and E$_{cusp}(60)=$ -0.43. Thus, the L=60 series has a downward cusp, a common feature of the type 1. The 2-body densities $\rho _2$ of the (60)$_1$ and (60)$_2$ are plotted in Fig.1a and 1b, where the 8-ring and 6-ring structures originate from the 8(1)- and 3(3)- accessibility, respectively. Since the 8-ring has more electrons in the ring, its moment of inertia is larger resulting in having a smaller rotation energy. Therefore it is lower than the 6-ring in the case of L=60. The L=36 series belongs to the type 1 with $\nu $=1. The $\nu $=1 states are special because they have only one basis function in the lowest Landau level, namely the Laughlin wave function (eq.(7)) with $j_o^{\prime }=1$ . Therefore the (36)$_1$ will be dominated by this function. In fact, in our calculation, this state has the weights of the lowest to the fourth lowest Landau levels to be 80.0%, 16.8%, 2.5%, and 0.7%, respectively. It is noted that the clear geometric features shown in Fig.1a and 1b arise from a coherent mixing of the basis functions. Although the (36)$_1$ is allowed by symmetry to get access to symmetric geometric configurations, this state is not able to possess a clear geometric feature as shown in Fig.1c due to the lack of coherent mixing. In fact, the feature of Fig.1c arises simply from the Laughlin wave function. Incidentally, the (36)$_1$ has a rather low energy E$_1$(36)=209.18 and a very large gap 4.98 lying between E$_1$(36) and E$_2$(36), thus this state is superior in stability. Fig.1d to 1f are examples of type 4. They do not have the 8-ring structure because this type is 8(1)-inaccessible. The (63)$_1$ and (81)$_1$ have a clear 6-ring originating from the 3(3) accessibility. To see clearer the structure of the core, the $\rho _3$ of the (81)$_1$ is plotted in Fig.2a, where a clear regular triangle is inside. However, instead of having a 6-ring, the (99)$_1$ has a 7-ring structure. It is noted that a 9-particle system does not contain the 7(k)-configuration. Hence, the 7-ring is not constrained by symmetry. Thus, it is not surprising that both the 6- and 7-ring emerge in the type 4. The 6-ring would be better than the 7-ring if L is smaller (e.g., the (63)$_2$ and (81)$_2$ are found to have a 7-ring ). However, the 7-ring would be better if L is larger due to having a larger moment of inertia. It is clear that , although the inherent nodal surfaces have imposed serious constraints on wave functions, the electronic structures are not uniquely determined by them. In addition to the pairwise interaction, the centrifugal barrier and the parabolic potential also play their role. The barrier leads to the preference for the configurations with a larger moment of inertia. Therefore, a critical value(s) of L denoted as L$_{crit}$ might exist for each type so that the first-states with L smaller than L$_{crit}$ and those with L$\geq $L$_{crit}$ are distinct in structure (e.g., the (81)$_1$ and (99)$_1$). The parabolic potential confines the number of effective basis functions taking part in coherent mixing. Thus the lower states with $\nu $ equal or close to 1 are insufficient in coherent mixing and therefore ambiguous in geometric feature (e.g., the (36)$_1$). In addition to the case of $\nu $ =1, an example of $\nu $ =18/19 is shown in Fig.1g to 1i belonging to type 7. The L=38 states have only two basis functions in the lowest Landau level. Consequently, both the (38)$_1$ and (38)$_2$ are ambiguous in geometric feature. However, the (38)$_3$ dominated by the second lowest levels has a clear 7-ring due to having a sufficient coherent mixing. There is a very large gap 3.15 lying between the E$_2$(38)and E$_3$(38). The two lower states have E$_1$(38)=220.61 and E$_2$(38)=221.25, much higher than the E$_1$(36). Noting that the type 7 is both 8(1)- and 3(3)-inaccessible. Therefore , the 7-ring is preferred.. Another example of type 7 is given in Fig. 1j. Since a number of the first Landau levels are contained in the L=50 series, instead of the third-state, the 7-ring appear first in the first-state. Fig.1k and 1$l$ show the similarity of the two first-states of type 2, both have a 6-ring. Fig.1m and 1n show the similarity of the two second-states, both have a square-structure. Evidently, these structures originates from the 3(3)- and 4(2)-accessibility. To see clearer the square, the $\rho _3$ of the (48)$_2$ is plotted in Fig.2b. Since these states are 8(1)-inaccessible, the inherent nodal surfaces at the 8(1) would spoil the stability of the square. Thus the square-structure is higher. Fig.1o and 1p are examples of type 5. Since this type is 8(1)-accessible but 3(3)-inaccessible, it is easy to understand why the octagon shape emerges. [**Classification of fermion states of N**]{}$\neq $[**9 dots**]{} For the classification of states of a general N-electron dot, we have first to figure out which $m(k)$ configurations with $mk$=N or N-1 will be contained in the domain of lower potential energy. Secondly, we have to make sure their accessibility to the $L-series$ . If N=6, the 5(1), 3(2), 2(3) configurations should be considered (the 6(1) is automatically taken into account due to the RULE 3). Then the type 1 has L=$j_o$15, which is the intersection of {L$\equiv $0 mod 5}and {L$\equiv $3 mod 6}. This is a well known result$^{14}$. When N is larger, the effect of the $m(k)-$accessibility might reduce, because in the coordinate space the domain of low energy is so broad that the wave function is easy to avoid the inaccessible configuration. However, even if N is as large as 19, the effect of the $m(k)$-accessibility is still explicit. When N=19, the 9(2), 3(6) and 2(9) have to be considered (the 6(3) is automatically taken into account due to the RULE 3). Then the classification is shown in TABLE 2. TABLE 2, Classification of states of a polarized 19-electron dot. ------ ------------------- ----------------------- Type accessible $m(k)$ $\{n_i\}$ 1 9(2),3(6),2(9) $n_2=0,n_3\geq 2$ 2 3(6),2(9) $n_2=0,n_3=1$ 3 9(2) $n_2\geq 1,n_3\geq 2$ 4 3(6) $n_2\geq 1,n_3=1$ 5 2(9) $n_2=0,n_3=0$ 6 $n_2\geq 1,n_3=0$ ------ ------------------- ----------------------- It was found that both the type 1 and type 2 are better in stability, they have downward cusps. For examples, we have E$_{cusp}$(177)=$-0.032$ and E$% _{cusp}$(183)=$-0.014$, both belong to type 2. Fig.2c show a 12-ring structure originating from the 6(3)-accessibility (similar to the fact that the 6-ring structure originates from the 3(3)-accessibility, cf. Fig.2a ). this 12-ring structure is common to the type 1 and 2. On the other hand, the geometric feature of the (184)$_1$ of type 6 is not very clear, but the ring-structure is explicit ( 13 electrons are found in the ring ). These figures support the ring structure proposed by other authors$^{10-12}$. In general, not matter how large N is, only the states with a superior stability are interesting. The stability is quite often associated with the geometric symmetry. Once the geometric symmetry is concerned, the effect of the $m(k)$-accessibility should be considered. [**Bosonic systems**]{} Since the above discussion is model-independent and is simply based on symmetry consideration, it can be generalized to bosonic systems as well. In this case the wave functions should be completely symmetric with respect to particle permutation. Thus, instead of eq.(4), we have the criterion $1-e^{i2\pi L/m}=0$(10) and accordingly the $m(k)$ configuration is accessible to the $L-series$ with $n_i\geq m_i$ disregarding $m$ and $k$ are even or odd. In particular, all the $m\left( k\right) $ are accessible to the L=0 states. In recent years the Bose-Einstein condensation has been extensively studied, and the trapped atoms gases have been shown to Bose condense$^{15,16}$. Cooper and Wilkin have studied the properties of rotating Bose-Einstein condensates in parabolic traps. When the rotation frequency is larger, they found the ground state angular momentum L for N=3 to 10- boson systems are 6, 12, 20, 30, 42, 56, 72, and 90, respectively$^{17}$. This can be explained based on the composite fermion model$^{17,18}.$ Alternatively, we now provide a model-independent explanation simply based on the $m(k)$-accessibility. For an example, the important $m(k)$ for the N=8 system is 7(1) and 4(2) (here the 8(1) is much less important than the 7(1), the former has only eight bonds to be optimized while the latter has fourteen ) , thus the type 1 has $n_2\geq 2$ and $n_7\geq 1$. Therefore the L=56 state would appear as a ground state when the rotation frequency lies in a specific region. The important $m(k)$ for the N=9 system is 8(1) and 3(3), thus the type 1 has $n_2\geq 3$ and $n_3\geq 1$, therefore the L=72 state would appear as a ground state. The important $m(k)$ for the N=10 system is 9(1), 5(2), 3(3), and 2(5), thus the type 1 has $n_2\geq 1$ , $n_3\geq 2,$ and $n_5\geq 1$, therefore the L=90 state would appear as a ground state. In fact, the downward cusps found in the ref.17 are closely related to the $% m(k)$-accessibility. For examples, from the TABLE I of the ref.17 , we know that the N=6 system has a cusp at L=6 and 12 which are associated with the 3(2) and 2(3) accessibility, a cusp at L=10 which is associated with the 5(1) and 2(3) accessibility, a cusp at L=15 which is associated with the 5(1) and 3(2) accessibility, etc. [**Concluding remarks**]{} An exact diagonalization of the Hamiltonian has been performed. Since higher Landau levels have been considered, the numerical results (even in the case of N=19) are very accurate in the qualitative sense. Since the $\omega $ of the basis functions is a variational parameter, the convergency is thereby greatly improved. Furthermore, in addition to the usually given $\rho _1$ and $\rho _2$ , $\rho _3$ have also been calculated to help the analysis. A scheme of classification according to the idempotent series $\{n_i\}$ of L has been proposed. In this scheme each type has its own $m(k)$-accessibility, or its own inherent nodal surfaces. The classification is objective and model-independent. Although the electronic structures are not uniquely determined by the $m(k)$-accessibility, its great effect has been confirmed by the numerical results. Since the type 1 is inherently nodeless in the domain of low potential energy, the first-states of this type are superior in stability and are the strongest candidates of the ground states. These noticeable states can be easily identified in our scheme. The analysis of this paper supports the Laughlin wave function and the composite fermion model from the aspect of symmetry. The $m(k)$-accessibility is an universal concept for all kinds of 2-dimensional systems with a center. The introduction of this concept would lead to a better understanding of these systems. The theory of number and quantum mechanics are two previously unrelated areas of science. Here we show a direct relation between the ultimate resolution of L as an integer and the accessibility of the $m(k)$ configuration. This finding might lead to a closer relation between these two important areas of science. Acknowledgment: This work is supported by the National Natural Science Foundation of China. 1, L. Jacak, P. Hawrylak, A. Wójs, [*Quantum Dots* ]{}(Springer, Berlin, 1998) 2, T. Chakraborty, [* Quantum Dots*]{} (Elsevier, Amsterdam, 1999)[* *]{} 3, G.W. Bryant, [* Phys. Rev. Lett*]{}. [**59**]{}, 1140 (1987) 4, C.G. Bao, [* Phys. Rev. Lett*]{}. [**79**]{}, 3475 (1997) 5, C.G. Bao, [*Few-Body Systems*]{}, [**13**]{}, 41 (1992) 6, K.V. Klitzing, G. Dorda, M. Pepper, [*Phys. Rev. Lett.*]{},[** 45**]{}, 494 (1980) 7, D.C. Tsui , H.L. Störmer, A.C. Gossard, [*Phys. Reev. Lett.*]{}, [**48**]{}, 1559(1982) 8, R.B. Laughlin, [* Phys. Rev*]{}. [**B23**]{}, 5632 (1981) 9, R.B. Laughlin, [*Phys. Rev. Lett.*]{} [**50**]{}, 1395 (1983) 10, C. de C. Chamon, X.G. Wen, [*Phys. Rev*]{}. [**B49**]{}, 8227 (1994) 11, H.M. Muller, S.E. Koonin, [*Phys. Rev*]{}. [** B54**]{}, 14532 (1996) 12, E. Goldmann, S.R. Renn, [* Phys. Rev.*]{} [**B60**]{}, 16611 (1999) 13, W.Y. Ruan, K.S. Chan, H.P. Ho, E.Y. B. Pun, [*J. Phys.: Condens. Matter*]{} [**12**]{}, 3911 (2000) 14, P.A. Maksym, H. Imamura, G.P. Mallon, H. Aoki, [*J. Phys.: Condens. Matter*]{}[** 12**]{}, R299 (2000) 15, M.H. Anderson et al., [*Science*]{} [**269**]{}, 198 (1995) 16, F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari,[* Rev. Mod Phys*]{}. [**71**]{}, 463 (1999) 17, N.R. Cooper, N.K.Wilkin, [*Phys. Rev*]{}. [**B60**]{}. R16279 (1999) 18, J.K. Jain, T. Kawamura, [*Europhys. Lett*]{}. [**29**]{}, 321 (1995)   FIGURE CAPTIONS Fig.1, Contour plots of $\rho _2$ of a 9-electron system. The given electron is marked by a black spot, its distance from the origin is given(in the unit $\sqrt{\hbar /(m^{*}\omega _o)}=194.7\AA $ ). This marked distance serves as a scale for both the X and Y directions (slightly different scales have been used for distinct states). The inmost contour (associated with the highest peak) is marked by a double-line. Fig.2, Contour plots of $\rho _3$ of 9- and 19-electron systems. Different scales have been used for different states.
--- abstract: 'We investigate the wake dynamics of an optimally designed wavy cylinder that completely suppresses the Kármán vortex shedding. Such a wavy cylinder is forced to oscillate with a sinusoidal motion in the crossflow direction. Examination of the lift force spectrum reveals that a critical forcing frequency exists, below which the flow control effectiveness of the wavy cylinder is retained, and beyond which the inherent vortex shedding resurrects. Moreover, the resurrected unsteady vortex shedding can persist even without sustained forcing, leading to the loss of control efficacy. This suggests that in addition to the steady state developed from uniform initial condition, an oscillatory state exists in the wake of wavy cylinder if the initial state is sufficiently perturbed. The discovery of the bistable states calls for examination of the flow control effectiveness of the wavy cylinder in more complicated inflow conditions.' author: - Kai Zhang - Dai Zhou - Hiroshi Katsuchi - Hitoshi Yamada - Zhaolong Han - Yan Bao bibliography: - 'refs.bib' title: Bistability in the wake of a wavy cylinder --- [^1] [^2] Introduction {#sec:intro} ============ Three-dimensional forcing techniques, which apply spanwise varying controls along nominally two-dimensional bluff body, have been recognized effective in controlling the vortex shedding in the wake [@choi2008control]. Among the various realizations of this class of methods, circular cylinder with spanwise varying diameter (referred to as wavy cylinder hereafter) has attracted a lot of attention due to its omnidirectional shape. Early experimental works by @ahmed1992transverse [@ahmed1993experimental] have revealed that pressure gradient exists in the spanwise direction, leading to the formation of non-uniform separation lines along the span and the development of the three-dimensional wake. The subsequent investigations discovered that such three-dimensional wake is associated with significant suppression of the Kármán vortex shedding and reduction in the drag and lift forces [@lam2004experimental; @lam2009effects; @xu2010large; @lam2008large; @lin2016effects]. Moreover, the spanwise-undulated geometry is found to resemble the whiskers of harbor seals. Such particular shape has been shown to exhibit superior hydrodynamic performance and to enhance the sensitivity of the whisker even in turbid water [@hanke2010harbor]. This finding has inspired the bio-mimicry innovations like the energy conserving flow sensors [@beem2012calibration]. The flow control effectiveness of the wavy cylinder is originated from the streamwise vortical structures in the near wake, as evidenced by a number of experimental and computational works [@zhang2005piv; @lam2009effects; @zhang2018large]. Embedded in the three-dimensional free shear layers, these vortical structures appear as counter-rotating pairs within each wavelength. The existence of such streamwise structures in the near wake tends to inhibit the roll-up of the vortex sheets along the spanwise direction, thus delaying the formation of the Kármán vortex shedding. With optimal shape parameters, the wake unsteadiness could even be completely concealed @lam2009effects. On the theoretical side, @hwang2013stabilization conducted linear stability analysis of the spanwise-wavy wake profiles. Using Floquet theory, they found that the introduction of the spanwise waviness attenuates the absolute instability of the two-dimensional wakes. Most of the previous studies on the wavy cylinders have focused on the simple configuration of uniform flow over a fixed body. However, configurations of practical interest exist, notably in bridge cables and deepwater risers, in which the slender cylindrical structures could be subjected to vibrations that are either self-induced or externally forced. Recently, @zhang2017numerical conducted numerical investigation for the vortex-induced vibration (VIV) of the wavy cylinder. Despite the complete suppression of vortex shedding in the fixed configuration, the wavy cylinder is able to develop large-amplitude crossflow vibrations when flexibly mounted, and the vibration response curve is similar to that of a two-dimensional circular cylinder at the same conditions. The numerical results were verified by @assi2018vortex, who conducted water tunnel experiment on the VIV of elliptical wavy cylinders. They have shown that once the cylinders start to oscillate, the separation lines straighten up, and the spanwise-coherent vortex filaments dominate the near wake, recovering a wide Kármán wake. The distinct behaviors of the wavy cylinder in the fixed configuration and in motion entail further examination of its flow control effectiveness, which is attempted in the current paper using direct numerical simulations. The wavy cylinder is optimally designed so that it completely suppresses the wake unsteadiness at a range of Reynolds numbers. To investigate the wake dynamics such a wavy cylinder, we force the cylinder to oscillate in the crossflow direction with prescribed motion. The behavior of the force coefficients in the perturbed flow, as well as the development of the wake, is analyzed. The current numerical results reveals the existence of the bistable state in the wake of the wavy cylinder, i.e., a steady state (fixed point) if the cylinder is slightly perturbed, and an oscillatory state (limited cycle) in highly perturbed flow. The rest of the paper is organized as follows. In §\[sec:setup\], we present the problem description and numerical setup. In §\[sec:static\], the flow over the fixed wavy cylinder is investigated to locate the control-effective Reynolds numbers. Then, we focus on the wake dynamics of the oscillating wavy cylinder in §\[sec:forced\]. Further in §\[sec:bistability\], we prove the existence of the bistability by letting the wake develop from different initial conditions. Finally, we conclude this paper by summarizing our findings in §\[sec:conclusion\]. Computational setup {#sec:setup} =================== The geometry of the wavy cylinder is schematically depicted in figure \[fig:scheme\]. The diameter of the wavy cylinder varies sinusoidally along the spanwise direction $z$ according to $$D(z)=D_m+2a\cos(2\pi z/\lambda),$$ where $D_m$ is the averaged diameter, $a$ and $\lambda$ are the geometric amplitude and wavelength, respectively. In the current paper, we assign $a=0.175D_m$ and $\lambda=2.5D_m$. This set of parameters has been shown by @lam2009effects to exhibit satisfactory flow control efficacy. The wavy cylinder is subjected to a uniform incoming flow $U_{\infty}$ in the $x$ direction. For non-dimensionalization, we normalize the spatial variables by the averaged diameter $D_m$, velocity by $U_{\infty}$, time by $D_m/U_{\infty}$, and frequency by $U_{\infty}/D_m$. The Reynolds number, defined as $Re\equiv U_{\infty}D_m/\nu$, where $\nu$ is the kinematic viscosity of the fluid, is kept below 160. ![Problem description and coordinate system.[]{data-label="fig:scheme"}](figures/wavyCylinderAndCoordinates3.eps) The wavy cylinder is forced to vibrate in the crossflow ($y$) direction with the prescribed motion $$y(t)=A_e\sin(2\pi f_e t),$$ in which $A_e$ is the nondimensional forcing amplitude, and $f_e$ is the dimensionless forcing frequency. The flow is governed by the incompressible Navier-Stokes equations, which are solved by the direct numerical simulations using the open-source software OpenFOAM. Both the time and space are discretized with second-order accurate schemes. The wavy cylinder is placed in the center of a circular computational domain of $30D_m$ in radius and $2.5D_m$ in height. Periodic boundary conditions is specified at the spanwise ends of the domain. The cylinder surface is treated as no-slip wall. A uniform flow condition with freestream velocity $U_{\infty}$ is specified at the inlet. An O-type mesh with resolution $N_c\times N_r\times N_z=140\times 140 \times 40$ (where $N_c$, $N_r$ and $N_z$ represent the grids in the circumferential, radial and spanwise directions) is used for the domain discretization. The mesh is concentrated in the vicinity of the cylinder to better resolve the near wake. The nondimensional time-step is set to be $\Delta t=0.02$. The convergence of the numerical results against the grid resolution has been verified through a mesh dependency test, as shown in table \[tab:kd\]. The mean drag coefficient $\overline{C_D}$ and the root-mean-squred lift coefficient $C_L^{\prime}$ are reported for three cases: flows over fixed wavy cylinders at $Re=100$ and $Re=150$, and flow over an oscillating wavy cylinder with $(A_e,f_e)=(0.2,0.28)$ at $Re=150$. Here, the drag and lift coefficients are defined as $C_D=2F_D/(\rho U_{\infty}^2D_mH)$ and $C_L=2F_L/(\rho U_{\infty}^2D_mH)$, in which $F_D$ and $F_L$ are the drag and lift forces, and $\rho$ the fluid density. It is observed that when the grid resolution is increased to \#2, the aerodynamic forces become converged upon further refinement of the mesh. Besides, the drag and lift coefficients of the fixed wavy cylinder at $Re=100$ are in agreement with the values reported in @lam2009effects for the same geometry (extracted from figure 4 of their paper). Thus, the mesh resolution \#2 is used throughout this paper. ------------------ ------ ------ ------ ----------------------- -- ------ ------ ------ -- ------ ------ ------ \#1 \#2 \#3 Ref [@lam2009effects] \#1 \#2 \#3 \#1 \#2 \#3 $\overline{C_D}$ 1.33 1.35 1.35 1.35 1.00 1.01 1.01 1.33 1.35 1.36 $C_L^{\prime}$ 0.20 0.21 0.21 0.21 0 0 0 1.05 1.09 1.09 ------------------ ------ ------ ------ ----------------------- -- ------ ------ ------ -- ------ ------ ------ [\[tab:kd\]]{} Static configuration {#sec:static} ==================== ![Mean drag, rms lift coefficients and the shedding frequency of flow past two-dimensional and wavy cylinders in the fixed configuration. Iso-surfaces of $\omega_z=-0.5$ (dark color) and $0.5$ (light color) at $Re=40,$ 100 and 150 are included. Shaded area indicates the control-effective regime.[]{data-label="fig:static"}](figures/static.eps) The flow around the static wavy cylinder subjected to uniform initial condition is examined at $Re=30-160$ to locate the control-effective regime. The mean drag coefficient $\overline{C_D}$, root-mean-squared lift coefficient $C_L^{\prime}$ and nondimensional shedding frequency $f_0$ are shown in figure \[fig:static\]. For $Re\lesssim 110$, the wakes of both the wavy and two-dimensional cylinders transit from steady state at low $Re$ to vortex shedding at higher $Re$, although the critical Reynolds number of the transition is slightly larger for the wavy cylinder. The drag, lift and Strouhal number of the two cylinders are also similar to each other. As the Reynolds number is increased to 120, great flow control efficacy is achieved by the wavy cylinder. The drag force suffers from a drastic decrease compared with the two-dimensional cylinder. The lift force drops to zero, suggesting the complete suppression of vortex shedding in the wake of the wavy cylinder. The steady flow achieved at $Re\gtrsim 120$ is due to the the formation of streamwise vortical structures in the near wake inhibiting the roll-up of the free shear layers, as elucidated in @lam2009effects [@hwang2013stabilization]. Forced vibration {#sec:forced} ================ ![Lift spectrum of the wavy cylinder undergoing forced vibration with $A_e=0.2$ at $Re=150$. Instantaneous vortical structures are presented at several forcing frequencies, with transparent gray standing for iso-surface of $\omega_z=\pm 0.5$, and red and blue for $\omega_x=0.3$ and -0.3, respectively.[]{data-label="fig:wavyForced"}](figures/forced4.eps) Now that the control-effective regime of the wavy cylinder is located, let us perturb the cylinder with sinusoidal oscillations in the crossflow direction and investigate its wake dynamics. The lift force spectrum of the oscillating wavy cylinder at $Re=150$ is shown in figure \[fig:wavyForced\](*a*). The amplitude of forcing is fixed at $A=0.2$ and the forcing frequency $f_e$ is varied from $0.05$ to $0.28$, at an interval of $0.01$. The vortical structures are visualized by iso-surfaces of $\omega_z$ and $\omega_x$ for selected cases. With small forcing frequency $f_e=0.05$, the lift coefficient is dominated by a single frequency at $f_e$. The vortical structures in the wake sway slightly with the motion of the cylinder. The streamwise vorticity remain in the near wake and play its role in suppressing the roll-up of the free shear layer. The vortical structure at $f_e=0.08$ appears more unsteady. Nevertheless, still a single peak at $f_e$ is observed in the lift spectrum. The situation is significantly different when it comes to $f_e \geq 0.09$. In the lift spectrum, apart from the forcing frequency, another peak reminiscent of the inherent shedding frequency as in the two dimensional case [@carberry2005controlled; @kumar2016lock] emerges at the nondimensional frequency of $f_s =0.15\sim 0.17$. This is also manifested in the corresponding wake vortical structures, where the roll-up of the free shear layers occurs much closer to the cylinder compared with $f_e=0.08$. Along with the spanwise vortical structures, the periodic shedding of the streamwise vorticity is also observed. With the revival of the inherent vortex shedding, the wake of the wavy cylinder is able to lock onto the forcing at $f_e=0.14\sim0.18$. Further increasing the forcing frequency reveals $f_s$ again, although its value is slightly smaller than that at smaller forcing frequencies. It has been confirmed that the resurrection of the inherent shedding frequency of a forced oscillating wavy cylinder also occurs at much higher Reynolds number of $Re=5000$ [@zhang2018numerical]. ![Classification of wake states of the wavy cylinder in forced vibration at $Re=150$. Solid square in shaded region: control-effective. Unshaded region: control effect lost. Solid circle: lock-on. Empty circle: lock-out. []{data-label="fig:regime"}](figures/regimeMap.eps) We also test the cases with different forcing amplitudes $A_e$ with varying forcing frequencies $f_e$. Based on the lift spectra of these cases, a classification of the wake states of the wavy cylinder undergoing forced vibration is presented in figure \[fig:regime\]. The wavy cylinder maintains its flow control effectiveness when the forcing frequency $f_e$ is smaller than a critical value. This critical value decreases with increasing forcing amplitude $A_e$. In this regime, only a single peak is found in the lift spectrum. As the forcing frequency exceeds the critical value, the inherent shedding frequency $f_s$ appears in the lift spectrum, siding with the forcing frequency $f_e$. Similar to a two-dimensional cylinder, the inherent vortex shedding can submit to the forcing when the two frequencies are close, giving rise to the lock-on region that features an Arnold tongue in the $f_e$-$A_e$ space. Bistability in the wake {#sec:bistability} ======================= More interestingly, with the resurrection of inherent shedding frequency $f_s$, the unsteady vortex shedding is able to persist by itself even without sustained forcing. To prove this, we manually turn off the forced vibration when the cylinder reaches the top or bottom position, at which the velocity of the cylinder becomes zero. This ensures a smooth transition from the dynamic simulation to a static one. We then focus on the evolution of the wake starting from the initial condition dictated by forced vibration. The time histories of $C_D$ obtained by this procedure are presented in figure \[fig:initial\](*a*) for $A_e=0.2$ with selected forcing frequencies. For cases with $f_e\lesssim 0.09$, the drag coefficients exhibit slight increase over time but eventually converge to a fixed value of $C_D=1.0$ (state I) as reported in §\[sec:static\]. On the other hand, at $f_e \gtrsim 0.09$, for which the inherent shedding in the forced wake has revived, the flow eventually arrives at state II with the drag coefficient oscillating at around $\overline{C_D}= 1.32$, signifying the loss of flow control efficacy. We further perform dynamic mode decomposition (DMD) to extract the coherent structures at both states. In the case of steady state I, DMD is conducted on snapshots in the linearly decaying regime, thus the modal structure, as shown in the bottom inset, is identical to the linear stability mode [@ferrer2014low; @tu2014dynamic; @theofilis2011global]. This mode features high three dimensionality with prominent streamwise strucutres. The DMD mode for the oscillatory state II is conducted on the periodical shedding regime and is plotted in the upper inset. In this mode, the streamwise structures that are prominent in the linear stability mode appear to be less active. This modal structure is similar to that of the fixed wavy cylinder at $Re\lesssim 110$. The comparison of the modal structures for the two states highlights the importance of streamwise vortical structures in maintaining the wake stability. ![Dependence of drag coefficients on the initial conditions at $Re=150$. The time at which the forced vibration is stopped is denoted as $t_0$. The bottom inset is the linear stability mode for the steady state I. The top inset is the DMD mode corresponding to the primary frequency of the wake. Transparent gray represents the iso-surfaces of $\tilde{\omega}_z=\pm 0.005$. Red and blue represents iso-surfaces of positive and negative $\tilde{\omega}_x$ of the same contour level. []{data-label="fig:initial"}](figures/initial.eps) A schematic diagram for the bistable states of the wavy cylinder wake is shown in figure \[fig:bistable\]. Both the steady state I and oscillatory state II are stable so that there exists a barrier between the two states. For weakly disturbed flow, the oscillations in the flow are damped and the wake eventually converges to the steady state I. However, the barrier is easily overcome when the initial condition is sufficiently perturbed, causing the flow to overshoot to the oscillatory state II. Such strong dependency of the long-term flow states on the initial condition is in glaring contrast to the conventional two-dimensional bluff body flows, for which the effect of initial condition is usually limited in time, and the long-term state depends only on the Reynolds number [@laroussi2014triggering]. ![Schematic diagram of the bi-stable states. State I represents the steady wake and state II the oscillatory wake.[]{data-label="fig:bistable"}](figures/bistable.eps) The bistability in the wake of the wavy cylinder is a result of the competition between the streamwise vortical structures that attempts to stabilize the flow [@lam2009effects; @hwang2013stabilization], and the absolute instability that tends to destabilize the wake [@zebib1987stability; @jackson1987finite]. While the latter mechanism is ever-present at super-critical Reynolds numbers, the streamwise vortices are susceptible to external disturbances. Once the inherent shedding is triggered, say, by structural oscillation, the steady streamwise vortical structures that are responsible for the wake stabilization are compelled to oscillate by the spanwise vortices and could no longer return to its initial state. As a result, the wake surrenders to the periodic Kármán vortex shedding and the flow control efficacy is lost. We note in passing that the existence of the oscillatory state is not the direct cause for the onset of vortex-induced vibration (VIV) of the wavy cylinders in uniform incoming flows [@zhang2017numerical; @assi2018vortex]. Instead, the destabilization of the wavy cylinder from steady flow is comparable to the VIV of a two-dimensional cylinder at subcritical Reynolds numbers of $Re\lesssim 47$ [@mittal2005vortex; @kou2017lowest]. Recent works have revealed that VIV occurs from the linear instability of the coupled fluid-structure system [@zhang2015mechanism; @mittal2016lock; @yao2017model]. On the other hand, the emergence of the oscillatory state in the wavy cylinder wake requires considerable forcing that is beyond the linear assumption. Once the flow is sufficiently perturbed by the vibration, the oscillatory state could be triggered and interact with the motion of the cylinder. Conclusion {#sec:conclusion} ========== Direct numerical simulations have been conducted to study the wake dynamics of a wavy cylinder at low Reynolds numbers. The wavy cylinder is optimally designed so that it completely suppresses wake unsteadiness in the fixed configuration. Deeper insights are obtained by perturbing the flow with sinusoidal structural oscillations with varying frequencies. It is disclosed that the control efficacy of the wavy cylinder could only be preserved with weak forcing. As the forcing frequency exceeds a critical value, the inherent shedding frequency that has been concealed in the fixed configuration revives, further leading the flow to lock-in. The resurrected inherent shedding vortices could persist even without sustained forcing, implying the existence of the bistable states in the wavy cylinder wake. In realistic applications, the transition from the steady state to the oscillatory state in the wavy cylinder wake could be triggered in many scenarios such as forced vibration, self-induced vibration, highly turbulent incoming flow, gusty winds, just to name a few. The discovery unveiled from this work calls for reexamination of the control effectiveness of the wavy cylinder in more complicated flow conditions. The financial support from the National Natural Science Foundation of China (Nos. 51679139,11772193and 51879160), the Innovation Program of Shanghai Municipal Education Commission (No.2019-01-07-00-02-E00066) ，the Shanghai Natural Science Foundation (No. 17ZR1415100 and 18ZR1418000) and the Program for Intergovernmental International S&T Cooperation Projects of Shanghai Municipality (No. 18290710600), are gratefully acknowledged. This research was also supported in part by the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (No. ZXDF010037), the Project of Thousand Youth Talents (No. BE0100002) and the Major Program of the National Natural Science Foundation of China (No. 51490674). KZ thanks Prof. Kunihiko Taira and Dr. Chi-An Yeh for their useful comments. [^1]: Current address: Department of Mechanical and Aerospace Engineering, University of California, Los Angeles. kzhang3@ucla.edu [^2]: Corresponding author: zhoudai@sjtu.edu.cn
--- abstract: 'The aim of this paper is to study the global existence of solutions to a coupled wave-Klein-Gordon system in space dimension two when initial data are small, smooth and mildly decaying at infinity. Some physical models strictly related to general relativity have shown the importance of studying such systems, but very few results are know at present in low space dimension. We study here a model two-dimensional system, in which the non-linearity writes in terms of “null forms”, and show the global existence of small solutions. Our goal is to prove some energy estimates on the solution when a certain number of Klainerman vector fields is acting on it, and some optimal uniform estimates. The former ones are obtained using systematically quasi-linear normal forms, in their para-differential version; the latter ones are recovered by deducing a new coupled system of a transport equation and an ordinary differential equation from the starting PDE system, by means of a semi-classical micro-local analysis of the problem. We expect the strategy developed here to be robust enough to enable us, in the future, to treat the case of the most general non-linearities.' author: - | A. Stingo [^1]\ Universit[é]{} Paris 13,\ Sorbonne Paris Cité, LAGA, CNRS (UMR 7539),\ 99, Avenue J.-B. Cl[é]{}ment,\ F-93430 Villetaneuse bibliography: - 'Biblio\_Anna.bib' date: 2018 title: ' Global existence of small amplitude solutions for a model quadratic quasi-linear coupled wave-Klein-Gordon system in two space dimension, with mildly decaying Cauchy data' --- Introduction {#introduction .unnumbered} ============ The result we present in this paper concerns the global existence of solutions to a quadratic quasi-linear coupled system of a wave equation and a Klein-Gordon equation in space dimension two, when initial data are small smooth and mildly decaying at infinity. We prove this result for a model non-linearity with the aim of extend it, in the future, to the most general case. Keeping this long term objective in mind, we shall try to develop a fairly general approach in spite of the fact that we are treating here a simple model. The Cauchy problem we consider is the following $$\label{eq:WKG_intro} \begin{cases} & (\partial^2_t - \Delta_x) u(t,x) = Q_0(v, \partial_1 v)\,, \\ & (\partial^2_t - \Delta_x + 1)v(t,x) = Q_0(v, \partial_1 u)\,, \end{cases} \qquad (t,x)\in ]1,+\infty[\times \mathbb{R}^2$$ with initial conditions $$\label{eq:data_intro} \begin{cases} & (u,v)(1,x) = \varepsilon(u_0(x),v_0(x))\,,\\ & (\partial_t u,\partial_t v)(1,x) = \varepsilon (u_1(x), v_1(x))\,, \end{cases}$$ where $\varepsilon>0$ is a small parameter, and $Q_0$ is the null form: $$Q_0(v, w) = (\partial_t v)(\partial_t w) - (\nabla_x v)\cdot (\nabla_x w)\,.$$ We also suppose that, for some $n\in\mathbb{N}$ sufficiently large, $(\nabla_x u_0, u_1)$ is in the unit ball of $H^n(\mathbb{R}^2,\mathbb{R})\times H^n(\mathbb{R}^2,\mathbb{R})$, $(v_0,v_1)$ in the unit ball of $H^{n+1}(\mathbb{R}^2,\mathbb{R})\times H^n(\mathbb{R}^2,\mathbb{R})$, and that $$\label{eq:condition_data_intro} \sum_{1\leq\abs{\alpha}\leq 3}\bigl(\norm{x^\alpha\nabla_xu_0}_{H^{\abs{\alpha}}} + \norm{x^\alpha v_0}_{H^{\abs{\alpha}+1}} + \norm{x^\alpha u_1}_{H^{\abs{\alpha}}} + \norm{x^\alpha v_1}_{H^{\abs{\alpha}}}\bigr)\leq 1.$$ Some physical models, especially related to general relativity, have shown the importance of studying such systems to which several recent works have been dedicated. Most of the results known at present concern wave-Klein-Gordon systems in space dimension 3. One of the first ones goes back to Georgiev [@Georgiev:system]. He observed that the vector fields’ method developed by Klainerman was not well adapted to handle at the same time massless and massive wave equations because of the fact that the scaling vector field $S = t\partial_t + x\cdot\nabla_x$ is not a Killing vector field for the Klein-Gordon equation. To overcome this difficulty he adapted Klainerman’s techniques, introducing a *strong null condition* to be satisfied by semi-linear nonlinearities that ensures global existence. In 2012 Katayama [@Katayama:coupled_systems] showed the global existence of small amplitude solutions to coupled systems of wave and Klein-Gordon equations under certain suitable conditions on the non-linearity that include the *null condition* of Klainerman ([@klainerman:null_condition]) on self-interactions between wave components, and are weaker than the *strong null condition* of Georgiev. Consequently, the result he obtains applies also to certain other physical systems such as Dirac-Klein-Gordon equations, Dirac-Proca equations and Klein-Gordon-Zakharov equations. Later, this problem was also studied by LeFloch, Ma [@LeFloch_Ma] and Wang [@Q.Wang] as a model for the full Einstein-Klein-Gordon system (E-KG) $$\begin{cases} & Ric_{\alpha\beta}=\textbf{D}_\alpha \psi \textbf{D}_\beta \psi + \frac{1}{2}\psi^2 g_{\alpha\beta} \\ & \Box_g \psi = \psi \end{cases}$$ The authors prove global existence of solutions to wave-Klein-Gordon systems with quasi-linear quadratic non-linearities satisfying suitable conditions, when initial data are small, smooth and compactly supported, using the so-called *hyperboloidal foliation method* introduced by Le Foch, Ma in [@LeFloch_Ma]. Global stability for the full (E-KG) has been then proved by LeFloch-Ma [@LeFloch-Ma:global_nl_stability; @lefloch-ma:global_stability_2] in the case of small smooth perturbations that agree with a Scharzschild solution outside a compact set (see also Wang [@Q.Wang:E-KG]). In a recent paper [@Ionescu_Pausader:WKG] Ionescu and Pausader prove global regularity and modified scattering in the case of small smooth initial data that decay at suitable rates at infinity, but not necessarily compactly supported. The quadratic quasi-linear problem they deal with is the following $$\begin{cases} & -\Box u = A^{\alpha\beta}\partial_\alpha v\partial_\beta v + D v^2 \\ & -(\Box + 1) v = u B^{\alpha\beta}\partial_\alpha \partial_b v \end{cases}$$ where $A^{\alpha\beta}, B^{\alpha\beta}, D$ are real constants. The system keeps the same linear structure as (E-KG) in harmonic gauge, but only keeps quadratic non-linearities that involve the massive scalar field $v$ (semilinear in the wave equation, quasi-linear in the Klein-Gordon equation). Moreover, the non-linearity they consider does not present a null structure but shows a particular resonant pattern. Their result relies, on the one hand, on a combination of energy estimates to control high Sobolev norms and weighted norms involving the admissible vector fields; on the other hand, on a Fourier analysis, in connection with normal forms and analysis of resonant sets, to prove dispersive estimates and decay in suitable lower regularity norms. The only results we know about global existence of small amplitude solutions in lower space dimension are due to Ma. In space dimension 2 he considers the case of compactly supported Cauchy data and adapts the hyperboloidal foliation method mentioned above to $2+1$ spacetime wave-Klein-Gordon systems (see [@ma:2D_tools]). In particular, in [@ma:2D_quasilinear] he combines this method with a normal form argument to treat some quasi-linear quadratic non-linearities, while in [@ma:2D_semilinear] he studies the case of some semi-linear quadratic interactions. In a very recent paper [@ma:1D_semilinear] he also tackles the one-dimensional problem, studying a model semi-linear cubic wave-Klein-Gordon system. In this work he finally overcomes the restriction on the support of initial data and generalizes the hyperboloidal foliation method, combining the hyperboloidal foliation of the translated light cone with the standard time-constant foliation outside of it. The analysis of the problem and the deduction of the estimates of interest is then made separately inside and outside the mentioned light cone. The result we prove in this paper is the following: \[thm:main\_intro\] There exists $\varepsilon_0>0$ such that for any $\varepsilon\in ]0,\varepsilon_0[$, system with initial data satisfying , admits a unique global solution defined on $[1,+\infty[$, with $\partial_{t,x}u\in C^0([1,+\infty[; H^n(\mathbb{R}^2))$ and $(v, \partial_tv)\in C^0([1,+\infty[; H^{n+1}(\mathbb{R}^2)\times H^n(\mathbb{R}^2))$. We describe below the strategy of the theorem’s proof. First of all, we rewrite system in terms of unkowns $$\label{eq:u_pm, v_pm_intro} u_\pm = \left(D_t \pm |D_x|\right)u, \quad v_\pm = \left(D_t \pm \langle D_x\rangle\right)v,$$ where $D_{t,x}=-i\partial_{t,x}$, and introduce the admissible Klainerman vector fields for this problem, i.e. $$\Omega = x_1\partial_2 -x_2\partial_1, \quad Z_j = x_j\partial_t + t\partial_j, \quad j=1,2.$$ We also denote by $\mathcal{Z}=\{\Gamma_1,\dots, \Gamma_5\}$ the family made by above vector fields together with the two spatial derivatives, and if $I=(i_1,\dots,i_p)$ is an element of $\{1,\dots,5\}^p$, $\Gamma^I w$ is the function obtained letting $\Gamma_{i_1},\dots,\Gamma_{i_p}$ act successively on $w$. We then set $$\label{eq:uI_pm, vI_pm_intro} u^I_\pm = \left(D_t \pm |D_x|\right)\Gamma^I u, \quad v^I_\pm = \left(D_t \pm \langle D_x\rangle\right)\Gamma^I v,$$ and introduce the following energies: $$E_0(t;u_\pm,v_\pm) = \int_{\R^2}\bigl(\abs{u_+(t,x)}^2 + \abs{u_-(t,x)}^2 + \abs{v_+(t,x)}^2 + \abs{v_-(t,x)}^2\bigr)\,dx,$$ then for $n\ge 3$, $$E_n(t;u_\pm,v_\pm) = \sum_{\abs{\alpha}\leq n}E_0(t;D_x^{\alpha}u_\pm, D_x^{\alpha}v_\pm),$$ which controls the $H^n$ regularity of $u_\pm, v_\pm$, and finally, for any integer $k$ between 0 and 2, $$E_3^k(t;u_\pm,v_\pm) = \sum_{\substack{\abs{\alpha}+\abs{I}\leq 3\\ \abs{I}\le 3-k}} E_0(t;D_x^\alpha u_\pm^I, D_x^\alpha v_\pm^I)$$ that takes into account the decay in space of $u_\pm, v_\pm$ and of at most three of their spatial derivatives. By a local existence argument, an a-priori estimate on $E_n$ on a certain time interval will be enough to ensure the extension of the solution to that interval. For this reason, we are led to prove a result as the following one, in which $\mathrm{R}=(\mathrm{R}_1,\mathrm{R}_2)$ denotes the Riesz transform: \[thm:boostrap\_intro\] Let $K_1,K_2$ two constants strictly bigger than 1. There exist two integers $n\gg \rho\gg 1$, $\varepsilon_0\in ]0,1[$ small enough, some small real $0<\delta\ll \delta_2\ll \delta_1\ll \delta_0\ll 1$ and two constants $A,B$ sufficiently large such that, if functions $u_\pm, v_\pm$, defined by from a solution to , satisfy $$\label{eq:boostrap_1_intro} \begin{split} &\norm{\absj{D_x}^{\rho+1} u_\pm(t,\cdot)}_{L^\infty} + \norm{\absj{D_x}^{\rho+1} \mathrm{R}u_\pm(t,\cdot)}_{L^\infty} \leq A\varepsilon t^{-\frac{1}{2}}\\ &\norm{\absj{D_x}^{\rho} v_\pm}_{L^\infty} \leq A\varepsilon t^{-1}\\ &E_n(t;u_\pm,v_\pm) \leq B^2\varepsilon^2 t^{2\delta}\\ &E_3^k(t; u_\pm,v_\pm) \leq B^2\varepsilon^2 t^{2\delta_{k}},\ 0\leq k\leq 2, \end{split}$$ for every $t\in [1,T]$, then on the same interval $[1,T]$ we have $$\label{eq:bootstrap_2_intro} \begin{split} &\norm{\absj{D_x}^{\rho+1} u_\pm(t,\cdot)}_{L^\infty} + \norm{\absj{D_x}^{\rho+1} \mathrm{R}u_\pm(t,\cdot)}_{L^\infty} \leq \frac{A}{K_1}\varepsilon t^{-\frac{1}{2}}\\ &\norm{\absj{D_x}^{\rho} v_\pm}_{L^\infty} \leq \frac{A}{K_1}\varepsilon t^{-1}\\ &E_n(t;u_\pm,v_\pm) \leq \frac{B^2}{K_2^2}\varepsilon^2 t^{2\delta}\\ &E_3^k(t; u_\pm,v_\pm) \leq \frac{B^2}{K_2^2}\varepsilon^2 t^{2\delta_{k}},\ 0\leq k\leq 2. \end{split}$$ The proof of the theorem consists, on the one hand, to prove that implies the latter two estimates in by means of an energy inequality. On the other hand, by reduction of the starting problem to a coupled system of an ordinary differential equation and a transport equation, we prove that implies the first two estimates in . In order to recover the mentioned energy inequality that allows us to propagate the a-priori energy estimates, we let family $\Gamma^I$ of vector fields act on and then pass to unknowns . We obtain a new system of the form $$\begin{split} (D_t\mp \abs{D_x})u^I_\pm &= \textit{NL}_\mathrm{w}(v_\pm^I,v_\pm^I)\\ (D_t\mp \abs{D_x})v^I_\pm &= \textit{NL}_\mathrm{kg}(v_\pm^I,u_\pm^I) \end{split}$$ where the non-linearities (whose explicit expression may be found in the right hand side of ) are bilinear quantities of their arguments. Because of the quasi-linear nature of our problem, the first step towards the derivation of the mentioned inequality is to highlight the very quasi-linear contribution to above non-linearities and make sure that it does not lead to a loss of derivatives. For this reason, we write the above system in a vectorial fashion by introducing vectors $$U^I = \begin{bmatrix} u_+^I\\0\\u_-^I\\0 \end{bmatrix}, \ V^I = \begin{bmatrix} 0\\v_+^I\\0\\v_-^I \end{bmatrix},\ W^I = U^I + V^I,$$ and successively *para-linearize* the vectorial equation satisfied by $W^I$ (using the tools introduced in subsection \[Subsection: Paradifferential Calculus\]) to stress out the quasi-linear contribution to the non-linearity. Finally, we *symmetrize* it (in the sense of subsection \[Subs: Symmetrization\]) by introducing some new unknown $W^I_s$ comparable to $W^I$. What we would need to show in order to prove the last two inequalities in is that, using the estimates in , the derivative in time of the $L^2$ norm to the square of $W^I_s$ is bounded by $\frac{C\varepsilon}{t}\|W^I\|_{L^2}$. By analysing the semi-linear contributions in the symmetrized equation satisfied by $W^I_s$, we find out that the $L^2$ norm of some of those ones can only be estimated making appear the $L^\infty$ norm on the wave factor and the $L^2$ norm on the Klein-Gordon one. Because of the very slow decay in time of the wave solution (the decay rate being $t^{-1/2}$, as assumed in the first inequality of ), we are hence very far away from the wished estimate. Consequently, the second step for the derivation of the right energy inequality consists in performing a normal form argument to get rid of those quadratic terms and replace them with cubic ones. For that, we first use a Shatah’ normal form adapted to quasi-linear equations (see subsection \[sub: a first normal form transformation\]) as already used by several authors (we cite [@ozawaTT:remarks; @D1; @D2; @D3] for quasi-linear Klein-Gordon equations, and [@HITW; @HIT; @IP1; @AD2; @IT1] for quasi-linear equations arising in fluids mechanics), but also a semi-linear normal form argument to treat some other terms on which we are allowed to lose some derivatives (see subsection \[sub: second normal form\]). These two normal forms’ steps lead us to define some new energies $\widetilde{E}_n(t;u_\pm, v_\pm), \widetilde{E}^k_3(t;u_\pm, v_\pm)$, equivalent to the starting ones $E_n(t;u_\pm, v_\pm)$, $\widetilde{E}^k_3(t;u_\pm, v_\pm)$, that we are able to propagate. That concludes the first part of the proof. The last thing that remains to prove is that implies the first two estimates in . The strategy we employ is very similar to the one developed in [@stingo:1D_KG]: we deduce from the starting system a new coupled one of an ordinary differential equation, coming from the Klein-Gordon equation, and of a transport equation, derived from the wave one. The study of this system will provide us with the wished $L^\infty$ estimates. We start our analysis by another normal form in order to replace almost all quadratic non-linear terms in the equations satisfied by $u_\pm, v_\pm$ with cubic ones. The only contributions that cannot be eliminated are those depending on $(v_+, v_{-})$ which are resonant and should be suitably treated. We do not use directly the normal forms obtained in the previous step. In fact, our aim is basically to obtain an $L^\infty$ estimate for at most $\rho$ derivatives of the solution, having a control on their $H^s$ norm for $s\gg\rho$. This permits us to lose some derivatives in the normal form reduction, so the fact that the system is quasi-linear is no longer important. We define two new unknowns $u^{NF}, v^{NF}$ by adding some quadratic perturbations to $u_{-}, v_{-}$, in such a way that they are solution to $$(D_t + \abs{D_x})\unf = q_w +c_w + \rnfw,\ (D_t + \abs{D_x})\vnf = \rnfkg,$$ where $\rnfw, c_w, \rnfkg$ are cubic terms, whereas $q_w$ is the mentioned bilinear expression in $v_+, v_{-}$ that cannot be eliminated by normal forms but whose structure will successively provide us with remainder terms. Then, if we define $$\ut(t,x) = t\unf(t,tx),\ \vt(t,x) = t\vnf(t,tx),$$ and introduce $h:=t^{-1}$ the *semi-classical parameter*, we obtain that $\ut, \vt$ verify $$\label{eq:ut_vt_intro} \begin{split} (D_t - \oph(x\cdot\xi - \abs{\xi}))\ut &= h^{-1}\left[ q_w(t,tx) + c_w(t,tx)+ \rnfw(t,tx)\right]\\ (D_t - \oph(x\cdot\xi - \absj{\xi}))\vt &= h^{-1}\rnfkg(t,tx) \end{split}$$ where $\oph$ is the Weyl quantization introduced, along with the semi-classical pseudo-differential calculus, in subsection \[Subsection: Paradifferential Calculus\]. We also consider the following operators $$\begin{split} \Mcal_j = \frac{1}{h}\Bigl(x_j\abs{\xi} -\xi_j\Bigr),\ \Lcal_j = \frac{1}{h}\Bigl(x_j -\frac{\xi_j}{\absj{\xi}}\Bigr), \end{split}$$ whose symbols are given respectively (up to the multiplication by $|\xi|$ for the former case) by the derivative with respect to $\xi$ of symbols $x\cdot\xi - |\xi|$ and $x\cdot\xi - \langle\xi\rangle$ in . Using the equation satisfied by $\unf$ (resp. $\vnf$), we can express $\mathcal{M}_j\ut$ (resp. $\Lcal_j\vt$) in terms of $Z_j\unf$ (resp. $Z_j \vnf$) and of $q_w, c_w, \rnfw$ (resp. $\rnfkg$). As done in [@stingo:1D_KG], we first introduce the lagrangian $$\Lkg = \Bigl\{(x,\xi): x - \frac{\xi}{\absj{\xi}} = 0\Bigr\}$$ which is the graph of $\xi =-d\phi(x)$, with $\phi(x)=\sqrt{1-|x|^2}$, and decompose $\vt$ into the sum of a contribution micro-localised on a neighbourhood of size $\sqrt{h}$ of $\Lkg$, and another one micro-localised out of that neighbourhood (in the spirit of [@ifrim_tataru:global_bounds]). The second contribution can be basically estimated in $L^\infty$ by $h^{\frac{1}{2}-0}$ times the $L^2$ norm of some iterates of operator $\Lcal$ acting on $\vt$ (which are controlled by the $L^2$ hypothesis in theorem \[thm:boostrap\_intro\]). The main contribution to $\vt$ is then represented by $\vt_\Lkg$, which appears to be solution to $$[D_t-\oph(x\cdot\xi - \absj{\xi})]\vt_\Lkg =\textrm{ controlled terms}.$$ Developing the symbol in the above left hand side on $\Lkg$ we finally obtain the wished ODE, which combined with the a-priori estimate of the “controlled terms” allows us to deduce from the second estimate in (with $\rho=0$, the general case being treated in the same way up to few more technicalities). The same strategy is employed to obtain some uniform estimates on $\ut$. We introduce the lagrangian $$\Lw = \Bigl\{(x,\xi): x - \frac{\xi}{\abs{\xi}} = 0\Bigr\}$$ which, differently from $\Lkg$, is not a graph but projects on the basis as an hypersurface. For this reason, the classical problem associated to the first equation in is rather a transport equation than an ordinary differential equation. It is obtained in a similar way by decomposing $\ut$ into two contributions: one denoted by $\ut_\Lw$ and micro-localised in a neighbourhood of size $h^{\frac{1}{2}-\sigma}$ (for some small $\sigma>0$) of $\Lw$; another one micro-localised away from this neighbourhood. As for the Klein-Gordon component, this latter contribution can be easily controlled thanks to the $L^2$ estimates that the last two inequalities in infer on the iterates of $\Mcal_j$ acting on $\ut$. By micro-localisation we derive that $\ut_\Lw$ satisfies $$[D_t-\oph(x\cdot\xi-\abs{\xi})]\ut_\Lw = \textrm{ controlled terms},$$ and by developing symbol $x\cdot\xi-\abs{\xi}$ on $\Lw$ we obtain the wished transport equation. Integrating this equation by the method of characteristics, we finally recover the first estimate in and conclude the proof of theorem \[thm:boostrap\_intro\]. Main Theorem and Preliminary Results ==================================== Statement of the main theorem {#sec: statement of the main results} ----------------------------- <span style="font-variant:small-caps;">Notations:</span> We warn the reader that, throughout the paper, we will often denote $\partial_t$ (resp. $\partial_{x_j}$, $j=1,2$) by $\partial_0$ (resp. $\partial_j$, $j=1,2$), while symbol $\partial$ without any subscript will stand for one of the three derivatives $\partial_a$, $a=0,1,2$. $\nabla_x f$ is the classical spatial gradient of $f$, $D:=-i\partial$ and $\mathrm{R}_j$ denotes the Riesz operator $D_j|D_x|^{-1}$, for $j=1,2$. We will also employ notation $\|\partial_{t,x}w\|$ with the meaning $\|\partial_tw\| + \|\partial_xw\|$ and $\|\mathrm{R}w\|=\sum_j\|\mathrm{R}_jw\|$. We consider the following quadratic, quasi-linear, coupled wave-Klein-Gordon system $$\label{wave KG system} \begin{cases} & (\partial^2_t - \Delta_x) u(t,x) = Q_0(v, \partial_1 v)\,, \\ & (\partial^2_t - \Delta_x + 1)v(t,x) = Q_0(v, \partial_1 u)\,, \end{cases} \qquad (t,x)\in ]1,+\infty[\times \mathbb{R}^2$$ with initial conditions $$\label{initial data} \begin{cases} & (u,v)(1,x) = \varepsilon(u_0(x),v_0(x))\,,\\ & (\partial_t u,\partial_t v)(1,x) = \varepsilon (u_1(x), v_1(x))\,, \end{cases}$$ where $\varepsilon>0$ is a small parameter, and $Q_0$ is the null form: $$\label{null form Q0} Q_0(v, w) = (\partial_t v)(\partial_t w) - (\nabla_x v)\cdot (\nabla_x w)\,.$$ Our aim is to prove that there is a unique solution to Cauchy problem - provided that $\varepsilon$ is sufficiently small and $u_0,v_0,u_1,v_1$ decay rapidly enough at infinity. The theorem we are going to demonstrate is the following: \[Thm: Main theorem\] There exist an integer $n$ sufficiently large and $\varepsilon_0\in]0,1[$ sufficiently small such that, for any $\varepsilon \in ]0,\varepsilon_0[$, any real valued $u_0,v_0,u_1,v_1$ satisfying: $$\label{condition_initial_data} \begin{gathered} \|\nabla_x u_0\|_{H^n}+ \|v_0\|_{H^{n+1}}+\|u_1\|_{H^n}+\|v_1\|_{H^n}\le 1,\\ \sum_{|\alpha|=1}^2 \left(\|x^\alpha \nabla_x u_0\|_{H^{|\alpha|}}+ \|x^\alpha v_0\|_{H^{|\alpha|+1}}+ \|x^\alpha u_1\|_{H^{|\alpha|}}+ \|x^\alpha v_1\|_{H^{|\alpha|}}\right)\le 1, \end{gathered}$$ system - admits a unique global solution $(u,v)$ with $\partial_{t,x}u\in C^0\left([1,\infty[;H^n(\mathbb{R}^2)\right)$, $v \in C^0\left([1,\infty[;H^{n+1}(\mathbb{R}^2)\right)\cap C^1\left([1,\infty[;H^n(\mathbb{R}^2)\right)$. The proof of the main theorem is based on the introduction of four new functions $u_+,u_{-},v_+,v_{-}$, defined in terms of $u,v$ as follows: $$\label{def u+- v+-} \begin{cases} & u_+ : = (D_t + |D_x|)u\,, \\ & u_{-} : = (D_t - |D_x|)u \,, \end{cases}\qquad \begin{cases} & v_+ : = (D_t +\langle D_x\rangle)v\,, \\ & v_{-} : = (D_t -\langle D_x\rangle)v\,, \end{cases}$$ and on the propagation of some a-priori estimates made on them in some interval $[1,T]$, for a fixed $T>1$. In order to state this result we consider the admissible Klainerman vector fields for the wave-Klein-Gordon system: $$\label{Omega, Zj} \Omega:=x_1\partial_2 - x_2 \partial_1\,, \quad Z_j := x_j\partial_t + t\partial_j\,, j=1,2$$ and denote by $\Gamma$ a generic vector field in $\mathcal{Z}=\{\Omega, Z_j, \partial_j, j=1,2\}$. If $\mathcal{Z}$ is assumed ordered, i.e. $$\label{order_Z} \begin{gathered} \mathcal{Z}=\{\Gamma_1,\dots, \Gamma_5\} \\ \text{with }\quad \Gamma_1=\Omega, \quad \Gamma_j= Z_{j-1} \ \text{for } j=2,3, \quad \Gamma_j=\partial_{j-3} \ \text{for } j=4,5, \end{gathered}$$ then for a multi-index $I=(i_1,\dots, i_n)$, $i_j\in\{1,\dots,5\}$ for $j=1,\dots,n$, we define the *length* of $I$ as $|I|:=n$, and $\Gamma^I:=\Gamma_{i_1}\cdots\Gamma_{i_n}$ the product of vector fields $\Gamma_{i_j}\in \mathcal{Z}$, $j=1,\dots,n$. Vector fields $\Gamma$ have two relevant properties: they act like derivations on non-linear terms; they exactly commute with the linear part of both wave and Klein-Gordon equation. This is the reason why we exclude of our consideration the scaling vector field $S=t\partial_t+\sum_j x_j \partial_j$, which is always considered in the so-called *Klainerman vector fields’ method* for the wave equation, as it does not commute with the Klein-Gordon operator. We also introduce the energy of $(u_+,u_{-},v_+,v_{-})$ at time $t\ge 1$ as $$E_0(t; u_\pm, v_\pm) :=\int \left(|u_+(t,x)|^2 + |u_{-}(t,x)|^2+ |v_+(t,x)|^2+ |u_{-}(t,x)|^2\right) dx,$$ together with the generalized energies \[def\_generalized\_energy\] $$E_n(t;u_\pm, v_\pm):= \sum_{|\alpha|\le n} E_0(t; D^\alpha_x u_\pm, D^\alpha_xv _\pm), \quad \forall n\in\mathbb{N}, n\ge 3,$$ and $$E^k_3(t; u_\pm, v_\pm) := \sum_{\substack{|\alpha|+|I|\le 3 \\ |I|\le 3-k}} E_0(t; D^\alpha_x u^I_\pm; D^\alpha_x v^I_\pm), \quad 0\le k\le 2,$$ where, for any multi-index $I$, $$\label{def uIpm vIpm} u^I_\pm := (D_t \pm |D_x|)\Gamma^Iu, \quad v^I_\pm := (D_t\pm \langle D_x\rangle)\Gamma^I v.$$ Energy $E_n(t;u_\pm, v_\pm)$, for $n\ge 3$, is introduced with the aim of controlling the Sobolev norm $H^n$ of $u_\pm, v_\pm$ for large values of $n$. The reason of dealing with $E^k_3(t;u_\pm, v_\pm)$ is, instead, to control the $L^2$ norm of $\Gamma^I u_\pm, \Gamma^I v_\pm$, for any general $\Gamma\in\mathcal{Z}$ and $|I|\le 3$. In particular, superscript $k$ indicates that we are considering only products $\Gamma^I$ containing at most $3-k$ vector fields in $\{\Omega, Z_m,m=1,2\}$. For instance, the $L^2$ norms of $\Omega^3 u_\pm, \Omega Z^2_1 v_\pm$ are bounded by $E^0_3(t;u_\pm, v_\pm)$ but not by $E^1_3(t;u_\pm, v_\pm)$, while the $L^2$ norms of $Z^2_1u_\pm, \partial_2\Omega Z_2v_\pm$ are controlled by both $E^1_3(t;u_\pm, v_\pm), E^0_3(t;u_\pm, v_\pm)$, etc. The interest of distinguishing between $k=0,1,2$, is to take into account the different growth in time of the $L^2$ norm of such terms depending on the number of vector fields $\Omega, Z_m$ acting on $u_\pm, v_\pm$, as emerges from a-priori estimate . \[Thm: bootstrap argument\] Let $K_1,K_2>1$ and $H^{\rho,\infty}$ be the space defined in \[def Sobolev spaces-NEW\] $(iii)$. There exist two integers $n\gg \rho$ sufficiently large, some $0<\delta\ll \delta_2 \ll \delta_1\ll \delta_0\ll 1$ small, two constants $A,B>1$ sufficiently large and $\varepsilon_0\in ]0,(2A+B)^{-1}[$ such that, for any $0<\varepsilon<\varepsilon_0$, if $(u,v)$ is solution to - on some interval $[1,T]$, for a fixed $T>1$, and $u_\pm, v_\pm$ defined in satisfy: \[est: bootstrap argument a-priori est\] $$\begin{gathered} \|u_\pm (t,\cdot)\|_{H^{\rho+1,\infty}}+ \|\mathrm{R} u_\pm(t,\cdot)\|_{H^{\rho+1,\infty}}\le A\varepsilon t^{-\frac{1}{2}}, \label{est: bootstrap upm} \\ \|v_\pm (t,\cdot)\|_{H^{\rho,\infty}}\le A\varepsilon t^{-1},\label{est: boostrap vpm} \\ E_n(t; u_\pm, v_\pm)^\frac{1}{2} \le B\varepsilon t^\frac{\delta}{2}, \label{est: bootstrap Enn}\\ E^k_3(t;u_\pm, v_\pm)^\frac{1}{2}\le B\varepsilon t^{\frac{\delta_k}{2}}, \quad \forall\ 0\le k\le 2, \label{est: bootstrap E02}\end{gathered}$$ for every $t\in [1,T]$, then in the same interval they verify also $$\begin{gathered} \|u_\pm (t,\cdot)\|_{H^{\rho+1,\infty}}+ \|\mathrm{R} u_\pm(t,\cdot)\|_{H^{\rho+1,\infty}}\le \frac{A}{K_1}\varepsilon t^{-\frac{1}{2}},\label{est:bootstrap enhanced upm} \\ \|v_\pm (t,\cdot)\|_{H^{\rho,\infty}}\le \frac{A}{K_1}\varepsilon t^{-1}, \label{est:bootstrap enhanced vpm}\\ E_n(t; u_\pm, v_\pm)^\frac{1}{2} \le \frac{B}{K_2}\varepsilon t^\frac{\delta}{2}, \label{est: bootstrap enhanced Enn}\\ E^k_3(t;u_\pm, v_\pm)^\frac{1}{2}\le \frac{B}{K_2}\varepsilon t^{\frac{\delta_k}{2}}, \quad \forall \ 0\le k\le 2.\label{est: boostrap enhanced E02}\end{gathered}$$ The a-priori estimates on the uniform norm of $u_\pm, \mathrm{R}u_\pm, v_\pm$ made in the above theorem translate in terms of $u_\pm, v_\pm$ the sharp decay in time we expect for the solution $(u,v)$ to starting problem . Indeed, from definitions it appears that $$\begin{gathered} D_tu = \frac{u_+ + u_{-}}{2}, \quad D_x u= \mathrm{R}\left(\frac{u_+-u_{-}}{2}\right), \\ D_t v= \frac{v_+ + v_{-}}{2}, \quad v= \langle D_x\rangle^{-1}\left(\frac{v_+-v_{-}}{2}\right), \end{gathered}$$ so , imply $$\|\partial_{t,x}u(t,\cdot)\|_{H^{\rho,\infty}}\le A\varepsilon t^{-\frac{1}{2}}, \quad \|\partial_t v(t,\cdot)\|_{H^{\rho,\infty}}+\|v(t,\cdot)\|_{H^{\rho+1,\infty}}\le A\varepsilon t^{-1}.$$ Furthermore, the following quantity $$\|\partial_t u(t,\cdot)\|_{H^n}+\|\nabla_x u(t,\cdot)\|_{H^n} + \|\partial_t v(t,\cdot)\|_{H^n}+\|\nabla_x v(t,\cdot)\|_{H^n}+\|v(t,\cdot)\|_{H^n}$$ is equivalent to the square root of $E_n(t;u_\pm, v_\pm)$, which implies that the propagation of a-priori energy estimate is equivalent to the propagation of a certain estimate on the above Sobolev norms. For this reason, the propagation of the a-priori estimate on $E_n(t;u_\pm, v_\pm)$ and a local existence argument will imply theorem \[Thm: Main theorem\]. Before ending this section and going into the core of the subject, we briefly remind the general definition of *null condition* for a multilinear form on $\mathbb{R}^{1+n}$ and a result by Hörmander (see [@hormander:the_analysis_III]). A $k$-linear form $G$ on $\mathbb{R}^{1+n}$ is said to satisfy the *null condition* if and only if, for all $\xi\in\mathbb{R}^n, \xi=(\xi_0, \dots, \xi_n)$ such that $\xi_0^2 - \sum_{j=1}^n \xi_j^2 =0$, $$\label{wave null condition} G(\underbrace{\xi,\dots,\xi}_{k}) = 0.$$ <span style="font-variant:small-caps;">Example:</span> The trilinear form $\xi_0^2 \xi_a - \displaystyle\sum_{j=1,2} \xi_j^2 \xi_a$ associated to $Q_0(v,\partial_a w)$ satisfies the null condition , for any $a=0,1,2$. This is the most common example of null form. \[Lemma : Vector Field on a null form\] Let $G$ be a $k$-linear form on $\mathbb{R}^{1+n}$, $k=k_1 + \dots + k_r$, with $k_j$ positive integers, and $\Gamma\in \mathcal{Z}$. For all $u_j\in C^{k+1}(\mathbb{R}^{1+n})$, all $\alpha_j\in \mathbb{N}^{1+n}$, $|\alpha_j|=k_j$, and $u_j^{(k_j)} := \partial^{\alpha_j} u_j$, $$\begin{split} \Gamma G(u_1^{(k_1)},\dots, u_r^{(k_r)}) & = G((\Gamma u_1)^{(k_1)},\dots, u_r^{(k_r)}) + \dots \\ & + G(u_1^{(k_1)},\dots, (\Gamma u_r)^{(k_r)}) + G_1(u_1^{(k_1)},\dots, u_r^{(k_r)})\,, \end{split}$$ where $G_1$ satisfies the *null condition*. \[Remark:Vector\_field\_on\_null\_structure\] Previous lemma simplifies when the multi-linear form $G$ satisfying the null condition is $Q_0(v, \partial_a w)$, for any $a=0,1,2$. Indeed, the structure of the null form is not modified by the action of vector field $\Gamma$ in the sense that $$\label{Gamma_nonlinearity} \Gamma Q_0(v,\partial_a w) = Q_0(\Gamma v, \partial_a w) + Q_0(v, \partial_a \Gamma w) + G_1(v,\partial w)\,,$$ where $G_1(v,\partial w)=0$ if $\Gamma=\partial_m$, $m=1,2$, and $$\label{def_G1} G_1(v,\partial w) = \begin{cases} -Q_0(v, \partial_m w), \ &\text{if } a=0, \Gamma=Z_m, m\in \{1,2\},\\ 0, \ &\text{if } a=0, \Gamma =\Omega,\\ -Q_0(v, \partial_t w), \ &\text{if } a\ne 0, \Gamma =Z_a, \\ 0, \ &\text{if } a\ne0, \Gamma=Z_m, m\in \{1,2\}\setminus\{a\},\\ (-1)^a Q_0(v, \partial_m w), \ &\text{with } m\in \{1,2\}\setminus \{a\}, \text{ if } a\ne0, \Gamma =\Omega. \end{cases}$$ If $\Gamma^I$ contains at least $k\le |I|$ space derivatives then $$\label{Gamma_I_nonlinearity} \Gamma^I Q_0(v,\partial_1 w) = \sum_{|I_1|+|I_2|=|I|} Q_0(\Gamma^{I_1} v, \partial_1\Gamma^{I_2} w) + \sum_{k\le |I_1|+|I_2|<|I|}c_{I_1,I_2} Q_0(\Gamma^{I_1} v, \partial\Gamma^{I_2} w),$$ with $c_{I_1,I_2}\in \{-1,0,1\}$. In the above equality we should think of multi-index $I_1$ (resp. $I_2$) as obtained by extraction of a $|I_1|$-tuple (resp. $|I_2|$-tuple) from $I=(i_1,\dots i_n)$, in such a way that each $i_j$ appearing in $I$ and corresponding to a spatial derivative (e.g. $\Gamma_{i_j}=D_m$, for $m\in\{1,2\}$), appears either in $I_1$ or in $I_2$, but not in both. For further references, we define $$\label{def mathcal(I)} \mathcal{I}(I):=\left\{(I_1,I_2)| I_1, I_2\text{ multi-indices obtained as described above}\right\}.$$ Preliminary Results ------------------- The aim of this section is to introduce most of the technical tools that will be used throughout the paper. In particular, subsections \[Subsection: Paradifferential Calculus\] and \[Subsection: Semiclassical Pseudodifferential Calculus\] are devoted to recall some definitions and results about paradifferential and pseudo-differential calculus respectively; subsection \[Subsection: Some Technical Estimates I\] and \[Subsection: Some Technical Estimates II\] are dedicated to the introduction of some special operators that we will frequently use when dealing with the wave and the Klein-Gordon component. Subsections \[Subsection: Paradifferential Calculus\], \[Subsection: Semiclassical Pseudodifferential Calculus\] barely contain proofs (we refer for that to [@bony:calcul_symbo], [@metivier:paradifferential], [@dimassi:spectral], [@zworski:semiclassical]), whereas subsections \[Subsection: Some Technical Estimates I\], \[Subsection: Some Technical Estimates II\] are much longer and richer in proofs and technicalities. ### Paradifferential calculus {#Subsection: Paradifferential Calculus} In the current subsection we recall some definitions and properties that will be useful in chapter \[Chap:Energy estimates\]. We first recall the definition of some spaces (Sobolev, Lipschitz and Hölder spaces) in dimension $d\ge 1$ and afterwards some results concerning symbolic calculus and the action of paradifferential operators on Sobolev spaces (see for instance [@metivier:paradifferential]). We warn the reader that we will use both notations $\hat{w}(\xi)$ and $\mathcal{F}_{x\mapsto \xi}w$ for the Fourier transform of a function $w=w(x)$. \[def Sobolev spaces-NEW\] - Let $s \in \mathbb{R}$. $H^s(\mathbb{R}^d)$ denotes the space of tempered distributions $w\in \mathcal{S}'(\mathbb{R}^d)$ such that $\hat{w}\in L^2_\textit{loc}(\mathbb{R}^d)$ and $$\|w\|^2_{H^s(\mathbb{R}^d)} := \frac{1}{(2\pi)^d}\int (1+|\xi|^2)^s |\hat{w}(\xi)|^2d\xi < +\infty;$$ - For $\rho\in\mathbb{N}$, $W^{\rho,\infty}(\mathbb{R}^d)$ denotes the space of distributions $w\in\mathcal{D}'(\mathbb{R}^d)$ such that $\partial^\alpha_x w\in L^\infty(\mathbb{R}^d)$, for any $\alpha\in\mathbb{N}^d$ with $|\alpha|\le \rho$, endowed with the norm $$\|w\|_{W^{\rho,\infty}} := \sum_{|\alpha|\le\rho}\|\partial^\alpha_x w\|_{L^\infty};$$ - For $\rho\in\mathbb{N}$, we also introduce $H^{\rho,\infty}(\mathbb{R}^d)$ as the space of tempered distributions $w\in\mathcal{S}'(\mathbb{R}^d)$ such that $$\|w\|_{H^{\rho,\infty}} :=\|\langle D_x\rangle^\rho w\|_{L^\infty} < +\infty .$$ An operator $T$ is said of order $\le m\in \mathbb{R}$ if it is a bounded operator from $H^{s+m}(\mathbb{R}^d)$ to $H^s(\mathbb{R}^d)$ for all $s\in\mathbb{R}$. Let $m\in \mathbb{R}$. - $S^m_0(\mathbb{R}^d)$ denotes the space of functions $a(x,\eta)$ on $\mathbb{R}^d\times \mathbb{R}^d$ which are $C^\infty$ with respect to $\eta$ and such that, for all $\alpha\in\mathbb{N}^d$, there exists a constant $C_\alpha>0$ and $$\|\partial^\alpha_\eta a(\cdot,\eta)\|_{L^\infty} \le C_\alpha (1+|\eta|)^{m-|\alpha|}, \qquad \forall \eta \in\mathbb{R}^d.$$ $\Sigma^m_0(\mathbb{R}^d)$ denotes the subclass of symbols $a\in S^m_0(\mathbb{R}^d)$ satisfying $$\label{spectral condition class Sigma^m} \exists \varepsilon<1\ : \mathcal{F}_{x\rightarrow \xi}a(\xi,\eta)=0 \quad \text{for}\ |\xi|>\varepsilon (1+|\eta|).$$ $S^m_0$ is equipped with seminorm $M^m_0(a;n)$ given by $$\label{def: seminorm Mmo} M^m_0(a ;n) = \sup_{|\beta|\le n}\sup_{\eta\in\mathbb{R}^2}\big\|(1+|\eta|)^{|\beta|-m}\partial^\beta_\eta a(\cdot,\eta)\big\|_{L^{\infty}}.$$ - For $r\in\mathbb{N}$, $S^m_r(\mathbb{R}^d)$ denotes more generally the space of symbols $a\in S^m_0(\mathbb{R}^d)$ such that, for all $\alpha\in \mathbb{N}^d$ and all $\eta \in \mathbb{R}^d$, function $x\rightarrow \partial^\alpha_\eta a(x,\eta)$ belongs to $W^{r,\infty}(\mathbb{R}^d)$ and there exists a constant $C_\alpha>0$ such that $$\|\partial^\alpha_\eta a(\cdot,\eta)\|_{W^{r,\infty}}\le C_\alpha (1+|\eta|)^{m-|\alpha|}, \qquad \forall\eta\in\mathbb{R}^d.$$ $\Sigma^m_r(\mathbb{R}^d)$ denotes the subclass of symbols $a\in S^m_r(\mathbb{R}^d)$ satisfying the spectral condition . $S^m_r$ is equipped with seminorm $M^m_r(a;n)$, given by $$\label{def: seminorm Mmr} M^m_r(a ;n) = \sup_{|\beta|\le n}\sup_{\eta\in\mathbb{R}^2}\big\|(1+|\eta|)^{|\beta|-m}\partial^\beta_\eta a(\cdot,\eta)\big\|_{W^{r,\infty}}.$$ These definitions extend to matrix valued symbols $a\in S^m_r$ ($a\in \Sigma^m_r$), $m\in\mathbb{R}$, $r\in\mathbb{N}$. If $a\in S^m_r$ (resp. $a\in \Sigma^m_r$), it is said *of order $m$*. \[Def: admissible cut off function\] An admissible cut-off function $\psi(\xi, \eta)$ is a $C^\infty$ function on $\mathbb{R}^d\times\mathbb{R}^d$ such that - there are $0<\varepsilon_1<\varepsilon_2<1$ and $$\label{adm_cut-off: property 1 } \begin{cases} & \psi(\xi,\eta) =1, \quad \text{for} \ |\xi|\le \varepsilon_1(1+|\eta|)\\ & \psi(\xi,\eta)=0, \quad \text{for} \ |\xi|\ge \varepsilon_2(1+|\eta|); \end{cases}$$ \[adm\_cut-off: property 2\] - for all $(\alpha,\beta)\in\mathbb{N}^d\times\mathbb{N}^d$ there is a constant $C_{\alpha,\beta}>0$ such that $$|\partial^\alpha_\xi \partial^\beta_\eta \psi(\xi,\eta)|\le C_{\alpha,\beta} (1+|\eta|)^{-|\alpha| - |\beta|}, \quad \forall (\xi, \eta).$$ <span style="font-variant:small-caps;">Example</span>: If $\chi$ is a smooth cut-off function such that $\chi(z)=1$ for $|z|\le \varepsilon_1$ and is supported in the open ball $B_{\varepsilon_2}(0)$, with $0<\varepsilon_1<\varepsilon_2<1$, function $\psi(\xi,\eta):= \chi\big(\frac{\xi}{\langle\eta\rangle}\big)$ is an admissible cut-off function in the sense of definition \[Def: admissible cut off function\]. We will only consider this type of admissible cut-off functions for the rest of the paper and refer (abusively) to $\chi$ itself as an admissible cut-off. \[Def: Paradiff\_operator\] Let $\chi$ be an admissible cut-off function and $a(x,\eta)\in S^m_r$, $m\in \mathbb{R}, r\in\mathbb{N}$. The *Bony quantization* (or *paradifferential quantization*) $Op^B(a(x,\eta))$ associated to symbol $a$ and acting on a test function $w$ is defined as $$\begin{split} & Op^B(a(x,\eta))w(x) :=\frac{1}{(2\pi)^d} \int_{\mathbb{R}^d} e^{i x\cdot\eta}\sigma^\chi_a(x,\eta) \hat{w}(\eta) d\eta\,, \\ & \text{with}\ \sigma^\chi_a(x,\eta) := \frac{1}{(2\pi)^d}\int_{\mathbb{R}^d} e^{i(x-y)\cdot\zeta}\chi\left(\frac{\zeta}{\langle\eta\rangle}\right)a(y,\eta) dyd\zeta\,. \end{split}$$ The operator defined above depends on the choice of the admissible cut-off function $\chi$. However, if $a\in S^m_r$ for some $m\in\mathbb{R}, r\in\mathbb{N}$, a change of $\chi$ modifies $Op^B(a)$ only by the addition of a $r$-smoothing operator (i.e. an operator which is bounded from $H^s$ to $H^{s+r}$, see [@bony:calcul_symbo]), so the choice of $\chi$ will be substantially irrelevant as long as we can neglect $r$-smoothing operators. For this reason, we will not indicate explicitly the dependence of $Op^B$ (resp. of $\sigma^\chi_a$) on $\chi$ to keep notations as light as possible. Let us also observe that, with such a definition, the Fourier transform of $Op^B(a)w$ has the following simple expression $$\label{fourier transform of paradiff op} \mathcal{F}_{x\rightarrow \xi}\Big(Op^B(a(x,\eta))w(x)\Big)(\xi) = \frac{1}{(2\pi)^d}\int \chi\left(\frac{\xi - \eta}{\langle\eta\rangle}\right)\hat{a}_y(\xi - \eta, \eta)\hat{w}(\eta) d\eta\,,$$ where $\hat{a}_y(\xi , \eta):=\mathcal{F}_{y\rightarrow\xi}\big(a(y,\eta)\big)$, and the product of two functions $u,v$ can be developed as $$\label{dev in paraproduct} uv = Op^B(u)v + Op^B(v)u + R(u,v),$$ where remainder $R(u,v)$ writes on the Fourier side as $$\label{fourier transform of R(v,w)} {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{R(u,v)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{R(u,v)}{\tmpbox}}(\xi) = \frac{1}{(2\pi)^d} \int \left(1 - \chi\left(\frac{\xi-\eta}{\langle\eta\rangle}\right) - \chi\left(\frac{\eta}{\langle\xi - \eta\rangle}\right)\right) \widehat{u}(\xi - \eta) \widehat{v}(\eta)d\eta.$$ We remark that frequencies $\eta$ and $\xi - \eta$ in the above integral are either bounded or equivalent, and $R(u,v) = R(v,u)$. With the aim of having uniform notations, we introduce the operator $Op^B_R$ associated to a symbol $a(x,\eta)$ and acting on a function $w$ as $$\label{operator OpB_R} \begin{split} & Op^B_R(a(x,\eta))w(x):=\frac{1}{(2\pi)^d}\int e^{ix\cdot\eta}\delta^{\chi}_a(x,\eta)\hat{w}(\eta)d\eta\,, \\ & \text{with}\ \delta^\chi_a(x,\eta):= \frac{1}{(2\pi)^d}\int e^{i(x-y)\cdot\zeta}\left(1 - \chi\left(\frac{\zeta}{\langle\eta\rangle}\right) - \chi\left(\frac{\eta}{\langle\zeta\rangle}\right)\right)a(y,\eta) dyd\zeta\,. \end{split}$$ For future references, we recall the definition of the Littlewood-Paley decomposition of a function $w$. \[Def: Littlewood Paley decomposition\] Let $\chi :\mathbb{R}^2\rightarrow [0,1]$ be a smooth decaying radial function, supported for $|x|\le 2-\frac{1}{10}$ and identically equal to 1 for $|x|\le 1+\frac{1}{10}$. Let also $\varphi(\xi) : =\chi(\xi) -\chi(2\xi)\in C^\infty_0(\mathbb{R}^2 \setminus \{0\})$, supported for $\frac{1}{2}<|\xi|<2$, and $\varphi_k(\xi):=\varphi(2^{-k}\xi)$ for all $k\in \mathbb{N}^*$, with the convention that $\varphi_0 : = \chi$. Then $\sum_{k\in\mathbb{N}} \varphi(2^{-k}\xi) = 1$, and for any $w\in \mathcal{S}'(\mathbb{R}^d)$ $$w = \sum_{k\in \mathbb{N}} \varphi_k(D_x) w$$ is the Littlewood-Paley decomposition of $w$. The following proposition is a classical result about the action of para-differential operators on Sobolev spaces (see [@bony:calcul_symbo] for further details). Proposition \[Prop: Paradiff action with non smooth symbols and R(u,v)\] shows, instead, that some results of continuity over $L^2$ hold also for operators whose symbol $a(x,\eta)$ is not a smooth function of $\eta$, and that map $(u,v)\mapsto R(u,v)$ is continuous from $H^{4,\infty}\times L^2$ to $L^2$. \[Prop : Paradiff action on Sobolev spaces-NEW\] Let $m\in \mathbb{R}$. For all $s\in\mathbb{R}$ and $a \in S^m_0$, $Op^B(a)$ is a bounded operator from $H^{s+m}(\mathbb{R}^d)$ to $H^s(\mathbb{R}^d)$. In particular, $$\|Op^B(a)w\|_{H^s} \lesssim M^m_0\Big(a ; \Big[\frac{d}{2}\Big]+1\Big) \|w\|_{H^{s+m}}.$$ \[Prop: Paradiff action with non smooth symbols and R(u,v)\] - Let $a(x, \eta) = a_1(x)b(\eta)$, with $a_1\in L^\infty(\mathbb{R}^2)$ and $b(\eta)$ bounded, supported in some ball centred in the origin and such that $|\partial^\alpha b(\eta)|\lesssim_\alpha |\eta|^{-|\alpha|+1}$ for any $\alpha\in\mathbb{N}^2$ with $|\alpha|\ge 1$. Then $Op^B(a(x,\eta)): L^2 \rightarrow L^2$ is bounded and for any $w\in L^2(\mathbb{R}^2)$ $$\|Op^B(a(x,\eta))w\|_{L^2}\lesssim \|a_1\|_{L^\infty} \|w\|_{L^2}.$$ The same result is true for $Op^B_R(a(x,\eta))$; - Map $(u,v)\in H^{4,\infty}\times L^2 \mapsto R(u,v)\in L^2$ is well defined and continuous. As concerns $(i)$ we have that $$Op^B(a(x, \eta))w(x) = \int K(x-z, x-y) a_1(y)w(z) dy dz$$ with $$K(x,y) := \frac{1}{(2\pi)^4}\int e^{i x\cdot\eta + i y\cdot\zeta} \chi\Big(\frac{\zeta}{\langle\eta\rangle}\Big) b(\eta)d\eta d\zeta$$ and $\chi$ is an admissible cut-off function. After the hypothesis on $b$ we have that for every $\alpha,\beta\in\mathbb{N}^2$, $$\begin{gathered} \Big|\partial^\beta_\zeta \Big[\chi\Big(\frac{\zeta}{\langle\eta\rangle}\Big)b(\eta)\Big]\Big|\lesssim \mathds{1}_{\{|\eta|\lesssim 1\}} |g_\beta(\zeta)|, \\ \Big|\partial^\alpha_\eta\partial^\beta_\zeta \Big[\chi\Big(\frac{\zeta}{\langle\eta\rangle}\Big)b(\eta)\Big]\Big|\lesssim \mathds{1}_{\{|\eta|\lesssim 1\}} |\eta|^{1-|\alpha|}|g_\beta(\zeta)|, \qquad |\alpha|\ge 1,\end{gathered}$$ for some bounded and compactly supported functions $g_\beta$. Lemma \[Lem\_appendix: Kernel with 1 function\] $(i)$ and corollary \[Cor\_appendix: decay of integral operators\] $(i)$ of appendix \[Appendix A\] imply that $|K(x,y)|\lesssim |x|^{-1}\langle x\rangle^{-2}\langle y\rangle^{-3}$ for any $(x,y)$, and statement $(i)$ follows by an inequality such as with $L=L^2$. In order to prove assertion $(ii)$ we consider a cut-off function $\psi\in C^\infty_0(\mathbb{R}^2)$ equal to 1 in some closed ball $\overline{B_C(0)}$, for a $C\gg 1$, and decompose $R(u,v)$ as follows, using : $$R(u,v) = \int K_0(x-y, y-z) u(y)v(z) dy dz + \int K_1(x-y,y-z)[\langle D_x\rangle^4 u](y)v(z) dy dz,$$ with $$\begin{aligned} K_0(x,y) &= \frac{1}{(2\pi)^2}\int e^{ix\cdot \xi + i y\cdot \eta } \left( 1-\chi\left(\frac{\xi -\eta}{\langle\eta\rangle}\right) - \chi\left(\frac{\eta}{\langle\xi-\eta\rangle}\right)\right)\psi(\eta) d\xi d\eta, \\ K_1(x,y)&= \frac{1}{(2\pi)^2}\int e^{ix\cdot \xi + i y\cdot \eta } \left( 1-\chi\left(\frac{\xi -\eta}{\langle\eta\rangle}\right) - \chi\left(\frac{\eta}{\langle\xi-\eta\rangle}\right)\right)(1-\psi)(\eta)\langle\xi-\eta\rangle^{-4} d\xi d\eta.\end{aligned}$$ Since frequencies $\xi, \eta$ are both bounded on the support of $\left( 1-\chi\left(\frac{\xi -\eta}{\langle\eta\rangle}\right) - \chi\left(\frac{\eta}{\langle\xi-\eta\rangle}\right)\right)\psi(\eta)$, one can show through some integration by parts that $|K_0(x,y)|\lesssim \langle x\rangle^{-3}\langle y \rangle^{-3}$ for any $(x,y)$, to then deduce that $$\left\| \int K_0(x-y, y-z) u(y)v(z) dy dz \right\|_{L^2(dx)}\lesssim \|u\|_{L^\infty}\|v\|_{L^2}.$$ Kernel $K_1(x,y)$ can be split using a Littlewood-Paley decomposition as follows $$K_1(x,y) =\sum_{k\ge 1}\frac{1}{(2\pi)^2} \underbrace{\int e^{ix\cdot \xi + i y\cdot \eta } \left( 1-\chi\left(\frac{\xi -\eta}{\langle\eta\rangle}\right) - \chi\left(\frac{\eta}{\langle\xi-\eta\rangle}\right)\right)(1-\psi)(\eta) \varphi(2^{-k}\eta)\langle\xi-\eta\rangle^{-4} d\xi d\eta}_{K_{1,k}(x,y)},$$for a suitable $\varphi \in C^\infty_0(\mathbb{R}^2\setminus\{0\})$. On the support of $ \left( 1-\chi\left(\frac{\xi -\eta}{\langle\eta\rangle}\right) - \chi\left(\frac{\eta}{\langle\xi-\eta\rangle}\right)\right)(1-\psi)(\eta)\varphi(2^{-k}\eta)$, frequencies $\eta, \xi-\eta$ are either bounded or equivalent and of size $2^k$ (which implies in particular that $\langle\xi-\eta\rangle^{-4}\lesssim \langle\xi\rangle^{-3}\langle\eta\rangle^{-1}$). After a change of coordinates and some integration by parts one can show that $|K_{1,k}(x,y)|\lesssim 2^k \langle x\rangle^{-3}\langle 2^ky\rangle^{-3}$, for any $k\ge 1$, and therefore that $$\begin{aligned} &\left\| \int e^{i(x-y)\cdot \xi + i(y-z)\cdot \eta } K_1(x-y,y-z)[\langle D_x\rangle^4 u](y)v(z) dy dz \right\|_{L^2(dx)}\\ \lesssim & \sum_{k\ge 1}2^k \left\| \int \langle x-y\rangle^{-3}\langle 2^k(y-z)\rangle^{-3}|\langle D_x\rangle^4 u(y)||w(z)| dydz \right\|_{L^2(dx)} \\ \lesssim & \sum_{k\ge 1} 2^k \int \langle y\rangle^{-3} \langle 2^k z\rangle^{-3} \|[\langle D_x\rangle^4u](\cdot-y) w(\cdot-y-z)\|_{L^2(dx)} dydz \lesssim \|u\|_{H^{4,\infty}}\|w\|_{L^2},\end{aligned}$$ which concludes the proof of statement $(ii)$. The last results of this subsection are stated without proofs. All the details can be found in chapter 6 of [@metivier:paradifferential] (see theorems 6.1.1, 6.1.4, 6.2.1, 6.2.4). \[Prop: paradifferential symbolic calculus\] Consider $a\in S^m_r$, $b\in S^{m'}_r$, $r\in\mathbb{N}^*$, $m, m'\in \mathbb{R}$. $(i)$ Symbol $a \sharp b := \displaystyle\sum_{|\alpha|<r }\frac{1}{\alpha !}\partial^\alpha_\xi a(x,\xi) D^\alpha_x b(x,\xi)$ is well defined in $\sum_{j<r}S^{m+m'-j}_{r-j}$; $(ii)$ $Op^B(a)Op^B(b) - Op^B(a \sharp b)$ is an operator of order $\le m + m' -r$, and for all $s\in\mathbb{R}$, there exists a constant $C>0$ such that, for all $a\in S^m_r(\mathbb{R}^d)$, $b\in S^{m'}_r(\mathbb{R}^d)$, and $w\in H^{s+m+m'-r}(\mathbb{R}^d)$, $$\begin{gathered} \|Op^B(a)Op^B(b)w - Op^B(a\sharp b)w\|_{H^s} \\ \le C \big(M^m_r(a;n)M^{m'}_0(b;n_0) + M^m_0(a;n)M^{m'}_r(b;n_0)\big) \|w\|_{H^{s+m+m'-r}},\end{gathered}$$ where $n_0=\left[\frac{d}{2}\right]+1$, $n=n_0 + r$. Moreover, $Op^B(a)Op^B(b)-Op^B(a\sharp b)=\widetilde{\sigma}_r(x,D_x)$ with $$\begin{gathered} \widetilde{\sigma}_r(x,\xi) = (\sigma_a\sharp \sigma_b)(x,\xi) -\sigma_{a\sharp b}(x,\xi)\\ + \sum_{|\alpha| = r}\frac{1}{r! (2\pi)^d}\int e^{i(x-y)\cdot\zeta}\left(\int_0^1 \partial^\alpha_\xi \sigma_a(x,\xi+t\zeta)(1-t)^{r-1}dt\right) \theta(\zeta,\xi) D^\alpha_x\sigma_b(y,\xi) dyd\zeta \end{gathered}$$ with $\theta\equiv 1$ in a neighbourhood of the support of $\mathcal{F}_{y\mapsto \eta}\sigma_b(\eta,\xi)$. These results extend to matrix valued symbols and operators. \[Remark:on\_symbol\_Bony\_calculus\] If symbol $a(x,\xi)$ only depends on $\xi$ then $\sigma_a \sharp \sigma_b - \sigma_{a\sharp b}=0$ and $\widetilde{\sigma}_r$ reduces to the only integral term. Moreover, $$\label{Fourier transform composition symbol} \mathcal{F}_{x\mapsto \eta}\widetilde{\sigma}_r(\eta,\xi) = \sum_{|\alpha|=r}\frac{1}{r!} \left(\int_0^1 \partial^\alpha_\xi a(\xi+t\eta)(1-t)^{[r]_+-1}dt\right) \chi\Big(\frac{\eta}{\langle\xi\rangle}\Big)\eta^\alpha \hat{b}_y(\eta,\xi),$$ where $\chi\big(\frac{\eta}{\langle\xi\rangle}\big)$ is the admissible cut-off function defining $\sigma_b$. \[Cor : paradiff symbolic calculus at order 1\] For $d=2$ and all $s\in\mathbb{R}$, there exists a constant $C>0$ such that, for $a\in S^m_r, b\in S^{m'}_r$, $r\in\mathbb{N}^*$, and $w\in H^{s+m+m'-1}$, $$\begin{gathered} \| Op^B(a)Op^B(b)w - Op(ab)w\|_{H^s} \\ \le C \big(M^m_1(a;3)M^{m'}_0(b;2) + M^m_0(a;3)M^{m'}_1(b;2)\big) \|w\|_{H^{s+m+m'-1}}.\end{gathered}$$ \[Paradiff Ajoint\] Consider $a\in S^m_r(\mathbb{R}^d)$, denote by $Op^B(a)^*$ the adjoint operator of $Op^B(a)$ and by $a^*(x,\xi)=\overline{a}(x,\xi)$ the complex conjugate of $a(x,\xi)$. $(i)$ Symbol $b(x,\xi): = \displaystyle\sum_{|\alpha|<r}\frac{1}{\alpha !}D^\alpha_x \partial^\alpha_\xi a^*(x,\xi)$ is well defined in $ \sum_{j<r}S^{m-j}_{r-j}$; $(ii)$ Operator $Op^B(a)^* - Op^B(b)$ is of order $\le m-r$. Precisely, for all $s\in \mathbb{R}$ there is a constant $C>0$ such that, for all $a\in S^m_r(\mathbb{R}^d)$ and $w\in H^{s + m -r}(\mathbb{R}^d)$, $$\left\| Op^B(a)^*w - Op^B(b)w\right\|_{H^s} \le C M^m_r(a;n)\|w\|_{H^{s+m -r}},$$ with $n_0 = \big[\frac{d}{2}\big]+1$, $n = n_0 + r$. These results extend to matrix valued symbols $a$, with $a^*(x,\xi)$ denoting the adjoint of matrix $a(x,\xi)$. \[Cor : paradiff ajoint at order 1\] For $d=2$ and all $s\in\mathbb{R}$, there exists a constant $C>0$ such that, for $a\in S^m_r$, $r\in\mathbb{N}^*$ and $w\in H^{s+m-1}$, $$\| Op^B(a)^*w - Op(a^*)w\|_{H^s} \le C M^m_1(a;3)\|w\|_{H^{s+m-1}}.$$ ### Semi-classical pseudodifferential calculus {#Subsection: Semiclassical Pseudodifferential Calculus} In this subsection we recall some definitions and results about semi-classical symbolic calculus in general space dimension $d\ge 1$ which will be used in section \[Sec: development of the PDE system\]. We refer the reader to [@dimassi:spectral] and [@zworski:semiclassical] for more details. An order function on $\mathbb{R}^d\times\mathbb{R}^d$ is a smooth map from $\mathbb{R}^d\times\mathbb{R}^d$ to $\mathbb{R}_+$ : $(x,\xi)\rightarrow M(x,\xi)$ such that there exist $N_0\in \mathbb{N}$, $C>0$ and for any $(x,\xi), (y,\eta)\in \mathbb{R}^d\times\mathbb{R}^d$ $$\label{def ineq order function} M(y,\eta) \le C \langle x-y\rangle^{N_0} \langle\xi-\eta\rangle^{N_0} M(x, \xi) \, ,$$ where $\langle x\rangle=\sqrt{1+|x|^2}$. Let *M* be an order function on $\mathbb{R}^d\times\mathbb{R}^d$, $\delta,\sigma \ge 0$. One denotes by $S_{\delta, \sigma}(M)$ the space of smooth functions $$\begin{aligned} (x,\xi, h) & \rightarrow a(x,\xi, h) \\ \mathbb{R}^d\times\mathbb{R}^d\times ]0,1] & \rightarrow \mathbb{C}\end{aligned}$$ satisfying for any $\alpha_1, \alpha_2 \in \mathbb{N}^d, k, N \in \mathbb{N}$ $$\label{symbol in S delta beta M} |\partial_x^{\alpha_1}\partial_{\xi}^{\alpha_2}(h\partial_h)^k a(x,\xi, h)| \lesssim M(x,\xi)\, h^{-\delta(|\alpha_1|+|\alpha_2|)}(1+\sigma h^{\sigma}|\xi|)^{-N}\, .$$ A key role in this paper will be played by symbols $a$ verifying with $M(x,\xi)= \langle \frac{x+f(\xi)}{\sqrt{h}} \rangle^{-N}$, for $N \in \mathbb{N}$ and a certain smooth function $f(\xi)$. This function $M$ is no longer an order function because of the term $h^{-\frac{1}{2}}$, but nevertheless we keep writing $a \in S_{\delta,\sigma}(\langle \frac{x+f(\xi)}{\sqrt{h}} \rangle^{-N})$. In the semi-classical setting we say that $a(x,\xi, h)$ is a symbol *of order $r$* if $a \in S_{\delta,\sigma}(\langle \xi \rangle^r)$, for some $\delta,\sigma \ge 0$. Let us observe that when $\sigma>0$ the symbol decays rapidly in $h^{\sigma}|\xi|$, which implies the following inclusion for $r\ge 0$: $$S_{\delta,\sigma}(\langle\xi\rangle^r) \subset h^{-\sigma r}S_{\delta, \sigma}(1).$$ This means that, up to a small loss in $h$, this type of symbols can be always considered as symbols of order zero. In the rest of the paper we will not indicate explicitly the dependence of symbols on $h$, referring to $a(x,\xi,h)$ simply as $a(x,\xi)$. \[Def: Weyl and standard quantization\] Let $a\in S_{\delta, \sigma}(M)$ for some order function $M$, some $\delta,\sigma\ge 0$. (i) We can define the *Weyl quantization* of $a$ to be the operator $\oph(a)=a^{w}(x, hD)$ acting on $u \in \mathcal{S}(\mathbb{R}^d)$ by the following formula: $$\oph(a(x,\xi))u(x) = \frac{1}{(2\pi h)^d}\int_{\mathbb{R}^d}\int_{\mathbb{R}^d} e^{\frac{i}{h}(x-y)\cdot\xi} a\Big(\frac{x+y}{2}, \xi\Big)\, u(y)\; dy d\xi \, ;$$ (ii) We define also the *standard quantization* of $a$: $$Op_h(a(x,\xi))u(x) = \frac{1}{(2 \pi h)^d} \int_{\mathbb{R}^d}\int_{\mathbb{R}^d} e^{\frac{i}{h}(x-y)\cdot\xi} a(x, \xi)\, u(y)\; dy d\xi \, .$$ It is clear from the definition that the two quantizations coincide when the symbol does not depend on $x$. We also introduce a semi-classical version of Sobolev spaces on which the above operators act naturally. \[def of h-Sobolev spaces\] (i) Let $\rho \in \mathbb{N}$. We define the semi-classical Sobolev space $H^{\rho, \infty}_h(\mathbb{R}^d)$ as the space of tempered distributions $w$ such that $\langle hD \rangle^\rho w := Op_h(\langle\xi\rangle^\rho)w \in L^\infty$, endowed with norm $$\|w\|_{H^{\rho,\infty}_h}=\|\langle hD \rangle^\rho w\|_{L^\infty};$$ (ii) \[def of H\^s\_h\] Let $s\in\mathbb{R}$. We define the semi-classical Sobolev space $H^s_h(\mathbb{R}^d)$ as the space of tempered distributions $w$ such that $\langle hD \rangle^s w := Op_h(\langle\xi\rangle^s)w\in L^2$, endowed with norm $$\|w\|_{H^s}=\|\langle hD \rangle^s w\|_{L^2}.$$ For future references, we write down the semi-classical Sobolev injection in space dimension 2: $$\label{semi-classical Sobolev injection} \|v_h\|_{H_h^{\rho, \infty}(\mathbb{R}^2)} \lesssim_{\sigma}h^{-1} \|v_h\|_{H^{\rho + 1+\sigma}_h(\mathbb{R}^2)}\, , \qquad \forall \sigma> 0\, .$$ The following two propositions are stated without proof. They concern the adjoint and the composition of pseudo-differential operators. All related details are provided in chapter 7 of [@dimassi:spectral] or in chapter 4 of [@zworski:semiclassical]. If $a(x,\xi)$ is a real symbol its Weyl quantization is self-adjoint, i.e. $$\big(\oph(a)\big)^*=\oph(a)\, .$$ \[Prop : Composition for Weyl quantization\] Let $a, b \in \mathcal{S}(\mathbb{R}^d)$. Then $$\oph(a)\circ \oph(b) = \oph (a\sharp b) \, ,$$ where $$\label{a sharp b integral formula} a \sharp b \,(x,\xi) := \frac{1}{(\pi h)^{2d}}\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}\int_{\mathbb{R}^d} e^{\frac{2i}{h}\sigma (y, \eta; \, z, \zeta)} a(x+z, \xi + \zeta) b(x+y, \xi +\eta) \; dy d\eta dz d\zeta ,$$ and $$\sigma (y, \eta; \, z, \zeta) = \eta\cdot z - y\cdot \zeta\, .$$ It is often useful to derive an asymptotic expansion for $a \sharp b$, as it allows easier computations than integral formula . This expansion is usually obtained by applying the stationary phase argument when $a, b \in S_{\delta, \sigma}(M)$, $\delta \in [0,\frac{1}{2}[$ (as shown in [@zworski:semiclassical]). Here we provide an expansion at any order even when one of two symbols belongs to $S_{\frac{1}{2},\sigma_1}(M)$ (still having the other in $S_{\delta,\sigma_2}(M)$ for $\delta<\frac{1}{2}$, and $\sigma_1,\sigma_2$ either equal or, if not, one of them equal to zero), whose proof is based on the Taylor development of symbols $a, b$, and can be found in the appendix of [@stingo:1D_KG] (for $d=1$). \[Prop: a sharp b\] Let $M_1,M_2$ be two order functions and $a\in S_{\delta_1, \sigma_1}(M_1)$, $b\in S_{\delta_2, \sigma_2}(M_2)$, $\delta_1, \delta_2 \in [0,\frac{1}{2}]$, $\delta_1 + \delta_2< 1$, $\sigma_1, \sigma_2 \ge 0$ such that $$\label{beta in symbolic calculus} \sigma_1=\sigma_2\ge 0 \qquad \mbox{or} \qquad \big[\sigma_1\ne\sigma_2 \,\mbox{and } \, \sigma_i=0\,,\sigma_j>0\,, i\ne j\in\{1,2\} \big]\,.$$ Then $a \sharp b \in S_{\delta, \sigma}(M_1 M_2)$, where $\delta = \max\{\delta_1, \delta_2 \}$, $\sigma=\max\{\sigma_1,\sigma_2\}$. Moreover, $$\label{a sharp b asymptotic formula} a \sharp b = \sum_{\substack{\alpha=(\alpha_1,\alpha_2) \\ |\alpha|=0, \dots, N-1}} \frac{(-1)^{|\alpha_1|}}{\alpha !}\Big(\frac{h}{2i}\Big)^{|\alpha|}\partial^{\alpha_1}_x\partial^{\alpha_2}_\xi a(x,\xi) \ \partial^{\alpha_2}_x\partial^{\alpha_1}_\xi b(x,\xi)+ r_N \,,$$ where $r_N \in h^{N(1-(\delta_1 + \delta_2))}S_{\delta, \sigma}(M_1 M_2)$ and $$\begin{gathered} \label{r_N 1} r_N(x,\xi) = \, \left(\frac{h}{2i}\right)^N \frac{N}{(\pi h)^{2d}} \sum_{\substack{\alpha= (\alpha_1, \alpha_2)\\ |\alpha|= N}} \frac{(-1)^{|\alpha_1|}}{\alpha!} \int_{\mathbb{R}^4}e^{\frac{2i}{h}(\eta\cdot z - y\cdot\zeta)} \\ \times \Big(\int_0^1 \partial_x^{\alpha_1}\partial_{\xi}^{\alpha_2}a(x+tz, \xi +t\zeta)(1-t)^{N-1} dt\Big) \partial_x^{\alpha_2}\partial_{\xi}^{\alpha_1}b(x+y, \xi + \eta)\, dy d\eta dz d\zeta \,,\end{gathered}$$ or $$\begin{gathered} \label{r_N 2} r_N(x,\xi) = \, \left(\frac{h}{2i}\right)^N \frac{N}{(\pi h)^{2d}} \sum_{\substack{\alpha= (\alpha_1, \alpha_2)\\ |\alpha|= N}} \frac{(-1)^{|\alpha_1|}}{\alpha!} \int_{\mathbb{R}^4}e^{\frac{2i}{h}(\eta\cdot z - y\cdot\zeta)} \partial_x^{\alpha_1}\partial_{\xi}^{\alpha_2}a(x+z, \xi +\zeta) \\ \times \Big(\int_0^1 \partial_x^{\alpha_2}\partial_{\xi}^{\alpha_1}b(x+ty, \xi + t\eta)(1-t)^{N-1} dt\Big)\, dy d\eta dz d\zeta \,.\end{gathered}$$ More generally, if $h^{N\delta_1}\partial^{\alpha}a \in S_{\delta_1,\sigma_1}(M^{N}_1)$, $h^{N\delta_2}\partial^{\alpha}b \in S_{\delta_2,\sigma_2}(M^{N}_2)$, for $|\alpha|=N$ and some order functions $M_1^N, M_2^N$, then $r_N \in h^{N(1-(\delta_1+\delta_2))}S_{\delta,\sigma}(M^N_1M^N_2)$. \[Remark:symbols\_with\_null\_support\_intersection\] From the previous proposition it follows that, if symbols $a\in S_{\delta_1,\sigma_1}(M_1)$, $b\in S_{\delta_2,\sigma_2}(M_2)$ are such that $\text{supp}a \cap \text{supp}b = \emptyset$, then $a\sharp b = O(h^\infty)$, meaning that, for every $N\in \mathbb{N}$, $a\sharp b =r_N$ with $r_N\in h^{N(1-(\delta_1+\delta_2))}S_{\delta,\sigma}(M_1M_2)$. We draw the reader’s attention to the fact that symbol $\sharp$ is used simultaneously in Bony calculus (see proposition \[Prop: paradifferential symbolic calculus\]) and in Weyl semi-classical calculus (as in ) with two different meaning. However, we avoid to introduce different notations as it will be clear by the context if we are dealing with the former or the latter one. The result of proposition \[Prop: a sharp b\] and remark \[Remark:symbols\_with\_null\_support\_intersection\] are still true even when one of the two order functions, or both, has the form $\langle\frac{x+f(\xi)}{\sqrt{h}}\rangle^{-1}$, for a smooth function $f(\xi)$, $\nabla f(\xi)$ bounded, as stated below (see the appendix of [@stingo:1D_KG]). \[Lem : a sharp b\] Let $f(\xi):\mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function, with $|\nabla f(\xi)|$ bounded. Consider $a \in S_{\delta_1,\sigma_1}(\langle \frac{x+f(\xi)}{\sqrt{h}}\rangle^{-m})$, $m\in\mathbb{N}$, and $b\in S_{\delta_2,\sigma_2}(M)$, for $M$ order function or $M(x,\xi)=\langle\frac{x+f(\xi)}{\sqrt{h}}\rangle^{-n}$, $n \in \mathbb{N}$, some $\delta_1 \in [0,\frac{1}{2}]$, $\delta_2 \in [0,\frac{1}{2}[$, $\sigma_1, \sigma_2\ge 0$ as in . Then $a\sharp b \in S_{\delta,\sigma}(\langle \frac{x+f(\xi)}{\sqrt{h}}\rangle^{-m}M)$, where $\delta = \max\{\delta_1,\delta_2\}$, $\sigma = \max\{\sigma_1,\sigma_2\}$, and the asymptotic expansion holds, with $r_N \in h^{N(1-(\delta_1+\delta_2))}S_{\delta,\sigma}(\langle \frac{x+f(\xi)}{\sqrt{h}}\rangle^{-m}M)$ given by (or equivalently ).\ More generally, if $h^{N\delta_1}\partial^{\alpha}a \in S_{\delta_1,\sigma_1}(\langle \frac{x+f(\xi)}{\sqrt{h}}\rangle^{-m'})$ and $h^{N\delta_2}\partial^{\alpha}b \in S_{\delta_2,\sigma_2}(M^N)$, $|\alpha|=N$, $M^N$ order function or $M^N(x,\xi)=\langle \frac{x+f(\xi)}{\sqrt{h}}\rangle^{-n'}$, for some $m',n' \in \mathbb{N}$, then remainder $r_N$ belongs to $h^{N(1-(\delta_1+\delta_2))}S_{\delta,\sigma}(\langle \frac{x+f(\xi)}{\sqrt{h}}\rangle^{-m'}M^N)$. ### Semi-classical Operators for the Wave Solution: Some Estimates {#Subsection: Some Technical Estimates I} From now on we place ourselves in space dimension $d=2$. This technical subsection focuses on the introduction and the analysis of some particular operators that we will use when dealing with the wave component in the semi-classical framework (subsection \[Subsection : The Derivation of the Transport Equation\]). More precisely, lemma \[Lemma on inequalities for Op(A)\] will be often recalled to prove that some operator belongs to $\mathcal{L}(L^2 ; L^\infty)$ and compute its norm; propositions \[Prop : continuity Op(gamma) L2 to L2\], \[Prop : continuity of Op(gamma1):X to L2\] concern the continuity of some important operators like $\Gamma^{w,k}$ defined in , while propositions \[Prop: L2 est of integral remainders\], \[Prop : Linfty est of integral remainders\] are devoted to prove the continuity of some other operators often arising when considering the quantization of symbolic integral remainders. Finally, lemmas \[Lemma : symbolic product development\] and \[Lemma : on the enhanced symbolic product\] deal with the development of some special symbolic products. While \[Lemma : symbolic product development\] will be used several times throughout the paper, lemma \[Lemma : on the enhanced symbolic product\] is stated explicitly on purpose to prove lemma \[Lem: preliminary on Op(e)\]. \[Lemma on inequalities for Op(A)\] There exists a constant $C>0$ such that, for any function $A(x,\xi)$ with $\partial^\alpha_x \partial^\beta_\xi A \in L^2(\mathbb{R}^2\times\mathbb{R}^2)$ for $|\alpha|, |\beta|\le 3$, and any function $w\in L^2(\mathbb{R}^2)$, $$\label{inequalities |Op(A)|} \left| \oph\big(A(x,\xi)\big)w(x) \right|\le C \|w\|_{L^2} \int_{\mathbb{R}^2} \langle x-y \rangle^{-3} \sum_{|\alpha|,|\beta|\le 3} \Big\|\partial_y^{\alpha}\partial_\xi^{\beta}\Big[A\Big(\frac{x+y}{2},h\xi\Big)\Big]\Big\|_{L^2(d\xi)} dy.$$ Moreover, if $A(x,\xi)$ is compactly supported in $x$ there exists a smooth function, supported in a neighbourhood of $supp A$, such that $$\label{inequality Op(A), A compactly supported} \left| \oph\big(A(x,\xi)\big)w(x) \right|\le C \|w\|_{L^2} \int_{\mathbb{R}^2} \Big|\theta'\Big(\frac{x+y}{2}\Big)\Big| \sum_{|\alpha|\le 3} \Big\|\partial_y^{\alpha}\Big[A\Big(\frac{x+y}{2},h\xi\Big)\Big]\Big\|_{L^2(d\xi)} dy.$$ Let us prove the statement for $A\in \mathcal{S}(\mathbb{R}^2\times \mathbb{R}^2)$ and $w\in\mathcal{S}(\mathbb{R}^2)$. The density of $\mathcal{S}(\mathbb{R}^2\times\mathbb{R}^2)$ into $\{A\in L^2(\mathbb{R}^2\times \mathbb{R}^2)| \partial^\alpha_x\partial^\beta_\xi A \in L^2(\mathbb{R}^2 \times \mathbb{R}^2), |\alpha|, |\beta|\le 3\}$ and of $\mathcal{S}(\mathbb{R}^2)$ into $L^2(\mathbb{R}^2)$ will then justify the definition of $\oph(A(x,\xi))w$ for $A$ and $w$ as in the statement, together with inequalities , . Using integration by parts, Cauchy-Schwarz inequality, and Young’s inequality for convolutions, we can write the following: $$\begin{aligned} & |\oph(A(x,\xi))w(x)| = \frac{1}{(2\pi)^2}\left|\int_{\mathbb{R}^4} e^{i(x-y)\cdot\xi} A\Big(\frac{x+y}{2},h\xi\Big)w(y)\ dy d\xi \right| \\ & = \frac{1}{(2\pi)^4} \left| \int_{\mathbb{R}^2} \widehat{w}(\eta) \int_{\mathbb{R}^2}\int_{\mathbb{R}^2} e^{i(x-y)\cdot\xi + i y\cdot \eta} A\Big(\frac{x+y}{2},h\xi\Big) \ dyd\xi\ d\eta\right| \\ & = \frac{1}{(2\pi)^4}\left| \int_{\mathbb{R}^2} \widehat{w}(\eta) \int_{\mathbb{R}^2}\int_{\mathbb{R}^2} \left(\frac{1-i(x-y)\cdot\partial_\xi}{1+|x-y|^2}\right)^3 \left(\frac{1+i(\xi - \eta)\cdot\partial_y}{1+|\xi - \eta|^2}\right)^3 e^{i(x-y)\cdot\xi + i y\cdot \eta} \right.\\ & \left. \hspace{10cm} \times A\Big(\frac{x+y}{2},h\xi\Big) \ dyd\xi\ d\eta\right| \\ & \lesssim \int_{\mathbb{R}^2} \left| \hat{w}(\eta)\right| \int_{\mathbb{R}^2}\int_{\mathbb{R}^2}\langle x- y\rangle^{-3} \langle \xi - \eta\rangle^{-3} \sum_{|\alpha|,|\beta|\le 3} \Big|\partial_y^{\alpha}\partial_\xi^{\beta}\Big[A\Big(\frac{x+y}{2},h\xi\Big)\Big]\Big| dy d\xi \ d\eta \\ & \lesssim \|\hat{w}\|_{L^2(d\eta)} \|\langle\eta\rangle^{-3}\|_{L^1(d\eta)}\int_{\mathbb{R}^2} \langle x-y \rangle^{-3} \sum_{|\alpha|,|\beta|\le 3} \Big\|\partial_y^{\alpha}\partial_\xi^{\beta}\Big[A\Big(\frac{x+y}{2},h\xi\Big)\Big]\Big\|_{L^2(d\xi)} dy \\ & \lesssim \|w\|_{L^2} \int_{\mathbb{R}^2} \langle x-y \rangle^{-3} \sum_{|\alpha|,|\beta|\le 3} \Big\|\partial_y^{\alpha}\partial_\xi^{\beta}\Big[A\Big(\frac{x+y}{2},h\xi\Big)\Big]\Big\|_{L^2(d\xi)} dy\,. \end{aligned}$$ If symbol $A(x,\xi)$ is compactly supported in $x$ we can consider a smooth function $\theta'\in C^\infty_0(\mathbb{R})$, identically equal to 1 on the support of $A(x,\xi)$, and write $$\begin{aligned} & |\oph(A(x,\xi))w(x)| = \frac{1}{(2\pi)^2}\left| \int_{\mathbb{R}^2} \widehat{w}(\eta) d\eta \int_{\mathbb{R}^2}\int_{\mathbb{R}^2} \left(\frac{1+i(\xi - \eta)\cdot\partial_y}{1+|\xi - \eta|^2}\right)^3 e^{i(x-y)\cdot\xi + i y\cdot \eta} \right.\\ & \left. \hspace{12cm} \times A\Big(\frac{x+y}{2},h\xi\Big) \ dyd\xi\right| \\ & \lesssim \int_{\mathbb{R}^2} \left| \hat{w}(\eta)\right| d\eta \int_{\mathbb{R}^2}\int_{\mathbb{R}^2}\Big|\theta'\Big(\frac{x+y}{2}\Big)\Big|\langle \xi - \eta\rangle^{-3} \sum_{|\alpha|\le 3} \big|\partial_y^{\alpha} \Big[A\Big(\frac{x+y}{2},h\xi\Big)\Big]\Big| dy d\xi \\ & \lesssim \|w\|_{L^2} \int_{\mathbb{R}^2}\Big|\theta'\Big(\frac{x+y}{2}\Big)\Big| \sum_{|\alpha| \le 3} \big\|\partial_y^{\alpha}\Big[A\Big(\frac{x+y}{2},h\xi\Big)\Big]\Big\|_{L^2(d\xi)} dy\,. \end{aligned}$$ A very important role in this subsection and in subsection \[Subsection : The Derivation of the Transport Equation\] will be played by functions of the form $\gamma\big(\frac{x|\xi| -\xi}{h^{1/2 - \sigma}}\big)\psi(2^{-k}\xi)$, where $\gamma \in C^\infty(\mathbb{R}^2)$ is such that $|\partial^\alpha\gamma(z)|\lesssim \langle z\rangle^{-|\alpha|}$, $\psi\in C^\infty_0(\mathbb{R}^2-\{0\})$, $\sigma>0$ is a small fixed constant and $k$ is an integer belonging to set $K$, with $$\label{set_frequencies_K} K :=\{k\in\mathbb{Z}\ :\ h\lesssim 2^k \lesssim h^{-\sigma}\}.$$ In various results, such as proposition \[Prop : continuity of Op(gamma1):X to L2\], we will need a more decaying smooth function $\gamma_1$ verifying that $|\partial^\alpha\gamma_1(z)|\lesssim \langle z \rangle^{-(1+|\alpha|)}$. We introduce here some notations we will keep throughout the whole paper: For any $n\in \mathbb{N}$, $\gamma_n$ denotes a smooth function in $\mathbb{R}^2$ such that $|\partial^\alpha \gamma_n(z)|\lesssim_\alpha \langle z\rangle^{-(n+|\alpha|)}$, for any $\alpha\in\mathbb{N}^2$. We use the simplest notation $\gamma$ for $\gamma_0$; For any integer $m\in \mathbb{Z}$, $b_m(\xi)$ will denote any function satisfying $|\partial^\beta b_m(\xi)|\lesssim_\beta |\xi|^{m-|\beta|}$, for any $\xi$ in its domain, any $\beta\in\mathbb{N}^2$. The following lemma is a useful reference when we need to deal with some derivatives of $\gamma\big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\big)$. \[Lem : est on gamma for wave\] Let $\sigma\in\mathbb{R}$ and $n\in\mathbb{N}$. For any multi-indices $\alpha, \beta\in \mathbb{N}^2$ we have that $$\label{derivatives of gamma_n 1} \partial^\alpha_x \partial^\beta_\xi \Big[\gamma_n\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\Big] = \sum_{k=0}^{|\beta|} h^{-(|\alpha| + k)(\frac{1}{2}-\sigma)}\gamma_{n+|\alpha|+k}\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big) b_{|\alpha| +k -|\beta|}(\xi).$$ Furthermore, if $\theta=\theta(x)\in C^\infty_0(\mathbb{R}^2)$, there exists a set $\{\theta_k(x)\}_{1\le k\le |\beta|}$ of smooth compactly supported functions such that $$\label{derivatives of gamma_n 2} \theta(x)\partial^\alpha_x \partial^\beta_\xi \Big[\gamma_n\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\Big] = \sum_{k=1}^{|\beta|} h^{-(|\alpha| + k)(\frac{1}{2}-\sigma)}\gamma_{n+|\alpha|+k}\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\theta_k(x) b_{|\alpha| +k -|\beta|}(\xi).$$ Let $\delta_{ij}$ be the Kronecker delta and ${\sum}'$ be a concise notation to indicate a linear combination. For $i=1,2$, $$\label{first derivative gamma_n} \begin{split} & \partial_{\xi_i}\Big[\gamma_n\Big(\frac{x|\xi| -\xi}{h^{1/2-\sigma}}\Big)\Big] = h^{-(\frac{1}{2}-\sigma)}\sum_{j=1}^2 (\partial_j\gamma_n)\Big(\frac{x|\xi| -\xi}{h^{1/2-\sigma}}\Big) (x_j\xi_i |\xi|^{-1} - \delta_{ij}) \\ & = \sum_{j=1}^2 (\partial_j\gamma_n)\Big(\frac{x|\xi| -\xi}{h^{1/2-\sigma}}\Big) \Big(\frac{x_j|\xi| - \xi_j}{h^{1/2-\sigma}}\Big)\ \xi_i|\xi|^{-2} + \sum_{j=1}^2 h^{-(\frac{1}{2}-\sigma)} (\partial_j\gamma_n)\Big(\frac{x|\xi| -\xi}{h^{1/2-\sigma}}\Big) [\xi_i\xi_j|\xi|^{-2} - \delta_{ij}], \end{split}$$ which can be summarized saying that $$\partial_{\xi_i}\Big[\gamma_n\Big(\frac{x|\xi| -\xi}{h^{1/2-\sigma}}\Big)\Big] = {\sum}' \gamma_n\Big(\frac{x|\xi| -\xi}{h^{1/2-\sigma}}\Big) b_{-1}(\xi) + h^{-(\frac{1}{2}-\sigma)}\gamma_{n+1}\Big(\frac{x|\xi| -\xi}{h^{1/2-\sigma}}\Big) b_{0}(\xi),$$ for some new functions $\gamma_n, \gamma_{n+1}, b_0, b_{-1}$. Iterating this argument one finds that, for all $\beta\in\mathbb{N}^2$, $$\partial^\beta_\xi \Big[\gamma_n\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\Big] = {\sum_{k=0, \dots, |\beta|}}' h^{-k(\frac{1}{2}-\sigma)} \gamma_{n+k}\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big) b_{k-|\beta|}(\xi),$$ and obtains using that, for any $m\in\mathbb{N}, \alpha\in\mathbb{N}^2$, $$\label{derivatives in x gamma_n} \partial^\alpha_x \Big[ \gamma_m\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\Big] = h^{-|\alpha|(\frac{1}{2}-\sigma)} (\partial^\alpha \gamma_m)\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big) |\xi|^{|\alpha|}.$$ Equality is obtained replacing with $$\begin{split} \theta(x)\partial_{\xi_i}\Big[\gamma_n\Big(\frac{x|\xi| -\xi}{h^{1/2-\sigma}}\Big)\Big] & = h^{-(\frac{1}{2}-\sigma)}\sum_{j=1}^2 (\partial_j\gamma_n)\Big(\frac{x|\xi| -\xi}{h^{1/2-\sigma}}\Big) (\theta(x) x_j\xi_i |\xi|^{-1} - \theta(x)\delta_{ij})\\ & = {\sum}' h^{-(\frac{1}{2}-\sigma)}\gamma_{n+1}\Big(\frac{x|\xi| -\xi}{h^{1/2-\sigma}}\Big) \theta_1(x)b_{0}(\xi), \end{split}$$ where $\theta_1(x)$ is a new compactly supported function. By iteration one finds that, for any $\beta\in\mathbb{N}^2$, there is a set of $|\beta|$ compactly supported functions $\theta_k(x)$, $1\le k\le |\beta|$, such that $$\begin{split} \theta(x) \partial^\beta_\xi \Big[\gamma_n\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\Big] & =\sum_{k=1}^{|\beta|} h^{-k(\frac{1}{2}-\sigma)}\gamma_{n+k}\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\theta_k(x) b_{k - |\beta|}(\xi), \end{split}$$ which combined with gives . In some of the following results we denote by $\Theta_h$ the operator of change of coordinates $$\Theta_hw(x) = w(\sqrt{h}x),$$ for any $h\in ]0,1]$, and use that for any symbol $a(x,\xi)$, $$\label{unitary_transf_Thetah} \oph\big(a(x,\xi)\big) = \Theta_h \oph\big(\widetilde{a}(x,\xi)\big)\Theta_h^{-1},$$ with $\widetilde{a}(x,\xi)=a\Big(\frac{x}{\sqrt{h}},\sqrt{h}\xi\Big)$. \[Prop : continuity Op(gamma) L2 to L2\] Let $\sigma>0$ be sufficiently small, $K$ be the set defined in , $k\in K$ and $p\in\mathbb{Z}$. Let also $\psi\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$ and $a(x)$ be a smooth function, bounded together with all its derivatives. Then $\oph\big(\gamma\big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\big)\psi(2^{-k}\xi)a(x)b_p(\xi)\big):L^2 \rightarrow L^2$ is bounded and $$\label{est L2-L2 Op(gamma1)} \Big\| \oph\Big(\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)\Big)\Big\|_{\mathcal{L}(L^2)} \lesssim 2^{kp}.$$ Let $A(x,\xi) = \gamma\big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\big)\psi(2^{-k}\xi)a(x)b_p(\xi)$. For indices $k\in K$ such that $h^{1/2-\sigma}\lesssim 2^k \lesssim h^{-\sigma}$ the statement follows from the fact that $A(x,\xi)\in 2^{kp}S_{\frac{1}{2},0}(1)$ and by theorem 7.11 of [@dimassi:spectral]. For $k\in K$ such that $h\le 2^k\le h^{1/2-\sigma}$, $\widetilde{A}(x,\xi) := A(\frac{x}{\sqrt{h}},\sqrt{h}\xi)\in 2^{kp}S_{\frac{1}{2},0}(1)$ and the result follows by theorem 7.11 of [@dimassi:spectral] and equality . \[Prop: L2 est of integral remainders\] Let $\sigma,k,p$ be as in the previous proposition. Let also $q\in \mathbb{Z}$, $\widetilde{\psi}\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$, $a'(x)$ be a smooth function, bounded together with all its derivatives, and $f\in C(\mathbb{R})$. Define $$\begin{gathered} \label{integral Ik} I^k_{p,q}(x,\xi):=\frac{1}{(\pi h)^4} \int e^{\frac{2i}{h}(\eta\cdot z - y \cdot\zeta)} \left[ \int_0^1 \Big(\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)\Big)|_{(x+tz,\xi +t\zeta)} f(t)dt \right. \\ \left. \times \widetilde{\psi}(2^{-k}(\xi + \eta))a'(x+y)b_q(\xi+\eta)\right]\ dydzd\eta d\zeta\end{gathered}$$ and $$\begin{gathered} \label{integral Jk} J^k_{p,q}(x,\xi):=\frac{1}{(\pi h)^4} \int e^{\frac{2i}{h}(\eta\cdot z - y \cdot\zeta)} \left[ \int_0^1 \widetilde{\psi}(2^{-k}(\xi + t\zeta))a'(x+tz)b_q(\xi+t\zeta) f(t)dt \right. \\ \left. \times \Big(\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)\Big)|_{(x+y, \xi + \eta)}\right]\ dydzd\eta d\zeta.\end{gathered}$$ Then $\oph(I^k_{p,q}(x,\xi))$ and $\oph(J^k_{p,q}(x,\xi))$ are bounded operators on $L^2$ and $$\left\| \oph(I^k_{p,q}(x,\xi))\right\|_{\mathcal{L}(L^2)} + \left\| \oph(J^k_{p,q}(x,\xi))\right\|_{\mathcal{L}(L^2)} \lesssim 2^{k(p+q)}.$$ The same results holds also if $q=0$ and $\widetilde{\psi}(2^{-k}\xi)b_q(\xi)\equiv 1$. We show the result for $\oph(I^k_{p,q})$, leaving the reader to check that a similar argument can be used for $\oph(J^k_{p,q})$. We distinguish between two ranges of frequencies. For indices $k\in K$ such that $h^{1/2-\sigma}\le 2^k \lesssim h^{-\sigma}$ we observe that $I^k_{p,q}(x,\xi)\in 2^{k(p+q)}S_{\frac{1}{2}, 0}(1)$. Indeed, $\gamma\big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\big)\psi(2^{-k}\xi)a(x)b_p(\xi)\in 2^{kp}S_{\frac{1}{2}, \sigma}(1)$ by lemma \[Lem : est on gamma for wave\] while $\widetilde{\psi}(2^{-k}\xi)a'(x)b_q(\xi)\in 2^{kq}S_{\frac{1}{2}-\sigma,\sigma}(1)$. Hence performing a change of variables $y\mapsto \sqrt{h}y$, $z\mapsto \sqrt{h}z$, $\eta \mapsto\sqrt{h}\eta$, $\zeta \mapsto \sqrt{h}\zeta$, writing $$\label{complex exponential} e^{2i (\eta\cdot z - y\cdot\zeta)}= \left(\frac{1+2iy\cdot \partial_\zeta}{1+4|y|^2}\right)^3 \left(\frac{1-2i z\cdot \partial_\eta}{1+4|z|^2}\right)^3 \left(\frac{1-2i\eta\cdot\partial_z}{1+4|\eta|^2}\right)^3 \left(\frac{1+2i\zeta\cdot \partial_y}{1+4|\zeta|^2}\right)^3 e^{2i (\eta\cdot z - y\cdot\zeta)},$$ and integrating by parts in all variables, we get that $$\left|I^k_{p,q}(x,\xi) \right| \lesssim 2^{k(p+q)} \int \langle y \rangle^{-3}\langle z\rangle^{-3} \langle \eta\rangle^{-3} \langle \zeta\rangle^{-3} \ dy dz d\eta d\zeta \lesssim 2^{k(p+q)},$$ without any loss in $h^{-\delta}$ due to the fact that we are considering symbols $A(x,\xi) \in S_{\delta, \sigma}(1)$ with $\delta\in \{0, 1/2-\sigma, 1/2\}$, and the derivatives of $A(x +\sqrt{h}y, \xi + \sqrt{h}\eta)$ (resp. of $A(x +t\sqrt{h}z, \xi +t\sqrt{h}\zeta)$) with respect to $y$ and $\eta$ (resp. with respect to $z$ and $\zeta$). In a similar way one can also prove that $|\partial^\alpha_x\partial^\beta_\xi I^k_{p,q}(x,\xi)|\lesssim_{\alpha,\beta} h^{-\frac{1}{2}(|\alpha| + |\beta|)} 2^{k(p+q)}$, for any $\alpha,\beta\in\mathbb{N}^2$. Theorem 7.11 of [@dimassi:spectral] implies then the statement in this case. For indices $k\in K$ such that $h \lesssim 2^k \le h^{1/2-\sigma}$ we observe that $$\begin{gathered} \gamma\Big(\frac{x|\xi|}{h^{1/2-\sigma}} - h^\sigma\xi\Big)\psi(2^{-k}\sqrt{h}\xi)a\Big(\frac{x}{\sqrt{h}}\Big)b_p(\sqrt{h}\xi) \in 2^{kp}S_{\frac{1}{2}, \sigma}(1), \\ \widetilde{\psi}(2^{-k}\sqrt{h}\xi)a'\Big(\frac{x}{\sqrt{h}}\Big)b_q(\sqrt{h}\xi) \in 2^{kq}S_{\frac{1}{2},\sigma}(1).\end{gathered}$$ Then $\widetilde{I}^k_{p,q}(x,\xi)=I^k_{p,q}\Big(\frac{x}{\sqrt{h}},\sqrt{h}\xi\Big) \in 2^{k(p+q)}S_{\frac{1}{2},0}(1)$ and theorem 7.11 of [@dimassi:spectral] along with equality imply that $\oph(I^k_{p,q}):L^2\rightarrow L^2$ is bounded, uniformly in $h$. The last part of the statement can be proved following an analogous scheme, after having previously made an integration in $dzd\eta$ (or in $dyd\zeta$ if dealing with $J^k_{p,0}$). \[Prop: Continuity\_Lp\_wave\] Let $1\le p\le +\infty$, $\gamma\in C^\infty_0(\mathbb{R}^2)$ be radial, $\psi\in C^\infty_0(\mathbb{R}^2\setminus \{0\})$, $a(x)$ be a smooth function, bounded together with all its derivatives. Let also $\sigma>0$ be small, $k\in K$ with $K$ given by and $q\in\mathbb{Z}$. Then $\oph\big(\gamma\big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\big)\psi(2^{-k}\xi)a(x)b_q(\xi)\big):L^p\rightarrow L^p$ is a bounded operator with $$\left\|\oph\Big(\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_q(\xi)\Big)\right\|_{\mathcal{L}(L^p)}\lesssim 2^{kq} .$$ In order to prove the result of the statement we need to show that kernel $K^k(x,y)$ associated to $\oph\big(\gamma\big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\big)\psi(2^{-k}\xi)a(x)b_q(\xi)\big)$, i.e. $$\label{kernel_Kk} K^k(x,y):=\frac{1}{(2\pi h)^2}\int e^{\frac{i}{h}(x-y)\cdot\xi} \gamma\Big(\frac{\big(\frac{x+y}{2}\big)|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a\Big(\frac{x+y}{2}\Big)b_q(\xi) d\xi,$$ is such that $$\sup_x \int |K^k(x,y)| dy \lesssim 2^{kq}, \quad \sup_y \int |K^k(x,y)|dx \lesssim 2^{kq}.$$ From the symmetry between variables $x,y$, it will be enough to show that one of the two above inequalities is satisfied. To do that we study $K^k$ separately in different spatial regions, distinguishing also between indices $k\in K$ such that $2^k\le h^{1/2-\sigma}$ and $2^k>h^{1/2-\sigma}$. We hence introduce three smooth cut-off functions $\theta_s, \theta_b, \theta$, supported respectively for $|x|\le m\ll 1$, $|x|\ge M\gg 1$, $0<m'\le |x|\le M'<+\infty$, for some constants $m, m', M, M'$, and such that $\theta_s+\theta_b+\theta\equiv 1$. Denoting concisely by $A^k(x,\xi)$ the multiplier in , we split it as follows $$A^k(x,\xi) = A^k_s(x,\xi) + A^k_b(x,\xi)+A^k_1(x,\xi),$$ with $A^k_s(x,\xi):=A^k(x,\xi)\theta_s(x)$, $A^k_b(x,\xi):=A^k(x,\xi)\theta_b(x)$ and $A^k_1(x,\xi):=A^k(x,\xi)\theta(x)$. Let us consider $k\in K$ such that $h\lesssim 2^k\le h^{1/2-\sigma}$. According to the above decomposition we have that $$K^k(x,y)=K^k_s(x,y)+K^k_b(x,y)+K^k_1(x,y),$$ with clear meaning of kernels $K^k_s,K^k_b,K^k_1$. Let us first prove that $$\label{L1norm_Ks_Kb} \sup_x \int |K^k_s(x,y)| dy + \sup_x \int |K^k_b(x,y)| dy \lesssim 2^{kq}.$$ For $|x|\ll 1$ (resp. $|x|\gg 1$), $\big|\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\big|\gtrsim h^{-1/2+\sigma}|\xi|$ (resp. $\big|\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\big|\gtrsim h^{-1/2+\sigma}|\xi||x|\gtrsim h^{-1/2+\sigma}|\xi|$) so by lemma \[Lem : est on gamma for wave\] $$\label{derivative_xi_gamma} \left| \partial^\beta_\xi\Big[\gamma\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\Big]\right|\lesssim \sum_{j=0}^{|\beta|}h^{-j(\frac{1}{2}-\sigma)}\left\langle \frac{x|\xi|-\xi}{h^{1/2-\sigma}}\right\rangle^{-j}|b_{j-|\beta|}(\xi)| \lesssim |\xi|^{-|\beta|}.$$ Therefore $$\label{partial_beta_Aks} \left|\partial^\beta_\xi A^k_s(x, 2^k\xi)\right|\lesssim \sum_{|\beta_1|\le |\beta|} 2^{k|\beta|}|2^k\xi|^{-|\beta_1|}2^{-k(|\beta|-|\beta_1|)+kq}\mathds{1}_{|\xi|\sim 1} \lesssim 2^{kq}\mathds{1}_{|\xi|\sim 1},$$ so making a change of coordinates $\xi\mapsto 2^k\xi$ and some integration by parts we derive that $$|K^k_s(x,y)|\lesssim 2^{kq}(2^kh^{-1})^2\left\langle 2^kh^{-1} (x-y)\right\rangle^{-3},$$ for every $(x,y)\in\mathbb{R}^2\times \mathbb{R}^2$. The same argument applies to $K^k_b(x,y)$, hence taking the $L^1$ norm we obtain . As concerns kernel $K^k_1(x,y)$, we deduce from lemma \[Lem : est on gamma for wave\], the fact that $\theta_1(x)$ is supported for $|x|\sim 1$, and that $2^k\lesssim h^{1/2-\sigma}$, the following inequality: $$\left| \partial^\beta_\xi \Big[A^k_1\Big(\frac{x+y}{2}, 2^k\xi\Big)\Big]\right|\lesssim 2^{k|\beta|}\Big[2^{k(q-|\beta|)}+\sum_{j=1}^{|\beta|}h^{-j(\frac{1}{2}-\sigma)}|b_{j-|\beta|+q}(2^k\xi)|\Big]\lesssim 2^{kq}.$$ Performing a change of coordinates $\xi\mapsto 2^k\xi$ and making some integration by parts one finds that $$|K^k_1(x,y)|\lesssim 2^{kq}(2^kh^{-1})^2 \left\langle 2^k h^{-1}(x-y) \right\rangle^{-3}, \qquad \forall (x,y),$$ and consequently that $$\sup_x \int |K^k(x,y)|dy\lesssim 2^{kq}.$$ Summing up with , this gives us that $$\oph(A^k(x,\xi))=\oph(A^k_s(x,\xi))+\oph(A^k_b(x,\xi))+\oph(A^k_1(x,\xi))$$ is a bounded operator on $L^p$, for every $1\le p\le +\infty$, with norm $O(2^{kq})$. Let us now suppose that $k\in K$ is such that $h^{1/2-\sigma}< 2^k\le h^{-\sigma}$. From we have that $\widetilde{A}^k_s(x,\xi) = A^k_s\big(\frac{x}{\sqrt{h}},\sqrt{h}\xi\big)$ satisfies $$\left|\partial^\beta_\xi \widetilde{A}^k_s(x,\xi)\right| \lesssim \sum_{|\beta_1|\le |\beta|} h^{\frac{|\beta|}{2}}|\sqrt{h}\xi|^{-|\beta_1|} 2^{-k(|\beta|-|\beta_1|)+kq} \mathds{1}_{|\xi|\sim 2^k h^{-1/2}},$$ for every $(x,\xi)\in\mathbb{R}^2\times\mathbb{R}^2$, and hence $$\left| \partial^\beta_\xi \widetilde{A}^k_s(x,2^kh^{-1/2}\xi)\right|\lesssim \sum_{|\beta_1|\le |\beta|}2^{k|\beta|}|2^k\xi|^{-|\beta_1|}2^{-k(|\beta|-|\beta_1|)+kq}\mathds{1}_{|\xi|\sim1}\lesssim 2^{kq}\mathds{1}_{|\xi|\sim1}.$$ By making a change of coordinates $\xi\mapsto 2^kh^{-1/2}\xi$, some integrations by parts and using the above inequality, one can show that kernel $\widetilde{K}^k_s(x,y)$ associated to $\oph(\widetilde{A}^k_s(x,\xi))$, i.e. $$\widetilde{K}^k_s(x,y)=\frac{1}{(2\pi h)^2}\int e^{\frac{i}{h}(x-y)\cdot\xi} \widetilde{A}^k_s\Big(\frac{x+y}{2},\xi\Big) d\xi,$$ is such that $$|\widetilde{K}^k_s(x,y)|\lesssim 2^{kq}(2^kh^{-\frac{3}{2}})^2\left\langle 2^kh^{-\frac{3}{2}}(x-y)\right\rangle^{-3} \qquad\forall (x,y),$$ which implies that $sup_x \int |\widetilde{K}^k_s(x,y)| dy\lesssim 2^{kq}$. The same argument and result hold for $\widetilde{K}^k_b(x,y)$ so $\oph(A^k_s)$ and $\oph(A^k_b)$ verify the statement. The last thing to prove is that $\oph(A_1(x,\xi))\in\mathcal{L}(L^p)$ for every $1\le p\le +\infty$. Let $K^k_1(x,y)$ be its associated kernel, i.e. $$K^k_1(x,y) =\frac{1}{(2\pi h)^2}\int e^{\frac{i}{h}(x-y)\cdot\xi} \gamma\Big(\frac{\big(\frac{x+y}{2}\big)|\xi|-\xi}{h^{1/2-\sigma}}\Big) \psi(2^{-k}\xi)a\Big(\frac{x+y}{2}\Big)b_q(\xi) d\xi,$$ and assume, without loss of generality, that $\gamma(x)=\gamma(|x|^2)$. Set $$\frac{x+y}{2} = r[\cos\theta, \sin\theta],$$ with $m'\le r\le M'$ on the support of $\theta_1\big(\frac{x+y}{2}\big)$, and for fixed $r,\theta$ let $$\label{change_of_coordinates} \xi= \rho[\cos\theta, \sin\theta] + r\Omega [-\sin\theta, \cos\theta].$$ We immediately notice that $[\frac{\partial(\xi_1,\xi_2)}{\partial(\rho,\Omega)}]=r\sim 1$ and that $|\xi|^2=\rho^2+r^2\Omega^2$. Moreover, $$\Big|\Big(\frac{x+y}{2}\Big)|\xi|-\xi \Big|^2 = \left[r\sqrt{\rho^2 + r^2\Omega^2}-\rho\right]^2 + r^2\Omega^2.$$ If the support of $\gamma$ is of size $0<\alpha\ll 1$ sufficiently small, from the above equality and the fact that $|\xi|\sim 2^k$ on the support of $\psi(2^{-k}\xi)$, with $h^{1/2-\sigma}< 2^k\lesssim h^{-\sigma}$, we deduce that $$r\Omega \le \sqrt{\alpha}h^{1/2-\sigma} \quad \text{and} \quad |\rho|\sim |\xi|\sim 2^k \quad \text{and} \quad \frac{r\Omega}{|\rho|}\le \sqrt{\alpha}.$$ Consequently $$\alpha h^{1-2\sigma}\ge \left[r\sqrt{\rho^2 + r^2\Omega^2}-\rho\right]^2 \gtrsim \rho^2|r-1|^2.$$ The above left inequality implies that $\rho>0$, inferring so the right one. Moreover $$\begin{split} \alpha h^{1-2\sigma}\ge \left[r\sqrt{\rho^2 + r^2\Omega^2}-\rho\right]^2 +r^2\Omega^2 &= \rho^2 \left[(r-1) + r \left[\sqrt{1+\frac{r^2\Omega^2}{\rho^2}}-1\right]\right]^2 + r^2\Omega^2\\ & = \rho^2|r-1|^2 + r^2\Omega^2\left[1+ a(r,\Omega,\rho)\right], \end{split}$$ with $a(r,\Omega,\rho)$ bounded such that, for any $l,m,n\in\mathbb{N}$, $$|\partial^l_r \partial^m_\Omega \partial^n_\rho a(r,\Omega, \rho)|=O(\rho^{-(m+n)}).$$ If $$\Gamma_h:=\gamma\Big(\frac{ \rho^2|r-1|^2}{h^{1-2\sigma}} + \frac{r^2\Omega^2}{h^{1-2\sigma}}\left[1+ a(r,\Omega,\rho)\right]\Big) \psi(2^{-k}\sqrt{\rho^2+r^2\Omega^2}) a(r,\theta)b_q(\rho),$$ from all the observations made above along with the fact that $h^{-1/2+\sigma}\lesssim \rho^{-1}$ we deduce that, for any $m,n\in\mathbb{N}$, $$\label{deriv_Gammah} \left|\partial^m_\rho \Gamma_h\right| =O(2^{kq}\rho^{-m}) \quad \text{and} \quad \left|\partial^n_\Omega \Gamma_h\right|=O(2^{kq}\rho^{-n}).$$ With the change of coordinates considered in , and setting $w:=x-y$, $e_\theta:=[\cos\theta, \sin\theta]$, kernel $K^k_1(x,y)$ transforms into $$\frac{1}{(2\pi h)^2}\int e^{\frac{i}{h}\rho w\cdot e_\theta + \frac{i}{h}r\Omega w\cdot e^\perp_\theta}\Gamma_h \, rd\rho d\Omega$$ and is restricted to $|\rho|\sim 2^k$, $|\Omega|\lesssim h^{1/2-\sigma}$, so by making some integrations by parts, using , and reminding that $|r-1|\ll 2^{-k}h^{1/2-\sigma}\ll 1$ on the support of $\Gamma_h$, we find that, for any $N\in\mathbb{N}$, $$|K^k_1(x,y)|\lesssim h^{-\frac{3}{2}-\sigma}2^k \left\langle \frac{2^k}{h}w\cdot e_\theta \right\rangle^{-N} \left\langle\frac{2^k}{h}w\cdot e^\perp_\theta \right\rangle^{-N} \mathds{1}_{||\frac{x+y}{2}|-1|\ll 1}.$$ Now, as $w=(x-y)$, $e_\theta = \frac{x+y}{|x+y|}$, and $|x+y|=2r\sim 1$ on the support of $\Gamma_h$, we have that $|w\cdot e_\theta|\sim ||x|^2-|y|^2|$, $|w\cdot e^\perp_\theta|\sim |(x+y)(x+y)^\perp|\sim 2|x\cdot y^\perp| = 2|x_1 y_2-x_2y_1|$, and consequently $$|K^k_1(x,y)|\lesssim h^{-\frac{3}{2}-\sigma}2^{k(1+q)} \left\langle \frac{2^k}{h}\big||x|^2-|y|^2\big| \right\rangle^{-N} \left\langle\frac{2^k}{h}(x_1y_2 - x_2y_1)\right\rangle^{-N} \mathds{1}_{||\frac{x+y}{2}|-1|\ll 1}.$$ Successively, taking the $L^1(dy)$ norm of $K^k_1(x,y)$ and using the above estimate we find that: $\bullet$ if $|x|\ll |y|$ or $|x|\gg |x|$, $$\left\langle \frac{2^k}{h}\big||x|^2-|y|^2\big| \right\rangle^{-N} \mathds{1}_{||\frac{x+y}{2}|-1|\ll 1}\lesssim h^{N(\frac{1}{2}+\sigma)},$$ as follows from the fact that $h2^{-k}< h^{1/2+\sigma}$. We obtain that $$\sup_x\int |K^k_1(x,y)|dy \lesssim h^{-\frac{3}{2}}2^{k(1+q)} h^{N(\frac{1}{2}+\sigma)} \lesssim 1$$ by taking $N\in\mathbb{N}$ sufficiently large (e.g. $N>3$) and $\sigma>0$ small. $\bullet$ if $|x|\sim |y|$, we deduce that $|x|\ge c>0$ from the fact that $\big| \big|\frac{x+y}{2}\big|-1\big|\le \sqrt{\alpha}h^{1/2-\sigma}2^{-k}$ on the support of $\Gamma_h$. Without loss of generality we can assume that $x=\lambda e_1$ (this always being possible by making a rotation) and $|\lambda|\ge c>0$. If $w:=x+y$, $$|x|^2-|y|^2 = w\cdot(x-y) = w\cdot (2x-w)=w\cdot (2\lambda e_1 -w) = 2\lambda w_1-w_1^2-w_2^2,$$ and then $$\frac{\big||x|^2-|y|^2\big|}{h} = -\frac{(w_1-\lambda)^2-\lambda^2}{h}+ \Big(\frac{w_2}{\sqrt{h}}\Big)^2$$ while $$x_1y_2-x_2y_1 = \lambda w_2.$$ This implies that $$|K^k_1(x,y)|\lesssim h^{-\frac{3}{2}-\sigma}2^{k(1+q)}\left\langle \frac{2^k}{h}\left((w_1-\lambda)^2-\lambda^2\right) \right\rangle^{-N} \left\langle \frac{2^k}{h} w_2\right\rangle^{-N}.$$ Since $\int |K^k_1(x,y)| dy = \int |K^k_1(x,y)| dw$, from the above estimate (with a fixed $N\in\mathbb{N}$ sufficiently large) this integral is bounded by $2^{kq}$ when restricted to $|x|\sim |y|$. Indeed, when the integral is taken in a neighbourhood of $w_1=0$ or $w_1=2\lambda$, $(w_1-\lambda^2)-\lambda^2$ can be considered as the variable of integration, and by a change of coordinates along with the fact that $2^{-k}<h^{-1/2+\sigma}$ one deduces that $$\int_{U_0\cup U_{2\lambda}} |K^k_1(x,y)| dw \lesssim h^{-\frac{3}{2}-\sigma}2^{k(1+q)}h^22^{-2k}\lesssim 2^{kq},$$ where $U_0$ (resp. $U_{2\lambda}$) is a neighbourhood of $w_1=0$ (resp. of $w_1=2\lambda$). Outside of $U_0\cup U_{2\lambda}$, $$\left\langle \frac{2^k}{h}\left((w_1-\lambda)^2-\lambda^2\right) \right\rangle^{-N}\lesssim (h2^{-k})^N \langle w_1\rangle^{-N}\lesssim h^{N(\frac{1}{2}+\sigma)}\langle w_1\rangle^{-N},$$ so $$\int_{(U_0\cup U_{2\lambda})^\complement} |K^k_1(x,y)| dw \lesssim h^{-\frac{3}{2}-\sigma}2^{k(1+q)}h2^{-k}h^{N(\frac{1}{2}+\sigma)}\lesssim 2^{kq}.$$ This finally proves that also $\oph(A^k_1(x,\xi))$ is a bounded operator on $L^p$ with norm $O(2^{kq})$. Let us introduce the Euclidean rotation in the semi-classical setting $$\label{def_Omega_h} \Omega_h :=x_1 hD_2 - x_2 hD_1 = \oph(x_1\xi_2-x_2\xi_1).$$ \[Prop : continuity of Op(gamma1):X to L2\] Under the same assumptions as in proposition \[Prop : continuity Op(gamma) L2 to L2\], with $\gamma$ replaced by $\gamma_1$, we have that for any $w\in L^2(\mathbb{R}^2)$ such that $\Omega_h w\in L^2_{loc}(\mathbb{R}^2)$ $$\label{est x-Linfty of Op(gamma1)} \Big\| \oph\Big(\gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)\Big)w\Big\|_{L^\infty} \lesssim 2^{kp} h^{-\frac{1}{2} -\sigma}\left(\|w\|_{L^2}+ \|\theta_0\Omega_hw\|_{L^2}\right),$$ where $\theta_0$ is a smooth function supported in some annulus centred in the origin. We prove the statement distinguishing between three spatial regions. For that, we introduce three cut-off functions: $\theta_s(x)$ supported for $|x|\le m \ll 1$; $\theta_b(x)$ supported for $|x| \ge M'\gg 1$; $\theta(x)$ supported for $m'\le |x| \le M'$, for some $0<m'\ll 1, M\gg 1$, such that $\theta_s + \theta_b + \theta \equiv 1$. We define respectively $A^k_s(x,\xi):= \gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi) \theta_s(x)$, $A^k_b(x,\xi):=\gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)\theta_b(x)$, and $A^k(x,\xi):=\gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)\theta(x)$, so that $$\gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi) = A^k_s(x,\xi) + A^k_b(x,\xi) + A^k(x,\xi).$$ The fact that $\oph(A^k_s), \oph(A^k_b)\in\mathcal{L}(L^2;L^\infty)$ and their norm is a $O(2^{kp}h^{-1/2-\sigma})$ follows from lemmas \[Lemma on inequalities for Op(A)\] and \[Lem : est on gamma for wave\]. Indeed, when $|x|\ll 1$ (resp. $|x|\gg 1$) we have that $\big|\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\big|\gtrsim h^{-1/2+\sigma}|\xi|$ (resp. $\big|\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\big|\gtrsim h^{-1/2+\sigma}|\xi||x|\gtrsim h^{-1/2+\sigma}|\xi|$), so from lemma \[Lem : est on gamma for wave\] we derive that $$\left| \partial^\alpha_x \partial^\beta_\xi\Big[\gamma_1\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\Big]\right|\lesssim \sum_{j=0}^{|\beta|}h^{-(|\alpha|+j)(\frac{1}{2}-\sigma)}\left| \frac{x|\xi|-\xi}{h^{1/2-\sigma}}\right|^{-1-|\alpha|-j}|b_{|\alpha|+j-|\beta|}(\xi)| \lesssim h^{\frac{1}{2}-\sigma}|\xi|^{-1-|\beta|}.$$ Consequently, as $2^{-k}h\le 1$, we deduce that $|\partial^\alpha_x \partial^\beta_\xi \big[A^k_s(\frac{x+y}{2}, h\xi)\big]|\lesssim 2^{kp}h^{-1/2-\sigma}|\xi|^{-1}$ for any $\alpha,\beta\in\mathbb{N}^2$. Therefore $$\left\|\partial_y^{\alpha}\partial_\xi^{\beta}\Big[A^k_s\Big(\frac{x+y}{2},h\xi\Big)\Big]\right\|_{L^2(d\xi)} \lesssim 2^{kp}h^{-\frac{1}{2}-\sigma} \left(\int_{|\xi|\sim 2^kh^{-1}}| \xi|^{-2}d\xi\right)^{\frac{1}{2}}\lesssim 2^{kp}h^{-\frac{1}{2}-\sigma}.$$ The same holds for $A^k_b(x,\xi)$ so, injecting these estimates in inequality , we derive that $\|\oph(A^k_s(x,\xi))w\|_{L^\infty} + \|\oph(A^k_b(x,\xi))w\|_{L^\infty} \le C 2^{kp}h^{-\frac{1}{2}-\sigma}\|w\|_{L^2}$. A different analysis is needed for $\oph(A^k(x,\xi))w$, since it is no longer true that there exists a positive constant $C$ such that $|x|\xi|-\xi|\ge C |\xi|$ on the support of $A^k(x,\xi)$. In this case we exploit the fact that $A^k(x,\xi)$ is supported in an annulus to perform a change of variables. If $\theta_0\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$ is a cut-off function equal to 1 on the support of $\theta$ we have that, for any $N\in\mathbb{N}$, $A^k(x,\xi) = \theta_0(x)\sharp A^k(x,\xi) + r_N^k(x,\xi)$ by means of proposition \[Prop: a sharp b\], where $$\begin{gathered} r^k_N(x,\xi) = \left(\frac{h}{2i}\right)^N \frac{N}{(\pi h)^4} \sum_{|\alpha|= N} \frac{(-1)^{|\alpha|}}{\alpha!}\int e^{\frac{2i}{h}(\eta\cdot z - y\cdot\zeta)} \int_0^1 \partial^\alpha_x \theta_0(x+tz)(1-t)^{N-1} dt\\ \times (\partial^\alpha_\xi A^k)(x, \xi+\eta) \,dy dz d\eta d\zeta.\end{gathered}$$ If we take $N$ sufficiently large it turns out that the quantization of $r^k_N$ satisfies a better estimate than . Indeed, using lemma \[Lem : est on gamma for wave\] and integrating in $dyd\zeta$, it can be rewritten as $$\begin{gathered} \label{rkN in prop 1.3} r^k_N(x,\xi) = \sum_{j\le N} \frac{h^{N-j(\frac{1}{2}-\sigma)}}{(\pi h)^2}\int e^{\frac{2i}{h}\eta\cdot z} \int_0^1 \theta_0(x+tz)(1-t)^{N-1} dt \\ \times \gamma_{1+j}\Big(\frac{x|\xi+\eta|-(\xi+\eta)}{h^{1/2-\sigma}}\Big)\psi(2^{-k}(\xi+\eta))\theta_j(x)a(x)b_{p+j-N}(\xi + \eta)\, dz d\eta,\end{gathered}$$ for some new functions $\theta_0, \gamma_{1+j}, \psi,\theta_j, a, b_{p+j-N}$. As it is compactly supported in $x$, by lemma \[Lemma on inequalities for Op(A)\] there is a new cut-off function (that we still call $\theta$) such that $$|\oph(r^k_N(x,\xi))w| \lesssim \|w\|_{L^2}\int \Big|\theta\Big(\frac{x+y}{2}\Big)\Big|\sum_{|\alpha'|\le 3}\Big\|\partial^{\alpha'}_y\Big[ r^k_N\Big(\frac{x+y}{2}, h\xi\Big)\Big]\Big\|_{L^2(d\xi)}dy.$$ One can check that the action of $\partial^{\alpha'}_y$ on $r^k_N(\frac{x+y}{2}, h\xi)$ makes appear factors $(h^{-1/2+\sigma}h|\xi+\eta|)^i$, for $i\le |\alpha'|$, without changing the underlining structure of $r^k_N$, and these are bounded by $(h^{-1/2+\sigma}2^k)^i$ on the support of $\psi(2^{-k}h(\xi+\eta))$. After a change of variables $\eta \mapsto h\eta$ in , we use that $e^{2i \eta\cdot z}=\left(\frac{1-2i\eta\cdot\partial_z}{1+4|\eta|^2}\right)^3\left(\frac{1-2i z\cdot \partial_\eta}{1+4|z|^2}\right)^3e^{2i \eta\cdot z}$, integrate by parts, apply Young’s inequality for convolutions, and fix $N>7$, in order to deduce the following chain of inequalities: $$\begin{gathered} \left\|\partial^{\alpha'}_y r^k_N\Big(\frac{x+y}{2}, h\xi\Big)\right\|_{L^2(d\xi)}^2\\ \lesssim \sum_{i\le |\alpha'|, j\le N} h^{2N-2j(\frac{1}{2}-\sigma)}\big( h^{-\frac{1}{2}+\sigma}2^k\big)^{2i} 2^{2k(p+j-N)}\int d\xi \left|\int \langle z \rangle^{-3} \langle \eta\rangle^{-3} |\psi(2^{-k}h(\xi+\eta))|dzd\eta\right|^2 \\ \lesssim \sum_{i\le |\alpha'|, j\le N} h^{2N-2j(\frac{1}{2}-\sigma)}\big( h^{-\frac{1}{2}+\sigma}2^k\big)^{2i} 2^{2k(p+j-N)} \int |\psi(2^{-k}h\xi)|^2 d\xi \\ \lesssim \sum_{i\le |\alpha'|, j\le N} h^{2N-2j(\frac{1}{2}-\sigma)}\big( h^{-\frac{1}{2}+\sigma}2^k\big)^{2i} 2^{2k(p+j-N)}\big( h^{-1}2^k\big)^2 \lesssim 2^{2kp},\end{gathered}$$ and that $\|\oph(r^k_N)\|_{\mathcal{L}(L^2;L^\infty)}\lesssim 2^{kp}$. We can then focus on the analysis of the $L^\infty$ norm of $\theta_0(x)\oph(A^k(x,\xi))w$. In polar coordinates $x=\rho e^{i\alpha}$ operator $\Omega_h$ reads as $D_\alpha$, so using the classical one-dimensional Sobolev injection with respect to variable $\alpha$, the one-dimensional semi-classical Sobolev injection with respect to variable $\rho$, and successively returning back to coordinates $x$, we deduce that $$\begin{aligned} \big| \theta_0(x)\oph(A^k(x,\xi))w \big| & \lesssim h^{-\frac{1}{2}} \Big[\|\oph(A^k)w\|_{L^2(dx)} + \|\oph(\xi)\oph(A^k)w\|_{L^2(dx)} \\ & \hspace{1cm}+ \|\Omega_h \theta_0 \oph(A^k)w\|_{L^2(dx)} + \|\oph(\xi)\Omega_h \theta_0 \oph(A^k)w\|_{L^2(dx)}\Big] \\ & \lesssim 2^{kp}h^{-\frac{1}{2}-\sigma}[\|w\|_{L^2} + \|\theta_0\Omega_h w\|_{L^2}]\,.\end{aligned}$$ The latter of above inequalities is derived observing that the commutator between $\Omega_h$ and $\oph(A^k)$ is a semi-classical pseudo-differential operator whose symbol is linear combination of terms of the form $$\gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)\theta(x)b_p(\xi),$$ for some new $\gamma_1, \psi, a, \theta, b_p$, and from the fact that operators $\oph(A^k(x,\xi))$, $\oph(\xi A^k(x,\xi))$ are bounded on $L^2$ (see proposition \[Prop : continuity Op(gamma) L2 to L2\]), with norm $O(2^{kp})$, $O(2^{k(p+1)})$ respectively, and that $2^k\le h^{-\sigma}$. \[Prop : Linfty est of integral remainders\] Under the same hypothesis as proposition \[Prop: L2 est of integral remainders\], $\oph(I^k_{p,q}(x,\xi))$ and $\oph(J^k_{p,q}(x,\xi))$ are bounded operators from $L^\infty$ to $L^2$, with $$\label{norm_Linfty-L2_OP(Ikpq)} \left\|\oph(I^k_{p,q}(x,\xi))\right\|_{\mathcal{L}(L^2;L^\infty)} + \left\|\oph(J^k_{p,q}(x,\xi))\right\|_{\mathcal{L}(L^2;L^\infty)} \lesssim \sum_{i\le 6} 2^{k(p+q)} (h^{-\frac{1}{2}+\sigma}2^k)^{i} (h^{-1}2^k).$$ The same result holds if $q=0$ and $\widetilde{\psi}(2^{-k}\xi)b_q(\xi)\equiv 1$. As in proposition \[Prop: L2 est of integral remainders\], we prove the statement only for $\oph(I^k_{p,q})$, leaving to the reader to check that the result is true also for $\oph(J^k_{p,q})$. Let $w\in L^2$. After lemma \[Lemma on inequalities for Op(A)\] we should prove that $\left\|\partial^\alpha_y \partial^\beta_\xi \Big[I^k_{p,q}(\frac{x+y}{2}, h\xi)\Big]\right\|_{L^2(d\xi)}$ is estimated by the right hand side of , for any $|\alpha|, |\beta|\le 3$. A change of variables $\eta \mapsto h\eta$, $\zeta \mapsto h\zeta$ allows us to write $I^k_{p,q}(\frac{x+y}{2}, h\xi)$ as $$\begin{gathered} \frac{1}{\pi^4} \int e^{2i (\eta\cdot z - y'\cdot\zeta)} \left[ \int_0^1 \Big(\gamma\big(h^{\frac{1}{2}+\sigma}(x|\xi|-\xi)\big)\psi(2^{-k}h\xi)a(x)b_p( h\xi)\Big)|_{(\frac{x+y}{2}+tz, \xi+t\zeta)} f(t)dt \right. \\ \left. \times \widetilde{\psi}(2^{-k}h(\xi + \eta))a'\Big(\frac{x+y}{2}+y'\Big)b_q( h(\xi+\eta))\right] dy' dz d\eta d\zeta.\end{gathered}$$ We observe that, while on the one hand the action of $\partial^\alpha_y$ on the above integral makes appear a factor $(h^{-\frac{1}{2}+\sigma} |h(\xi+t\zeta)|)^i$, with $i\le |\alpha|$, on the other hand that of $\partial^\beta_\xi$ has basically no effect on the $L^2$ norm that we want to estimate as one can check using lemma \[Lem : est on gamma for wave\] and the fact that $2^{-k}h\le 1$. With this in mind, we can reduce to the study of the $L^2(d\xi)$ norm of an integral function as$$\begin{gathered} \sum_{i \le 3} (h^{-\frac{1}{2}+\sigma}2^k)^i \int e^{2i (\eta\cdot z - y'\cdot\zeta)} \left[ \int_0^1 \Big(\gamma\Big(h^{\frac{1}{2}+\sigma}(x|\xi|-\xi)\Big)\psi(2^{-k}h\xi)a(x)b_p(h\xi)\Big)|_{(\frac{x+y}{2}+tz, \xi +t\zeta)}f(t) dt\right. \\ \times \widetilde{\psi}(2^{-k}h(\xi + \eta))a'\Big(\frac{x+y}{2}+y'\Big)b_q(h(\xi + \eta))\Big] dy' dz d\eta d\zeta,\end{gathered}$$for some new functions $\gamma, \psi, a, b_p, \widetilde{\psi}, a', b_q$, with the same properties as their previous homonyms. We use that $$e^{2i (\eta z - y'\cdot\zeta)}=\left(\frac{1+2iy'\cdot\partial_\zeta}{1+4|y'|^2}\right)^3\left(\frac{1-2i\eta\cdot\partial_z}{1+4|\eta|^2}\right)^3\left(\frac{1-2i z\cdot \partial_\eta}{1+4|z|^2}\right)^3\left(\frac{1+2i\zeta\cdot\partial_{y'}}{1+4|\zeta|^2}\right)^3e^{2i (\eta\cdot z - y'\cdot\zeta)}$$ and make some integration by parts to obtain the integrability in $dy'dz d\eta d\zeta$, up to new factors $(h^{-\frac{1}{2}+\sigma} |h(\xi+t\zeta)|)^{j}$, with $j\le 3$, coming out from the derivation of the integrand with respect to $z$. Then, using that functions $h^jb_{p-j}(h(\xi+t\zeta))$ (resp. $h^j b_{q-j}(h(\xi+\eta))$), $j\le 3$, appearing from the derivation of $b_p(h(\xi+t\zeta))$ with respect to $\zeta$ (resp. the derivation of $b_q(h(\xi+\eta))$ with respect to $\eta$), are such that $|h^jb_{p-j}(h(\xi+t\zeta))|\le h^j2^{k(p-j)}\lesssim 2^{kp}$ on the support of $\psi(2^{-k}h(\xi+t\zeta))$ (resp. $|h^jb_{q-j}(h(\xi +\eta))|\le 2^{kq}$ on the support of $\widetilde{\psi}(2^{-k}h(\xi+\eta))$), and the fact that $$\Big\| \int \langle\eta\rangle^{-3} |\widetilde{\psi}(2^{-k}h(\xi + \eta))|d\eta\Big\|_{L^2(d\xi)}\le \|\widetilde{\psi}(2^{-k}h\cdot)\|_{L^2} \lesssim h^{-1}2^k,$$ we obtain the result of the statement. The last part of the statement can be proved following an analogous scheme, after having previously made an integration in $dzd\eta$ (or in $dyd\zeta$ if dealing with $J^k_{p,0}$). \[Lem : remainder r\^k\_N\] Let $\sigma>0$ be sufficiently small, $k\in K$ with $K$ given by and $p,q\in\mathbb{N}$. Let also $\psi, \widetilde{\psi}\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$, $a(x)$ be either a smooth compactly supported function or $a\equiv 1$, and $f\in C(\mathbb{R})$. For a fixed integer $N> 2(p+q)+9$ we define $$\begin{gathered} \label{integral rkN} r_{N,p}^k(x,\xi) := \frac{h^N}{(\pi h)^4} \sum_{|\alpha|=N} \int e^{\frac{2i}{h}(\eta\cdot z - y\cdot \zeta)} \left[\int_0^1 \partial^{\alpha}_x\Big( \gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)\Big)|_{(x+tz,\xi+t\zeta)} \right.\\ \times f(t)dt\Big] \partial^{\alpha}_\xi\big(b_q(\xi)\widetilde{\psi}(2^{-k}\xi)\big)|_{(x+y,\xi + \eta)}\, dy dz d\eta d\zeta,\end{gathered}$$and $$\begin{gathered} \label{integral rtilde kN} \widetilde{r}^k_{N,p}(x,\xi) := \frac{h^N}{(\pi h)^4} \sum_{|\alpha_1|+|\alpha_2|=N} \int e^{\frac{2i}{h}(\eta\cdot z - y\cdot \zeta)} \left[\int_0^1 \partial^{\alpha_1}_x\partial^{\alpha_2}_\xi\Big( \gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)\Big)|_{(x+tz,\xi+t\zeta)}\right. \\ \times f(t)dt \Big] \partial^{\alpha_2}_x \partial^{\alpha_1}_\xi\big(x_n b_q(\xi)\widetilde{\psi}(2^{-k}\xi)\big)|_{(x+y,\xi + \eta)}\, dy dz d\eta d\zeta\,.\end{gathered}$$ Then $$\label{est L2 and Linfty rkN r'kN} \|\oph(r^k_{N,p})\|_{\mathcal{L}(L^2)} +\|\oph(\widetilde{r}^k_{N,p})\|_{\mathcal{L}(L^2)} + \|\oph(r^k_{N,p})\|_{\mathcal{L}(L^2;L^\infty)}+ \|\oph(\widetilde{r}^k_{N,p})\|_{\mathcal{L}(L^2;L^\infty)} \lesssim h^{p+q}.$$ We remind definition of integral $I^k_{p,q}(x,\xi)$ for general $k\in K, p,q\in\mathbb{Z}$. After an explicit development of the derivatives appearing in we find that $r^k_{N,p}(x,\xi)$ may be written as $$\sum_{j\le N}h^{N-j(\frac{1}{2}-\sigma)}I^k_{p+j,q-N}(x,\xi)$$ where $\gamma$ is replaced with $\gamma_1$ and $a'\equiv 1$ in $I^k_{p+j,q-N}$. Propositions \[Prop: L2 est of integral remainders\] and \[Prop : Linfty est of integral remainders\], combined with the fact that $h\le 2^k\le h^{-\sigma}$, imply respectively that $$\begin{aligned} &\|\oph(r^k_{N,p})\|_{\mathcal{L}(L^2)} \lesssim \sum_{j\le N}h^{N-j(\frac{1}{2}-\sigma)}2^{k(p+j+q-N)}\\ &\lesssim \sum_{\substack{j\le N \\ p+j+q\le N}} h^{N-j(\frac{1}{2}-\sigma) +p+j+q-N} + \sum_{\substack{j\le N \\ p+j+q> N}} h^{N-j(\frac{1}{2}-\sigma) -\sigma(p+j+q-N)} \lesssim h^{p+q}\end{aligned}$$ and $$\begin{gathered} \|\oph(r^k_{N,p})\|_{\mathcal{L}(L^2;L^\infty)}\lesssim \sum_{i\le 6, j\le N}h^{N-j(\frac{1}{2}-\sigma)}2^{k(p+j+q-N)}(h^{-\frac{1}{2}+\sigma}2^k)^i(h^{-1}2^k)\\ \lesssim \sum_{\substack{i\le 6, j\le N \\ p+i+j+q\le N-1}} h^{N-1- (i+j)(\frac{1}{2}-\sigma)+p+i+j+q-N+1} + \sum_{\substack{i\le 6, j\le N \\ p+i+j+q> N-1}} h^{N-1 - (i+j)(\frac{1}{2}-\sigma)-\sigma(p+i+j+q-N+1)} \\ \lesssim h^{p+q},\end{gathered}$$ as $N>2(p+q)+9$. As regards , we first observe that index $\alpha_2$ is such that $|\alpha_2|\le 1$ since $x_nb_q(\xi)\widetilde{\psi}(2^{-k}\xi)$ is linear in $x_n$. An explicit development of derivatives in , combined with lemma \[Lem : est on gamma for wave\], shows that $\widetilde{r}^k_{N,p}(x,\xi)$ splits into two contributions: $$\begin{aligned} J_0(x,\xi) = \frac{h^N}{(\pi h)^4}\sum_{i\le N} h^{-i(\frac{1}{2}-\sigma)} \int e^{\frac{2i}{h}(\eta\cdot z -y\cdot\zeta)}\int_0^1\Big( \gamma_{1+i}\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_{p+i}(\xi)\Big)|_{(x+tz, \xi+t\zeta)} f(t)dt\\ \times (x_n+y_n)b_{q-N}(\xi+\eta)\widetilde{\psi}(2^{-k}(\xi+\eta))\, dydzd\eta d\zeta,\end{aligned}$$ for some new functions $a, \psi, \widetilde{\psi}$ and clear meaning for $\gamma_i, b_{p+i}$, $b_{q-N}$, coming out when $|\alpha_2|=0$; $$\begin{gathered} J_1(x,\xi) = \frac{h^N}{(\pi h)^4}\sum_{i\le N-1, j\le 1} h^{-(i+j)(\frac{1}{2}-\sigma)} \int e^{\frac{2i}{h}(\eta\cdot z -y\cdot\zeta)}\\ \times \int_0^1\Big( \gamma_{1+i+j}\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_{p+i+j-1}(\xi)\Big)|_{(x+tz, \xi+t\zeta)} f(t)dt \\ \times b_{q-N+1}(\xi+\eta)\widetilde{\psi}(2^{-k}(\xi+\eta))\, dydzd\eta d\zeta,\end{gathered}$$for some new other $a, \psi, \widetilde{\psi}$, corresponding instead to $|\alpha_2|=1$. One has that $$J_1(x,\xi) = \sum_{i\le N-1, j\le 1}h^{N - (i+j)(\frac{1}{2}-\sigma)}I^k_{p+i+j-1,q-N+1}(x,\xi),$$ with $\gamma$ replaced with $\gamma_1$ and $a'\equiv 1$, so propositions \[Prop: L2 est of integral remainders\] and \[Prop : Linfty est of integral remainders\], along with the fact that $N>2(p+q)+9$, imply $$\| \oph(J_1(x,\xi))\|_{\mathcal{L}(L^2)} \lesssim \sum_{i\le N-1, j\le 1} h^{N-(i+j)(\frac{1}{2}-\sigma)}2^{k(p+i+j+q-N)} \lesssim h^{p+q},$$ $$\| \oph(J_1(x,\xi))\|_{\mathcal{L}(L^2; L^\infty)} \lesssim \sum_{\substack{i\le N-1, j\le 1\\ l\le 6}} h^{N-(i+j)(\frac{1}{2}-\sigma)}2^{k(p+i+j+q-N)}(h^{-\frac{1}{2}+\sigma}2^k)^l (h^{-1}2^k) \lesssim h^{p+q}.$$ In order to derive the same estimates for $J_0(x,\xi)$ we split the sum $x_n+y_n$ and analyse separately the two out-coming integrals, that we denote $J_{0,x}(x,\xi), J_{0,y}(x,\xi)$. In the latter one, we use that $y_ne^{-\frac{2i}{h}y\cdot\zeta} = -\frac{h}{2i}\partial_{\zeta_n} e^{-\frac{2i}{h}y\cdot\zeta}$ and successively integrate by parts in $d\zeta_n$ obtaining, with the help of lemma \[Lem : est on gamma for wave\], that $$\begin{gathered} \label{integral J0y} J_{0,y}(x,\xi) = \sum_{i\le N, j\le 1}h^{N+1-(i+j)(\frac{1}{2}-\sigma)}\int e^{\frac{2i}{h}(\eta\cdot z - y\cdot\zeta)} \\ \times \int_0^1 \Big( \gamma_{1+i+j}\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_{p+i+j-1}(\xi)\Big)|_{(x+tz, \xi +t\zeta)}f(t) dt \\ \times b_{q-N}(\xi-\eta)\widetilde{\psi}(2^{-k}(\xi+\eta))\ dydzd\eta d\zeta\end{gathered}$$ for some new functions $a, \psi, \widetilde{\psi}, f$. Again by propositions \[Prop: L2 est of integral remainders\], \[Prop : Linfty est of integral remainders\] and the fact that $h\le 2^k\le h^{-\sigma}$, $N>2(p+q)+9$, we deduce that: \[est J0y\] $$\| \oph(J_{0,y}(x,\xi))\|_{\mathcal{L}(L^2)} \lesssim \sum_{i\le N, j\le 1}h^{N+1-(i+j)(\frac{1}{2}-\sigma)}2^{k(p+i+j+q-N-1)} \lesssim h^{p+q},$$ $$\| \oph(J_{0,y}(x,\xi))\|_{\mathcal{L}(L^2; L^\infty)} \lesssim \sum_{\substack{i\le N, j\le 1\\ l\le 6}}h^{N+1-(i+j)(\frac{1}{2}-\sigma)}2^{k(p+i+j+q-N-1)} (h^{-\frac{1}{2}-\sigma}2^k)^l (h^{-1}2^k) \lesssim h^{p+q}.$$ In $J_{0,x}(x,\xi)$ we first integrate in $dyd\zeta$ and then we split the occurring integral into two other contributions, called $J_{0, x+tz}(x,\xi), J_{0,tz}(x,\xi)$, by writing $x_n= (x_n+tz_n) - tz_n$. Similarly to what done above, we use that $z_n e^{\frac{2i}{h}\eta\cdot z} = \frac{h}{2i}\partial_{\eta_n} e^{\frac{2i}{h}\eta\cdot z}$ in $J_{0,tz}$, and successively integrate by parts in $d\eta_n$: as $2^{-k}h\le 1$, we obtain that $J_{0, tz}$ has the same form as for some new $b_{q-N},\widetilde{\psi}$, and verifies . Finally, using that $x_n+tz_n = h^{\frac{1}{2}-\sigma}\big(\frac{(x_n+tz_n)|\xi| - \xi_n}{h^{1/2-\sigma}}\big)|\xi|^{-1} + \xi_n |\xi|^{-1}$, we derive that $$\begin{split} & J_{0, x+tz}(x,\xi) \\ & = \sum_{i\le N} h^{N-(i-1)(\frac{1}{2}-\sigma)}\int e^{\frac{2i}{h}\eta\cdot z}\int \Big(\gamma_{i}\Big(\frac{x|\xi| -\xi}{h^{1/2-\sigma}}\Big) \psi(2^{-k}\xi) a(x)b_{p+i-1}(\xi)\Big)|_{(x+tz, \xi)} f(t) dt \\ & \hspace{10cm} \times b_{q-N}(\xi+\eta) \widetilde{\psi}(2^{-k}(\xi +\eta)) dz d\eta, \\ & + \sum_{i\le N} h^{N-i(\frac{1}{2}-\sigma)}\int e^{\frac{2i}{h}\eta\cdot z}\int \Big(\gamma_{1+i}\big(\frac{x|\xi| -\xi}{h^{1/2-\sigma}}\Big) \psi(2^{-k}\xi) a(x)b_{p+i}(\xi)\Big)|_{(x+tz, \xi)}f(t) dt \\ & \hspace{10cm}\times b_{q-N}(\xi+\eta) \widetilde{\psi}(2^{-k}(\xi +\eta)) dz d\eta, \end{split}$$ so by propositions \[Prop: L2 est of integral remainders\] and \[Prop : Linfty est of integral remainders\] $$\|\oph(J_{0,x+tz}(x,\xi))\|_{\mathcal{L}(L^2)} \lesssim \sum_{i\le N} h^{N-i(\frac{1}{2}-\sigma)}2^{k(p+i+q -N)}\lesssim h^{p+q},$$ $$\|\oph(J_{0,x+tz}(x,\xi))\|_{\mathcal{L}(L^2; L^\infty)} \lesssim \sum_{i\le N, l\le 3} h^{N-i(\frac{1}{2}-\sigma)}2^{k(p+i+q -N)}(h^{-\frac{1}{2}+\sigma}2^k)^l(h^{-1}2^k)\lesssim h^{p+q}.$$ That concludes the proof as $\widetilde{r}^k_{N,p} = J_{0, x+tz}+J_{0, tz} + J_{0,y} + J_1$. We introduce the following operator: $$\label{def Mj} \mathcal{M}_j:=\frac{1}{h}\oph(x_j|\xi| - \xi_j), \quad j=1,2$$ and use the notation $\|\mathcal{M}^\gamma w\| = \|\mathcal{M}^{\gamma_1}_1 \mathcal{M}^{\gamma_2}_2 w\|$ for any $\gamma=(\gamma_1,\gamma_2)\in\mathbb{N}^2$. We have now all the ingredients to state and prove the following two results. \[Lemma : symbolic product development\] Let $\sigma,k,p,\psi,a$ be as in lemma \[Lem : remainder r\^k\_N\] and $\widetilde{a}(x)$ such that $$\begin{gathered} (a\equiv 1) \Rightarrow (\widetilde{a}\equiv 1),\\ (a \text{ compactly supported }) \Rightarrow [(\widetilde{a} \equiv 1) \text{ or } (\widetilde{a} \text{ compactly supported and } \widetilde{a}a\equiv a )].\end{gathered}$$ We have that $$\begin{gathered} \label{symbolic dev 1} \oph\Big(\gamma_1\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)(x_n|\xi| - \xi_n)\Big) \\ = \oph\Big(\gamma_1\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)\Big)\widetilde{a}(x) h\mathcal{M}_n + \oph( r^k_p(x,\xi)),\end{gathered}$$ where \[est Op(rkp)\] $$\label{est L2 Op(rkp)} \big\| \oph(r^k_p (x,\xi))w\big\|_{L^2} \lesssim h^{1-\beta}\|w\|_{L^2} ,$$ $$\label{est Linfty Op(rkp)} \big\| \oph(r^k_p(x,\xi))w\big\|_{L^\infty} \lesssim h^{\frac{1}{2}-\beta}(\|w\|_{L^2}+\|\theta_0 \Omega_h w\|_{L^2}),$$ for some $\theta\in C^\infty_0(\mathbb{R}^2\setminus \{0\})$ and a small $\beta>0$, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. Moreover \[est: L2 Linfty with L\] $$\label{est: L2 of gamma1 with L} \Big\|\oph\Big(\gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)(x_n|\xi| - \xi_n)\Big)w\Big\|_{L^2} \lesssim h^{1-\beta} \big(\|w\|_{L^2} + \|\mathcal{M}_nw\|_{L^2} \big),$$ $$\begin{gathered} \label{est: Linfty of gamma1 with L} \Big\|\oph\Big(\gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi) (x_n|\xi| - \xi_n) \Big)w\Big\|_{L^\infty} \\ \lesssim h^{\frac{1}{2}-\beta}\sum_{\mu=0}^1\Big(\|(\theta_0\Omega_h)^\mu w\|_{L^2}+ \| (\theta_0\Omega_h)^\mu \mathcal{M}_nw\|_{L^2}\Big).\end{gathered}$$ The proof of the statement is basically made of tedious calculations and the application of propositions \[Prop : continuity Op(gamma) L2 to L2\], \[Prop : continuity of Op(gamma1):X to L2\] along with lemma \[Lem : remainder r\^k\_N\]. Let $\widetilde{\psi}\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$ such that $\widetilde{\psi}\equiv 1$ on the support of $\psi$. From formulas , and the hypothesis of the statement we derive that for a fixed $N\in\mathbb{N}$, and up to negligible multiplicative constants, $$\label{dev of gamma1 with his argument} \begin{split} &\left[\gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big) \psi(2^{-k}\xi) a(x)b_p(\xi) \right] \sharp \left[(x_n|\xi| - \xi_n) \widetilde{a}(x)\widetilde{\psi}(2^{-k}\xi)\right]\\ &= \gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big) \psi(2^{-k}\xi) a(x)b_p(\xi) (x_n|\xi| - \xi_n) \\ &+ \, h \left\{ \gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi) , (x_n|\xi| - \xi_n)\right\} \\ &+ \sum_{\substack{2\le |\alpha|<N\\ |\alpha_1| + |\alpha_2| = |\alpha|}} h^{|\alpha|}\partial^{\alpha_1}_x\partial^{\alpha_2}_\xi\Big[ \gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)\Big]\partial^{\alpha_2}_x \partial^{\alpha_1}_\xi \big[(x_n|\xi|-\xi_n)\big] + r_{N,p}^k(x,\xi), \end{split}$$ with $$\begin{gathered} \label{rkNp} r_{N,p}^k(x,\xi) = \frac{h^N}{(\pi h)^4} \sum_{|\alpha_1|+|\alpha_2|=N} \int e^{\frac{2i}{h}(\eta\cdot z - y\cdot \zeta)} \left[\int_0^1 \partial^{\alpha_1}_x\partial^{\alpha_2}_\xi\Big[ \gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)\Big]|_{(x+tz,\xi+t\zeta)}\right.\\ \times (1-t)^{N-1}dt\Big] \partial^{\alpha_2}_x\partial^{\alpha_1}_\xi\big[(x_n|\xi|-\xi_n)\widetilde{a}(x)\widetilde{\psi}(2^{-k}\xi)\big]|_{(x+y,\xi + \eta)}\, dy dz d\eta d\zeta\,.\end{gathered}$$If $\widetilde{a}\equiv 1$ above $r^k_{N,p}$ can be decomposed into the sum of integrals of the form and with $q=1$, so $$\label{norms_Op_rkNp} \left\|\oph(r^k_{N,p})\right\|_{\mathcal{L}(L^2)}+ \left\|\oph(r^k_{N,p})\right\|_{\mathcal{L}(L^2;L^\infty)} \lesssim h^{1+p}$$ if $N$ is taken sufficiently large (e.g. $N>2p+11$). The same is true if functions $a,\widetilde{a}$ are compactly supported as follows by propositions \[Prop: L2 est of integral remainders\] and \[Prop : Linfty est of integral remainders\], since from lemma \[Lem : est on gamma for wave\] and definition of $I^k_{p,q}$ for general $k\in K, p,q\in \mathbb{Z}$ $$r^k_{N,p}(x,\xi) = \sum_{\substack{|\alpha_1|+|\alpha_2| = N \\ i\le |\alpha_1|, 1\le j\le |\alpha_2|}} h^{N-(i+j)(\frac{1}{2}-\sigma)}I^k_{p+i+j -|\alpha_2|, 1-|\alpha_1|}(x,\xi).$$ An explicit computation of the Poisson bracket in shows that it is equal to $$\begin{gathered} \label{first order term symb dev} h (\partial \gamma_1)\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\Big(\frac{x_1\xi_2 - x_2\xi_1}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi) \\ +{\sum}' h \gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi),\end{gathered}$$ where $\sum'$ is a concise notation to indicate a linear combination, and $\psi, a, b_p$ are some new functions with the same features of their homonyms. After writing $$\label{x1 xi2 - x2 xi1} (x_1\xi_2 - x_2\xi_1) = (x_1|\xi| - \xi_1)\xi_2|\xi|^{-1} - (x_2|\xi| - \xi_2)\xi_1|\xi|^{-1},$$ we recognize that the quantization of verifies estimates thanks to propositions \[Prop : continuity Op(gamma) L2 to L2\], \[Prop : continuity of Op(gamma1):X to L2\] and the fact that $2^{kp}\le h^{-\sigma p}$. Let us denote concisely by $t^k_\alpha$ the $|\alpha|$-order contributions in , for $2\le |\alpha|<N$. As factor $x_n|\xi| -\xi_n$ is affine in $x_n$, the length of multi-index $\alpha_2$ is less or equal than 1 and, using lemma \[Lem : est on gamma for wave\], $t^k_\alpha$ appears to be the sum of two terms. The first one corresponds to $|\alpha_2| = 0$ and has the form $${\sum_{\substack{i\le |\alpha|\\ \mu=0,1}}}'h^{|\alpha| - i(\frac{1}{2}-\sigma)} \gamma_{1+i}\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_{p+i+1 - |\alpha| }(\xi)\, x^\mu_n ,$$ for some new functions $\psi, a$. Observe that $\mu=0$ if $a\equiv 1$ because the derivation of $\gamma_1\big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\big)$ $|\alpha_1|$-times with respect to $x$ makes appear, inter alia, a factor $|\xi|^{|\alpha_1|}$ that allows us to rewrite $\partial^{\alpha_1}_\xi(x_n|\xi|-\xi_n)$ from $(x_n|\xi|-\xi_n)+b_0(\xi)$, for some new $b_0$, and $\partial^{\alpha_1} _z\gamma_1(z)z_n$ is of the form $\gamma_{|\alpha_1|}(z)$). The second term, corresponding instead to $|\alpha_2|=1$, is given by $${\sum_{i\le |\alpha|-1, j\le 1}}'h^{|\alpha| - (i+j)(\frac{1}{2}-\sigma)} \gamma_{1+i+j}\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_{p+i+j+1 - |\alpha| }(\xi),$$ for some new other functions $\psi, a$. From propositions \[Prop : continuity Op(gamma) L2 to L2\], \[Prop : continuity of Op(gamma1):X to L2\] we then deduce that $$\begin{gathered} \|\oph(t^k_\alpha)w\|_{L^2} \lesssim (h^{\frac{|\alpha|}{2}-\beta} + h^{1+p})\|w\|_{L^2},\label{tk_alpha} \\ \|\oph(t^k_\alpha)w\|_{L^\infty} \lesssim (h^{\frac{|\alpha|-1}{2}-\beta} + h^{\frac{1}{2}+p})(\|w\|_{L^2}+\| \theta \Omega_h w\|_{L^2}),\end{gathered}$$ which concludes that $$\begin{gathered} \left[\gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big) \psi(2^{-k}\xi) a(x)b_p(\xi) \right] \sharp \left[(x_n|\xi| - \xi_n) \widetilde{a}(x)\widetilde{\psi}(2^{-k}\xi)\right] \\ = \gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big) \psi(2^{-k}\xi) a(x)b_p(\xi) (x_n|\xi| - \xi_n) + r^k_p(x,\xi),\end{gathered}$$ with $r^k_p$ satisfying . Finally, by symbolic calculus we have that, up to some multiplicative constants, $$\begin{aligned} \oph\big((x_n|\xi| - \xi_n)\widetilde{a}(x)\widetilde{\psi}(2^{-k}\xi)\big) &= \widetilde{a}(x) \oph\big((x_n|\xi| - \xi_n)\widetilde{\psi}(2^{-k}\xi)\big) + \oph(r^k(x,\xi)) \\ &=\oph(\widetilde{\psi}(2^{-k}\xi))\widetilde{a}(x) h\mathcal{M}_n + h\widetilde{a}(x) \oph((\partial\widetilde{\psi})(2^{-k}\xi)(2^{-k}|\xi|)) \\ &+ \oph(\widetilde{r}^k(x,\xi)) h\mathcal{M}_n + \oph(r^k(x,\xi)),\end{aligned}$$ where $$\begin{aligned} r^k(x,\xi)&=\frac{h}{(2\pi)^2}\int e^{\frac{2i}{h}\eta\cdot z}\int \partial_x\widetilde{a}(x+tz) dt\ \partial_\xi\big[(x_n|\xi| -\xi_n)\widetilde{\psi}(2^{-k}\xi)\big]|_{(x,\xi+\eta)} dzd\eta, \\ \widetilde{r}^k(x,\xi)&=\frac{h2^{-k}}{(2\pi)^2}\int e^{\frac{2i}{h}\eta\cdot z}\int \partial_x\widetilde{a}(x+tz) dt\ (\partial_\xi\widetilde{\psi})(2^{-k}(\xi+\eta)) dzd\eta,\end{aligned}$$ are such that $\|\oph(r^k_1)\|_{\mathcal{L}(L^2)}=O(h)$, $\|\oph(\widetilde{r}^k_1)\|_{\mathcal{L}(L^2)}=O(1)$. An explicit computation shows also that $\|[\Omega_h, \oph(r^k)]\|_{\mathcal{L}(L^2)}=O(h)$ and $\|[\Omega_h, \oph(\widetilde{r}^k)]\|_{\mathcal{L}(L^2)}=O(1)$. Therefore, since $\widetilde{\psi}\equiv 1$ on the support of $\psi$, $\widetilde{a}\equiv 1$ on the support of $a$, one can use remark \[Remark:symbols\_with\_null\_support\_intersection\] together with propositions \[Prop: L2 est of integral remainders\], \[Prop : Linfty est of integral remainders\], and also propositions \[Prop : continuity Op(gamma) L2 to L2\], \[Prop : continuity of Op(gamma1):X to L2\], to show that $$\begin{gathered} \oph\Big(\gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big) \psi(2^{-k}\xi) a(x)b_p(\xi)\Big)\oph\big((x_n|\xi| - \xi_n) \widetilde{a}(x)\widetilde{\psi}(2^{-k}\xi)\big)\\ = \oph\Big(\gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big) \psi(2^{-k}\xi) a(x)b_p(\xi)\Big)\widetilde{a}(x)h\mathcal{M}_n +\oph(r^k_p(x,\xi)),\end{gathered}$$ for a new $\oph(r^k_p(x,\xi))$ satisfying . This concludes the proof of and of the entire statement by applying propositions \[Prop : continuity Op(gamma) L2 to L2\], \[Prop : continuity of Op(gamma1):X to L2\] to the first operator in the above right hand side. \[Lem: Gamma with double argument-wave\] Let $\sigma>0$ be small, $k\in K$ with $K$ given by and $p\in \mathbb{N}$. Let also $\gamma\in C^\infty_0(\mathbb{R}^2)$ be equal to 1 in a neighbourhood of the origin, $\psi\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$, and $a\in C^\infty_0(\mathbb{R}^2)$. For any function $w\in L^2(\mathbb{R}^2)$ such that $\mathcal{M}w\in L^2(\mathbb{R}^2)$, any $m,n=1,2$, we have that $$\begin{gathered} \oph\Big(\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)(x_m|\xi|-\xi_m)(x_n|\xi|-\xi_n) \Big)w\\= \oph\Big(\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)(x_m|\xi|-\xi_m) \Big)[h\mathcal{M}_n w] +O_{L^2}\big(h^{2-\beta}(\|w\|_{L^2}+\|\mathcal{M}w\|_{L^2})\big),\end{gathered}$$ with $\beta>0$ small, $\beta\rightarrow 0$ as $\sigma\rightarrow0$. Let $\widetilde{\gamma}(z):=\gamma(z)z_m$ and $\widetilde{\psi}\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$ be identically equal to 1 on the support of $\psi$. We saw in the proof of the previous lemma that the symbolic product $$\left[\widetilde{\gamma}\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)\right] \sharp [(x_n|\xi|-\xi_n)\widetilde{\psi}(2^{-k}\xi)]$$ develops as in , , with $\gamma_1$ replaced with $\widetilde{\gamma}$ and $\widetilde{a}\equiv 1$. From , the fact that $$\{x_m|\xi|-\xi_m, x_n|\xi|-\xi_n\}= \begin{cases} 0 \quad &\text{if } m=n,\\ (-1)^{m+1} (x_1\xi_2-\xi_2x_1) &\text{if } m\ne n, \end{cases}$$ and that $(x_1\xi_2-\xi_2x_1) = (x_1|\xi|-\xi_1)\xi_2|\xi|^{-1}-(x_2|\xi|-\xi_2)\xi_1|\xi|^{-1}$, we derive that the first order term of the mentioned symbolic development is a linear combination of products of the form $$h^{\frac{3}{2}}\gamma \Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)(x_j|\xi|-\xi_j),$$ for some new functions $\gamma, \psi, a$, and its quantization acting on $w$ is a remainder as in the statement after estimate . The second order term is given, up to some negligible multiplicative constants, by $$\begin{gathered} h^{1+2\sigma}\sum_{|\alpha|=2}(\partial^\alpha\gamma)\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a_1(x)b_{p+1}(\xi)(x_m|\xi|-\xi_m)\\ + h^{\frac{3}{2}+\sigma}\sum_{|\alpha|=1}(\partial^\alpha\gamma)\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi_2(2^{-k}\xi)a_2(x)b_{p+1}(\xi)\\ + h^2{\sum}' \gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi_3(2^{-k}\xi)a_3(x)b_{p+1}(\xi),\end{gathered}$$ for some new smooth compactly supported $\psi_2,\psi_3, a_1, a_2, a_3$, and as the derivatives of $\gamma$ vanish in a neighbourhood of the origin we can replace $(\partial^\alpha \gamma)(z)$ with $\sum_j \gamma_1^j(z)z_j$, $\gamma_j^1(z):=(\partial^\alpha\gamma)(z)z_j|z|^{-2}$, when $|\alpha|=1$. The third order one is given by $$\begin{gathered} h^{\frac{3}{2}+3\sigma}\sum_{|\alpha|=3}(\partial^\alpha\gamma)\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a_1(x)b_{p+1}(\xi)(x_m|\xi|-\xi_m)\\ + h^2{\sum}' \gamma_1\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi_1(2^{-k}\xi)a_2(x)b_{p+1}(\xi),\end{gathered}$$ for some other $\psi_1, a_1, a_2$ and a new $\gamma_1\in C^\infty_0(\mathbb{R}^2)$. Using estimate for the summations in $\alpha$ and proposition \[Prop : continuity Op(gamma) L2 to L2\] for the remaining terms in the above expressions we obtain that the quantizations of the second and third order term are also a $O_{L^2}\big(h^{2-\beta}(\|w\|_{L^2}+\|\mathcal{M}w\|_{L^2})\big)$ when acting on $w$, for a small $\beta>0$, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. In all the other $|\alpha|$-order terms, with $4\le |\alpha|\le N-1$, and in integral remainder $r^k_{N,p}$, we look at $\gamma\big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\big)\psi(2^{-k}\xi)a(x)b_p(\xi)(x_m|\xi|-\xi_m)$ as a symbol of the form $$\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_{p+1}(\xi)$$ for a new $a_1\in C^\infty_0(\mathbb{R}^2)$. From and when $N>11$, we derive that the quantizations of these terms are also a $O_{L^2}\big(h^{2-\beta}(\|w\|_{L^2}+\|\mathcal{M}w\|_{L^2})\big)$ when acting on $w$. We finally obtain that $$\begin{gathered} \oph\Big(\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)(x_m|\xi|-\xi_m)(x_n|\xi|-\xi_n) \Big)w\\= \oph\Big(\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)a(x)b_p(\xi)(x_m|\xi|-\xi_m) \Big)\oph\big((x_n|\xi|-\xi_n)\widetilde{\psi}(2^{-k}\xi)\big) \\ +O_{L^2}\big(h^{2-\beta}(\|w\|_{L^2}+\|\mathcal{M}w\|_{L^2})\big),\end{gathered}$$ and the conclusion of the proof comes then from the fact that, by symbolic calculus, $$\oph\big((x_n|\xi| - \xi_n)\widetilde{\psi}_1(2^{-k}\xi)\big)= h \oph(\widetilde{\psi}_1(2^{-k}\xi))\mathcal{M}_n - \frac{h}{2i} \oph\big((\partial\widetilde{\psi}_1)(2^{-k}\xi)\cdot(2^{-k}\xi)\big),$$ and by remark \[Remark:symbols\_with\_null\_support\_intersection\] as all derivatives of $\widetilde{\psi}$ vanish on the support of $\psi$. The following lemma is introduced especially for the proof of lemma \[Lem: preliminary on Op(e)\]. Even if quite similar to lemma \[Lemma : symbolic product development\], we are going to see that the particular structure of symbolic product in the left hand side of allows for a remainder $r^k_p$ satisfying enhanced estimate rather than . \[Lemma : on the enhanced symbolic product\] Let us take $\sigma>0$ sufficiently small, $k\in K$ and $p,q\in\mathbb{N}$. Let also $\gamma\in C^\infty_0(\mathbb{R}^2)$ such that $\gamma\equiv 1$ in a neighbourhood of the origin, $\psi, \widetilde{\psi}\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$ such that $\psi \equiv 1$ on the support of $\widetilde{\psi}$, $a(x)$ be a smooth compactly supported function. Then $$\begin{gathered} \label{symboli_dev_enhanced} \Big[ (x_n|\xi| - \xi_n)\widetilde{\psi}(2^{-k}\xi)a(x)b_p(\xi)\Big] \sharp \, \Big[\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)\Big]\\ = \gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\widetilde{\psi}(2^{-k}\xi)a(x)b_p(\xi)(x_n|\xi| - \xi_n) + r^k_p(x,\xi),\end{gathered}$$ where \[est Op(rkp) enhanced\] $$\label{est L2 Op(rkp) enhanced} \left\|\oph(r^k_p(x,\xi))w \right\|_{L^2}\lesssim h^{\frac{3}{2}-\beta} (\|w\|_{L^2} + \| \mathcal{M}w\|_{L^2}) +h^{1+p}\|w\|_{L^2},$$ $$\label{est Linfty Op(rkp) enhanced} \left\|\oph(r^k_p(x,\xi))w \right\|_{L^\infty} \lesssim h^{1-\beta}\sum_{\mu=0}^1\Big(\|(\theta_0\Omega_h)^\mu w\|_{L^2}+ \| (\theta_0\Omega_h)^\mu \mathcal{M}w\|_{L^2}\Big),$$ for some $\theta\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$, and a small $\beta>0$, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. Using proposition \[Prop: a sharp b\], for a fixed $N\in\mathbb{N}$ and up to multiplicative constants independent of $h, k,$ we have the following symbolic development: $$\label{symb dev 2} \begin{split} &\Big[ (x_n|\xi| - \xi_n) \widetilde{\psi}(2^{-k}\xi)a(x)b_p(\xi)\Big] \sharp \, \Big[\gamma\Big( \frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)\Big] \\ & = \gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\widetilde{\psi}(2^{-k})a(x)b_p(\xi)(x_n|\xi| - \xi_n) \\ & + h \left\{(x_n|\xi| - \xi_n)\widetilde{\psi}(2^{-k}\xi)a(x)b_p(\xi), \gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\right\} \\ & + \sum_{\substack{\alpha=(\alpha_1,\alpha_2)\\ 2\le |\alpha| <N}}h^{|\alpha|}\partial^{\alpha_1}_x \partial^{\alpha_2}_\xi\Big[(x_n|\xi| - \xi_n)\widetilde{\psi}(2^{-k}\xi)a(x)b_p(\xi) \Big] \partial^{\alpha_2}_x \partial^{\alpha_1}_\xi\Big[\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\Big]+ r^k_{N,p}(x,\xi), \end{split}$$ with $$\begin{gathered} r^k_{N,p}(x,\xi) = \frac{h^N}{(\pi h)^4}\sum_{|\alpha_1|+|\alpha_2| =N}\int e^{\frac{2i}{h}(\eta\cdot z - y\cdot \zeta)}\left[\int_0^1 \partial^{\alpha_1}_x \partial^{\alpha_2}_\xi\big[(x_n|\xi| - \xi_n)a(x)b_p(\xi)\widetilde{\psi}(2^{-k}\xi) \big]|_{(x+tz, \xi+t\zeta)}\right.\\ \times (1-t)^{N-1}dt \Big] \partial^{\alpha_2}_x \partial^{\alpha_1}_\xi\Big[ \gamma\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)\Big]|_{(x+y,\xi+\eta)} \ dydz d\eta d\zeta.\end{gathered}$$ For sake of simplicity, we denote by $t^k_1$ (resp. $t^k_\alpha$, $|\alpha|=2,\dots, N-1$) the Poisson brackets (resp. the $|\alpha|$-th contribution) in . An explicit computation of $t^k_1$, combined with the fact that $x_1\xi_2-x_2\xi_1 = (x_1|\xi|-\xi_1)\xi_2|\xi|^{-1} - (x_2|\xi|-\xi_2)\xi_1|\xi|^{-1}$, shows that it is linear combination of terms of the form $$h(\partial\gamma)\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\Big(\frac{x_j|\xi| - \xi_j}{h^{1/2-\sigma}}\Big)\widetilde{\psi}(2^{-k}\xi)a(x)b_p(\xi),$$ for $j\in\{1,2\}$ and some new functions $\widetilde{\psi}, a, b_p$, so by inequalities we derive that \[est L2 Linfty Op(tk1)\] $$\left\|\oph(t^k_1)w \right\|_{L^2}\lesssim h^{\frac{3}{2}-\beta} \left(\|w\|_{L^2}+\|\mathcal{M}w\|_{L^2}\right),$$ $$\left\|\oph(t^k_1)w \right\|_{L^\infty}\lesssim h^{1-\beta}\sum_{\mu=0}^1(\|(\theta_0\Omega_h)^\mu w\|_{L^2} + \|(\theta_0\Omega_h)^\mu \mathcal{M}w\|_{L^2}).$$ The improvement of these estimates with respect to is attributable to the choice of $\psi$ identically equal to 1 on the support of $\widetilde{\psi}$. All derivatives of $\psi$ vanish against $\widetilde{\psi}$, so in the development of $t^k_1$ we avoid terms like $\gamma\big(\frac{x|\xi|-\xi|}{h^{1/2-\sigma}}\big)\widetilde{\psi}(2^{-k}\xi)a(x)b_p(\xi)(\partial\psi)(2^{-k}\xi)(2^{-k}|\xi|)$, coming out from $\{x_n|\xi| -\xi_n, \psi(2^{-k}\xi)\}\gamma\big(\frac{x|\xi|-\xi|}{h^{1/2-\sigma}}\big)\widetilde{\psi}(2^{-k}\xi)a(x)b_p(\xi)$, that do not enjoy estimates like . Using formula and looking at $(x_n|\xi|-\xi_n)\widetilde{\psi}(2^{-k}\xi)a(x)b_p(\xi)$ as a linear combination of terms $\widetilde{\psi}(2^{-k}\xi)a(x)b_{p+1}(\xi)$, for some new $\widetilde{\psi},a,b_{p+1}$, we realize that, for any $2\le |\alpha|<N$, $$t^k_\alpha = \sum_{\substack{|\alpha_1| + |\alpha_2| = |\alpha| \\ 1\le j \le |\alpha_1|}} h^{|\alpha| - (j+|\alpha_2|)(\frac{1}{2}-\sigma)}\gamma_{j+|\alpha_2|}\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\widetilde{\psi}(2^{-k}\xi)a_j(x) b_{p+j+1-|\alpha_1|}(\xi),$$ for some new other $\widetilde{\psi}, a_j$, with $a_j$ compactly supported, and then that $$\|\oph(t^k_\alpha)w\|_{L^2}\lesssim \sum_{\substack{|\alpha_1| + |\alpha_2| = |\alpha| \\ 1\le j\le |\alpha_1|}} h^{|\alpha| - (j+|\alpha_2|)(\frac{1}{2}-\sigma)}2^{k(p+j+1 - |\alpha_1|)} \|w\|_{L^2},$$ $$\begin{gathered} \|\oph(t^k_\alpha)w\|_{L^\infty}\\ \lesssim \sum_{\substack{|\alpha_1| + |\alpha_2| = |\alpha| \\ 1\le j\le |\alpha_1|}} h^{|\alpha| - (j+|\alpha_2| )(\frac{1}{2}-\sigma)}2^{k(p+j+1 - |\alpha_1|)}h^{-\frac{1}{2}-\sigma} \sum_{\mu=0}^1(\|(\theta_0\Omega_h)^\mu w\|_{L^2} + \|(\theta_0\Omega_h)^\mu \mathcal{M}w\|_{L^2}),\end{gathered}$$ after propositions \[Prop : continuity Op(gamma) L2 to L2\], \[Prop : continuity of Op(gamma1):X to L2\]. For $|\alpha|\ge 3$, the above estimates imply $\|\oph(t^k_\alpha)\|_{\mathcal{L}(L^2)}\lesssim h^{\frac{3}{2}-\beta}$ and $\|\oph(t^k_\alpha)w\|_{L^\infty}\lesssim h^{1-\beta} \sum_{\mu=0}^1(\|(\theta_0\Omega_h)^\mu w\|_{L^2} + \|(\theta_0\Omega_h)^\mu \mathcal{M}w\|_{L^2})$. For $|\alpha| =2$, we exploit the fact that functions $\gamma_{j+|\alpha_2|}$ vanish in a neighbourhood of the origin, as they come from $\gamma$’s derivatives, and define $\gamma^i_{j+|\alpha_2| }(z):= \gamma_{j+|\alpha_2|}(z) z_i|z|^{-2}$, $i=1,2$, so that $$t^k_\alpha = \sum_{\substack{|\alpha_1| + |\alpha_2| = |\alpha| \\ 1\le j \le |\alpha_1|, i=1,2}} h^{|\alpha| - (j+|\alpha_2|)(\frac{1}{2}-\sigma)}\gamma^i_{j+|\alpha_2|}\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\Big(\frac{x_i|\xi| - \xi_i}{h^{1/2-\sigma}}\Big)\widetilde{\psi}(2^{-k}\xi)a_j(x) b_{p+j+1-|\alpha_1|}(\xi),$$to which we can then apply lemma \[Lemma : symbolic product development\]. After inequalities , $\oph(t^k_\alpha)$ with $|\alpha|=2$ also satisfies . Finally, reminding definition of $J^k_{p,q}(x,\xi)$ for general $k\in K, p,q\in\mathbb{Z}$, and developing derivatives in $r^k_{N,p}$ using lemma \[Lem : est on gamma for wave\], we observe that $$r^k_{N,p} = \sum_{\substack{|\alpha_1| + |\alpha_2| = N \\ 0\le j\le |\alpha_1|}} h^{N-(|\alpha_2| +j)(\frac{1}{2}-\sigma)}J^k_{p+1-|\alpha_2|, |\alpha_2|+j-|\alpha_1|}(x,\xi),$$ hence propositions \[Prop: L2 est of integral remainders\] and \[Prop : Linfty est of integral remainders\] give that $$\|\oph(r^k_{N,p})\|_{\mathcal{L}(L^2)}\lesssim \sum_{\substack{|\alpha_1| + |\alpha_2| = N \\ 0\le j\le |\alpha_1|}} h^{N-(|\alpha_2| +j)(\frac{1}{2}-\sigma)}2^{k(p+1 +j- |\alpha_1|)}\lesssim h^{1+p},$$ $$\|\oph(r^k_{N,p})\|_{\mathcal{L}(L^2;L^\infty)}\lesssim \sum_{\substack{|\alpha_1| + |\alpha_2| = N\\ 0\le j\le |\alpha_1|, i\le 6}} h^{N-(|\alpha_2| +j)(\frac{1}{2}-\sigma)}2^{k(p+1+ j- |\alpha_1|)}(h^{-\frac{1}{2}+\sigma}2^k)^i(h^{-1}2^k)\lesssim h^{1+p},$$ if $N$ is chosen sufficiently large (e.g. $N>10+2p$). We should also highlight the fact that, at the difference of , does not improve : if we get a $h^{\frac{3}{2}-\beta}$ factor in front of the first term in the right hand side, the second term $h^{1+p}\|w\|_{L^2}$ is just a $O(h^{1-\beta})$ in the case $p=0$, coming from $|\alpha_1|=N$, $j=|\alpha_2|=0$, $p=0$ above. ### Operators for the Klein-Gordon solution: some estimates {#Subsection: Some Technical Estimates II} This subsection is mostly devoted to the introduction of some symbols and operators, along with their properties, that we will often use in the paper when dealing with the Klein-Gordon component of the solution to starting system . From now on we will use the notation $p(\xi):=\sqrt{1+|\xi|^2}$ (thus, $p'(\xi)$ denotes the gradient of $p(\xi)$, $p''(\xi)=(\partial^2_{ij}p(\xi))_{ij}$ the $2\times 2$ Hessian matrix of $p(\xi)$). Proposition \[Prop : Continuity on H\^s\] is a general result about continuity on spaces $H^s_h(\mathbb{R}^2)$ of operators with symbols of order $r\in\mathbb{R}$ and generalises theorem 7.11 in [@dimassi:spectral]. Proposition \[Prop : Continuity from $L^2$ to L\^inf\] is a result of continuity from $L^2$ to $H_h^{\rho, \infty}$ of a particular class of operators that will act on the Klein-Gordon component. In the spirit of [@ifrim_tataru:global_bounds] for the Schrödinger equation, it allows to pass from uniform norms to the $L^2$ norm losing only a power $h^{-\frac{1}{2}-\beta}$ for a small $\beta>0$ instead of a $h^{-1}$ as for the semi-classical Sobolev injection. Proposition \[Prop:Continuity Lp-Lp\] is, instead, a result of uniform $L^p-L^p$ continuity of such operators, for every $1\le p\le +\infty$. \[Prop : Continuity on H\^s\] Let $s\in \mathbb{R}$. Let $a \in S_{\delta,\sigma}(\langle\xi\rangle^r)$, $r\in\mathbb{R}$, $\delta \in [0, \frac{1}{2}]$, $\sigma \ge 0$. Then $\oph(a)$ is uniformly bounded : $H^s_h(\mathbb{R}^2)\rightarrow H^{s-r}_h(\mathbb{R}^2)$ and there exists a positive constant $C$ independent of $h$ such that $$\|\oph(a)\|_{\mathcal{L}(H^s_h;H^{s-r}_h)}\le C\, , \qquad \forall h\in ]0,1]\, .$$ \[Continuity from $L^2$ to $H_h^{\rho ,\infty}$\]\[Prop : Continuity from $L^2$ to L\^inf\] Let $\rho \in \mathbb{N}$. Let $a \in S_{\delta,\sigma}(\langle \frac{x-p'(\xi)}{\sqrt{h}}\rangle^{-1})$, $\delta \in [0, \frac{1}{2}]$, $\sigma>0$. Then $\oph(a)$ is bounded : $L^2(\mathbb{R}^2)\rightarrow H_h^{\rho, \infty}(\mathbb{R}^2)$ and there exists a positive constant $C$ independent of $h$ such that $$\|\oph(a)\|_{\mathcal{L}(L^2;H_h^{\rho, \infty})}\le C h^{-\frac{1}{2}-\beta}\, , \qquad \forall h\in ]0,1]\, ,$$ where $\beta>0$ depends linearly on $\sigma$. We first remark that, after definition \[def of h-Sobolev spaces\] $(i)$ of the $H_h^{\rho,\infty}$ norm, $$\|\oph(a)w\|_{H^{\rho,\infty}_h} = \|\langle hD_x\rangle^\rho \oph(a)w\|_{L^\infty},$$ and that, by symbolic calculus of lemma \[Lem : a sharp b\], $\langle \xi\rangle^\rho \sharp a(x,\xi)$ belongs to $S_{\delta,\sigma}(\langle\xi\rangle^\rho \big\langle \frac{x-p'(\xi)}{\sqrt{h}}\big\rangle^{-1})\subset h^{-\rho\sigma } S_{\delta,\sigma}( \big\langle \frac{x-p'(\xi)}{\sqrt{h}}\big\rangle^{-1})$. This means that estimating the $H_h^{\rho,\infty}$ norm of an operator whose symbol is rapidly decaying in $|h^{\sigma}\xi|$ corresponds actually to estimate the $L^{\infty}$ norm of an operator associated to another symbol (namely, $\tilde{a}(x,\xi)= \langle \xi \rangle^\rho \sharp a(x,\xi) $) which is still in the same class as $a$, up to a small loss $h^{-\rho\sigma}$. From definition \[Def: Weyl and standard quantization\] $(i)$ of $\oph(a)w$, and using a change of coordinates $y\mapsto \sqrt{h}y$, $\xi\mapsto \sqrt{h}\xi$, integration by part, Cauchy-Schwarz inequality, and Young’s inequality for convolutions, we derive what follows: $$\label{form 3.19} \begin{split} & |\oph(a)w| = \\ & = \left| \frac{1}{(2\pi)^2}\int\int e^{i(\frac{x}{\sqrt{h}}-y)\cdot\xi}a\Big(\frac{x+\sqrt{h} y}{2},\sqrt{h}\xi\Big) w(\sqrt{h}y ) \, dyd\xi \right|\\ & = \left|\frac{1}{(2\pi)^4 h}\int\hat{w}\Big(\frac{\eta}{\sqrt{h}}\Big)d\eta \int\int e^{i(\frac{x}{\sqrt{h}}-y)\cdot\xi + i\eta\cdot y}a\Big(\frac{x+\sqrt{h} y}{2},\sqrt{h}\xi\Big)\, dyd\xi \right| \\ & = \left|\frac{1}{(2\pi)^4 h}\int \hat{w}\Big(\frac{\eta}{\sqrt{h}}\Big) \int\int\left(\frac{1-i\big(\frac{x}{\sqrt{h}}-y\big)\cdot\partial_{\xi}}{1+|\frac{x}{\sqrt{h}}-y|^2}\right)^{3}\left(\frac{1+i(\xi -\eta)\cdot\partial_y}{1+|\xi -\eta|^2}\right)^{3}e^{i(\frac{x}{\sqrt{h}}-y)\cdot\xi + i\eta\cdot y} \right. \\ & \left. \hspace{0.5 cm} \times \, a\Big(\frac{x+\sqrt{h} y}{2},\sqrt{h}\xi\Big)\, dyd\xi d\eta \right| \\ & \lesssim \frac{1}{h} \int \left|\hat{w}\Big(\frac{\eta}{\sqrt{h}}\Big)\right| \int\int \Big\langle \frac{x}{\sqrt{h}}- y\Big\rangle^{-3} \langle \xi - \eta \rangle^{-3}\langle h^{\sigma}\sqrt{h}\xi \rangle^{-N} \Big\langle\frac{\frac{x+\sqrt{h} y}{2}- p'(\sqrt{h}\xi)}{\sqrt{h}}\Big\rangle^{-1} dy d\xi d\eta\\ & \lesssim \frac{1}{h}\left\|\hat{w}\Big(\frac{\cdot}{\sqrt{h}}\Big)\right\|_{L^2} \|\langle \eta \rangle^{-3}\|_{L^1(\eta)}\, \left\| \int \Big\langle \frac{x}{\sqrt{h}}-y\Big\rangle^{-3}\langle h^{\sigma}\sqrt{h}\xi \rangle^{-N} \Big\langle \frac{\frac{x+ \sqrt{h}y}{2}-p'(\sqrt{h}\xi)}{\sqrt{h}}\Big\rangle^{-1} dy \right\|_{L^2(d\xi)} \\ & \lesssim h^{-\frac{1}{2}}\|w\|_{L^2} \int \left\langle \frac{x}{\sqrt{h}}-y \right\rangle^{-3} \Big\|\langle h^{\sigma}\sqrt{h}\xi \rangle^{-N}\Big\langle \frac{\frac{x+ \sqrt{h}y}{2}-p'(\sqrt{h}\xi)}{\sqrt{h}}\Big\rangle^{-1} \Big\|_{L^2(\xi)} dy \, , \end{split}$$ where $N>0$ will be properly chosen later. We draw attention to two facts: in the third equality in we use that $$\left(\frac{1-i(\frac{x}{\sqrt{h}}-y)\cdot\partial_{\xi}}{1+(\frac{x}{\sqrt{h}}-y)^2}\right)^{3}\left(\frac{1+i(\xi -\eta)\cdot\partial_y}{1+(\xi -\eta)^2}\right)^{3} \left[e^{i(\frac{x}{\sqrt{h}}-y)\cdot\xi + i\eta\cdot y}\right]=e^{i(\frac{x}{\sqrt{h}}-y)\cdot\xi + i\eta\cdot y}$$ so, integrating by part, derivatives $\partial_y, \partial_\xi$ fall on $\langle \frac{x}{\sqrt{h}}-y\rangle^{-1}$, $\langle \xi - \eta\rangle^{-1}$ (giving rise to more decreasing factors) and/or on $a\left(\frac{x+\sqrt{h}y}{2},\sqrt{h}\xi\right)$; symbol $a$ belongs to $S_{\delta,\sigma}(1)$ with $\delta \le \frac{1}{2}$, but the loss of $h^{-\delta}$ is offset by the factor $\sqrt{h}$ coming from the derivation of $a(\frac{x+\sqrt{h}y}{2},\sqrt{h}\xi)$ with respect to $y$ and $\xi$. In order to estimate $\big\|\langle h^{\sigma}\sqrt{h}\xi \rangle^{-N}\big\langle\frac{\frac{x+\sqrt{h} y}{2}- p'(\sqrt{h}\xi)}{\sqrt{h}}\big\rangle^{-1}\big\|_{L^2_\xi}$ we first introduce a smooth cut-off function $\chi(\frac{x+\sqrt{h}y}{2})$, with $\chi$ supported in some ball $B_C(0)$, to distinguish between the case when $\frac{x+\sqrt{h}y}{2}$ is bounded from the one where $|\frac{x+\sqrt{h}y}{2}|\rightarrow +\infty$. In the latter situation, say for $|\frac{x + \sqrt{h}y}{2}|>2$, we have $\big\langle\frac{\frac{x+\sqrt{h} y}{2}-p'(\sqrt{h}\xi)}{\sqrt{h}}\big\rangle^{-1} \lesssim \sqrt{h}$ and $$\Big|(1-\chi)\Big(\frac{x+\sqrt{h}y}{2}\Big) \Big|\Big\|\langle h^{\sigma}\sqrt{h}\xi \rangle^{-N}\Big\langle\frac{\frac{x+\sqrt{h} y}{2}-p'(\sqrt{h}\xi)}{\sqrt{h}}\Big\rangle^{-1}\Big\|_{L^2(d\xi)} \lesssim h^{-\sigma}.$$ On the other hand, when $\frac{x+\sqrt{h}y}{2}$ is bounded we consider a Littlewood-Paley decomposition and write $$\label{summation over k} \begin{split} \left\|\langle h^{\sigma}\sqrt{h}\xi \rangle^{-N}\Big\langle\frac{\frac{x+\sqrt{h} y}{2}-p'(\sqrt{h}\xi)}{\sqrt{h}}\Big\rangle^{-1}\right\|_{L^2(\xi)}^2 & = h^{-1}\sum_{k\ge 0} \int \langle h^{\sigma}\xi \rangle^{-2N}\Big\langle\frac{\frac{x+\sqrt{h} y}{2}-p'(\xi)}{\sqrt{h}}\Big\rangle^{-2} \varphi_k(\xi) d\xi \\ &= h^{-1}\sum_{k\ge 0} I_k \end{split}$$ where $$I_0= \int \langle h^{\sigma}\xi \rangle^{-2N} \Big\langle\frac{\frac{x+\sqrt{h} y}{2}-p'(\xi)}{\sqrt{h}}\Big\rangle^{-2} \varphi_0(\xi) d\xi$$ and $$\label{I_k} \begin{split} I_k &= \int \langle h^{\sigma}\xi \rangle^{-2N} \Big\langle\frac{\frac{x+\sqrt{h} y}{2}-p'(\xi)}{\sqrt{h}}\Big\rangle^{-2} \varphi(2^{-k}\xi) d\xi \\ & = 2^{2k} \int \langle h^{\sigma}2^k \xi \rangle^{-2N}\Big\langle\frac{\frac{x+\sqrt{h} y}{2}-p'(2^k\xi)}{\sqrt{h}}\Big\rangle^{-2} \varphi(\xi) d\xi \\ & \lesssim 2^{(-2N+2)k}h^{-2\sigma N} \int \Big\langle\frac{\frac{x+\sqrt{h} y}{2}-p'(2^k\xi)}{\sqrt{h}}\Big\rangle^{-2} \varphi(\xi) d\xi \, . \end{split} \qquad k\ge 1$$ For a fixed $k_0$ and any $k\le k_0$, $|\det(p''(2^k\xi))|\ge C> 0$ on the support of $\varphi$. For $k\ge k_0$, function $\xi \rightarrow g_k(\xi)= 2^{3k}(\frac{x+\sqrt{h} y}{2})- 2^{3k}p'(2^k\xi)$ is such that $\text{det}(g_k'(\xi)) = \frac{2^{4k}}{(1+|2^k\xi|^2)^2}$ and $|\text{det}(g_k'(\xi))|\sim 1$ on the support of $\varphi$. We may thus split the $d\xi$ integral in a finite number (independent of $k$) of integrals, computed on compact domains, on which $\xi \mapsto g_k(\xi)$ is a change of variables with Jacobian of size 1. We are then reduced to estimate $2^{(-2N+2)k} h^{-2\sigma N}\int_{|z|\le C} \langle \frac{z+g_k(\xi_0)}{2^{3k}\sqrt{h}}\rangle^{-2} dz$, where $C$ is a positive constant and $\xi_0$ is in $supp\varphi$. Since we assumed that $\frac{x+\sqrt{h} y}{2}$ is bounded, $|g_k(\xi_0)| = O(2^{3k})$ and we get $$\begin{split} I_k &\lesssim 2^{(-2N+2)k} h^{-2\sigma N}\int_{|z|\lesssim 2^{3k}} \Big\langle\frac{z}{2^{3k}\sqrt{h}}\Big\rangle^{-2} dz \\ & \lesssim 2^{(-2N+8)k}h^{-2\sigma N}h \int_{|z|\lesssim h^{-1/2}} \langle z \rangle^{-2} dz \\ & \lesssim 2^{(-2N+8)k}h^{-2\sigma N + 1} \log (h^{-1})\, . \end{split}$$ Taking the sum of all $I_k$ for $k\ge 0$ we then deduce that $$\left\|\langle h^{\sigma}\sqrt{h}\xi\rangle^{-N}\Big\langle\frac{\frac{x+\sqrt{h} y}{2}-p'(\sqrt{h}\xi)}{\sqrt{h}}\Big\rangle^{-1}\right\|_{L^2(\xi)}\lesssim h^{-\sigma N -\delta}\Big(\sum_{k\ge 0}2^{(-2N +8)k}\Big)^\frac{1}{2} \lesssim h^{-\sigma N - \delta} \, ,$$ for $\delta>0$ as small as we want, if we choose $N>0$ such that $-2N +8<0$ (e.g. $N=5$). Finally $$\|\oph(a)\|_{\mathcal{L}(L^2; H_h^{\rho, \infty})} = O( h^{-\frac{1}{2}- \beta})\,,$$ where $\beta(\sigma) = (N + \rho)\sigma + \delta$. The following lemma is as simple as useful and will be largely recalled from subsection \[Subsection : The Derivation of the ODE Equation\] on. It is also useful to introduce now the following manifold $$\label{def_Lkg} \Lkg:=\{(x,\xi)\in\mathbb{R}^2\times\mathbb{R}^2 : x-p'(\xi)=0\}$$ which appears to be the graph of function $\xi=-d\phi(x)$, with $\phi(x)=\sqrt{1-|x|^2}$ (see picture \[picture: Lkg\]). \[Lem:family\_thetah\] Let $\gamma,\chi\in C^\infty_0(\mathbb{R}^2)$ be equal to 1 in a neighbourhood of the origin and with sufficiently small support, and $\sigma>0$ be small. There exists a family of smooth functions $\theta_h(x)$, real valued, equal to 1 for $|x|\le 1-ch^{2\sigma}$ and supported for $|x|\le 1-c_1h^{2\sigma}$, for some $0<c_1<c$, with $\|\partial^{\alpha}\theta_h\|_{L^{\infty}}=O(h^{-2|\alpha|\sigma})$ and $(h\partial_h)^k\theta_h$ bounded for every $k\in\mathbb{N}$, such that $$\label{cut-off-thetah} \gamma\left(\frac{x-p'(\xi)}{\sqrt{h}}\right)\chi(h^\sigma\xi)= \theta_h(x) \gamma\left(\frac{x-p'(\xi)}{\sqrt{h}}\right)\chi(h^\sigma\xi).$$ Straightforward after observing that function $\gamma\big(\frac{x-p'(\xi)}{\sqrt{h}}\big)\chi(h^\sigma\xi)$ is localized around manifold $\Lkg$, meaning that its support is included in $\{(x,\xi) | |\xi| \lesssim h^{-\sigma}, |x| \le 1 - ch^{2\sigma}\}$, for a small $c>0$. \[Prop:Continuity Lp-Lp\] Let $\gamma,\chi\in C^\infty_0(\mathbb{R}^2)$ be equal to 1 in a neighbourhood of the origin and with sufficiently small support, $\Sigma(\xi)=\langle\xi\rangle^\rho$ with $\rho\in\mathbb{N}$, and $\sigma>0$. Then $\oph\big(\gamma\big(\frac{x-p'(\xi)}{\sqrt{h}}\big)\chi(h^\sigma\xi)\Sigma(\xi)\big):L^p\rightarrow L^p$ is bounded and its $\mathcal{L}(L^p)$ norm is estimated by $h^{-\sigma\rho-\beta}$, for a small $\beta>0$, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$, for every $1\le p\le +\infty$. From lemma \[Lem:family\_thetah\] and the fact that $\gamma\left(\frac{x-p'(\xi)}{\sqrt{h}}\right)\chi(h^\sigma\xi)$ is supported in a neighbourhood of $\Lambda_{kg}$ introduced above, we can find a new smooth cut-off function $\gamma_1$, suitably supported, so that$$\oph\left(\gamma\left(\frac{x-p'(\xi)}{\sqrt{h}}\right)\chi(h^\sigma\xi)\Sigma(\xi)\right)= \oph\left( \gamma\left(\frac{x-p'(\xi)}{\sqrt{h}}\right)\chi(h^\sigma\xi)\Sigma(\xi) \gamma_1\Big(\frac{\xi+d\phi(x)}{h^{1/2-\beta}}\Big)\theta_h(x)\right)$$where $\beta>0$ is a small constant, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$, that takes into account the degeneracy of the equivalence between the two equations of $\Lambda_{kg}$ when approaching the boundary of $supp \theta_h$. Denoting $\gamma\left(\frac{x-p'(\xi)}{\sqrt{h}}\right)\chi(h^\sigma\xi)\Sigma(\xi)$ concisely by $A(x,\xi)$ and looking at the kernel associated to the above operator $$\begin{split} K(x,y)&:= \frac{1}{(2\pi h)^2}\int e^{\frac{i}{h}(x-y)\cdot\xi} A\left(\frac{x+y}{2},\xi \right) \gamma_1\Big(\frac{\xi+d\phi(\frac{x+y}{2})}{h^{1/2-\beta}}\Big)\theta_h\Big(\frac{x+y}{2}\Big) d\xi\\ &= \frac{e^{-\frac{i}{h}(x-y)\cdot d\phi(\frac{x+y}{2})}}{(2 \pi h)^2}\theta_h\Big(\frac{x+y}{2}\Big) \int e^{\frac{i}{h}(x-y)\cdot\xi} A\left(\frac{x+y}{2},\xi-d\phi\Big(\frac{x+y}{2}\Big) \right) \gamma_1\Big(\frac{\xi}{h^{1/2-\beta}}\Big) d\xi, \end{split}$$ we observe that, since $$\Big(\frac{x}{\sqrt{h}}\Big)^\alpha e^{\frac{i}{h}(x-y)\cdot\xi} = \Big(\frac{\sqrt{h}}{i}\Big)^{|\alpha|}\partial^\alpha_\xi e^{\frac{i}{h}(x-y)\cdot\xi}$$ and $h^{|\alpha|/2}\partial^\alpha_\xi A(\frac{x+y}{2},\xi)$ is bounded by $h^{-\sigma\rho}$ for any $\alpha\in\mathbb{N}^2$, by making some integration by parts $$\left|\Big(\frac{x}{\sqrt{h}}\Big)^\alpha K(x,y) \right| \lesssim h^{-2-\sigma\rho}\int_{|\xi|\lesssim h^{1/2-\beta}} d\xi \lesssim h^{-1-\sigma\rho-2\beta}, \quad \forall (x,y)\in\mathbb{R}^2\times\mathbb{R}^2.$$ This means in particular that $$|K(x,y)|\lesssim h^{-1-\sigma\rho-2\beta}\Big\langle \frac{x}{\sqrt{h}}\Big\rangle^{-3}, \quad |K(x,y)|\lesssim h^{-1-\sigma\rho-2\beta}\Big\langle \frac{y}{\sqrt{h}}\Big\rangle^{-3},\quad \forall (x,y)$$ implying that $$\sup_x \int |K(x,y)| dy \lesssim h^{-\sigma\rho-2\beta}, \quad \sup_y\int |K(x,y)|dx \lesssim h^{-\sigma\rho-2\beta}.$$ The operator associated to $K(x,y)$ is hence bounded on $L^p$ with norm $O(h^{-\sigma\rho-2\beta})$, for every $1\le p\le +\infty$. The following lemma shows that we have nice upper bounds for operators whose symbol is supported for large frequencies $|\xi|\ge h^{-\sigma}$, $\sigma >0$, when acting on functions $w$ that belong to $H^s_h$, for some large $s$. We state it in space dimension 2 but it is clear that it holds true in general space dimension $d\ge 1$. This result is useful when we want to reduce to symbols rapidly decaying in $|h^{\sigma}\xi|$, for example in the intention of using proposition \[Prop : Continuity from $L^2$ to L\^inf\] or when we want to pass from a symbol of a certain positive order to another one of order zero, up to small losses of order $O(h^{-\beta})$, $\beta>0$ depending linearly on $\sigma$. We can always split a symbol using that $1= \chi(h^{\sigma}\xi) + (1-\chi)(h^{\sigma}\xi)$, for a smooth $\chi$ equal to 1 close to the origin, and consider as remainders all contributions coming from the latter. \[Lem : new estimate 1-chi\] Let $s'\ge 0$ and $\chi \in C^{\infty}_0(\mathbb{R}^2)$, $\chi \equiv 1$ in a neighbourhood of zero. Then $$\|\oph((1-\chi)(h^{\sigma}\xi))w\|_{H^{s'}_h}\le C h^{\sigma(s-s')}\|w\|_{H^s_h} \, , \qquad \qquad \forall s> s'\, .$$ The result is a simple consequence of the fact that $(1-\chi)(h^{\sigma}\xi)$ is supported for $|\xi|\gtrsim h^{-\sigma}$, because $$\begin{split} \|\oph((1-\chi)(h^{\sigma}\xi))w \|_{H^{s'}_h}^2 & = \int (1+ |h\xi|^2)^{s'}|(1-\chi)(h^{\sigma}h\xi)|^2 |\hat{w}(\xi)|^2 d\xi \\ & = \int (1+ |h\xi|^2)^s (1+ |h\xi|^2)^{s'-s}|(1-\chi)(h^{\sigma}h\xi)|^2 |\hat{w}(\xi)|^2 d\xi \\ & \le C h^{2\sigma(s-s')}\|w\|^2_{H^s_h}\, , \end{split}$$ where the last inequality follows from an integration on $|h\xi|\gtrsim h^{-\sigma}$ and from the fact that $s'-s< 0$, $(1+|h\xi|^2)^{s'-s}\le C h^{-2\sigma(s'-s)}$. We introduce the following operator: $$\label{def Lj} \mathcal{L}_j : = \frac{1}{h}\oph(x - p'_j(\xi)), \quad j=1,2,$$ and use the notation $\|\mathcal{L}^\gamma w\| = \|\mathcal{L}^{\gamma_1}_1 \mathcal{L}^{\gamma_2}_2 w\|$ for any $\gamma=(\gamma_1,\gamma_2)\in\mathbb{N}^2$. \[Lem:class of gamma c(x,xi)\] Let $\gamma\in C^\infty_0(\mathbb{R}^2)$ be equal to 1 in a neighbourhood of the origin, $c(x,\xi)\in S_{\delta,\sigma}(1)$ with $\delta\in [0,\frac{1}{2}[$ and $\sigma>0$. Then $\gamma(\frac{x-p'(\xi)}{\sqrt{h}})c(x,\xi)$ belongs to $S_{\frac{1}{2},\sigma}(1)\big(\langle\frac{x-p'(\xi)}{\sqrt{h}}\rangle^{-N}\big)$, for all $N\ge 0$. Straightforward. \[Lem : composition gamma-1 and its argument\] Let $n\in\mathbb{N}$ and $\gamma_n(z)$ be a smooth function such that $|\partial^{\alpha}\gamma_n(z)| \lesssim \langle z\rangle^{-|\alpha|-n}$ for all $\alpha\in\mathbb{N}^2$. Let also $c(x,\xi) \in S_{\delta,\sigma}(1)$, with $\delta \in [0,\frac{1}{2}[$, $\sigma>0$, be supported for $|\xi|\lesssim h^{-\sigma}$. Up to some multiplicative constants independent of $h$, we have the following equality: $$\begin{gathered} \label{sharp gammatilde and its argument} \left[c(x,\xi)\gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big) \right]\sharp \big(x_j- p_j'(\xi)\big) =c(x,\xi)\gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big) \big(x_j- p_j'(\xi)\big) \\ + h \gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\big[(\partial_{\xi_j} c) + (\partial_xc)\cdot (\partial_\xi p'_j)\big] + h \sum_{|\alpha|=2}(\partial^\alpha\gamma_n)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)c(x,\xi)(\partial^\alpha_\xi p'_j)(\xi) + r(x,\xi),\end{gathered}$$ with $r \in h^{3/2-\delta} S_{\frac{1}{2},\sigma}\big(\langle\frac{x-p'(\xi)}{\sqrt{h}}\rangle^{-n}\big)$, and if $\chi \in C^\infty_0(\mathbb{R}^2)$ is such that $\chi(h^\sigma\xi)\equiv 1$ on the support of $c(x,\xi)$, \[est L2 Linfty for GammaTilde\] $$\label{est L2 for GammaTilde} \left\|\oph\Big(c(x,\xi)\gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)(x_j-p'_j(\xi))\Big)\widetilde{v} \right\|_{L^2} \lesssim \sum_{|\gamma|=0}^1 h^{1-\beta} \|\oph(\chi(h^\sigma\xi))\mathcal{L}^\gamma\widetilde{v}\|_{L^2} \, ,$$ $$\label{est Linf for GammaTilde} \left\|\oph\Big(c(x,\xi)\gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)(x_j-p'_j(\xi))\Big)\widetilde{v} \right\|_{L^{\infty}} \lesssim \sum_{|\gamma|=0}^1 h^{\frac{1}{2}\delta_n-\beta} \|\oph(\chi(h^\sigma\xi))\mathcal{L}^\gamma\widetilde{v}\|_{L^2} \, ,$$ where $\delta_n=1$ if $n>0$, 0 otherwise, and $\beta>0$ is small, $\beta\rightarrow 0$ as $\delta,\sigma\rightarrow 0$. Moreover, if $n\in\mathbb{N}^*$ and $\partial^\alpha\gamma_n$ vanishes in a neighbourhood of the origin whenever $|\alpha|\ge 1$, we also have that \[est:L2 Linfty Op(gamma) LLv\] $$\begin{gathered} \label{est:L2 Op(gamma)LLv} \left\|\oph\Big(c(x,\xi)\gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)(x_i-p'_i(\xi))(x_j-p'_j(\xi))\Big)\widetilde{v} \right\|_{L^2} \lesssim\\ \sum_{0\le |\gamma| \le 2} h^{2-\beta} \|\oph(\chi(h^\sigma\xi))\mathcal{L}^\gamma \widetilde{v}\|_{L^2},\end{gathered}$$ $$\begin{gathered} \left\|\oph\Big(c(x,\xi)\gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)(x_i-p'_i(\xi))(x_j-p'_j(\xi))\Big)\widetilde{v} \right\|_{L^{\infty}} \lesssim\\ \sum_{0\le |\gamma|\le 2} h^{\frac{3}{2}-\beta}\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\gamma\widetilde{v}\|_{L^2}.\end{gathered}$$ As $c(x,\xi)\gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big) \in S_{\frac{1}{2},\sigma}\big(\langle\frac{x-p'(\xi)}{\sqrt{h}}\rangle^{-n}\big)$ and $\partial^\alpha_{x,\xi}( x_j-p'_j(\xi))\in S_{0,0}(1)$ for any $|\alpha|\ge 1$, equality follows from the last part of lemma \[Lem : a sharp b\] and symbolic development until order 2, after having observed that $$\label{poisson brackets gamma and its symbol} \left\{c(x,\xi)\gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big), x_j-p'_j(\xi)\right\} = \gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big) \big[(\partial_{\xi_j}c) + (\partial_x c)\cdot (\partial_\xi p'_j)\big],$$ and that, up to some multiplicative negligible, $$\begin{gathered} h^2\sum_{|\alpha|=2}\partial^\alpha_x\left[c(x,\xi)\gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\right](\partial^\alpha_\xi p'_j)(\xi) = h \sum_{|\alpha|=2}(\partial^\alpha\gamma_n)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)c(x,\xi)(\partial^\alpha_\xi p'_j)(\xi) \\ +\underbrace{ h^{\frac{3}{2}}\sum_{\substack{|\alpha|=2\\ |\alpha_1|,|\alpha_2|= 1}} (\partial^{\alpha_1}\gamma_n)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)(\partial^{\alpha_2}_xc)(x,\xi)(\partial^\alpha_\xi p'_j)(\xi) + h^2 \sum_{|\alpha|=2}\gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)(\partial^\alpha_xc)(x,\xi)(\partial^\alpha_\xi p'_j)(\xi) }_{\in h^{\frac{3}{2}-\delta}S_{\frac{1}{2},\sigma}\big(\big\langle\frac{x-p'(\xi)}{\sqrt{h}}\big\rangle^{-n}\big)}.\end{gathered}$$ If $\chi$ is a cut-off function as in the statement, its derivatives vanish on the support of $c(x,\xi)$, and from remark \[Remark:symbols\_with\_null\_support\_intersection\] $$\label{gamma_n-sharp chi} c(x,\xi)\gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big) = \left[c(x,\xi)\gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\right]\sharp \chi(h^\sigma\xi) + r_\infty(x,\xi)$$ with $r_\infty\in h^NS_{\frac{1}{2},\sigma}(\langle \frac{x-p'(\xi)}{\sqrt{h}}\rangle^{-n})$, $N\in\mathbb{N}$ as large as we want. Estimates follow then as a straight consequence of , definition of $\mathcal{L}_j$, proposition \[Prop : Continuity on H\^s\] and semi-classical Sobolev’s injection (resp. proposition \[Prop : Continuity from $L^2$ to L\^inf\]) when $n=0$ (resp. $n>0$). In order to prove the last part of the statement (estimates ) we use equality with $\gamma_n$ replaced by $\widetilde{\gamma}_{n-1}(z)=\gamma_n(z)z_i$, where $|\partial^\alpha \widetilde{\gamma}_{n-1}(z)|\lesssim \langle z\rangle^{-|\alpha|-(n-1)}$, which gives that $$\begin{aligned} c(x,\xi)\gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)(x_i-p'_i(\xi))(x_j-p'_j(\xi)) = \left[c(x,\xi)\gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)(x_i-p'_i(\xi))\right]\sharp (x_j-p'_j(\xi))\\ - h\gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)(x_i-p'_i(\xi))\left[(\partial_{\xi_j}c) + (\partial_xc)\cdot(\partial_\xi p'_j)\right] \\ - h^\frac{3}{2}\sum_{|\alpha|=2}(\partial^\alpha \widetilde{\gamma}_{n-1})\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)c(x,\xi)(\partial^\alpha_\xi p'_j)(\xi) - \sqrt{h}r(x,\xi),\end{aligned}$$ with $r\in h^{\frac{3}{2}-\delta}S_{\frac{1}{2},\sigma}\big(\langle\frac{x-p'(\xi)}{\sqrt{h}}\rangle^{-(n-1)}\big)$. As $\partial^\alpha \widetilde{\gamma}_{n-1}$ vanishes in a neighbourhood of the origin for $|\alpha|=2$ by the hypothesis made on $\gamma_n$, we can rewrite it as $\sum_{l=1}^2 \widetilde{\gamma}^l_{n+2}(z) z_l$, where $\widetilde{\gamma}^l_{n+2}(z):=(\partial^\alpha \widetilde{\gamma}_{n-1})(z)z_l|z|^{-2}$ is such that $|\partial^\beta \widetilde{\gamma}^l_{n+2}(z)|\lesssim \langle z\rangle^{-|\beta|-(n+2)}$. Then, using again equality for all products different from $r(x,\xi)$ in the above right hand side (with $c$ replaced with $h^\delta[(\partial_{\xi_j}c) - (\partial_xc)\cdot(\partial_\xi p'_j)]$ in the second addend, and $\gamma_n$ and $c$ replaced with $\widetilde{\gamma}^l_{n+2}$ and $c(\partial^\alpha_\xi p'_j)$ respectively in the third one, $l=1,2$) we find that $$\begin{gathered} c(x,\xi)\gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)(x_i-p'_i(\xi))(x_j-p'_j(\xi)) = \\ \left[c(x,\xi)\gamma_n\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\right] \sharp (x_i-p'_i(\xi))\sharp (x_j-p'_j(\xi)) + h r_1(x,\xi)\sharp (x_j-p'_j(\xi)) - \sqrt{h}r(x,\xi),\end{gathered}$$ for a new $r_1\in h^{-\delta}S_{\frac{1}{2},\sigma}\big(\langle\frac{x-p'(\xi)}{\sqrt{h}}\rangle^{-n}\big)$. Estimates are then obtained using and propositions \[Prop : Continuity on H\^s\], \[Prop : Continuity from $L^2$ to L\^inf\]). We will also need the following result, which is detailed in lemma 1.2.6 in [@delort:semiclassical] for the one-dimensional case. \[Lem : on e and etilde\] Let $\gamma \in C^{\infty}_0(\mathbb{R}^2)$, and $\phi(x)=\sqrt{1-|x|^2}$. If the support of $\gamma$ is sufficiently small, $$\begin{gathered} (x_k-p'_k(\xi))\gamma\big(\langle\xi\rangle^2(x-p'(\xi))\big) = \sum_{l=1}^2 e^k_l(x,\xi)(\xi_l + d_l\phi(\xi)), \\ (\xi_k + d_k\phi(x)) \gamma\big(\langle\xi\rangle^2(x-p'(\xi))\big) = \sum_{l=1}^2 \widetilde{e}^k_l(x,\xi)(x_l-p'_l(\xi)),\end{gathered}$$ for any $k=1,2$, where functions $e^k_l(x,\xi), \widetilde{e}^k_l(x,\xi)$ are such that, for any $\alpha,\beta\in\mathbb{N}^2$, $$\begin{aligned} |\partial^{\alpha}_x \partial_{\xi}^{\beta}e^k_l(x,\xi)| &\lesssim_{\alpha\beta} \langle\xi\rangle^{-3+2|\alpha|-|\beta|}\,,\\ |\partial^{\alpha}_x \partial_{\xi}^{\beta}\widetilde{e}^k_l(x,\xi)| &\lesssim_{\alpha\beta} \langle\xi\rangle^{3+2|\alpha|-|\beta|} \,,\label{etildekj}\end{aligned}$$ for any $k,l=1,2$. Energy Estimates {#Chap:Energy estimates} ================ The aim of this chapter is to write an energy inequality for $E_n(t;u_\pm, v_\pm)$ and $E^k_3(t;u_\pm, v_\pm)$ respectively, which allows us to propagate the a-priori energy estimates made in theorem \[Thm: bootstrap argument\], i.e. to pass from to , . Such an inequality is in general derived by computing and estimating the derivative in time of the energy, i.e. of the $L^2$ norm to the square of $u^I_\pm, v^I_\pm$. As this computation makes use of the system of equations satisfied by $(u^I_\pm, v^I_\pm)$ (see ), two main difficulties arise due to the quasi-linear nature of the starting problem and the very slow decay in time of the wave solution. On the one hand, among all quadratic terms appearing in the right hand side of we find the quasi-linear ones $Q^\mathrm{w}_0(v_\pm, D_1 v^I_\pm)$ and $Q^{\mathrm{kg}}_0(v_\pm, D_1u^I_\pm)$, whose $L^2$ norm is bounded by $\|v_\pm(t,\cdot)\|_{H^{1,\infty}}(\|u^I_\pm(t,\cdot)\|_{H^1}+ \|v^I_\pm(t,\cdot)\|_{H^1})$, as usual for this kind of terms. This means that they are at the wrong energy level, in the sense that they cannot be controlled in $L^2$ by $E_n(t;u_\pm, v_\pm)$ or $E^k_3(t;u_\pm, v_\pm)$. This causes a “loss of derivatives” in the energy inequality if we roughly estimate $$\frac{1}{2}\partial_t \Big(\|u^I_\pm(t,\cdot)\|^2_{L^2}+ \|v^I_\pm(t,\cdot)\|^2_{L^2}\Big) = -\Im \Big[\langle Q^\mathrm{w}_0(v_\pm, D_1 v^I_\pm), u^I_\pm\rangle + \langle Q^{\mathrm{kg}}_0(v_\pm, D_1u^I_\pm), v^I_\pm\rangle + \dots\Big]$$ using the Cauchy-Schwarz inequality. This issue is however only technical. In fact, by writing system in a vectorial fashion and para-linearising it in order to stress out the very troublesome terms (see subsection \[Subsection: Paralinearization\]) we are able to *symmetrize* it, i.e. to derive an equivalent system in which the quasi-linear contribution is represented by a self-adjoint operator of order 1 (see subsection \[Subs: Symmetrization\], proposition \[Prop: equation of WIs\]). As this operator is self-adjoint it essentially disappears in the energy inequality, replaced with an operator of order 0 whose action on $u^I_\pm, v^I_\pm$ is bounded in $L^2$ by $E_n(t;u_\pm, v_\pm)$ or $E^k_3(t;u_\pm, v_\pm)$, depending on the multi-index $I$ we are dealing with. On the other hand, the $L^2$ norm of some semi-linear contributions to the right hand side of decays very slowly in time. It is the case, for instance, of $Q^\mathrm{kg}_0(v^I_\pm, D_1u_\pm)$, whose $L^2$ norm is bounded by $\|u_\pm(t,\cdot)\|_{H^{2,\infty}}\|v^I_\pm(t,\cdot)\|_{L^2}$ and only has the slow decay of the wave component $u_\pm$. Since we want to prove that $$\partial_t E_n(t;u_\pm, v_\pm) = O\big(\varepsilon t^{-1+\frac{\delta}{2}} E_n(t;u_\pm, v_\pm)^\frac{1}{2}\big), \quad \partial_t E^k_3(t;u_\pm, v_\pm) = O\big(\varepsilon t^{-1+\frac{\delta_k}{2}} E^k_3(t;u_\pm, v_\pm)^\frac{1}{2}\big)$$ we need to get rid of such terms by means of normal forms (see section \[Section : Normal Forms for system\]). Because of the quasi-linear nature of our problem, some of them will be eliminated by an adapted quasi-linear normal form argument (see subsection \[sub: a first normal form transformation\]), while the remaining ones can be treated with an usual semi-linear one (see subsection \[sub: second normal form\]). At that point we will be able to prove proposition \[Prop: Propagation of the energy estimate\] and to derive estimates , . Paralinearization and Symmetrization {#Section: Paralinearization and Symmetrization} ------------------------------------ As anticipated above, the first step towards the derivation of the right energy inequality is to handle the quasi-linear terms appearing in the right hand side of in order to avoid any loss of derivatives. We realize that the very quasi-linear contribution to our system appears in equation through a para-differential operator whose symbol is a real *non symmetric* matrix. As we need this operator to be self-adjoint (up to an operator of order 0), we *symmetrize* equation by defining a new function $W^I_s$ in terms of $W^I$, that will be solution to a new equation in which the symbol of the quasi-linear contribution is a real symmetric matrix (see subsection \[Subs: Symmetrization\]). Also, we set aside subsection \[Subs: Estimate of quadratic terms\] to the estimate of the $L^2$ norms of the non-linear terms in the right hand side of . ### Paralinearization {#Subsection: Paralinearization} Let us remind definitions and . Since admissible vector fields considered in $\mathcal{Z}=\{\Omega, Z_j, \partial_j, j=1,2\}$ exactly commute with the linear part of system , we deduce from remark \[Remark:Vector\_field\_on\_null\_structure\] and that, for any multi-index $I$, $(\Gamma^Iu, \Gamma^Iv)$ is solution to $$\begin{cases} \left(\partial^2_t - \Delta_x\right)\Gamma^Iu = \displaystyle\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|+|I_2|=|I|}} Q_0(\Gamma^{I_1}v, \partial_1\Gamma^{I_2}v)+ \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|+|I_2|<|I|}} c_{I_1,I_2}Q_0(\Gamma^{I_1}v, \partial\Gamma^{I_2}v) ,\\ \left(\partial^2_t - \Delta_x+ 1\right)\Gamma^Iv = \displaystyle\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|+|I_2|=|I|}} Q_0(\Gamma^{I_1}v, \partial_1\Gamma^{I_2}u)+ \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|+|I_2|<|I|}} c_{I_1,I_2}Q_0(\Gamma^{I_1}v, \partial\Gamma^{I_2}u) , \end{cases}$$ with coefficients $c_{I_1,I_2}\in \{-1,0,1\}$ such that $c_{I_1,I_2}=1$ for $|I_1|+|I_2|=|I|$, in which case the derivative $\partial$ acting on $\Gamma^{I_2}v$ (resp. on $\Gamma^{I_2}u$) is equal to $\partial_1$, and $\partial$ representing one of the partial derivatives $\partial_a$, $a\in \{0,1,2\}$. Let us remind that, if $\Gamma^I$ contains at least $k$ ($\le |I|$) space derivatives, above summations are taken over indices $I_1,I_2$ such that $k \le |I_1|+|I_2|\le |I|$. Hence, introducing from , , $$\label{Q0_pm} \begin{split} Q^{\mathrm{w}}_0(v_\pm, D_a v_\pm) & := \frac{i}{4}\left[(v_+ + v_{-})D_a(v_+ + v_{-}) - \frac{D_x}{\langle D_x \rangle}(v_+ - v_{-})\cdot\frac{D_x D_a}{\langle D_x \rangle}(v_+ - v_{-})\right] ,\\ Q^{\mathrm{kg}}_0(v_\pm, D_a u_\pm) & := \frac{i}{4} \left[(v_+ + v_{-})D_a(u_+ + u_{-}) - \frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\cdot\frac{D_x D_a}{|D_x|}(u_+ - u_{-})\right], \end{split}$$ for any $a=0,1,2$, we deduce that $(u^I_+, v^I_+, u^I_{-}, v^I_{-})$ is solution to $$\label{system for uI+-, vI+-} \begin{cases} & (D_t - |D_x|)u^I_+(t,x) = \displaystyle\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1| + |I_2| = |I| }}Q_0^{\mathrm{w}}(v^{I_1}_\pm, D_1 v^{I_2}_\pm) + \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\|I_1| + |I_2| < |I|} } c_{I_1, I_2}Q_0^{\mathrm{w}}(v^{I_1}_\pm, D v^{I_2}_\pm) \\ & (D_t - \langle D_x\rangle)v^I_+(t,x) = \displaystyle\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\|I_1| + |I_2| = |I|} } Q_0^{\mathrm{kg}}(v^{I_1}_\pm, D_1 u^{I_2}_\pm) + \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\|I_1| + |I_2| < |I| }} c_{I_1, I_2}Q_0^{\mathrm{kg}}(v^{I_1}_\pm, D u^{I_2}_\pm) \\ & (D_t + |D_x|)u^I_{-}(t,x) = \displaystyle\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1| + |I_2| = |I| }}Q_0^{\mathrm{w}}(v^{I_1}_\pm, D_1 v^{I_2}_\pm) + \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\|I_1| + |I_2| < |I|} } c_{I_1, I_2}Q_0^{\mathrm{w}}(v^{I_1}_\pm, D v^{I_2}_\pm) \\ & (D_t + \langle D_x \rangle)v^I_{-}(t,x) = \displaystyle\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\|I_1| + |I_2| = |I|} } Q_0^{\mathrm{kg}}(v^{I_1}_\pm, D_1 u^{I_2}_\pm) + \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\|I_1| + |I_2| < |I| }} c_{I_1, I_2}Q_0^{\mathrm{kg}}(v^{I_1}_\pm, D u^{I_2}_\pm) \end{cases}$$ The quasi-linear structure of the above system can be emphasized by using and decomposing $Q_0^{\mathrm{w}}(v_\pm, D_1 v^I_\pm)$, $Q_0^{\mathrm{kg}}(v_\pm, D_1 u^I_\pm)$ as follows: $$\label{dec quasi-linear term} Q_0^{\mathrm{w}}(v_\pm, D_1 v^I_\pm) = (QL)_1 + (SL)_1,\quad Q_0^{\mathrm{kg}}(v_\pm, D_1 u^I_\pm) =(QL)_2 + (SL)_2 ,$$ with $$\begin{split} & (QL)_1 := \frac{i}{4}\left[Op^B\big((v_+ + v_{-})\eta_1\big)(v^I_+ + v^I_{-}) - Op^B\Big(\frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\cdot\frac{\eta \eta_1}{\langle\eta\rangle}\Big)(v^I_+- v^I_{-}) \right] ,\\ & (SL)_1 := \frac{i}{4}\left[Op^B\big(D_1(v^I_+ + v^I_{-})\big)(v_+ + v_{-}) - Op^B\Big(\frac{D_x D_1}{\langle D_x\rangle}(v^I_+ - v^I_{-})\cdot\frac{\eta}{\langle\eta\rangle}\Big)(v_+ - v_{-}) \right. \\ & \left. \hspace{2cm }+ Op^B_R\big((v_+ + v_{-})\eta_1\big)(v^I_+ + v^I_{-})\big) - Op^B_R\Big(\frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\cdot\frac{\eta \eta_1}{\langle \eta\rangle}\Big)(v^I_+ - v^I_{-})\right], \\ & (QL)_2 := \frac{i}{4}\left[Op^B\big((v_+ + v_{-})\eta_1\big)(u^I_+ + u^I_{-}) - Op^B\Big(\frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\cdot\frac{\eta \eta_1}{|\eta|}\Big)(u^I_+- u^I_{-}) \right] ,\\ & (SL)_2 := \frac{i}{4}\left[Op^B\big(D_1(u^I_+ + u^I_{-})\big)(v_+ + v_{-}) - Op^B\Big(\frac{D_x D_1}{|D_x|}(u^I_+ - u^I_{-})\cdot\frac{\eta}{\langle\eta\rangle}\Big)(v_+ - v_{-}) \right. \\ & \left. \hspace{2cm }+ Op^B_R\big((v_+ + v_{-})\eta_1\big)(u^I_+ + u^I_{-}) - Op^B_R\Big(\frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\cdot\frac{\eta \eta_1}{|\eta|}\Big)(u^I_+- u^I_{-}) \right]\,, \end{split}$$ where the Bony quantization $Op^B$ (resp. $Op^B_R$) has been defined in \[Def: Paradiff\_operator\] (resp. in ). We do a similar decomposition also for the semi-linear contribution $Q^{\mathrm{kg}}_0(v^I_\pm, D_1 u_\pm)$, for this term will thereafter be the object of the two normal forms mentioned at the beginning of this section: $$\label{dec semi-linear term} \begin{split} Q_0^{\mathrm{kg}}(v^I_\pm, D_1 u_\pm) &= \frac{i}{4}\left[Op^B\big((v^I_+ + v^I_{-})\eta_1\big)(u_+ + u_{-}) - Op^B\Big(\frac{D_x}{\langle D_x\rangle}(v^I_+ - v^I_{-})\cdot\frac{\eta \eta_1}{|\eta|}\Big)(u_+- u_{-}) \right] \\ & +\frac{i}{4}\left[Op^B\big(D_1(u_+ + u_{-})\big)(v^I_+ + v^I_{-}) - Op^B\Big(\frac{D_x D_1}{|D_x|}(u_+ - u_{-})\cdot\frac{\eta}{\langle\eta\rangle}\Big)(v^I_+ - v^I_{-}) \right] \\ & +\frac{i}{4}\left[ Op^B_R\big((v^I_+ + v^I_{-})\eta_1\big)(u_+ + u_{-}) - Op^B_R\Big(\frac{D_x}{\langle D_x\rangle}(v^I_+ - v^I_{-})\cdot\frac{\eta \eta_1}{|\eta|}\Big)(u_+- u_{-}) \right]\,. \end{split}$$ In order to handle system in the most efficient way we proceed to write it in a vectorial fashion. To this purpose, we introduce the following matrices: $$\label{matrices A A'} A(\eta)= \begin{bmatrix} |\eta| & 0 & 0 & 0 \\ 0 & \langle\eta\rangle & 0 & 0\\ 0 & 0 & -|\eta| & 0 \\ 0 & 0 & 0 & - \langle\eta\rangle \end{bmatrix}, \quad A'(V;\eta) := \begin{bmatrix} 0 & a_k\eta_1 & 0 & b_k\eta_1 \\ a_0\eta_1 & 0 & b_0\eta_1 & 0 \\ 0 & a_k\eta_1 & 0 & b_k\eta_1\\ a_0\eta_1 & 0 & b_0\eta_1 & 0 \end{bmatrix},$$ $$\label{matrix A''(VI)} A''(V^I;\eta) := \begin{bmatrix} 0 & 0 & 0 & 0 \\ a^I_0\eta_1 & 0 & b^I_0 \eta_1 & 0 \\ 0 & 0 & 0 & 0\\ a^I_0\eta_1 & 0 & b^I_0 \eta_1 & 0 \end{bmatrix},$$ $$\label{matrices C' C''} C'(W^I ;\eta):= \begin{bmatrix} 0 & c^I_0 & 0 & d^I_0 \\ 0 & e_0^I & 0 & f_0^I \\ 0 & c^I_0 & 0 & d^I_0 \\ 0 & e_0^I & 0 & f_0^I \end{bmatrix}, \quad C''(U ;\eta):= \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & e_0 & 0 & f_0 \\ 0 & 0 & 0 & 0 \\ 0 & e_0 & 0 & f_0 \end{bmatrix}$$ where $$\label{def ak, bk, a0, b0} \begin{cases} & a_k=a_k(v_{\pm}; \eta) := \frac{i}{4}\big[(v_+ + v_{-}) - \frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\cdot \frac{\eta}{\langle\eta\rangle}\big] \\ & b_k=b_k(v_{\pm}; \eta) := \frac{i}{4}\big[(v_+ + v_{-}) + \frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\cdot \frac{\eta}{\langle\eta\rangle}\big] \\ & a_0=a_0(v_{\pm}; \eta) := \frac{i}{4}\big[(v_+ + v_{-}) - \frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\cdot \frac{\eta}{|\eta|}\big] \\ & b_0=b_0(v_{\pm}; \eta) := \frac{i}{4}\big[(v_+ + v_{-}) + \frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\cdot \frac{\eta}{|\eta|}\big] \end{cases}$$ $$\label{def c0 d0 e0 f0} \begin{cases} & c_0=c_0(v_{\pm};\eta) := \frac{i}{4}\big[D_1(v_+ + v_{-}) - \frac{D_x D_1}{\langle D_x\rangle}(v_+ - v_{-})\cdot\frac{\eta}{\langle\eta\rangle}\big] \\ & d_0=d_0(v_{\pm};\eta) := \frac{i}{4}\big[D_1(v_+ + v_{-}) + \frac{D_x D_1}{\langle D_x\rangle}(v_+ - v_{-})\cdot\frac{\eta}{\langle\eta\rangle}\big] \\ & e_0=e_0(u_{\pm};\eta) := \frac{i}{4}\big[D_1(u_+ + u_{-}) - \frac{D_x D_1}{|D_x|}(u_+ - u_{-})\cdot \frac{\eta}{\langle\eta \rangle}\big] \\ & f_0=f_0(u_{\pm}; \eta) := \frac{i}{4}\big[D_1(u_+ + u_{-}) + \frac{D_x D_1}{|D_x|}(u_+ - u_{-})\cdot\frac{\eta}{\langle\eta\rangle}\big] \end{cases}$$ $$\label{def_aI0 bI0 cI0 dI0 eI0 fI0} \begin{gathered} a^I_0 = a_0(v^I_\pm;\eta), \quad b^I_0 = b_0(v^I_\pm; \eta), c^I_0 = c_0(v^I_\pm; \eta), \quad d^I_0 = d_0(v^I_\pm;\eta), \\ e^I_0 = e_0(u^I_\pm;\eta), \quad f_0^I(u^I_\pm;\eta), \end{gathered}$$ vectors $W, U, V$: $$\label{def W,V, U} W:= \begin{bmatrix} u_+ \\ v_+ \\ u_{-} \\ v_{-} \end{bmatrix}, \quad V:= \begin{bmatrix} 0 \\ v_+ \\ 0 \\ v_ {-} \end{bmatrix}, \quad U:= \begin{bmatrix} u_+ \\ 0 \\ u_{-} \\ 0 \end{bmatrix},$$ along with $W^I$ (resp. $V^I, U^I$) defined from $W$ (resp. $V,U$) by replacing $u_\pm, v_\pm$ with $u^I_\pm, v^I_\pm$; and finally $$\label{matrix QI} Q^I_0(V,W) = \begin{bmatrix} \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I) \\ |I_2|<|I|}} c_{I_1, I_2} Q^{\mathrm{w}}_0(v^{I_1}_\pm, D v^{I_2}_\pm) \\ \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I) \\ |I_1|, |I_2|<|I|}} c_{I_1, I_2} Q^{\mathrm{kg}}_0(v^{I_1}_\pm, D u^{I_2}_\pm) \\ \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I) \\ |I_2|<|I|}} c_{I_1, I_2} Q^{\mathrm{w}}_0(v^{I_1}_\pm, D v^{I_2}_\pm) \\ \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I) \\ |I_1|, |I_2|<|I|}} c_{I_1, I_2} Q^{\mathrm{kg}}_0(v^{I_1}_\pm, D u^{I_2}_\pm) \end{bmatrix}$$ The quantization $Op^B$ (resp. $Op^B_R$) of a matrix $A=(a_{ij})_{1\le i,j\le n}$ is meant as a matrix of operators $Op^B(A) = (Op^B(a_{ij}))_{1\le i,j\le n}$ (resp. $Op^B_R(A) = (Op^B_R(a_{ij}))_{1\le i,j\le n}$), and for a vector $Y=[y_1, \dots, y_n]$, $$Op^B(A)Y^\dagger = \begin{bmatrix} \displaystyle\sum_{j=1}^n Op^B(a_{1j})y_j \\ \vdots \\ \displaystyle\sum_{j=1}^n Op^B(a_{nj})y_j \end{bmatrix},$$ $Y^\dagger$ being the transpose of $Y$. We also remind that $$\|A\|_{L^2}=\Big(\sum_{i,j}|a_{ij}|^2\Big)^{\frac{1}{2}}, \quad \|A\|_{L^\infty}=\sup_{ij}|a_{ij}|.$$ With notations introduced above, system writes in the following compact fashion which has the merit to well highlight, among all non-linear terms, the very quasi-linear contributions $(QL)_1, (QL)_2$, represented below by $Op^B(A'(V;\eta))W^I$: $$\label{equation WI} \begin{split} D_t W^I & = A(D) W^I + Op^B(A'(V;\eta))W^I + Op^B(C'(W^I;\eta))V + Op^B_R(A'(V;\eta))W^I \\ & + Op^B(A''(V^I;\eta))U + Op^B(C''(U;\eta))V^I + Op^B_R(A''(V^I;\eta))U + Q^I_0(V,W). \end{split}$$ The energies defined in take the form \[energy Ekm(t,W)\] $$\begin{gathered} E_n(t;u_\pm, v_\pm) = \sum_{|\alpha|\le n} \|D^\alpha_x W(t,\cdot)\|_{L^2}, \quad \forall\, n\in\mathbb{N}, n\ge 3, \\ E^k_3(t;u_\pm, v_\pm) = \sum_{\substack{ |\alpha|+|I|\le 3\\ |I|\le 3-k}}\|D^\alpha_x W^I(t,\cdot)\|^2_{L^2}, \quad\forall \, 0\le k \le 2, \label{energy_Ek2}\end{gathered}$$ and we can refer to them, respectively, as $E_n(t;W), E^k_3(t;W)$. We also notice that, since \[commutators\_Z\] $$\label{commutator_Z_Dt-|D|} [\Gamma, D_t \pm |D_x|]= \begin{cases} 0 \quad &\text{if } \Gamma\in \{\Omega, \partial_j, j=1,2\} ,\\ \mp \frac{D_m}{|D_x|}(D_t \pm |D_x|) \quad &\text{if } \Gamma =Z_m, m=1,2, \end{cases}$$ $$\label{commutator_Z_Dt-<D>} [\Gamma, D_t \pm \langle D_x\rangle]= \begin{cases} 0 \quad &\text{if } \Gamma\in \{\Omega, \partial_j, j=1,2\} ,\\ \mp \frac{D_m}{\langle D_x\rangle}(D_t \pm \langle D_x\rangle) \quad &\text{if } \Gamma =Z_m, m=1,2, \end{cases}$$ and operators $D_m|D_x|^{-1}, D_m\langle D_x\rangle^{-1}$ are continuous on $L^2$ for $m=1,2$, there exists a constant $C>0$ such that $$\label{equivalence GammaIW-Ekn} C^{-1}\sum_{I\in\mathcal{I}^k_3}\|\Gamma^I W(t,\cdot)\|^2_{L^2}\le E^k_3(t;W)\le C\sum_{I\in\mathcal{I}^k_3}\|\Gamma^I W(t,\cdot)\|^2_{L^2},$$ where, for any integer $0\le k \le 2$, $$\label{set_Ik3} \mathcal{I}^k_3:=\left\{|I|\le 3: \Gamma^I = D^\alpha_x \Gamma^J \text{ with } |\alpha|+|J|=|I|, |J|\le 3-k\right\}.$$ For convenience, we also introduce the following set: $$\label{set_In} \mathcal{I}_n:=\left\{|I|\le n : \Gamma^I= D^\alpha_x \text{ with } |\alpha|=|I|\right\}, \quad n\in\mathbb{N}, n\ge 3.$$ Matrices $A(\eta), A'(V;\eta), A''(V^I;\eta)$ are of order 1 and $A'(V;\eta), A''(V^I;\eta)$ are singular at $\eta = 0$ (i.e. some of their elements are singular at $\eta = 0$), while $C'(W^I;\eta), C''(U;\eta)$ are of order 0. Since we will need to do some symbolic calculus on $A'(V;\eta)$, we need to isolate the mentioned singularity. We hence define $$\label{matrices A'1 A'-1} A'_1(V;\eta) := \begin{bmatrix} 0 & a_0\eta_1 & 0 & b_0\eta_1 \\ a_0\eta_1 & 0 & b_0\eta_1 & 0 \\ 0 & a_0\eta_1 & 0 & b_0\eta_1\\ a_0\eta_1 & 0 & b_0\eta_1 & 0 \end{bmatrix}, \quad A'_{-1}(V;\eta) : = \begin{bmatrix} 0 & (a_k-a_0)\eta_1 & 0 & (b_k-b_0)\eta_1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 &0\\ 0 & (a_k-a_0)\eta_1 & 0 & (b_k-b_0)\eta_1 \end{bmatrix},$$ $A'_1(V;\eta)$ being a matrix of order 1, $A'_{-1}(V;\eta)$ of order $-1$, both singular at $\eta = 0$, and write $A'_1(V;\eta) = A'_1(V;\eta)(1-\chi)(\eta) + A'_1(V;\eta)\chi(\eta)$, where $\chi\in C^\infty_0(\mathbb{R}^2)$ is equal to 1 in the unit ball. Equation can be the rewritten as follows $$\label{equation WI-1} \begin{split} D_t W^I & = A(D) W^I + Op^B(A'_1(V;\eta)(1-\chi)(\eta))W^I + Op^B(A'_1(V;\eta)\chi(\eta))W^I \\ & + Op^B(A'_{-1}(V;\eta))W^I + Op^B(C'(W^I;\eta))V + Op^B_R(A'(V;\eta))W^I + Op^B(A''(V^I;\eta))U \\& + Op^B(C''(U;\eta))V^I + Op^B_R(A''(V^I;\eta))U + Q^I_0(V,W), \end{split}$$ and the symbol $A'_1(V;\eta)(1-\chi)(\eta)$ associated to the quasi-linear contribution is no longer singular at $\eta=0$. We observe that this matrix is real since $i(v_+ + v_{-}) = 2\partial_t v$, $i\frac{D_x}{\langle D_x\rangle}(v_+ - v_{-}) = 2\partial_x v$ and $v$ is a real solution, but it is not symmetric and such a lack of symmetry could lead to a loss of derivatives when writing an energy inequality for $W^I$. The issue is however only technical, in the sense that $A_1(V;\eta)(1-\chi)(\eta)$ can be replaced with a real, symmetric matrix, as explained in subsection \[Subs: Symmetrization\] (see proposition \[Prop: equation of WIs\]). Before proving such result, we need to derive some $L^2$ estimates for the semi-linear terms in the right hand side of . ### Estimates of quadratic terms {#Subs: Estimate of quadratic terms} In this subsection we recover some estimates for the $L^2$ norm of the non-linear terms in the right hand side of equation . \[Lemma: L2 estimate of semilinear terms\] Let $I$ be a fixed multi-index and $\chi\in C^\infty_0(\mathbb{R}^2)$ equal to 1 in a neighbourhood of the origin. The following estimates hold: $$\label{L2 est on Op(A'1 + A'0)WI} \left\|\Big[Op^B\big(A'_1(V;\eta)\chi(\eta)\big) + Op^B\big(A'_{-1}(V;\eta)\big)\Big]W^I(t,\cdot)\right\|_{L^2} \lesssim \|V(t,\cdot)\|_{H^{1,\infty}} \|W^I(t,\cdot)\|_{L^2};$$ $$\label{L2 est on Op(C(WI,eta))V} \left\|Op^B(C'(W^I;\eta))V(t,\cdot) \right\|_{L^2}\lesssim \|V(t,\cdot)\|_{H^{6,\infty}} \|W^I(t,\cdot)\|_{L^2} ;$$ $$\label{L2 est on OpBR(A')WI} \| Op^B_R(A'(V;\eta))W^I(t,\cdot)\|_{L^2} \lesssim \|V(t,\cdot)\|_{H^{7,\infty}} \|W^I(t,\cdot)\|_{L^2};$$ $$\begin{gathered} \label{L2 est on OpBR(A'')W} \| Op^B(A''(V^I;\eta))U(t,\cdot)\|_{L^2} + \| Op^B_R(A''(V^I;\eta))U(t,\cdot)\|_{L^2} \\ \lesssim \big( \| R_1U (t,\cdot)\|_{H^{6,\infty}} + \|U(t,\cdot)\|_{H^{6,\infty}}\big)\|V^I(t,\cdot)\|_{L^2} ;\end{gathered}$$ $$\label{L2 est on OpB(C")VI} \| Op^B (C''(U;\eta))V^I(t,\cdot)\|_{L^2} \lesssim \big( \| R_1U (t,\cdot)\|_{H^{2,\infty}} + \|U(t,\cdot)\|_{H^{2,\infty}}\big)\|W^I(t,\cdot)\|_{L^2} ;$$ $\bullet$ Inequality follows applying proposition \[Prop : Paradiff action on Sobolev spaces-NEW\] to $Op^B\left(A'_{-1}(V;\eta)(1-\chi)(\eta)\right)W^I$ whose symbol $A'_{-1}(V;\eta)(1-\chi)(\eta)$ is of order $-1$ and has $M^{-1}_0$ seminorm bounded from above by $\|V(t,\cdot)\|_{H^{1,\infty}}$, after definitions , and . $\bullet$ Since from definition of matrix $C'(W^I;\eta)$ $$\begin{gathered} \left\|Op^B(C'(W^I;\eta))V \right\|_{L^2}\lesssim \left\|Op^B(D_1(v^I_+ + v^I_{-}))v_\pm\right\|_{L^2} + \left\|Op^B\Big(\frac{D_xD_1}{\langle D_x\rangle}(v^I_+ -v^I_{-})\cdot\frac{\eta}{\langle\eta\rangle}\Big)v_\pm\right\|_{L^2} \\ + \left\|Op^B\big(D_1(u^I_+ + u^I_{-})\big)v_\pm\right\|_{L^2} + \left\|Op^B\Big(\frac{D_xD_1}{| D_x|}(u^I_+ -u^I_{-})\cdot\frac{\eta}{\langle\eta\rangle}\Big)v_\pm\right\|_{L^2},\end{gathered}$$ we reduce to prove inequality for $Op^B\big(\frac{D_xD_1}{\langle D_x\rangle}(v^I_+ -v^I_{-})\cdot\frac{\eta}{\langle\eta\rangle}\big)v_+$, the same argument being applicable to all other $L^2$ norms appearing in the above right hand side. Using equality , and considering a new admissible cut-off function $\chi_1$ identically equal to 1 on the support of $\chi$, we first derive that $$\begin{aligned} & {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Op^B\Big(\frac{D_x D_1}{\langle D_x \rangle}(v^I_+ + v^I_{-})\cdot\frac{\eta}{\langle\eta\rangle}\Big)v_+}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Op^B\Big(\frac{D_x D_1}{\langle D_x \rangle}(v^I_+ + v^I_{-})\cdot\frac{\eta}{\langle\eta\rangle}\Big)v_+}{\tmpbox}}(\xi) = \frac{1}{(2\pi)^2}\int \chi\Big(\frac{\xi-\eta}{\langle\eta\rangle}\Big) {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\frac{D_x D_1}{\langle D_x \rangle}(v^I_+ + v^I_{-})}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\frac{D_x D_1}{\langle D_x \rangle}(v^I_+ + v^I_{-})}{\tmpbox}}(\xi-\eta) \cdot {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\frac{D_x}{\langle D_x \rangle} v_+}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\frac{D_x}{\langle D_x \rangle} v_+}{\tmpbox}}(\eta) d\eta \\ & = \frac{1}{(2\pi)^2}\int \chi\Big(\frac{\xi-\eta}{\langle\eta\rangle}\Big)\Big(\frac{\xi_1 - \eta_1}{\langle\eta\rangle}\Big) {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\frac{D_x}{\langle D_x\rangle}(v^I_+ + v^I_{-})}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\frac{D_x}{\langle D_x\rangle}(v^I_+ + v^I_{-})}{\tmpbox}}(\xi-\eta) \cdot{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{D_x v_+}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{D_x v_+}{\tmpbox}}(\eta) d\eta \\ & = \frac{1}{(2\pi)^2}\int \chi_1\Big(\frac{\xi-\eta}{\langle\eta\rangle}\Big){\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{ \left[\chi\Big(\frac{D_x}{\langle\eta\rangle}\Big)\frac{D_1}{\langle\eta\rangle}\frac{D_x}{\langle D_x\rangle}(v^I_+ + v^I_{-})\right]}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{ \left[\chi\Big(\frac{D_x}{\langle\eta\rangle}\Big)\frac{D_1}{\langle\eta\rangle}\frac{D_x}{\langle D_x\rangle}(v^I_+ + v^I_{-})\right]}{\tmpbox}}(\xi-\eta)\cdot {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{D_x v_+}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{D_x v_+}{\tmpbox}}(\eta) d\eta \\ &= {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Op^B\Big(\chi\Big(\frac{D_x}{\langle\eta\rangle}\Big)\frac{D_1}{\langle\eta\rangle}\frac{D_x}{\langle D_x\rangle}( v^I_+ + v^I_{-})\Big) D_x v_+}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Op^B\Big(\chi\Big(\frac{D_x}{\langle\eta\rangle}\Big)\frac{D_1}{\langle\eta\rangle}\frac{D_x}{\langle D_x\rangle}( v^I_+ + v^I_{-})\Big) D_x v_+}{\tmpbox}}(\xi).\end{aligned}$$ Successively, by decomposition and the fact that $R(u,v)$ is symmetric in $(u,v)$, we have that $$\begin{gathered} Op^B\Big(\chi\Big(\frac{D_x}{\langle\eta\rangle}\Big)\frac{D_1}{\langle\eta\rangle}\frac{D_x}{\langle D_x\rangle}( v^I_+ + v^I_{-})\Big) D_x v_+ = \chi\Big(\frac{D_x}{\langle\eta\rangle}\Big)\frac{D_1}{\langle\eta\rangle}\frac{D_x}{\langle D_x \rangle}(v^I_+ + v^I_{-}) \cdot D_x v_+\\ - \left[Op^B(D_x v_+) + Op^B_R(D_x v_+)\right]\left[\chi\Big(\frac{D_x}{\langle\eta\rangle}\Big)\frac{D_1}{\langle\eta\rangle}\frac{D_x}{\langle D_x\rangle}(v^I_+ + v^I_{-})\right],\end{gathered}$$ so propositions \[Prop : Paradiff action on Sobolev spaces-NEW\], \[Prop: Paradiff action with non smooth symbols and R(u,v)\] $(ii)$, and the fact that $\chi\Big(\frac{D_x}{\langle\eta\rangle}\Big)\frac{D_1}{\langle\eta\rangle}\frac{D_x}{\langle D_x\rangle}$ is an operator uniformly bounded on $L^2$, imply that $$\left\| Op^B\Big(\frac{D_x D_1}{\langle D_x \rangle}(v^I_+ + v^I_{-})\cdot\frac{\eta}{\langle\eta\rangle}\Big)v_+\right\|_{L^2} \lesssim \|V(t,\cdot)\|_{H^{6,\infty}} \|V^I(t,\cdot)\|_{L^2}.$$ $\bullet$ By definition of $A'(V;\eta)$, $$\begin{gathered} \left\| Op^B_R\big(A'(V;\eta)\big)W^I(t,\cdot) \right\|_{L^2} \lesssim \left\| Op^B_R(v_++v_{-})v^I_\pm \right\|_{L^2} + \left\| Op^B_R\Big(\frac{D_x}{\langle D_x\rangle}(v_+-v_{-})\cdot\frac{\eta}{\langle\eta\rangle}\Big)v^I_\pm \right\|_{L^2}\\ + \left\|Op^B_R(v_++v_{-})u^I_\pm \right\|_{L^2} + \left\| Op^B_R\Big(\frac{D_x}{\langle D_x\rangle}(v_+-v_{-})\cdot\frac{\eta}{|\eta|}\Big)u^I_\pm\right\|_{L^2}.\end{gathered}$$ Let us only show that inequality holds for $Op^B\big(\frac{D_x}{\langle D_x\rangle}(v_+-v_{-})\cdot\frac{\eta \eta_1}{|\eta|}\big)u^I_+$. For a smooth cut-off function $\phi$ equal to 1 in the unit ball we write $$\begin{gathered} Op^B_R\left(\frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\cdot \frac{\eta\eta_1}{|\eta|}\right)u^I_+ = Op^B_R\Big(\frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\cdot \frac{\eta\eta_1}{|\eta|} \phi(\eta)\Big)u^I_+ \\ + Op^B_R\Big(\frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\cdot \frac{\eta\eta_1}{|\eta|}(1-\phi)(\eta)\Big)u^I_+,\end{gathered}$$ where by proposition \[Prop: Paradiff action with non smooth symbols and R(u,v)\] $(i)$ $$\begin{gathered} \left\| Op^B_R\Big(\frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\cdot \frac{\eta\eta_1}{|\eta|} \phi(\eta)\Big)u^I_+\right\|_{L^2}\lesssim \left\| \frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})(t,\cdot)\right\|_{L^\infty} \|u^I_+(t,\cdot)\|_{L^2}\\ \lesssim \|V(t,\cdot)\|_{H^{1,\infty}} \|W^I(t,\cdot)\|_{L^2}.\end{gathered}$$ On the other hand $$Op^B_R\Big(\frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\cdot \frac{\eta\eta_1}{|\eta|}(1-\phi)(\eta)\Big)u^I_+ = \int e^{ix\cdot\xi} m(\xi,\eta) \left[\langle D_x\rangle^7(\hat{v}_+ - \hat{v}_{-})(\xi-\eta)\right] \hat{u}^I_+(\eta) d\xi d\eta,$$ where $$m(\xi,\eta):= \frac{1}{(2\pi)^2} \left(1 - \chi\left(\frac{\xi-\eta}{\langle\eta\rangle}\right) - \chi\left(\frac{\eta}{\langle\xi - \eta\rangle}\right)\right)(1-\phi)(\eta) \frac{\xi-\eta}{\langle\xi -\eta\rangle^8}\cdot \frac{\eta\eta_1}{|\eta|}$$ and frequencies $\xi-\eta$ and $\eta$ are either bounded or equivalent on the support of $m(\xi,\eta)$. Therefore $m(\xi,\eta)$ satisfies the hypothesis of lemma \[Lem\_appendix: Kernel with 1 function\] $(i)$ $|\partial^\alpha_\xi \partial^\beta_\eta m(\xi,\eta)|\lesssim \langle\xi\rangle^{-3}\langle\eta\rangle^{-3}$ for any $\alpha,\beta\in\mathbb{N}^2$, and by inequality $$\left\| Op^B_R\Big(\frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\cdot \frac{\eta\eta_1}{|\eta|}(1-\phi)(\eta)\Big)u^I_+\right\|_{L^2}\lesssim \|V(t,\cdot)\|_{H^{7,\infty}}\|W^I(t,\cdot)\|_{L^2}.$$ $\bullet$ From definition of $A''(V;\eta)$, $$\left\| Op^B\big(A''(V;\eta)\big)U(t,\cdot) \right\|_{L^2}\lesssim \left\|Op^B\big((v^I_+ + v^I_{-})\eta_1\big)u_\pm \right\|_{L^2} + \left\|Op^B\Big(\frac{D_x}{\langle D_x\rangle}(v^I_+ -v^I_{-})\cdot\frac{\eta\eta_1}{|\eta|}\Big)u_\pm\right\|_{L^2},$$ (the same inequality holds evidently when $Op^B$ is replaced by $Op^B_R$). As done for previous cases, we reduce to show for $Op^B\big(\frac{D_x}{\langle D_x\rangle}(v^I_+ - v^I_{-} )\cdot \frac{\eta\eta_1}{|\eta|}\big)u_+$ (resp. for $Op^B$ replaced with $Op^B_R$). Using decomposition and the fact that $R(u,v)$ is symmetric in $(u,v)$ we find that $$\begin{gathered} Op^B\Big(\frac{D_x}{\langle D_x\rangle}(v^I_+ - v^I_{-} )\cdot \frac{\eta\eta_1}{|\eta|}\Big)u_+ = \frac{D_x}{\langle D_x\rangle}(v^I_+ - v^I_{-}) \cdot\frac{D_xD_1}{|D_x|}u_+\\ - Op^B\Big(\frac{D_x D_1}{|D_x|}u_+\cdot\frac{\eta}{\langle\eta\rangle}\Big)(v^I_+ - v^I_{-}) - Op^B_R\Big(\frac{D_x D_1}{|D_x|}u_+\cdot\frac{\eta}{\langle\eta\rangle}\Big)(v^I_+ - v^I_{-}),\end{gathered}$$ and $$Op^B_R\Big(\frac{D_x}{\langle D_x\rangle}(v^I_+ - v^I_{-} )\cdot \frac{\eta\eta_1}{|\eta|}\Big)u_+ = Op^B_R\Big(\frac{D_x D_1}{|D_x|}u_+\cdot\frac{\eta}{\langle\eta\rangle}\Big)(v^I_+ - v^I_{-}),$$ so a direct application of propositions \[Prop : Paradiff action on Sobolev spaces-NEW\] and \[Prop: Paradiff action with non smooth symbols and R(u,v)\] $(ii)$ gives that the $L^2$ norm of the above right hand sides is bounded by $\left\| \frac{D_x D_1}{|D_x|}u_+\right\|_{H^{4,\infty}} \|V^I(t,\cdot)\|_{L^2}$, and hence by $\|R_1U(t,\cdot)\|_{H^{6,\infty}}\|V^I(t,\cdot)\|_{L^2}$, which gives inequality . $\bullet$ From definition of matrix $C''(U;\eta)$, $$\begin{gathered} \|Op^B(C''(U;\eta))V^I\|_{L^2} \lesssim \\ \left\|Op^B(D_1(u_+ + u_{-}))(v^I_+ + v^I_{-})\right\|_{L^2} + \left\| Op^B\left(\frac{D_x D_1}{|D_x|}(u_+ - u_{-})\cdot\frac{\eta}{\langle\eta\rangle}\right)(v^I_+ - v^I_{-})\right\|_{L^2},\end{gathered}$$ so estimate follows immediately from proposition \[Prop : Paradiff action on Sobolev spaces-NEW\]. Lemmas \[Lem: L2 est nonlinearities\] and \[Lem:L2 est nonlinearity Dt\] below are introduced with the aim of deriving an estimate of the $L^2$ norm of vector $Q^I_0(V,W)$ defined in (see corollary \[Cor: L2 est QI0(V,W)\]). We remind that the summations defining $Q^I_0(V,W)$ come from the action of family $\Gamma^I$ of admissible vector fields on the quadratic non-linearities $Q_0(v, \partial_1v)$ and $Q_0(v, \partial_1u)$ in (or, in terms of $u_\pm, v_\pm$, on $Q^\mathrm{w}_0(v_\pm, D_1v_\pm)$ and $ Q^\mathrm{kg}_0(v_\pm, D_1u_\pm)$). According to remark \[Remark:Vector\_field\_on\_null\_structure\], if $I\in \mathcal{I}_n$ and $\Gamma^I$ is a product of spatial derivatives only the action of $\Gamma^I$ on $Q^\mathrm{w}_0(v_\pm, D_1v_\pm)$ (resp. on $Q^\mathrm{kg}_0(v_\pm, D_1 u_\pm)$) “distributes” entirely on its factors, meaning that $$\Gamma^I Q^\mathrm{w}_0(v_\pm, D_1v_\pm) = \sum_{\substack{(I_1,I_2)\in \mathcal{I}(I)\\ |I_1|+|I_2|=|I|}}Q^\mathrm{w}_0(v^{I_1}_\pm, D_1 v^{I_2}_\pm),$$ (the same for $\Gamma^I Q^\mathrm{kg}_0(v_\pm, D_1u_\pm)$), and all coefficients $c_{I_1,I_2}$ in the right hand side of are equal to 0. On the contrary, if $I\in\mathcal{I}^k_3$ for $0\le k\le 2$ and $\Gamma^I$ contains some Klainerman vector fields $\Omega, Z_m, m=1,2$, the commutation between $\Gamma^I$ and the null structure gives rise to new quadratic contributions in which the derivative $D_1$ is eventually replaced with $D_2, D_t$. As already seen in , in this case we have $$\label{GammaI_Q0} \Gamma^I Q^\mathrm{w}_0(v_\pm, D_1v_\pm) = \sum_{\substack{(I_1,I_2)\in \mathcal{I}(I)\\ |I_1|+|I_2|=|I|}}Q^\mathrm{w}_0(v^{I_1}_\pm, D_1 v^{I_2}_\pm) + \sum_{\substack{(I_1,I_2)\in \mathcal{I}(I)\\ |I_1|+|I_2|<|I|}}c_{I_1,I_2} Q^\mathrm{w}_0(v^{I_1}_\pm, D v^{I_2}_\pm),$$ with some of the coefficients $c_{I_1,I_2}$ being equal to $1$ or $-1$, and $D\in\{D_1,D_2,D_t\}$ depending on the addend we are considering (similarly for $\Gamma^I Q^\mathrm{kg}_0(v_\pm, D_1u_\pm)$). For our scopes, there will be no difference between the case $D=D_1$ and $D=D_2$, the two associated quadratic contributions enjoying the same $L^2$ and $L^\infty$ estimates. When $D=D_t$, we should make use of the equation satisfied by $v^{I_2}_\pm$ (resp. by $u^{I_2}_\pm$) in system to replace $Q^\mathrm{w}_0(v^{I_1}_\pm, D_tv^{I_2}_\pm)$ (resp. $Q^\mathrm{kg}_0(v^{I_1}_\pm, D_tu^{I_2}_\pm)$) with $$\label{Q(Dt)} \begin{gathered} Q^\mathrm{w}_0(v^{I_1}_\pm, \langle D_x\rangle v^{I_2}_\pm) + Q^\mathrm{w}_0\left(v^{I_1}_\pm, \Gamma^{I_2} Q^\mathrm{kg}_0(v_\pm, D_1u_\pm)\right), \\ \left(\text{resp. with } Q^\mathrm{kg}_0(v^{I_1}_\pm, |D_x| u^{I_2}_\pm) + Q^\mathrm{kg}_0\left(v^{I_1}_\pm, \Gamma^{I_2} Q^\mathrm{w}_0(v_\pm, D_1v_\pm)\right)\right), \end{gathered}$$ where the left hand side quadratic terms are given by $$\label{Qw0-Qk0- |Dx|} \begin{gathered} Q^\mathrm{w}_0(v^{I_1}_\pm, \langle D_x\rangle v^{I_2}_\pm) = (v^{I_1}_+ + v^{I_1}_{-})\langle D_x\rangle(v^{I_2}_+ - v^{I_2}_{-}) - \frac{D_x}{\langle D_x\rangle}(v^{I_1}_+ - v^{I_1}_{-}) \cdot D_x(v^{I_2}_+ + v^{I_2}_{-}),\\ \left(\text{resp. } Q^\mathrm{kg}_0(v^{I_1}_\pm, |D_x| u^{I_2}_\pm) = (v^{I_1}_+ + v^{I_1}_{-})| D_x| (u^{I_2}_+ - u^{I_2}_{-}) - \frac{D_x}{\langle D_x\rangle}(v^{I_1}_+ - v^{I_1}_{-}) \cdot D_x(u^{I_2}_+ + u^{I_2}_{-})\right), \end{gathered}$$ while the right hand side ones in are cubic. On the Fourier side, these new quadratic contributions write as $$\begin{gathered} \sum_{j_1,j_2\in \{+,-\}}\int j_2\left(1-j_1j_2\frac{\xi-\eta}{\langle \xi-\eta\rangle}\cdot\frac{\eta}{\langle\eta\rangle}\right)\langle\eta\rangle \hat{v}^{I_1}_{j_1}(\xi-\eta)\hat{v}^{I_2}_{j_2}(\eta) d\xi d\eta, \\ \left(\text{resp. } \sum_{j_1,j_2\in \{+,-\}}\int j_2\left(1-j_1j_2\frac{\xi-\eta}{\langle \xi-\eta\rangle}\cdot\frac{\eta}{|\eta|}\right)|\eta| \hat{v}^{I_1}_{j_1}(\xi-\eta)\hat{u}^{I_2}_{j_2}(\eta) d\xi d\eta\right), \end{gathered}$$ and have basically the same nature of the starting ones, as $$\begin{gathered} {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q^\mathrm{w}_0(v^{I_1}_\pm, D_1 v^{I_2}_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q^\mathrm{w}_0(v^{I_1}_\pm, D_1 v^{I_2}_\pm)}{\tmpbox}}(\xi)= \sum_{j_1,j_2\in \{+,-\}}\int \left(1-j_1j_2\frac{\xi-\eta}{\langle \xi-\eta\rangle}\cdot\frac{\eta}{\langle\eta\rangle}\right)\eta_1 \hat{v}^{I_1}_{j_1}(\xi-\eta)\hat{v}^{I_2}_{j_2}(\eta) d\xi d\eta, \\ \left(\text{resp. } {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q^\mathrm{kg}_0(v^{I_1}_\pm, D_1 u^{I_2}_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q^\mathrm{kg}_0(v^{I_1}_\pm, D_1 u^{I_2}_\pm)}{\tmpbox}}(\xi)=\sum_{j_1,j_2\in \{+,-\}}\int \left(1-j_1j_2\frac{\xi-\eta}{\langle \xi-\eta\rangle}\cdot\frac{\eta}{|\eta|}\right)\eta_1 \hat{v}^{I_1}_{j_1}(\xi-\eta)\hat{u}^{I_2}_{j_2}(\eta) d\xi d\eta\right). \end{gathered}$$ For this reason, as long as we can neglect the cubic terms in , we will not pay attention to the value of $D\in \{D_1,D_2,D_t\}$ in the second sum in the right hand side of . Lemma \[Lem:L2 est nonlinearity Dt\] is meant to show that the mentioned cubic terms are, indeed, remainders. Before proving lemmas \[Lem: L2 est nonlinearities\], \[Lem:L2 est nonlinearity Dt\], we need to introduce a new set of indices. According to the order established in $\mathcal{Z}$ at the beginning of section \[sec: statement of the main results\] (see ), we define $$\label{set_K} \mathcal{K}:=\{I=(i_1,i_2) : i_1,i_2 =1,2, 3\}$$ as the set of indices $I$ such that $\Gamma^I$ is the product of two Klainerman vector fields only, together with $$\label{set_V} \mathcal{V}^k:=\{I\in \mathcal{I}^k_3 : \exists (I_1,I_2)\in\mathcal{I}(I) \text{ with } I_1\in\mathcal{K} \},$$ which is evidently empty when $k=2$. We also warn the reader that, in inequality with $k=2$, $E^3_3(t;W)$ stands for $E_3(t;W)$, this double notation allowing us to combine in one line all cases $k=0,1,2$. \[Lem: L2 est nonlinearities\] $(i)$ Let $n\in\mathbb{N}, n\ge 3$ and $I\in \mathcal{I}_n$. Then $$\label{est: L2 Qw0 (vI1 vI2)-only derivatives} \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_2|<n}}\left\|Q^\mathrm{w}_0(v^{I_1}_\pm, D_x v^{I_2}_\pm)\right\|_{L^2}+\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|\le [\frac{n}{2}], |I_2|<n}}\left\|Q^\mathrm{kg}_0(v^{I_1}_\pm, D_xu^{I_2}_\pm)\right\|_{L^2} \lesssim \|V(t,\cdot)\|_{H^{[\frac{n}{2}]+2, \infty}}E_n(t;W)^\frac{1}{2},$$ $$\label{est: L2 Qkg0 (vI1 vI2)-only derivatives} \sum_{\substack{(I_1,I_2)\in \mathcal{I}(I)\\ |I_1|> [\frac{n}{2}]}}\left\|Q^\mathrm{kg}_0(v^{I_1}_\pm, D_xu^{I_2}_\pm)\right\|_{L^2} \lesssim \left(\|U(t,\cdot)\|_{H^{[\frac{n}{2}]+2, \infty}}+ \|\mathrm{R}_1 U(t,\cdot)\|_{H^{[\frac{n}{2}]+2,\infty}}\right)E_n(t;W)^\frac{1}{2}.$$ $(ii)$ Let $0\le k \le 2$ and $I\in \mathcal{I}^k_3$. There exists a constant $C>0$ such that, if we assume a-priori estimates , satisfied and $0<\varepsilon_0<(2A+B)^{-1}$ small, for any $\chi\in C^\infty_0(\mathbb{R}^2)$ equal to 1 in a neighbourhood of the origin and $\sigma>0$ small we have \[decomposition\_lemma\_Qwo-Qkg0\] $$\begin{gathered} \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_2|<3}} Q^\mathrm{w}_0(v^{I_1}_\pm, D_x v^{I_2}_\pm) = \mathfrak{R}^k_3(t,x),\label{decomposition_Qw0} \\ \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\|I_1|, |I_2|<3}} Q^\mathrm{kg}_0(v^{I_1}_\pm, D_x u^{I_2}_\pm) = \delta_{\mathcal{V}^k}\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ I_1\in\mathcal{K}, |I_2|\le 1}} Q^\mathrm{kg}_0\left(v^{I_1}_\pm, \chi(t^{-\sigma}D_x) D_x u^{I_2}_\pm\right) +\mathfrak{R}^k_3(t,x), \label{decomposition_Qkg0}\end{gathered}$$ where $\delta_{\mathcal{V}^k}=1$ if $I\in\mathcal{V}^k$, 0 otherwise, and $$\label{est:L2_norm_Rk3(t,x)} \|\mathfrak{R}^k_3(t,\cdot)\|_{L^2}\le C(A+B)\varepsilon t^{-1} E^k_3(t,W)^\frac{1}{2} + CB\varepsilon t^{-\frac{5}{4}},$$ with $\beta>0$ small, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$, for all $t\in[1,T]$. The same result holds with $D_x v^{I_2}_\pm$ (resp. $D_x u^{I_2}_\pm$) replaced with $\langle D_x\rangle v^{I_2}_\pm$ (resp. $|D_x| u^{I_2}_\pm$). $(i)$ The proof of follows straightly from with $a=1,2$, by bounding the $L^2$ norm of each product with the $L^\infty$ norm of the factor indexed in $J\in \{I_1,I_2\}$ such that $|J|\le \big[\frac{|I|}{2}\big]$, times the $L^2$ norm of the remaining one. $(ii)$ Let $I\in\mathcal{I}^k_3$. One immediately sees that: $$\begin{gathered} \label{ineq:I1,0- 0,I2} \sum_{(J,0)\in\mathcal{I}(I)} \|Q^\mathrm{w}_0(v^J_\pm, D_x v_\pm)\|_{L^2} + \sum_{\substack{(J,0)\in\mathcal{I}(I)\\ |J|<3 }}\Big(\|Q^\mathrm{w}_0(v_\pm, D_x v^J_\pm)\|_{L^2}+ \|Q^\mathrm{kg}_0(v_\pm, D_x u^J_\pm)\|_{L^2}\Big) \\ \lesssim \|V(t,\cdot)\|_{H^{2,\infty}}E^k_3(t;W)^\frac{1}{2};\end{gathered}$$ if $(I_1,I_2)\in\mathcal{I}(I)$ is such that $|I_2|<3$ and either $\Gamma^{I_1}$ or $\Gamma^{I_2}$ is a product of spatial derivatives only $$\label{est:quadratic-terms-derivative1} \|Q^\mathrm{w}_0(v^{I_1}_\pm, D_x v^{I_2}_\pm)\|_{L^2}\lesssim \|V(t,\cdot)\|_{H^{4,\infty}}E^k_3(t;W)^\frac{1}{2};$$ if $(I_1,I_2)\in\mathcal{I}(I)$ is such that $|I_2|<3$ and $\Gamma^{I_1}$ is a product of spatial derivatives only $$\label{est:quadratic-term-derivative2} \|Q^\mathrm{kg}_0(v^{I_1}_\pm, D_x u^{I_2}_\pm)\|_{L^2}\lesssim \|V(t,\cdot)\|_{H^{3,\infty}}E^k_3(t;W)^\frac{1}{2}.$$ Hence, the remaining quadratic contributions to be estimated are those corresponding to indices $(I_1,I_2)\in\mathcal{I}(I)$, with $|I_2|<3$, such that: both $\Gamma^{I_1}$ and $\Gamma^{I_2}$ contain at least one Klainerman vector field, in the left hand side of ; $\Gamma^{I_1}$ contains one or two Klainerman vector fields, in the left hand side of . The idea to estimate the $L^2$ norm of the $Q^\mathrm{w}_0(v^{I_1}_\pm, D v^{I_2}_\pm)$, for indices $I_1,I_2$ just mentioned above, is to decompose the Klein-Gordon component carrying exactly one Klainerman vector field in frequencies, by means of a truncation $\chi(t^{-\sigma}D_x)$ for some smooth cut-off function $\chi$ and $\sigma>0$ small. Basically, the $L^\infty$ norm of the contribution truncated for large frequencies $|\xi|\gtrsim t^\sigma$ can be bounded by making appear a power of $t$ as negative as we want, while that of the remaining one, localized for $|\xi|\lesssim t^{\sigma}$, enjoys the sharp Klein-Gordon decay $t^{-1}$ as proved in lemma \[Lem\_appendix: sharp\_est\_VJ\] in appendix \[Appendix B\]. The same argument can be applied to $Q^\mathrm{kg}_0(v^{I_1}_\pm, D_xu^{I_2}_\pm)$ with $I_1$ such that $\Gamma^{I_1}$ contains exactly one Klainerman vector field. Then, by lemma \[Lem\_app:products\_Gamma\] in appendix \[Appendix B\] with $L=L^2$ we find that, for some $\chi\in C^\infty_0(\mathbb{R}^2)$, the following: if $\Gamma^{I_1}$ contains exactly one Klainerman vector field, $$\begin{gathered} \left\|Q^\mathrm{w}_0(v^{I_1}_\pm, D_x v^{I_2}_\pm)(t,\cdot)\right\|_{L^2}\lesssim \left\|\chi(t^{-\sigma}D_x)v^{I_1}_\pm(t,\cdot)\right\|_{H^{1,\infty}}\|v^{I_2}_\pm(t,\cdot)\|_{H^1}\\ + t^{-N(s)}\left(\|v_\pm(t,\cdot)\|_{H^s}+\|D_t v_\pm(t,\cdot)\|_{H^s}\right)\Big(\sum_{|\mu|=0}^1\|x^\mu v^{I_2}_\pm(t,\cdot)\|_{H^1}+t\|v^{I_2}_\pm(t,\cdot)\|_{H^1}\Big)\end{gathered}$$ and $$\begin{gathered} \left\|Q^\mathrm{kg}_0(v^{I_1}_\pm, D_xu^{I_2}_\pm)(t,\cdot)\right\|_{L^2}\lesssim \left\|\chi(t^{-\sigma}D_x)v^{I_1}_\pm(t,\cdot)\right\|_{H^{1,\infty}}\|u^{I_2}_\pm(t,\cdot)\|_{H^1}\\ + t^{-N(s)}\left(\|v_\pm(t,\cdot)\|_{H^s}+\|D_t v_\pm(t,\cdot)\|_{H^s}\right)\Big(\sum_{|\mu|=0}^1\|x^\mu D_xu^{I_2}_\pm(t,\cdot)\|_{L^2}+t\|u^{I_2}_\pm(t,\cdot)\|_{H^1}\Big);\end{gathered}$$ if $\Gamma^{I_2}$ contains exactly one Klainerman vector field, $$\begin{gathered} \left\|Q^\mathrm{w}_0(v^{I_1}_\pm, D_xv^{I_2}_\pm)(t,\cdot)\right\|_{L^2}\lesssim \left\|\chi(t^{-\sigma}D_x)v^{I_2}_\pm(t,\cdot)\right\|_{H^{2,\infty}}\|v^{I_1}_\pm(t,\cdot)\|_{L^2}\\ + t^{-N(s)}\left(\|v_\pm(t,\cdot)\|_{H^s}+\|D_t v_\pm(t,\cdot)\|_{H^s}\right)\Big(\sum_{|\mu|=0}^1\|x^\mu v^{I_1}_\pm(t,\cdot)\|_{L^2}+t\|v^{I_1}_\pm(t,\cdot)\|_{L^2}\Big),\end{gathered}$$ where, in all above inequalities, $N(s)\ge 3$ if $s>0$ is large enough. From inequalities , , estimates , lemma \[Lem\_appendix: sharp\_est\_VJ\] and the boostrap assumptions , together with the fact $\delta,\delta_j\ll 1$ are small, for $j=0,1,2$, we derive that there is a positive constant $C$ such that, for multi-indices $I_1,I_2$ considered in above inequalities, $$\left\|Q^\mathrm{w}_0(v^{I_1}_\pm, Dv^{I_2}_\pm)(t,\cdot)\right\|_{L^2} + \left\|Q^\mathrm{kg}_0(v^{I_1}_\pm, Du^{I_2}_\pm)(t,\cdot)\right\|_{L^2}\le CB\varepsilon t^{-1}E^k_3(t;W)^\frac{1}{2} + CB\varepsilon t^{-\frac{5}{4}}.$$ The remaining quadratic terms are $Q^\mathrm{kg}_0(v^{I_1}_\pm, D_x u^{I_2}_\pm)$ with $I_1\in \mathcal{K}$ (and hence $|I_2|\le 1$) if $\mathcal{V}^k$ is non empty. Applying lemma \[Lem\_app:products\_Gamma\] with $L=L^2$, $w=u$ and the same $s$ as before, and making use of estimates , , together with inequality , we see that $$\begin{gathered} \left\|Q^\mathrm{kg}_0(v^{I_1}_\pm, D_x u^{I_2}_\pm)(t,\cdot)\right\|_{L^2}\lesssim \left\|Q^\mathrm{kg}_0\left(v^{I_1}_\pm, \chi(t^{-\sigma}D_x) D_xu^{I_2}_\pm\right)(t,\cdot)\right\|_{L^2} \\ + t^{-3}\Big(\sum_{|\mu|=0}^1 \|x^\mu v^{I_1}_\pm(t,\cdot)\|_{L^2}+t\|v^{I_1}_\pm(t,\cdot)\|_{L^2}\Big)\left(\|u_\pm(t,\cdot)\|_{H^s}+\|D_tu_\pm(t,\cdot)\|_{H^s}\right)\\ \lesssim \left\|Q^\mathrm{kg}_0\left(v^{I_1}_\pm, \chi(t^{-\sigma}D_x) D_xu^{I_2}_\pm\right)(t,\cdot)\right\|_{L^2} + CB\varepsilon t^{-\frac{5}{4}},\end{gathered}$$ which hence concludes the proof of $(ii)$. We should highlight the fact that the quadratic contribution in the above left hand side is treated differently from the previous ones, because we do not have a sharp decay $O(t^{-1})$ for $v^{I_1}_\pm$ when $I_1\in\mathcal{K}$ (neither when truncated for moderate frequencies), but only a control in $O(t^{-1+\beta'})$, for some small $\beta'>0$ (see lemma \[Lem\_appendix:est vI I=2\]). Moreover, the decay enjoyed by the uniform norm of $\chi(t^{-\sigma}D_x)D_x u^{I_2}_\pm$, appearing in the quadratic term in the above right hand side, is very weak (only $t^{-1/2+\beta'}$, see lemma \[Lem\_appendix: est UJ\]). Such terms, that contribute to the energy and decay slowly in time, will be successively eliminated by a normal form argument (see subsection \[sub: second normal form\]). \[Lem:L2 est nonlinearity Dt\] Let $0\le k \le 2$ and $I\in\mathcal{I}^k_3$. For any $\chi\in C^\infty_0(\mathbb{R}^2)$ equal to 1 in a neighbourhood of the origin and $\sigma>0$ small $$\begin{gathered} \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|+|I_2|\le 2}}Q^\mathrm{w}_0(v^{I_1}_\pm, D_tv^{I_2}_\pm) = \mathfrak{R}^k_3(t,x),\label{eq:Qw0(Dt)-statement} \\ \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|+|I_2|\le 2}}Q^\mathrm{kg}_0(v^{I_1}_\pm, D_tu^{I_2}_\pm) =\delta_{\mathcal{V}^k}\sum_{\substack{(J,0)\in\mathcal{I}(I)\\ J\in\mathcal{K}}}Q^\mathrm{kg}_0(v^J_\pm, \chi(t^{-\sigma}D_x) |D_x| u_\pm)+ \mathfrak{R}^k_3(t,x), \label{eq:Qkg0(Dt)-statement}\end{gathered}$$ with $\delta_{\mathcal{V}^k}=1$ if $I\in\mathcal{V}^k$, 0 otherwise, and $\mathfrak{R}^k_3(t,x)$ satisfying . Using the equation satisfied by $v^{I_2}_\pm$ and $u^{I_2}_\pm$ respectively in system with $I=I_2$ we see that \[eq:non-lin-Dt\] $$\begin{gathered} \label{eq:Qw0(Dt)} \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|+|I_2|\le 2}}Q^\mathrm{w}_0(v^{I_1}_\pm, D_tv^{I_2}_\pm) = \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|+|I_2|\le 2}}Q^\mathrm{w}_0(v^{I_1}_\pm, \langle D_x\rangle v^{I_2}_\pm)\\ +\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|+|I_2|\le 2}} \sum_{(J_1,J_2)\in\mathcal{I}(I_2)}c_{J_1,J_2} Q^\mathrm{w}_0\left(v^{I_1}_\pm, Q^\mathrm{kg}_0(v^{J_1}_\pm, D u^{J_2}_\pm)\right),\end{gathered}$$ $$\begin{gathered} \label{eq:Qkg0(Dt)} \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|+|I_2|\le 2}}Q^\mathrm{kg}_0(v^{I_1}_\pm, D_tu^{I_2}_\pm) = \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|+|I_2|\le 2}}Q^\mathrm{kg}_0(v^{I_1}_\pm, | D_x| u^{I_2}_\pm)\\ +\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|+|I_2|\le 2}} \sum_{(J_1,J_2)\in\mathcal{I}(I_2)}c_{J_1,J_2} Q^\mathrm{kg}_0\left(v^{I_1}_\pm, Q^\mathrm{w}_0(v^{J_1}_\pm, D v^{J_2}_\pm)\right),\end{gathered}$$ with coefficients $c_{J_1,J_2}\in \{-1,0,-1\}$ such that $c_{J_1,J_2}=1$ whenever $|J_1|+|J_2|=|I_2|$, and $Q^\mathrm{w}_0(v^{I_1}_\pm, \langle D_x\rangle v^{I_2}_\pm)$ (in which case $D=D_1$), $Q^\mathrm{kg}_0(v^{I_1}_\pm, |D_x| u^{I_2}_\pm)$ given explicitly by . After lemma \[Lem: L2 est nonlinearities\] $(ii)$ we know that $$\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|+|I_2|\le 2}}\left[Q^\mathrm{w}_0(v^{I_1}_\pm, \langle D_x\rangle v^{I_2}_\pm) + Q^\mathrm{kg}_0(v^{I_1}_\pm, | D_x| u^{I_2}_\pm)\right]= \sum_{\substack{(J,0)\in \mathcal{I}(I)\\ J\in\mathcal{K}}}Q^\mathrm{kg}_0(v^J_\pm, | D_x| u_\pm) + \mathfrak{R}^k_3(t,x),$$ with $\mathfrak{R}^k_3$ verifying . The only thing to prove is that the cubic terms in the right hand side of are remainders $\mathfrak{R}^k_3$. We focus on those in the right hand side of as the same argument applies to the ones in . First, let us consider cubic terms corresponding to indices $I_1,I_2$ such that $|I_1|=2$ and $|I_2|=0$. In this case we evidently have that $|J_1|=|J_2|=0$, and by with $s=1$ and $\theta\ll 1$ small, together with a-priori estimate , $$\left\|Q^\mathrm{w}_0\left(v^{I_1}_\pm , Q^\mathrm{kg}_0(v_\pm, D_1 u_\pm)\right)\right\|_{L^2}\lesssim \|v^{I_1}_\pm(t,\cdot) \|_{L^2} \|Q^\mathrm{kg}_0(v_\pm, D_1 u_\pm)\|_{H^{1,\infty}}\le CB\varepsilon t^{-\frac{3}{2}+\beta'},$$ for some $\beta'>0$ small as long as $\sigma, \delta_0$ are small. Let us now consider indices $I_1,I_2$ such that $\Gamma^{I_1}\in\{\Omega, Z_m, m=1,2\}$. As we also require that $(I_1,I_2)\in\mathcal{I}(I)$ with $|I_2|\le 2$, we have in this case that $|I_2|\le 1$ and consequently, for each $(J_1,J_2)\in\mathcal{I}(I_2)$, either $|J_1|=0$ or $|J_2|=0$. Using lemma \[Lem\_app:products\_Gamma\] in appendix \[Appendix B\] with $L=L^2$ and $w=v$, we derive that for any $\chi\in C^\infty_0(\mathbb{R}^2)$ as in the statement and $\sigma>0$ small $$\begin{aligned} \sum_{(J_1,J_2)\in\mathcal{I}(I_2)}&\left\|Q^\mathrm{w}_0\left(v^{I_1}_\pm, Q^\mathrm{kg}_0(v^{J_1}_\pm, D u^{J_2}_\pm)\right)(t,\cdot)\right\|_{L^2}\\ & \lesssim \sum_{(J_1,J_2)\in\mathcal{I}(I_2)}\left\|\chi(t^{-\sigma}D_x)v^{I_1}_\pm(t,\cdot)\right\|_{L^\infty}\left\|Q^\mathrm{kg}_0(v^{J_1}_\pm, D u^{J_2}_\pm)(t,\cdot)\right\|_{L^2}\\ &+\sum_{(J_1,J_2)\in\mathcal{I}(I_2)}t^{-N(s)}\left(\|v_\pm(t,\cdot)\|_{H^s} + \|D_tv_\pm(t,\cdot)\|_{H^s}\right)\\ &\hspace{2cm}\times\Big(\sum_{|\mu|=0}^1\left\|x^\mu Q^\mathrm{kg}_0(v^{J_1}_\pm, D u^{J_2}_\pm)(t,\cdot)\right\|_{L^2}+ t \left\|Q^\mathrm{kg}_0(v^{J_1}_\pm, D u^{J_2}_\pm)(t,\cdot)\right\|_{L^2}\Big),\end{aligned}$$ with $N(s)\ge 3$ is $s>0$ is sufficiently large. Here $$\begin{aligned} & \sum_{(J_1,J_2)\in\mathcal{I}(I_2)} \left\|x Q^\mathrm{kg}_0(v^{J_1}_\pm, D u^{J_2}_\pm)(t,\cdot)\right\|_{L^2} \\ &\hspace{0.5cm}\lesssim \sum_{\substack{|\mu|=0,1 \\ |J|\le 1}}\left[\left\|x\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty}\left(\|u^J_\pm(t,\cdot)\|_{H^1} + \|D_tu^J_\pm(t,\cdot)\|_{L^2}\right)\right. \\ &\hspace{1cm} \left. + \|xv^J_\pm(t,\cdot)\|_{L^2}\left(\|\mathrm{R}^\mu u_\pm(t,\cdot)\|_{H^{2,\infty}}+ \|D_t\mathrm{R}^\mu u_\pm(t,\cdot)\|_{H^{1,\infty}}\right)\right] \le C(A+B)B\varepsilon^2 t^{\frac{1}{2}+\frac{\delta_2}{2}}\end{aligned}$$ and $$\begin{gathered} \label{est_J1J2_less 2} \sum_{(J_1,J_2)\in\mathcal{I}(I_2)}\left\|Q^\mathrm{kg}_0(v^{J_1}_\pm, D u^{J_2}_\pm)(t,\cdot)\right\|_{L^2}\\ \lesssim \sum_{|J|\le 1}\Big[ \|v^J_\pm(t,\cdot)\|_{L^2}\Big(\sum_{|\mu|=0}^1\|\mathrm{R}^\mu u_\pm(t,\cdot)\|_{H^{2,\infty}}+ \|D_t\mathrm{R}^\mu u_\pm(t,\cdot)\|_{H^{1,\infty}}\Big) \\+ \|v_\pm(t,\cdot)\|_{H^{1,\infty}}\big(\|u^J_\pm(t,\cdot)\|_{H^1}+ \|D_tu^J_\pm(t,\cdot)\|_{L^2}\big)\Big]\le C(A+B)B\varepsilon^2 t^{-\frac{1}{2}+\frac{\delta_2}{2}}\end{gathered}$$ by , , , and estimates , , , so together with lemma \[Lem\_appendix: sharp\_est\_VJ\] and , these inequalities give $$\sum_{(J_1,J_2)\in\mathcal{I}(I_2)}\left\|Q^\mathrm{w}_0\left(v^{I_1}_\pm, Q^\mathrm{kg}_0(v^{J_1}_\pm, D u^{J_2}_\pm)\right)(t,\cdot)\right\|_{L^2} \le CB\varepsilon t^{-\frac{3}{2}+\beta'},$$ for some new $\beta'>0$ small, $\beta'\rightarrow 0$ as $\sigma,\delta_0\rightarrow 0$. Finally, for indices $I_1,I_2$ such that $\Gamma^{I_1}\in \{D^\alpha_x , |\alpha|\le 1\}$ $$\label{last_J1J2} \sum_{(J_1,J_2)\in\mathcal{I}(I_2)}\left\|Q^\mathrm{w}_0\left(v^{I_1}_\pm, Q^\mathrm{kg}_0(v^{J_1}_\pm, D u^{J_2}_\pm)\right)\right\|_{L^2} \lesssim \sum_{(J_1,J_2)\in\mathcal{I}(I_2)}\|v_\pm(t,\cdot)\|_{H^{2,\infty}}\left\|Q^\mathrm{kg}_0(v^{J_1}_\pm, D u^{J_2}_\pm)\right\|_{L^2}.$$ For $(J_1,J_2)\in\mathcal{I}(I_2)$ such that $|J_1|+|J_2|=|I_2|$ we have by lemma \[Lem: L2 est nonlinearities\] $(ii)$ and a-priori estimates that $$\begin{split} \| Q^\mathrm{kg}_0(v^{J_1}_\pm, D_1 u^{J_2}_\pm)\|_{L^2} &\lesssim \|\mathfrak{R}^k_3(t,\cdot)\|_{L^2} + \sum_{J\in\mathcal{K}}\|Q^\mathrm{kg}_0(v^J_\pm, D_1 \chi(t^{-\sigma}D_x)u_\pm)\|_{L^2}\\ &\lesssim \|\mathfrak{R}^k_3(t,\cdot)\|_{L^2} +t^\beta \sum_{|\mu|=0}^1\|\mathrm{R}_1^\mu u_\pm(t,\cdot)\|_{L^\infty} E^1_3(t;W)^\frac{1}{2}\\ &\le CB\varepsilon t^{-\frac{1}{2}+\beta +\frac{\delta_1}{2}}, \end{split}$$ with $\beta>0$ small, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$, while for $(J_1,J_2)\in\mathcal{I}(I_2)$ such that $|J_1|+|J_2|<|I_2|$ (hence $<2$) an estimate such as holds. These estimates, together with , imply that the right hand side of is bounded by $CAB\varepsilon^2 t^{-\frac{3}{2}+\beta'}$, for a new small $\beta'>0$, $\beta'\rightarrow 0$ as $\sigma, \delta_0\rightarrow 0$, and that concludes the proof of the statement. \[Cor: L2 est QI0(V,W)\] Let $Q^I_0(V,W)$ be the vector defined in . There exists a constant $C>0$ such that, if we assume that a-priori estimates are satisfied in interval $[1,T]$, for some fixed $T>1$, with $\varepsilon_0<(2A+B)^{-1}$ small: $(i)$ if $I\in\mathcal{I}_n$ with $n\ge 3$: $$\label{est:QI0-In} \|Q^I_0(V,W)\|_{L^2}\le C A\varepsilon t^{-\frac{1}{2}+\frac{\delta}{2}};$$ $(ii)$ if $I\in\mathcal{I}^k_3$, with $0\le k\le 2$, $$\label{est:QI0-Ik3} \|Q^I_0(V,W)\|_{L^2} \le C(A+B)\varepsilon t^{-\frac{1}{2}+\frac{\delta_k}{2}}.$$ $(i)$ Inequality is straightforward after definition (all coefficients $c_{I_1,I_2}$ are equal to 0 when $I\in\mathcal{I}_n$), lemma \[Lem: L2 est nonlinearities\] $(i)$, and a-priori estimates , . $(ii)$ If $I\in\mathcal{I}^k_3$ for a fixed $0\le k\le 2$ we have by definition and lemmas \[Lem: L2 est nonlinearities\], \[Lem:L2 est nonlinearity Dt\] that $$\label{dec_1} \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_2|<|I|}} Q^\mathrm{w}_0(v^{I_1}_\pm, D_x v^{I_2}_\pm) + \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\|I_1|+|I_2|\le 2, |I_2|<|I|}} Q^\mathrm{w}_0(v^{I_1}_\pm, D_t v^{I_2}_\pm)=\mathfrak{R}^k_3(t,x),$$ with $\mathfrak{R}^k_3(t,x)$ satisfying . Moreover, for some smooth $\chi\in C^\infty_0(\mathbb{R}^2)$, equal to 1 in a neighbourhood of the origin and $\sigma>0$ small, $$\label{dec_Qkg0} \begin{gathered} \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_2|<|I|}} Q^\mathrm{kg}_0(v^{I_1}_\pm, D_x u^{I_2}_\pm)=\delta_{\mathcal{V}^k}\sum_{\substack{(I_1,I_2)\in \mathcal{I}(I)\\ I_1\in\mathcal{K}, |I_2|\le 1}}Q^\mathrm{kg}_0\left(v^{I_1}_\pm, \chi(t^{-\sigma}D_x)D_x u^{I_2}_\pm\right)+ \mathfrak{R}^k_3(t,x), \\ \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\|I_1|+|I_2|\le 2, |I_2|<|I|}} Q^\mathrm{kg}_0(v^{I_1}_\pm, D_t u^{I_2}_\pm)=\delta_{\mathcal{V}^k}\sum_{\substack{(J,0)\in \mathcal{I}(I)\\J\in\mathcal{K}}}Q^\mathrm{kg}_0\left(v^J_\pm, \chi(t^{-\sigma}D_x)|D_x| u_\pm\right)+ \mathfrak{R}^k_3(t,x), \end{gathered}$$ with sets $\mathcal{K}, \mathcal{V}^k$ given, respectively, by , , $\delta_{\mathcal{V}^k}=1$ if $I\in\mathcal{V}^k$, 0 otherwise (remind that $\mathcal{V}^2$ is empty). Observe that, if $k=0,1$, $I\in\mathcal{I}^k_3$ and $(I_1,I_2)\in\mathcal{I}(I)$ with $I_1\in\mathcal{K}$, two situations may occur: if $\Gamma^{I_2}\in\{D^\alpha_x, |\alpha|\le 1\}$ then product $\Gamma^{I_1}$ contains exactly the same number of Klainerman vector fields as in $\Gamma^I$ and $V^{I_1}$ would be at the same energy level as $V^I$ (i.e. its $L^2$ norm being controlled by $E^k_3(t;W)^{1/2}$). In this case, from a-priori estimates $$\label{product vI1, uI2_1} \begin{split} \|v^{I_1}_\pm(t,\cdot)\|_{L^2}\left(\|\chi(t^{-\sigma}D_x)u^{I_2}_\pm(t,\cdot)\|_{H^{\rho,\infty}}+ \|\chi(t^{-\sigma}D_x)\mathrm{R} u^{I_2}_\pm(t,\cdot)\|_{H^{\rho,\infty}}\right) &\le A\varepsilon t^{-\frac{1}{2}}E^k_3(t;W)^\frac{1}{2}. \end{split}$$ If instead $I_2$ is such that $\Gamma^{I_2}\in \{\Omega, Z_m, m=1,2\}$ is a Klainerman vector field, we automatically have that $\Gamma^I$ is a product of three Klainerman vector fields and that $V^{I_1}$ is at an energy level strictly lower than $V^I$ (i.e. its $L^2$ norm is controlled by $E^1_3(t;W)^{1/2}$ whereas that of $V^I$ is bounded by $E^0_3(t;W)^{1/2}$). From lemma \[Lem\_appendix: est UJ\] we deduce that $$\begin{gathered} \label{product_vI1-uI2_2} \|v^{I_1}_\pm(t,\cdot)\|_{L^2}\left(\|\chi(t^{-\sigma}D_x)u^{I_2}_\pm(t,\cdot)\|_{H^{\rho,\infty}}+ \|\chi(t^{-\sigma}D_x)\mathrm{R}u^{I_2}_\pm(t,\cdot)\|_{H^{\rho,\infty}}\right)\\ \le C(A+B)\varepsilon t^{-\frac{1}{2}+\beta+\frac{\delta_1}{2}}E^1_3(t;W)^\frac{1}{2},\end{gathered}$$ for a small $\beta>0$, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. Summing up to and using we obtain that there is a positive constant $C$ such that $$\label{ineq_QI0_L2} \|Q^I_0(V,W)\|_{L^2} \le \delta_k C(A+B)\varepsilon t^{-\frac{1}{2}}\Big[E^k_3(t;W)^\frac{1}{2} +\delta_0 t^{\beta+\frac{\delta_1}{2}}E^1_3(t;W)^\frac{1}{2}\Big]+ CB\varepsilon t^{-\frac{5}{4}},$$ with $\delta_k=1$ for $k=0,1$, equal to 0 when $k=2$, and $\delta_0=1$ only when $k=0$, 0 otherwise. Finally, taking $\sigma>0$ small so that $\beta+\delta_1/2\ll \delta_0/2$ and using a-priori estimates we deduce estimate from . ### Symmetrization {#Subs: Symmetrization} \[Prop: equation of WIs\] As long as $H^{1,\infty}$ norm of $V(t,\cdot)$ is sufficiently small, there exists a real matrix $P(V;\eta)$ of order 0 and a real symmetric matrix $\widetilde{A}_1(V;\eta)$ of order 1, vanishing at order 1 at $V=0$, such that $$\label{def_WIs} W^I_s:=Op^B\big(P(V;\eta)\big)W^I$$ is solution to $$\label{equation WIs} \begin{split} D_t W^I_s & = A(D) W^I_s + Op^B(\widetilde{A}_1(V;\eta))W^I_s + Op^B(A''(V^I;\eta))U \\ &+ Op^B(C''(U;\eta))V^I + Op^B_R(A''(V^I;\eta))U + Q^I_0(V,W) + \mathfrak{R}(U, V), \end{split}$$ where $\mathfrak{R}(U, V)$ satisfies, for any $\theta\in ]0,1[$, $$\label{L2 est of R(U,V)} \begin{split} \|\mathfrak{R}(U, V) (t,\cdot)\|_{L^2} &\lesssim \Big[\|V(t,\cdot)\|_{H^{7,\infty}} + \|V(t,\cdot)\|^{1-\theta}_{H^{1,\infty}} \|V(t,\cdot)\|^\theta_{H^{3}}\left(\| U(t,\cdot)\|_{H^{2,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}\right)\\ &+ \|V(t,\cdot)\|_{L^\infty}\left(\| U(t,\cdot)\|^{1-\theta}_{H^{2,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|^{1-\theta}_{H^{2,\infty}}\right)\|U(t,\cdot)\|^\theta_{H^{4}} \Big] \|W^I(t,\cdot)\|_{L^2} \\ & + \|V(t,\cdot)\|_{H^{1,\infty}} \Big(\|W(t,\cdot)\|_{H^{7,\infty}} + \|\mathrm{R}U(t,\cdot)\|_{H^{6,\infty}}\Big) \|W^I(t,\cdot)\|_{L^2} \\ & + \|V(t,\cdot)\|_{H^{1,\infty}}\|Q^I_0(V,W)\|_{L^2}. \end{split}$$ Moreover, for any $n,r\in\mathbb{N}$, $$\begin{gathered} M^0_r\left(P(V;\eta)-I_4;n\right)\lesssim \|V(t,\cdot)\|_{H^{1+r,\infty}}, \label{seminorm P-I4} \\ M^1_r\big(\widetilde{A}_1(V;\eta);n\big)\lesssim \|V(t,\cdot)\|_{H^{1+r,\infty}}, \label{seminorm Atilde 1}\end{gathered}$$ and as long as the $H^{2,\infty}$ norm of $V(t,\cdot)$ is small there is a constant $C>0$ such that $$\label{equivalence WIs WI} C^{-1}\|W^I(t,\cdot)\|_{L^2}\le \|W^I_s(t,\cdot)\|_{L^2}\le C \|W^I(t,\cdot)\|_{L^2}.$$ In order to prove proposition \[Prop: equation of WIs\], we first need to introduce the following lemma. \[Lemma: existence of P(alpha, beta)\] Let $\alpha,\beta\in\mathbb{R}$, $L\in M_2(\mathbb{R})$ and $M_0, N(\alpha,\beta)\in M_4(\mathbb{R})$ given by $$L = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix},\quad M_0 = \begin{bmatrix} I_2 & 0 \\ 0 & -I_2 \end{bmatrix}, \quad N(\alpha,\beta)= \begin{bmatrix} \alpha L & \beta L \\ \alpha L & \beta L \end{bmatrix} = \begin{bmatrix} 0 & \alpha & 0 & \beta \\ \alpha & 0 & \beta & 0 \\ 0 & \alpha & 0 & \beta \\ \alpha & 0 & \beta & 0 \end{bmatrix} .$$ There exist a small $\delta>0$ and a smooth function defined on open ball $B_\delta(0)$ of radius $\delta$, $$(\alpha,\beta)\in B_\delta(0)\rightarrow P(\alpha,\beta)\in Sym_4(\mathbb{R}),$$ with values in the space of real, symmetric, $4\times 4$ matrices $Sym_4(\mathbb{R})$, such that $P(0,0)=I_4$, $P(\alpha,\beta)=I_4 + O(|\alpha|+|\beta|)$ and $P(\alpha,\beta)^{-1}\big(M_0 + N(\alpha, \beta)\big)P(\alpha,\beta)$ is symmetric for any $(\alpha, \beta)\in B_\delta(0)$. Furthermore $P^{-1}(\alpha,\beta)= I_4 + O(|\alpha| + |\beta|)$. Let $\mathcal{E}$ be the vector space of $2\times 2$ matrices $B(\alpha, \beta)=\alpha I_2 + \beta L$ and $\mathcal{F}$ be the set of $4\times 4$ matrices of the form $$\begin{bmatrix} F_{11} & F_{12} \\ F_{21} & F_{22} \end{bmatrix}$$ with $F_{ij}\in \mathcal{E}$. We look for a matrix $P$ of the form $$\label{matrix P(B)} P(B) = (I_2 - B^2)^{-\frac{1}{2}} \begin{bmatrix} I_2 & -B \\ -B & I_2 \end{bmatrix}$$ with $B\in \mathcal{E}$ close to zero (so that in particular $(I_2-B^2)^{1/2}$ is well defined). We remark that matrix $P(B)^{-1}$ has the form $$P(B)^{-1}= (I_2 - B^2)^{-\frac{1}{2}} \begin{bmatrix} I_2 & B \\ B & I_2 \end{bmatrix}$$ and that $P(0)=P^{-1}(0)=I_4$. We consider $\Phi : \mathbb{R}^2\times \mathcal{E}\rightarrow \mathcal{F}$ defined by $\Phi(\alpha,\beta, B):= P(B)^{-1}\big[M_0 + N(\alpha, \beta)\big]P(B) = \big(\Phi_{ij}(\alpha, \beta, B)\big)_{1\le i,j\le 2}$, where $\Phi_{ij}\in \mathcal{E}$ as $\mathcal{E}$ is a commutative sub-algebra of $M_2(\mathbb{R})$. We also define $\Psi(\alpha, \beta, B):= \Phi_{12}(\alpha, \beta, B) - \Phi_{21}^\dagger(\alpha,\beta, B)$ with $\Phi_{21}^\dagger$ denoting the transpose of $\Phi_{21}$. We have that $\Psi(0,0, 0)=0$ and $$D_B\Phi(0,0,0)\cdot B = \begin{bmatrix} 0 & B \\ B & 0 \end{bmatrix}M_0 - M_0 \begin{bmatrix} 0 & B \\ B & 0 \end{bmatrix} = 2 \begin{bmatrix} 0 & -B \\ B & 0 \end{bmatrix}$$ from which follows that $D_B\Psi(0,0,0)\cdot B = -4B$, i.e. $D_B\Psi(0,0,0) = -4I$. Therefore, there exist a small $\delta>0$ and a smooth function $(\alpha, \beta)\in B_\delta(0)\rightarrow B(\alpha,\beta)\in \mathcal{E}$ such that $B(0,0)=0$ (which implies $P(B(0,0))=I_4$), and $\Psi(\alpha, \beta, B(\alpha, \beta)) = 0$ $\forall (\alpha,\beta)\in B_\delta(0)$. This is equivalent to say that $\Phi(\alpha,\beta, B(\alpha,\beta))$ is symmetric and moreover $P(B(\alpha,\beta)), P(B(\alpha,\beta))^{-1} = I_4 + O(|\alpha|+|\beta|)$. Defining $P(\alpha,\beta):=P(B(\alpha,\beta))$ concludes the proof of the statement. With notations introduced in lemma \[Lemma: existence of P(alpha, beta)\] and in , , $A(\eta) = \langle \eta\rangle M_0 + S(\eta)$ and $A'_1(V;\eta)(1-\chi)(\eta) = \langle \eta\rangle N(\alpha,\beta)$, with $$S(\eta) = \begin{bmatrix} |\eta| -\langle\eta\rangle & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & -(|\eta| - \langle\eta\rangle) & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \text{whose elements are $O(|\eta|^{-1}), |\eta|\rightarrow +\infty$},$$ and $\alpha = a_0(v_{\pm};\eta)\frac{\eta_1}{\langle\eta\rangle}(1-\chi)(\eta)$, $\beta= b_0(v_{\pm};\eta)\frac{\eta_1}{\langle\eta\rangle}(1-\chi)(\eta)$, $a_0, b_0$ defined in . Since $\sup_\eta\big(|\alpha| + |\beta|\big) \lesssim \|V(t,\cdot)\|_{H^{1,\infty}}$, by lemma \[Lemma: existence of P(alpha, beta)\] we have that, as long as $\|V(t,\cdot)\|_{H^{1,\infty}}$ is sufficiently small, there exists a real symmetric matrix $P=P(V;\eta)$ of the form such that $P(V;\eta)^{-1}\big[M_0 + N(\alpha, \beta)\big]P(V;\eta)$ is real and symmetric. Moreover $P = I_4 + Q(V;\eta)$ and $P^{-1} = I_4 + Q'(V;\eta)$, where $Q(V;\eta)$, $Q'(V;\eta)$ are matrices depending smoothly on $\alpha,\beta$ (which are symbols of order 0), null at order 1 at $V=0$, verifying for any $n,r\in\mathbb{N}$ $$M^0_r\left(Q(V;\eta);n\right)+ M^0_r\left(Q'(V;\eta);n\right)\lesssim \|V(t,\cdot)\|_{H^{1+r,\infty}}.$$ We define the following matrix of order 1 $$\widetilde{A}_1(V;\eta) := P(V;\eta)^{-1}\big[\langle\eta\rangle\big(M_0 + N(\alpha, \beta)\big)\big]P(V;\eta) - \langle\eta\rangle M_0$$ and $W^I_s:= Op^B(P^{-1}(V;\eta))W^I$. From the fact that $\widetilde{A}_1(V;\eta)$ also writes as $$\langle\eta\rangle \left[Q'(V;\eta)M_0 + P^{-1}(V;\eta)M_0Q(V;\eta) + P^{-1}(V;\eta)N(\alpha,\beta)P(V;\eta)\right]$$ we see that it vanishes at order 1 at $V=0$ and is such that $M^1_r(\widetilde{A}_1(V;\eta);n)\lesssim \|V(t,\cdot)\|_{H^{1+r,\infty}}$. Moreover, from proposition \[Prop: paradifferential symbolic calculus\] $(ii)$ with $r=1$ it follows that $$\label{dec_operator_I} I = Op^B(P(V;\eta))Op^B(P^{-1}(V;\eta)) + T_{-1}(V),$$ where operator $T_{-1}(V)$ is of order less or equal than $-1$ whose $\mathcal{L}(L^2)$ norm is a $O(\|V(t,\cdot)\|_{H^{2,\infty}})$. Therefore $W^I = Op^B(P(V;\eta))W^I_s + T_{-1}(V)W^I$ and from proposition \[Prop : Paradiff action on Sobolev spaces-NEW\] the $L^2$ norms of $W^I, W^I_s$ are equivalent as long as the $H^{2,\infty}$ norm of $V$ is small. Using equation we find that: $$\label{eq: preliminary equation WIs} \begin{split} D_t W^I_s &= Op^B(P^{-1}(V;\eta))Op^B\big(A(\eta) + A'_1(V;\eta)(1-\chi)(\eta)\big)W^I \\ & + Op^B(P^{-1}(V;\eta))\Big[Op^B\big(A'_1(V;\eta)\chi(\eta)\big) + Op^B\big(A'_{-1}(V;\eta)\big)\Big]W^I \\ & + Op^B(P^{-1}(V;\eta))\Big[Op^B(C'(W^I;\eta))V + Op^B_R(A'(V;\eta))W^I \Big]\\ & + Op^B(P^{-1}(V;\eta))\Big[ Op^B(A''(V^I;\eta))U + Op^B(C''(U;\eta))V^I + Op^B_R(A''(V^I;\eta))U\Big] \\ & + Op^B(P^{-1}(V;\eta))Q^I_0(V,W) + Op^B(D_tP^{-1}(V;\eta))W^I \end{split}$$ where $$\label{equation WIs - calculations} \begin{split} & Op^B(P^{-1}(V;\eta))Op^B\big(A(\eta) + A'_1(V;\eta)(1-\chi)(\eta)\big)W^I \\ &= Op^B(P^{-1}(V;\eta))Op^B\left(\langle\eta\rangle\big(M_0 + N(\alpha,\beta)\big)\right)W^I + Op^B(S(\eta))W^I + Op^B(Q'(V;\eta))Op^B(S(\eta))W^I \\ &= Op^B(P^{-1}(V;\eta))Op^B\left(\langle\eta\rangle\big(M_0 + N(\alpha,\beta)\big)\right)Op^B(P(V;\eta))W^I_s \\ & + Op^B(P^{-1}(V;\eta))Op^B\left(\langle\eta\rangle\big(M_0 + N(\alpha,\beta)\big)\right)T_{-1}(V)W^I + Op^B(S(\eta))W^I_s \\ & + Op^B(S(\eta))Op^B(Q(V;\eta))W^I_s+ Op^B(S(\eta))T_{-1}(V)W^I + Op^B(Q'(V;\eta))Op^B(S(\eta))W^I \\ = & Op^B(A(\eta) + \widetilde{A}_1(V;\eta))W^I_s + \widetilde{T}_0(V)W^I_s + \widetilde{T'}_0(V)W^I \end{split}$$ with $\widetilde{T}_0(V), \widetilde{T'}_0(V)$ operators of order 0 and $\mathcal{L}(L^2)$ norm $O(\|V(t,\cdot)\|_{H^{2,\infty}})$. Last equality follows indeed from the fact that, by proposition \[Prop: paradifferential symbolic calculus\] $(ii)$ with $r=1$ and proposition \[Prop : Paradiff action on Sobolev spaces-NEW\], $$\begin{gathered} Op^B(P^{-1}(V;\eta))Op^B\big[\langle\eta\rangle\big(M_0 + N(\alpha,\beta)\big)\big]Op^B(P(V;\eta)) \\ = Op^B\big(P(V;\eta)^{-1}\big[\langle\eta\rangle\big(M_0 + N(\alpha, \beta)\big)\big]P(V;\eta)\big) + \widetilde{T}_0(V)\end{gathered}$$ and $Op^B(S(\eta))Op^B(Q(V;\eta))$, $Op^B(Q'(V;\eta))Op^B(S(\eta))$ are operator of order 0, too (the former of the form $\widetilde{T}_0(V)$, the latter of the form $T_0(V)$), while $Op^B(S(\eta))T_{-1}(V)$ is of order $-1$ (and can be included in $T_0(V)$). The equivalence between the $L^2$ norms of $W^I_s$ and $W^I$ implies that $\widetilde{T}_0(V)W^I_s + \widetilde{T'}_0(V)W^I$ in is a remainder $\mathfrak{R}(U, V)$. All operators appearing in the second and third line of are also remainders $\mathfrak{R}(U,V)$ because, from proposition \[Prop : Paradiff action on Sobolev spaces-NEW\], the fact that $M^0_0(P^{-1}(V;\eta);2)=O(1)$ and lemma \[Lemma: L2 estimate of semilinear terms\], their $L^2$ norm is bounded by $\|V(t,\cdot)\|_{H^{7,\infty}}\|W^I(t,\cdot)\|_{L^2}$. Last term in also contributes to $\mathfrak{R}(U,V)$ for matrix $D_tP^{-1}(V;\eta)$ is of order 0, its $M^0_0(\cdot,2)$ seminorm is bounded by $\|D_tV(t,\cdot)\|_{H^{1,\infty}}$ and for any $\theta \in[0,1]$ $$\begin{split} \|D_tV(t,\cdot)\|_{H^{1,\infty}}&\lesssim \|V(t,\cdot)\|_{H^{2,\infty}} + \|V(t,\cdot)\|^{1-\theta}_{H^{1,\infty}} \|V(t,\cdot)\|^\theta_{H^{3}}\left(\| U(t,\cdot)\|_{H^{2,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}\right)\\ &+ \|V(t,\cdot)\|_{L^\infty}\left(\| U(t,\cdot)\|^{1-\theta}_{H^{2,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|^{1-\theta}_{H^{2,\infty}}\right)\|U(t,\cdot)\|^\theta_{H^{4}}, \end{split}$$ as follows from with $s=1$. Finally, in remaining contributions in we replace $Op^B(P^{-1}(V;\eta))$ with $I+Op^B(Q'(V;\eta))$ and observe that the terms on which $Op^B(Q'(V;\eta))$ acts are remainders $\mathfrak{R}(U,V)$ after proposition \[Prop : Paradiff action on Sobolev spaces-NEW\], the fact that $M^0_0\left(Q'(V;\eta);2\right) = O(\|V(t,\cdot)\|_{H^{1,\infty}})$ and lemma \[Lemma: L2 estimate of semilinear terms\]. Interchanging the notation of $P(V;\eta)$ and $P^{-1}(V;\eta)$, we obtain the result of the statement. Normal forms and energy estimates {#Sec: Quasi-linear normal forms and energy estimates} --------------------------------- Before going further in writing an energy inequality for $W^I_s$ we should make few remarks. As we previously anticipated, the $L^2$ norm of some of the semi-linear terms appearing in equation have a very slow decay in time. On the one hand, it is the case of $Op^B(A''(V^I;\eta))U$, $Op^B(C''(U;\eta))V^I$ and $Op^B_R(A''(V;\eta))U$, whose $L^2$ norms are estimated in , in terms of the uniform norms of $U, \mathrm{R}_1U$. On the other hand, also some of the contributions to $Q^I_0(V,W)$ are only a $O_{L^2}(t^{-1/2+\beta'})$, for some small $\beta'>0$, after corollary \[Cor: L2 est QI0(V,W)\]. Nevertheless, we are going to see that $Op^B(A''(V^I;\eta))U$, $Op^B_R(A''(V;\eta))U$ and the mentioned contributions to $Q^I_0(V,W)$ can be easily eliminated by performing a semi-linear normal form argument in the energy inequality (see subsection \[sub: second normal form\]). Such an argument is however not well adapted to handle $Op^B(C''(U;\eta))V^I$, for it leads to a loss of derivatives linked to the quasi-linear nature of the problem, i.e. to the fact that matrix $\widetilde{A}_1(V;\eta)$ in the right hand side of is of order 1. This latter contribution should instead be eliminated through a suitable normal form applied directly on equation , which is the object of the subsection \[sub: a first normal form transformation\]. ### A first normal forms transformation and the energy inequality {#sub: a first normal form transformation} First of all, we replace $Op^B(C''(U;\eta))V^I$ in equation with $Op^B(C''(U;\eta))V^I_s$, having defined $V^I_s:= Op^B(P^{-1}(V;\eta))V^I$, and remind that from with $r=0$ and the $L^2$ norm of $V^I$ and $V^I_s$ are equivalent as long as the $H^{2,\infty}$ norm of $V(t,\cdot)$ is small (assumption compatible with if $\rho\ge 2$). We will rather deal with $$\label{equation WIs_1} \begin{split} (D_t - A(D)) W^I_s & = Op^B(\widetilde{A}_1(V;\eta))W^I_s + Op^B(A''(V^I;\eta))U + Op^B(C''(U;\eta))V^I_s\\ & + Op^B_R(A''(V^I;\eta))U + Q^I_0(V,W) + \mathfrak{R}(U, V), \end{split}$$ for a new $\mathfrak{R}(U,V)$ satisfying and show how to get rid of $Op^B(C''(U;\eta))V^I_s$ in the above right hand side. More precisely, we are going to prove the following result: \[Prop: a first normal form\] Let $N\in\mathbb{N}^*$. There exist three matrices $E^0_d(U;\eta), E^{-1}_d(U;\eta)$, $E_{nd}(U;\eta)$ linear in $(u_+,u_{-})$, with $E^0_d(U;\eta)$ real diagonal of order 0 and $E^{-1}_d(U;\eta), E_{nd}(U;\eta)$ of order -1, and, as long as $\|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}$ is small, a real diagonal matrix $F^0_d(U;\eta)$ of order 0 such that, if $$\label{def_WtildeI} \begin{gathered} \widetilde{W}^I_s:= Op^B(I_4 + E(U;\eta))W^I_s ,\\ \text{with } E(U;\eta) := E^0_d(U;\eta)+ E^{-1}_d(U;\eta) + E_{nd}(U;\eta), \end{gathered}$$ then $$\label{equation Wtilde-Is} \begin{split} &(D_t -A(D))\widetilde{W}^I_s = Op^B\left((I_4+ E^0_d(U;\eta))\widetilde{A}_1(V;\eta)(I_4 + F^0_d(U;\eta))\right)\widetilde{W}^I_s \\ & + Op^B(A''(V^I;\eta))U + Op^B_R(A''(V^I;\eta))U + Q^I_0(V,W) + T_{-N}(U)W^I_s + \mathfrak{R}'(U, V). \end{split}$$ In the above right hand side $T_{-N}(U)=(\sigma_{ij}(U,D_x))_{ij}$ is a pseudo-differential operator of order less or equal than $-N$, with $$\label{norm of T-3 in propositon} \|T_{-N}(U)\|_{\mathcal{L}(H^{s-N};H^s)}\lesssim \|\mathrm{R}_1U(t,\cdot)\|_{H^{N+2,\infty}} + \|U(t,\cdot)\|_{H^{N+6,\infty}},$$ for any $s\in\mathbb{R}$ and such that \[sigmaij in proposition Normal Forms\] $$\mathcal{F}_{x\mapsto\xi}(\sigma_{ij}(U,\eta))(\xi) = \begin{cases} \sigma^+_{ij}(\xi, \eta)\hat{u}_+(\xi) + \sigma^{-}_{ij}(\xi,\eta)\hat{u}_{-}(\xi), \qquad & i,j \in\{2,4\}, \\ 0, &\text{otherwise}, \end{cases}$$ with $\sigma^\pm_{ij}(\xi,\eta)$ supported for $|\xi|\le \varepsilon\langle\eta\rangle$ for a small $\varepsilon>0$, and for any $\alpha,\beta\in\mathbb{N}^2$ $$|\partial^\alpha_\xi \partial^\beta_\eta \sigma^\pm_{ij}(\xi,\eta)|\lesssim_{\alpha,\beta} |\xi|^{N+1-|\alpha|}\langle\eta\rangle^{-N-|\beta|}, \quad i,j \in \{2,4\}.$$ Also, $\mathfrak{R}'(U,V)$ is a remainder satisfying, for any $\theta\in ]0,1[$ $$\label{L2 norm R'(U,V)} \begin{split} \|\mathfrak{R}' (U, V)& (t,\cdot)\|_{L^2} \lesssim (1 + \|U(t,\cdot)\|_{H^{5,\infty}})\|\mathfrak{R}(U,V)\|_{L^2} \\ &+ \left(\|\mathrm{R}_1U(t,\cdot)\|_{H^{1,\infty}} + \|U(t,\cdot)\|_{H^{5,\infty}} \right) \Big[ \|Q^I_0(V,W)\|_{L^2} \\ & +\left(\|\mathrm{R} U(t,\cdot)\|_{H^{6,\infty}}+ \|U(t,\cdot)\|_{H^{6,\infty}}\right)\|W^I(t,\cdot)\|_{L^2} \Big]\\ & + \|V(t,\cdot)\|^{2-\theta}_{H^{5,\infty}} \|V(t,\cdot)\|^\theta_{H^7} \|W^I(t,\cdot)\|_{L^2}, \end{split}$$ with $\mathfrak{R}(U,V)$ verifying . For any $n,r\in\mathbb{N}$, any $\chi\in C^\infty_0(\mathbb{R}^2)$ equal to 1 close to the origin and supported in open ball $B_{\varepsilon}(0)$, with $\varepsilon>0$ sufficiently small, we have that \[seminorms E\] $$\label{seminorm E0d} M^0_r\left(E^0_d\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right) U;\eta\right);n\right) \lesssim \|\mathrm{R}_1 U(t,\cdot)\|_{H^{1+r,\infty}},$$ $$\label{seminorm E-1d} M^{-1}_r\left(E^{-1}_d\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right) U;\eta\right);n\right) \lesssim \|U(t,\cdot)\|_{H^{5+r,\infty}},$$ $$\label{seminorm End} M^{-1}_r\left(E_{nd}\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right) U;\eta\right);n\right) \lesssim \|U(t,\cdot)\|_{H^{5+r,\infty}};$$ and $$\label{seminorm F0} M^0_r\left(F^0_d\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right) U;\eta\right);n\right) \lesssim \|\mathrm{R}_1 U(t,\cdot)\|_{H^{1+r,\infty}}.$$ Finally, as long as $\|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}+\|U(t,\cdot)\|_{H^{5,\infty}}$ is small, there is a constant $C>0$ such that $$\label{equivalence WtildeIs WIs} C^{-1}\|W^I_s(t,\cdot)\|_{L^2}\le \|\widetilde{W}^I_s(t,\cdot)\|_{L^2}\le C\|W^I_s(t,\cdot)\|_{L^2}.$$ From propositions \[Prop: equation of WIs\] and \[Prop: a first normal form\] it follows that, as long as $\|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}$, $\|U(t,\cdot)\|_{H^{5,\infty}}$ and $\|V(t,\cdot)\|_{H^{2,\infty}}$ are small, there is a constant $C>0$ such that $$\label{remark on equivalence between L2 norms} C^{-1}\|W^I(t,\cdot)\|_{L^2}\le \|\widetilde{W}^I_s(t,\cdot)\|_{L^2}\le C\|W^I(t,\cdot)\|_{L^2}.$$ This implies that, if \[modified\_energies\_Etilde\] $$\begin{gathered} \widetilde{E}_n(t;W):= \sum_{|\alpha|\le n} \left\|Op^B(I_4+E(U;\eta))Op^B(P(V;\eta))D^\alpha_x W(t,\cdot)\right\|_{L^2}, \quad \forall \, n\in\mathbb{N}, n \ge 3,\label{energy E_tilde_n}\\ \widetilde{E}^k_3(t;W):=\sum_{\substack{|\alpha|+|I|\le 3\\ |I|\le 3-k}} \left\|Op^B(I_4+E(U;\eta))Op^B(P(V;\eta))D^\alpha_x W^I(t,\cdot)\right\|_{L^2}, \forall\, 0\le k\le 2,\label{energy E_tilde_k2}\end{gathered}$$ there exists a constant $C_1>0$ such that $$\label{energy_equivalence Ekm Etilde_km} \begin{gathered} C_1^{-1}E_n(t;W)\le \widetilde{E}_n(t;W)\le C_1 E_n(t;W), \quad \forall \, n\ge 3, \\ C_1^{-1}E^k_3(t;W)\le \widetilde{E}^k_3(t;W)\le C_1 E^k_3(t;W), \quad \forall \, 0\le k\le 2. \end{gathered}$$ Thanks to the above equivalence, the propagation of some suitable estimates on $\widetilde{E}_n(t;W)$ and $\widetilde{E}^k_3(t;W)$ will provide us with and respectively, so we can rather focus on the derivation of an energy inequality for $\widetilde{E}_n(t;W), \widetilde{E}^k_3(t;W)$. In order to get rid of $Op^B(C''_d(U;\eta))V^I_s$ in we introduce matrices $$C''_d(U;\eta) = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & e_0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & f_0 \end{bmatrix}, \quad C''_{nd}(U;\eta) = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & f_0 \\ 0 & 0 & 0 & 0 \\ 0 & e_0 & 0 & 0 \end{bmatrix}$$ so that $$C''(U;\eta) = C''_d(U;\eta) + C''_{nd}(U;\eta),$$ and proceed to eliminate $Op^B(C''_d(U;\eta))V^I_s$ and $Op^B(C''_{nd}(U;\eta))V^I_s$ separately. \[Lem: Normal Forms on C”d(U,eta)\] Let $N\in\mathbb{N}^*$. There exists a diagonal matrix $E_d(U;\eta)$ of order 0, linear in $(u_+, u_{-})$, such that $$\begin{gathered} \label{equation for Op(Ed)} Op^B(C''_d(U;\eta))V^I_s + Op^B(D_t E_d(U;\eta))W^I_s - [A(D),Op^B(E_d(U;\eta))]W^I_s \\ = T_{-N}(U)W^I_s + \mathfrak{R}'(V,V),\end{gathered}$$ where $\mathfrak{R}'(V,V)$ satisfies, for any $\theta\in ]0,1[$, $$\label{L2 est R'(V,V)} \|\mathfrak{R}'(V,V)(t,\cdot)\|_{L^2}\lesssim \|V(t,\cdot)\|^{2-\theta}_{H^{5,\infty}}\|V(t,\cdot)\|^\theta_{H^7}\|V^I(t,\cdot)\|_{L^2},$$ and $T_{-N}(U)$ is a pseudo-differential operator of order less or equal than $-N$ such that, for any $s\in\mathbb{R}$, $$\label{norm of T-3WI} \|T_{-N}(U) \|_{\mathcal{L}(H^{s-N};H^s)}\lesssim \|\mathrm{R}_1U(t,\cdot)\|_{H^{N+2,\infty}} + \|U(t,\cdot)\|_{H^{N+6,\infty}},$$ whose symbol $\sigma(U,\eta) = \left(\sigma_{ij}(U,\eta)\right)_{1\le i,j\le 4}$ is such that \[sigma ij of operator T-3\] $$\mathcal{F}_{x\mapsto \xi}(\sigma_{ij}(U,\eta))(\xi) = \begin{cases} \sigma^+_{ii}(\xi, \eta)\hat{u}_+(\xi) + \sigma^{-}_{ii}(\xi,\eta)\hat{u}_{-}(\xi), \qquad & i=j \in\{2,4\}, \\ 0, &\text{otherwise}, \end{cases}$$ with $\sigma^\pm_{ii}(\xi,\eta)$ supported for $|\xi|\le \varepsilon\langle\eta\rangle$ for a small $\varepsilon>0$, and verifying, for any $\alpha,\beta\in\mathbb{N}^2$, $$|\partial^\alpha_\xi \partial^\beta_\eta \sigma^\pm_{ii}(\xi,\eta)|\lesssim_{\alpha,\beta} |\xi|^{N+1-|\alpha|}\langle\eta\rangle^{-N-|\beta|}, \quad \text{for } i=2,4.$$ Moreover, if $\chi\in C^\infty_0(\mathbb{R}^2)$ is equal to 1 close to the origin and has a sufficiently small support, $$\label{decomposition Ed} E_d\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right)U;\eta\right) = E^0_d\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right)U;\eta\right) + E^{-1}_d\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right)U;\eta\right),$$ the former matrix in the above right hand side being real of order 0 and satisfying , the latter being of order $-1$ and verifying . Because of the diagonal structure of $A(\eta)$ and $C''_d(U;\eta)$ we look for a matrix $E_d = (e_{ij})_{1\le i,j\le 4}$ satisfying such that $e_{ij}=0$ for all $i,j$ but $i=j\in\{2,4\}$, and we also require symbols $e_{22}, e_{44}$ to be of order 0 and linear in $(u_+, u_{-})$. If we remind that matrix $A(\eta)$ in is of order 1 and make the ansatz that $E_d$ is of order 0, then by symbolic calculus of proposition \[Prop: paradifferential symbolic calculus\] we have that $$\label{commutator A(D) Ed} -[A(D), Op^B(E_d(U;\eta))] = - \sum_{ |\alpha|=1}^N\frac{1}{\alpha!} Op^B\left(\partial^\alpha_\eta A(\eta)D^\alpha_xE_d(U;\eta)\right) + T_{-N}(U)$$ with $T_{-N}(U)$ pseudo-differential operator of order less or equal than $-N$ such that, for any $s\in\mathbb{R}$, $$\label{norm T-N} \|T_{-N}(U)\|_{\mathcal{L}(H^{s-N}; H^s)} \lesssim M^1_{N+1}(A(\eta);N+3)M^0_0(E_d(U;\eta);2) + M^1_{0}(A(\eta);N+3)M^0_{N+1}(E_d(U;\eta);2)$$and whose symbol $\sigma(U,\eta)=\left(\sigma_{ij}(U,\eta)\right)_{ij}$ is such that $\sigma_{ij}(U,\eta)=0$ for all $i,j$ but $i=j\in \{2,4\}$. Therefore, for any fixed $\chi\in C^\infty_0(\mathbb{R}^2)$ equal to 1 in $\overline{B_{\varepsilon_1}(0)}$ and supported in $B_{\varepsilon_2}(0)$, for some $0<\varepsilon_1<\varepsilon_2\ll 1$, we look for $E_d(U;\eta)$ such that $$\chi\left(\frac{D_x}{\langle\eta\rangle}\right)\left[C''_d(U;\eta) + D_tE_d(U;\eta) - \sum_{ |\alpha| =1}^N \frac{1}{\alpha!} \partial^\alpha_\eta A(\eta) D^\alpha_x E_d(U;\eta)\right] = 0.$$ Since $E_d(U;\eta)$ is required to be linear in $(u_+, u_{-})$, we should write it rather as $E_d(u_+, u_{-}; \eta)$ to then realize that, as $u_+$ (resp. $u_{-}$) is solution to the first (resp. to the third) equation in with $|I|=0$, $$\begin{aligned} D_t E_d(u_+, u_{-};\eta) &= E_d(|D_x| u_+, - |D_x|u_{-};\eta) + E_d\big(Q^{\mathrm{w}}_0(v_\pm, D_1 v_\pm), Q^{\mathrm{w}}_0(v_\pm, D_1 v_\pm);\eta\big), \\ D^\alpha_x E_d(u_+, u_{-};\eta) & = E_d(D^\alpha_x u_+, D^\alpha_x u_{-};\eta), \quad \forall \alpha\in\mathbb{N}^2.\end{aligned}$$ If we temporarily neglecting contribution $E_d\big(Q_0^{\mathrm{w}}(v_\pm, D_1 v_\pm), Q_0^{\mathrm{w}}(v_\pm, D_1 v_\pm);\eta\big)$, we are lead to solve the following equation $$\chi\left(\frac{D_x}{\langle\eta\rangle}\right)\left[C''_d(U;\eta) + E_d(|D_x| u_+, - |D_x|u_{-};\eta) - \sum_{ |\alpha| =1}^N \frac{1}{\alpha!} \partial^\alpha_\eta A(\eta) E_d(D^\alpha_x u_+, D^\alpha_x u_{-};\eta)\right] = 0,$$ which is equivalent to system $$\begin{cases} e_{22}\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right)\Big(|D_x| - \displaystyle\sum_{|\alpha|=1}^N\frac{1}{\alpha!}\partial^\alpha_\eta(\langle\eta\rangle)D^\alpha_x\Big) u_+, -\chi\left(\frac{D_x}{\langle\eta\rangle}\right)\Big(|D_x| + \sum_{|\alpha|=1}^N\frac{1}{\alpha!}\partial^\alpha_\eta(\langle\eta\rangle) D^\alpha_x\Big)u_{-} ; \eta \right) \\ \hspace{11cm} = -\chi\left(\frac{D_x}{\langle\eta\rangle}\right)e_0 \\ e_{44}\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right)\Big(|D_x| + \displaystyle\sum_{|\alpha|=1}^N\frac{1}{\alpha!}\partial^\alpha_\eta(\langle\eta\rangle) D^\alpha_x\Big) u_+, -\chi\left(\frac{D_x}{\langle\eta\rangle}\right)\Big(|D_x| - \sum_{|\alpha|=1}^N\frac{1}{\alpha!}\partial^\alpha_\eta(\langle\eta\rangle) D^\alpha_x\Big)u_{-}; \eta \right) \\ \hspace{11cm} = -\chi\left(\frac{D_x}{\langle\eta\rangle}\right) f_0, \end{cases}$$ with $e_0, f_0$ defined in . Then, if we look for $e_{ii}$ of the form $$\label{form of eii} e_{ii}(u_+, u_{-};\eta) = \int e^{ix\cdot\xi} \alpha_{ii}(\xi, \eta)\hat{u}_+(\xi) d\xi + \int e^{ix\cdot\xi} \beta_{ii}(\xi, \eta)\hat{u}_{-}(\xi) d\xi,$$ this system implies, inter alia, that $$\begin{gathered} \int e^{ix\cdot\xi}\chi\left(\frac{\xi}{\langle\eta\rangle}\right)\Bigl( |\xi| - \sum_{|\alpha|=1}^N \frac{1}{\alpha!}\partial^\alpha_\eta(\langle\eta\rangle)\xi^\alpha\Bigr)\alpha_{22}(\xi,\eta)\hat{u}_+(\xi)d\xi =\\ - \frac{i}{4}\int e^{ix\cdot\xi}\chi\left(\frac{\xi}{\langle\eta\rangle}\right)\Bigl(1-\frac{\eta}{\langle\eta\rangle}\cdot\frac{\xi}{|\xi|}\Bigr)\xi_1 \hat{u}_+(\xi) d\xi.\end{gathered}$$ As $$\left(1 \mp \sum_{|\alpha|=1}^N \frac{1}{\alpha!}\partial^\alpha_\eta (\langle\eta\rangle) \frac{\xi^\alpha}{|\xi|}\right) = 1\mp \sum_{k=1}^N\frac{1}{k!}(\xi\cdot\nabla_\eta)^k(\langle\eta\rangle)$$ and $$(\partial_{\eta_1}\xi_1 + \partial_{\eta_2}\xi_2)^k \langle\eta\rangle = \frac{|\xi|^k}{\langle \eta\rangle^{k-1}}\left(1- \left(\frac{\eta}{\langle\eta\rangle}\cdot\frac{\xi}{|\xi|}\right)^2\right)b_k(\xi,\eta), \quad 2\le k\le N,$$ with $b_k(\xi,\eta)$ polynomial of degree $k-2$ in $\frac{\eta}{\langle\eta\rangle}\cdot\frac{\xi}{|\xi|}$, we derive that $$\label{formula_1} \left(1 \mp \sum_{|\alpha|=1}^N \frac{1}{\alpha!}\partial^\alpha_\eta (\langle\eta\rangle) \frac{\xi^\alpha}{|\xi|}\right) = \left(1\mp \frac{\eta}{\langle\eta\rangle}\cdot\frac{\xi}{|\xi|}\right)\left(1 \mp b_\pm(\xi,\eta)\right)$$ with $$\label{derivatives bpm} \begin{gathered} b_\pm(\xi,\eta):= \sum_{k=2}^N \frac{1}{k!}|\xi|^{k-1}\langle\eta\rangle^{-(k-1)}\Big(1\pm \frac{\eta}{\langle\eta\rangle}\cdot\frac{\xi}{|\xi|}\Big)b_k(\xi,\eta),\\ |\partial^\mu_\xi \partial^\nu_\eta b_\pm(\xi,\eta)|\lesssim_{\mu, \nu} |\xi|^{1-|\mu|}\langle\eta\rangle^{-1-|\nu|}, \quad \forall \mu,\nu\in\mathbb{N}^2, \end{gathered}$$ and we can then choose $\alpha_{22}(\xi,\eta)$ in such that, when $|\xi|\le \varepsilon_2\langle\eta\rangle$, $$\alpha_{22}(\xi,\eta) = - \frac{i}{4} \left(1 - b_+(\xi,\eta)\right)^{-1}\frac{\xi_1}{|\xi|}.$$ Similarly, we choose multipliers $\beta_{22}, \alpha_{44}, \beta_{44}$ such that, as long as $|\xi|\le \varepsilon_2\langle\eta\rangle$, $$\begin{aligned} &\beta_{22}(\xi, \eta)= \frac{i}{4} \left(1 + b_{-}(\xi,\eta)\right)^{-1}\frac{\xi_1}{|\xi|}, \\ &\alpha_{44}(\xi, \eta) = -\frac{i}{4} \left(1 + b_{-}(\xi,\eta)\right)^{-1}\frac{\xi_1}{|\xi|}, \quad \beta_{44}(\xi, \eta) = \frac{i}{4} \left(1 - b_+(\xi,\eta)\right)^{-1}\frac{\xi_1}{|\xi|}.\end{aligned}$$ These multipliers are all well defined for $|\xi|\le \varepsilon_2\langle\eta\rangle$ as $b_\pm(\xi,\eta)=O(|\xi|\langle\eta\rangle^{-1})$. Moreover, using that $(1\pm b_\mp(\xi,\eta))^{-1}=1\mp b_\mp(\xi,\eta) + O(|\xi|^2\langle\eta\rangle^{-2})$ as long as $|\xi|\le \varepsilon_2\langle\eta\rangle$, we have that $$\alpha_{22}(\xi,\eta) = -\frac{i}{4} \frac{\xi_1}{|\xi|} + \alpha^{-1}_{22}(\xi, \eta), \quad \beta_{22}(\xi,\eta) = \frac{i}{4} \frac{\xi_1}{|\xi|} + \beta^{-1}_{22}(\xi, \eta),$$ $$\alpha_{44}(\xi,\eta) = -\frac{i}{4} \frac{\xi_1}{|\xi|} + \alpha^{-1}_{44}(\xi, \eta), \quad \beta_{44}(\xi,\eta) = \frac{i}{4} \frac{\xi_1}{|\xi|} + \beta^{-1}_{44}(\xi, \eta),$$ with $|\partial^\mu_\xi\partial^\nu_\eta \alpha^{-1}_{ii}| + |\partial^\mu_\xi\partial^\nu_\eta \beta^{-1}_{ii}| \lesssim_{\mu,\nu} |\xi|^{1-|\mu|}\langle\eta\rangle^{-1-|\nu|}$ for any $\mu,\nu\in\mathbb{N}^2$. Injecting the above $\alpha_{ii},\beta_{ii}$, $i\in\{2,4\}$, in we find that $$\begin{aligned} e_{22}\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right) u_+, \chi\left(\frac{D_x}{\langle\eta\rangle}\right)u_{-};\eta\right) &= -\frac{i}{4}R_1(u_+- u_{-}) + e^{-1}_{22}\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right) u_+, \chi\left(\frac{D_x}{\langle\eta\rangle}\right)u_{-};\eta\right), \\ e_{44}\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right) u_+, \chi\left(\frac{D_x}{\langle\eta\rangle}\right)u_{-};\eta\right) &= -\frac{i}{4}R_1(u_+ - u_{-})+ e^{-1}_{44}\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right) u_+, \chi\left(\frac{D_x}{\langle\eta\rangle}\right)u_{-};\eta\right),\end{aligned}$$ where, for $i\in\{2,4\}$, $$\begin{gathered} e^{-1}_{ii}\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right) u_+, \chi\left(\frac{D_x}{\langle\eta\rangle}\right)u_{-};\eta\right) =\\ \int e^{ix\cdot\xi} \chi\left(\frac{\xi}{\langle\eta\rangle}\right)\alpha^{-1}_{ii}(\xi,\eta) \hat{u}_+(\xi) d\xi + \int e^{ix\cdot\xi} \chi\left(\frac{\xi}{\langle\eta\rangle}\right)\beta^{-1}_{ii}(\xi,\eta) \hat{u}_{-}(\xi) d\xi.\end{gathered}$$ After lemma \[Lem\_appendix: Kernel with 1 function\] $(i)$ and above estimates for $\alpha^{-1}_{ii}, \beta^{-1}_{ii}$, kernels $$K^i_+(x,\eta):= \int e^{ix\cdot\xi} \chi\left(\frac{\xi}{\langle\eta\rangle}\right)\alpha^{-1}_{ii}(\xi,\eta) \langle\xi\rangle^{-4}d\xi, \quad K^i_{-}(x,\eta):=\int e^{ix\cdot\xi} \chi\left(\frac{\xi}{\langle\eta\rangle}\right)\beta^{-1}_{ii}(\xi,\eta) \langle\xi\rangle^{-4}d\xi$$ are such that, for any $\beta\in\mathbb{N}^2$, $|\partial^\beta_\eta K^i_\pm(x,\eta)|\lesssim |x|^{-1}\langle x \rangle^{-2}\langle\eta\rangle^{-1-|\beta|}$ for every $(x,\eta)$. This implies that $$\begin{gathered} \left|\partial^\beta_\eta e^{-1}_{ii}\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right) u_+, \chi\left(\frac{D_x}{\langle\eta\rangle}\right)u_{-};\eta\right)\right| \le \\ \left|\int \partial^\beta_\eta K^i_+(x-y,\eta)[\langle D_x\rangle^4 u_+](y)dy\right| + \left|\int \partial^\beta_\eta K^i_{-}(x-y,\eta) [\langle D_x\rangle^4 u_{-}](y)dy\right| \lesssim \|U(t,\cdot)\|_{H^{4,\infty}}\langle\eta\rangle^{-1-|\beta|}\end{gathered}$$ and $e^{-1}_{ii}$ is a symbol of order $-1$, for $i=2,4$. Moreover, using definition and the fact that space $W^{r,\infty}$ injects in $H^{r+1,\infty}$, one can check that for any $r,n\in\mathbb{N}$, $$M^{-1}_r\left( e^{-1}_{ii}\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right) u_+, \chi\left(\frac{D_x}{\langle\eta\rangle}\right)u_{-};\eta\right);n \right) \lesssim \|U(t,\cdot)\|_{H^{5+r,\infty}}$$ and therefore that $$M^0_r\left( e_{ii}\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right) u_+, \chi\left(\frac{D_x}{\langle\eta\rangle}\right)u_{-};\eta\right);n \right)\lesssim \|\mathrm{R}_1U(t,\cdot)\|_{H^{1+r,\infty}}+\|U(t,\cdot)\|_{H^{5+r,\infty}}.$$ Defining $$E^0_d(U;\eta) = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & -\frac{i}{4}R_1(u_+-u_{-}) & 0 & 0 \\ 0 & 0 & 0& 0 \\ 0 & 0 & 0 & -\frac{i}{4}R_1(u_+-u_{-}) \end{bmatrix}, \quad E^{-1}_d(U;\eta) = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & e^{-1}_{22} & 0 & 0\\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0& e^{-1}_{44} \end{bmatrix},$$ decomposition and estimate , hold. Consequently, as $$E_d\big(Q_0^{\mathrm{w}}(v_\pm, D_1 v_\pm), Q_0^{\mathrm{w}}(v_\pm, D_1 v_\pm);\eta\big) = E^{-1}_d\big(Q_0^{\mathrm{w}}(v_\pm, D_1 v_\pm), Q_0^{\mathrm{w}}(v_\pm, D_1 v_\pm);\eta\big)$$ for any $n\in\mathbb{N}$ and $\theta\in ]0,1[$, we derive from with $s=4$ that $$\begin{gathered} M^0_0\left(E_d\big(Q_0^{\mathrm{w}}(v_\pm, D_1 v_\pm), Q_0^{\mathrm{w}}(v_\pm, D_1 v_\pm);\eta\big);n\right) \lesssim \|Q_0^{\mathrm{w}}(v_\pm, D_1 v_\pm)\|_{H^{4,\infty}}\\ \lesssim \|V(t,\cdot)\|^{2-\theta}_{H^{5,\infty}}\|V(t,\cdot)\|^\theta_{H^7},\end{gathered}$$ and hence that the quantization of $E_d\big(Q_0^{\mathrm{w}}(v_\pm, D_1 v_\pm), Q_0^{\mathrm{w}}(v_\pm, D_1 v_\pm);\eta\big)$ acting on $V^I_s$ verifies after proposition \[Prop : Paradiff action on Sobolev spaces-NEW\]. Also, is deduced from while properties are obtained using essentially . \[Lem: Normal Form on C”nd\] Let $N\in\mathbb{N}^*$. There exists a purely imaginary matrix $E_{nd}(U;\eta)$, linear in $(u_+, u_{-})$ and of order $-1$, satisfying estimate , such that $$\begin{gathered} \label{equation for Op(End)} Op^B(C''_{nd}(U;\eta))V^I_s + Op^B(D_t E_{nd}(U;\eta))W^I_s - [A(D),Op^B(E_{nd}(U;\eta))]W^I_s \\ = T_{-N}(U)W^I_s + \mathfrak{R}'(V,V),\end{gathered}$$ where $ \mathfrak{R}'(V,V)$ is a remainder satisfying and $T_{-N}(U)$ is a pseudo-differential operator of order less or equal than $-N$ such that, for any $s\in\mathbb{R}$, $$\label{norm_T-N_lemmaCnd} \|T_{-N}(U)\|_{\mathcal{L}(H^{s-N};H^s)} \lesssim \|U(t,\cdot)\|_{H^{N+6,\infty}}.$$ Moreover, its symbol $\sigma(U,\eta) = \left(\sigma_{ij}(U,\eta)\right)_{1\le i,j\le 4}$ is such that \[sigma ij of operator T-3nd\] $$\mathcal{F}_{x\mapsto\xi}(\sigma_{ij}(U,\eta))(\xi) = \begin{cases} \sigma^+_{ij}(\xi, \eta)\hat{u}_+(\xi) + \sigma^{-}_{ij}(\xi,\eta)\hat{u}_{-}(\xi), \qquad & (i,j)\in \{(2,4), (4,2)\}, \\ 0, &\text{otherwise}, \end{cases}$$ with $\sigma^\pm_{ij}$ supported for $|\xi|\le \varepsilon\langle\eta\rangle$ for a small $\varepsilon>0$, and verifying, for any $\alpha,\beta\in\mathbb{N}^2$, $$|\partial^\alpha_\xi \partial^\beta_\eta \sigma^\pm_{ij}(\xi,\eta)|\lesssim_{\alpha,\beta} |\xi|^{N+2-|\alpha|}\langle\eta\rangle^{-N-1-|\beta|},$$ for $(i,j)\in \{(2,4), (4,2)\}$. Because of the structure of $C''_{nd}(U;\eta)$, we seek for a matrix $E_{nd}(U;\eta)$ satisfying , of the form $E_{nd}(U;\eta) = (e_{ij})_{1\le i,j\le 4}$ with $e_{ij}=0$ for all $i,j$, except $(i,j)\in\{ (2,4), (4,2)\}$. If we make the ansatz that $E_{nd}(U;\eta)$ is linear in $(u_+,u_{-})$, of order $-1$, and remind that $A(\eta)$ in is of order 1, from symbolic calculus of proposition \[Prop: paradifferential symbolic calculus\] we have that $$\begin{aligned} - [A(D),Op^B(E_{nd}(U;\eta))] =& - Op^B(A(\eta)E_{nd}(U;\eta) - E_{nd}(U;\eta)A(\eta)) \\ &- \sum_{|\alpha|=1}^N \frac{1}{\alpha !}Op^B(\partial^\alpha_\eta A(\eta) \cdot D^\alpha_x E_{nd}(U;\eta)) + T_{-N}(U),\end{aligned}$$ where $T_{-N}(U)$ is a pseudo-differential operator of order less or equal than $-N$, such that, for any $s\in\mathbb{R}$,$$\label{norm Tnd-N} \|T_{-N}(U)\|_{\mathcal{L}(H^{s-N}; H^s)} \lesssim M^1_{N+1}(A(\eta);N+3)M^{-1}_0(E_{nd}(U;\eta);2) + M^1_{0}(A(\eta);N+3)M^{-1}_{N+1}(E_{nd}(U;\eta);2),$$and whose symbol $\sigma(U,\eta)=(\sigma_{ij}(U,\eta))_{ij}$ is such that $\sigma_{ij}=0$ for all $i,j$ but $(i,j)\in\{(2,4), (4,2)\}$. Hence, for any fixed $\chi\in \mathbb{R}^2$ equal to 1 in $\overline{B_{\varepsilon_1}(0)}$ and supported in $B_{\varepsilon_2}(0)$, for some $0<\varepsilon_1<\varepsilon_2\ll 1$, we look for $E_{nd}(U;\eta)$ such that $$\begin{gathered} \label{equation for End} \chi\left(\frac{D_x}{\langle\eta\rangle}\right)\Big[C''_{nd}(U;\eta) + D_tE_{nd}(U;\eta) - A(\eta)E_{nd}(U;\eta) + E_{nd}(U;\eta)A(\eta) \\ - \sum_{|\alpha|=1}^N \frac{1}{\alpha!} \partial^\alpha_\eta A(\eta)\cdot D^\alpha_x E_{nd}(U;\eta)\Big] = 0.\end{gathered}$$ Furthermore, as $E_{nd}(U;\eta) = E_{nd}(u_+, u_{-};\eta)$ is linear in $(u_+, u_{-})$ and $u_+$ (resp. $u_{-}$) is solution to the first (resp. the third) equation in with $|I|=0$, we have that $$\begin{aligned} D_t E_{nd}(u_+, u_{-};\eta) &= E_{nd}(|D_x| u_+, - |D_x|u_{-};\eta) + E_{nd}\big(Q^{\mathrm{w}}_0(v_\pm, D_1 v_\pm), Q^{\mathrm{w}}_0(v_\pm, D_1 v_\pm);\eta\big), \\ D^\alpha_x E_{nd}(u_+, u_{-};\eta) & = E_{nd}(D^\alpha_x u_+, D^\alpha_x u_{-};\eta), \quad \forall \alpha\in\mathbb{N}^2\end{aligned}$$ while $$- A(\eta)E_{nd}(U;\eta) + E_{nd}(U;\eta)A(\eta) = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -2\langle\eta\rangle e_{24} \\ 0 & 0 & 0 & 0 \\ 0 & 2\langle\eta\rangle e_{42} & 0 & 0 \end{bmatrix}.$$ Then we rather search for symbols $e_{24}$ and $e_{42}$ such that $$\begin{cases} & \chi\left(\frac{D_x}{\langle\eta\rangle}\right) e_{2,4}\left( \Big(|D_x| - \displaystyle\sum_{|\alpha|=1}^N \frac{1}{\alpha!}\partial^\alpha(\langle\eta\rangle) D^\alpha_x - 2\langle\eta\rangle\Big) u_+, - \Big(|D_x|+ \sum_{|\alpha|=1}^N\frac{1}{\alpha!}\partial^\alpha(\langle\eta\rangle) D^\alpha_x + 2\langle\eta\rangle\Big)u_{-};\eta\right)\\ & \hspace{11cm} = -\chi\left(\frac{D_x}{\langle\eta\rangle}\right)f_0, \\ & \chi\left(\frac{D_x}{\langle\eta\rangle}\right)e_{4,2}\left( \Big(|D_x| + \displaystyle\sum_{|\alpha|=1}^N\frac{1}{\alpha!}\partial^\alpha(\langle\eta\rangle) D^\alpha_x + 2\langle\eta\rangle\Big) u_+, - \Big(|D_x| - \sum_{|\alpha|=1}^N\frac{1}{\alpha!}\partial^\alpha(\langle\eta\rangle)D^\alpha_x - 2\langle\eta\rangle\Big)u_{-};\eta\right)\\ & \hspace{11cm} = -\chi\left(\frac{D_x}{\langle\eta\rangle}\right)e_0, \end{cases}$$with $e_0, f_0$ given by , neglecting contribution $E_{nd}\big(Q^{\mathrm{w}}_0(v_\pm, D_1 v_\pm), Q^{\mathrm{w}}_0(v_\pm, D_1 v_\pm);\eta\big)$ whose quantization acting on $W^I_s$ gives rise to a remainder $\mathfrak{R}'(V,V)$, as we will see at the end of the proof. We look for $e_{ij}$ of the form $$e_{ij}(u_+, u_{-};\eta) = \int e^{ix\cdot\xi} \alpha_{ij}(\xi, \eta)\hat{u}_+(\xi) d\xi + \int e^{ix\cdot\xi} \beta_{ij}(\xi, \eta)\hat{u}_{-}(\xi) d\xi,$$ for $(i,j)\in\{(2,4), (4,2)\}$, and reminding , we choose the above multipliers such that, as long as $|\xi|\le \varepsilon_2\langle\eta\rangle$, $$\begin{aligned} & \alpha_{24}(\xi,\eta)= -\frac{i}{4}\left(1+ \frac{\eta}{\langle\eta\rangle}\cdot\frac{\xi}{|\xi|}\right)\left(\left(1- \frac{\eta}{\langle\eta\rangle}\cdot\frac{\xi}{|\xi|}\right)(1-b_+(\xi,\eta)) - 2\frac{\langle\eta\rangle}{|\xi|}\right)^{-1} \frac{\xi_1}{|\xi|}, \\ & \beta_{24}(\xi,\eta)=-\frac{i}{4} \left(1-\frac{\eta}{\langle\eta\rangle}\cdot\frac{\xi}{|\xi|}\right)\left(\left(1+ \frac{\eta}{\langle\eta\rangle}\cdot\frac{\xi}{|\xi|}\right)(1+b_{-}(\xi,\eta)) + 2\frac{\langle\eta\rangle}{|\xi|}\right)^{-1} \frac{\xi_1}{|\xi|} ,\end{aligned}$$ $$\alpha_{42}(\xi,\eta) =\beta_{24}, \quad \beta_{42}(\xi,\eta) =\alpha_{24}(\xi,\eta).$$ One can check that, on the support of $\chi\big(\frac{\xi}{\langle\eta\rangle}\big)$ and for any $\mu,\nu\in\mathbb{N}^2$, $|\partial^\mu_\xi \partial^\nu_\eta \alpha_{ij}|+|\partial^\mu_\xi \partial^\nu_\eta \beta_{ij}|\lesssim_{\mu,\nu}|\xi|^{1-|\mu|}\langle\eta\rangle^{-1-|\nu|}$, and then that, if $$K^{ij}_+(x,\eta):=\int e^{ix\cdot\eta} \chi\Big(\frac{\xi}{\langle\eta\rangle}\Big)\alpha_{ij}(\xi,\eta)\langle\xi\rangle^{-4} d\xi , \quad K^{ij}_{-}(x,\eta):=\int e^{ix\cdot\eta} \chi\Big(\frac{\xi}{\langle\eta\rangle}\Big)\beta_{ij}(\xi,\eta)\langle\xi\rangle^{-4} d\xi,$$ for $(i,j)\in \{(2,4), (4,2)\}$, $|\partial^\beta_\eta K^{ij}_\pm(x,\eta)|\lesssim |x|^{-1}\langle x \rangle^{-2}\langle\eta\rangle^{-1-|\beta|}$, for any $\beta\in\mathbb{N}^2$ and $(x,\eta)\in\mathbb{R}^2\times\mathbb{R}^2$, as a consequence of lemma \[Lem\_appendix: Kernel with 1 function\]. Therefore $$\begin{gathered} \left|\partial^\beta_\eta e_{ij}\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right) u_+, \chi\left(\frac{D_x}{\langle\eta\rangle}\right)u_{-};\eta\right)\right| \le \\ \left|\int \partial^\beta_\eta K^{ij}_+(x-y,\eta)[\langle D_x\rangle^4 u_+](y)dy\right| + \left|\int \partial^\beta_\eta K^{ij}_{-}(x-y,\eta) [\langle D_x\rangle^4 u_{-}](y)dy\right| \lesssim \|U(t,\cdot)\|_{H^{4,\infty}}\langle\eta\rangle^{-1-|\beta|},\end{gathered}$$ which implies that $e_{24}, e_{42}$ are symbols of order $-1$. Also, for $(i,j)\in\{(2,4), (4,2)\}$ and any $n,r\in\mathbb{N}$, one can prove that $$M^{-1}_r\left(e_{ij}\left(\chi\left(\frac{D_x}{\langle\eta\rangle}\right) u_+, \chi\left(\frac{D_x}{\langle\eta\rangle}\right)u_{-};\eta\right);n\right) \lesssim \|U(t,\cdot)\|_{H^{5+r,\infty}}$$ using definition and the fact that space $W^{r,\infty}$ injects in $H^{r+1}$ for any $r\in\mathbb{N}$. Estimate follows from and symbol $\sigma(U;\eta)$ associated to $T_{-N}(U)$ satisfies , as one can check using and the estimates derived above for $\alpha_{ij},\beta_{ij}$. Finally, from with $s=4$ we deduce that, for any $\theta \in ]0,1[$, $$M^{-1}_0\big(E_{nd}\big(Q^{\mathrm{w}}_0(v_\pm, D_1 v_\pm), Q^{\mathrm{w}}_0(v_\pm, D_1 v_\pm);\eta\big);n\big)\lesssim \|V(t,\cdot)\|^{2-\theta}_{H^{5,\infty}}\|V(t,\cdot)\|^\theta_{H^7},$$ and the quantization of $E_{nd}\big(Q^{\mathrm{w}}_0(v_\pm, D_1 v_\pm), Q^{\mathrm{w}}_0(v_\pm, D_1 v_\pm)$ acting on $W^I_s$ is a remainder verifying by proposition \[Prop : Paradiff action on Sobolev spaces-NEW\]. Lemmas \[Lem: Normal Forms on C”d(U,eta)\] and \[Lem: Normal Form on C”nd\] show that there exist two matrices $E_d(U;\eta)$ and $E_{nd}(U;\eta)$, linear in $(u_+, u_{-})$, satisfying equations and respectively. After definition of $\widetilde{W}^I_s$ and equalities , and we deduce that $$\begin{aligned} & (D_t - A(D))\widetilde{W}^I_s = Op^B(\widetilde{A}_1(V;\eta))W^I_s + Op^B(A''(V^I;\eta))U + Op^B_R(A''(V^I;\eta))U \\ & + Q^I_0(V,W) + \mathfrak{R}(U,V) + Op^B(E(U;\eta))\Big[ Op^B(\widetilde{A}_1(V;\eta))W^I_s + Op^B(A''(V^I;\eta))U \\ & + Op^B(C''(U;\eta))V^I_s + Op^B_R(A''(V^I;\eta))U + Q^I_0(V,W) + \mathfrak{R}(U,V)\Big] + T_{-N}(U)W^I_s + \mathfrak{R}'(V,V)\end{aligned}$$ where $\mathfrak{R}(U,V)$ satisfies , $\mathfrak{R}'(V,V)$ satisfies , and $T_{-N}(U)$ is a pseudo-differential operator of order less or equal than $-N$ verifying , . Contribution $$\begin{gathered} Op^B(E(U;\eta))\Big[ Op^B(A''(V^I;\eta))U + Op^B(C''(U;\eta))V^I_s + Op^B_R(A''(V^I;\eta))U\\ + Q^I_0(V,W) + \mathfrak{R}(U,V)\Big] \end{gathered}$$ is a remainder of the form $\mathfrak{R}'(U,V)$ satisfying estimate as a consequence of proposition \[Prop : Paradiff action on Sobolev spaces-NEW\], estimates with $r=0$, lemma \[Lemma: L2 estimate of semilinear terms\], and the fact that the $L^2$ norms of $V^I_s$ and $V^I$ are equivalent as long as $\|V(t,\cdot)\|_{H^{2,\infty}}$ is small. According to the definition of $E(U;\eta)$ and decomposition $$\begin{aligned} Op^B(E(U;\eta))Op^B(\widetilde{A}_1(V;\eta)) &= Op^B(E^0_d(U;\eta))Op^B(\widetilde{A}_1(V;\eta)) \\& + Op^B\big(E^{-1}_d(U;\eta) +E_{nd}(U;\eta) \big)Op^B(\widetilde{A}_1(V;\eta)).\end{aligned}$$ Proposition \[Prop : Paradiff action on Sobolev spaces-NEW\] and estimates , , with $r=0$, imply that the latter addend in the above right hand side is a bounded operator on $L^2$ whose $\mathcal{L}(L^2)$ norm is estimated by $\|U(t,\cdot)\|_{H^{5,\infty}}\|V(t,\cdot)\|_{H^{1,\infty}}$. The former one writes instead as $Op^B(E^0_d(U;\eta)\widetilde{A}_1(V;\eta))+ T_0(U,V)$, for an operator $T_0(U,V)$ of order less or equal than 0 and $\mathcal{L}(L^2)$ norm controlled by $\|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}\|V(t,\cdot)\|_{H^{2,\infty}}$, as follows from corollary \[Cor : paradiff symbolic calculus at order 1\] and estimates , with $r=1$. Hence, $$Op^B(E(U;\eta))Op^B(\widetilde{A}_1(V;\eta))W^I_s = Op^B(E^0_d(U;\eta)\widetilde{A}_1(V;\eta))W^I_s + \mathfrak{R}'(U,V),$$ for a new $\mathfrak{R}'(U,V)$ satisfying . After matrix $I_4+ E^0_d\big(\chi(\frac{D_x}{\langle\eta\rangle})U;\eta\big)$ is invertible as long as $\|\mathrm{R}_1U(t,\cdot)\|_{H^{1,\infty}}$ is small and $F^0_d(U;\eta):=\big[I_4+ E^0_d\big(\chi(\frac{D_x}{\langle\eta\rangle})U;\eta\big)\big]^{-1}-I_4$ is such that, for any $n,r\in\mathbb{N}$, $$M^0_r\left(F^0_d\Bigl(\chi\Bigl(\frac{D_x}{\langle\eta\rangle}\Bigr)U;\eta\Bigr);n\right)\lesssim \|\mathrm{R}_1U(t,\cdot)\|_{H^{1+r,\infty}}.$$ Moreover, $F^0_d(U;\eta)$ is a real diagonal matrix of order 0, and by corollary \[Cor : paradiff symbolic calculus at order 1\] with $r=1$ $$Op^B(I_4 + F^0_d(U;\eta)) Op^B(I_4+E^0_d(U;\eta))= Id + T_{-1}(U),$$ with $T_{-1}(U)$ of order less or equal than 0 and $\mathcal{L}(H^{s-1};H^s)$ norm bounded by $\|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}$, for any $s\in\mathbb{R}$. This implies that $$\begin{gathered} Op^B(I_4+F^0_d(U;\eta))\widetilde{W}^I_s = W^I_s + \widetilde{T}_{-1}(U)W^I_s, \widetilde{T}_{-1}(U)=T_{-1}(U) + Op^B(E^{-1}_d(U;\eta) + E_{nd}(U;\eta)) \end{gathered}$$ with $\widetilde{T}_{-1}(U)$ of order less or equal than $-1$ and $$\label{norm_Ttilde-1U} \|\widetilde{T}_{-1}(U)\|_{\mathcal{L}(H^{s-1};H^s)}\lesssim \|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}+ \|U(t,\cdot)\|_{H^{5,\infty}}$$ for any $s\in\mathbb{R}$. Hence, as long as this quantity is small, there exists a positive constant $C$ such that holds. Also, $$\begin{aligned} Op^B(I_4+E^0_d(U;\eta))&Op^B(\widetilde{A}_1(V;\eta))W^I_s \\ &= Op^B(I_4+E^0_d(U;\eta))Op^B(\widetilde{A}_1(V;\eta)) Op^B(I_4+F^0_d(U;\eta))\widetilde{W}^I_s\\ &- Op^B(I_4+E^0_d(U;\eta))Op^B(\widetilde{A}_1(V;\eta))\widetilde{T}_{-1}(U)W^I_s,\end{aligned}$$ where from proposition \[Prop : Paradiff action on Sobolev spaces-NEW\], , , and the $L^2$ norm of the latter term in the above right hand side is estimated by $$\label{est_norm_Ttilde0} \|V(t,\cdot)\|_{H^{1,\infty}}\left(\|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}+\|U(t,\cdot)\|_{H^{5,\infty}}\right)\|W^I(t,\cdot)\|_{L^2}.$$ On the other hand, by corollary \[Cor : paradiff symbolic calculus at order 1\] with $r=1$ we get that $$\begin{aligned} Op^B(I_4+E^0_d(U;\eta))Op^B(\widetilde{A}_1(V;\eta)) &Op^B(I_4+F^0_d(U;\eta))\widetilde{W}^I_s \\ &= Op^B\big((I_4+E^0_d(U;\eta))\widetilde{A}_1(V;\eta)(I_4+F^0_d(U;\eta))\big)\widetilde{W}^I_s\\ & + Op^B(I_4+E^0_d(U;\eta))T_0(U,V)\widetilde{W}^I_s + \widetilde{T}_0(U,V)\widetilde{W}^I_s,\end{aligned}$$ with $T_0(U,V), \widetilde{T}_0(U,V)$ operators of order less or equal than 0 and $\mathcal{L}(L^2)$ norm controlled, respectively, by $\|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}\|V(t,\cdot)\|_{H^{2,\infty}}$ and , so the last two terms in above right hand side are also remainders $\mathfrak{R}'(U,V)$ by proposition \[Prop : Paradiff action on Sobolev spaces-NEW\] and estimate . That concludes the proof of the statement. ### A second normal forms transformation. {#sub: second normal form} In proposition \[Prop: a first normal form\] in previous subsection we showed that one can get rid of the slow-decaying-in-time semi-linear contribution $Op^B\big(C''(U;\eta)\big)V^I_s$ in by introducing a new function $\widetilde{W}^I_s$, defined in in terms of $W^I_s$ and solution to equation . That naturally led us to the introduction of new energies $\widetilde{E}_n(t;W)$, for $n\in\mathbb{N}, n\ge 3$, and $\widetilde{E}^k_3(t;W)$, for $k\in\mathbb{N}, 0\le k\le 2$, (see ) which are respectively equivalent to starting $E_n(t;W)$ and $E^k_3(t;W)$ whenever some uniform norms of $U,V$ are sufficiently small. However, these new energies do not allow us yet to recover enhanced estimates and as it is not true that $$\label{wished_energy_estimate} \left|\partial_t \widetilde{E}_n(t;W)\right| = O\left(\varepsilon t^{-1+\frac{\delta}{2}}E_n(t;W)^\frac{1}{2}\right), \quad \left|\partial_t \widetilde{E}^k_3(t;W)\right| = O\left(\varepsilon t^{-1+\frac{\delta}{2}}E^k_3(t;W)^\frac{1}{2}\right).$$ This is do to the fact that we still have to deal with semi-linear slow-decaying contributions $Op^B(A''(V^I;\eta))U$, $Op^B_R(A''(V^I;\eta))U$, $Q^I_0(V,W)$ to the right hand side of , together with the new $T_{-N}(U)W^I_s$ whose $L^2$ norm is also a $O(t^{-\frac{1}{2}}\|W^I(t,\cdot)\|_{L^2})$ after and . The aim of the current subsection is hence to perform a new normal form argument to replace the mentioned terms with more decaying ones. This is actually done at the energy level, meaning that we are going to add some suitable cubic perturbations to $\widetilde{E}_n(t;W)$ and $\widetilde{E}^k_3(t;W)$ so that the new energies so defined satisfy estimates as in . Let us first focus on the slow decaying terms that appear when computing $$\partial_t \widetilde{E}_n(t;W) = \sum_{I\in\mathcal{I}_n}\left\langle \partial_t \widetilde{W}^I_s, \widetilde{W}^I_s\right\rangle$$ for any integer $n\ge 3$. Using equation and rewriting $\widetilde{W}^I_s$ in terms of $W^I$ we find, on the one hand, the contribution $$\label{term_getridoff_Etilden} -\sum_{I\in \mathcal{I}_n}\Im\Bigl[\langle Op^B(A''(V^I;\eta))U + Op^B_R(A''(V^I;\eta))U , W^I\rangle + \langle T_{-N}(U)W^I, W^I\rangle\Bigr],$$ which is a $O(\varepsilon t^{-1/2}E_n(t;W))$ after Cauchy-Schwarz inequality, lemma \[Lemma: L2 estimate of semilinear terms\] and a-priori estimates . But we also have $$\label{contribution-to-get-ridoff} -\sum_{I\in \mathcal{I}_n}\sum_{\substack{(I_1,I_2)\in \mathcal{I}(I) \\ [\frac{|I|}{2}]<|I_1|<|I|}}\Im \left[\langle Q^{\mathrm{kg}}_0(v^{I_1}_\pm, D_1u^{I_2}_\pm), v^I_+ + v^I_{-}\rangle \right],$$ which enjoys the same decay as the previous one, as can be immediately seen using again Cauchy-Schwarz inequality along with and . From definition of matrix $A''(V^I,\eta)$, Plancherel’s formula, and the fact that $\overline{v^I_+} = -v^I_{-}$ $$\begin{aligned} &\langle Op^B(A''(V^I;\eta))U, W^I \rangle = \langle Op^B(a_0(v^I_\pm;\eta)\eta_1)u_+ + Op^B(b_0(v^I_\pm;\eta)\eta_1)u_{-}, v^I_+ + v^I_{-}\rangle \\ & = -\frac{i}{4(2\pi)^2} \int \chi\left(\frac{\xi - \eta}{\langle\eta\rangle}\right) \left[({\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{v_+^{I}+v_{-}^{I}}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{v_+^{I}+v_{-}^{I}}{\tmpbox}})(\xi-\eta)({\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{u_+ + u_{-}}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{u_+ + u_{-}}{\tmpbox}})(\eta) - \frac{\xi-\eta}{\langle\xi-\eta\rangle}\cdot\frac{\eta}{|\eta|}({\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{v_+^{I}-v_{-}^{I}}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{v_+^{I}-v_{-}^{I}}{\tmpbox}})(\xi-\eta) \right. \\ & \left. \hspace{8,5cm} \times ({\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{u_+ - u_{-}}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{u_+ - u_{-}}{\tmpbox}})(\eta)\right]\eta_1( {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{v^I_{-} + v^I_{-}}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{v^I_{-} + v^I_{-}}{\tmpbox}})(-\xi) d\xi d\eta,\end{aligned}$$ with $\chi$ denoting a smooth function equal to 1 in $\overline{B_{\varepsilon_1}(0)}$ and supported in $B_{\varepsilon_2}(0)$, for some $0<\varepsilon_1<\varepsilon_2\ll 1$. Hence $$\label{sum of CIk} - \Im\left[\langle Op^B(A''(V^I;\eta))U, W^I \rangle\right]= \sum_{j_k\in\{+,-\}} C^I_{(j_1,j_2,j_3)}$$ with $$\label{integral_CI} C^I_{(j_1,j_2,j_3)}= \frac{1}{4(2\pi)^2}\displaystyle\int \chi\Big(\frac{\xi-\eta}{\langle\eta\rangle}\Big)\left(1-j_1j_2\frac{\xi-\eta}{\langle\xi-\eta\rangle}\cdot\frac{\eta}{|\eta|}\right)\eta_1 \hat{v}^I_{j_1}(\xi-\eta) \hat{u}_{j_2}(\eta) \hat{v}^I_{j_3}(-\xi) d\xi d\eta,$$ for any $j_1,j_2,j_3\in\{+,-\}$. Analogously, from equality $$\label{sum of CIRk} - \Im\left[\langle Op^B_R(A''(V^I;\eta))U, W^I \rangle\right]=\sum_{j_k\in\{+,-\}} C^{I,R}_{(j_1,j_2,j_3)}$$ with $$\begin{gathered} \label{integral_CIR} C^{I,R}_{(j_1,j_2,j_3)}= \frac{1}{4(2\pi)^2}\displaystyle\int \left[1-\chi\Big(\frac{\xi-\eta}{\langle\eta\rangle}\Big) - \chi\Big(\frac{\eta}{\langle\xi-\eta\rangle}\Big)\right]\left(1-j_1j_2\frac{\xi-\eta}{\langle\xi-\eta\rangle}\cdot\frac{\eta}{|\eta|}\right)\eta_1 \\ \times \hat{v}^I_{j_1}(\xi-\eta) \hat{u}_{j_2}(\eta) \hat{v}^I_{j_3}(-\xi) d\xi d\eta.\end{gathered}$$ After proposition \[Prop: a first normal form\], $T_{-N}(U)=(\sigma_{ij}(U,D_x))_{ij}$ with symbols $\sigma_{ij}(U,\eta)$ satisfying . Introducing $\rho:\{+,-\}\rightarrow \{2,4\}$ such that $\rho(+)=2, \rho(-) =4$ and using the convention that $-j_k\in \{+,-\}\setminus \{j_k\}$, we have that $$\label{expression T-N in normal forms} \begin{split} \langle T_{-N}(U)W^I, W^I\rangle & = \sum_{i,j\in\{+,-\}} \langle \sigma_{\rho(i) \rho(j)}(U,D_x) v^I_j, v^I_i\rangle \\ & = - \frac{1}{(2\pi)^2}\sum_{j_k\in \{+,-\}}\int \sigma^{j_2}_{\rho(j_3), \rho(j_1)}(\eta, \xi-\eta) \hat{v}^I_{j_1}(\xi-\eta)\hat{u}_{j_2}(\eta)\hat{v}^I_{-j_3}(-\xi)d\xi d\eta , \end{split}$$ where multipliers $\sigma^{j_2}_{\rho(j_3), \rho(j_1)}(\eta, \xi-\eta)$ are supported for $|\eta|\le\varepsilon |\xi-\eta|$ and such that, for any $\alpha,\beta\in\mathbb{N}^2$, $$\left|\partial^\alpha_\xi \partial^\beta_\eta \sigma^{j_2}_{\rho(j_3), \rho(j_1)}(\eta, \xi-\eta)\right|\lesssim_{\alpha,\beta} |\eta|^{N+1-|\beta|}\langle \xi-\eta\rangle^{-N-|\alpha|},$$ for any $(\xi,\eta)\in\mathbb{R}^2\times\mathbb{R}^2$, any $j_1,j_2,j_3\in \{+,-\}$. Moreover, by we have that $$\begin{gathered} \label{sum CI1I2} -\Im \left[\langle Q^{\mathrm{kg}}_0(v^{I_1}_\pm, D_1u^{I_2}_\pm), v^I_+ + v^I_{-}\rangle \right] = \sum_{j_k\in\{+,-\}} C^{I_1,I_2}_{(j_1,j_2,j_3)}\end{gathered}$$ with $$\label{integral_CI1I2} C^{I_1,I_2}_{(j_1,j_2,j_3)}:= \frac{1}{4(2\pi)^2}\displaystyle\int \left(1-j_1j_2\frac{\xi-\eta}{\langle\xi-\eta\rangle}\cdot\frac{\eta}{|\eta|}\right)\eta_1\ \hat{v}^{I_1}_{j_1}(\xi-\eta) \hat{u}^{I_2}_{j_2}(\eta) \hat{v}^I_{j_3}(-\xi) d\xi d\eta.$$ The above equalities lead us to introduce the following multipliers $$\label{def of B(i1,i2,i3)} B^k_{(j_1,j_2,j_3)}(\xi,\eta):= \frac{1}{j_1\langle\xi-\eta\rangle+j_2|\eta| + j_3\langle\xi\rangle}\left(1-j_1j_2\frac{\xi-\eta}{\langle\xi-\eta\rangle}\cdot\frac{\eta}{|\eta|}\right)\eta_k, \quad k=1,2$$ and $$\label{def_multiplier_sigmatildeN} \widetilde{\sigma}^N_{(j_1,j_2,j_3)}(\xi,\eta):= \frac{ \sigma^{j_2}_{\rho(j_3), \rho(j_1)}(\eta, \xi-\eta)}{j_1\langle\xi-\eta\rangle + j_2|\eta| -j_3\langle\xi\rangle},$$ together with the following integrals \[integral Dk DR\_k DT\] $$D^I_{(j_1,j_2,j_3)} := \frac{i}{4(2\pi)^2}\displaystyle\int \chi\left(\frac{\xi-\eta}{\langle\eta\rangle}\right) B^1_{(j_1,j_2,j_3)}(\xi, \eta)\hat{v}^I_{j_1}(\xi-\eta) \hat{u}_{j_2}(\eta)\hat{v}^I_{j_3}(-\xi)\, d\xi d\eta, \label{def of D_k}$$ $$\begin{gathered} D^{I,R}_{(j_1, j_2, j_3)} := \frac{i}{4(2\pi)^2}\displaystyle\int\left[1 - \chi\left(\frac{\xi-\eta}{\langle\eta\rangle}\right) - \chi\left(\frac{\eta}{\langle\xi-\eta\rangle}\right)\right] B^1_{(j_1,j_2,j_3)}(\xi, \eta )\\ \times \hat{v}^I_{j_1}(\xi-\eta) \hat{u}_{j_2}(\eta)\hat{v}^I_{j_3}(-\xi)d\xi d\eta , \label{def of DR_k} \end{gathered}$$ $$D^{I,T_{-N}}_{(j_1, j_2, j_3)} := \Re \left[\frac{1}{(2\pi)^2}\int\widetilde{\sigma}^N_{(j_1,j_2,j_3)}(\xi, \eta)\hat{v}^I_{j_1}(\xi-\eta)\hat{u}_{j_2}(\eta)\hat{v}^I_{-j_3}(-\xi)d\xi d\eta\right] \label{def of DIT}$$ and $$\label{def of DI1I2} D^{I_1,I_2}_{(j_1,j_2,j_3)} := \frac{i}{4(2\pi)^2}\displaystyle\int B^1_{(j_1,j_2,j_3)}(\xi, \eta)\hat{v}^{I_1}_{j_1}(\xi-\eta) \hat{u}^{I_2}_{j_2}(\eta)\hat{v}^I_{j_3}(-\xi)\, d\xi d\eta$$ for any triplet $(j_1,j_2,j_3)\in \{+,-\}^3$. We warn the reader that definitions and are given here for any general multi-indices $I, I_1, I_2$. \[def\_Edag\_n\] For every integer $n\ge 3$ we define the second modified energy $\widetilde{E}^\dagger_n(t;W)$ as $$\begin{gathered} \label{energy_dag_En} \widetilde{E}^\dagger_n(t;W):= \widetilde{E}_n(t;W)+ \sum_{\substack{I\in\mathcal{I}_n \\ j_i\in\{+,-\}}} \left(D^I_{(j_1,j_2,j_3)} + D^{I,R}_{(j_1,j_2,j_3)} + D^{I,T_{-N}}_{(j_1,j_2,j_3)}\right)\\ + \sum_{\substack{I\in\mathcal{I}_n \\ j_i\in\{+,-\}}}\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ [\frac{|I|}{2}]<|I_1|<|I|}} D^{I_1,I_2}_{(j_1,j_2,j_3)}.\end{gathered}$$ Let us now analyse the time derivative of $\widetilde{E}^k_3(t;W)$ for integers $0\le k\le 2$. As in the previous case, from equation we see appear the same contribution as in , but with the sum over $\mathcal{I}_n$ replaced with that on $\mathcal{I}^k_3$. We also find $$\label{term_getridoff_QI0} -\sum_{I\in\mathcal{I}^k_3}\Im[\langle Q^I_0(V,W), W^I\rangle]$$ which is a $O(\varepsilon t^{-(1+\delta_k)/2}E^k_3(t;W)^{1/2})$ from Cauchy-Schwarz inequality and estimate . To be more precise, the slow decay in time of the above scalar product is due to some particular quadratic term appearing in $Q^I_0(V,W)$. In fact, according to definition and to , , , , for any $I\in\mathcal{I}^k_3$ $$\begin{gathered} \label{est_Qw0} \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_2|<|I|}} \left|\langle Q^\mathrm{w}_0(v^{I_1}_\pm, D_x v^{I_2}_\pm), u^I_+ + u^I_{-}\rangle\right| + \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|+|I_2|\le 2, |I_2|<|I|}} \left|\langle Q^\mathrm{w}_0(v^{I_1}_\pm, D_t v^{I_2}_\pm), u^I_+ + u^I_{-}\rangle\right| \\ \lesssim \|\mathfrak{R}^k_3(t,\cdot)\|_{L^2}\|U^I(t,\cdot)\|_{L^2} \le C(A+B)\varepsilon t^{-1+\frac{\delta_k}{2}} E^k_3(t;W)^\frac{1}{2}.\end{gathered}$$ Also, after and we have that for all $I\notin \mathcal{V}^k$, with $\mathcal{V}^k$ defined in , $$\begin{gathered} \label{est_Qkg0-Ik3} \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|,|I_2|<|I|}}\left|\left\langle Q^\mathrm{kg}_0(v^{I_1}_\pm, D_xu^{I_2}_\pm), v^I_+ +v^I_{-}\right\rangle\right| +\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|+|I_2|\le 2, |I_1|,|I_2|<|I|}}\left|\left\langle Q^\mathrm{kg}_0(v^{I_1}_\pm, D_t u^{I_2}_\pm), v^I_+ +v^I_{-}\right\rangle\right| \\ \lesssim \|\mathfrak{R}^k_3(t,\cdot)\|_{L^2}\|V^I(t,\cdot)\|_{L^2} \le C(A+B)\varepsilon t^{-1+\frac{\delta_k}{2}} E^k_3(t;W)^\frac{1}{2}.\end{gathered}$$ Observe that the decay rate $O(t^{-1+\delta_k/2})$ in the right hand side of the two above inequalities is the slowest one that allows us to propagate a-priori estimate and it gives us back exactly the slow growth in time $t^{\delta_k/2}$ enjoyed by $E^k_3(t;W)^{1/2}$, for $0\le k\le 2$. On the other hand, for $I\in\mathcal{V}^k$ with $k=0,1,$ we have that, for some smooth cut-off function $\chi$ and some $\sigma>0$ small, $$\begin{gathered} \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|,|I_2|<|I|}}c_{I_1,I_2} Q^\mathrm{kg}_0(v^{I_1}_\pm, D_xu^{I_2}_\pm) = \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ I_1\in \mathcal{K}, |I_2|\le 1}}c_{I_1,I_2} Q^\mathrm{kg}_0\left( v^{I_1}_\pm,\chi(t^{-\sigma}D_x) D_x u^{I_2}_\pm\right) + \mathfrak{R}^k_3(t,x),\\ \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\|I_1|+|I_2|\le 2, |I_1|,|I_2|<|I|}}c_{I_1,I_2} Q^\mathrm{kg}_0(v^{I_1}_\pm, D_tu^{I_2}_\pm) =\sum_{\substack{(J,0)\in\mathcal{I}(I)\\ J\in\mathcal{K}}} c_{J,0} Q^\mathrm{kg}_0(v^J_\pm, \chi(t^{-\sigma}D_x) |D_x| u_\pm)+ \mathfrak{R}^k_3(t,x). \end{gathered}$$ The $L^2$ norms of the summations in the above right hand sides are bounded by $$\sum_{J|\le 1}\left( \|\chi(t^{-\sigma} D_x) u^J_\pm(t,\cdot)\|_{H^{2,\infty}} + \|\chi(t^{-\sigma} D_x)\mathrm{R} u^J_\pm(t,\cdot)\|_{H^{2,\infty}}\right) E^k_3(t;W)^\frac{1}{2}$$ and hence by $\varepsilon t^{-1/2}E^k_3(t;W)^{1/2}$ as follows by sharp estimate derived in appendix \[Appendix B\]. Therefore, the very contribution to that has to be eliminated from $\partial_t \widetilde{E}^k_3(t;W)$ appears only for $k=0,1$ and is $$\begin{gathered} \label{decomposition-Qkg0-complete} -\sum_{I\in \mathcal{V}^k}\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ I_1\in\mathcal{K}, |I_2|\le 1}}c_{I_1,I_2} \Im\left[\left\langle Q^\mathrm{kg}_0\left( v^{I_1}_\pm,\chi(t^{-\sigma}D_x) D_x u^{I_2}_\pm\right), v^I_+ + v^I_{-}\right\rangle\right] \\ -\sum_{I\in \mathcal{V}^k}\sum_{\substack{(J,0)\in\mathcal{I}(I)\\ J\in\mathcal{K}}}c_{J,0} \Im\left[\left\langle Q^\mathrm{kg}_0\left( v^J_\pm,\chi(t^{-\sigma}D_x) |D_x| u_\pm\right), v^I_+ + v^I_{-}\right\rangle\right] .\end{gathered}$$ As $$\label{sum_FI1I2} \begin{gathered} -\Im \left[\langle Q^\mathrm{kg}_0(v^{I_1}_\pm, \chi(t^{-\sigma}D_x)D_l u^{I_2}_\pm), v^I_+ + v^I_{-} \rangle\right] =\sum_{j_i\in \{+,-\}} F^{I_1,I_2,l}_{(j_1,j_2,j_3)}, \quad l=1,2\\ -\Im \left[\langle Q^\mathrm{kg}_0(v^{I_1}_\pm, \chi(t^{-\sigma}D_x)|D_x| u^{I_2}_\pm), v^I_+ + v^I_{-} \rangle\right] =\sum_{j_i\in \{+,-\}} F^{I_1,I_2,3}_{(j_1,j_2,j_3)}, \end{gathered}$$ with $$\label{integral_FI1I2} F^{I_1,I_2, l}_{(j_1,j_2,j_3)} = \frac{1}{4(2\pi)^2}\displaystyle\int \left(1-j_1j_2\frac{\xi-\eta}{\langle\xi-\eta\rangle}\cdot\frac{\eta}{|\eta|}\right)\eta_l\ \hat{v}^{I_1}_{j_1}(\xi-\eta) {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\chi(t^{-\sigma}D_x) u^{I_2}_{j_2}}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\chi(t^{-\sigma}D_x) u^{I_2}_{j_2}}{\tmpbox}}(\eta) \hat{v}^I_{j_3}(-\xi) d\xi d\eta,$$ for any $j_i\in\{+,-\}$, $l=1,2,3$, and $\eta_3:=j_2|\eta|$, we introduce a new multiplier $$\label{multiplier_B3} B^3_{(j_1,j_2,j_3)}(\xi,\eta):= \frac{j_2}{j_1\langle\xi-\eta\rangle+j_2|\eta| + j_3\langle\xi\rangle}\left(1-j_1j_2\frac{\xi-\eta}{\langle\xi-\eta\rangle}\cdot\frac{\eta}{|\eta|}\right)|\eta|$$ together with integrals $$\label{integral_GI1I2} G^{I_1,I_2,l}_{(j_1,j_2,j_3)} = \frac{i}{4(2\pi)^2}\displaystyle\int B^l_{(j_1,j_2,j_3)}(\xi,\eta)\ \hat{v}^{I_1}_{j_1}(\xi-\eta) {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\chi(t^{-\sigma}D_x) u^{I_2}_{j_2}}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\chi(t^{-\sigma}D_x) u^{I_2}_{j_2}}{\tmpbox}}(\eta) \hat{v}^I_{j_3}(-\xi) d\xi d\eta$$ for any $l=1,2,3$, $(j_1,j_2,j_3)\in\{+,-\}^3$, with multipliers $B^l_{(j_1,j_2,j_3)}$ given by when $l=1,2$, and by when $l=3$. We warn the reader that in what follows we will sometimes refer to multipliers $B^l_{(j_1,j_2,j_3)}$ (resp. integrals $F^{I_1,I_2, l}_{(j_1,j_2,j_3)}$ and $G^{I_1,I_2,l}_{(j_1,j_2,j_3)}$) simply as $B_{(j_1,j_2,j_3)}$ (resp. $F^{I_1,I_2}_{(j_1,j_2,j_3)}$ and $G^{I_1,I_2}_{(j_1,j_2,j_3)}$) forgetting about superscript $l$. This choice reveals to be convenient when we do not need to distinguish between $l=1,2,3$. \[def\_Edag\_k3\] For every integer $0\le k\le 2$ we define the second modified energy $\widetilde{E}^{k,\dagger}_3(t;W)$ as $$\begin{gathered} \label{energy_dag_Ek2} \widetilde{E}^{k,\dagger}_3(t;W):= \widetilde{E}^k_3(t;W)+ \sum_{\substack{I\in\mathcal{I}^k_3\\ j_i\in \{+,-\}}}\left(D^I_{(j_1,j_2,j_3)} + D^{I,R}_{(j_1,j_2,j_3)} + D^{I,T_{-N}}_{(j_1,j_2,j_3)}\right) \\ +\delta_{k<2}\sum_{\substack{I\in\mathcal{V}^k\\ j_i\in \{+,-\}}}\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ I_1\in\mathcal{K}, |I_2|\le 1}}c_{I_1,I_2} G^{I_1,I_2}_{(j_1,j_2,j_3)},\end{gathered}$$ with $\delta_{k<2}=1$ if $k=0,1,$ 0 otherwise, and coefficients $c_{I_1,I_2}\in \{-1,0,1\}$. In view of the lemmas to follow it is useful to remind that, after system , for any multi-index $I$ vector $(\hat{u}^I_+, \hat{v}^I_+, \hat{u}^I_{-}, \hat{v}^I_{-})$ is solution to $$\label{system for hat(u)I+-, hat(v)I+-} \begin{cases} & (D_t - |\xi|)\hat{u}^I_+(t,\xi) = \sum_{|I_1| + |I_2| = |I| }{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q_0^{\mathrm{w}}(v^{I_1}_\pm, D_1 v^{I_2}_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q_0^{\mathrm{w}}(v^{I_1}_\pm, D_1 v^{I_2}_\pm)}{\tmpbox}} + \sum_{|I_1| + |I_2| < |I| } c_{I_1, I_2}{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q_0^{\mathrm{w}}(v^{I_1}_\pm, D v^{I_2}_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q_0^{\mathrm{w}}(v^{I_1}_\pm, D v^{I_2}_\pm)}{\tmpbox}} \\ & (D_t - \langle\xi\rangle)\hat{v}^I_+(t,\xi) = \sum_{|I_1| + |I_2| = |I| } {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q_0^{\mathrm{kg}}(v^{I_1}_\pm, D_1 u^{I_2}_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q_0^{\mathrm{kg}}(v^{I_1}_\pm, D_1 u^{I_2}_\pm)}{\tmpbox}} + \sum_{|I_1| + |I_2| < |I| } c_{I_1, I_2}{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q_0^{\mathrm{kg}}(v^{I_1}_\pm, D u^{I_2}_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q_0^{\mathrm{kg}}(v^{I_1}_\pm, D u^{I_2}_\pm)}{\tmpbox}} \\ & (D_t + |\xi|)\hat{u}^I_{-}(t,x) = \sum_{|I_1| + |I_2| = |I| }{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q_0^{\mathrm{w}}(v^{I_1}_\pm, D_1 v^{I_2}_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q_0^{\mathrm{w}}(v^{I_1}_\pm, D_1 v^{I_2}_\pm)}{\tmpbox}} + \sum_{|I_1| + |I_2| <|I| } c_{I_1, I_2}{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q_0^{\mathrm{w}}(v^{I_1}_\pm, D v^{I_2}_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q_0^{\mathrm{w}}(v^{I_1}_\pm, D v^{I_2}_\pm)}{\tmpbox}} \\ & (D_t + \langle \xi \rangle)\hat{v}^I_{-}(t,x) = \sum_{|I_1| + |I_2| = |I| } {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q_0^{\mathrm{kg}}(v^{I_1}_\pm, D_1 u^{I_2}_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q_0^{\mathrm{kg}}(v^{I_1}_\pm, D_1 u^{I_2}_\pm)}{\tmpbox}} + \sum_{|I_1| + |I_2| < |I| } c_{I_1, I_2}{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q_0^{\mathrm{kg}}(v^{I_1}_\pm, D u^{I_2}_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q_0^{\mathrm{kg}}(v^{I_1}_\pm, D u^{I_2}_\pm)}{\tmpbox}} \end{cases}$$ with coefficients $c_{I_1, I_2}\in \{-1,0,1\}$ and indices $I_1,I_2$ in above right hand side such that $(I_1,I_2)\in\mathcal{I}(I)$. In lemmas \[Lem:Quartic\_terms\_1\] and \[Lem:Quartic\_Terms\_II\] we will check that, with definitions \[def\_Edag\_n\], \[def\_Edag\_k3\], the slow decaying contributions highlighted in are replaced in $\partial_t \widetilde{E}^\dagger_n(t;W)$, $\partial_t\widetilde{E}^{k,\dagger}_3(t;W)$ by some new quartic terms. These latter ones are obtained from integrals by replacing each factor $\hat{v}^I_{j_1}, \hat{u}_{j_2},\hat{v}^I_{j_3}$ at a time with the non-linearity appearing in the equation that factor satisfies in . Lemma \[Lem:Quartic\_terms\_III\] (resp. lemma \[Lem:Analysis quartic terms IV\]) shows that the same is for troublesome contributions in $\partial_t\widetilde{E}^\dagger_n(t;W)$ (resp. for in $\partial_t\widetilde{E}^{k,\dagger}_3(t;W)$). We are also going to see that, if $N\in\mathbb{N}^*$ is chosen sufficiently large (e.g. $N=18$), all these quartic terms suitably decay in time, and that modified energies $\widetilde{E}^\dagger_n(t;W), \widetilde{E}^{k,\dagger}_3(t;W)$ are equivalent, respectively, to $E_n(t;W), E^k_3(t,W)$. We point out the fact that the normal form’s step performed in previous section was necessary to avoid here some problematic quartic contributions coming from quasi-linear terms in and that could lead to some loss of derivatives. Before proving the mentioned lemmas, we need to introduce two preliminary results, that will be useful in the proof of lemmas \[Lem:Quartic\_terms\_1\], \[Lem:Quartic\_terms\_III\]. \[Lem:Est\_integrals\_quartic-terms\] For any $j_i\in\{+,-\}$, $i=1,2,3$, let $B^k_{(j_1,j_2,j_3)}(\xi,\eta)$ be the multiplier defined in when $k=1,2$ and in when $k=3$, and $\psi_1,\psi_2, \psi_3$ be three smooth cut-off functions such that $\psi_1(x)$ is supported for $|x|\le c$, $\psi_2(x)$ is supported for $c'\le |x|\le C'$, $\psi_3(x)$ is supported for $|x|\ge C$, for some $0<c,c'\ll 1$, $C,C'\gg 1$, and $\psi_1+\psi_2+\psi_3 \equiv 1$. Let also $\delta_k$ be equal to 1 for $k=1,2$, 0 for $k=3$. $(i)$ For any $j_1,\dots,j_5\in\{+,-\}$, $i=1,2$, and any $u_1,u_2,u_3,u_4$ such that $u_1\in H^{4,\infty}(\mathbb{R}^2)$, $u_2,u_4\in L^2(\mathbb{R}^2)$, $u_3\in H^{11,\infty}(\mathbb{R}^2)$ and $\delta_k\mathrm{R}_ku_3\in H^{7,\infty}(\mathbb{R}^2)$, $$\begin{gathered} \label{ineq:psi1-psi2} \left| \int \psi_i\Big(\frac{\xi-\eta}{\langle\eta\rangle}\Big)B^k_{(j_1,j_2,j_3)}(\xi,\eta) \left(1-j_4j_5\frac{\xi-\eta-\zeta}{\langle\xi-\eta-\zeta\rangle}\cdot\frac{\zeta}{|\zeta|}\right)\zeta_1 \hat{u}_1(\xi-\eta-\zeta)\hat{u}_2(\zeta)\hat{u}_3(\eta) \hat{u}_4(-\xi) d\xi d\eta d\zeta \right|\\ \lesssim \|u_1\|_{H^{4,\infty}}\|u_2\|_{L^2}\left(\|u_3\|_{H^{11,\infty}}+ \delta_k\|\mathrm{R}_ku_3\|_{H^{7,\infty}}\right)\|u_4\|_{L^2};\end{gathered}$$$(ii)$ For any $j_1,\dots,j_5\in\{+,-\}$, and any $u_1,u_2,u_3,u_4$ such that $u_1\in H^{7,\infty}(\mathbb{R}^2)$, $u_2\in H^1(\mathbb{R}^2)$, $u_4\in L^2(\mathbb{R}^2)$, $u_3\in H^{4,\infty}(\mathbb{R}^2)$ and $\delta_k\mathrm{R}_k u_3\in L^\infty(\mathbb{R}^2)$, $$\begin{gathered} \label{ineq:psi3} \left| \int \psi_3\Big(\frac{\xi-\eta}{\langle\eta\rangle}\Big)B^k_{(j_1,j_2,j_3)}(\xi,\eta) \left(1-j_4j_5\frac{\xi-\eta-\zeta}{\langle\xi-\eta-\zeta\rangle}\cdot\frac{\zeta}{|\zeta|}\right)\zeta_1 \hat{u}_1(\xi-\eta-\zeta)\hat{u}_2(\zeta)\hat{u}_3(\eta) \hat{u}_4(-\xi) d\xi d\eta d\zeta \right|\\ \lesssim \|u_1\|_{H^{7,\infty}}\|u_2\|_{H^1}\left(\|u_3\|_{H^{4,\infty}}+ \delta_k\|\mathrm{R}_ku_3\|_{L^\infty}\right)\|u_4\|_{L^2}.\end{gathered}$$Let $k=1,2$. We are going to refer to $B^k_{(j_1,j_2,j_3)}$ (resp. $\eta_k$ and $\mathrm{R}_k$) simply as $B_{(j_1,j_2,j_3)}$ (resp. $\eta$ and $\mathrm{R}$) and rather use a superscript to define a decomposition of this multiplier (see ) Let us observe that, as $$B_{(j_1,j_2,j_3)}(\xi,\eta) =\frac{j_1\langle\xi-\eta\rangle+j_2|\eta| - j_3\langle\xi\rangle}{2j_1j_2\langle\xi-\eta\rangle |\eta|}\eta,$$ we can write $$\label{B=B0+B1} B_{(j_1,j_2,j_3)}(\xi,\eta) = B^0_{(j_1,j_2,j_3)}(\xi,\eta)\frac{\eta}{|\eta|} + B^1_{(j_1,j_2,j_3)}(\xi,\eta)\langle\eta\rangle^4,$$ where for any smooth cut-off function $\phi$, equal to 1 in a neighbourhood of the origin, $$\label{def_B0-B1} \begin{gathered} B^0_{(j_1,j_2,j_3)}(\xi,\eta) :=\frac{j_1\langle\xi-\eta\rangle+j_2|\eta| - j_3\langle\xi\rangle}{2j_1j_2\langle\xi-\eta\rangle }\phi(\eta),\\ B^1_{(j_1,j_2,j_3)}(\xi,\eta) :=\frac{j_1\langle\xi-\eta\rangle+j_2|\eta| - j_3\langle\xi\rangle}{2j_1j_2\langle\xi-\eta\rangle |\eta|}\eta\langle\eta\rangle^{-4}(1-\phi)(\eta). \end{gathered}$$ According to decomposition we have that, for any $i=1,2,3$, $$\begin{split} &\int \psi_i\Big(\frac{\xi-\eta}{\langle\eta\rangle}\Big)B_{(j_1,j_2,j_3)}(\xi,\eta) \left(1-j_4j_5\frac{\xi-\eta-\zeta}{\langle\xi-\eta-\zeta\rangle}\cdot\frac{\zeta}{|\zeta|}\right)\zeta_1 \hat{u}_1(\xi-\eta-\zeta)\hat{u}_2(\zeta)\hat{u}_3(\eta) \hat{u}_4(-\xi) d\xi d\eta d\zeta \\ &= \int \psi_i\Big(\frac{\xi-\eta}{\langle\eta\rangle}\Big)B^0_{(j_1,j_2,j_3)}(\xi,\eta) \left(1-j_4j_5\frac{\xi-\eta-\zeta}{\langle\xi-\eta-\zeta\rangle}\cdot\frac{\zeta}{|\zeta|}\right)\zeta_1 \hat{u}_1(\xi-\eta-\zeta)\hat{u}_2(\zeta)\widehat{\mathrm{R} u}_3(\eta) \hat{u}_4(-\xi) d\xi d\eta d\zeta\\ & + \int \psi_i\Big(\frac{\xi-\eta}{\langle\eta\rangle}\Big)B^1_{(j_1,j_2,j_3)}(\xi,\eta) \left(1-j_4j_5\frac{\xi-\eta-\zeta}{\langle\xi-\eta-\zeta\rangle}\cdot\frac{\zeta}{|\zeta|}\right)\zeta_1 \hat{u}_1(\xi-\eta-\zeta)\hat{u}_2(\zeta)\widehat{\langle D_x\rangle^4 u}_3(\eta) \hat{u}_4(-\xi) \\ & \hspace{14cm} d\xi d\eta d\zeta\\ & =: I^0_i + I^1_i. \end{split}$$$(i)$ The first thing we observe concerning integral $I^k_i$ for $k=0,1$, $i=1,2$, is that $|\xi-\eta|, |\xi|\lesssim \langle\eta\rangle$ on the support of $\psi_i\big(\frac{\xi-\eta}{\langle\eta\rangle}\big)$ and that $|\zeta|\le \langle \xi-\eta-\zeta\rangle \langle\eta\rangle$. Therefore, introducing the following multipliers $$B^{i,k}_{(j_1,\dots,j_5)}(\xi,\eta,\zeta):=\psi_i\Big(\frac{\xi-\eta}{\langle\eta\rangle}\Big) B^k_{(j_1,j_2,j_3)}(\xi,\eta)\left(1-j_4j_5\frac{\xi-\eta-\zeta}{\langle\xi-\eta-\zeta\rangle}\cdot\frac{\zeta}{|\zeta|}\right)\zeta_1 \langle\eta\rangle^{-7}\langle\xi-\eta-\zeta\rangle^{-4},$$ for any $j_1,\dots j_5\in\{+,-\}$, $k=0,1$, $i=1,2,$ a straight computation shows that, for any $\alpha,\beta,\gamma\in\mathbb{N}^2$, $$\label{est_multipliers_Bik} \begin{gathered} \left|\partial^\alpha_\xi \partial^\beta_\eta B^{i,k}_{(j_1,\dots,j_5)}(\xi,\eta,\zeta)\right| \lesssim \langle\zeta\rangle^{-3}|g_{\alpha,\beta}(\xi,\eta)|,\\ \left|\partial^\alpha_\xi \partial^\beta_\eta \partial^\gamma_\zeta B^{i,k}_{(j_1,\dots,j_5)}(\xi,\eta,\zeta)\right| \lesssim (|\zeta|\langle\zeta\rangle^{-1})^{1-|\gamma|}\langle\zeta\rangle^{-3}|g_{\alpha,\beta}(\xi,\eta)|, \ |\gamma|\ge 1,\\ \end{gathered}$$ with $$\label{est_multipliers_g-alpha-beta} \begin{gathered} |g_{\alpha,0}(\xi,\eta)|\lesssim_{\alpha} \langle\eta\rangle^{-3}\langle\xi\rangle^{-3},\\ |g_{\alpha,\beta}(\xi,\eta)|\lesssim_{\alpha,\beta} (|\eta|\langle\eta\rangle^{-1})^{1-|\beta|}\langle\eta\rangle^{-3}\langle\xi\rangle^{-3}, \ |\beta|\ge 1. \end{gathered}$$ If $$K^{i,k}_{(j_1,\dots, j_5)}(x,y,z):=\int e^{ix\cdot\xi + iy\cdot\eta + iz\cdot\zeta} B^{i,k}_{(j_1,\dots,j_5)}(\xi,\eta,\zeta) d\xi d\eta d\zeta,$$ by lemma \[Lem\_appendix: Kernel with 1 function\] $(i)$ we first find that, for any $\alpha,\beta\in\mathbb{N}^2$, $$\left|\partial^\alpha_\xi\partial^\beta_\eta \int e^{iz\cdot\zeta}B^{i,k}_{(j_1,\dots,j_5)}(\xi,\eta,\zeta) d\zeta\right|\lesssim |z|^{-1}\langle z\rangle^{-2}|g_{\alpha,\beta}(\xi,\eta)|$$ and successively that $$\left|\partial^\alpha_\xi \int e^{iy\cdot\eta+ iz\cdot\zeta}B^{i,k}_{(j_1,\dots,j_5)}(\xi,\eta,\zeta)d\eta d\zeta\right|\lesssim |y|^{-1}\langle y\rangle^{-2} |z|^{-1}\langle z\rangle^{-2}\langle \xi\rangle^{-3},$$ for every $\xi\in\mathbb{R}^2$, $(y,z)\in\mathbb{R}^2\times\mathbb{R}^2$. Corollary \[Cor\_appendix: decay of integral operators\] $(i)$ hence implies that $$|K^{i,k}_{(j_1,\dots,j_5)}(x,y,z)|\lesssim \langle x\rangle^{-3} |y|^{-1}\langle y\rangle^{-2} |z|^{-1}\langle z\rangle^{-2}, \quad\forall (x,y,z)\in(\mathbb{R}^2)^3.$$ As for $i=1,2$ $$\begin{split} I^0_i & = \int B^{i,0}_{(j_1,\dots,j_5)}(\xi,\eta,\zeta) \widehat{\langle D_x\rangle^4 u_1}(\xi-\eta-\zeta) \hat{u}_2(\zeta) {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\langle D_x\rangle^7 \mathrm{R}u_3}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\langle D_x\rangle^7 \mathrm{R}u_3}{\tmpbox}}(\eta) \hat{u}_4(-\xi)\, d\xi d\eta d\zeta, \\ & = \int K^{i,0}_{(j_1,\dots,j_5)}(t-x,x-z,x-y) [\langle D_x\rangle^4 u_1](x) u_2(y) [\langle D_x\rangle^7 \mathrm{R}u_3](z) u_4(t) dxdydzdt, \end{split}$$ $$\begin{split} I^1_i &= \int B^{i,1}_{(j_1,\dots,j_5)}(\xi,\eta,\zeta) \widehat{\langle D_x\rangle^4 u_1}(\xi-\eta-\zeta) \hat{u}_2(\zeta) \widehat{\langle D_x\rangle^{11}u_3}(\eta) \hat{u}_4(-\xi)\, d\xi d\eta d\zeta \\ & = \int K^{i,1}_{(j_1,\dots,j_5)}(t-x,x-z,x-y) [\langle D_x\rangle^4 u_1](x) u_2(y) [\langle D_x\rangle^{11}u_3](z) u_4(t) dxdydzdt, \end{split}$$ inequality follows by the fact that, for any $\widetilde{u}_1, \dots \widetilde{u}_4\in L^2\cap L^\infty$, any $f,g,h\in L^1$, integrals such as $$\label{example_integral_u1-u4} \int f(t-x)g(x-z)h(x-y) |\widetilde{u}_1(x)| |\widetilde{u}_2(y)| |\widetilde{u}_3(z)| |\widetilde{u}_4(t)| dxdydzdt$$ can be bounded from above by the product of the $L^2$ norm of any two functions $\widetilde{u}_k$ times the $L^\infty$ norm of the remaining ones. $(ii)$ For a cut-off function $\phi$ as the one introduced at the beginning of the proof we decompose integral $I^k_3$, $k=0,1,$ distinguishing between $|\zeta|\lesssim 1$ and $|\zeta|\gtrsim 1$. On the one hand, for any $j_1,\dots,j_5, k=0,1,$ we consider $$B^{3,k}_{(j_1,\dots,j_5)}(\xi,\eta,\zeta):=\psi_3\Big(\frac{\xi-\eta}{\langle\eta\rangle}\Big)\phi(\zeta)B^k_{(j_1,j_2,j_3)}(\xi,\eta)\left(1-j_4j_5\frac{\xi-\eta-\zeta}{\langle\xi-\eta-\zeta\rangle}\cdot\frac{\zeta}{|\zeta|}\right)\zeta_1 \langle\xi-\eta-\zeta\rangle^{-3}$$ and observe that, since $|\xi|\le \langle\xi-\eta-\zeta\rangle$ on the support of $\psi_3\big(\frac{\xi-\eta}{\langle\eta\rangle}\big)\phi(\zeta)$, the above multiplier satisfies estimates , . From the same argument as before this implies that $$\begin{gathered} \label{ineq: J03} \left|J_3^0:= \int B^{3,0}_{(j_1,\dots,j_5)}(\xi,\eta,\zeta) \widehat{\langle D_x\rangle^3u_1}(\xi-\eta-\zeta) \hat{u}_2(\zeta) \widehat{\mathrm{R}u_3}(\eta) \hat{u}_4(-\xi) d\xi d\eta d\zeta \right| \\ \lesssim \|u_1\|_{H^{3,\infty}}\|u_2\|_{L^2}\|\mathrm{R}u_3\|_{L^\infty}\|u_4\|_{L^2}\end{gathered}$$ together with $$\begin{gathered} \label{ineq: J13} \left|J_3^1:= \int B^{3,1}_{(j_1,\dots,j_5)}(\xi,\eta,\zeta) \widehat{\langle D_x\rangle^3u_1}(\xi-\eta-\zeta) \hat{u}_2(\zeta) \widehat{\langle D_x\rangle^4 u_3}(\eta) \hat{u}_4(-\xi) d\xi d\eta d\zeta \right| \\ \lesssim \|u_1\|_{H^{3,\infty}}\|u_2\|_{L^2}\|u_3\|_{H^{4,\infty}}\|u_4\|_{L^2}.\end{gathered}$$ On the other hand, we make a further decomposition on the integral restricted to $|\zeta|\gtrsim 1$ by means of functions $\psi_i, i=1,2,3$, distinguishing between three regions: for $|\zeta|\le c \langle\xi-\eta\rangle$, for $c'\langle \xi-\eta\rangle\le |\zeta|\le C'\langle\xi-\eta\rangle$ and $|\zeta|>C\langle\xi-\eta\rangle$. For any $j_1,\dots,j_5\in\{+,-\}$, $k=0,1$, we hence introduce the following multipliers $$\begin{gathered} \widetilde{B}^{3,i,k}_{(j_1,\dots,j_5)}(\xi,\eta,\zeta):= \psi_3\Big(\frac{\xi-\eta}{\langle\eta\rangle}\Big)(1-\phi)(\zeta)\psi_i\Big(\frac{\zeta}{\langle\xi-\eta\rangle}\Big)\\ \times B^k_{(j_1,j_2,j_3)}(\xi,\eta)\left(1-j_4j_5\frac{\xi-\eta-\zeta}{\langle\xi-\eta-\zeta\rangle}\cdot\frac{\zeta}{|\zeta|}\right)\zeta_1 \langle\xi-\eta-\zeta\rangle^{-7};\end{gathered}$$ for $i=1m,3$, and $$\begin{gathered} \label{multiplier_Btilde,3,2,k} \widetilde{B}^{3,2,k}_{(j_1,\dots,j_5)}(\xi,\eta,\zeta):= \psi_3\Big(\frac{\xi-\eta}{\langle\eta\rangle}\Big)(1-\phi)(\zeta)\psi_2\Big(\frac{\zeta}{\langle\xi-\eta\rangle}\Big)\\ \times B^k_{(j_1,j_2,j_3)}(\xi,\eta)\left(1-j_4j_5\frac{\xi-\eta-\zeta}{\langle\xi-\eta-\zeta\rangle}\cdot\frac{\zeta}{|\zeta|}\right)\zeta_1 \langle\zeta\rangle^{-1}.\end{gathered}$$ Since $|\xi|\sim |\xi-\eta|\sim|\xi-\eta-\zeta|$ on the support of $\psi_3\big(\frac{\xi-\eta}{\langle\eta\rangle}\big)(1-\phi)(\zeta)\psi_1\big(\frac{\zeta}{\langle\xi-\eta\rangle}\big)$ (resp. $|\xi|\sim |\xi-\eta|\lesssim |\zeta|\sim |\xi-\eta-\zeta|$ on the support of $\psi_3\big(\frac{\xi-\eta}{\langle\eta\rangle}\big)(1-\phi)(\zeta)\psi_3\big(\frac{\zeta}{\langle\xi-\eta\rangle}\big)$), a straight computation shows that above multipliers verify , , from which follows that $$\begin{gathered} \label{ineq: Jtilde-i0} \left|\widetilde{J}^{i,0}_3:= \int \widetilde{B}^{3,i, 0}_{(j_1,\dots,j_5)}(\xi,\eta,\zeta) \widehat{\langle D_x\rangle^7u_1}(\xi-\eta-\zeta) \hat{u}_2(\zeta) \widehat{\mathrm{R}u_3}(\eta) \hat{u}_4(-\xi) d\xi d\eta d\zeta \right| \\ \lesssim \|u_1\|_{H^{7,\infty}}\|u_2\|_{L^2}\|\mathrm{R}u_3\|_{L^\infty}\|u_4\|_{L^2}\end{gathered}$$ along with $$\begin{gathered} \label{ineq: Jtilde-i1} \left|\widetilde{J}^{i,1}_3:= \int \widetilde{B}^{3,i, 1}_{(j_1,\dots,j_5)}(\xi,\eta,\zeta) \widehat{\langle D_x\rangle^7u_1}(\xi-\eta-\zeta) \hat{u}_2(\zeta) \widehat{\langle D_x\rangle^4 u_3}(\eta) \hat{u}_4(-\xi) d\xi d\eta d\zeta \right| \\ \lesssim \|u_1\|_{H^{7,\infty}}\|u_2\|_{L^2}\|u_3\|_{H^{4,\infty}}\|u_4\|_{L^2},\end{gathered}$$ for $i=1,3$. Finally, on the support of $\psi_3\big(\frac{\xi-\eta}{\langle\eta\rangle}\big)(1-\phi)(\zeta)\psi_2\big(\frac{\zeta}{\langle\xi-\eta\rangle}\big)$ we have that $|\xi|\sim |\xi-\eta|\sim |\zeta|$ and $|\xi-\eta-\zeta|\lesssim |\zeta|$. Replacing $\zeta$ with $\xi-\zeta$ by a change of coordinates we find that, for any $\alpha,\beta,\gamma\in\mathbb{N}^2$, $$\label{est_Btilde-3,2k} \begin{gathered} \left|\partial^\alpha_\xi \partial^\gamma_\zeta \widetilde{B}^{3,2,k}_{(j_1,\dots,j_5)}(\xi,\eta,\xi-\zeta) \right|\lesssim_{\alpha,\gamma} \langle\eta\rangle^{-3}\langle\xi\rangle^{-|\alpha|}, \\ \left|\partial^\alpha_\xi \partial^\beta_\eta \partial^\gamma_\zeta \widetilde{B}^{3,2,k}_{(j_1,\dots,j_5)}(\xi,\eta,\xi-\zeta) \right|\lesssim (|\eta|\langle\eta\rangle^{-1})^{1-|\beta|}\langle\eta\rangle^{-3}\langle\xi\rangle^{-|\alpha|}, \ |\beta|\ge 1. \end{gathered}$$ If we introduce a Littlewood-Paley decomposition such that $$\widetilde{B}^{3,2,k}_{(j_1,\dots,j_5)}(\xi,\eta,\xi-\zeta) =\sum_{l\ge 1} \widetilde{B}^{3,2,k}_{(j_1,\dots,j_5)}(\xi,\eta,\xi-\zeta)\varphi(2^{-l}\xi),$$ one can check, using lemma \[Lem\_appendix: Kernel with 1 function\] $(i)$ to obtain the decay in $y$, making a change of coordinates $\xi\mapsto 2^l\xi$, some integration by parts, and using inequalities , that $$K^{k,l}_{(j_1,\dots,j_5)}(x,y,z):=\int e^{ix\cdot\xi + iy\cdot\eta + iz\cdot\zeta}\widetilde{B}^{3,2,k}_{(j_1,\dots,j_5)}(\xi,\eta,\xi-\zeta) \varphi(2^{-l}\xi) d\xi d\eta d\zeta$$ is such that $$\label{est_kernel_Kkl} |K^{k,l}_{(j_1,\dots,j_5)}(x,y,z)|\lesssim 2^{2l}\langle 2^l x\rangle^{-3}|y|^{-1}\langle y\rangle^{-2}\langle z\rangle^{-3}, \quad \forall (x,y,z)\in (\mathbb{R}^2)^3.$$ Moreover, since $|\xi|\sim|\xi-\zeta|$ on the support of $\widetilde{B}^{3,2,k}_{(j_1,\dots,j_5)}(\xi,\eta,\zeta)$ there are two other suitably supported cut-off functions $\varphi_1, \varphi_2$ such that $\varphi(2^{-l}\xi)=\varphi(2^{-l}\xi)\varphi_1(2^{-l}\xi)\varphi_2(2^{-l}(\xi-\zeta))$, for any $l\ge 1$. If $\Delta^l_j w:=\varphi_j(2^{-l}D_x)w$, we finally obtain that $$\begin{split} \widetilde{J}^{2,0}_3& :=\int \widetilde{B}^{3,2,0}_{(j_1,\dots,j_5)}(\xi,\eta,\zeta) \hat{u}_1(\xi-\eta-\zeta) \widehat{\langle D_x\rangle u}_2(\zeta) \widehat{\mathrm{R}u}_3(\eta)\hat{u}_4(-\xi) d\xi d\eta d\zeta \\ & = \int \widetilde{B}^{3,2,0}_{(j_1,\dots,j_5)}(\xi,\eta,\xi-\zeta) \hat{u}_1(\zeta-\eta) \widehat{\langle D_x\rangle u}_2(\xi-\zeta) \widehat{\mathrm{R}u}_3(\eta)\hat{u}_4(-\xi) d\xi d\eta d\zeta\\ & =\sum_{l\ge 1}\int K^{0,l}_{(j_1,\dots,j_5)}(t-y,x-z,y-x) u_1(x) [\Delta^l_1\langle D_x\rangle u_2](y) [\mathrm{R}u_3](z) [\Delta^l_2u_4](t) dxdydzdt, \end{split}$$ and by together with Cauchy-Schwarz inequality we derive that $$\label{ineq: Jtilde-20} | \widetilde{J}^{2,0}_3|\lesssim \|u_1\|_{L^\infty}\|\mathrm{R}_1u_3\|_{L^\infty}\sum_{l\ge 1}\|\Delta^l_1 \langle D_x\rangle u_2\|_{L^2}\|\Delta^l_2 u_4\|_{L^2}\lesssim \|u_1\|_{L^\infty}\|u_2\|_{H^1} \|\mathrm{R}_1u_3\|_{L^\infty} \|u_4\|_{L^2}.$$ In a similar way we obtain that $$\widetilde{J}^{2,1}_3 :=\int \widetilde{B}^{3,2,1}_{(j_1,\dots,j_5)}(\xi,\eta,\xi-\zeta) \hat{u}_1(\zeta-\eta) \widehat{\langle D_x\rangle u}_2(\xi-\zeta) \widehat{\langle D_x\rangle^4 u}_3(\eta)\hat{u}_4(-\xi) d\xi d\eta d\zeta$$ satisfies $$\label{ineq: Jtilde-21} | \widetilde{J}^{2,1}_3|\lesssim \|u_1\|_{L^\infty}\|u_2\|_{H^1} \|u_3\|_{H^{4,\infty}} \|u_4\|_{L^2}.$$ The result of statement $(ii)$ follows then from inequalities , , , , , , after having recognized that $$\begin{gathered} \int \psi_3\Big(\frac{\xi-\eta}{\langle\eta\rangle}\Big)B_{(j_1,j_2,j_3)}(\xi,\eta) \left(1-j_4j_5\frac{\xi-\eta-\zeta}{\langle\xi-\eta-\zeta\rangle}\cdot\frac{\zeta}{|\zeta|}\right)\zeta_1 \hat{u}_1(\xi-\eta-\zeta)\hat{u}_2(\zeta)\hat{u}_3(\eta) \hat{u}_4(-\xi) d\xi d\eta d\zeta \\ = \sum_{k=0}^1J^k_3 + \sum_{k=0}^1\sum_{i=1}^3 \widetilde{J}^{i,k}_3.\end{gathered}$$In conclusion, the same proof of above applies to multiplier $B^3_{(j_1,j_2,j_3)}$ introduced in , which can be decomposed as $$j_2 B^0_{(j_1,j_2,j_3)}(\xi,\eta) + \widetilde{B}^1_{(j_1,j_2,j_3)}(\xi,\eta)\langle\eta\rangle^4$$ with the same $B^0_{(j_1,j_2,j_3)}$ as in and $$\widetilde{B}^1_{(j_1,j_2,j_3)}(\xi,\eta):=\frac{j_1\langle\xi-\eta\rangle+j_2|\eta| - j_3\langle\xi\rangle}{2j_1\langle\xi-\eta\rangle }\langle\eta\rangle^{-4}(1-\phi)(\eta).$$ The lack of factor $\eta_1|\eta|^{-1}$ against $B^0_{(j_1,j_2,j_3)}$, in comparison to decomposition , is the reason why inequality (resp. ) holds with $\|u_3\|_{H^{11,\infty}}+\|\mathrm{R}u_3\|_{H^{7,\infty}}$ (resp. $\|u_3\|_{H^{4,\infty}}+\|\mathrm{R}u_3\|_{L^\infty}$) replaced with $\|u_3\|_{H^{11,\infty}}$ (resp. with $\|u_3\|_{H^{4,\infty}}$). \[Lem:Est\_integrals\_quartic-terms-2\] Under the same assumptions as in lemma \[Lem:Est\_integrals\_quartic-terms\] we have that: $(i)$ for any $j_1,\dots,j_5\in\{+,-\}$, $i=1,2$, and any $u_1,u_2,u_3,u_4$ such that $u_1\in H^{4,\infty}(\mathbb{R}^2)$, $u_2,u_4\in L^2(\mathbb{R}^2)$, $u_3\in H^{11,\infty}(\mathbb{R}^2)$ and $\delta_k\mathrm{R}_ku_3\in H^{7,\infty}(\mathbb{R}^2)$, $$\begin{gathered} \label{ineq:psi1-psi2-second lemma} \left| \int \psi_i\Big(\frac{\xi-\eta}{\langle\eta\rangle}\Big)B^k_{(j_1,j_2,j_3)}(\xi,\eta) \left(1+j_4j_5\frac{\xi+\zeta}{\langle\xi+\zeta\rangle}\cdot\frac{\zeta}{|\zeta|}\right)\zeta_1 \hat{u}_1(-\xi-\zeta)\hat{u}_2(\zeta)\hat{u}_3(\eta) \hat{u}_4(\xi-\eta) d\xi d\eta d\zeta \right|\\ \lesssim \|u_1\|_{H^{4,\infty}}\|u_2\|_{L^2}\left(\|u_3\|_{H^{11,\infty}}+\delta_k \|\mathrm{R}_ku_3\|_{H^{7,\infty}}\right)\|u_4\|_{L^2};\end{gathered}$$$(ii)$ for any $j_1,\dots,j_5\in\{+,-\}$, and any $u_1,u_2,u_3,u_4$ such that $u_1\in H^{7,\infty}(\mathbb{R}^2)$, $u_2\in L^2(\mathbb{R}^2)$, $u_4\in H^1(\mathbb{R}^2)$, $u_3\in H^{4,\infty}(\mathbb{R}^2)$ and $\delta_k\mathrm{R}_ku_3\in L^\infty(\mathbb{R}^2)$, $$\begin{gathered} \label{integral_psi3} \left| \int \psi_3\Big(\frac{\xi-\eta}{\langle\eta\rangle}\Big)B^k_{(j_1,j_2,j_3)}(\xi,\eta) \left(1+j_4j_5\frac{\xi+\zeta}{\langle\xi+\zeta\rangle}\cdot\frac{\zeta}{|\zeta|}\right)\zeta_1 \hat{u}_1(-\xi-\zeta)\hat{u}_2(\zeta)\hat{u}_3(\eta) \hat{u}_4(\xi-\eta) d\xi d\eta d\zeta \right|\\ \lesssim \|u_1\|_{H^{7,\infty}}\|u_2\|_{L^2}\left(\|u_3\|_{H^{4,\infty}}+ \delta_k \|\mathrm{R}_k u_3\|_{L^\infty}\right)\|u_4\|_{H^1}.\end{gathered}$$The proof of the statement is analogous to that of lemma \[Lem:Est\_integrals\_quartic-terms\] after a change of coordinates $-\xi\mapsto \xi-\eta$. In we take the $H^1$ norm on $u_4$ instead of $u_2$, as done in , by replacing multiplier $\widetilde{B}^{3,2,k}_{(j_1,j_2,j_3)}$ in with $$\psi_3\Big(\frac{\xi-\eta}{\langle\eta\rangle}\Big)(1-\phi)(\zeta)\psi_2\Big(\frac{\zeta}{\langle\xi-\eta\rangle}\Big) B^k_{(j_1,j_2,j_3)}(\xi,\eta)\left(1-j_4j_5\frac{\xi-\eta-\zeta}{\langle\xi-\eta-\zeta\rangle}\cdot\frac{\zeta}{|\zeta|}\right)\zeta_1 \langle\xi\rangle^{-1}.$$ \[Analysis of quartic terms. I\] \[Lem:Quartic\_terms\_1\] For any general multi-index $I$, any $j_k\in \{+,-\}$, $k=1,2,3$, let $C^I_{(j_1,j_2,j_3)}$, $C^{I,R}_{(j_1,j_2,j_3)}$ be the integrals defined in , respectively, and $D^I_{(j_1,j_2,j_3)}$, $D^{I,R}_{(j_1,j_2,j_3)}$ introduced in , . Then $$\label{derivative DIs} \partial_t \left[D^I_{(j_1,j_2,j_3)}+D^{I,R}_{(j_1,j_2,j_3)} \right] = -C^I_{(j_1,j_2,j_3)}-C^{I,R}_{(j_1,j_2,j_3)} + \mathfrak{D}^{I}_{\text{quart}},$$ where $\mathfrak{D}^{I}_{\text{quart}}$ satisfies $$\label{est_DI1_quart} \begin{split} & \left|\mathfrak{D}^{I}_{\text{quart}}(t)\right| \\ &\lesssim \left[\|V(t,\cdot)\|^{\frac{7}{4}}_{H^{10,\infty}}\|V(t,\cdot)\|^{\frac{1}{4}}_{H^{12}}+ \|V(t,\cdot)\|_{H^{4,\infty}}\left(\|U(t,\cdot)\|_{H^{11,\infty}}+\|\mathrm{R}_1U(t,\cdot)\|_{H^{7,\infty}}\right) \right]\|W^I(t,\cdot)\|^2_{L^2}\\ &+ \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_2|<|I|}} \|Q^{\mathrm{kg}}_0(v^{I_1}_\pm, Du^{I_2}_\pm)(t,\cdot)\|_{L^2} \left(\|U(t,\cdot)\|_{H^{11,\infty}}+\|\mathrm{R}_1U(t,\cdot)\|_{H^{7,\infty}}\right) \|V^I(t,\cdot)\|_{L^2}. \end{split}$$Using definitions , , with $k=1$, and system , we find that $$\label{dt DI} \begin{split} &-4(2\pi)^2\left[ \partial_t D^I_{(j_1, j_2, j_3)} + C^I_{(j_1, j_2, j_3)}\right] \\ & = \int \chi\left(\frac{\xi-\eta}{\langle\eta\rangle}\right)B^1_{(j_1,j_2,j_3)}(\xi,\eta) \left[\sum_{(I_1,I_2)\in\mathcal{I}(I)}c_{I_1,I_2}{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q^{\mathrm{kg}}_0(v^{I_1}_\pm, D u^{I_2}_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q^{\mathrm{kg}}_0(v^{I_1}_\pm, D u^{I_2}_\pm)}{\tmpbox}}\right](\xi-\eta) \hat{u}_{j_2}(\eta) \hat{v}^I_{j_3}(-\xi) d\xi d\eta \\ & + \int \chi\left(\frac{\xi-\eta}{\langle\eta\rangle}\right)B^1_{(j_1,j_2,j_3)}(\xi,\eta)\ \hat{v}^I_{j_1}(\xi-\eta) {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q^{\mathrm{w}}_0(v_\pm, D_1v_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q^{\mathrm{w}}_0(v_\pm, D_1v_\pm)}{\tmpbox}}(\eta) \hat{v}^I_{j_3}(-\xi) d\xi d\eta \\ &+ \int\chi\left(\frac{\xi-\eta}{\langle\eta\rangle}\right) B^1_{(j_1,j_2,j_3)}(\xi,\eta) \hat{v}^I_{j_1}(\xi-\eta)\hat{u}_{j_2}(\eta) \left[\sum_{(I_1,I_2)\in\mathcal{I}(I)}c_{I_1,I_2}{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q^{\mathrm{kg}}_0(v^{I_1}_\pm, Du^{I_2}_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q^{\mathrm{kg}}_0(v^{I_1}_\pm, Du^{I_2}_\pm)}{\tmpbox}}\right](-\xi) d\xi d\eta \\ & =: S_1 + S_2 + S_3, \end{split}$$ where coefficients $c_{I_1,I_2}\in \{-1,0,1\}$ are such that $c_{I_1,I_2}=1$ when $|I_1| + |I_2| = |I|$ (in which case $D=D_1$) and $\chi\in C^\infty_0(\mathbb{R}^2)$ is equal to 1 close to the origin and has a sufficiently small support. All integrals in the above right hand side are quartic terms for they involve the quadratic non-linearities of . The fact that $S_2$ is a remainder $\mathfrak{D}^{I}_{\text{quart}}$ satisfying follows by inequalities , with $s=7$, and the fact that $$\label{est_R1Qw0} \|\mathrm{R}_1 Q^{\mathrm{w}}_0(v_\pm, D_1v_\pm)\|_{H^{7,\infty}} \lesssim \|V(t,\cdot)\|^{2-(2-\theta)\theta}_{H^{10,\infty}} \|V(t,\cdot)\|^{(2-\theta)\theta}_{H^{12}},$$ for any $\theta\in ]0,1[$. The above inequality is justified by the fact that, for any function $w\in W^{1,\infty}\cap H^1$, $\rho\in\mathbb{N}$ and any $\theta \in ]0,1[$, setting $p=\frac{2}{\theta}\in ]2,\infty[$, $$\begin{gathered} \label{injection_R1w} \|\langle D_x\rangle^\rho\mathrm{R}_1 w\|_{L^\infty}\lesssim \|\langle D_x\rangle^\rho\mathrm{R}_1 w\|_{W^{1,p}} \lesssim \|\langle D_x\rangle^\rho w\|_{W^{1,p}} \lesssim \|\langle D_x\rangle^\rho w\|^{1-\theta}_{W^{1,\infty}}\|\langle D_x\rangle^\rho w\|^\theta_{H^1}\\ \lesssim \|\langle D_x\rangle^\rho w\|^{1-\theta}_{H^{2,\infty}}\|\langle D_x\rangle^\rho w\|^\theta_{H^1},\end{gathered}$$ as a consequence of Morrey’s inequality, continuity of $\mathrm{R}_1:L^p\rightarrow L^p$ for $p<+\infty$, interpolation inequality, and the injection of $W^{1,\infty}$ into $H^{2,\infty}$. This implies that $$\label{ineq_for R1Qw0} \|\mathrm{R}_1 Q^{\mathrm{w}}_0(v_\pm, D_1v_\pm)\|_{H^{\rho,\infty}}\lesssim \| Q^{\mathrm{w}}_0(v_\pm, D_1v_\pm)\|^{1-\theta}_{H^{\rho+2,\infty}} \| Q^{\mathrm{w}}_0(v_\pm, D_1v_\pm)\|^{\theta}_{H^{\rho+1}},$$ for any $\rho\in\mathbb{N}$, and gives when $\rho=7$ after inequalities with $s=8$, with $s=9$. Therefore, for any $\theta\in ]0,1[$, $$|S_2|\lesssim \left(\|V(t,\cdot)\|^{2-\theta}_{H^{8,\infty}}\|V(t,\cdot)\|^\theta_{H^{10}} + \|V(t,\cdot)\|^{2-(2-\theta)\theta}_{H^{10,\infty}} \|V(t,\cdot)\|^{(2-\theta)\theta}_{H^{12}}\right) \|V^I(t,\cdot)\|^2_{L^2},$$ so choosing $\theta\ll 1$ small (e.g. $\theta\le 1/8$) and keeping in mind estimates , we deduce that $S_2$ is controlled by the first term in the right hand side of . Inequality allows also to estimate all integrals in summations $S_1, S_3$ corresponding to indices $(I_1,I_2)\in\mathcal{I}(I)$ with $|I_2|<|I|$, and to bound them with the latter term in the right hand side of . This is not the case for integrals with $I_2=I$ involving quasi-linear term $Q^{\mathrm{kg}}_0(v_\pm, D_1u^I_\pm)$, because a straight application of that inequality would give a bound at the wrong energy level $n+1$, as $\|Q^{\mathrm{kg}}_0(v_\pm, D_1u^{I}_\pm)\|_{L^2}\lesssim \|V(t,\cdot)\|_{H^{1,\infty}}\|D_1U^I(t,\cdot)\|_{L^2}$. Instead, since $$\label{Fourier transform Qkg0} {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q^{\mathrm{kg}}_0(v_\pm, D_1u^I_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q^{\mathrm{kg}}_0(v_\pm, D_1u^I_\pm)}{\tmpbox}}(\xi) = \frac{i}{4}\sum_{j_4,j_5\in\{+,-\}}\int \Big(1 - j_4j_5 \frac{\xi-\zeta}{\langle\xi - \zeta\rangle}\cdot\frac{\zeta}{|\zeta|} \Big)\zeta_1\hat{v}_{j_4}(\xi - \zeta) \hat{u}^I_{j_5}(\zeta)d\zeta,$$ we can rather write those integrals as the sum over $j_k\in \{+,-\}, k=1,\dots 4$, of the following: \[integrals B1 B2\] $$\label{integral B-1} \int \chi\Big(\frac{\xi-\eta}{\langle\eta \rangle}\Big)B^1_{(j_1,j_2,j_3)}(\xi,\eta) \Big(1 - j_4j_5 \frac{\xi-\eta-\zeta}{\langle\xi-\eta - \zeta\rangle}\cdot\frac{\zeta}{|\zeta|} \Big)\zeta_1 \hat{v}_{j_4}(\xi-\eta - \zeta) \hat{u}^I_{j_5}(\zeta) \hat{u}_{j_2}(\eta)\hat{v}^I_{j_3}(-\xi)\ d\xi d\eta d\zeta,$$ $$\label{integral B-2} \int \chi\Big(\frac{\xi-\eta}{\langle\eta \rangle}\Big)B^1_{(j_1,j_2,j_3)}(\xi,\eta) \Big(1 + j_4j_5 \frac{\xi+\zeta}{\langle\xi+\zeta\rangle}\cdot\frac{\zeta}{|\zeta|} \Big)\zeta_1 \hat{v}^I_{j_1}(\xi-\eta) \hat{u}_{j_2}(\eta) \hat{v}_{j_4}(-\xi-\zeta) \hat{u}^I_{j_5}(\zeta)\ d\xi d\eta d\zeta,$$ and estimate them by using inequalities and respectively. We hence obtain that $$\begin{split} |S_1|+|S_3|&\lesssim \|V(t,\cdot)\|_{H^{4,\infty}}\left(\|U(t,\cdot)\|_{H^{11,\infty}}+\|\mathrm{R}_1U(t,\cdot)\|_{H^{7,\infty}}\right) \|W^I(t,\cdot)\|^2_{L^2} \\ &+ \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_2|<|I|}} \|Q^{\mathrm{kg}}_0(v^{I_1}_\pm, Du^{I_2}_\pm)(t,\cdot)\|_{L^2} \left(\|U(t,\cdot)\|_{H^{11,\infty}}+\|\mathrm{R}_1U(t,\cdot)\|_{H^{7,\infty}}\right) \|V^I(t,\cdot)\|_{L^2}, \end{split}$$ and, since the same argument applies to $\partial_t D^{I,R}_{(j_1,j_2,j_3)}$, this also concludes the proof of the statement. \[Lem:Quartic\_Terms\_II\] For any general multi-index $I$, any $j_k\in \{+,-\}$, $k=1,2,3$, let $D^{I,T_{-N}}_{(j_1,j_2,j_3)}$ be defined as in . Then $$\label{derivative DIT} \partial_t D^{I,T_{-N}}_{(j_1,j_2,j_3)} = \Im\left[\langle T_{-N}(U)W^I, W^I\right] + \mathfrak{D}^{I,N}_{\text{quart}}$$ and if $N\ge 18$ $\mathfrak{D}^{I,N}_{\text{quart}}$satisfies $$\label{est_DIN_quart} \begin{split} \left|\mathfrak{D}^{I,N}_{\text{quart}} \right| &\lesssim \|V(t,\cdot)\|_{H^{N+4,\infty}}^{\frac{7}{4}}\|V(t,\cdot)\|^{\frac{1}{4}}_{H^{N+6}}\|W^I(t,\cdot)\|^2_{L^2}\\ & +\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_2|<|I|}}\left\| Q^{\mathrm{kg}}_0(v^{I_1}_\pm, D u^{I_2}_\pm)\right\|_{L^2} \|U(t,\cdot)\|_{H^{N+3,\infty}}\|V^I(t,\cdot)\|_{L^2}. \end{split}$$ For any triplet $(j_1,j_2,j_3)$, we compute the time derivative of $D^{I,T_{-N}}$ by making use of system . Recalling and , we find that $$\begin{split} & \partial_t \left[\sum_{j_k\in\{+,-\}} D^{I,T_{-N}}_{(j_1,j_2,j_3)}\right] - \Im[\langle T_{-N}(U)W^I,W^I\rangle] = \\ & =\Re \left[ \frac{1}{(2\pi)^2} \int \widetilde{\sigma}^N_{(j_1,j_2,j_3)}(\xi,\eta) \left[\sum_{(I_1,I_2)\in \mathcal{I}(I)}c_{I_1,I_2}{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q^{\mathrm{kg}}_0(v^{I_1}_\pm, D u^{I_2}_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q^{\mathrm{kg}}_0(v^{I_1}_\pm, D u^{I_2}_\pm)}{\tmpbox}}(\xi-\eta)\right] \hat{u}_{j_2}(\eta) \hat{v}^I_{-j_3}(-\xi) d\xi d\eta \right. \\ & \hspace{0,8cm}\left.+ \frac{1}{(2\pi)^2} \int \widetilde{\sigma}^N_{(j_1,j_2,j_3)}(\xi,\eta) \hat{v}^I_{j_1}(\xi-\eta) {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q^{\mathrm{w}}_0(v_\pm, D_1v_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q^{\mathrm{w}}_0(v_\pm, D_1v_\pm)}{\tmpbox}}(\eta) \hat{v}^I_{-j_3}(-\xi) d\xi d\eta \right. \\ & \hspace{0,8cm}\left. + \frac{1}{(2\pi)^2} \int \widetilde{\sigma}^N_{(j_1,j_2,j_3)}(\xi,\eta) \hat{v}^I_{j_1}(\xi-\eta) \hat{u}_{j_2}(\eta) \left[\sum_{(I_1,I_2)\in \mathcal{I}(I)}c_{I_1,I_2}{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q^{\mathrm{kg}}_0(v^{I_1}_\pm, D u^{I_2}_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q^{\mathrm{kg}}_0(v^{I_1}_\pm, D u^{I_2}_\pm)}{\tmpbox}}(-\xi)\right] d\xi d\eta \right] \\ & =: S^{T_{-N}}_1 + S^{T_{-N}}_2 + S^{T_{-N}}_3, \end{split}$$ with coefficients $c_{I_1,I_2}\in\{-1,0,1\}$ such that $c_{I_1,I_2}=1$ whenever $|I_1|+|I_2|=|I|$ (in which case $D=D_1$). After lemma \[Lem\_appendix: integral sigma\_tilde\_N\] and inequality with $s=N+3$ we deduce that, if $N\ge 15$, for any $\theta\in ]0,1[$ $$|S^{T_{-N}}_2|\lesssim \|V(t,\cdot)\|_{H^{N+4,\infty}}^{2-\theta}\|V(t,\cdot)\|^\theta_{H^{N+6}}\|V^I(t,\cdot)\|^2_{L^2}.$$ Choosing $\theta\ll 1$ small (e.g. $\theta\le 1/8$) we then obtain that $S^{T_{-N}}_2$ is a remainder $\mathfrak{D}^{I,N}_{\text{quart}}$ satisfying . Also, the same lemma implies that each contribution in $S^{T_{-N}}_1, S^{T_{-N}}_3$ corresponding to $(I_1,I_2)\in \mathcal{I}(I)$ with $|I_2|<|I|$ is bounded by $$\left\| Q^{\mathrm{kg}}_0(v^{I_1}_\pm, D u^{I_2}_\pm)\right\|_{L^2} \|U(t,\cdot)\|_{H^{N+3,\infty}}\|V^I(t,\cdot)\|_{L^2}.$$ Reminding instead , we find that the remaining contribution to $S^{T_{-N}}_1$, corresponding to $I_2=I$, is equal to the sum over $j_1,\dots,j_5\in \{+,-\}$ of the (imaginary part) of the following integrals: $$\label{integral sigma_tilde_N-1} \int \widetilde{\sigma}^N_{(j_1,j_2,j_3)}(\xi,\eta)\Big(1 - j_4j_5 \frac{\xi-\eta-\zeta}{\langle\xi-\eta - \zeta\rangle}\cdot\frac{\zeta}{|\zeta|} \Big)\zeta_1 \hat{v}_{j_4}(\xi-\eta - \zeta) \hat{u}^I_{j_5}(\zeta) \hat{u}_{j_2}(\eta)\hat{v}^I_{j_3}(-\xi)\ d\xi d\eta d\zeta.$$ Analogously, the contribution corresponding to $I_2=I$ in $S^{T_{-N}}_3$ is the sum over $j_k\in\{+,-\}, k=1,\dots,5$ of $$\label{integral sigma_tilde_N-2} \int \widetilde{\sigma}^N_{(j_1,j_2,j_3)}(\xi,\eta) \Big(1 + j_4j_5 \frac{\xi+\zeta}{\langle\xi+\zeta\rangle}\cdot\frac{\zeta}{|\zeta|} \Big)\zeta_1 \hat{v}^I_{j_1}(\xi-\eta) \hat{u}_{j_2}(\eta) \hat{v}_{j_4}(-\xi-\zeta) \hat{u}^I_{j_5}(\zeta)\ d\xi d\eta d\zeta.$$ Since $\widetilde{\sigma}^N_{(j_1,j_2,j_3)}(\xi,\eta)$ satisfies and is supported for $|\eta|\le \varepsilon |\xi-\eta|$, for a small $0<\varepsilon\ll 1$, we rewrite above integrals, respectively, as $$\begin{gathered} \label{new_integral sigma_tilde_N-1} \int \widetilde{\sigma}^N_{(j_1,j_2,j_3)}(\xi,\eta) \langle\eta\rangle^{-N-3}\Big(1 - j_4j_5 \frac{\xi-\eta-\zeta}{\langle\xi-\eta - \zeta\rangle}\cdot\frac{\zeta}{|\zeta|} \Big)\zeta_1 \langle\xi-\eta-\zeta\rangle^{-4}\\ \times {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\langle D_x\rangle^4 v}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\langle D_x\rangle^4 v}{\tmpbox}}_{j_4}(\xi-\eta - \zeta) \hat{ u}^I_{j_5}(\zeta) {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\langle D_x\rangle^{N+3} u}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\langle D_x\rangle^{N+3} u}{\tmpbox}}_{j_2}(\eta)\hat{v}^I_{j_3}(-\xi)\ d\xi d\eta d\zeta,\end{gathered}$$ and $$\begin{gathered} \label{new_integral sigma_tilde_N-2} \int \widetilde{\sigma}^N_{(j_1,j_2,j_3)}(\xi,\eta)\langle\eta\rangle^{-N-7} \Big(1 + j_4j_5 \frac{\xi+\zeta}{\langle\xi+\zeta\rangle}\cdot\frac{\zeta}{|\zeta|} \Big)\zeta_1\langle\xi+\zeta\rangle^{-4}\\ \times \hat{ v}^I_{j_1}(\xi-\eta) {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\langle D_x\rangle^{N+7} u}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\langle D_x\rangle^{N+7} u}{\tmpbox}}_{j_2}(\eta) {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\langle D_x\rangle^4v}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\langle D_x\rangle^4v}{\tmpbox}}_{j_4}(-\xi-\zeta) \hat{u}^I_{j_5}(\zeta)\ d\xi d\eta d\zeta.\end{gathered}$$ With such a choice, the new multipliers, that we denote concisely by $\widetilde{\sigma}^{N,k}_{(j_1,\dots, j_5)}(\xi,\eta,\zeta)$, $k=0,1$, verify, for any $\alpha,\beta,\gamma\in\mathbb{N}^2$, $$\begin{gathered} \left|\partial^\alpha_\xi\partial^\beta_\eta \widetilde{\sigma}^{N,k}_{(j_1,\dots, j_5)}(\xi,\eta,\zeta)\right| \lesssim \langle\zeta\rangle^{-3} |g^N_{\alpha,\beta}(\xi)|, \\ \left|\partial^\alpha_\xi\partial^\beta_\eta \partial^\gamma_\zeta \widetilde{\sigma}^{N,k}_{(j_1,\dots, j_5)}(\xi,\eta,\zeta)\right| \lesssim (|\zeta|\langle \zeta\rangle^{-1})^{1-|\gamma|}\langle\zeta\rangle^{-3} |g^N_{\alpha,\beta}(\xi)|, \quad |\gamma|\ge 1,\end{gathered}$$ with $g^N_{\alpha,\beta}(\xi,\eta)$ supported for $|\eta|\le \varepsilon |\xi-\eta|$ and such that $$\begin{gathered} |g^N_{\alpha,\beta}(\xi,\eta)|\lesssim \langle \xi-\eta\rangle^{6-N+|\alpha| +2|\beta|} |\eta|^{N-|\beta|}\langle \eta\rangle^{-N-3}, \quad \forall (\xi,\eta)\in\mathbb{R}^2\times\mathbb{R}^2.\end{gathered}$$ If $N\in\mathbb{N}^*$ is sufficiently large (e.g. $N\ge 18$), the above estimate implies that, for any $\alpha,\beta\in\mathbb{N}^2$ of length less or equal than 3, $$|g^N_{\alpha,\beta}(\xi,\eta)|\lesssim \langle\eta\rangle^{-3}\langle\xi\rangle^{-3},$$ so by lemma \[Lem\_appendix: Kernel with 1 function\] $(i)$ together with corollary \[Cor\_appendix: decay of integral operators\] $(i)$ we obtain that, for any $k=0,1$, $$K^{N,k}_{(j_1,\dots,j_5)}(x,y,z):=\int e^{ix\cdot\xi + i y\cdot \eta + iz\cdot\zeta} \widetilde{\sigma}^{N,k}_{(j_1,\dots, j_5)}(\xi,\eta,\zeta) d\xi d\eta d\zeta$$ is such that $$\label{est_KNk} |K^{N,k}_{(j_1,\dots,j_5)}(x,y,z)|\lesssim \langle x\rangle^{-3}|y|^{-1}\langle y \rangle^{-2}|z|^{-1}\langle z \rangle^{-2}, \quad \forall (x,y,z)\in(\mathbb{R}^2)^3.$$ By , , integrals , are respectively equal to $$\int K^{N,0}_{(j_1,\dots,j_5)}(t-x, x-z, x-y) [\langle D_x\rangle^4v_{j_4}](x) u^I_{j_5}(y) [\langle D_x\rangle^{N+3} u_{j_2}](z) v^I_{j_3}(t) dxdydzdt$$ and $$\int K^{N,1}_{(j_1,\dots,j_5)}(z-x, x-y, z-t) v^I_{j_1}(x) [\langle D_x\rangle^{N+7} u_{j_2}](y) [\langle D_x\rangle^4v_{j_4}](z) u^I_{j_5}(t) dxdydzdt.$$ Using and the fact that integrals such as can be bounded from above by the product of the $L^2$ norm of any two functions $\widetilde{u}_k$ times the $L^\infty$ norm of the remaining ones, they are estimated by $$\|V(t,\cdot)\|_{H^{4,\infty}}\|U(t,\cdot)\|_{H^{N+7,\infty}}\|W^I(t,\cdot)\|^2_{L^2},$$ which concludes the proof of the statement. \[Lem:Quartic\_terms\_III\] Let $n\in\mathbb{N}, n\ge 3$, $I\in\mathcal{I}_n$ and $(I_1,I_2)\in\mathcal{I}(I)$ be such that $[\frac{|I|}{2}]<|I_1|<|I|$. Let also $C^{I_1,I_2}_{(j_1,j_2,j_3)}$, $D^{I_1,I_2}_{(j_1,j_2,j_3)}$ be the integrals defined, respectively, in , , for any $j_k\in \{+,-\}, k=1,2,3$. Then $$\label{derivative DI1I2} \partial_t D^{I_1,I_2}_{(j_1,j_2,j_3)} =- C^{I_1,I_2}_{(j_1,j_2,j_3)}+ \mathfrak{D}^{I_1,I_2}_{\text{quart}},$$ where $\mathfrak{D}^{I_1,I_2}_{\text{quart}}$ satisfies $$\begin{gathered} \label{est_DI1I2_quart} \left|\mathfrak{D}^{I_1,I_2}_{\text{quart}}(t) \right|\lesssim \left[\left(\|W(t,\cdot)\|_{H^{[\frac{n}{2}]+12,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|_{H^{[\frac{n}{2}]+8,\infty}}\right)^2 \right. \\ \left. + \|V(t,\cdot)\|^{\frac{7}{4}}_{H^{[\frac{n}{2}]+11,\infty}} \|V(t,\cdot)\|^{\frac{1}{4}}_{H^{[\frac{n}{2}]+12}}\right] E_n(t;W).\end{gathered}$$ We compute the time derivative of $D^{I_1,I_2}_{(j_1,j_2,j_3)}$ by making use of system . We remind that, after remark \[Remark:Vector\_field\_on\_null\_structure\] and definition , if $\Gamma^I$ is a product of spatial derivatives then all couples of indices $(I_1,I_2)$ belonging to $\mathcal{I}(I)$ are such that $|I_1|+|I_2|=|I|$ and $\Gamma^{I_1},\Gamma^{I_2}$ are also products of spatial derivatives. Therefore, all coefficients $c_{I_1,I_2}$ appearing in the right hand side of are equal to 0. By definitions with $k=1$, , , we find that $$\begin{split} -4(2\pi)^2&\Big[ \partial_t D^{I_1,I_2}_{(j_1, j_2, j_3)}+ C^{I_1,I_2}_{(j_1, j_2, j_3)} \Big]= \\ & \int B^1_{(j_1,j_2,j_3)}(\xi,\eta) \left[\sum_{(J_1,J_2)\in\mathcal{I}(I_1)}{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q^{\mathrm{kg}}_0(v^{J_1}_\pm, D_1 u^{J_2}_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q^{\mathrm{kg}}_0(v^{J_1}_\pm, D_1 u^{J_2}_\pm)}{\tmpbox}}(\xi-\eta)\right]\hat{u}^{I_2}_{j_2}(\eta) \hat{v}^I_{j_3}(-\xi) d\xi d\eta \\ + & \int B^1_{(j_1,j_2,j_3)}(\xi,\eta)\ \hat{v}^{I_1}_{j_1}(\xi-\eta) \left[\sum_{(J_1,J_2)\in\mathcal{I}(I_2)}{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q^{\mathrm{w}}_0(v^{J_1}_\pm, D_1v^{J_2}_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q^{\mathrm{w}}_0(v^{J_1}_\pm, D_1v^{J_2}_\pm)}{\tmpbox}}(\eta)\right] \hat{v}^I_{j_3}(-\xi) d\xi d\eta \\ + & \int B^1_{(j_1,j_2,j_3)}(\xi,\eta) \hat{v}^{I_1}_{j_1}(\xi-\eta)\hat{u}^{I_2}_{j_2}(\eta) \left[\sum_{(J_1,J_2)\in\mathcal{I}(I)}{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q^{\mathrm{kg}}_0(v^{J_1}_\pm, D_1u^{J_2}_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q^{\mathrm{kg}}_0(v^{J_1}_\pm, D_1u^{J_2}_\pm)}{\tmpbox}}\right](-\xi) d\xi d\eta \\ =: & S^{I_1,I_2}_1 + S^{I_1,I_2}_2 + S^{I_1,I_2}_3. \end{split}$$ Since $|J_1|+|J_2|=|I_1|<|I|\le n$ in $S^{I_1,I_2}_1$, we can estimate all its contributions using inequality . Using lemma \[Lem: L2 est nonlinearities\] $(i)$, the fact that $|I_2|\le [\frac{n}{2}]$ by the hypothesis and, hence, that $$\|u^{I_2}_\pm(t,\cdot)\|_{H^{7,\infty}}+\|\mathrm{R}_1u^{I_2}_\pm(t,\cdot)\|_{H^{7,\infty}}\lesssim \|U(t,\cdot)\|_{H^{[\frac{n}{2}]+8,\infty}},$$ we deduce that $$\left|S^{I_1,I_2}_1\right|\lesssim \left(\|W(t,\cdot)\|_{H^{[\frac{n}{2}]+2}} + \|\mathrm{R}_1U(t,\cdot)\|_{H^{[\frac{n}{2}]+2,\infty}}\right)\|U(t,\cdot)\|_{H^{[\frac{n}{2}]+8,\infty}}E_n(t;W),$$ and above estimate holds also for all integrals in $S^{I_1,I_2}_3$ corresponding to $|J_2|<|I|$. The same inequality , combined with applied to $Q^\mathrm{w}_0(v^{J_1}_\pm, D_1v^{J_2}_\pm)$ and with corollary \[Cor\_appendixA:Hs-Hsinfty norm of bilinear expressions\] in appendix \[Appendix A\], gives that, for any $\theta\in ]0,1[$, $$\begin{split} &|S^{I_1,I_2}_2| \\ &\lesssim \sum_{|J_1|+|J_2|= |I_2|}\left[\left\|Q^\mathrm{w}_0(v^{J_1}_\pm, D_1v^{J_2}_\pm) \right\|_{H^{7, \infty}} + \left\|Q^\mathrm{w}_0(v^{J_1}_\pm, D_1v^{J_2}_\pm) \right\|^{1-\theta}_{H^{9, \infty}}\left\|Q^\mathrm{w}_0(v^{J_1}_\pm, D_1v^{J_2}_\pm) \right\|^\theta_{H^8} \right] E_n(t;W)\\ &\lesssim \|V(t,\cdot)\|^{2-(2-\theta)\theta}_{H^{[\frac{n}{2}]+11,\infty}} \|V(t,\cdot)\|^{(2-\theta)\theta}_{H^{[\frac{n}{2}]+12}} E_n(t;W). \end{split}$$ Finally, the last remaining integral in $S^{I_1,I_2}_3$, corresponding to indices $J_1=0,J_2=I$, can be written using as $$\sum_{j_4, j_4\in \{+,-\}}\int B^1_{(j_1,j_2,j_3)}(\xi,\eta)\left(1+j_4j_5\frac{\xi+\zeta}{\langle\xi+\zeta\rangle}\cdot\frac{\zeta}{|\zeta|}\right)\zeta_1 \hat{v}^{I_1}_{j_1}(\xi-\eta)\hat{u}^{I_2}_{j_2}(\eta)\hat{v}_{j_4}(-\xi-\zeta)\hat{u}^I_{j_5}(\zeta) d\xi d\eta d\zeta,$$ and is estimated, after lemma \[Lem:Est\_integrals\_quartic-terms-2\] and the fact that $|I_1|<|I|$, by $$\|V(t,\cdot)\|_{H^{7,\infty}}\left(\|U(t,\cdot)\|_{H^{[\frac{n}{2}]+12,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|_{H^{[\frac{n}{2}]+8,\infty}}\right)E_n(t;W).$$ This gives that $$\left| S^{I_1,I_2}_3\right|\lesssim \left(\|W(t,\cdot)\|_{H^{[\frac{n}{2}]+12,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|_{H^{[\frac{n}{2}]+8,\infty}}\right)^2 E_n(t;W)$$ and concludes the proof of the statement. \[Lem:Analysis quartic terms IV\] Let $k=0,1$, $\mathcal{K},\mathcal{V}_k$ be the sets introduced in , respectively, $I\in \mathcal{V}^k$ and $(I_1,I_2)\in\mathcal{I}(I)$ be such that $I_1\in\mathcal{K}$, $|I_2|\le 1$. Let also $F^{I_1,I_2,l}_{(j_1,j_2,j_3)}$, $G^{I_1,I_2,l}_{(j_1,j_2,j_3)}$ be the integrals defined in , , for any $l=1,2,3$, $j_i\in\{+,-\}, i=1,2,3$. For any $l=1,2,3$, any triplet $(j_1,j_2,j_3)$, we have that $$\label{derivative_GI1I2} \partial_t G^{I_1,I_2,l}_{(j_1,j_2,j_3)} = -F^{I_1,I_2,l}_{(j_1,j_2,j_3)} + \mathfrak{G}^{I_1,I_2}_{\text{quart}},$$ and there is a constant $C>0$ such that, if a-priori estimates are satisfied in interval $[1,T]$ for a fixed $T>1$, with $\varepsilon_0<(2A+B)^{-1}$ small, $$\label{est_GI1I2quart} |\mathfrak{G}^{I_1,I_2}_{\text{quart}}(t)|\le C(A+B)^2\varepsilon^2 t^{-1+\frac{\delta_k}{2}}\left[E^k_3(t;W)^\frac{1}{2}+\delta_{\mathcal{V}^0}t^{\beta+\frac{\delta_1}{2}}E^1_3(t;W)^\frac{1}{2}+ t^{-\frac{1}{4}-\frac{\delta_k}{2}}\right],$$ for every $t\in [1,T]$, with $\delta_{\mathcal{V}^0}=1$ if $I\in\mathcal{V}^0$, 0 otherwise, and $\beta>0$ as small as we want. First of all, it is useful to remind that from , and a-priori estimate , for any $k=0,1$, $I\in\mathcal{I}^k_3$, $(I_1,I_2)\in\mathcal{I}(I)$ such that $I_1\in\mathcal{K}$, $|I_2|\le 1$, and $\sigma>0$ sufficiently small $$\label{est_VI1UI2} \|V^{I_1}(t,\cdot)\|_{L^2}\left(\|\chi(t^{-\sigma}D_x)U^{I_2}(t,\cdot)\|_{H^{\rho,\infty}}+ \|\chi(t^{-\sigma}D_x)\mathrm{R} U^{I_2}(t,\cdot)\|_{H^{\rho,\infty}}\right) \le C(A+B)B\varepsilon^2 t^{-\frac{1}{2}+\frac{\delta_k}{2}},$$ for every $t\in [1,T]$. For any fixed $(j_1,j_2,j_3)$, any $l=1,2,3$, we compute $\partial_t G^{I_1,I_2,l}_{(j_1,j_2,j_3)}$ recurring to system along with its compact form $$\begin{cases} (D_t \mp \langle D_x\rangle)v^I_\pm = \Gamma^I Q^\mathrm{w}_0(v_\pm, D_1 v_\pm),\\ (D_t \mp | D_x|)u^I_\pm = \Gamma^I Q^\mathrm{kg}_0(v_\pm, D_1 u_\pm), \end{cases}$$ and using that $[D_t, \chi(t^{-\sigma}D_x)]=t^{-1}\chi_1(t^{-\sigma}D_x)$ with $\chi_1(\xi):=i\sigma (\partial\chi)(\xi)\cdot\xi$. We find that$$\begin{split} &-4(2\pi)^2 \left[\partial_t G^{I_1,I_2,l}_{(j_1,j_2,j_3)} + F^{I_1,I_2,l}_{(j_1,j_2,j_3)}\right] \\ &= \int B^l_{(j_1,j_2,j_3)}(\xi,\eta) \Big[{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\Gamma^{I_1}Q^\mathrm{kg}_0(v_\pm, D_1u_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\Gamma^{I_1}Q^\mathrm{kg}_0(v_\pm, D_1u_\pm)}{\tmpbox}}(\xi-\eta)\Big] {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\chi(t^{-\sigma}D_x)u^{I_2}_{j_2}}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\chi(t^{-\sigma}D_x)u^{I_2}_{j_2}}{\tmpbox}}(\eta)\hat{v}^I_{j_3}(-\xi) d\xi d\eta \\ &+ \int B^l_{(j_1,j_2,j_3)}(\xi,\eta) \hat{v}^{I_1}_{j_1}(\xi-\eta) \Big[{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\chi(t^{-\sigma}D_x)\Gamma^{I_2}Q^\mathrm{w}_0(v_\pm, D_1 v_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\chi(t^{-\sigma}D_x)\Gamma^{I_2}Q^\mathrm{w}_0(v_\pm, D_1 v_\pm)}{\tmpbox}}(\eta) + t^{-1}{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\chi_1(t^{-\sigma}D_x)u^{I_2}_{j_2}}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\chi_1(t^{-\sigma}D_x)u^{I_2}_{j_2}}{\tmpbox}}(\eta)\Big]\hat{v}^I_{j_3}(-\xi) d\xi d\eta \\ & + \int B^l_{(j_1,j_2,j_3)}(\xi,\eta) \hat{v}^{I_1}_{j_1}(\xi-\eta) {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\chi(t^{-\sigma}D_x)u^{I_2}_{j_2}}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\chi(t^{-\sigma}D_x)u^{I_2}_{j_2}}{\tmpbox}}(\eta)\Big[\sum_{(J_1,J_2)\in\mathcal{I}(I)}c_{J_1,J_2}{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{Q^\mathrm{kg}_0(v^{J_1}_\pm, Du^{J_2}_\pm)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{Q^\mathrm{kg}_0(v^{J_1}_\pm, Du^{J_2}_\pm)}{\tmpbox}}(-\xi)\Big] d\xi d\eta\\ &=: S^{I_1,I_2,l}_1 + S^{I_1,I_2,l}_2 + S^{I_1,I_2,l}_3, \end{split}$$with $B^l_{(j_1,j_2,j_3)}$ given by when $l=1,2$ or when $l=3$. Applying to $S^{I_1,I_2,l}_2$, using with $w=\Gamma^{I_2}Q^\mathrm{w}_0(v_\pm, D_1v_\pm)$ and $\rho=7$, together with the fact that operators $\chi(t^{-\sigma}D_x)$, $\chi_1(t^{-\sigma}D_x)$ are bounded from $L^\infty$ to $H^{\rho,\infty}$ for any $\rho\ge 0$ with norm $O(t^{\sigma\rho})$, and from $L^2$ to $H^s$ for any $s\ge0$ with norm $O(t^{\sigma s})$, we deduce that, for any $\theta\in ]0,1[$, $$\begin{gathered} \label{preliminary_SI1I2_2} |S^{I_1,I_2,l}_2|\lesssim t^\beta \|V^{I_1}(t,\cdot)\|_{L^2}\|V^I(t,\cdot)\|_{L^2}\\ \times \left[\|\Gamma^{I_2}Q^\mathrm{w}_0(v_\pm, D_1v_\pm)\|_{L^\infty}+ \delta_l \|\Gamma^{I_2}Q^\mathrm{w}_0(v_\pm, D_1v_\pm)\|_{L^\infty}^{1-\theta}\|\Gamma^{I_2}Q^\mathrm{w}_0(v_\pm, D_1v_\pm)\|^\theta_{L^2} \right.\\ \left.+ t^{-1} \left(\|\chi_1(t^{-\sigma}D_x)u^{I_2}_\pm(t,\cdot)\|_{L^\infty}+\|\chi_1(t^{-\sigma}D_x)\mathrm{R} u^{I_2}_\pm(t,\cdot)\|_{L^\infty}\right)\right],\end{gathered}$$ for some $\beta>0$ small, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$, and with $\delta_l=1$ if $l=1,2$, 0 otherwise. When $|I_2|=0$ the above right hand side can be estimated using , and a-priori estimates . When $|I_2| = 1$ we derive from that $$\Gamma^{I_2}Q^\mathrm{w}_0(v_\pm, D_1v_\pm) = Q^\mathrm{w}_0(v^{I_2}_\pm, D_1v_\pm) + Q^\mathrm{w}_0(v_\pm, D_1v^{I_2}_\pm) + G^w_1(v_\pm, D v_\pm)$$ with $G^w_1(v_\pm, Dv_\pm) = G_1(v, \partial v)$ given by . Using lemma \[Lem\_app:products\_Gamma\] in appendix \[Appendix B\] with $L=L^\infty$ to estimate the $L^\infty$ norm of the first two quadratic terms in the above right hand side, we find that, for some new $\chi\in C^\infty_0(\mathbb{R}^2)$ and $\sigma>0$ small, there is a constant $C>0$ such that $$\begin{split} \|\Gamma^{I_2}& Q^\mathrm{w}_0(v_\pm, D_1v_\pm)\|_{L^\infty}\lesssim \left\|\chi(t^{-\sigma}D_x)v^{I_2}_\pm(t,\cdot)\right\|_{H^{2,\infty}}\|v_\pm(t,\cdot)\|_{H^{2,\infty}} \\ &+ t^{-N(s)}\left(\|v_\pm(t,\cdot)\|_{H^s}+ \|D_tv_\pm(t,\cdot)\|_{H^s}\right)\Big(\sum_{|\mu|=0}^1\|x^\mu v_\pm(t,\cdot)\|_{H^1}+ t\|v_\pm(t,\cdot)\|_{H^1}\Big) \\ & + \|v_\pm(t,\cdot)\|_{H^{1,\infty}}\left(\|v_\pm(t,\cdot)\|_{H^{2,\infty}}+\|D_tv_\pm(t,\cdot)\|_{H^{1,\infty}}\right)\\ &\le CAB\varepsilon^2 t^{-2}, \end{split}$$ where last inequality is obtained by picking $s>0$ sufficiently large so that $N(s)\ge 4$ and using , , , lemma \[Lem\_appendix: sharp\_est\_VJ\], together with a-priori estimates. Also, by with $s=0$ and a-priori estimates $$\|\Gamma^{I_2}Q^\mathrm{w}_0(v_\pm, D_1v_\pm)\|_{L^2} \lesssim \|V(t,\cdot)\|_{H^{2,\infty}}\left(\|V^{I_2}(t,\cdot)\|_{H^1}+\|D_tV(t,\cdot)\|_{L^2}\right)\le CAB\varepsilon^2t^{-1+\frac{\delta_2}{2}}.$$ Therefore, using lemma \[Lem\_appendix: est UJ\] and taking $\theta,\sigma>0$ sufficiently small we deduce from and the above estimates that, for any $l=1,2,3$ and a new $C>0$, $$\label{est_SI1I2_2} |S^{I_1,I_2,l}_2|\le CAB\varepsilon^2 t^{-\frac{5}{4}}E^k_3(t;W)^\frac{1}{2}.$$ We make use of inequality to estimate $S^{I_1,I_2,l}_1$, too. From we have that $$\Gamma^{I_1}Q^\mathrm{kg}_0(v_\pm, D_1u_\pm) = Q^\mathrm{kg}_0(v^{I_1}_\pm, D_1u_\pm)+ \sum_{\substack{(J_1,J_2)\in\mathcal{I}(I_1)\\ |J_1|<|I_1|}}c_{J_1,J_2} Q^\mathrm{kg}_0(v^{J_1}_\pm, Du^{J_2}_\pm)$$ with $c_{J_1,J_2}\in \{-1,0,1\}$, and then from , and the fact that $I_1\in\mathcal{K}$, $$\Gamma^{I_1}Q^\mathrm{kg}_0(v_\pm, D_1u_\pm)= Q^\mathrm{kg}_0(v^{I_1}_\pm, \chi(t^{-\sigma}D_x)D_1u_\pm) +\mathfrak{R}^k_3(t,x),$$ with $\mathfrak{R}^k_3$ satisfying and $$\| Q^\mathrm{kg}_0(v^{I_1}_\pm, \chi(t^{-\sigma}D_x)D_1 u_\pm)\|_{L^2}\le \left(\|U(t,\cdot)\|_{H^{2,\infty}}+\|\mathrm{R}U(t,\cdot)\|_{H^{2,\infty}}\right)\|V^{I_1}(t,\cdot)\|_{L^2}.$$ So from , , lemma \[Lem\_appendix: est UJ\] and priori estimates $$\label{est_SI1I1_1} \begin{split} |S^{I_1,I_2,l}_1|&\lesssim \left[\left(\|U(t,\cdot)\|_{H^{2,\infty}}+\|\mathrm{R}U(t,\cdot)\|_{H^{2,\infty}}\right)\|V^{I_1}(t,\cdot)\|_{L^2} + \|\mathfrak{R}^k_3(t,\cdot)\|_{L^2}\right]\\ &\times \left(\|\chi(t^{-\sigma}D_x)U^{I_2}(t,\cdot)\|_{H^{7,\infty}}+ \|\chi(t^{-\sigma}D_x)\mathrm{R} U^{I_2}(t,\cdot)\|_{H^{7,\infty}} \right)\|V^I(t,\cdot)\|_{L^2}\\ & \le CAB\varepsilon^2 t^{-1+\frac{\delta_k}{2}}E^k_{3}(t;W)^\frac{1}{2}. \end{split}$$ Let us now consider all the addends in $S^{I_1,I_2,l}_3$ with $|J_2|<|I|$, which by inequality are bounded by $$\|V^{I_1}(t,\cdot)\|_{L^2}\Big(\sum_{|\mu|=0}^1 \|\chi(t^{-\sigma}D_x)\mathrm{R}^\mu U^{I_2}(t,\cdot)\|_{H^{7,\infty}}\Big)\sum_{\substack{(J_1,J_2)\in\mathcal{I}(I)\\ |J_2|<|I|}}\Big\|c_{J_1,J_2}Q^\mathrm{kg}_0(v^{J_1}_\pm, Du^{J_2}_\pm)\Big\|_{L^2}.$$ As the latter above factor is bounded by the $L^2$ norm of $Q^I_0(V,W)$ (see definition ), inequalities and imply that those integrals are remainders $\mathfrak{G}^{I_1,I_2}_{\text{quart}}$ satisfying . Finally, the last contribution to $S^{I_1,I_2,l}_3$, corresponding to $|J_1|=0, J_2=I$, for which $D=D_1$, can be rewritten using as the sum over $j_4,j_5\in \{+,-\}$ of $$\int B^1_{(j_1,j_2,j_3)}(\xi,\eta) \left(1+j_4j_5\frac{\xi+\zeta}{\langle\xi+\zeta\rangle}\cdot\frac{\zeta}{|\zeta|}\right)\zeta_1 \hat{v}_{4}(-\xi-\zeta) \hat{u}^I_{j_5}(\zeta) {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\chi(t^{-\sigma}D_x)u^{I_2}_{j_2}}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\chi(t^{-\sigma}D_x)u^{I_2}_{j_2}}{\tmpbox}}(\eta) \hat{v}^{I_1}_{j_1}(\xi-\eta) d\xi d\eta.$$ By means of lemma \[Lem:Est\_integrals\_quartic-terms-2\] it is bounded by $$\|V(t,\cdot)\|_{H^{7,\infty}}\Big(\sum_{|\mu|=0}^1\|\chi(t^{-\sigma}D_x)D_1\mathrm{R}^\mu U^{J_2}(t,\cdot)\|_{H^{11,\infty}} \Big)\|V^{I_1}(t,\cdot)\|_{H^1}\|U^I(t,\cdot)\|_{L^2}$$ for every $t\in [1,T]$, and hence by $CA(A+B)\varepsilon^2 t^{-\frac{3}{2}+\beta'} E^k_3(t;W)$, with $\beta'>0$ small as long as $\sigma,\delta_0$ are small, as follows by a-priori estimate and lemma \[Lem\_appendix: est UJ\]. ### Propagation of the energy estimate \[Prop: Propagation of the energy estimate\] Let us fix $K_2>0$. There exist two integers $n\gg \rho\gg 1$ sufficiently large, two constants $A,B>1$ sufficiently large, $\varepsilon_0\in ]0,(2A+B)^{-1}[$ sufficiently small, and some $0<\delta \ll \delta_2\ll \delta_1\ll \delta_0\ll 1$ small such that, for any $0<\varepsilon<\varepsilon_0$, if $(u,v)$ is solution to - in some interval $[1,T]$ for a fixed $T>1$, and $u_\pm, v_\pm$ defined in satisfy a-priori estimates for every $t\in [1,T]$, then they also verify , on the same interval $[1,T]$. For any integer $k,n\in\mathbb{N}$, with $n\ge 3$ and $0\le k\le 2$, let $\widetilde{E}_n(t;W)$, $\widetilde{E}^k_3(t;W)$ be the first modified energies introduced in and $\widetilde{E}^\dagger_n(t;W)$, $\widetilde{E}^{k,\dagger}_3(t;W)$ be the second modified energies, introduced in and respectively. Let also $D^I_{(j_1,j_2,j_3)}, D^{I,R}_{(j_1,j_2,j_3)}$, $D^{I,T_{-N}}_{(j_1,j_2,j_3)}$ be the integrals defined in , $D^{I_1,I_2}_{(j_1,j_2,j_3)}$ in , and $G^{I_1,I_2,l}_{(j_1,j_2,j_3)}$ in . Fix $N=18$. The first thing we observe is that, as long as estimates , are satisfied and $\rho\in\mathbb{N}$ is sufficiently large (e.g. $\rho\ge \max\{[\frac{n}{2}]+8,21\}$), there is a constant $C>0$ such that for every $t\in [1,T]$ $$\begin{gathered} C^{-1}E_n(t;W)\le \widetilde{E}^{\dagger}_n(t;W)\le C E_n(t;W),\label{equivalence_En-Edaggern}\\ C^{-1}E^k_3(t;W)\le \widetilde{E}^{k,\dagger}_3(t;W)\le C E^k_3(t;W).\label{equivalence_Ek2-Edaggerk2}\end{gathered}$$ Above equivalences follow from , a-priori estimates , , the fact that for a general multi-index $I$ ($I\in\mathcal{I}_n$ or $I\in\mathcal{I}^k_3$ for $0\le k\le 2$) $$\sum_{j_i\in\{+,-\}} \left| D^I_{(j_1,j_2,j_3)}\right| + \left| D^{I,R}_{(j_1,j_2,j_3)}\right| \lesssim \left(\|U(t,\cdot)\|_{H^{7,\infty}} + \|\mathrm{R}_1U(t,\cdot)\|_{H^{7,\infty}}\right) \|V^I(t,\cdot)\|^2_{L^2}$$ by inequality , $$\sum_{j_k\in\{+,-\}} \left| D^{I,T_{-18}}_{(j_1,j_2,j_3)}\right| \lesssim \|U(t,\cdot)\|_{H^{21,\infty}} \|W^I(t,\cdot)\|^2_{L^2}$$ by inequality , and: $\bullet$ as concerns especially , from the fact that for any $I\in\mathcal{I}_n$, any $(I_1,I_2)\in\mathcal{I}(I)$ with $[\frac{|I|}{2}]<|I_1|<|I|$, by $$\begin{split} \sum_{j_i\in\{+,-\}} \left| D^{I_1,I_2}_{(j_1,j_2,j_3)}\right|& \lesssim \left(\|U^{I_2}(t,\cdot)\|_{H^{7,\infty}}+\|\mathrm{R}_1U^{I_2}(t,\cdot)\|_{H^{7,\infty}}\right)\|V^{I_1}(t,\cdot)\|_{L^2}\|V^I(t,\cdot)\|_{L^2}\\ &\lesssim \left(\|U(t,\cdot)\|_{H^{[\frac{n}{2}]+8,\infty}} + \|\mathrm{R}_1U(t,\cdot)\|_{H^{[\frac{n}{2}]+8,\infty}}\right)E_n(t;W);\\ \end{split}$$ $\bullet$ as concerns especially , the fact that for any $I\in\mathcal{V}^k$ (see definition ), any $(I_1,I_2)\in\mathcal{I}(I)$ with $I_1\in\mathcal{K}$ (see ) and $|I_2|\le 1$, and any $l=1,2,3$, by and $$\begin{split} \sum_{j_i\in\{+,-\}}\left| G^{I_1,I_2,l}_{(j_1,j_2,j_3)}\right| & \lesssim \sum_{|\mu|=0}^1\|\chi(t^{-\sigma}D_x)\mathrm{R}^\mu U^{I_2}(t,\cdot)\|_{H^{7,\infty}} \|V^{I_1}(t,\cdot)\|_{L^2}\|V^I(t,\cdot)\|_{L^2}\\ &\le C(A+B)B\varepsilon^2 t^{-\frac{1}{2}+\frac{\delta_k}{2}}E^k_3(t;W)^\frac{1}{2}. \end{split}$$ Let us now consider a general multi-index $I$. From equation we deduce the following equality: $$\label{partial t || WIs ||} \begin{split} & \frac{1}{2}\partial_t \|\widetilde{W}^I_s(t,\cdot)\|^2_{L^2}= -\Im\left[\langle D_t \widetilde{W}^I_s,\widetilde{W}^I_s\rangle\right]\\ & = -\Im \left[\langle A(D)\widetilde{W}^I_s, \widetilde{W}^I_s\rangle + \Bigl\langle Op^B\Bigl((I_4+ E^0_d(U;\eta))\widetilde{A}_1(V;\eta)(I_4 + F^0_d(U;\eta))\Bigr)\widetilde{W}^I_s , \widetilde{W}^I_s \Bigr\rangle \right.\\ & \hspace{0.7cm} \left. +\langle Op^B(A''(V^I;\eta))U + Op^B_R(A''(V^I;\eta))U , \widetilde{W}^I_s\rangle + \langle Q^I_0(V,W) , \widetilde{W}^I_s\rangle \right. \\ & \hspace{0.7cm} \left. + \langle T_{-18}(U)W^I_s, \widetilde{W}^I_s\rangle + \langle\mathfrak{R}'(U, V),\widetilde{W}^I_s\rangle\right] \end{split}$$ and immediately notice that $\Im[\langle A(D)\widetilde{W}^I_s, \widetilde{W}^I_s\rangle] = 0$ because of the fact that $A(\eta)$, introduced in , is real diagonal matrix and its quantization is a self-adjoint operator. Matrix $(I_4+ E^0_d(U;\eta))\widetilde{A}_1(V;\eta)(I_4 + F^0_d(U;\eta))$ is real, symmetric, of order 1, with semi-norm $$M^1_1\Bigl(\big(I_4+ E^0_d(U;\eta))\widetilde{A}_1(V;\eta)(I_4 + F^0_d(U;\eta)\big), 3\Bigr) \lesssim (1+ \|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}})^2\|V(t,\cdot)\|_{H^{2,\infty}}$$ as follows by estimate on $E^0_d$, of $F^0_d$, and on $\widetilde{A}_1(V;\eta)$. Corollary \[Cor : paradiff ajoint at order 1\] and a-priori estimates , imply then that the second term in the right hand side of reduces to $\langle T_0(U,V) \widetilde{W}^I_s, \widetilde{W}^I_s\rangle$, with $T_0(U,V)$ operator of order less or equal than 0 such that $$\|T_0(U,V)\|_{\mathcal{L}(L^2)}\lesssim M^1_1\Bigl(\big(I_4+ E^0_d(U;\eta))\widetilde{A}_1(V;\eta)(I_4 + F^0_d(U;\eta)\big), 3\Bigr) \le CA\varepsilon t^{-1},$$ so after Cauchy-Schwarz inequality and equivalence it is a remainder $R(t)$ satisfying, for every $t\in [1,T]$ $$\label{remainder_R(t)} \left| R(t) \right|\le CA\varepsilon t^{-1}\|W^I(t,\cdot)\|^2_{L^2}.$$ Observe that, by the definition of $\widetilde{W}^I_s$ in and of $W^I_s$ in , we have that $$\label{est_L2_Witildes-WI} \begin{split} \left\|(\widetilde{W}^I_s - W^I)(t,\cdot) \right\|_{L^2}&\le \|Op^B(P(V;\eta)-I_4)W^I\|_{L^2} + \|Op^B(E(U;\eta))W^I_s\|_{L^2}\\ &\lesssim \left(\|V(t,\cdot)\|_{H^{1,\infty}}+\|U(t,\cdot)\|_{H^{5,\infty}}+\|\mathrm{R}_1U(t,\cdot)\|_{H^{1,\infty}}\right)\|W^I(t,\cdot)\|_{L^2}, \end{split}$$ the latter inequality following from proposition \[Prop : Paradiff action on Sobolev spaces-NEW\], estimate , the fact that $E(U;\eta)$ verifies, after and for any admissible cut-off function $\chi$, $$M^0_0\left(E\Big(\chi\Big(\frac{D_x}{\langle \eta\rangle}\Big) U;\eta\Big);n \right)\lesssim \|U(t,\cdot)\|_{H^{5,\infty}}+\|\mathrm{R}_1U(t,\cdot)\|_{H^{1,\infty}},$$ and equivalence . Therefore, third and fifth brackets in the right hand side of can be replaced with $$\langle Op^B(A''(V^I;\eta))U + Op^B_R(A''(V^I;\eta))U , W^I\rangle + \langle T_{-18}(U)W^I, W^I\rangle$$ up to some new remainders $R(t)$, satisfying after Cauchy-Schwarz inequality, estimates , , and , . Summing up, equality reduces to: $$\label{partial WIs-2} \begin{split} \frac{1}{2}\partial_t \|\widetilde{W}^I_s(t,\cdot)\|_{L^2} = -&\Im\Big[ \langle Op^B(A''(V^I;\eta))U + Op^B_R(A''(V^I;\eta))U , W^I\rangle\\ &+ \langle Q^I_0(V,W), \widetilde{W}^I_s\rangle + \langle T_{-18}(U)W^I_s, \widetilde{W}^I_s\rangle + \langle\mathfrak{R}'(U,V), \widetilde{W}^I_s\rangle \Big] +R(t). \end{split}$$ In order to analyse the behaviour of the second and fourth brakets in above right hand side we need, at this point, to distinguishing between indices $I\in \mathcal{I}_n$ and $I\in\mathcal{I}^k_3$. : Let us suppose that $I\in\mathcal{I}_n$. Using and estimate we find that $$\label{QI0,WtildeIS} \langle Q^I_0(V,W), \widetilde{W}^I_s\rangle = \langle Q^I_0(V,W), W^I\rangle +R_n(t)$$ where, for a new constant $C>0$ and every $t\in [1,T]$, $$\label{remainder_Rn(t)} |R_n(t)|\le CA\varepsilon t^{-1+\frac{\delta}{2}}E_n(t;W)^\frac{1}{2}.$$ Reminding definition of $Q^I_0(V,W)$ and the fact that coefficients $c_{I_1,I_2}$ are all equal to 0 when $I\in\mathcal{I}_n$, we notice that some of the contributions to the scalar product in the right hand side of are also remainders $R_n(t)$. These are precisely the following ones: $$\sum_{(I_1,I_2)\in\mathcal{I}(I)}\langle Q^\mathrm{w}_0(v^{I_1}_\pm, D_1v^{I_2}_\pm), u^I_+ + u^I_{-}\rangle + \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|\le [\frac{|I|}{2}]}}\langle Q^\mathrm{kg}_0(v^{I_1}_\pm, D_1u^{I_2}_\pm), v^I_+ + v^I_{-}\rangle$$ in consequence of Cauchy-Schwarz inequality and estimates , , . Moreover, $\langle \mathfrak{R}'(U,V), \widetilde{W}^I\rangle$ in the right hand side of is also a remainder $R_n(t)$ because of Cauchy-Schwarz, , a-priori estimates , , and the fact that $$\|\mathfrak{R}'(U,V)\|_{L^2}\le CA\varepsilon t^{-1+\frac{\delta}{2}},$$ which follows choosing $\theta\ll 1$ in , using and -. Since remainder $R(t)$ in (verifying ) can be enclosed in $R_n(t)$ after , we obtain that equality can be further reduced to $$\begin{gathered} \frac{1}{2}\partial_t \|\widetilde{W}^I_s(t,\cdot)\|^2_{L^2} = -\Im\Big[\langle Op^B(A''(V^I;\eta))U + Op^B_R(A''(V^I;\eta))U , W^I\rangle\\ + \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ [\frac{|I|}{2}]<|I_1|<|I|}} \langle Q^\mathrm{kg}_0(v^{I_1}_\pm, D_1u^{I_2}_\pm), v^I_+ + v^I_{-}\rangle + \langle T_{-18}(U)W^I, W^I\rangle \Big] + R_n(t).\end{gathered}$$ From definition , equalities , , with $N=18$, , together with , with $N=18$, , we deduce that $$\begin{gathered} \frac{1}{2}\left| \partial_t \widetilde{E}^{\dagger}_n(t;W) \right| \lesssim |R_n(t)| + \sum_{I\in\mathcal{I}_n}\left( \left|\mathfrak{D}^{I}_{\text{quart}}(t)\right| + \left|\mathfrak{D}^{I,18}_{\text{quart}}(t)\right| \right) + \sum_{I\in\mathcal{I}_n}\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ [\frac{|I|}{2}]<|I_1|<|I|}} \left| \mathfrak{D}^{I_1, I_2}_{\text{quart}}(t)\right|,\end{gathered}$$ where quartic terms $\mathfrak{D}^{I}_{\text{quart}}, \mathfrak{D}^{I,18}_{\text{quart}}, \mathfrak{D}^{I_1, I_2}_{\text{quart}}$ satisfy, respectively, , with $N=18$, . These latter ones can also be considered as remainders $R_n(t)$ thanks to lemma \[Lem: L2 est nonlinearities\] $(i)$ and a-priori estimates , which implies that, for some new $C>0$ and every $t\in [1,T]$, $$\left|\partial_t \widetilde{E}^{\dag}_n(t;W)\right| \le CA\varepsilon t^{-1+\frac{\delta}{2}} E_n(t;W)^\frac{1}{2}.$$ Then $$\widetilde{E}^{\dagger}_n(t;W)^\frac{1}{2}\le \widetilde{E}^{\dagger}_n(1;W)^\frac{1}{2} + \int_1^t C A\varepsilon \tau^{-1+\frac{\delta}{2}} d\tau ,$$ so after equivalence and a-priori estimate $$\begin{split} E_n(t;W)^\frac{1}{2}& \le C E_n(1;W)^\frac{1}{2} + \int_1^t C A\varepsilon\tau^{-1+\frac{\delta}{2}} d\tau\\ & \le C E_n(1;W)^\frac{1}{2} + \frac{2C A\varepsilon}{\delta}t^\frac{\delta}{2}, \end{split}$$ again for a new $C>0$. Taking $B>1$ sufficiently large so that $E_n(1;W)^\frac{1}{2}\le \frac{B\varepsilon}{2C K_2}$ and $\frac{2CA}{\delta}<\frac{B}{2K_2}$ we finally obtain . : Let us now suppose that $I\in\mathcal{I}^k_3$ for $0\le k\le 2$. After and we have that $$\langle Q^I_0(V,W), \widetilde{W}^I_s\rangle = \langle Q^I_0(V,W), W^I\rangle +R^k_3(t)$$ with $$|R^k_3(t)|\le CA(A+B)\varepsilon^2 t^{-1+\frac{\delta_k}{2}}E^k_3(t;W)^\frac{1}{2},$$ and moreover $$\begin{split} -\Im\left[\langle Q^I_0(V,W), W^I\rangle \right] = & -\delta_{\mathcal{V}^k}\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ I_1\in\mathcal{K}, |I_2|\le 1}}c_{I_1,I_2} \Im\left[\left\langle Q^\mathrm{kg}_0\left( v^{I_1}_\pm,\chi(t^{-\sigma}D_x) D_x u^{I_2}_\pm\right), v^I_+ + v^I_{-}\right\rangle\right]\\ & -\delta_{\mathcal{V}^k} \sum_{\substack{(J,0)\in\mathcal{I}(I)\\ J\in\mathcal{K}}}c_{J,0} \Im\left[\left\langle Q^\mathrm{kg}_0\left( v^J_\pm,\chi(t^{-\sigma}D_x) |D_x| u_\pm\right), v^I_+ + v^I_{-}\right\rangle\right] +R^k_3(t), \end{split}$$ with $\delta_{\mathcal{V}^k}=1$ if $I\in\mathcal{V}^k$, 0 otherwise, as already seen in . Also, $\langle \mathfrak{R}'(U,V), \widetilde{W}^I_s\rangle$ in the right hand side of and $R(t)$ are remainders $R^k_3(t)$ in consequence of the same argument used in the previous case, but with estimate replaced with . Therefore, we can further reduce to the following equality: $$\begin{split} \frac{1}{2}\partial_t \|\widetilde{W}^I_s(t,\cdot)\|^2_{L^2} =& -\Im\left[\langle Op^B(A''(V^I;\eta))U + Op^B_R(A''(V^I;\eta))U , \widetilde{W}^I\rangle + \langle T_{-18}(U)W^I, W^I\rangle\right]\\ & -\delta_{\mathcal{V}^k}\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ I_1\in\mathcal{K}, |I_2|\le 1}}c_{I_1,I_2} \Im\left[\left\langle Q^\mathrm{kg}_0\left( v^{I_1}_\pm,\chi(t^{-\sigma}D_x) D u^{I_2}_\pm\right), v^I_+ + v^I_{-}\right\rangle\right] \\ & -\delta_{\mathcal{V}^k} \sum_{\substack{(J,0)\in\mathcal{I}(I)\\ J\in\mathcal{K}}}c_{J,0} \Im\left[\left\langle Q^\mathrm{kg}_0\left( v^J_\pm,\chi(t^{-\sigma}D_x) |D_x| u_\pm\right), v^I_+ + v^I_{-}\right\rangle\right] +R^k_3(t), \end{split}$$ and deduce from definition , equalities , , with $N=18$, , together with , with $N=18$, and , that $$\left| \partial_t \widetilde{E}^{k,\dagger}_3(t;W) \right| \lesssim |R^k_3(t)| + \sum_{I\in\mathcal{I}^k_3}\left( \left|\mathfrak{D}^{I}_{\text{quart}}(t)\right| + \left|\mathfrak{D}^{I,18}_{\text{quart}}(t)\right| \right) + \delta_{k<2}\sum_{\substack{I\in\mathcal{V}^k\\ j_i\in \{+,-\}}}\sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ I_1\in\mathcal{K}, |I_2|\le 1}} \left|\mathfrak{G}^{I_1,I_2}_{(j_1,j_2,j_3)}\right|$$ with $\delta_{k<2}=1$ for $k<2$, 0 otherwise. On the one hand, quartic terms $\mathfrak{D}^{I}_{\text{quart}}, \mathfrak{D}^{I,18}_{\text{quart}}$ satisfy, respectively, and with $N=18$, and are remainders $R^k_3(t)$ after and a-priori estimates. On the other hand, $\mathfrak{G}^{I_1,I_2}_{(j_1,j_2,j_3)}$ verifies estimate . Consequently, there is a constant $C>0$ such that $$\begin{gathered} \widetilde{E}^{k,\dagger}_3(t;W)\le \widetilde{E}^{k,\dagger}_3(1;W) + C(A+B)^2\varepsilon^2 \int_1^t \tau^{-1+\frac{\delta_k}{2}}E^k_3(t\tau;W)^\frac{1}{2} d\tau \\ + \delta_{k<2}\, C(A+B)^2\varepsilon^2 \left[ \delta_{k=0}\int_1^t \tau^{-1+\frac{\delta_0}{2}+\beta+\frac{\delta_1}{2}}E^1_3(\tau;W)^\frac{1}{2}d\tau+\int_1^t \tau^{-\frac{5}{4}}d\tau \right]\end{gathered}$$ with $\delta_{k=0}=1$ if $k=0$, 0 otherwise, $\beta>0$ as small as we want, and after equivalence $$\begin{gathered} E^k_3(t;W)\le C E^k_3(1;W) + C(A+B)^2\varepsilon^2 \int_1^t \tau^{-1+\frac{\delta_k}{2}}E^k_3(\tau;W)^\frac{1}{2} d\tau \\ + \delta_{k<2}\, C(A+B)^2\varepsilon^2 \left[ \delta_{k=0}\int_1^t \tau^{-1+\frac{\delta_0}{2}+\beta+\frac{\delta_1}{2}}E^1_3(\tau;W)^\frac{1}{2}d\tau +\int_1^t \tau^{-\frac{5}{4}}d\tau \right] ,\end{gathered}$$ for a new $C>0$. Injecting in the above inequality and integrating in $d\tau$, we obtain that $$E^k_3(t;W)\le CE^k_3(1;W) +C(A+B)^2B\varepsilon^3\left[ \frac{1}{\delta_k} t^{\delta_k}+\delta_{k=0}\ \frac{1}{\frac{\delta_0}{2}+\beta+\delta_1}t^{\frac{\delta_0}{2}+\beta+\delta_1}\right],$$ and taking $\beta$ sufficiently small so that $\beta+ \delta_1\le\delta_0/2$, $B>1$ sufficiently large so that $E^k_3(1;W)\le \frac{B^2\varepsilon^2}{2CK_2^2}$ and $B\ge A$, and $\varepsilon_0>0$ sufficiently small so that $$\varepsilon_0 \le \frac{1}{8BCK^2_2}\Big[\frac{1}{\delta_k}+\delta_{k=0}\frac{1}{\frac{\delta_0}{2}+\beta+\delta_1}\Big]^{-1},$$ we finally derive enhanced estimate and the conclusion of the proof. Uniform Estimates ================= Semilinear normal forms {#Section : Normal Forms for system} ----------------------- In proposition \[Prop: Propagation of the energy estimate\] of the previous chapter we proved the propagation of the a-priori the energy estimates, i.e. that there exist some constants $A,B>1$ large and $\varepsilon_0>0$ small, such that implies , . To conclude the proof of theorem \[Thm: bootstrap argument\] it only remains to show that also implies , . In particular, as $u_+ = -\overline{u_{-}}$ and $v_+ =-\overline{u_{-}}$, it will be enough to prove this result for $(u_{-},v_{-})$, which is solution to $$\label{wave-KG for u- v-} \begin{cases} & \left(D_t + |D_x|\right) u_{-} = Q_0^\mathrm{w}(v_\pm, D_1 v_\pm), \\ & \left(D_t + \langle D_x\rangle\right) v_{-} = Q_0^{\mathrm{kg}}(v_\pm, D_1 u_\pm), \end{cases}$$ with $Q_0^\mathrm{w}(v_\pm, D_1 v_\pm), Q_0^{\mathrm{kg}}(v_\pm, D_1 u_\pm)$ given by . As for the simpler case of the one-dimensional Klein-Gordon equation (see [@stingo:1D_KG]), the main idea is to reformulate system in terms of two new functions $\widetilde{u}, \widetilde{v}$, defined from $u_{-}, v_{-}$ and living in a new framework (the *semi-classical framework*), and to deduce a new simpler system, made of a transport equation and an ODE. Through this new system we will be able to recover the required enhanced estimates , . Before introducing the semi-classical framework in which we will work for the rest of the paper, we need to replace almost all quadratic non-linearities in with cubic ones by a normal forms. This is the object of the following two subsections. We highlight the fact that we do not make use directly of the normal forms obtained in the proof of the energy inequality, because our goals and constraints are henceforth different. In fact, we want to obtain a $L^\infty$ estimate for essentially $\rho$ derivatives of our solution, having a control on its $H^s$ norm for $s\gg \rho$. Therefore, we are allowed to lose some derivatives in the normal form reduction, which means that we do not care any more about the quasi-linear nature of our problem. We warn the reader that, for seek of compactness, we will often use the notation $\Nlw$ (resp. $\Nlkg$) when referring to $Q^\mathrm{w}_0(v_\pm, D_1 v_\pm)$ (resp. to $Q^\mathrm{kg}_0(v_\pm, D_1 u_\pm)$). ### Normal forms for the Klein-Gordon equation The aim of this subsection is to introduce a new unknown $\vnf$, defined in terms of $v_{-}$, in such a way it is solution to a cubic half Klein-Gordon equation instead of the quadratic one satisfied by $v_{-}$ in . This normal form is motivated by the fact that the $L^2$ norm of $Q^\mathrm{kg}_0(v_\pm, D_1u_\pm)$ decays too slowly in time (only $t^{-1+\delta/2}$), as follows from and a-priori estimates , , and this decay is not enough in view of proposition \[Prop:propagation\_unif\_est\_V\] (the required one being strictly faster than $t^{-3/2}$). It is fundamental to observe that, after and inequality below with $\theta\ll 1$ small enough (e.g. $\theta<(2+\delta)^{-1}$), $\vnf$ and $v_{-}$ are comparable, in the sense that there is a positive constant $C$ such that $$\label{equivalence v- vnf} \left| \|v_{-}(t,\cdot)\|_{H^{\rho,\infty}} - \|\vnf(t,\cdot)\|_{H^{\rho,\infty}}\right| \le C\varepsilon^2 t^{-1}.$$ Then bootstrap assumption implies that the new unknown $\vnf$ disperses in time at the same rate $t^{-1}$ as $v_{-}$, and the propagation of a suitable estimate of the $H^{\rho,\infty}$ norm of $\vnf$ will provide us with enhanced . \[Prop: normal forms on KG\] Assume that $(u,v)$ is solution to in $[1,T]$ for a fixed $T>1$, consider $(u_+, v_+, u_{-}, v_{-})$ defined in and solution to with $|I|=0$, and remind definition of vectors $U,V$. Let $$\label{def vNF} v^{NF}:= v_{-} - \frac{i}{4(2\pi)^2}\sum_{j_1,j_2\in\{+,-\}} \int e^{ix\cdot\xi} B^1_{(j_1,j_2,+)}(\xi,\eta) \hat{v}_{j_1}(\xi-\eta) \hat{u}_{j_2}(\eta) d\xi d\eta,$$ with $B^1_{(j_1,j_2,+)}(\xi,\eta)$ given by with $k=1$ and $j_3=+$. Then for every $t\in [1,T]$ $v^{NF}$ is solution to $$\label{KG equation vNF} \left(D_t + \langle D_x\rangle\right) v^{NF}(t,x) = r^{NF}_{kg}(t,x),$$ where $$\begin{gathered} \label{def rNF-kg} r^{NF}_{kg}(t,x) = -\frac{i}{4(2\pi)^2}\sum_{j_1,j_2\in\{+,-\}} \int e^{ix\cdot\xi} B^1_{(j_1,j_2,+)}(\xi,\eta) \\ \times \left[\widehat{\textit{NL}_{kg}}(\xi - \eta) \hat{u}_{j_2}(\eta)+ \hat{v}_{j_1}(\xi-\eta)\widehat{\textit{NL}_{w}}(\eta) \right]d\xi d\eta\end{gathered}$$ satisfies \[est L2 Linfty (cut-off) rNF-kg-new\] $$\label{est_rnfkg_L2} \begin{split} \|r^{NF}_{kg}(t,\cdot)\|_{L^2}& \lesssim \sum_{\mu=0}^1 \|V(t,\cdot)\|_{H^{1,\infty}}\|\mathrm{R}^\mu_1 U(t,\cdot)\|_{L^\infty}\|U(t,\cdot)\|_{H^1} + \|V(t,\cdot)\|^2_{H^{2,\infty}}\|V(t,\cdot)\|_{H^2}, \end{split}$$$$\label{est_rnfkg_Linfty} \begin{split} & \|\chi(t^{-\sigma}D_x)r^{NF}_{kg}(t,\cdot)\|_{L^\infty} \lesssim \|V(t,\cdot)\|_{H^{1,\infty}}\Big(\sum_{\mu=0}^1\|\mathrm{R}^\mu_1 U(t,\cdot)\|_{H^{2,\infty}}\Big)^2 + t^\sigma\|V(t,\cdot)\|^3_{H^{2,\infty}}, \end{split}$$ for any $\chi\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$. Moreover, for every $s,\rho\ge 0$, any $\theta \in ]0,1[$, \[est\_vNF-v-\] $$\label{est_Hs_vnf-v} \left\| (\vnf- v_{-})(t,\cdot)\right\|_{H^s}\lesssim \sum_{\mu=0}^1\|V(t,\cdot)\|_{H^s} \|\mathrm{R}_1^\mu U(t,\cdot)\|_{L^\infty} + \|V(t,\cdot)\|_{L^\infty}\|U(t,\cdot)\|_{H^{s+1}},$$ $$\label{est_Hsinfty_vnf-v} \begin{split} \left\| (\vnf- v_{-})(t,\cdot)\right\|_{H^{s, \infty}}& \lesssim \sum_{\mu=0}^1\|V(t,\cdot)\|^{1-\theta}_{H^{s,\infty}}\|V(t,\cdot)\|^\theta_{H^{s+2}}\|\mathrm{R}^\mu_1U(t,\cdot)\|_{L^\infty}\\ & +\sum_{\mu=0}^1 \|V(t,\cdot)\|_{L^\infty}\|\mathrm{R}^\mu_1 U(t,\cdot)\|^{1-\theta}_{H^{s+1,\infty}}\|U(t,\cdot)\|^\theta_{H^{s+3}}, \end{split}$$ $$\label{est_Omega_vnf-v} \begin{split} \left\| \Omega(\vnf- v_{-})(t,\cdot)\right\|_{L^2}&\lesssim \sum_{\mu,\nu=0}^1 \left[\|\Omega^\mu V(t,\cdot)\|_{L^2}\|\mathrm{R}^\nu_1U(t,\cdot)\|_{L^\infty} + \|V(t,\cdot)\|_{L^\infty}\|\Omega^\mu U(t,\cdot)\|_{H^1}\right]\\ & + \|\Omega V(t,\cdot)\|_{H^2}\|U(t,\cdot)\|_{H^1} + \|V(t,\cdot)\|_{L^2}\|\Omega U(t,\cdot)\|_{H^2}, \end{split}$$ and $$\label{est_chi_vnf-v} \left\|\chi(t^{-\sigma}D_x)(\vnf - v_{-})(t,\cdot)\right\|_{L^2}\lesssim t^\sigma \|V(t,\cdot)\|_{H^{1,\infty}}\|U(t,\cdot)\|_{L^2},$$ $$\begin{gathered} \label{est_chi_Omega_vnf-v} \left\|\chi(t^{-\sigma}D_x)\Omega(\vnf - v_{-})(t,\cdot)\right\|_{L^2}\\ \lesssim t^\sigma \left[\sum_{\mu=0}^1 \|\Omega V(t,\cdot)\|_{L^2}\|\mathrm{R}^\mu_1 U(t,\cdot)\|_{L^\infty} + \|V(t,\cdot)\|_{H^{1,\infty}}\|\Omega^\mu U(t,\cdot)\|_{L^2}\right].\end{gathered}$$ From definition of $v^{NF}$, system with $|I|=0$, and the fact that $$Q^{\mathrm{kg}}_0(v_\pm, D_1u_\pm) = \frac{i}{4(2\pi)^2}\sum_{j_1,j_2\in \{+,-\}}\int e^{ix\cdot\xi}\Big(1-j_1j_2 \frac{\xi-\eta}{\langle \xi-\eta\rangle}\cdot\frac{\eta}{|\eta|}\Big)\eta_1 \hat{v}_{j_1}(\xi-\eta) \hat{u}_{j_2}(\eta) d\xi d\eta,$$ it immediately follows that $v^{NF}$ is solution to with $r^{NF}_{kg}$ given by . We observe that, after formula , we have the following explicit expressions: $$\label{explicit_vNF-v_chapter5} \begin{split} v^{NF}-v_{-} =-\frac{i}{8}&\Big[(v_+ +v_{-})\mathrm{R}_1(u_+ - u_{-}) - \frac{D_1}{\langle D_x\rangle}(v_+ - v_{-}) (u_+ +u_{-}) \\ &+ D_1\big[[\langle D_x\rangle^{-1}(v_+-v_{-})](u_+ + u_{-})\big] - \langle D_x\rangle \big[[\langle D_x\rangle^{-1}(v_+ -v_{-})]\mathrm{R}_1 (u_+ - u_{-})\big]\Big] \end{split}$$ and $$\label{explicit_rnfkg_chapter5} \rnfkg = -\frac{i}{4}\left[\Nlkg\, \mathrm{R}_1(u_+-u_{-}) - \frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\, \Nlw + D_1\big[\langle D_x\rangle^{-1}(v_+-v_{-})\, \Nlw\big] \right].$$ Inequalities , are straightforward from and corollary \[Cor\_appendixA:Hs-Hsinfty norm of bilinear expressions\] in appendix \[Appendix A\]. Inequality is also obtained from corollary \[Cor\_appendixA:Hs-Hsinfty norm of bilinear expressions\], after having applied $\Omega$ to and used the Leibniz rule, and from bounding the $L^\infty$ norm of $\Omega u_\pm, \Omega v_\pm$ with their $H^2$ norm by means of the classical Sobolev injection. Inequalities , are also straightforward if one observes that operator $\chi(t^{-\sigma}D_x)$, with $\chi\in C^\infty_0(\mathbb{R}^2)$ and $\sigma>0$, is $L^2-H^1$ continuous with norm $O(t^\sigma)$. As concerns $\rnfkg$, from and corollary \[Cor\_appendixA:Hs-Hsinfty norm of bilinear expressions\] we find that $$\begin{split} \|r^{NF}_{kg}(t,\cdot)\|_{L^2} &\lesssim \sum_{\mu=0}^1\|\textit{NL}_{kg}(t,\cdot)\|_{L^2}\|\mathrm{R}^\mu_1U(t,\cdot)\|_{L^\infty}+ \|V(t,\cdot)\|_{L^2}\|\Nlw(t,\cdot)\|_{L^\infty}\\ & + \|V(t,\cdot)\|_{L^\infty}\|\textit{NL}_w(t,\cdot)\|_{H^1} \end{split}$$ and $$\|\chi(t^{-\sigma}D_x) r^{NF}_{kg}(t,\cdot)\|_{L^\infty}\lesssim \sum_{\mu=0}^1\|\textit{NL}_{kg}(t,\cdot)\|_{L^\infty}\|\mathrm{R}^\mu_1U(t,\cdot)\|_{L^\infty} + t^\sigma \|V(t,\cdot)\|_{H^{1,\infty}}\|\Nlw(t,\cdot)\|_{L^\infty}.$$ Inequalities and follow then by with $s=1$, , and . ### Normal forms for the wave equation {#Subsection: Section : Normal Forms for the Wave Equation} We now focus on the wave equation satisfied by $u_{-}$: $$(D_t + |D_x| )u_{-}(t,x) = Q^{\mathrm{w}}_0(v_\pm, D_1 v_\pm),$$ and perform a normal form argument in order to replace (a part of) the quadratic non-linearity in the above right hand side with a cubic non-local one. The fundamental reason for that is to be found in lemma \[Lemma : estimate of e(x,xi)\], where we have to prove that the $L^2$ norm of some operator, acting on the non-linearity of the equation satisfied by $u_{-}$, decays like $t^{-1/2+\beta}$, for a small $\beta>0$. That decay is obtained if the $L^2$ norm of the mentioned non-linearity is a $O(t^{-3/2+\beta'})$, for some new small $\beta'>0$, which is not the case for $Q^{\mathrm{w}}_0(v_\pm, D_1 v_\pm)$, as follows from , , . This normal form can be actually performed only on contributions depending on $(v_+, v_+)$ and $(v_{-}, v_{-})$ but not on $(v_+,v_{-})$, which are resonant. Nevertheless, the structure of these latter contributions allows us to recover the right mentioned time decay for their $L^2$ norm (see lemmas \[Lem: hD|V|2\] and \[Lem: (xi+dphi)Op(gamma)\]). Thanks to inequalities , and a-priori estimates , the new unknown $\unf$ we define in below is equivalent to the former $u_{-}$, meaning that there exists a positive constant $C$ such that $$\label{equivalence u- unf} \sum_{\kappa=0}^1 \left| \|\mathrm{R}^\kappa_1 u_{-}(t,\cdot)\|_{H^{\rho+1,\infty}} - \|\mathrm{R}^\kappa_1\unf(t,\cdot)\|_{H^{\rho,\infty}} \right| \le C\varepsilon^2 t^{-1+\frac{\delta}{2}}.$$ After this means that $\unf$ and $\mathrm{R}_1\unf$ decay in the $H^{\rho+1,\infty}$ norm at the same rate $t^{-1/2}$ as $u_{-},\mathrm{R}_1u_{-}$, and the propagation of a suitable estimate of this norm will provide us with enhanced . Let us rewrite $Q^{\mathrm{w}}_0(v_\pm, D_1 v_\pm)$ as follows $$\label{decomposition Q0w} \begin{split} Q_0^{\mathrm{w}}(v_\pm, D_1 v_\pm) &= -\frac{1}{2}\Im\left[v_+\, D_1v_{-} +\frac{D_x}{\langle D_x\rangle}v_+ \cdot \frac{D_xD_1}{\langle D_x\rangle}v_{-}\right]\\ & + \frac{i}{4(2\pi)^2} \sum_{j\in\{+,-\}}\int e^{ix\cdot\xi}\left(1-\frac{\xi-\eta}{\langle\xi-\eta\rangle}\cdot\frac{\eta}{\langle\eta\rangle}\right)\eta_1 \hat{v}_j(\xi-\eta)\hat{v}_j(\eta) d\xi d\eta, \end{split}$$ and introduce, for any $j\in\{+,-\}$, $$\label{def Dj1j2(xi,eta)} D_j(\xi,\eta):= \frac{\left(1-\frac{\xi-\eta}{\langle\xi-\eta\rangle}\cdot\frac{\eta}{\langle\eta\rangle}\right)\eta_1}{j\langle\xi-\eta\rangle + j\langle\eta\rangle + |\xi|}.$$ \[Prop: NF on wave\] Assume that $(u,v)$ is solution to in $[1,T]$ for a fixed $T>1$, consider $(u_+, v_+, u_{-}, v_{-})$ defined in and solution to with $|I|=0$, remind definition of vectors $U,V$ and of $\vnf$. Let $$\label{def uNF} u^{NF} := u_{-} - \frac{i}{4(2\pi)^2}\sum_{j\in\{+,-\}} \int e^{i x\cdot \xi}D_j(\xi,\eta)\hat{v}_j(\xi - \eta)\hat{v}_j(\eta)d\xi d\eta,$$ with multiplier $D_j$ defined in . For every $t\in [1, T]$ $u^{NF}$ is solution to $$\label{wave equation uNF} (D_t + |D_x|)u^{NF}(t,x) = q_w(t,x)+c_w(t,x)+ r^{NF}_w(t,x),$$ where quadratic term $q_w$ is given by $$\label{def_qw} q_w(t,x) = \frac{1}{2}\Im\left[\overline{\vnf}\, D_1\vnf - \overline{\frac{D_x}{\langle D_x\rangle}\vnf}\cdot\frac{D_xD_1}{\langle D_x\rangle}\vnf\right],$$ while cubic terms $c_w, r^{NF}_w$ are equal, respectively, to $$\label{def_cw} \begin{split} c_w(t,x)=\frac{1}{2}\Im& \left[\overline{(v_{-}-\vnf)}\, D_1v_{-} + \overline{\vnf}\, D_1(v_{-}-\vnf) \right.\\ &\left. - \overline{\frac{D_x}{\langle D_x\rangle}(v_{-}-\vnf)}\cdot\frac{D_xD_1}{\langle D_x\rangle}v_{-} - \overline{\frac{D_x}{\langle D_x\rangle}\vnf}\cdot \frac{D_xD_1}{\langle D_x\rangle}(v_{-} - \vnf)\right], \end{split}$$ and $$\label{def rNF} r^{NF}_w(t,x) = -\frac{i}{4(2\pi)^2}\sum_{j\in\{+,-\}}\int e^{ix\cdot \xi} D_j(\xi, \eta) \left[\widehat{\textit{NL}_{kg}}(\xi - \eta) \hat{v}_j(\eta)+ \hat{v}_j(\xi-\eta)\widehat{\textit{NL}_{kg}}(\eta) \right]d\xi d\eta.$$ For any $s,\rho\ge 0$, any $t\in [1,T]$, \[norms uNF - u-\] $$\label{Hs norm uNF- u-} \|u^{NF}(t,\cdot) - u_{-}(t,\cdot)\|_{H^s} \lesssim \|V(t,\cdot)\|_{L^\infty}\|V(t,\cdot)\|_{H^{s+15}},$$ $$\label{Hrho-infty norm uNF- u-} \|u^{NF}(t,\cdot) - u_{-}(t,\cdot)\|_{H^{\rho+1,\infty}} \lesssim \|V(t,\cdot)\|_{L^\infty}\|V(t,\cdot)\|_{H^{\rho+18}} ,$$ $$\label{Hrho-infty norm R(uNF-u-)} \|\mathrm{R}_j u^{NF}(t,\cdot) -\mathrm{R}_j u_{-}(t,\cdot)\|_{H^{\rho+1,\infty}} \lesssim \|V(t,\cdot)\|_{L^\infty}\|V(t,\cdot)\|_{H^{\rho+8}}, \quad j=1,2.$$ Moreover, for any cut-off function $\chi\in C^\infty_0(\mathbb{R}^2)$ and $\sigma>0$ there exists some $\chi_1\in C^\infty_0(\mathbb{R}^2)$ and $s>0$ such that \[est\_cw\] $$\label{est_cw_L2} \begin{split} \left\|\chi(t^{-\sigma}D_x)c_w(t,\cdot)\right\|_{L^2}&\lesssim t^\sigma \left\|\chi_1(t^{-\sigma}D_x)(\vnf-v_{-})(t,\cdot)\right\|_{L^2}\left(\|V(t,\cdot)\|_{H^{2,\infty}}+\|\vnf(t,\cdot)\|_{H^{1,\infty}}\right) \\ & + t^{-N(s)}\left\| (\vnf - v_{-})(t,\cdot)\right\|_{H^1}\left(\|V(t,\cdot)\|_{H^s}+\|\vnf(t,\cdot)\|_{H^s}\right), \end{split}$$ $$\label{est_cw_Linfty} \begin{split} \left\|\chi(t^{-\sigma}D_x)c_w(t,\cdot)\right\|_{L^\infty} & \lesssim t^\sigma \left\|\chi_1(t^{-\sigma}D_x)\left(\vnf-v_{-}\right)(t,\cdot)\right\|_{L^\infty}\left(\|V(t,\cdot)\|_{H^{2,\infty}}+\|\vnf(t,\cdot)\|_{H^{1,\infty}}\right) \\ & + t^{-N(s)}\left\| (\vnf - v_{-})(t,\cdot)\right\|_{H^1}\left(\|V(t,\cdot)\|_{H^s}+\|\vnf(t,\cdot)\|_{H^s}\right) \end{split}$$ $$\label{est_Omega_cw} \begin{split} \left\|\chi(t^{-\sigma}D_x)\Omega c_w(t,\cdot)\right\|_{L^2}&\lesssim t^\sigma \left\|\chi_1(t^{-\sigma}D_x)\Omega (\vnf - v_{-})(t,\cdot)\right\|_{L^2} \left(\|V(t,\cdot)\|_{H^{2,\infty}}+\|\vnf(t,\cdot)\|_{H^{1,\infty}}\right)\\ & + t^{-N(s)}\left\|\Omega (\vnf - v_{-})(t,\cdot)\right\|_{L^2} \left(\|V(t,\cdot)\|_{H^s}+\|\vnf(t,\cdot)\|_{H^s}\right)\\ & + t^\sigma \left\|(\vnf - v_{-})(t,\cdot)\right\|_{H^{1,\infty}}\times \sum_{\mu=0}^1\left(\|\Omega^\mu V(t,\cdot)\|_{H^1}+\|\Omega^\mu \vnf(t,\cdot)\|_{L^2}\right) \end{split}$$ with $N(s)>0$ as large as we want as long as $s>0$ is large, and \[est rNF\] $$\label{est L2 rNF} \|\chi(t^{-\sigma} D_x) r^{NF}_w(t,\cdot)\|_{L^2} \lesssim \|V(t,\cdot)\|^2_{H^{13,\infty}}\|U(t,\cdot)\|_{H^1},$$ $$\label{est Linfty rNF} \|\chi(t^{-\sigma} D_x) r^{NF}_w(t,\cdot)\|_{L^\infty}\lesssim \|V(t,\cdot)\|^2_{H^{13,\infty}}\left(\|U(t,\cdot)\|_{H^{2,\infty}}+\|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}\right),$$ and for any $\theta\in ]0,1[$, $$\label{est phi(D) Omega rNF-new} \begin{split} \|\chi(t^{-\sigma} D_x)& \Omega r^{NF}_w(t,\cdot)\|_{L^2} \lesssim t^\beta\Big[ \|V(t,\cdot)\|^{1-\theta}_{H^{15,\infty}} \|V(t,\cdot)\|^\theta_{H^{17}}\left(\| U(t,\cdot)\|_{H^{1,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|_{H^{1,\infty}}\right)\\ &+ \|V(t,\cdot)\|_{L^\infty}\left(\| U(t,\cdot)\|^{1-\theta}_{H^{16,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|^{1-\theta}_{H^{16,\infty}}\right)\|U(t,\cdot)\|^\theta_{H^{18}}\Big]\|\Omega V(t,\cdot)\|_{L^2}\\ &+ t^\beta \Big[\|V(t,\cdot)\|_{H^{1,\infty}}\left(\|U(t,\cdot)\|_{H^1} + \|\Omega U(t,\cdot)\|_{H^1}\right)\\ & + \left(\|U(t,\cdot)\|_{H^{2,\infty}}+\| \mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}\right)\left(\|V(t,\cdot)\|_{L^2}+\|\Omega V(t,\cdot)\|_{L^2}\right)\Big] \|V(t,\cdot)\|_{H^{17,\infty}}. \end{split}$$ By definition of $u^{NF}$, system with $|I|=0$, and , it follows that $u^{NF}$ is solution to $$(D_t + |D_x|)u^{NF}(t,x) =-\frac{1}{2}\Im\left[v_+\, D_1v_{-} +\frac{D_x}{\langle D_x\rangle}v_+ \cdot \frac{D_xD_1}{\langle D_x\rangle}v_{-}\right]+ r^{NF}_w(t,x),$$ with $r^{NF}_w$ given by . Reminding that $v_+=-\overline{v_{-}}$ and replacing each occurrence of $v_{-}$ in the quadratic contribution to the above right hand side, we find that $\unf$ is solution to . The first part of lemma \[Lem\_Appendix: est on Dj1j2\] and the fact that any $H^{\rho+1,\infty}$ injects into $H^{\rho+3}$ by Sobolev inequality immediately imply estimates and $$\|\chi(t^{-\sigma} D_x) r^{NF}_w(t,\cdot)\|_{L^2}\lesssim \|\textit{NL}_{kg}(t,\cdot)\|_{L^2}\|V(t,\cdot)\|_{H^{13,\infty}},$$ $$\|\chi(t^{-\sigma} D_x) r^{NF}_w(t,\cdot)\|_{L^\infty}\lesssim \|\textit{NL}_{kg}(t,\cdot)\|_{L^\infty}\|V(t,\cdot)\|_{H^{13,\infty}},$$ for any $s,\rho\ge 0$. Moreover, from we derive that $$\begin{gathered} \|\chi(t^{-\sigma} D_x) \Omega r^{NF}_w(t,\cdot)\|_{L^2} \lesssim t^\beta \left(\|\textit{NL}_{kg}(t,\cdot)\|_{L^2}+ \|\Omega \textit{NL}_{kg}(t,\cdot)\|_{L^2}\right)\|V(t,\cdot)\|_{H^{17,\infty}} \\ + t^\beta \|\textit{NL}_{kg}(t,\cdot)\|_{H^{15,\infty}}\|\Omega V(t,\cdot)\|_{L^2},\end{gathered}$$ so estimates are obtained using , with $s=15$, and . Finally, inequality (resp. ) is obtained using lemma \[Lem\_appendix:L\_estimate of products\] in appendix \[Appendix B\] with $L=L^2$ (resp. $L=L^\infty$), $w=v_{-}-\vnf$, and the fact that $\chi_1(t^{-\sigma}D_x)$ is continuous from $L^2$ to $H^1$ (resp. from $L^\infty$ to $H^{1,\infty}$) with norm $O(t^\sigma)$. Inequality is deduced applying $\Omega$ to and using the Leibniz rule. The $L^2$ norm of products in which $\Omega$ is acting on $v_{-} -\vnf$ is estimated by means of lemma \[Lem\_appendix:L\_estimate of products\] with $L=L^2$, $w=v_{-}-\vnf$, whereas the $L^2$ norm of the remaining products is simply estimated by taking the $L^\infty$ norm on $v_{-} - \vnf$ times the $L^2$ norm of the remaining factor. From PDEs to ODEs {#Sec: development of the PDE system} ----------------- In the previous section we showed that, if $(u_{-},v_{-})$ is solution to system in some interval $[1,T]$, for a fixed $T>1$, one can define two new functions, $v^{NF}$ as in and $u^{NF}$ as in , respectively comparable to $v_{-}$ and $u_{-}$ in the sense of and , such that $(u^{NF}, v^{NF})$ is solution to a new wave-Klein-Gordon system: $$\label{wave_KG NF system} \begin{cases} \left(D_t + |D_x|\right)u^{NF}(t,x) =q_w(t,x)+c_w(t,x)+ r^{NF}_w(t,x),\\ \left(D_t + \langle D_x\rangle\right)v^{NF}(t,x) = r^{NF}_{kg}(t,x), \end{cases}$$ for every $(t,x)\in [1,T]\times\mathbb{R}^2$, where quadratic inhomogeneous term $q_w$ is given by and cubic ones $c_w$, $r^{NF}_w$ and $r^{NF}_{kg}$ respectively by , and . As anticipated before, our aim is to deduce from a system made of a transport equation and an ODE, from which it will be possible to deduce suitable estimates on $(u^{NF},v^{NF})$ (and consequently on $(u_{-},v_{-})$). Thanks to and these estimates will allow us to close the bootstrap argument and prove theorem \[Thm: bootstrap argument\]. In subsection \[Subsection : The Derivation of the ODE Equation\] we focus on the deduction of the mentioned ODE starting from the Klein-Gordon equation satisfied by $v^{NF}$, while in subsection \[Subsection : The Derivation of the Transport Equation\] we show how to derive a transport equation from the wave equation satisfied by $u^{NF}$. The framework in which this plan takes place is the *semi-classical framework*, introduced below. Let us introduce the *semi-classical parameter* $h:=t^{-1}$ together with the following two new functions: $$\label{def utilde vtilde} \widetilde{u}(t,x) := t u^{NF}(t,tx), \qquad \widetilde{v}(t,x) := t v^{NF}(t,tx),$$ and observe that, from definition and inequalities , , a-priori estimates , are equivalent respectively to $$\begin{gathered} \|\widetilde{u}(t,\cdot)\|_{H^{\rho+1,\infty}_h} + \left\|\oph(\xi|\xi|^{-1})\widetilde{u}(t,\cdot)\right\|_{H^{\rho+1,\infty}_h}\le C\varepsilon h^{-\frac{1}{2}}, \label{est:a-priori_ut}\\ \|\vt(t,\cdot)\|_{H^{\rho,\infty}_h}\le C\varepsilon,\label{est:a-priori_vt}\end{gathered}$$ for some positive constant $C$. A suitable propagation of the above estimates will therefore provide us with and . A straight computation shows that $(\widetilde{u}, \widetilde{v})$ satisfies the following coupled system of semi-classical pseudo-differential equations: $$\label{wave-KG system in semi-classical coordinates} \begin{cases} & \big[D_t - \oph(x\cdot\xi - |\xi|)\big]\widetilde{u}(t,x) = h^{-1}\left[q_w(t,tx)+c_w(t,tx)+r^{NF}_w(t, tx) \right]\\ & \big[D_t - \oph(x\cdot\xi - \langle\xi\rangle)\big]\widetilde{v}(t,x) = h^{-1}r^{NF}_{kg}(t,tx), \end{cases}$$ where $\oph$ denotes the semi-classical Weyl quantization introduced in \[Def: Weyl and standard quantization\] $(i)$. Moreover, if $\mathcal{M}_j$ (resp. $\mathcal{L}_j$), $j=1,2$, is the operator introduced in (resp. ), $\mathcal{M}_j\widetilde{u}$ (resp. $\mathcal{L}_j\widetilde{v}$) can be expressed in term of $Z_ju^{NF}$ (resp. $Z_j v^{NF}$). We have the following general result: \[Lem: relation between Z and M/L\] $(i)$ Let $w(t,x)$ be a solution to the inhomogeneous half wave equation $$\label{half-wave-w} \left[D_t + |D_x| \right] w(t,x) = f(t,x),$$ and $\widetilde{w}(t,x)=tw(t,tx)$. For any $j=1,2$, $$\label{relation_Zjw_Mjwidetilde(w)} Z_jw(t,y) = ih \left[-\mathcal{M}_j\widetilde{w}(t,x) + \frac{1}{2i}\oph\left(\frac{\xi_j}{|\xi|}\right)\widetilde{w}(t,x) \right]|_{x=\frac{y}{t}} + iy_jf(t,y);$$ $(ii)$ If $w(t,x)$ is solution to the inhomogeneous half Klein-Gordon equation $$\label{half KG} \left[D_t + \langle D_x\rangle \right] w(t,x) = f(t,x),$$ then $$\label{relation_Zjw_Ljwidetilde(w)} Z_j w(t,y) = ih\left[-\oph(\langle\xi\rangle)\mathcal{L}_j\widetilde{w}(t,x)+\frac{1}{i}\oph\Big(\frac{\xi_j}{\langle\xi\rangle}\Big)\widetilde{w}(t,x)\right]|_{x=\frac{y}{t}}+ iy_jf(t,y).$$ $(i)$ If $w$ is solution to half wave equation then $\widetilde{w}(t,x)$ satisfies $$\big[D_t - \oph(x\cdot\xi - |\xi|)\big]\widetilde{w}(t,x) = h^{-1}f(t, tx),$$ so, for any $i=1,2$, $$\begin{split} &Z_j w(t,y) =\\ & i h^{-1}\left[x_j D_t + \oph(\xi_j - x_j x\cdot\xi) + \frac{3h}{2i}x_j\right]\left(\frac{1}{t}\widetilde{w}(t,x)\right)\Big|_{x=\frac{y}{t}} \\ & = i \left[ x_j D_t + \oph(\xi_j - x_jx\cdot\xi) + \frac{h}{2i}x_j\right]\widetilde{w}(t,x)\Big|_{x=\frac{y}{t}} \\ & = i \left[x_j \oph(x\cdot\xi - |\xi|)\widetilde{w}(t,x) + \oph(\xi_j - x_j x\cdot \xi)\widetilde{w}(t,x) + \frac{h}{2i}x_j\widetilde{u}(t,x) + h^{-1}x_j f(t,tx)\right]\big|_{x=\frac{y}{t}} \\ & = ih \left[-\mathcal{M}_j\widetilde{w}(t,x) + \frac{1}{2i}\oph\left(\frac{\xi_j}{|\xi|}\right)\widetilde{w}(t,x)\right]|_{x=\frac{y}{t}}+ iy_j f(t,y). \end{split}$$ We should specify that last equality is obtained by a trivial version of symbolic calculus , that applies also to symbols $b(\xi)$ singular at $\xi=0$. Indeed, if symbol $a=a(x,\xi)$ is linear in $x$, and $b(\xi)$ is lipschitz, the development $a\sharp b$ is actually finite: $$a\sharp b(x,\xi) = a(x,\xi)b(\xi) - \frac{h}{2i}\partial_x a(x,\xi)\cdot\partial_\xi b(\xi).$$ $(ii)$ The result is analogous to the previous one, after observing that $\widetilde{w}$ satisfies $$\big[D_t - \oph(x\cdot\xi - \langle \xi\rangle)\big]\widetilde{w}(t,x) = h^{-1}f(t, tx).$$ As a straight consequence of lemma \[Lem: relation between Z and M/L\] and system we have that $$\label{relation between Zju and Mj utilde-new} Z_j u^{NF}(t,y) = ih \left[-\mathcal{M}_j\widetilde{u}(t,x) + \frac{1}{2i}\oph\left(\frac{\xi_j}{|\xi|}\right)\widetilde{u}(t,x)\right]|_{x=\frac{y}{t}} + iy_j\left[q_w+c_w+r^{NF}_w\right](t,y),$$ $$\label{relation between Zjv and Lj vtilde} Z_jv^{NF}(t,y) = ih \left[ - \oph(\langle\xi\rangle)\mathcal{L}_j\widetilde{v}(t,x) + \frac{1}{i}\oph\Big(\frac{\xi_j}{\langle\xi\rangle}\Big)\widetilde{v}(t,x) \right]\Big|_{x=\frac{y}{t}} + iy_jr^{NF}_{kg}(t,y).$$ In view of lemma \[Lemma : estimate of e(x,xi)\], it is also useful to write down the analogous relation between $(Z_m Z_nu)_{-}$ and $\mathcal{M}[t(Z_nu)_{-}(t,tx)]$. As $(Z_nu)_{-}$ is solution to $$\big(D_t + |D_x|\big)(Z_nu)_{-} = Z_n\Nlw(t,x),$$ from equality with $w=(Z_nu)_{-}$ and the commutation between $Z_m$ and $D_t-|D_x|$ (see ) we find that $$\begin{gathered} \label{relation ZmZnu Mutilde Zn-new} (Z_mZ_nu)_{-}(t,y) = ih\Big[-\mathcal{M}_m\widetilde{u}^J(t,x) + \frac{1}{2i}\oph\Big(\frac{\xi_m}{|\xi|}\Big)\widetilde{u}^J(t,x)\Big]\big|_{x=\frac{y}{t}} + iy_mZ_n \Nlw(t,y) \\ -\frac{D_m}{|D_y|}(Z_nu)_{-}(t,y),\end{gathered}$$ where $J$ is the index such that $\Gamma^J=Z_n$ and $\widetilde{u}^J(t,x):= t(Z_nu)_{-}(t,tx)$. Also, observe that from , , and $$Z_n\Nlw = Q^{\mathrm{w}}_0\big((Z_nv)_\pm, D_1v_\pm\big) + Q^{\mathrm{w}}_0\big(v_\pm, D_1(Z_nv)_\pm\big) - \delta_n^1 Q^{\mathrm{w}}_0(v_\pm, D_tv_\pm)$$ with $\delta_n^1=1$ for $n=1$, and that from inequality with $s=0$, $$\begin{gathered} \label{L2 est NLwZn} \|Z_n \Nlw(t,\cdot)\|_{L^2}\lesssim \|Z_nV(t,\cdot)\|_{H^1}\|V(t,\cdot)\|_{H^{2,\infty}}+ \big[\|V(t,\cdot)\|_{H^1} \\ + \|V(t,\cdot)\|_{L^2}\left(\|U(t,\cdot)\|_{H^{1,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|_{H^{1,\infty}}\right) + \|V(t,\cdot)\|_{L^\infty}\|U(t,\cdot)\|_{H^1}\big]\|V(t,\cdot)\|_{H^{1,\infty}}.\end{gathered}$$ Moreover, from the definition of $\Mcal_j$ and $\Lcal_j$ we see that $$\begin{gathered} h\mathcal{M}_j\widetilde{w}(t,x) = \left[y_j|D_y| - tD_j + \frac{1}{2i}\frac{D_j}{|D_y|}\right]w(t,y)|_{y=tx},\\ h\oph(\langle \xi\rangle)\mathcal{L}_j \widetilde{w}(t,x) = \left[y_j\langle D_y\rangle - tD_j -i \frac{D_j}{\langle D_y\rangle}\right]w(t,y)|_{y=tx},\end{gathered}$$ so lemma \[Lem: relation between Z and M/L\] implies that, if $w$ is solution to half wave equation (resp. to half Klein-Gordon ), \[relation\_w\_Zjw\] $$\begin{gathered} \left[y_j|D_y| - tD_j + \frac{1}{2i}\frac{D_j}{|D_y|}\right]w(t,y)= iZ_jw(t,y)+ \frac{1}{2i}\frac{D_j}{|D_y|}w(t,y)+ y_jf(t,y),\label{relation_w_wave_Zjw-new} \\ \left(\text{resp. }\left[\langle D_y\rangle y_j - tD_j \right]w(t,y) =iZ_jw(t,y)-i \frac{D_j}{\langle D_y\rangle}w(t,y)+y_jf(t,y)\right). \label{relation_w_KG_Zjw}\end{gathered}$$ ### Derivation of the ODE and propagation of the uniform estimate on the Klein-Gordon component {#Subsection : The Derivation of the ODE Equation} Let us firstly deal with the semi-classical Klein-Gordon equation satisfied by $\widetilde{v}$: $$\label{semi-classical KG equation} \big[D_t - \oph(x\cdot\xi - p(\xi))\big]\widetilde{v}(t,x) = h^{-1}r^{NF}_{kg}(t,tx),$$ where $p(\xi) = \langle\xi\rangle$ and $r^{NF}_{kg}$ is given by and satisfies . We remind that $p'(\xi)$ denotes the gradient of $p(\xi)$ while $p''(\xi)$ is its $2\times 2$ Hessian matrix, and that $\Lcal_j$ is the operator introduced in for $j=1,2$. We also remind definition of manifold $\Lkg$, represented in dimension 1 by picture \[picture: Lkg\] below, and decompose $\vt$ into the sum of two contributions: one localized in a neighbourhood of $\Lkg$ of size $\sqrt{h}$ (in the spirit of [@ifrim_tataru:global_bounds]), the other localized out of this neighbourhood. (0,-1.7) – (0,1.7); (-2,0) – (2,0); at (1.9,0) [$x$]{}; at (0,1.6) [$\xi$]{}; (-1,-1.7) – (-1,1.7); (1,-1.7) – (1,1.7); at (-1,0) [$-1$]{}; at (1,0) [$1$]{}; plot(, [/((1-\^2)\^(1/2))]{}); plot(-, [-/((1-\^2)\^(1/2))]{}); at (-0.8,-1.6) [$\Lkg$]{}; The contribution localized away from $\Lkg$ appears to be a $O(h^{1/2-0})$ if we assume a moderate growth for the $L^2$ norm of $\mathcal{L}^\mu\vt$, with $0\le |\mu|\le 2$, and has hence a better decay in time than the one expected for $\vt$ (remind $h=t^{-1}$). Thus the main contribution to $\vt$ is the one localized around $\Lkg$. We are going to show that this latter one is solution to an ODE (see proposition \[Prop: ODE for vtilde Sigma\_Lambda\]) and that its $H^{\rho,\infty}_h$ norm is uniformly bounded in time, which will finally enable us to propagate and obtain (see proposition \[Prop:propagation\_unif\_est\_V\]). For any fixed $\rho\in\mathbb{Z}$ let $\Sigma(\xi):=\langle\xi\rangle^{\rho}$, and for some $\gamma, \chi\in C^\infty_0(\mathbb{R}^2)$ equal to 1 close to the origin, $\sigma>0$ small (e.g. $\sigma<\frac{1}{4}$) let $$\label{def Gamma_kg} \Gamma^{kg} := \oph\left(\gamma\left(\frac{x - p'(\xi)}{\sqrt{h}}\right)\chi(h^\sigma\xi)\right).$$ We also introduce the following notations: $$\widetilde{v}^\Sigma:= \oph(\Sigma(\xi))\widetilde{v},$$ together with \[def\_both\_vtilde\_Sigma\_Lambda\] $$\label{def vtilde_Sigma_Lambda} \widetilde{v}^\Sigma_{\Lambda_{kg}} := \Gamma^{kg}\widetilde{v}^\Sigma,$$ $$\label{def vtilde_Lambda,c} \widetilde{v}^\Sigma_{\Lambda^c_{kg}}:= \oph\left(1 -\gamma\left(\frac{x-p'(\xi)}{\sqrt{h}}\right)\chi(h^\sigma\xi)\right)\widetilde{v}^\Sigma,$$ so that $\widetilde{v}^\Sigma = \widetilde{v}^\Sigma_{\Lambda_{kg}} + \widetilde{v}^\Sigma_{\Lambda^c_{kg}}$, and remind that $\|\mathcal{L}^\gamma w\| = \|\mathcal{L}^{\gamma_1}_1\mathcal{L}^{\gamma_2}_2w\|$, for any $\gamma=(\gamma_1,\gamma_2)\in\mathbb{N}^2$. \[Lem:Op(gammatilde)v\] Let $\widetilde{\gamma}\in C^\infty(\mathbb{R}^2)$ vanish in a neighbourhood of the origin and be such that $|\partial^\alpha_z\widetilde{\gamma}(z)|\lesssim \langle z\rangle^{-|\alpha|}$. Let $c(x,\xi)\in S_{\delta,\sigma}(1)$ with $\delta\in [0,\frac{1}{2}]$, $\sigma>0$, be supported for $|\xi|\lesssim h^{-\sigma}$. For any $\chi\in C^\infty_0(\mathbb{R}^2)$ such that $\chi(h^\sigma\xi)\equiv 1$ on the support of $c(x,\xi)$, \[est\_1L\] $$\begin{gathered} \left\| \oph\Big(\widetilde{\gamma}\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)c(x,\xi)\Big)w\right\|_{L^2} \lesssim \sum_{|\mu|=0}^1 h^{\frac{1}{2}-\beta}\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu w \right\|_{L^2}, \label{est_1L-L2}\\ \left\| \oph\Big(\widetilde{\gamma}\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)c(x,\xi)\Big)w\right\|_{L^\infty} \lesssim \sum_{|\mu|=0}^1 h^{-\beta}\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu w \right\|_{L^2}, \label{est_1L-Linfty}\end{gathered}$$ and \[est\_2L\] $$\begin{gathered} \left\| \oph\Big(\widetilde{\gamma}\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)c(x,\xi)\Big)w\right\|_{L^2} \lesssim \sum_{|\mu|=0}^2 h^{1-\beta}\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu w \right\|_{L^2}, \label{est_2L-L2}\\ \left\| \oph\Big(\widetilde{\gamma}\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)c(x,\xi)\Big)w\right\|_{L^\infty} \lesssim \sum_{|\mu|=0}^2 h^{\frac{1}{2}-\beta}\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu w \right\|_{L^2}, \label{est_2L-Linfty}\end{gathered}$$ for a small $\beta>0$, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. The proof of (resp. of ) follows straightly by inequalities (resp. ), after observing that, as $\widetilde{\gamma}$ vanishes in a neighbourhood of the origin, $$\widetilde{\gamma}\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)c(x,\xi) = \sum_{j=1}^2\widetilde{\gamma}_1^j\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\Big(\frac{x_j-p'_j(\xi)}{\sqrt{h}}\Big)c(x,\xi),$$ where $\widetilde{\gamma}_1^j(z):=\widetilde{\gamma}(z)z_j|z|^{-2}$ is such that $|\partial^\alpha_z\widetilde{\gamma}_1^j(z)|\lesssim \langle z\rangle^{-1-|\alpha|}$ (resp. $$\widetilde{\gamma}\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)c(x,\xi) = \sum_{j=1}^2\widetilde{\gamma}_2\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)^2c(x,\xi),$$ where $\widetilde{\gamma}_2(z):=\widetilde{\gamma}(z)|z|^{-2}$ is such that $|\partial^\alpha_z\widetilde{\gamma}(z)|\lesssim\langle z\rangle^{-2-|\alpha|}$). \[Prop : est on vtilde\_Lambda,c\] There exists $s>0$ sufficiently large such that \[est widetildev\_Lambda,c\] $$\label{est H1h widetildev_Lambda,c} \left\|\widetilde{v}^\Sigma_{\Lambda^c_{kg}}(t,\cdot)\right\|_{L^2} \lesssim h^{1-\beta}\left(\|\widetilde{v}(t,\cdot)\|_{H^s_h} +\sum_{1\le |\mu|\le 2}\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu\widetilde{v}(t,\cdot)\|_{L^2}\right),$$ $$\label{est Linfty of widetildev_Lambda,c} \left\|\widetilde{v}^\Sigma_{\Lambda^c_{kg}}(t,\cdot)\right\|_{L^\infty} \lesssim h^{\frac{1}{2}-\beta}\left(\|\widetilde{v}(t,\cdot)\|_{H^s_h} + \sum_{1\le |\mu|\le 2}\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu\widetilde{v}(t,\cdot)\|_{L^2}\right).$$ for a small $\beta>0$, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. Since symbol $1- \gamma\big(\frac{x-p'(\xi)}{\sqrt{h}}\big)\chi(h^\sigma\xi)$ is supported for $|\frac{x-p'(\xi)}{\sqrt{h}}|\ge d_1>0$ or $|h^\sigma \xi|\ge d_2>0$, for some small $d_1,d_2>0$, we may consider a smooth cut-off function $\widetilde{\chi}$ equal to 1 close to the origin and such that $\widetilde{\chi}\chi \equiv \widetilde{\chi}$, so that $1- \gamma\big(\frac{x-p'(\xi)}{\sqrt{h}}\big)\chi(h^\sigma\xi)$ writes as $$\left[1-\gamma\left(\frac{x-p'(\xi)}{\sqrt{h}}\right)\right]\widetilde{\chi}(h^\sigma\xi) + \left[1- \gamma\left(\frac{x-p'(\xi)}{\sqrt{h}}\right)\chi(h^\sigma\xi)\right](1-\widetilde{\chi})(h^\sigma\xi),$$ the first symbol being supported in $\{(x,\xi): |\frac{x-p'(\xi)}{\sqrt{h}}|\ge d_1, |\xi|\lesssim h^{-\sigma}\}$, the second one for large frequencies $|\xi|\gtrsim h^{-\sigma}$. Using lemma \[Lem : a sharp b\] and the fact that $\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)\in S_{\frac{1}{2},\sigma}\big(\big\langle \frac{x-p'(\xi)}{\sqrt{h}} \big\rangle^{-M}\big)$, for any $M\in\mathbb{N}$, we have that, for a fixed $N\in\mathbb{N}^*$, $$\begin{aligned} \left[1-\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)\right]\big(1-\widetilde{\chi}(h^\sigma\xi)\big)&= \big(1-\widetilde{\chi}(h^\sigma\xi)\big)\sharp \left[1-\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)\right]\\ & + \sum_{1\le j<N}\widetilde{\chi}_j(h^\sigma\xi)\sharp a_j(x,\xi) + r_N(x,\xi),\end{aligned}$$ where function $\widetilde{\chi}_j(h^\sigma\xi)$ is still supported for large frequencies $|\xi|\gtrsim h^{-\sigma}$, for every $1\le j<N$, up to negligible multiplicative constants, $$a_j(x,\xi) = h^{j(\frac{1}{2}+\sigma)}\sum_{|\alpha|=j}(\partial^\alpha\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi) \in h^{j(\frac{1}{2}+\sigma)}S_{\frac{1}{2},\sigma}\big(\big\langle \frac{x-p'(\xi)}{\sqrt{h}} \big\rangle^{-M}\big),$$ and $r_N\in h^{N(\frac{1}{2}+\sigma)}S_{\frac{1}{2},\sigma}\big(\big\langle \frac{x-p'(\xi)}{\sqrt{h}} \big\rangle^{-M}\big)$. Lemma \[Lem : new estimate 1-chi\], proposition \[Prop : Continuity on H\^s\], and the semi-classical Sobolev injection imply that $$\begin{gathered} \left\|\oph\Big( \Big[1- \gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)\Big](1-\widetilde{\chi})(h^\sigma\xi)\Big)\widetilde{v}^\Sigma(t,\cdot)\right\|_{L^2} \lesssim h^{N(s)}\|\widetilde{v}(t,\cdot)\|_{H^s_h}, \\ \left\|\oph\Big( \Big[1- \gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)\Big](1-\widetilde{\chi})(h^\sigma\xi)\Big)\widetilde{v}^\Sigma (t,\cdot)\right\|_{L^\infty} \lesssim h^{N'(s)} \|\widetilde{v}(t,\cdot)\|_{H^s_h},\end{gathered}$$ where $N(s), N'(s)\ge 1$ if $s>2$ is sufficiently large. On the other hand, as function $(1-\gamma)\big(\frac{x-p'(\xi)}{\sqrt{h}}\big)$ vanishes in a neighbourhood of the origin and is such that $|\partial^\alpha_z (1-\gamma)(z)|\lesssim \langle z\rangle^{-|\alpha|}$, by inequalities and the fact that, using symbolic calculus to commute $\mathcal{L}$ with $\Sigma(\xi)$, $$\label{Lgamma vSigma} \|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu \widetilde{v}^\Sigma(t,\cdot)\|_{L^2}\lesssim h^{-\nu}\sum_{|\mu_1|\le |\mu|} \|\oph(\chi(h^\sigma\xi))\mathcal{L}^{\mu_1} \widetilde{v}(t,\cdot)\|_{L^2}$$ with $\nu = \rho\sigma$ if $\rho\ge 0$, 0 otherwise, we have that $$\begin{gathered} \left\|\oph\Big((1-\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)\Big)\widetilde{v}^\Sigma(t,\cdot)\right\|_{L^2}\lesssim \sum_{|\mu|\le 2}h^{1-\beta} \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu \widetilde{v}(t,\cdot)\right\|_{L^2},\\ \left\|\oph\Big((1-\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)\Big)\widetilde{v}^\Sigma(t,\cdot)\right\|_{L^\infty}\lesssim \sum_{|\mu|\le 2}h^{\frac{1}{2}-\beta} \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu \widetilde{v}(t,\cdot)\right\|_{L^2},\end{gathered}$$ for a small $\beta>0$, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. In the following lemma we show how to develop the symbol $a(x,\xi)$ associated to an operator acting on $\Gamma^{kg}w$, for some suitable function $w$, at $\xi = -d\phi(x)$, where $\phi(x)=\sqrt{1-|x|^2}$. \[Lem:dev of a symbol at xi = -dvarphi(x)\] Let $a(x,\xi)$ be a real symbol in $S_{\delta,0}(\langle\xi\rangle^q)$, $q \in\mathbb{R}$, for some $\delta>0$ small, $\Sigma(\xi)=\langle\xi\rangle^\rho$ for some fixed $\rho\in\Z$, $\Gamma^{kg}$ the operator introduced in and $w=w(t,x)$ such that $\Lcal^\mu w(t,\cdot)\in L^2(\R^2)$ for any $|\mu|\le 2$. Let us also introduce $w^\Sigma_{\Lkg}:=\Gamma^{kg}\oph(\Sigma)w$. There exists a family $(\theta_h(x))_h$ of $C^{\infty}_0$ functions real valued, equal to 1 on the closed ball $\overline{B_{1-ch^{2\sigma}}(0)}$ and supported in $\overline{B_{1-c_1h^{2\sigma}}(0)}$, for some small $0<c_1<c,\sigma>0$, with $\|\partial^\alpha_x\theta_h\|_{L^\infty}=O(h^{-2|\alpha|\sigma})$ and $(h\partial_h)^k\theta_h$ bounded for every $k$, such that $$\label{Op(a)v development} \oph(a)w^\Sigma_{\Lkg}= \theta_h(x)a(x,-d\phi(x))w^\Sigma_{\Lkg}+ R_1(w) \,,$$ where $R_1(w)$ satisfies \[est L2 Linfty R1(widetildev)\] $$\label{est L2 of R1(widetildev)} \|R_1(w)(t,\cdot)\|_{L^2}\lesssim h^{1-\beta}\left(\|w(t,\cdot)\|_{H^s_h} + \sum_{|\gamma|=1}\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\gamma w(t,\cdot)\|_{L^2}\right),$$ $$\label{est Linfty of R1(widetildev)} \|R_1(w)(t,\cdot)\|_{L^\infty}\lesssim h^{\frac{1}{2}-\beta}\left(\|w(t,\cdot)\|_{H^s_h} + \sum_{|\gamma|=1}\|\oph(\chi(h^\sigma\xi)) \mathcal{L}^\gamma w(t,\cdot)\|_{L^2}\right),$$ with $\beta=\beta(\sigma,\delta)>0$, $\beta\rightarrow 0$ as $\sigma, \delta \rightarrow 0$. Moreover, if $\partial_\xi a(x,\xi)$ vanishes at $\xi=-d\phi(x)$, the above estimates can be improved and $R_1(w)$ is rather a remainder $R_2(w)$ such that \[est L2 Linfty of R2(widetildev)\] $$\label{est L2 of R2(widetildev)} \|R_2(w)(t,\cdot)\|_{L^2}\lesssim h^{2-\beta}\left(\|w(t,\cdot)\|_{H^s_h} +\sum_{1\le |\gamma|\le 2 } \|\oph(\chi(h^\sigma\xi))\mathcal{L}^\gamma w(t,\cdot)\|_{L^2}\right),$$ $$\label{est Linfty of R2(widetildev)} \|R_2(w)(t,\cdot)\|_{L^\infty}\lesssim h^{\frac{3}{2}-\beta}\left(\|w(t,\cdot)\|_{H^s_h} + \sum_{1\le |\gamma|\le 2}\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\gamma w(t,\cdot)\|_{L^2}\right).$$ After lemma \[Lem:family\_thetah\] we know that there exists a family of functions $\theta_h(x)$ as in the statement such that equality holds. We highlight the fact that any derivative of $\theta_h$ vanishes on the support of $\gamma\big(\frac{x-p'(\xi)}{\sqrt{h}}\big)\chi(h^\sigma\xi)$ and its derivatives. After remark \[Remark:symbols\_with\_null\_support\_intersection\], this implies that $$w^\Sigma_{\Lkg}= \theta_h(x)w^\Sigma_{\Lkg}+ r_\infty, \quad r_\infty\in h^NS_{\frac{1}{2},\sigma}(\langle x \rangle^{-\infty})$$ and hence that $$\oph(a)w^\Sigma_{\Lkg}= \oph(a)\theta_h(x)w^\Sigma_{\Lkg}+ \oph( r^a_\infty)w^\Sigma_{\Lkg},$$ with $r^a_\infty=a\sharp r_\infty \in h^{N-\gamma}S_{\frac{1}{2},\sigma}(\langle x\rangle^{-\infty})$ and $\gamma=q\sigma$ if $q\ge 0$, 0 otherwise. From proposition \[Prop : Continuity on H\^s\] and the semi-classical Sobolev injection it follows at once that $\oph( r^a_\infty)w^\Sigma_{\Lkg}$ satisfies enhanced estimates if $N$ is taken sufficiently large. Up to negligible multiplicative constants, a further application of symbolic calculus gives also that $$\begin{gathered} \oph(a(x,\xi))\theta_h(x)w^\Sigma_{\Lkg}= \oph(a(x,\xi)\theta_h(x))w^\Sigma_{\Lkg}+\sum_{|\alpha|=1}^{N-1}h^{|\alpha|} \oph\big(\partial^\alpha_\xi a(x,\xi)\partial^\alpha_x\theta_h(x)\big)w^\Sigma_{\Lkg} \\ + \oph(r_N(x,\xi))w^\Sigma_{\Lkg},\end{gathered}$$ where $r_N\in h^{N-\beta} S_{\delta',0}(\langle\xi\rangle^{q-N}\langle x \rangle^{-\infty})$ for a new small $\beta=\beta(\delta,\sigma)$ and $\delta'=\max\{\delta,\sigma\}$. From the same argument as above $\oph(r_N)w^\Sigma_{\Lkg}$ verifies enhanced estimates if $N$ is suitably chosen. Also, since the support of $\partial^\alpha_\xi a(x,\xi)\cdot\partial^\alpha_x\theta_h(x)$ has empty intersection with that of $\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)$ for any $|\alpha|\ge 1$, all the $|\alpha|$-order terms in the above equality are remainders $R_1(w)$. Now, as symbol $a(x,\xi)\theta_h(x)$ is supported for $|x|\le 1-c_1h^{2\sigma}<1$, we are allowed to develop it at $\xi = -d\phi(x)$: $$\begin{aligned} \label{dev of a at xo = -dvarphi} a(x,\xi)\theta_h(x) &= a(x,-d\phi(x))\theta_h(x) + \sum_{|\alpha| =1}\int_0^1 (\partial^\alpha_\xi a)(x, t\xi + (1-t)d\phi(x))dt\, \theta_h(x) (\xi + d\phi(x))^\alpha \\ & = a(x,-d\phi(x))\theta_h(x) + \sum_{j=1}^2 b_j(x,\xi) (x_j - p'_j(\xi)), {\addtocounter{equation}{1}\tag{\theequation}}\end{aligned}$$ with $$\label{def bj in development} b_j(x,\xi) = \sum_{|\alpha| =1}\int_0^1 (\partial^\alpha_\xi a)(x, t\xi + (1-t)d\phi(x)) dt\, \theta_h(x)\frac{(\xi + d\phi(x))^\alpha (x_j -p'_j(\xi))}{|x-p'(\xi)|^2}, \quad j=1,2.$$ If $\chi_1\in C^\infty_0(\mathbb{R}^2)$ is a new cut-off function equal to 1 close to the origin, we can reduce ourselves to the analysis of symbol $b_j(x,\xi)(x_j - p'_j(\xi))\chi_1(h^\sigma \xi)$. In fact, as $b_j(x,\xi)(x_j - p'_j(\xi))(1-\chi_1)(h^\sigma\xi)$ is supported for large frequencies, one can prove that its operator acting on $w^\Sigma_{\Lkg}$ is a $O_{L^2\cap L^\infty}(h^N\|w(t,\cdot)\|_{H^s_h})$ with $N>0$ large as long as $s>0$ is large, by using the semi-classical Sobolev injection, symbolic calculus of proposition \[Prop: a sharp b\], lemma \[Lem : new estimate 1-chi\] and proposition \[Prop : Continuity on H\^s\]. Furthermore, if we consider a smooth cut-off function $\widetilde{\gamma}\in C^\infty_0(\mathbb{R}^2)$, equal to 1 close to the origin and such that $\widetilde{\gamma}\big(\langle\xi\rangle^2 (x-p'(\xi))\big)\equiv 1$ on the support of $\gamma\big(\frac{x-p'(\xi)}{\sqrt{h}}\big)\chi(h^\sigma\xi)$ (which is possible if $\sigma<1/4$), we have that $$\begin{aligned} b_j(x,\xi)(x_j - p'_j(\xi))\chi_1(h^\sigma \xi) &= b_j(x,\xi)(x_j - p'_j(\xi))\chi_1(h^\sigma \xi) \widetilde{\gamma}\big(\langle\xi\rangle^2 (x-p'(\xi))\big) \\ &+ b_j(x,\xi)(x_j - p'_j(\xi))\chi_1(h^\sigma \xi)(1-\widetilde{\gamma})\big(\langle\xi\rangle^2 (x-p'(\xi))\big).\end{aligned}$$ Since $b_j(x,\xi)(x_j - p'_j(\xi))\chi_1(h^\sigma \xi)(1-\widetilde{\gamma})\big(\langle\xi\rangle^2 (x-p'(\xi))\big)\in h^{-\beta} S_{\delta,\sigma}(1)$, for some new small $\beta, \delta>0$, and its support has empty intersection with that of $\gamma\big(\frac{x-p'(\xi)}{\sqrt{h}}\big)$ (which instead belongs to class $S_{\frac{1}{2},0}(\langle \frac{x-p'(\xi)}{\sqrt{h}}\rangle^{-M})$, for $M\in\mathbb{N}$ as large as we want), its quantization acting on $w^\Sigma_{\Lkg}$ is also an enhanced remainder $R_2(w)$. The very contribution that only enjoys estimates is $\oph\big(c(x,\xi)(x_j - p'_j(\xi))\big)w^\Sigma_{\Lkg}$, with $c(x,\xi):=b_j(x,\xi)\chi_1(h^\sigma \xi) \widetilde{\gamma}\big(\langle\xi\rangle^2 (x-p'(\xi))\big)\in h^{-\beta}S_{2\sigma,\sigma}(1)$ and $\beta$ depending linearly on $\sigma$. In fact, if we assume that the support of $\chi_1$ is sufficiently small so that $\chi_1\chi \equiv \chi_1$ and all derivatives of $\chi$ vanish on that support, by using symbolic development until a sufficiently large order $N$ and observing that $$\begin{gathered} \left\{ c(x,\xi)(x_j-p'_j(\xi)), \gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big) \right\}= \left\{c(x,\xi), \gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big) \right\} (x_j-p'_j(\xi)) \\ =\left[(\partial_\xi c)\cdot (\partial \gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)+(\partial_x c)\cdot (\partial \gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big) p''(\xi)\right]\Big(\frac{x_j-p'_j(\xi)}{\sqrt{h}}\Big)\end{gathered}$$ does not lose any power $h^{-1/2}$, we derive that, up to negligible constants, $$\begin{gathered} \Big[c(x,\xi)(x_j - p'_j(\xi))\Big] \sharp \Big[\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)\Big] = \gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)c(x,\xi)(x_j - p'_j(\xi))\\ + {\sum}'h\widetilde{\gamma}\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\widetilde{c}(x,\xi)+ r_N(x,\xi).\end{gathered}$$ In the above equality ${\sum}'$ is a concise notation to indicate a linear combination, $\widetilde{\gamma}\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$, $\widetilde{c}\in h^{-\beta}S_{\delta,\sigma}(1)$ for some new small $\beta,\delta>0$, and $r_N\in h^{(N+1)/2-\beta}S_{\frac{1}{2},\sigma}\big(\big\langle\frac{x-p'(\xi)}{\sqrt{h}}\big\rangle^{-(M-1)}\big)$ as $c(x,\xi)(x_j-p'_j(\xi))\in h^{1/2-\beta}S_{2\sigma,\sigma}\big(\langle \frac{x-p'(\xi)}{\sqrt{h}}\rangle\big)$. From inequalities and we deduce that $\oph\big(\gamma\big(\frac{x-p'(\xi)}{\sqrt{h}}\big)\chi(h^\sigma\xi)c(x,\xi)(x_j - p'_j(\xi))\big)\oph(\Sigma)w$ is a remainder $R_1(w)$ satisfying . The quantization of all the addends in ${\sum}'$ acting on $\oph(\Sigma)w$ is estimated by using that $\widetilde{\gamma}(z)$ vanishes in a neighbourhood of the origin and can be rewritten as $\sum_{j=1,2}\widetilde{\gamma}_2(z)z^2_j$, with $\widetilde{\gamma}_2(z):=\widetilde{\gamma}(z)|z|^{-2}$ such that $|\partial^\alpha_z \widetilde{\gamma}_2(z)|\lesssim \langle z\rangle^{-2-|\alpha|}$. Inequalities and the successive commutation of $\mathcal{L}^\gamma$ with $\Sigma$, for $|\gamma|=1,2$, give then that $h\oph\big(\widetilde{\gamma}\big(\frac{x-p'(\xi)}{\sqrt{h}}\big)\widetilde{c}(x,\xi)\big)\oph(\Sigma)w$ is a remainder $R_2(w)$. Finally, as $$r_N(x,\xi)\sharp \Sigma(\xi)\in h^{\frac{N}{2}-\beta-\mu}S_{\frac{1}{2},\sigma}(\langle\frac{x-p'(\xi)}{\sqrt{h}}\rangle^{-(M-1)})$$ with $\mu=\sigma\rho$ if $\rho\ge 0$, 0 otherwise, $\oph(r_N)\oph(\Sigma)w$ is also a remainder $R_2(w)$ just from \[Prop : Continuity on H\^s\], \[Prop : Continuity from $L^2$ to L\^inf\], fixing $N\in\mathbb{N}$ sufficiently large (e.g. $N=3$). If symbol $a(x,\xi)$ is such that $\partial_\xi a|_{\xi = -d\phi}=0$, instead of equality with $b_j$ given by , we have $$a(x,\xi)\theta_h(x) = a(x, -d\phi(x)) \theta_h(x)+ \sum_{j=1,2} b(x,\xi)(x_j - p'_j(\xi))^2,$$ with $$b(x,\xi) = \sum_{|\alpha|=2}\frac{2}{\alpha!}\int_0^1 (\partial^\alpha_\xi a)(t\xi - (1-t)d\phi(x))(1-t) dt\, \theta_h(x)\frac{(\xi +d\phi(x))^\alpha}{|x-p'(\xi)|^2}.$$ The same argument as before can be applied to $\oph\big(b(x,\xi)\theta_h(x)(x_j - p'_j(\xi))^2\big)w^\Sigma_{\Lkg}$ to show that it reduces to $$\oph\Big(b(x,\xi)\theta_h(x)(x_j - p'_j(\xi))^2\chi_1(h^\sigma \xi) \widetilde{\gamma}\big(\langle\xi\rangle^2 (x-p'(\xi))\big)\Big)w^\Sigma_{\Lkg} + R_2(w),$$ with $R_2(w)$ satisfying . If $$B(x,\xi) := b(x,\xi)\theta_h(x)\chi_1(h^\sigma \xi) \widetilde{\gamma}\big(\langle\xi\rangle^2 (x-p'(\xi))\big)$$ then $B(x,\xi)(x_j-p'_j(\xi))^2 \in h^{-\beta}S_{\delta',\sigma}(1)$ by lemma \[Lem : on e and etilde\], for some new small $\beta, \delta'>0$ depending on $\sigma,\delta$. Using lemma \[Lem : a sharp b\], symbolic development until order 4, and assuming that the support of $\chi_1$ is sufficiently small so that $\chi\chi_1\equiv \chi$, we derive that $$\begin{gathered} \Big[B(x,\xi)(x_j - p'_j(\xi))^2\Big] \sharp \Big[\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)\Big] = B(x,\xi) \gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)(x_j-p'_j(\xi))^2 \\ + \frac{h}{i} \sum_{i=1}^2(\partial_i\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\Big(\frac{x_j-p'_j(\xi)}{\sqrt{h}}\Big)\left[(\partial_{\xi_i}B) +\sum_j (\partial_{x_j}B) p''_{ij}(\xi)\right](x_j-p'_j(\xi))\\ + {\sum_{2\le |\alpha|\le 3}}' h^{\frac{|\alpha|}{2}-2\delta'-\beta} \gamma_\alpha\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)B_\alpha(x,\xi)+r_4(x,\xi),\end{gathered}$$ where $\gamma_\alpha\in C^\infty_0(\mathbb{R}^2\setminus \{0\})$, $B_\alpha(x,\xi)\in S_{\delta',\sigma}(1)$, and $r_4(x,\xi)\in h^{2-4\delta'-\beta}S_{\frac{1}{2},\sigma}\big(\langle\frac{x-p'(\xi)}{\sqrt{h }}\rangle^{-M}\big)$. As $r_4(x,\xi)\sharp \Sigma(\xi) \in h^{2-\beta'}S_{\frac{1}{2},\sigma}\big(\langle\frac{x-p'(\xi)}{\sqrt{h }}\rangle^{-M}\big)$, for $\beta'=2-4\delta'-\beta-\rho\sigma$ if $\rho\ge 0$, $\beta'=2-4\delta'-\beta$ otherwise, it immediately follows from propositions \[Prop : Continuity on H\^s\] and \[Prop : Continuity from $L^2$ to L\^inf\] that $\oph(r_4)\widetilde{v}^\Sigma$ is a remainder $R_2(w)$. After inequalities with $\gamma_n=\gamma$ and $c=B$ (resp. inequalities with $\gamma_n(z) = \partial_i\gamma(z) z_j$ and $c = h^{\delta'}[(\partial_{\xi_i}B) + (\partial_x B)\cdot (\partial_\xi p'_1 + \partial_\xi p'_2)]\in S_{\delta',\sigma}(1)$, for $i,j=1,2$), and , we deduce that the quantization of the first (resp. the second) contribution in above symbolic development is a remainder $R_2(w)$, when acting on $\oph(\Sigma)w$. Finally, as $\gamma_\alpha$ vanishes in a neighbourhood of the origin, we write $$\begin{gathered} \gamma_\alpha\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big) = \sum_{k=1}^2 h^{-1}\underbrace{ \gamma_\alpha\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\Big|\frac{x-p'(\xi)}{\sqrt{h}}\Big|^{-2}}_{\widetilde{\gamma}_\alpha\big(\frac{x-p'(\xi)}{\sqrt{h}}\big)}\times (x_k-p'_k(\xi))^2, \quad |\alpha|=2,\\ \gamma_\alpha\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big) = \sum_{k=1}^2 h^{-\frac{1}{2}} \underbrace{\gamma_\alpha\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\Big(\frac{x_k-p'_k(\xi)}{\sqrt{h}}\Big)\Big|\frac{x-p'(\xi)}{\sqrt{h}}\Big|^{-2}}_{\widetilde{\gamma}^k_\alpha\big(\frac{x-p'(\xi)}{\sqrt{h}}\big)} \times (x_k-p'_k(\xi)), \quad |\alpha| =3\end{gathered}$$ and obtain that the quantization of $\alpha$-th order term with $|\alpha|=2$ (resp. $|\alpha|=3$) is a remainder $R_2(w)$ when acting on $\oph(\Sigma)w$, after inequalities (resp. ) with $\gamma_n = \widetilde{\gamma}_\alpha$ (resp. $\gamma_n=\widetilde{\gamma}^k_\alpha$, $k=1,2$) and $c=B_\alpha$. The following two results allow us to finally derive the ODE satisfied by $\widetilde{v}^\Sigma_{\Lambda_{kg}}$. \[Lem: Commutator Gamma-kg\] We have that $$\label{commutator Gamma kg with linear part} \big[D_t - \oph(x\cdot\xi - p(\xi)), \Gamma^{kg}\big] = \oph(b),$$ where $$\label{symbol of commutator} \begin{split} b(x,\xi) &= -\frac{h}{2i} (\partial\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\cdot\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big) \chi(h^\sigma\xi) - \frac{\sigma h}{i}\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big) (\partial\chi)(h^\sigma\xi)\cdot(h^\sigma\xi) \\ &+\frac{i}{24}h^\frac{3}{2}\sum_{|\alpha|=3} (\partial^\alpha\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)(\partial^\alpha_\xi p'(\xi))\chi(h^\sigma\xi)+ r(x,\xi) \end{split}$$ and $r \in h^{5/2} S_{\frac{1}{2},\sigma}(\langle \frac{x-p'(\xi)}{\sqrt{h}}\rangle^{-N})$ for any $N\ge 0.$ Therefore, function $\widetilde{v}^\Sigma_{\Lambda_{kg}}$ is solution to $$\label{equation for vtilde Sigma Lambda} \big[D_t - \oph(x\cdot\xi - p(\xi))\big]\widetilde{v}^\Sigma_{\Lambda_{kg}} = \Gamma^{kg}\oph(\Sigma(\xi))\big[h^{-1}r^{NF}_{kg}(t,tx)\big] + R_2(\widetilde{v})$$ with $R_2(\widetilde{v})$ satisfying estimates . Recalling the definition of $\Gamma^{kg}$, one can prove by a straight computation that $$\begin{gathered} \big[D_t, \Gamma^{kg}\big] = \frac{h}{i} \oph\Big((\partial\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\cdot\frac{p''(\xi)\xi}{\sqrt{h}}\chi(h^\sigma\xi)\Big) \\ + \frac{h}{2i}\oph\Big((\partial\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\cdot\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big) \chi(h^\sigma\xi)\Big) - \frac{(1+\sigma)h}{i}\oph\Big(\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)(\partial\chi)(h^\sigma\xi)\cdot(h^\sigma\xi)\Big).\end{gathered}$$Since the development of a commutator’s symbol only contains odd-order terms, lemma \[Lem : a sharp b\] gives that the symbol associated to $\big[\Gamma^{kg}, \oph(x\cdot\xi - p(\xi))\big]$ writes as $$\frac{h}{i}\Big\{\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi), x\cdot\xi - p(\xi)\Big\}+\frac{i}{24}h^\frac{3}{2}\sum_{|\alpha|=3} (\partial^\alpha\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)(\partial^\alpha_\xi p(\xi)) + r_5(x,\xi)$$ with $r_5\in h^{5/2}S_{\frac{1}{2},\sigma}(\langle \frac{x-p'(\xi)}{\sqrt{h}}\rangle^{-N})$ for any $N\ge 0$. Developing the above Poisson bracket one finds that $$\begin{gathered} \big[\Gamma^{kg}, \oph(x\cdot\xi - p(\xi))\big]= -\frac{h}{i}\oph\Big((\partial\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\cdot\frac{p''(\xi)\xi}{\sqrt{h}}\chi(h^\sigma\xi)\Big) \\ - \frac{h}{i}\oph\Big((\partial\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\cdot\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big) \chi(h^\sigma\xi)\Big) + \frac{h}{i}\oph\Big(\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)(\partial\chi)(h^\sigma\xi)\cdot(h^\sigma\xi)\Big) \\+\frac{i}{24}h^\frac{3}{2}\sum_{|\alpha|=3} \oph\Big((\partial^\alpha\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)(\partial^\alpha_\xi p'(\xi))\chi(h^\sigma\xi)\Big) + \oph(r_5(x,\xi)),\end{gathered}$$ which summed to the previous commutator gives . The last part of the statement follows applying to equation operators $\oph(\Sigma(\xi))$ (which commutes exactly with the linear part of the equation, evident in non semi-classical coordinates) and $\Gamma^{kg}$. Since $$\begin{gathered} h\oph\Big((\partial\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\cdot\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big) \chi(h^\sigma\xi)\Big)\widetilde{v}^\Sigma \\= \sum_{k=1}^2\oph\Big(\gamma^k\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\cdot (x-p'(\xi))(x_k-p'_k(\xi)) \Big)\widetilde{v}^\Sigma\end{gathered}$$ with $\gamma^k(z):=(\partial\gamma)(z)z_k|z|^{-2}$, and $$h^\frac{3}{2}\oph\Big((\partial^\alpha\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)(\partial^\alpha_\xi p'(\xi))\Big) = h \oph\Big(\gamma^k_\alpha\Big(\frac{\xi-p'(\xi)}{\sqrt{h}}\Big)(\partial^\alpha_\xi p'(\xi))(x_k-p'_k(\xi))\Big)\widetilde{v}^\Sigma$$ with $\gamma^k_\alpha(z):=(\partial^\alpha\gamma)(z)z_k|z|^{-2}$, we obtain from inequalities (resp. ) and that $h\oph\big((\partial\gamma)\big(\frac{x-p'(\xi)}{\sqrt{h}}\big)\cdot\big(\frac{x-p'(\xi)}{\sqrt{h}}\big) \chi(h^\sigma\xi)\big)\widetilde{v}^\Sigma$ (resp. $h^{3/2}\oph\big((\partial^\alpha\gamma)\big(\frac{x-p'(\xi)}{\sqrt{h}}\big)(\partial^\alpha_\xi p'(\xi))\big)$, $|\alpha|=3$) is a remainder $R_2(\widetilde{v})$. The same holds true for $\oph\big(\gamma\big(\frac{x-p'(\xi)}{\sqrt{h}}\big) (\partial\chi)(h^\sigma\xi)\cdot(h^\sigma\xi) \big)\widetilde{v}^\Sigma$, as follows combining symbolic calculus and lemma \[Lem : new estimate 1-chi\], because its symbol is supported for large frequencies $|\xi|\gtrsim h^{-\sigma}$. From propositions \[Prop : Continuity on H\^s\] and \[Prop : Continuity from $L^2$ to L\^inf\] it immediately follows that $\oph(r_5)\widetilde{v}^\Sigma$ satisfies and . \[Prop: ODE for vtilde Sigma\_Lambda\] There exists a family $(\theta_h(x))_h$ of $C^{\infty}_0$ functions, real valued, equal to 1 on the closed ball $\overline{B_{1-ch^{2\sigma}}(0)}$ and supported in $\overline{B_{1-c_1h^{2\sigma}}(0)}$, for some small $0<c_1<c$, $\sigma>0$, with $\|\partial^\alpha_x\theta_h\|_{L^\infty}=O(h^{-2|\alpha|\sigma})$ and $(h\partial_h)^k\theta_h$ bounded for every $k$, such that $$\label{dev linear part widetildev Sigma Lambda} \oph(x\cdot\xi - p(\xi))\widetilde{v}^\Sigma_{\Lambda_{kg}} = -\phi(x)\theta_h(x)\widetilde{v}^\Sigma_{\Lambda_{kg}} + R_2(\widetilde{v}),$$ where $\phi(x)=\sqrt{1-|x|^2}$ and $R_2(\widetilde{v})$ satisfies estimates . Therefore, $\widetilde{v}^\Sigma_{\Lambda_{kg}}$ is solution of the following non-homogeneous ODE: $$\label{ODE for widetildev Sigma Lambda} D_t \widetilde{v}^\Sigma_{\Lambda_{kg}} = -\phi(x) \theta_h(x) \widetilde{v}^\Sigma_{\Lambda_{kg}} + \Gamma^{kg}\oph(\Sigma(\xi))\big[h^{-1}r^{NF}_{kg}(t,tx)\big] + R_2(\widetilde{v}),$$ with $r^{NF}_{kg}$ given by . The proof of the statement follows directly from lemma \[Lem:dev of a symbol at xi = -dvarphi(x)\] if we observe that $\partial_\xi (x\cdot\xi - p(\xi)) = 0$ at $\xi = -d\phi(x)$ and $x\cdot (-d\phi(x))- p(-d\phi(x))=-\phi(x)$. Therefore, holds and, injecting it in , we obtain . \[Prop:propagation\_unif\_est\_V\] Let us fix $K_1>0$. There exist two integers $n\gg \rho\gg 1$ sufficiently large, two constants $A,B>1$ sufficiently large, $\varepsilon_0\in ]0,(2A+B)^{-1}[$ sufficiently small, and $0\ll \delta\ll \delta_2\ll \delta_1\ll \delta_0\ll 1$ small, such that, for any $0<\varepsilon<\varepsilon_0$, if $(u,v)$ is solution to - in some interval $[1,T]$ for a fixed $T>1$, and $u_\pm, v_\pm$ defined in satisfy a-priori estimates for every $t\in [1,T]$, then it also verify in the same interval $[1,T]$. We warn the reader that, throughout the proof, we will denote by $C$, $\beta$ (resp. $\beta'$) two positive constants such that $\beta\rightarrow 0$ as $\sigma\rightarrow0$ (resp. $\beta'\rightarrow 0$ as $\delta_0,\sigma\rightarrow 0$). These constants may change line after line. We also remind that $h=1/t$. In proposition \[Prop: normal forms on KG\] we introduced function $v^{NF}$, defined from $v_{-}$ through , and proved that its $H^{\rho,\infty}$ norm differs from that of $v_{-}$ by a quantity satisfying . Hence, from a-priori estimates , , and for $\theta\in ]0,1[$ sufficiently small (e.g. $\theta<1/4$) $$\label{Hrho_infty_norm_v- in function of vNF} \|v_{-}(t,\cdot)\|_{H^{\rho,\infty}}\le \|v^{NF}(t,\cdot)\|_{H^{\rho,\infty}}+ CA^{2-\theta}B^\theta\varepsilon^2t^{-\frac{5}{4}}.$$ We successively introduced $\widetilde{v}$ in and decomposed it into the sum of functions $\widetilde{v}^\Sigma_{\Lambda_{kg}}$ and $\widetilde{v}^\Sigma_{\Lambda^c_{kg}}$ (see ). We will show in lemma \[Lem: from energy to norms in sc coordinates-KG\] of appendix \[Appendix B\] that, for any $s\le n$, $$\label{est_vtilde_energy} \|\widetilde{v}(t,\cdot)\|_{H^s_h}+ \sum_{|\gamma|=1}^2\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\gamma\widetilde{v}(t,\cdot) \right\|_{L^2}\le CB\varepsilon h^{-\beta'}$$ for all $t\in [1,T]$, so inequality gives that $$\label{est_vSigma_Lambdakg} \|\widetilde{v}^\Sigma_{\Lambda^c_{kg}}(t,\cdot)\|_{L^\infty}\le CB\varepsilon h^{\frac{1}{2}-\beta'}.$$ As concerns $\vt^\Sigma_\Lkg$, we proved in proposition \[Prop: ODE for vtilde Sigma\_Lambda\] that it is solution to ODE , with $r^{NF}_{kg}$ given by and satisfying , and $R_2(\widetilde{v})$ verifying . From , we then have that $$\|R_2(\widetilde{v})(t,\cdot)\|_{L^\infty}\le CB\varepsilon t^{-\frac{3}{2}+\beta'}.$$ We also have that $$\label{est_Gammakg_rNFkg} \left\| \Gamma^{kg}\oph(\Sigma(\xi))[tr^{NF}_{kg}(t,tx)]\right\|_{L^\infty(dx)}\le C(A+B)AB\varepsilon^3 t^{-\frac{3}{2}+\beta'}.$$ In fact, by symbolic calculus of lemma \[Lem : a sharp b\] we derive that, for a fixed $N\in\mathbb{N}$ (e.g. $N>\rho$) and up to negligible multiplicative constants, $$\Gamma^{kg}\oph(\Sigma(\xi)) = \sum_{|\alpha|=0}^{N-1}h^\frac{|\alpha|}{2} \oph\Big((\partial^\alpha\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)(\partial^\alpha\Sigma)(\xi)\Big) + \oph(r_N(x,\xi)),$$ where $r_N\in h^\frac{N}{2}S_{\frac{1}{2},\sigma}(\langle\frac{x-p'(\xi)}{\sqrt{h}}\rangle^{-1})$. Choosing $N$ sufficiently large, we deduce from proposition \[Prop : Continuity from $L^2$ to L\^inf\], the fact that $\|tw(t,t\cdot)\|_{L^2}=\|w(t,\cdot)\|_{L^2}$, inequality and a-priori estimates, that for every $t\in [1,T]$ $$\Big\|\oph(r_N(x,\xi))[tr^{NF}_{kg}(t,tx)]\Big\|_{L^\infty(dx)}\le CA^2B\varepsilon^3 t^{-2}.$$ Using, instead, proposition \[Prop:Continuity Lp-Lp\] with $p=+\infty$, inequality in appendix \[Appendix B\], and that $h=t^{-1}$, we deduce that $$\begin{gathered} \sum_{|\alpha|=0}^{N-1}h^{\frac{|\alpha|}{2}}\left\|\oph\Big((\partial^\alpha\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)(\partial^\alpha\Sigma)(\xi)\Big) \oph(\chi_1(h^\sigma\xi))[tr^{NF}_{kg}(t,tx)]\right\|_{L^\infty} \\ \lesssim t^{1+\beta}\left\|\chi(t^{-\sigma}D_x) r^{NF}_{kg}(t,\cdot)\right\|_{L^\infty}\le C(A+B)AB\varepsilon^3 t^{-\frac{3}{2}+\beta'}.\end{gathered}$$ Summing up, $\Gamma^{kg}\oph(\Sigma(\xi))[t^{-1}r^{NF}_{kg}(t,tx)]+ R_2(\widetilde{v}) = F_{kg}(t,x)$ with $$\|F_{kg}(t,\cdot)\|_{L^\infty}\le [C(A+B)AB\varepsilon^3 +CB\varepsilon] t^{-\frac{3}{2}+\beta'},$$ Using equation we deduce that $$\frac{1}{2}\partial_t |\widetilde{v}^\Sigma_{\Lambda_{kg}}(t,x)|^2 = \Im\left(\widetilde{v}^\Sigma_{\Lambda_{kg}} \overline{D_t \widetilde{v}^\Sigma_{\Lambda_{kg}}}\right)\le |\widetilde{v}^\Sigma_{\Lambda_{kg}}(t,x)| |F_{kg}(t,x)|$$ and hence that $$\begin{split} \|\widetilde{v}^\Sigma_{\Lambda_{kg}}(t,\cdot)\|_{L^\infty}&\le \|\widetilde{v}^\Sigma_{\Lambda_{kg}}(1,\cdot)\|_{L^\infty} + \int_1^t \|F_{kg}(\tau, \cdot)\|_{L^\infty} d\tau \\ & \le \|\widetilde{v}^\Sigma_{\Lambda_{kg}}(1,\cdot)\|_{L^\infty} + C(A+B)AB\varepsilon^3 +CB\varepsilon. \end{split}$$ As $\|\widetilde{v}^\Sigma_{\Lambda_{kg}}(1,\cdot)\|_{L^\infty}\lesssim \|\widetilde{v}(1,\cdot)\|_{L^2}\le CB\varepsilon$ by proposition \[Prop : Continuity from $L^2$ to L\^inf\] and a-priori estimate , the above inequality together with and definition of $\widetilde{v}$, gives that $$\|v^{NF}(t,\cdot)\|_{L^\infty}\le (C(A+B)AB\varepsilon^3 +CB\varepsilon)t^{-1},$$ which injected in leads finally to if we take $A>1$ sufficiently large such that $CB<\frac{A}{3K_1}$, and $\varepsilon_0>0$ sufficiently small to verify $C(A+B)B\varepsilon^2_0 + CA^{1-\theta}B^\theta\varepsilon_0\le \frac{1}{3K_1}$. ### The derivation of the transport equation {#Subsection : The Derivation of the Transport Equation} We now focus on the semi-classical wave equation satisfied by $\widetilde{u}$: $$\label{wave for utilde} \big[D_t - \oph(x\cdot\xi - |\xi|)\big]\widetilde{u}(t,x) = h^{-1}\left[q_w(t,tx)+c_w(t,tx) + r^{NF}_w(t,tx)\right],$$ with $q_w, c_w, r^{NF}_w$ given by , , respectively, and on the derivation of the mentioned transport equation. As we will make use several times of proposition \[Prop : continuity of Op(gamma1):X to L2\] and inequalities , we remind the reader about definition of $\Omega_h$ and of $\Mcal_j$. Also, $\theta_0(x)$ denotes a smooth radial cut-off function (often coming with operator $\Omega_h$) while $\chi\in C^\infty_0(\mathbb{R}^2)$ is equal to 1 in a neighbourhood of the origin and suitably supported. In order to recover a sharp estimate for $\widetilde{u}$ such as , we study the behaviour of this function separately in different regions of the phase space $(x,\xi)\in\mathbb{R}^2\times\mathbb{R}^2$. We start by fixing $\rho \in\mathbb{Z}$, and by introducing $$\label{Sigma_j} \Sigma_j(\xi):= \begin{cases} \langle\xi\rangle^\rho, \quad &\text{for } j=0,\\ \langle\xi\rangle^\rho \xi_j|\xi|^{-1}, \quad &\text{for } j=1,2. \end{cases}$$ Taking a smooth cut-off function $\chi_0$ equal to 1 in a neighbourhood of the origin, a Littlewood-Paley decomposition, and a small $\sigma>0$, we write the following for any $j\in \{0,1,2\}$: $$\begin{gathered} \label{decomposition frequencies utilde} \oph(\Sigma_j(\xi))\widetilde{u} =\oph(\Sigma_j(\xi)\chi_0(h^{-1}\xi))\widetilde{u} + \sum_k \oph\big(\Sigma_j(\xi)(1-\chi_0)(h^{-1}\xi)\varphi(2^{-k}\xi)\chi_0(h^\sigma\xi)\big)\widetilde{u}\\ + \oph(\Sigma_j(\xi)(1-\chi_0)(h^\sigma\xi))\widetilde{u},\end{gathered}$$observing that the sum over $k$ is actually finite and restricted to set of indices $K:=\{k\in \mathbb{Z}: h\lesssim 2^k\lesssim h^{-\sigma}\}$. From the classical Sobolev injection and the continuity on $L^2$ of the Riesz operator $$\label{utilde small frequencies} \|\oph(\Sigma_j(\xi)\chi_0(h^{-1}\xi))\widetilde{u}(t,\cdot)\|_{L^\infty} = \|\Sigma_j(hD)\chi_0(D)\widetilde{u}(t,\cdot)\|_{L^\infty}\lesssim \|\widetilde{u}(t,\cdot)\|_{L^2},$$ while from the semi-classical Sobolev injection along with lemma \[Lem : new estimate 1-chi\] $$\label{utilde large frequencies} \|\oph(\Sigma_j(\xi)(1-\chi_0)(h^\sigma\xi))\|_{L^\infty}\lesssim h^N \|\widetilde{u}(t,\cdot)\|_{H^s_h},$$ where $N=N(s)\ge 0$ if $s>0$ is sufficiently large. The remaining terms in the right hand side of , localised for frequencies $|\xi|\sim 2^k$, need a sharper analysis because a direct application of semi-classical Sobolev injection only gives that $$\left\|\oph\big(\Sigma_j(\xi)(1-\chi_0)(h^{-1}\xi)\varphi(2^{-k}\xi)\chi_0(h^\sigma\xi)\big)\widetilde{u} \right\|_{L^\infty} \le 2^kh^{-1-\mu}\|\widetilde{u}\|_{L^2},$$ with $\mu=\sigma\rho$ if $\rho\ge 0$, 0 otherwise, and factor $2^k h^{-1-\mu}$ may grow too much when $h\rightarrow 0$. For any fixed $k\in K$, $\rho\in\mathbb{Z}$ and $j\in \{0,1,2\}$, let us introduce $$\label{def utilde-Sigma,k} \widetilde{u}^{\Sigma_j,k}(t,x) := \oph\big(\Sigma_j(\xi)(1-\chi_0)(h^{-1}\xi)\varphi(2^{-k}\xi)\chi_0(h^\sigma\xi)\big)\widetilde{u}(t,x)$$ and observe that, from the commutation of the above operator with the linear part of equation , we get that $\widetilde{u}^{\Sigma_j,k}$ is solution to $$\label{wave equation u^k} \begin{split} & [D_t - \oph(x\cdot\xi - |\xi|)]\widetilde{u}^{\Sigma_j,k}(t,x)\\ &= h^{-1}\oph\big(\Sigma_j(\xi)(1-\chi_0)(h^{-1}\xi)\varphi(2^{-k}\xi)\chi_0(h^\sigma\xi)\big)\left[q_w(t,tx)+c_w(t,tx)+ r^{NF}_w(t,tx)\right]\\ &- i h \,\oph\big(\Sigma_j(\xi)(\partial\chi_0)(h^{-1}\xi)\cdot(h^{-1}\xi)\varphi(2^{-k}\xi)\big)\widetilde{u}-i\sigma h \, \oph\big(\Sigma_j(\xi)\varphi(2^{-k}\xi)(\partial\chi_0)(h^\sigma\xi))\cdot(h^\sigma\xi)\big)\widetilde{u}. \end{split}$$ We introduce the following manifold (see picture \[picture: Lw\]) $$\label{def_Lw} \Lw :=\left\{(x,\xi) : x-\frac{\xi}{|\xi|} = 0\right\},$$ together with operator $$\label{def of Gamma_wk} \Gamma^{w,k} := \oph\Big(\gamma\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)\Big),$$ for some $\gamma\in C^\infty_0(\mathbb{R}^2)$ equal to 1 close to the origin and $\psi\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$ equal to 1 on $supp\varphi$, whose symbol is localized in a neighbourhood of $\Lw \cap \{|\xi|\sim 2^k\}$ of size $h^{1/2-\sigma}$. (0,-1.7) – (0,1.7); (-2,0) – (2,0); at (1.9,0) [$x$]{}; at (0,1.6) [$\xi$]{}; (-1,-1.7) – (-1,1.7); (1,-1.7) – (1,1.7); at (-1,0) [$-1$]{}; at (1,0) [$1$]{}; at (1,1) [$\Lw$]{}; plot(, [/((1-\^2)\^(1/2))]{}); plot(-, [-/((1-\^2)\^(1/2))]{}); We also define \[def\_ukLambda-ukLambdac\] $$\label{def of uk Lambda} \widetilde{u}^{\Sigma_j,k}_{\Lambda_w} := \Gamma^{w,k} \widetilde{u}^{\Sigma_j,k}\,,$$ $$\label{def of uk Lambda complementary} \widetilde{u}^{\Sigma_j,k}_{\Lambda^c_w} := \oph\Big(\big(1- \gamma\big)\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)\Big)\widetilde{u}^{\Sigma_j,k},$$ so that $\widetilde{u}^{\Sigma_j,k} = \widetilde{u}^{\Sigma_j,k}_{\Lambda_w} + \widetilde{u}^{\Sigma_j,k}_{\Lambda^c_w}$. We are going to prove that, if we suitably control the $L^2$ norm of $(\theta_0\Omega_h)^\mu\mathcal{M}^\nu\widetilde{u}^{\Sigma_j, k}$, for any $\mu, |\nu|\le 1$, then $\ut^{\Sigma_j,k}_{\Lw^c}$ is a $O_{L^\infty}(h^{-0})$ (see proposition \[Prop : est on widetildeu Lambda,c\]). As $h=t^{-1}$, this means that $\ut^{\Sigma_j,k}_{\Lambda^c_w}$ grows in time at a rate slower than the one expected for $\ut^{\Sigma_j,k}$ (that is $t^{1/2}$ after ). Analogously to the Klein-Gordon case discussed in the previous subsection, the main contribution to $\ut^{\Sigma_j,k}$ is hence the one localized around $\Lw$ and represented by $\ut^{\Sigma_j,k}_\Lw$. We will show that this function is solution to a transport equation (see proposition \[Prop: transport equation for uLambda\]), from which we will be able to derive a suitable estimate of its uniform norm and to finally propagate (see proposition \[Prop: Propagation uniform estimate U,RU\]). \[Prop : est on widetildeu Lambda,c\] There exists a constant $C>0$ such that, for any $h\in ]0,1]$, $k\in K$, \[est L2 Linfty utilde Lambda,c\] $$\|\widetilde{u}^{\Sigma_j,k}_{\Lambda^c_w}(t,\cdot)\|_{L^2} \le C h^{\frac{1}{2}-\beta} \big(\|\widetilde{u}^{\Sigma_j, k}(t,\cdot)\|_{L^2} + \|\mathcal{M}\widetilde{u}^{\Sigma_j, k}(t,\cdot)\|_{L^2} \big)\,,$$ $$\label{Linfy_norm_utilde_Lambdac} \|\widetilde{u}^{\Sigma_j,k}_{\Lambda^c_w}(t,\cdot)\|_{L^\infty} \le C h^{-\beta}\sum_{\mu=0}^1\big(\|(\theta_0\Omega_h)^\mu\widetilde{u}^{\Sigma_j, k}(t,\cdot)\|_{L^2}+ \|(\theta_0\Omega_h)^\mu\mathcal{M}\widetilde{u}^{\Sigma_j, k}(t,\cdot)\|_{L^2}\big)\,,$$ for a small $\beta>0$, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. The proof is straightforward if one writes $$\widetilde{u}^{\Sigma_j,k}_{\Lambda^c_w} = \sum_{j=1}^2\oph\Big(\gamma^j_1\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\Big(\frac{x_j|\xi| - \xi_j}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)\Big)\widetilde{u}^{\Sigma_j,k},$$ where $\gamma^j_1(z):= \frac{(1-\gamma)(z)z_j}{|z|^2}$ is such that $|\partial^\alpha_z\gamma^j_1(z)|\lesssim \langle z \rangle^{-(|\alpha|+1)}$, and uses inequalities with $a(x)=b_p(\xi)\equiv 1$. \[Lem: commutator Gamma-wk with linear part\] Let $\widetilde{\varphi}\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$ be such that $\widetilde{\varphi}\equiv 1$ on $supp\varphi$ and have a sufficiently small support so that $\psi\widetilde{\varphi}\equiv \psi$. Then for any $k\in K$ $$\label{commutator_Gammawk_equation} \left[\Gamma^{w,k}, D_t - \oph\big((x\cdot \xi - |\xi|)\widetilde{\varphi}(2^{-k}\xi)\big)\right] \oph(\varphi(2^{-k}\xi)) = \oph(b(x,\xi)),$$ where, for any $w\in L^2$ such that $\theta_0\Omega_h w, (\theta_0\Omega_h)^\mu\mathcal{M}w\in L^2(\mathbb{R}^2)$, for $\mu=0,1$, $$\label{L2 norm Op(b)} \| \oph(b(x,\xi))w\|_{L^2} \lesssim h^{1-\beta}\left(\|w\|_{L^2}+\|\mathcal{M}w\|_{L^2}\right),$$ $$\label{Linfty norm Op(b)} \| \oph(b(x,\xi))w\|_{L^\infty} \lesssim h^{1-\beta}\sum_{\mu=0}^1\big(\|(\theta_0\Omega_h)^\mu w\|_{L^2}+\|(\theta_0\Omega_h)^\mu\mathcal{M}w\|_{L^2}\big),$$ with $\beta>0$ small, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. We warn the reader that most of the terms arising from the development of the commutator in the left hand side of satisfy a better $L^2$ estimate than , namely $$\label{better L2 fwk} \| \cdot\|_{L^2}\lesssim h^{\frac{3}{2}-\beta}\big(\|w\|_{L^2} + \|\mathcal{M}w\|_{L^2} \big).$$ The only contribution whose $L^2$ norm is only a $O(h\|w\|_{L^2})$ is the integral remainder called $\widetilde{r}^k_N$, appearing in symbolic development . Since $\partial_t = -h^2\partial_h$, an easy computation shows that $$\label{commutator Dt Gamma^wk} \begin{split} [\Gamma^{w,k}, D_t] = & \Big(\frac{1}{2}+\sigma\Big)\frac{h}{i}\oph\Big((\partial\gamma)\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\cdot\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\psi(2^{-k}\xi)\Big) \\ & + \frac{h}{i} \oph\Big(\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)(\partial\psi)(2^{-k}\xi)\cdot(2^{-k}\xi)\Big). \end{split}$$ The first term in the above right hand side satisfies and after inequalities . The same estimates hold also for the latter one when it acts on $\oph(\varphi(2^{-k}\xi))w$, for the derivatives of $\psi$ vanish on the support of $\widetilde{\varphi}$ (and then of $\varphi$) as a consequence of our assumptions. In fact, if we introduce a smooth function $\widetilde{\psi}\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$, equal to 1 on the support of $\partial\psi$ and such that $supp \widetilde{\psi}\cap supp\varphi = \emptyset$, and use symbolic calculus we find that, for any fixed $N\in\mathbb{N}$, $$\begin{gathered} \oph\left(\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)(\partial\psi)(2^{-k}\xi)\cdot(2^{-k}\xi)\right)\oph(\varphi(2^{-k}\xi))\\ = \oph\left(\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)(\partial\psi)(2^{-k}\xi)\cdot(2^{-k}\xi)\right) \oph\big(\widetilde{\psi}(2^{-k}\xi)\varphi(2^{-k}\xi)\big)- \oph(r^k_N),\end{gathered}$$ where the first term in the above right hand side is 0, and integral remainder $r^k_N$ is given by $$\begin{gathered} r^k_N = \Big(\frac{h}{2i}\Big)^N\sum_{|\alpha| = N}\frac{N (-1)^{|\alpha|}}{\alpha! (\pi h )^4}\int e^{\frac{2i}{h}(\eta \cdot z - y\cdot\zeta)}\int_0^1 \partial^\alpha_x \Big[\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)(\partial\psi)(2^{-k}\xi)\cdot (2^{-k}\xi)\Big]|_{(x+tz, \xi + t\zeta)} dt \\ \times \partial^\alpha_\xi \big(\widetilde{\psi}(2^{-k}\xi)\big)|_{(\xi + \eta)}\, dydz d\eta d\zeta.\end{gathered}$$Developing explicitly the above derivatives and reminding definition of integrals $I^k_{p,q}$, for general $k\in K$, $p,q\in\mathbb{Z}$, one recognizes that, up to some multiplicative constants, $r^k_N$ has the form $$h^{N-N(\frac{1}{2}-\sigma)}2^{-kN}I^k_{N,0}(x,\xi),$$ with $a,a',b_q\equiv 1$, $p=N$ and $\psi(2^{-k}\xi)$ replaced with $(\partial\psi)(2^{-k}\xi)\cdot(2^{-k}\xi)$. Propositions \[Prop: L2 est of integral remainders\] and \[Prop : Linfty est of integral remainders\] imply then that $$\|\oph(r^k_N)\|_{\mathcal{L}(L^2)} + \|\oph(r^k_N)\|_{\mathcal{L}(L^2;L^\infty)} \lesssim h$$ if $N\in\mathbb{N}$ is chosen sufficiently large (e.g. $N>9$), which implies that the $\mathcal{L}(L^2)$ and $\mathcal{L}(L^2;L^\infty)$ norms of the latter operator in the right hand side of is bounded by $h^2$. As regards $[\Gamma^{w,k}, \oph((x\cdot\xi - |\xi|)\widetilde{\varphi}(2^{-k}\xi))]$, we first remind that the symbolic development of a commutator’s symbol only contains odd order terms. Consequently, for a new fixed $N\in\mathbb{N}$ and up to multiplicative constants independent of $h, k$, the symbol of the considered commutator writes as $$\begin{gathered} \label{formula: symb dev} h\Big\{\gamma\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big), (x\cdot\xi - |\xi|)\widetilde{\varphi}(2^{-k}\xi)\Big\} \\ + \sum_{\substack{3\le |\alpha|< N \\ |\alpha|=|\alpha_1|+|\alpha_2|}} h^{|\alpha|} \partial^{\alpha_1}_x \partial^{\alpha_2}_{\xi}\Big[\gamma\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\Big]\partial^{\alpha_2}_x \partial^{\alpha_1}_{\xi}\big[(x\cdot\xi - |\xi|)\widetilde{\varphi}(2^{-k}\xi)\big] + \widetilde{r}^k_N(x,\xi),\end{gathered}$$ with $$\begin{gathered} \widetilde{r}^k_N(x, \xi) = \Big(\frac{h}{2i}\Big)^N\sum_{|\alpha_1|+|\alpha_2| = N}\frac{N (-1)^{|\alpha_1|}}{\alpha! (\pi h )^4}\int e^{\frac{2i}{h}(\eta\cdot z - y\cdot\zeta)}\int_0^1 \partial^{\alpha_1}_x \partial^{\alpha_2}_{\xi}\Big[\gamma\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big) \psi(2^{-k}\xi)\Big]\big|_{(x+tz,\xi+t\zeta)} dt \\ \times \partial^{\alpha_2}_x \partial^{\alpha_1}_{\xi}\big[(x\cdot\xi - |\xi|)\widetilde{\varphi}(2^{-k}\xi)\big]|_{(x+y,\xi +\eta)} \ dydzd\eta d\zeta \,.\end{gathered}$$Since $\big\{\gamma\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big), x\cdot \xi - |\xi|\big\} =0$ the Poisson braket in the above sum reduces to $$h\sum_{j,l}(\partial_j\gamma)\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)(\partial_j\widetilde{\varphi})(2^{-k}\xi)\big(\frac{x_l|\xi|-\xi_l}{h^{1/2-\sigma}}\big)(2^{-k}\xi_l)$$ and its quantization acting on $\oph(\varphi(2^{-k}\xi))w$ satisfies , because $\partial\widetilde{\varphi}$ vanishes on the support of $\varphi$. An explicit calculation of terms of order $3\le |\alpha|<N$, with the help of lemma \[Lem : est on gamma for wave\] and the observation that $|\alpha_2|\le 1$ because $(x\cdot\xi-|\xi|)\widetilde{\varphi}(2^{-k}\xi)$ is affine in $x$, shows that they are linear combination of products $$h^{|\alpha| - |\alpha|(\frac{1}{2}-\sigma)}\gamma_{|\alpha|}\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\widetilde{\varphi}(2^{-k}\xi)x^\nu b_1(\xi)$$ and $$h^{|\alpha| - (|\alpha|-1)(\frac{1}{2}-\sigma)}\widetilde{\gamma}\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\widetilde{\varphi}(2^{-k}\xi)b_0(\xi)$$ for two new cut-off functions $\widetilde{\gamma},\widetilde{\varphi}$, $|\partial^\beta b_0(\xi)|\lesssim_\beta |\xi|^{-|\beta|}$, and $\nu\in\mathbb{N}^2$ of length at most 1. Furthermore, for $j=1,2$, $$\begin{gathered} h^{|\alpha| - |\alpha|(\frac{1}{2}-\sigma)}\gamma_{|\alpha|}\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\widetilde{\varphi}(2^{-k}\xi)x_jb_1(\xi) = h^{|\alpha| - (|\alpha|-1)(\frac{1}{2}-\sigma)}\widetilde{\gamma}^j_{|\alpha|}\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\widetilde{\varphi}(2^{-k}\xi)b_0(\xi)\\ + h^{|\alpha| - |\alpha|(\frac{1}{2}-\sigma)}\gamma_{|\alpha|}\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\widetilde{\varphi}(2^{-k}\xi)\xi_jb_0(\xi),\end{gathered}$$ with $\widetilde{\gamma}^j_{|\alpha|}(z):=\gamma_{|\alpha|}(z)z_j$. From propositions \[Prop : continuity Op(gamma) L2 to L2\], \[Prop : continuity of Op(gamma1):X to L2\], the fact that $|\alpha|\ge 3$ and $2^k\le h^{-\sigma}$ we deduce that the quantization of these $|\alpha|$-order terms acting on $\oph(\varphi(2^{-k}\xi))w$ satisfies , . Finally, we notice that integral remainder $\widetilde{r}^k_N$ can be actually seen as the sum of two contributions, one of the form , the other like , with $a\equiv 1$ and $p=1$. Lemma \[Lem : remainder r\^k\_N\] implies then that the $\mathcal{L}(L^2)$ and $\mathcal{L}(L^2;L^\infty)$ norms of $\oph(\widetilde{r}^k_N)$ are bounded by $h$ as foretold, which concludes the proof of the statement. Function $\widetilde{u}^{\Sigma_j,k}_{\Lambda_w}$ is solution to the following equation: $$\label{wave equation uk-Lambda} \begin{split} &\big[D_t - \oph\big((x\cdot \xi - |\xi|)\widetilde{\varphi}(2^{-k}\xi)\big)\big] \widetilde{u}^{\Sigma_j,k}_{\Lambda_w}(t,x) = f^w_k(t,x) \\ & +h^{-1} \Gamma^{w,k} \oph\big(\Sigma_j(\xi)(1-\chi_0)(h^{-1}\xi)\varphi(2^{-k}\xi)\chi_0(h^\sigma\xi)\big)\left[q_w(t,tx)+c_w(t,tx)+r^{NF}_w(t,tx)\right]\\ &- i h \,\Gamma^{w,k}\oph\big(\Sigma_j(\xi)(\partial\chi_0)(h^{-1}\xi)\cdot(h^{-1}\xi)\varphi(2^{-k}\xi)\big)\widetilde{u}\\ &-i\sigma h \, \Gamma^{w,k}\oph\big(\Sigma_j(\xi)\varphi(2^{-k}\xi)(\partial\chi_0)(h^\sigma\xi))\cdot(h^\sigma\xi)\big)\widetilde{u}, \end{split}$$ where $\widetilde{\varphi}\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$ is equal to 1 on $supp\varphi$, and there exist two constants $C, C'>0$ such that, for any $h\in]0,1], k\in K$, \[est L2 Linfty fk\] $$\label{est L2 of fk} \|f^w_k(t,\cdot)\|_{L^2} \le C h^{1-\beta} \big(\|\widetilde{u}^{\Sigma_j,k}(t,\cdot)\|_{L^2} + \|\mathcal{M}\widetilde{u}^{\Sigma_j,k}(t,\cdot)\|_{L^2} \big)\,,$$ $$\label{est Linfty of fk} \|f^w_k(t,\cdot)\|_{L^\infty} \le C' h^{1-\beta}\sum_{\mu=0}^1\big(\|(\theta_0\Omega_h)^\mu\widetilde{u}^{\Sigma_j,k}(t,\cdot)\|_{L^2} + \|(\theta_0\Omega_h)^\mu\mathcal{M}\widetilde{u}^{\Sigma_j,k}(t,\cdot)\|_{L^2} \big)\,,$$ with $\beta>0$ small, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. If we consider a cut-off function $\widetilde{\varphi}\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$ such that $\widetilde{\varphi}\equiv 1$ on the support of $\varphi$ ($\varphi$ being the truncation on $\widetilde{u}^{\Sigma_j,k}$’s frequencies), we have the exact equality $$\oph(x\cdot\xi - |\xi|)\widetilde{u}^{\Sigma_j,k} = \oph((x\cdot\xi - |\xi|)\widetilde{\varphi}(2^{-k}\xi))\widetilde{u}^{\Sigma_j,k}.$$ Moreover, if we assume that its support is sufficiently small so that $\psi\widetilde{\varphi}\equiv \widetilde{\varphi}$, and apply operator $\Gamma^{w,k}$ to equation , lemma \[Lem: commutator Gamma-wk with linear part\] gives us the result of the statement. The transport equation we talked about at the beginning of this section will be deduced from equation by suitably developing symbol $(x\cdot \xi - |\xi|)\widetilde{\varphi}(2^{-k}\xi)$. To do that, we first need to restrict the support of that symbol to bounded values of $x$ through the introduction of a new cut-off function $\theta(x)$. We remind that $\Sigma'$ is a concise notation that we use to indicate a linear combination of a finite number of terms of the same form. \[Lem: PDE equation for utilde-k\] Let $0<D_1<D_2$ and $\theta=\theta(x)$ be a smooth function equal to 1 for $|x|\le D_1$ and supported for $|x|\le D_2$. Then, $$\begin{gathered} \label{linear op with theta} \oph\big((x\cdot\xi - |\xi|)\widetilde{\varphi}(2^{-k}\xi)\big) = \oph\big(\theta(x)(x\cdot\xi - |\xi|)\widetilde{\varphi}(2^{-k}\xi)\big) + (1-\theta)(x)\oph((x\cdot\xi - |\xi|)\widetilde{\varphi}(2^{-k}\xi))\\ + {\sum}' \widetilde{\theta}(x)\oph(\widetilde{\varphi}_1(2^{-k}\xi)) + \oph(r(x,\xi)), \end{gathered}$$ where $\widetilde{\theta}$ is a smooth function supported for $D_1<|x|<D_2$, $\widetilde{\varphi}_1\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$ and $$\|\oph(r)\|_{\mathcal{L}(L^2)} + \|\oph(r)\|_{\mathcal{L}(L^2;L^\infty)}\lesssim h.$$ Therefore, $\widetilde{u}^{\Sigma_j,k}_{\Lambda_{kg}}$ verifies $$\label{PDO equation for utilde-k} \begin{split} &\Big[D_t - \oph\big(\theta(x)(x\cdot\xi - |\xi|)\widetilde{\varphi}(2^{-k}\xi)\big)\Big]\widetilde{u}^{\Sigma_j,k}_{\Lambda_w}(t,x) = f^w_k(t,x) \\ & + (1-\theta)(x)\oph((x\cdot\xi - |\xi|)\widetilde{\varphi}(2^{-k}\xi))\widetilde{u}^{\Sigma_j,k}_{\Lambda_w} + {\sum}'\widetilde{\theta}(x)\oph(\widetilde{\varphi}_1(2^{-k}\xi))\widetilde{u}^{\Sigma_j,k}_{\Lambda_w}\\ &+ h^{-1} \Gamma^{w,k} \oph\big(\Sigma(\xi)(1-\chi_0)(h^{-1}\xi)\varphi(2^{-k}\xi)\chi_0(h^\sigma\xi)\big)\left[q_w(t,tx)+c_w(t,tx)+r^{NF}_w(t,tx)\right]\\ &- i h \,\Gamma^{w,k}\oph\big(\Sigma(\xi)(\partial\chi_0)(h^{-1}\xi)\cdot(h^{-1}\xi)\varphi(2^{-k}\xi)\big)\widetilde{u}\\ &-i\sigma h \, \Gamma^{w,k}\oph\big(\Sigma(\xi)\varphi(2^{-k}\xi)(\partial\chi_0)(h^\sigma\xi))\cdot(h^\sigma\xi)\big)\widetilde{u}, \end{split}$$ where $f^w_k$ satisfies estimates . Let $\theta(x)$ be the cut-off function of the statement. By proposition \[Prop: a sharp b\] we have that$$\label{dev with 1-theta} \begin{split} & (1-\theta)(x) (x\cdot\xi - |\xi|)\widetilde{\varphi}(2^{-k}\xi)= (1-\theta)(x) \sharp \, \big[(x\cdot\xi - |\xi|)\widetilde{\varphi}(2^{-k}\xi) \big]\\ & - \frac{h}{2i} \partial\theta(x)\cdot \Big(x - \frac{\xi}{|\xi|}\Big) \widetilde{\varphi}(2^{-k}\xi) -\frac{2^{-k}h}{2i} (x\cdot\xi - |\xi|)\partial\theta(x)\cdot(\partial \widetilde{\varphi})(2^{-k}\xi) + r_2^k(x,\xi) \\ &= \, (1-\theta)(x) \sharp \, \big[(x\cdot\xi - |\xi|) \widetilde{\varphi}(2^{-k}\xi)\big] - \frac{h}{2i}\Big[\partial\theta(x)\cdot x \Big] \sharp \,\widetilde{\varphi}(2^{-k}\xi) + \frac{h}{2i}\sum_{l=1}^2\partial_l\theta(x)\sharp \Big[\frac{\xi_l}{|\xi|}\widetilde{\varphi}(2^{-k}\xi) \Big]\\ & -\frac{h}{2i}\sum_{j,l=1}^2\Big[\partial_j\theta(x)x_l \Big]\sharp \,\Big[(\partial_j\widetilde{\varphi})(2^{-k}\xi)(2^{-k}\xi_l)\Big] + \frac{h}{2i}\sum_{l=1}^2\partial_l\theta(x)\sharp \Big[(2^{-k}|\xi|)(\partial_l\widetilde{\varphi})(2^{-k}\xi)\Big] + r^k_2(x,\xi)+ \widetilde{r}^k_2(x,\xi), \end{split}$$where $\partial\theta$ is supported for $D_1<|x|<D_2$, and $r^k_2(t,x)$ (resp. $\widetilde{r}^k_2(t,x)$) is a linear combination of integrals of the form $$\frac{h^22^{-k}}{(\pi h)^2}\int e^{\frac{2i}{h}\eta\cdot z}\int_0^1 \theta(x+tz) (1-t)^2 dt \ x^\nu \widetilde{\varphi}(2^{-k}(\xi+\eta)) dzd\eta,$$ with $|\nu|=0,1$ (resp. $|\nu|=0$), for some new $\theta, \widetilde{\varphi} \in C^\infty_0(\mathbb{R}^2\setminus\{0\})$. By writing $x$ as $(x+tz)-tz$, using that $ze^{\frac{2i}{h}\eta\cdot z}= \big(\frac{h}{2i}\big)\partial_\xi e^{\frac{2i}{h}\eta\cdot z}$, and making an integration by parts, one can express$r^k_2(t,x)$ as the sum over $|\nu|=0,1$ of integrals such as $$\frac{h^22^{-k}(h2^{-k})^\nu}{(\pi h)^2}\int e^{\frac{2i}{h}\eta\cdot z}\int_0^1 \theta(x+tz) f(t) dt \ \widetilde{\varphi}(2^{-k}(\xi+\eta)) dzd\eta,$$ for some new smooth $\theta, f, \widetilde{\varphi}$, and show that for any $\alpha,\beta\in\mathbb{N}^2$ $$\big|\partial^\alpha_x\partial^\beta_\xi \big[(r^k_2+\widetilde{r}^k_2)(x,h\xi)\big]\big|\lesssim_{\alpha,\beta} h^22^{-k}\lesssim_{\alpha,\beta}h.$$ Thus $(r^k_2+ \widetilde{r}^k_2)(x,h\xi) \in hS_{0}(1)$, which means, by classical results on pseudo-differential operators (see for instance [@hormander:the_analysis_III]), that $$\oph((r^k_2 + \widetilde{r}^k_2)(x,\xi)) = Op^w((r^k_2+\widetilde{r}^k_2)(x, h\xi))\in \mathcal{L}(L^2)$$ with norm $O(h)$. Furthermore, one can also show that $\|\oph(r^k_2+\widetilde{r}^k_2)\|_{\mathcal{L}(L^2;L^\infty)}\lesssim h$ using lemma \[Lemma on inequalities for Op(A)\] and the fact that, by making some integrations by parts, for any multi-indices $\alpha, \beta\in\mathbb{N}^2$ and a new $\widetilde{\varphi}\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$ $$\begin{aligned} \left\|\partial^\alpha_y\partial^\beta_\xi \left[(r^k_2+ \widetilde{r}^k_2)\Big(\frac{x+y}{2},h\xi\Big)\right]\right\|_{L^2(d\xi)} \lesssim h^22^{-k} \left\|\int \langle \eta\rangle^{-3} |\widetilde{\varphi}(2^{-k}h(\xi+\eta))| d\eta \right\|_{L^2(d\xi)}\lesssim h.\end{aligned}$$ These considerations, along with the continuity of $\Gamma^{w,k}$ on $L^2$, uniformly in $h$ and $k$ (see proposition \[Prop : continuity Op(gamma) L2 to L2\]), imply that $\oph(r^k_2 + \widetilde{r}_2^k)\widetilde{u}^{\Sigma_j,k}_{\Lambda_w}$ is a remainder $f^w_k$. \[Lemma : development of linear part\] We have that $$|\xi| - x\cdot\xi = \frac{1}{2}(1-|x|^2)x\cdot\xi + e(x,\xi)$$ with $$\label{def of e(x,xi)} e(x,\xi) = \frac{1}{2}|\xi|\Big|x - \frac{\xi}{|\xi|}\Big|^2 + \frac{1}{2}\Big(\Big(x-\frac{\xi}{|\xi|}\Big)\cdot\xi\Big)\Big(x- \frac{\xi}{|\xi|}\Big)\cdot\Big(x + \frac{\xi}{|\xi|}\Big).$$ $$\begin{split} |\xi| -x\xi & = \frac{1}{2}|\xi| \left|x-\frac{\xi}{|\xi|}\right|^2 + \frac{1}{2}|\xi| (1-|x|^2) \\ & = \frac{1}{2}|\xi| \left|x-\frac{\xi}{|\xi|}\right|^2 + \frac{1}{2}(|\xi| - x\cdot\xi) (1-|x|^2) + \frac{1}{2}(1-|x|^2)x\cdot\xi \\ & = \underbrace{\frac{1}{2}|\xi| \left|x-\frac{\xi}{|\xi|}\right|^2 + \frac{1}{2}\left(\left(\frac{\xi}{|\xi|}-x\right)\cdot\xi\right)\left(\frac{\xi}{|\xi|} -x\right)\cdot\left(\frac{\xi}{|\xi|} + x\right)}_{e(x,\xi)} + \frac{1}{2}(1-|x|^2)x\cdot\xi\,. \end{split}$$ \[Lem: preliminary on Op(e)\] Let $\gamma, \theta\in C^\infty_0(\mathbb{R}^2)$ and $\widetilde{\varphi}\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$ be such that $\widetilde{\varphi}\equiv 1$ on the support of $\varphi$ and have a sufficiently small support so that $\psi \widetilde{\varphi}\equiv \widetilde{\varphi}$. Let also $$\label{def B(x,xi)} B(x,\xi):=\gamma\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\widetilde{\varphi}(2^{-k}\xi)\theta(x)\Big(x_m - \frac{\xi_m}{|\xi|}\Big), \qquad m\in \{1,2\}.$$ For any function $w\in L^2(\mathbb{R}^2)$ such that $\mathcal{M}w\in L^2(\mathbb{R}^2)$, any $m,n\in \{1,2\}$, $$\label{est L2 Op(e)} \left\|\oph\Big(\theta(x)\widetilde{\varphi}(2^{-k}\xi)\Big(x_m - \frac{\xi_m}{|\xi|}\Big)(x_n|\xi|-\xi_n)\Big)\Gamma^{w,k}w\right\|_{L^2}\lesssim h^{1-\beta} \big(\|w\|_{L^2} + \|\mathcal{M}w\|_{L^2}\big),$$ $$\begin{gathered} \label{est Linfty Op(e)} \left\|\oph\Big(\theta(x)\widetilde{\varphi}(2^{-k}\xi)\Big(x_m - \frac{\xi_m}{|\xi|}\Big)(x_n|\xi|-\xi_n)\Big)\Gamma^{w,k}w\right\|_{L^\infty}\lesssim\\ h^{1-\beta} \big(\|w\|_{L^2} + \|\mathcal{M}w\|_{L^2}\big) + h^{-\beta}\|\oph\big(B(x,\xi)\xi\big)\mathcal{M}w\|_{L^2},\end{gathered}$$ with $\beta>0$ small, $\beta\rightarrow 0$ as $\sigma\rightarrow 0.$ After lemma \[Lemma : on the enhanced symbolic product\] with $p=0$ we have that $$\oph\left(\theta(x)\widetilde{\varphi}(2^{-k}\xi)\Big(x_m - \frac{\xi_m}{|\xi|}\Big)(x_n|\xi|-\xi_n)\right)\Gamma^{w,k}w= \oph\left(B(x,\xi)(x_n|\xi| - \xi_n)\right)w+ \oph(r^k_0(x,\xi))w,$$and the $L^2$ (resp. $L^\infty$) norm of the latter term in the above right hand side is bounded by the right hand side of (resp. of ) after inequality (resp. ). Moreover, the $L^2$ norm of $\oph\big(B(x,\xi)(x_n|\xi| - \xi_n)\big)w$ is also bounded by the right hand side of as straightly follows from emma \[Lemma : symbolic product development\]. It only remains to prove that the $L^\infty$ norm of this term is bounded by the right hand side of . We first consider a new cut-off function $\widetilde{\varphi}_1 \in C^\infty_0(\mathbb{R}^2\setminus\{0\})$, equal to 1 on $supp\widetilde{\varphi}$ so that its derivatives vanish against $\varphi$, and use symbolic calculus to write $$\oph\big(B(x,\xi)(x_n|\xi| - \xi_n)\big) = \oph(\widetilde{\varphi}_1(2^{-k}\xi))\oph\big(B(x,\xi)(x_n|\xi| - \xi_n)\big) + \oph(r^k_{N,1}(x,\xi)),$$ where $r^k_{N,1}(x,\xi)$ is obtained using . Up to interchange the role of variables $y$ and $z$ (resp. $\eta$ and $\zeta$) and to consider $e^{\frac{2i}{h}(y\cdot\zeta-\eta\cdot\zeta)}$ instead of $e^{\frac{2i}{h}(\eta\cdot z-y\cdot\zeta)}$ (which does not affect estimate ), $r^k_{N,1}$ is analogous to integral with $p=1$. Therefore, if $N\in\mathbb{N}$ is chosen sufficiently large (e.g. $N> 11$), lemma \[Lem : remainder r\^k\_N\] implies that $\|\oph(r^k_{N,1})\|_{\mathcal{L}(L^2;L^\infty)}=O(h).$ Since $\widetilde{\varphi}_1$ localises frequencies $\xi$ in an annulus, the classical Sobolev injection gives that $$\begin{gathered} \left\|\oph(\widetilde{\varphi}_1(2^{-k}\xi))\oph\big(B(x,\xi)(x_n|\xi| - \xi_n)\big)w\right\|_{L^\infty} \\ \lesssim \log h \left\|\oph\big(B(x,\xi)(x_n|\xi| - \xi_n)\big)w\right\|_{L^2} + \left\|D_x \oph\big(B(x,\xi)(x_n|\xi| - \xi_n)\big)w\right\|_{L^2}.\end{gathered}$$ As previously said, the former norm in the above right hand side satisfies inequality . As concerns the latter one, we remark that thanks to the specific structure of symbol $B(x,\xi)$ its first derivative with respect to $x$ does not lose any factor $h^{-1/2+\sigma}$, as $$\label{partial x B(x,xi)} \partial_x \left[\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\right]\widetilde{\varphi}(2^{-k}\xi)\theta(x)\Big(x_m - \frac{\xi_m}{|\xi|}\Big) = (\partial\gamma)\Big(\frac{x|\xi| -\xi}{h^{1/2-\sigma}}\Big)\widetilde{\varphi}(2^{-k}\xi)\theta(x)\Big(\frac{x_m|\xi| - \xi_m}{h^{1/2-\sigma}}\Big).$$ Consequently, by using symbolic calculus we derive that $$\begin{gathered} D_x \oph\big(B(x,\xi)(x_n|\xi| - \xi_n)\big)w = h^{-1}\oph\big(B(x,\xi)(x_n|\xi| - \xi_n)\xi\big)w \\ + {\sum}' \oph\Big(\gamma\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)\widetilde{\varphi}(2^{-k}\xi)a(x)b_0(\xi)(x_j|\xi| - \xi_j) \Big)w,\end{gathered}$$ where ${\sum}'$ is a concise notation to indicate linear combinations, $j\in\{m,n\}$ and $\gamma, \widetilde{\varphi}, a$ are some new smooth functions with $a(x)$ compactly supported. Again by lemma \[Lemma : symbolic product development\] the $L^2$ norms of latter contributions in the above right hand side are bounded by $h^{1-\beta}(\|w\|_{L^2}+\|\mathcal{M}w\|_{L^2})$. Finally, we observe that symbol $B(x,\xi)\xi$ can be seen as $$\label{B(x,xi)xi 1} \gamma\Big(\frac{x|\xi| - \xi}{h^{1/2-\sigma}}\Big)(x_m|\xi| - \xi_m)\widetilde{\varphi}(2^{-k}\xi)\theta(x)b_0(x),$$ which implies, after lemma \[Lem: Gamma with double argument-wave\], that $$h^{-1}\oph\big(B(x,\xi)(x_n|\xi|-\xi_n)\xi\big)w =\oph\big(B(x,\xi)\xi\big)\mathcal{M}_nw+ O_{L^2}(h^{1-\beta}(\|w\|_{L^2} + \|\mathcal{M}w\|_{L^2})).$$ \[Lemma : estimate of e(x,xi)\] Let $e(x,\xi)$ be the symbol defined in , $\theta\in C^\infty_0(\mathbb{R}^2)$, and $\widetilde{\varphi}\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$ with sufficiently small support so that $\psi \widetilde{\varphi}\equiv \widetilde{\varphi}$. If a-priori estimates are satisfied for every $t\in [1,T]$, for some fixed $T>1$, there exists a constant $C>0$ such that$$\label{est L2 Linfty e(x,xi)} \left\|\oph\Big(\theta(x)\widetilde{\varphi}(2^{-k}\xi)e(x,\xi)\Big)\widetilde{u}^{\Sigma_j,k}_{\Lambda_w}(t,\cdot)\right\|_{L^2} + \left\|\oph\Big(\theta(x)\widetilde{\varphi}(2^{-k}\xi)e(x,\xi)\Big)\widetilde{u}^{\Sigma_j,k}_{\Lambda_w}(t,\cdot)\right\|_{L^\infty}\le CB\varepsilon h^{1-\beta'}$$for every $t\in[1,T]$, with $\beta>0$ small, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. We warn the reader that, throughout this proof, $C, \beta$ and $\beta'$ will denote three positive constants that may change line after line, with $\beta\rightarrow 0$ as $\sigma\rightarrow 0$ (resp. $\beta'\rightarrow 0$ as $\sigma, \delta_1\rightarrow 0$). Since symbol $e(x,\xi)$ writes as $$e(x,\xi) = \frac{1}{2}\sum_{m=1}^2 \Big(x_m - \frac{\xi_m}{|\xi|}\Big)(x_m|\xi|-\xi_m) + \frac{1}{2}\sum_{m,n=1}^2 \Big(x_m - \frac{\xi_m}{|\xi|}\Big)(x_n|\xi|-\xi_n)\Big(\frac{\xi_m}{|\xi|}\frac{\xi_n}{|\xi|} + x_n \frac{\xi_m}{|\xi|} \Big)\,,$$ it follows that the $L^2$ norm of $\oph\big(\theta(x)\widetilde{\varphi}(2^{-k}\xi)e(x,\xi)\big)\widetilde{u}^{\Sigma_j,k}_{\Lambda_w}$ satisfies inequality after lemmas \[Lem: preliminary on Op(e)\] and \[Lem: from energy to norms in sc coordinates-WAVE\] in appendix \[Appendix B\]. Moreover, from lemma \[Lem: preliminary on Op(e)\] $$\begin{gathered} \left\|\oph\Big(\theta(x)\widetilde{\varphi}(2^{-k}\xi)e(x,\xi)\Big)\widetilde{u}^{\Sigma_j,k}_{\Lambda_w}\right\|_{L^\infty} \lesssim h^{1-\beta}\left(\|\widetilde{u}^{\Sigma_j,k}(t,\cdot)\|_{L^2} + \|\mathcal{M}\widetilde{u}^{\Sigma_j,k}(t,\cdot)\|_{L^2}\right)\\ + h^{-\beta}\|\oph\big(B(x,\xi)\xi\big)\mathcal{M}\widetilde{u}^{\Sigma_j,k}(t,\cdot)\|_{L^2},\end{gathered}$$ with $B(x,\xi)$ defined in . The aim of the proof is then to show that the $L^2$ norm of $\oph\big(B(x,\xi)\xi\big)\mathcal{M}\widetilde{u}^{\Sigma_j,k}$ is estimated by the right hand side of . First of all, we remind that $B(x,\xi)\xi$ can be seen as a symbol of the form . From proposition \[Prop : continuity Op(gamma) L2 to L2\] we hence have that $$\label{norm_operator B(x,xi)xi-1} \|\oph\big(B(x,\xi)\xi\big)\|_{\mathcal{L}(L^2)}=O( h^{\frac{1}{2}-\beta}),$$ while from inequality $$\label{norm_operator B(x,xi)xi-2} \|\oph\big(B(x,\xi)\xi\big)w\|_{L^2} \lesssim h^{1-\beta}(\|w\|_{L^2}+\|\mathcal{M}w\|_{L^2}).$$ We also recall definition of $\widetilde{u}^{\Sigma_j,k}$, use the concise notation $\phi^j_k(\xi)$ for its symbol $\Sigma_j(\xi)(1-\chi_0)(h^{-1}\xi)\varphi(2^{-k}\xi)\chi_0(h^\sigma\xi)$, and observe that $$\label{comm_Mcal_phik} \begin{gathered} \Big[\Mcal_n, \oph(\phi^j_k(\xi))\Big] = -\frac{1}{2i}\oph\big(|\xi|\partial_n\phi^j_k(\xi)\big),\\ \Big\| \Big[\Mcal_n, \oph(\phi^j_k(\xi))\Big] \Big\|_{\Lcal(L^2)} = O(h^{-\sigma}), \end{gathered}$$ after propositions \[Prop: a sharp b\] and \[Prop : continuity Op(gamma) L2 to L2\]. Using and recalling relation , we find that for any $n=1,2$, $$\begin{split} & \|\oph\big(B(x,\xi)\xi\big)\mathcal{M}_n\widetilde{u}^{\Sigma_j,k}(t,\cdot)\|_{L^2}\lesssim \|\oph\big(B(x,\xi)\xi\big)\oph(\phi^j_k(\xi))[t (Z_n u^{NF})(t,tx)]\|_{L^2(dx)}\\ &+ \left\|\oph\big(B(x,\xi)\xi\big)\oph\big(\xi_n|\xi|^{-1}\phi^j_k(\xi))\widetilde{u}(t,\cdot) \right\|_{L^2} + \left\|\oph\big(B(x,\xi)\xi\big)\oph\big(|\xi|\partial_n\phi^j_k(\xi)\big)\widetilde{u}(t,\cdot) \right\|_{L^2}\\ &+ \left\|\oph\big(B(x,\xi)\xi\big)\oph(\phi^j_k(\xi))\left[t(tx_n)\left[q_w(t,tx)+c_w(t,tx)+\rnfw(t,tx)\right]\right]\right\|_{L^2(dx)}, \end{split}$$ with $u^{NF}$ defined in , $q_w$, $c_w$ and $\rnfw$ given by , and respectively. Evidently, after and a further commutation of $\mathcal{M}$ with $\oph\big(\xi_n|\xi|^{-1}\phi^j_k(\xi)\big)$ and $\oph\big(|\xi|\partial_n\phi^j_k(\xi)\big)$ respectively, the second and third $L^2$ norm in the above right hand side are estimated by $$h^{1-\beta}(\|\widetilde{u}(t,\cdot)\|_{L^2}+ \|\oph(\chi(h^\sigma\xi))\mathcal{M}\widetilde{u}(t,\cdot)\|_{L^2}),$$ for some $\chi\in C^\infty_0(\mathbb{R}^2)$. They are hence bounded by $CB\varepsilon h^{1-\beta'}$ by lemma \[Lem: from energy to norms in sc coordinates-WAVE\]. This $L^2$ norm is basically estimated in terms of the $L^2$ norm of $(Z^\mu u)_{-}$, for $|\mu|\le 2$. In fact, after definition and equality $$\begin{gathered} \label{dev_Znu} (Z_nu^{NF})(t,tx) = (Z_nu)_{-}(t,tx)+\Big(\frac{D_n}{|D_x|}u_{-}\Big)(t,tx)\\ -\frac{i}{4(2\pi)^2}\sum_{l\in\{+,-\}} \Big[Z_n\int e^{iy\cdot\xi} D_l(\xi,\eta) \hat{v}_l(\xi-\eta)\hat{v}_l(\eta) d\xi d\eta\Big]\big|_{y=tx},\end{gathered}$$ with $D_l$ given by . On the one hand, taking a new smooth cut-off function $\theta_1$ equal to 1 on the support of $\theta$, using with $\widetilde{a}=\theta_1$, together with , proposition \[Prop : continuity Op(gamma) L2 to L2\], and , we deduce that $$\begin{gathered} \|\oph\big(B(x,\xi)\xi)\oph(\phi^j_k(\xi))[t(Z_nu)_{-}(t,tx)]\|_{L^2(dx)}\\ \lesssim \sum_{m=1}^2 h\|\theta_1(x)\oph(\phi^j_k(\xi))\mathcal{M}_m[t(Z_nu)_{-}(t,tx)]\|_{L^2(dx)}+ h^{1-\beta}\|(Z_nu)_{-}(t,\cdot)\|_{L^2}.\end{gathered}$$ After relation , $$\begin{gathered} \|\theta_1(x)\oph(\phi^j_k(\xi))\mathcal{M}_m[t(Z_nu)_{-}(t,tx)]\|_{L^2} \lesssim \|(Z_mZ_nu)_{-}(t,\cdot)\|_{L^2}+\|(Z_nu)_{-}(t,\cdot)\|_{L^2} \\ + \Big\|\theta_1\Big(\frac{x}{t}\Big)\phi^j_k(D_x)\left[x_m Z_n\Nlw\right](t,\cdot) \Big\|_{L^2}.\end{gathered}$$ Moreover, $$\theta_1\Big(\frac{x}{t}\Big)\phi^j_k(D_x)x_m =t \theta_{1,m}\Big(\frac{x}{t}\Big)\phi^j_k(D_x) + \theta_1\Big(\frac{x}{t}\Big)[\phi^j_k(D_x),x_m],$$ where $\theta_{1,m}(z)=\theta_1(z)z_m$, and $[\phi^j_k(D_x),x_m]$ is a bounded operator on $L^2$ with norm $O(t)$, as one can check computing its associated symbol and using that $2^{-k}\lesssim h^{-1}=t$. Therefore, using also inequality together with a-priori estimates we deduce that $$\label{est_A} \begin{split} &\left\| \oph\big(B(x,\xi)\xi)\oph(\phi^j_k(\xi))\Big[t(Z_nu)_{-}(t,tx)\Big]\right\|_{L^2(dx)} \\ &\lesssim \sum_{|\mu|=1}^2h \|(Z^\mu u)_{-}(t,\cdot)\|_{L^2} + \|Z_nV(t,\cdot)\|_{H^1}\|V(t,\cdot)\|_{H^{2,\infty}}+ \big[\|V(t,\cdot)\|_{H^1} \\ &+ \|V(t,\cdot)\|_{L^2}\left(\|U(t,\cdot)\|_{H^{1,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|_{H^{1,\infty}}\right) + \|V(t,\cdot)\|_{L^\infty}\|U(t,\cdot)\|_{H^1}\big]\|V(t,\cdot)\|_{H^{1,\infty}}\\ & \le CB\varepsilon h^{1-\frac{\delta_1}{2}}. \end{split}$$ On the other hand, it is a straight consequence of , and lemma \[Lem: from energy to norms in sc coordinates-WAVE\] that $$\begin{gathered} \label{est_B} \left\| \oph\big(B(x,\xi)\xi)\oph(\phi^j_k(\xi)) [t (D_n|D_x|^{-1}u)_{-}(t,tx)]\right\|_{L^2}\\ \lesssim h^{1-\beta}(\|\widetilde{u}(t,\cdot)\|_{L^2}+\|\oph(\chi(h^\sigma\xi))\mathcal{M}\widetilde{u}(t,\cdot)\|_{L^2})\le CB\varepsilon h^{1-\frac{\delta_2}{2}}.\end{gathered}$$ Finally, by symbolic calculus and we have that $$\label{Op(B)hD} \oph(B(x,\xi)\xi)=\oph(B(x,\xi))(hD_x) + \frac{h}{2i} \oph\big(\partial_xB(x,\xi)\big),$$ where $\partial_xB$ is of the form $$\label{derivative_B(x,xi)xi} \gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\widetilde{\varphi}(2^{-k}\xi)\theta(x)b_0(\xi)$$ for some new $\gamma, \theta\in C^\infty_0(\mathbb{R}^2)$. Consequently, by proposition \[Prop : continuity Op(gamma) L2 to L2\] $$\begin{gathered} \label{Op(Bxi)integral_Zn} \Big\|\oph(B(x,\xi)\xi)\oph(\phi^j_k(\xi))\Big[t Z_n\int e^{i y\cdot\xi} D_l(\xi,\eta) \hat{v}_l(\xi-\eta)\hat{v}_l(\eta) d\xi d\eta\Big]\big|_{y=tx}\Big\|_{L^2(dx)} \\ \lesssim\Big\|\chi(t^{-\sigma}D_x) D_xZ_n\int e^{ix\cdot\xi} D_l(\xi,\eta) \hat{v}_l(\xi-\eta)\hat{v}_l(\eta) d\xi d\eta\Big\|_{L^2(dx)}\\+ h\Big\|\chi(t^{-\sigma}D_x) Z_n\int e^{ix\cdot\xi} D_l(\xi,\eta) \hat{v}_l(\xi-\eta)\hat{v}_l(\eta) d\xi d\eta\Big\|_{L^2(dx)}\end{gathered}$$ and the above right hand side is bounded by $$\begin{gathered} h^{-\beta} \left(\|V(t,\cdot)\|_{H^1}+ \|V(t,\cdot)\|_{L^2}(\|U(t,\cdot)\|_{H^{1,\infty}}+\|\mathrm{R}_1U(t,\cdot)\|_{H^{1,\infty}})+ \|V(t,\cdot)\|_{L^\infty}\|U(t,\cdot)\|_{H^1}\right)\\ \times \left(\|V(t,\cdot)\|_{H^{14,\infty}}+h\|V(t,\cdot)\|_{H^{13}}\right) + h^{-\beta}\|Z_nV(t,\cdot)\|_{L^2}\|V(t,\cdot)\|_{H^{17,\infty}}\end{gathered}$$ after inequalities , and with $s=0$. From a-priori estimates we then deduce that the left hand side of is bounded by $CB\varepsilon h^{1-\beta'}$, which implies, together with equality and estimates, , that the $L^2$ norm of contribution $ \oph\big(B(x,\xi)\xi)\oph(\phi^j_k(\xi))[t(Z_nu^{NF})(t,tx)]$ is estimated with the right hand side of . : After definition of $q_w(t,x)$ and of $\widetilde{v}$, we first notice that $$\label{def_qtilde_w} tq_w(t,tx) = \frac{h}{2} \Im \left[\overline{\widetilde{v}}\, \oph(\xi_1)\vt - \overline{\oph\Big(\frac{\xi_1}{\langle \xi\rangle}\Big)\vt}\cdot \oph\Big(\frac{\xi\xi_1}{\langle \xi\rangle}\Big)\vt\right](t,x) =:\widetilde{q}_w(t,x),$$ where $$\label{norm_L2_qtilde_w} \|\widetilde{q}_w(t,\cdot)\|_{L^2}\lesssim h\|\vt(t,\cdot)\|_{H^{1,\infty}}\|\vt(t,\cdot)\|_{H^1}.$$ Then $$\|\oph(B(x,\xi)\xi)\oph(\phi^j_k(\xi)) \left[t(tx_n)q_w(t,tx)\right]\|_{L^2(dx)} = h^{-1}\|\oph(B(x,\xi)\xi)\oph(\phi^j_k(\xi))\left[x_n\widetilde{q}_w(t,x)\right]\|_{L^2(dx)}.$$ Since $B(x,\xi)$ is compactly supported in $x$ and $$\Big\|\Big[\oph\big(B(x,\xi)\xi\big)\oph(\phi^j_k(\xi)), x_n\Big]\Big\|_{\Lcal(L^2)} = O(h^{\frac{1}{2}-\beta}),$$ as follows from symbolic calculus, , equality and proposition \[Prop : continuity Op(gamma) L2 to L2\], we can morally reduce ourselves to the study of the $L^2$ norm of $$h^{-1} \oph(B(x,\xi)\xi)\oph(\phi^j_k(\xi))\widetilde{q}_w(t,x)$$ up to a $O_{L^2}(h^{-1/2-\beta}\|\widetilde{q}_w\|_{L^2})$. Using , , together with proposition \[Prop : continuity Op(gamma) L2 to L2\], we deduce that $$h^{-1}\left\|\oph(B(x,\xi)\xi)\oph(\phi^j_k(\xi))\widetilde{q}_w(t,\cdot)\right\|_{L^2}\lesssim h^{-1}\| \oph(\phi^j_k(\xi)) (hD_x)\widetilde{q}_w(t,\cdot)\|_{L^2}+ \|\widetilde{q}_w(t,\cdot)\|_{L^2},$$ so from lemma \[Lem: hD|V|2\] below, estimates , , and lemmas \[Lem: from energy to norms in sc coordinates-KG\], \[Lem\_appendix: estimate L2vtilde\] in appendix \[Appendix B\], we conclude that $$\begin{gathered} \label{norm_(hD)_qtilde} h^{-1}\| \oph(\phi^j_k(\xi)) (hD_x)\widetilde{q}_w(t,\cdot)\|_{L^2}\\ \lesssim h^{1-\beta}\Big(\|\vt(t,\cdot)\|_{H^s}+\sum_{|\mu|=1}^2\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu \vt(t,\cdot)\|_{L^2}\Big) \|\vt(t,\cdot)\|_{H^{1,\infty}}\le CB\varepsilon h^{1-\beta'};\end{gathered}$$ : As for the previous estimate, we can reduce to the study of the $L^2$ norm of $$\oph(B(x,\xi)\xi)\oph(\phi^j_k(\xi))[t^2 c_w(t,tx)],$$ up to a $O_{L^2}\big(h^{-1/2-\beta}\|\oph(\chi(h^\sigma\xi))[tc_w(t,tx)]\|_{L^2(dx)}\big)$ for some $\chi\in C^\infty_0(\mathbb{R}^2)$. So using , the fact that $\|tw(t,t\cdot)\|_{L^2}=\|w(t,\cdot)\|_{L^2}$, and with $s>0$ sufficiently large so that $N(s)>2$, we obtain that for a new $\chi_1\in C^\infty_0(\mathbb{R}^2)$ $$\begin{split} & \|\oph(B(x,\xi)\xi)\oph(\phi^j_k(\xi))[t^2c_w(t,t\cdot)]\|_{L^2}\lesssim h^{-\frac{1}{2}-\beta}\left\|\chi(t^{-\sigma}D_x)c_w(t,\cdot)\right\|_{L^2}\\ &\lesssim h^{-\frac{1}{2}-\beta}\left\|\chi_1(t^{-\sigma}D_x)(\vnf-v_{-})(t,\cdot)\right\|_{L^2}\left(\|V(t,\cdot)\|_{H^{2,\infty}}+\|\vnf(t,\cdot)\|_{H^{1,\infty}}\right) \\ & + h^{\frac{3}{2}}\left\| (\vnf - v_{-})(t,\cdot)\right\|_{H^1}\left(\|V(t,\cdot)\|_{H^s}+\|\vnf(t,\cdot)\|_{H^s}\right). \end{split}$$ Then inequalities with $s=1$ and , together with a-priori estimates, give that $$\|\oph(B(x,\xi)\xi)\oph(\phi^j_k(\xi))[t^2c_w(t,t\cdot)]\|_{L^2}\le CB\varepsilon h^{1-\beta'}.$$ : Analogously, from and we obtain that $$\begin{gathered} \label{Op(B)rNFw} \|\oph(B(x,\xi)\xi)\oph(\phi^j_k(\xi))[t^2 r^{NF}_w(t,t\cdot)]\|_{L^2}\lesssim h^{-\frac{1}{2}-\beta}\|\chi(t^{-\sigma}D_x) r^{NF}_w(t,\cdot)\|_{L^2}\\ \lesssim h^{-\frac{1}{2}-\beta}\|V(t,\cdot)\|^2_{H^{13,\infty}}\|U(t,\cdot)\|_{H^1}\le CB\varepsilon h^{\frac{3}{2}-\beta'}.\end{gathered}$$ \[Lem: hD|V|2\] Let $\varphi\in C^\infty_0(\mathbb{R}^2\setminus \{0\})$, $k\in K$ and $a_j(\xi)$ be two smooth real symbols of order $j=0,1$. Then $$\begin{gathered} \label{(hD)|V|2} \left\|\oph(\varphi(2^{-k}\xi))(hD_x)\left[ \overline{a_0(hD_x)\vt} \, a_1(hD_x)\vt\right](t,\cdot)\right\|_{L^2}\\ \lesssim h^{1-\beta}\Big(\|\vt(t,\cdot)\|_{H^s_h}+\sum_{|\mu|=1}^2\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu \vt(t,\cdot)\|_{L^2}\Big)\|\vt(t,\cdot)\|_{H^{1,\infty}_h}.\end{gathered}$$ Let us split both $\vt$ in the left hand side of into the sum $\vtl+\vtlc$, with $\vtl, \vtlc$ introduced in with $\Sigma_j\equiv 1$. Remind that $\vtlc$ satisfies inequality and that $$\|a_0(hD_x)\vt(t,\cdot)\|_{L^\infty}+ \|a_0(hD_x)\vt_{\Lambda_{kg}}(t,\cdot)\|_{L^\infty} \lesssim h^{-\beta}\|\vt(t,\cdot)\|_{H^{1,\infty}_h},$$ for a small $\beta>0$, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$, as follows from lemma \[Prop:Continuity Lp-Lp\] with $p=+\infty$ and the uniform continuity of $a_0(hD_x)$ from $H^{1,\infty}$ to $L^\infty$. Therefore, using the continuity on $L^2$ of $\oph(\varphi(2^{-k}\xi))(hD_x)$ with norm $O(2^k)$ and the fact that $2^k\lesssim h^{-\sigma}$ we deduce that, for any $w_1,w_2\in \{\vt, \vtl,\vtlc\}$ with at least one $w_j$ equal to $\vtlc$, $$\begin{gathered} \left\|\oph(\varphi(2^{-k}\xi))(hD_x)\left[\overline{a_0(hD_x)w_1} a_1(hD_x)w_2\right]\right\|_{L^2}\\ \lesssim h^{1-\beta}\Big(\|\vt(t,\cdot)\|_{H^s_h}+\sum_{|\mu|=1}^2\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu \vt(t,\cdot)\|_{L^2}\Big)\|\vt(t,\cdot)\|_{H^{1,\infty}_h}.\end{gathered}$$ We are thus reduced to proving inequality for $$\left\|\oph(\varphi(2^{-k}\xi))(hD_x)\left[ \overline{a_0(hD_x)\vt_{\Lambda_{kg}}} \, a_1(hD_x)\vt_{\Lambda_{kg}}\right](t,\cdot)\right\|_{L^2}.$$ Furthermore, by means of lemma \[Lem:dev of a symbol at xi = -dvarphi(x)\] we can replace the action of $a_j(hD_x)$ in the above $L^2$ norm, for $j=0,1$, with the multiplication operator by a real function, up to new remainders bounded in $L^2$ by the right hand side of . In fact, $$a_j(hD_x)\vt_{\Lambda_{kg}}=\theta_h(x)a_j(-d\phi(x))\vt_{\Lambda_{kg}}+R_1(\vt), \quad j=0,1,$$ where $\theta_h$ is a smooth cut-off function as in the statement of lemma \[Lem:dev of a symbol at xi = -dvarphi(x)\] and $R_1(\vt)$ satisfies . Now $$hD_x |\vt_{\Lambda_{kg}}|^2 = \left[\oph(\xi+d\phi(x)\theta_h(x))\vt_{\Lambda_{kg}}\right]\overline{\vt_{\Lambda_{kg}}} - \vt_{\Lambda_{kg}}\left[\overline{\oph(\xi+d\phi(x)\theta_h(x))\vt_{\Lambda_{kg}}}\right],$$ and from lemma \[Lem: (xi+dphi)Op(gamma)\] below $$\|\oph(\xi+d\phi(x)\theta_h(x))\vt_{\Lambda_{kg}}(t,\cdot)\|_{L^2}\lesssim h^{1-\beta}\sum_{|\mu|=0}^1\|\oph(\chi(h^\sigma\xi))\mathcal{L}^{\mu} \vt(t,\cdot)\|_{L^2}.$$ This implies, after having applied the Leibniz rule and proposition \[Prop:Continuity Lp-Lp\], that $$\begin{gathered} \left\|hD_x \left[a_0(-d\phi(x))a_1(-d\phi(x))\theta^2_h(x)|\vt_{\Lambda_{kg}}|^2(t,\cdot)\right]\right\|_{L^2} \\ \lesssim h^{1-\beta}\Big(\|\vt(t,\cdot)\|_{H^s_h}+\sum_{|\mu|=1}^2\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu \vt(t,\cdot)\|_{L^2}\Big)\|\vt(t,\cdot)\|_{L^\infty}\end{gathered}$$ and the conclusion of the statement. \[Lem: (xi+dphi)Op(gamma)\] Let $\gamma, \chi\in C^\infty_0(\mathbb{R}^2)$ be equal to 1 in a neighbourhood of the origin, $\sigma>0$ small, $(\theta_h(x))_h$ be a family of $C^{\infty}_0(B_1(0))$ functions, equal to 1 on the support of $\gamma\big(\frac{x-p'(\xi)}{\sqrt{h}}\big)\chi(h^\sigma\xi)$, with $\|\partial^\alpha_x\theta_h\|_{L^\infty}=O(h^{-2|\alpha|\sigma})$ and $(h\partial_h)^k\theta_h$ bounded for every $k$. Let also $\phi(x)=\sqrt{1-|x|^2}$. Then for every $j=1,2$ $$\begin{gathered} \left\|\oph(\xi_j+ d_j\phi(x)\theta_h(x))\oph\Big(\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)\Big)\widetilde{v}(t,\cdot) \right\|_{L^2}\\ \lesssim h^{1-\beta}\sum_{|\mu|=0}^2\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu \widetilde{v}(t,\cdot)\|_{L^2},\end{gathered}$$ with $\beta>0$ small, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. By symbolic calculus of lemma \[Lem : a sharp b\] and the fact that $\theta_h\equiv 1$ on the support of $\gamma\big(\frac{x-p'(\xi)}{\sqrt{h}}\big)\chi(h^\sigma\xi)$, we have that, for any $j=1,2$, $$\label{(xi-dphi)V} \begin{split} \oph(\xi_j+ &d_j\phi(x)\theta_h(x))\oph\Big(\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)\Big)\widetilde{v} = \oph\Big(\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)(\xi_j+d_j\phi(x))\Big)\widetilde{v} \\ &+\frac{\sqrt{h}}{2i}\oph\Big((\partial_j\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)\Big)\widetilde{v} \\ &-\frac{\sqrt{h}}{2i}\sum_{k,l=1}^2\oph\Big((\partial_l\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)p''_{k,l}(\xi)\partial_k( d_j\phi(x)\theta_h(x)) \chi(h^\sigma\xi)\Big)\widetilde{v}\\ & +\frac{h^{1+\sigma}}{2i}\sum_{k=1}^2\oph\Big(\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\partial_k( d_j\phi(x)\theta_h(x)) (\partial_k\chi)(h^\sigma\xi)\Big)\widetilde{v} + \oph(r_2(x,\xi))\widetilde{v}, \end{split}$$ with $r_2\in h^{1-4\sigma}S_{\frac{1}{2},\sigma}(\langle \frac{x-p'(\xi)}{\sqrt{h}}\rangle^{-1})$. On the one hand, as $$\oph\Big(\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)(\xi_j-d_j\phi(x))\Big)\widetilde{v} = \sum_{k=1}^2\oph\Big(\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)\widetilde{e}^j_k(x,\xi)(x_k-p'_k(\xi))\Big)\widetilde{v},$$ with $\widetilde{e}^j_k$ satisfying on the support of $\gamma\big(\frac{x-p'(\xi)}{\sqrt{h}}\big)\chi(h^\sigma\xi)$, the $L^2$ norm of the first term in the right hand side of can be estimated using . On the other hand, as $\partial\gamma$ vanishes in a neighbourhood of the origin, the $L^2$ norm of the second and third term in the right hand side of can be estimated using . The two remaining contributions to the right hand side of , that already carry the right power of $h$, can be estimated with $h^{1-\beta}\|\vt(t,\cdot)\|_{L^2}$ simply by proposition \[Prop : Continuity on H\^s\]. We can finally state the following result: \[Prop: transport equation for uLambda\] For any fixed $T>1$, $D>0$, let $\mathcal{C}^T_D:=\{(t,x):1\le t \le T, |x|\le D\}$ be the truncated cylinder, and assume that estimates are satisfied in time interval $[1,T]$. Then function $$\label{def_u_Sigma_Lw} \widetilde{u}^{\Sigma_j}_{\Lambda_w}(t,x):=\sum_k\widetilde{u}^{\Sigma_j,k}_{\Lambda_w}(t,x)$$ is solution to the following transport equation: $$\label{eq: transport equation for uSigmaLambda} \left[D_t +\frac{1}{2}(1-|x|^2)x\cdot (hD_x) + \frac{h}{2i}(1-2|x|^2)\right]\widetilde{u}^{\Sigma}_{\Lambda_w}(t,x) = F_w(t,x) , \quad \forall (t,x)\in \mathcal{C}^T_D,$$ and there exists some constant $C>0$ such that $$\label{Linfty_norm_Fw} \|F_w(t,\cdot)\|_{L^\infty} \le CB\varepsilon h^{1-\beta'}$$ for some $\beta'>0$ small, $\beta'\rightarrow 0$ as $\sigma,\delta_1\rightarrow 0$. By the assumption in the statement, all that we are going to say is to be meant in time interval $[1,T]$. We remind the reader that, by the definition of $\widetilde{u}^{\Sigma_j,k}_{\Lambda_w}$ in and of $\widetilde{u}^{\Sigma_j,k}$ in , the sum defining $\widetilde{u}^{\Sigma_j}_{\Lambda_w}$ is finite and restricted to indices $k\in K:=\{k\in\mathbb{Z}: h\lesssim 2^k\lesssim h^{-\sigma}\}$. Also, we warn the reader that, throughout the proof, $C$ and $\beta$ will denote two positive constants that may change line after line, with $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. In lemma \[Lem: PDE equation for utilde-k\] we proved that function $\widetilde{u}^{\Sigma_j,k}_{\Lambda_w}$ is solution to with $f^w_k$ verifying . Hence, by lemma \[Lem: from energy to norms in sc coordinates-WAVE\] we derive that $f^w_k$ is a remainder of the form $F_w$ satisfying . For seek of compactness, we denote symbol $\Sigma_j(\xi)(1-\chi_0)(h^{-1}\xi)\varphi(2^{-k}\xi)\chi_0(h^\sigma\xi)$ in the right hand side of by $\phi^j_k(\xi)$. On the one hand, reminding and using the $L^\infty-L^\infty$ continuity of operator $\Gamma^{w,k}$ (see proposition \[Prop: Continuity\_Lp\_wave\]), together with the classical Sobolev injection, the fact that $$\label{norm_O(mu)} \left\|\oph(\phi^j_k(\xi))\right\|_{\Lcal(L^2)}=O(h^{-\mu}),$$ with $\mu=\sigma\rho$ if $\rho\ge 0$, 0 otherwise, estimates , and , we find that $$\label{nl_term_1} \begin{split} &\left\|\Gamma^{w,k}\oph(\Sigma_j(\xi)(1-\chi_0)(h^{-1}\xi)\varphi(2^{-k}\xi)\chi_0(h^\sigma\xi))\left[t q_w(t,tx)\right]\right\|_{L^\infty}\\ &\lesssim h^{-\beta}\|\widetilde{q}_w(t,\cdot)\|_{L^2}+ h^{-1-\beta}\|\oph(\varphi(2^{-k}\xi))(hD_x)\widetilde{q}_w(t,\cdot)\|_{L^2}\le CB\varepsilon h^{1-\beta'}. \end{split}$$ On the other hand, using proposition \[Prop : continuity of Op(gamma1):X to L2\], estimate , the fact that the commutator between $\oph(\phi^j_k(\xi))$ and $\Omega_h$ is also continuous on $L^2$ with norm $O(h^{-\mu})$, equality $\|tw(t,t\cdot)\|_{L^2}=\|w(t,\cdot)\|_{L^2}$, and , (in which we choose $s>0$ large enough to have, say, $N(s)\ge 2$), we deduce that there is a $\chi\in C^\infty_0(\mathbb{R}^2)$ such that $$\label{nl_term_1.5} \begin{split} &\|\Gamma^{w,k}\oph\big(\phi^j_k(\xi)\big)(h^{-1}c_w(t,tx))\|_{L^\infty(dx)}\\ & \lesssim t^{\frac{1}{2}+\beta}\left\|\chi(t^{-\sigma}D_x)(\vnf-v_{-})(t,\cdot)\right\|_{L^2}\left(\|V(t,\cdot)\|_{H^{2,\infty}}+\|\vnf(t,\cdot)\|_{H^{1,\infty}}\right) \\ & + t^{-\frac{3}{2}+\beta}\left\| (\vnf - v_{-})(t,\cdot)\right\|_{H^1}\left(\|V(t,\cdot)\|_{H^s}+\|\vnf(t,\cdot)\|_{H^s}\right) \\ & + t^{\frac{1}{2}+\beta}\left\|\chi(t^{-\sigma}D_x)\Omega (\vnf - v_{-})(t,\cdot)\right\|_{L^2} \left(\|V(t,\cdot)\|_{H^{2,\infty}}+\|\vnf(t,\cdot)\|_{H^{1,\infty}}\right)\\ & + t^{-\frac{3}{2}+\beta}\left\|\Omega (\vnf - v_{-})(t,\cdot)\right\|_{L^2} \left(\|V(t,\cdot)\|_{H^s}+\|\vnf(t,\cdot)\|_{H^s}\right)\\ & + t^{\frac{1}{2}+\beta} \left\|(\vnf - v_{-})(t,\cdot)\right\|_{H^{1,\infty}}\times \sum_{\mu=0}^1\left(\|\Omega^\mu V(t,\cdot)\|_{H^1}+\|\Omega ^\mu\vnf(t,\cdot)\|_{L^2}\right). \end{split}$$ Also, from , we get that for every $\theta\in ]0,1[$ $$\label{nl_term_2} \begin{split} &\|\Gamma^{w,k}\oph\big(\phi^j_k(\xi)\big)(h^{-1}r^{NF}_w(t,tx))\|_{L^\infty} \lesssim t^{\frac{1}{2}+\beta}\|V(t,\cdot)\|^2_{H^{13,\infty}}\|U(t,\cdot)\|_{H^1} \\ &+t^{\frac{1}{2}+\beta}\Big[ \|V(t,\cdot)\|^{1-\theta}_{H^{15,\infty}} \|V(t,\cdot)\|^\theta_{H^{17}}\left(\| U(t,\cdot)\|_{H^{1,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|_{H^{1,\infty}}\right)\\ &\hspace{20pt}+ \|V(t,\cdot)\|_{L^\infty}\left(\| U(t,\cdot)\|^{1-\theta}_{H^{16,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|^{1-\theta}_{H^{16,\infty}}\right)\|U(t,\cdot)\|^\theta_{H^{18}}\Big]\|\Omega V(t,\cdot)\|_{L^2}\\ &+ t^{\frac{1}{2}+\beta} \Big[\|V(t,\cdot)\|_{H^{1,\infty}}\left(\|U(t,\cdot)\|_{H^1} + \|\Omega U(t,\cdot)\|_{H^1}\right)\\ &\hspace{20pt} + \left(\|U(t,\cdot)\|_{H^{2,\infty}}+\| \mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}\right)\|\Omega V(t,\cdot)\|_{L^2}\Big] \|V(t,\cdot)\|_{H^{17,\infty}}. \end{split}$$ Therefore, using with $s=1$, , , and choosing $\theta\ll 1$ sufficiently small, we derive that $h^{-1}\Gamma^{w,k}\oph\big(\phi^j_k(\xi)\big)(c_w(t,tx)+r^{NF}_w(t,tx))$ is a remainder $F_w(t,x)$ satisfying . Since function $(\partial\chi_0)(h^{-1}\xi)$ is localized for frequencies of size $h$, its product with $\psi(2^{-k}\xi)$ is non-zero only for values of $k\in\mathbb{Z}$ such that $2^k\sim h$. In that case, by commutating $\Gamma^{w,k}$ with $\oph\big((\partial\chi_0)(h^{-1}\xi)\cdot(h^{-1}\xi)\psi(2^{-k}\xi)\big)$ and using the classical Sobolev injection, together with proposition \[Prop : continuity Op(gamma) L2 to L2\], we find that $$\label{nl_term_3} \left\|i h \,\Gamma^{w,k}\oph\big((\partial\chi_0)(h^{-1}\xi)\cdot(h^{-1}\xi)\psi(2^{-k}\xi)\big)\widetilde{u}(t,\cdot) \right\|_{L^\infty}\lesssim h \|\widetilde{u}(t,\cdot)\|_{L^2}.$$ Since $(\partial\chi_0)(h^\sigma\xi)$ is, instead, localized for frequencies larger than $h^{-\sigma}$, by applying the semi-classical Sobolev injection and lemma \[Lem : new estimate 1-chi\] we find that $$\label{nl_term_4} \left\| i\sigma h \, \Gamma^{w,k}\oph\big(\psi(2^{-k}\xi)(\partial\chi_0)(h^\sigma\xi))\cdot(h^\sigma\xi)\big)\widetilde{u}(t,\cdot)\right\|_{L^\infty}\lesssim h^N \|\widetilde{u}(t,\cdot)\|_{H^s_h},$$ with $N=N(s)>1$ as long as $s>0$ is sufficiently large. By lemma \[Lem: from energy to norms in sc coordinates-WAVE\] we obtain that also the fifth and sixth addend in the right hand side of are remainders $F_w(t,x)$. Finally, after lemma \[Lemma : development of linear part\] $$\begin{gathered} -\oph\big(\theta(x)(x\cdot\xi - |\xi|)\widetilde{\varphi}(2^{-k}\xi)\big)\widetilde{u}^{\Sigma_j,k}_{\Lambda_w} = \frac{1}{2}\oph\big(\theta(x)(1-|x|^2)x\cdot\xi \widetilde{\varphi}(2^{-k}\xi)\big)\widetilde{u}^{\Sigma_j,k}_{\Lambda_w} \\ + \oph(\theta(x)e(x,\xi)\widetilde{\varphi}(2^{-k}\xi))\widetilde{u}^{\Sigma_j,k}_{\Lambda_w}\end{gathered}$$ with $e(x,\xi)$ given by , and latter term in the above right hand side satisfies . Using symbolic calculus of proposition \[Prop: a sharp b\] until order $N\in\mathbb{N}$ we find that $$\begin{gathered} \frac{1}{2}\oph\big(\theta(x)(1-|x|^2)x\cdot\xi \widetilde{\varphi}(2^{-k}\xi)\big)\widetilde{u}^{\Sigma_j,k}_{\Lambda_w} = \theta(x)\Big[\frac{1}{2}(1-|x|^2)x\cdot(hD_x) + \frac{h}{2i}(1 -2|x|^2)\Big]\oph(\widetilde{\varphi}(2^{-k}\xi))\widetilde{u}^{\Sigma_j,k}_{\Lambda_w} \\ + \frac{h}{4i}(\partial\theta)(x)\cdot x(1-|x|^2)\oph(\widetilde{\varphi}(2^{-k}\xi))\widetilde{u}^{\Sigma_j,k}_{\Lambda_w} + {\sum}'h\theta_1(x) \oph(\widetilde{\varphi}_1(2^{-k}\xi))\widetilde{u}^{\Sigma_j,k}_{\Lambda_w} + \oph(r(_Nx,\xi))\widetilde{u}^{\Sigma_j,k}_{\Lambda_w},\end{gathered}$$with ${\sum}'$ being a concise notation to indicate a linear combination, $\partial\theta (x)$ supported for $|x|>D_1$, $\theta_1\in C^\infty_0(\mathbb{R}^2)$, $\widetilde{\varphi}_1\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$ coming out from the derivatives of $\widetilde{\varphi}$, and $r_N(x,\xi)$ integral remainder of the form $$\frac{h^N}{(\pi h)^2}\int e^{\frac{2i}{h}\eta\cdot z}\int_0^1 \theta_N(x+tz)(1-t)^{N-1}dt\ \widetilde{\varphi}_N(2^{-k}(\xi+\eta)) dzd\eta,$$ for some other $\theta_N\in C^\infty_0(\mathbb{R}^2)$, $ \widetilde{\varphi}_N\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$, verifying that $$\label{norm_Op(r)} \|\oph(r_N(x,\xi))\|_{\mathcal{L}(L^2;L^\infty)}=O(h)$$ if $N$ is taken sufficiently large. Therefore, from proposition \[Prop : continuity Op(gamma) L2 to L2\], and $$\left\| \oph(r(x,\xi))\widetilde{u}^{\Sigma_j,k}_{\Lambda_w}(t,\cdot)\right\|_{L^\infty}\lesssim h^{1-\beta}\|\widetilde{u}(t,\cdot)\|_{L^2}\le CB\varepsilon h^{1-\beta'}.$$ Moreover, since $\widetilde{\varphi}\equiv 1$ on the support of $\varphi$ (which defines $\widetilde{u}^{\Sigma_j,k}$), by commutating $\oph(\widetilde{\varphi}(2^{-k}\xi))$ with $\Gamma^{w,k}$ and using remark \[Remark:symbols\_with\_null\_support\_intersection\] we find that, for any $N\in\mathbb{N}$ as large as we want, $$\oph(\widetilde{\varphi}(2^{-k}\xi))\widetilde{u}^{\Sigma_j,k}_{\Lambda_w} = \widetilde{u}^{\Sigma_j,k}_{\Lambda_w} + O_{L^\infty}(h^N\|\widetilde{u}\|_{L^2}).$$ Also, since $\widetilde{\varphi}_1$ is obtained from the derivatives of $\widetilde{\varphi}$ and vanishes on the support of $\varphi$, $$\theta_1(x)\oph(\widetilde{\varphi}_1(2^{-k}\xi))\widetilde{u}^{\Sigma_j,k}_{\Lambda_w} =O_{L^\infty}(h^N\|\widetilde{u}\|_{L^2}).$$ Therefore, again from we deduce that $$\begin{gathered} -\oph\big(\theta(x)(x\cdot\xi - |\xi|)\widetilde{\varphi}(2^{-k}\xi)\big)\widetilde{u}^{\Sigma_j,k}_{\Lambda_w}= \theta(x)\Big[\frac{1}{2}(1-|x|^2)x\cdot(hD_x) + \frac{h}{2i}(1 -2|x|^2)\Big]\widetilde{u}^{\Sigma_j,k}_{\Lambda_w} \\ + \frac{h}{4i}(\partial\theta)(x)\cdot x(1-|x|^2)\widetilde{u}^{\Sigma_j,k}_{\Lambda_w} + \oph\big(\theta(x)\widetilde{\varphi}(2^{-k}\xi)e(x,\xi)\big)\widetilde{u}^{\Sigma_j,k}_{\Lambda_w} + O_{L^\infty}(h^{1-\beta'}),\end{gathered}$$ which implies, summed up with estimates from to , that $\widetilde{u}^{\Sigma_j,k}_{\Lambda_w}$ is solution to $$\begin{gathered} \left[D_t + \theta(x)\frac{1}{2}(1-|x|^2)x\cdot (hD_x) + \theta(x)\frac{h}{2i}(1-2|x|^2)\right]\widetilde{u}^{\Sigma_j,k}_{\Lambda_w}(t,x) = F^k_w(t,x) \\ + \Big[(1-\theta)(x)\oph((x\cdot\xi - |\xi|)\widetilde{\varphi}(2^{-k}\xi))+ \widetilde{\theta}(x)\oph(\widetilde{\varphi}_1(2^{-k}\xi))- \frac{h}{4i}(\partial\theta)(x)\cdot x(1-|x|^2)\Big]\widetilde{u}^{\Sigma_j,k}_{\Lambda_w}(t,x)\,,\end{gathered}$$ where $F^k_w(t,x)$ satisfies . Choosing $D_1=D$, we obtain that $\widetilde{u}^{\Sigma_j}_{\Lambda_{w}}$ is solution to in cylinder $\mathcal{C}^T_D$, with $F_w(t,x):=\sum_k F^k_w(t,x)$ (this sum being finite and restricted to indices $k\in K$) satisfying the same $L^\infty$ estimate as $F^k_w$, up to an additional factor $h^{-\sigma}$. Analysis of the transport equation and end of the proof ------------------------------------------------------- In previous section (see proposition \[Prop:propagation\_unif\_est\_V\]) we firstly showed how to propagate a-priori uniform estimate on the Klein-Gordon component $v_{-}$, in the sense of deducing from estimates . We then passed to the study of the wave equation and proved that, if $(u_{-},v_{-})$ is solution to in some interval $[1,T]$, function $\widetilde{u}^{\Sigma_j}_{\Lambda_w}$ defined in is solution to transport equation in truncated cylinder $\mathcal{C}^T_D:=\{(t,x):1\le t\le T, |x|\le D\}$, for any $D>0$. The aim of this section is to study such a transport equation in order to deduce some information on the uniform norm of its solutions. This will allow us to finally propagate a-priori estimate on the wave component $u_{-}$ and to close the bootstrap argument. A short proof of main theorem \[Thm: Main theorem\] is given at the end of this section. ### The inhomogeneous transport equation The aim of this subsection is to study the behaviour of a solution $w$ to the following transport equation $$\left[D_t + \frac{1}{2}(1-|x|^2)x\cdot (hD_x) - \frac{i}{2t}(1-2|x|^2)\right]w = f\,,$$ in a cylinder $\mathcal{C} = \{(t,x) : t\ge 1, |x|\le D\}$ for a large constant $D\gg 1$, where the inhomogeneous term $f$ is a $O_{L^\infty}(\varepsilon t^{-1+\beta})$, for some $\varepsilon>0$ small and $0\le \beta<1/2$. We distinguish in $\mathcal{C}$ two subregions: $$I_1 := \Big\{(t,x) : t\ge 1, |x|< \Big(\frac{t}{t-1}\Big)^{\frac{1}{2}}, |x|\le D\Big\}\,, \qquad I_2: = \Big\{(t,x) : t> 1, \Big(\frac{t}{t-1}\Big)^{\frac{1}{2}}\le |x|\le D\Big\},$$ and denote by $I_{1,t}, I_{2,t}$ their sections at a fixed time $t\ge 1$, $$I_{1,t} := \Big\{x : |x|< \Big(\frac{t}{t-1}\Big)^{\frac{1}{2}}, |x|\le D\Big\}\,, \qquad I_{2,t}: = \Big\{x : \Big(\frac{t}{t-1}\Big)^{\frac{1}{2}}\le |x|\le D\Big\}.$$ (0,-1) – (0,6); (-1,0) – (10,0); plot([((2.5)\*/(-1))\^(1/2)]{},); plot(, 1); plot(8.92, ); plot(2.5, ); at (2.5,0) [$1$]{}; at (0,1) [1]{}; at (8.92,0) [$D$]{}; at (5,4) [$I_2$]{}; at (1, 2) [$I_1$]{}; at (9.8,0) [$x$]{}; at (0,5.9) [$t$]{}; The result we prove is the following. \[Prop : estimate of solution of Tr in a cylinder\] Let $\varepsilon>0$ be small and $w$ be the solution to the following Cauchy problem $$\label{Tr} \begin{cases} & \left[D_t + \frac{1}{2}(1-|x|^2)x\cdot (hD_x) - \frac{i}{2t}(1-2|x|^2)\right]w = f\,, \\ & w(1,x) = \varepsilon w_0(x)\,, \end{cases}$$ with $f =O_{L^\infty}(\varepsilon t^{-1+\beta})$, for some fixed $0\le \beta<1/2$. Let us suppose that $|w_0(x)|\lesssim \langle x \rangle^{-2}$ and that $|w(t,x)|\lesssim\varepsilon t^{\beta'}$ for some $\beta'>0$ whenever $|x|>D\gg 1$. Therefore, $$\label{est of w in the cylinder} |w(t,x)|\lesssim \varepsilon \|w_0\|_{L^\infty} t^{\beta''} (1+|x|)^{-\frac{1}{2}}(t^{-1} + |1-|x||)^{-\frac{1}{2}+\beta''}\,,$$ for every $(t,x)\in \mathcal{C}_D=\{(t,x) | t\ge 1, |x|\le D\}$, with $\beta'' = \max\{\beta, \beta'\}$. We observe that, if $W(t,x)=t^{-1}w(t,t^{-1}x)$, the above inequality implies that $$|W(t,x)|\lesssim \varepsilon\|w_0\|_{L^\infty}(t+|x|)^{-\frac{1}{2}}(1+|t-|x||)^{-\frac{1}{2}+\beta''},$$ showing that the uniform norm of $W(t,\cdot)$ decays in time at a rate $t^{-1/2}$, enhanced to $t^{-1+\beta''}$ out of the light cone $t=|x|$. In order to prove the result of proposition \[Prop : estimate of solution of Tr in a cylinder\] we fix $T\ge 1$, $x\in B_D(0)$, and look for the characteristic curve of with initial point $(T,x)$, i.e. map $t\mapsto X(t;T,x)$ solution of $$\label{eq of characteristics in I1} \begin{cases} \frac{d}{dt}X(t;T,x) = \frac{1}{2t}\big(1 - |X(t;T,x)|^2\big) X(t;T,x) \\ X(T;T,x) = x \end{cases}\qquad t\ge T.$$ \[Lem: characteristic equ in I1\] Solution $X(t;T,x)$ to writes explicitly as $$\label{expression of X(t;1,x) in I1} X(t;T,x) = \frac{\sqrt{t}x}{(T - (T-t)|x|^2)^{\frac{1}{2}}}$$ and it is well defined for all $t>T(1-|x|^{-2})$. Moreover, for any fixed $t>T$, map $x\in\mathbb{R}^2 \mapsto X(t;T,x) \in \Big\{|x|< \big(\frac{t}{t-T}\big)^{\frac{1}{2}}\Big\}$ is a diffeomorphism of inverse $Y(t,y) = \frac{\sqrt{T} y}{(t + (T-t)|y|^2)^\frac{1}{2}}$. Multiplying equation by $2X(t;T,x)$ we deduce that $|X(t;T,x)|^2$ satisfies the equation $$\frac{d}{dt}|X(t;T,x)|^2 = \frac{1}{t}\big(1-|X(t;T,x)|^2\big)|X(t;T,x)|^2\,,$$ from which follows that $1- |X(t;T,x)|^2 = \frac{T(1-|x|^2)}{T- (T-t)|x|^2}$. Injecting this result in and integrating in time, we obtain expression and observe that the obtained map is well defined for all $t>T(1-|x|^{-2})$. In order to prove the second part of the statement, we fix $t>T$, $y \in \Big\{|x|\le\big(\frac{t}{t-T}\big)^{\frac{1}{2}}\Big\}$ and look for $Y(t,y)$ such that $X(t;T, Y(t,y)) = y$. In other words, $$y = \frac{\sqrt{t}Y(t,y)}{(T - (T-t)|Y(t,y)|^2)^{\frac{1}{2}}},$$ which implies that $Y(t,y) = \frac{\sqrt{T} y}{(t + (T-t)|y|^2)^\frac{1}{2}}$. This map is well defined as long as $|y|<\big(\frac{t}{t-T}\big)^\frac{1}{2}$. Along the characteristic curve $X(t;T,x)$ function $w$ satisfies $$\begin{split} \frac{d}{dt}w\big(t, X(t;T,x)\big) & = -\frac{1}{2t}\big(1 - 2|X(t;T,x)|^2\big)\ w\big(t, X(t;T,x)\big) + i f\big(t, X(t;T,x)\big) \\ & = - \frac{1}{2t}\frac{T- T|x|^2 - t|x|^2}{T-(T-t)|x|^2}\ w\big(t, X(t;T,x)\big) + if\big(t, X(t;T,x)\big) \end{split}$$ and hence $$\begin{gathered} \label{dt exp w(t, X(t,T,x))} \frac{d}{dt}\left[\left(\exp \int_T^t \frac{1}{2\tau}\frac{T- T|x|^2 - \tau |x|^2}{T-(T-\tau)|x|^2}d\tau\right)w\big(t, X(t;T,x)\big)\right] \\ =i \left(\exp \int_T^t \frac{1}{2\tau}\frac{T- T|x|^2 - \tau |x|^2}{T-(T-\tau)|x|^2} d\tau\right)f\big(t, X(t;T,x)\big)\,.\end{gathered}$$ $$\label{calculation exp of integral} \exp\int_T^t \frac{1}{2\tau}\frac{T- T|x|^2 - \tau |x|^2}{T-(T-\tau)|x|^2}\, d\tau = \Big(\frac{t}{T}\Big)^\frac{1}{2}\Big(\frac{T - T|x|^2 + t|x|^2}{T}\Big)^{-1}.$$ The result follows writing $$\frac{1}{2\tau}\frac{T- T|x|^2 - \tau |x|^2}{T-(T-\tau)|x|^2} = \frac{1}{2\tau} - \frac{|x|^2}{T -T|x|^2 + \tau |x|^2}\,,$$ taking the integral over $\tau \in [T,t]$ and then passing to its exponential. Let us first study the behaviour of $w$, solution to , in region $I_1$. We fix $T=1$ and, integrating equality over $[1,t]$, we find that $$\begin{gathered} \label{expression of w along X} \left( \exp \int_1^t \frac{1}{2\tau}\frac{1- |x|^2 - \tau |x|^2}{1-(1-\tau)|x|^2} d\tau\right) w\big(t, X(t;1,x)\big) \\ = w(1,x) + i \int_1^t \left(\exp \int_1^s \frac{1}{2\tau}\frac{1- |x|^2 - s |x|^2}{1-(1-s)|x|^2}ds \right)f\big(s, X(s;1 ,x)\big)ds.\end{gathered}$$ Using and the fact that $f = O_{L^\infty}(\varepsilon t^{-1+\beta})$, we then obtain that $$\begin{gathered} \label{inequality for w along charact} \big|w(t, X(t;1,x))\big| \le t^{-\frac{1}{2}}(1-|x|^2 + t|x|^2)|w(1,x)| \\ + C\varepsilon t^{-\frac{1}{2}}(1-|x|^2 + t|x|^2) \int_1^t \frac{ds}{(1-|x|^2+s|x|^2)s^{\frac{1}{2}-\beta}}\,,\end{gathered}$$ for some positive constant $C$. For any fixed $0\le \beta<1/2$ $$\label{integral inequality} \int_1^t \frac{ds}{(1-|x|^2+s|x|^2)s^{\frac{1}{2}-\beta}} \lesssim \frac{t^{\frac{1}{2}+\beta}}{(1 + \sqrt{t}|x|)^{1+2\beta}}(1+|x|)^{-1+2\beta + \beta'}\,,$$ for all $t\ge 1$ and $\beta'>0$ as small as we want. For $\sqrt{t} |x|\le 1$, we have that $$\int_1^t \frac{ds}{(1-|x|^2+s|x|^2)s^{\frac{1}{2}-\beta}} \lesssim t^{\frac{1}{2}+\beta} \lesssim \frac{t^{\frac{1}{2}+\beta}}{(1 + \sqrt{t}|x|)^{1+2\beta}}(1+|x|)^{-1+2\beta + \beta'},$$ for any $\beta'\ge 0$. Suppose then that $\sqrt{t}|x| > 1$. For $t\le 2$ $$\begin{gathered} \displaystyle\int_1^t \frac{ds}{(1-|x|^2+s|x|^2)s^{\frac{1}{2}-\beta}} \lesssim (1+|x|)^{-2}\log(1+ |x|^2) \\ \text{and}\quad |x|^{-2}\log(1+ |x|^2) \frac{(1+\sqrt{t}|x|)^{1+2\beta}}{t^{\frac{1}{2}+\beta}}\lesssim (1+|x|)^{-1+2\beta}\log(1+|x|^2), \end{gathered}$$ which immediately implies inequality . For $t\ge 2$ $$\int_1^t \frac{ds}{(1-|x|^2+s|x|^2)s^{\frac{1}{2}-\beta}} = \int_1^2 \frac{ds}{(1-|x|^2+s|x|^2)s^{\frac{1}{2}-\beta}} + \int_2^t \frac{ds}{(1-|x|^2+s|x|^2)s^{\frac{1}{2}-\beta}}\, ,$$ where the first integral is bounded from the right hand side of . The second one is less or equal than $\int_1^{t-1}\frac{ds}{(1+s|x|^2)s^{\frac{1}{2}-\beta}}$, so for $|x|\ge 1$ it follows that $$\int_1^{t-1}\frac{ds}{(1+s|x|^2)s^{\frac{1}{2}-\beta}} \le |x|^{-2}\int_1^{t-1}\frac{ds}{s^{\frac{3}{2}-\beta}}\lesssim (1+|x|)^{-2}.$$ Since $\frac{(1+ \sqrt{t}|x|)^{1+2\beta}}{t^{\frac{1}{2}+\beta}}\le (1+|x|)^{1+2\beta}$, from the above inequality we deduce the right bound of the statement. For $|x|< 1$, a change of variables gives that $$\int_1^{t-1}\frac{ds}{(1+s|x|^2)s^{\frac{1}{2}-\beta}} = |x|^{-1-2\beta}\int_{|x|^2}^{(t-1)|x|^2}\frac{ds}{(1+s)s^{\frac{1}{2}-\beta}}\lesssim |x|^{-1-2\beta}\frac{(t|x|^2)^{\frac{1}{2}+\beta}}{(1+t|x|^2)^{\frac{1}{2}+\beta}}\le \frac{t^{\frac{1}{2}+\beta}}{(1+t|x|^2)^{\frac{1}{2}+\beta}}.$$ If initial condition $w_0(x)$ is sufficiently decaying in space, e.g. $|w_0(x)|\lesssim \langle x\rangle^{-2}$, we deduce from inequalities and the following bound for $w$ along the characteristic curve $X(t;1,x)$: $$\label{est for w along characteristic in I1} \big|w(t, X(t;1,x))\big| \lesssim\varepsilon \|w_0\|_{L^\infty} t^\beta (1+\sqrt{t}|x|)^{1-2\beta}(1+|x|)^{-1+2\beta +\beta'}\,,$$ for any $\beta'>0$ as small as we want. (0,-1) – (0,7); (-1,0) – (12,0); at (11.8,0) [$x$]{}; at (0,6.9) [$t$]{}; plot(, 1); at (0,1) [1]{}; plot([((2.5/2)\*\^(1/2))/((9/8)+(1/4)\*)\^(1/2)]{},); at (1.25, 1.5) [$\bullet$]{}; at (1.25, 1.5) ; plot(2.5, ); plot([((2.5)\*5\*\^(1/2))/((3/2)-(75/2)+(25)\*)\^(1/2)]{},); at (8.35, 1.58) [$\bullet$]{}; at (8.35, 1.58) ; at (2.5,0) [$1$]{}; plot([((2.5)\*/(-1))\^(1/2)]{},); at (10,2) ; \[Prop: estimate for transport sol n I1\] Let $w$ be the solution to transport equation , with $\|f(t,\cdot)\|_{L^\infty}\lesssim \varepsilon t^{-1+\beta}$ for some fixed $0\le \beta<1/2$, and initial condition $|w_0(x)|\lesssim \langle x \rangle^{-2}$, $\forall x\in\mathbb{R}^2$. Then $$\label{est of w in I1} |w(t,x)|\lesssim\varepsilon t^\beta \big[t^{-1} + |1-|x||\big]^{-\frac{1}{2}+\beta}$$ for every $(t,x)\in I_1=\{(t,x) : t\ge 1, |x|< \big(\frac{t}{t-1}\big)^{\frac{1}{2}}, |x|\le D\}$. In lemma \[Lem: characteristic equ in I1\] we proved that, for any fixed $t>T=1$, map $x\in\mathbb{R}^2\mapsto X(t;1,x)\in \big\{x : |x|<(\frac{t}{t-1})^\frac{1}{2}\big\}$ is a diffeomorphism with inverse $Y(t,y) = y(t + (1-t)|y|^2)^{-1/2}$. From inequality we hence deduce that, for any $y$ such that $|y|<\big(\frac{t}{t-1}\big)^\frac{1}{2}$, $$|w(t,y)|\lesssim\varepsilon t^\beta \big(1 + \sqrt{t}|Y(t,y)|\big)^{1-2\beta}\big(1 + |Y(t,y)|\big)^{-1+2\beta+\beta'}.$$ In particular, as $t(1-|y|^2)+|y|^2)\sim t|1-|y|^2| + |y|^2$ when $|y|<\big(\frac{t}{t-1}\big)^\frac{1}{2}$ and $t\ge t_0>1$, and $t|1-|y|^2| + |y|^2 \sim t|1-|y|| + |y|$ when $|y|\le D$, we find for those values of $(t,y)$ that $$|w(t,y)| \lesssim\varepsilon t^\beta \left(1 + \frac{\sqrt{t}|y|}{(t|1-|y|| + |y|)^{\frac{1}{2}}}\right)^{1-2\beta} \lesssim\varepsilon t^\beta\big[t^{-1} + |1-|y||\big]^{-\frac{1}{2}+\beta}\,,$$ simply using that $(1+|Y(t,y)|)^{-1+2\beta+\beta'}\le 1$. Moreover, for $t\rightarrow 1$ and $|y|\le D$, $$|w(t,y)|\lesssim \varepsilon \lesssim \varepsilon t^\beta \big[t^{-1} + |1-|y||\big]^{-\frac{1}{2}+\beta}.$$ (0,-1) – (0,6); (-1,0) – (10,0); plot([((2.5)\*/(-1))\^(1/2)]{},); plot([1.60\*(2\*\^(1/2))/(-6+4\*)\^(1/2)]{},); plot([1.60\*(2\*\^(1/2))/(-6+4\*)\^(1/2)]{},); plot(, 1); plot(8.92, ); plot(2.5, ); at (2.5,0) [$1$]{}; at (7.02,2.55) [$\bullet$]{}; at (7.02,2.55) ; at (8.92, 2.27) ; at (8.92, 2.32) [$\star$]{}; at (0,1) [1]{}; at (8.92,0) [$D$]{}; at (9.8,0) [$x$]{}; at (0,5.9) [$t$]{}; \[Prop : est for transport solution in I2\] Let $\varepsilon>0$ be small and $w$ be the solution to transport equation , with $\|f(t,\cdot)\|_{L^\infty}\lesssim \varepsilon t^{-1+\beta}$ for some fixed $0\le \beta<1/2$, and suppose that $|w(t,x)|\lesssim\varepsilon t^{\beta'}$ for some $\beta'>0$ whenever $|x|\ge D$. Then $$|w(t,x)| \lesssim\varepsilon t^{\beta''} (|x|^2 - 1)^{\beta''-\frac{1}{2}}\,,$$ for every $(t,x)\in I_2 = \{(t,x) : t>1, \big(\frac{t}{t-1}\big)^\frac{1}{2}\le |x| \le D\}$, where $\beta'' = \max\{\beta, \beta'\}$. For a fixed $(T,x)\in I_2$ we look at $X(t; T,x)$, solution to and given by the explicit expression . We observe that there exists a time $T^*$, $1< T^*<T$, such that $X(t;T,x)$ hits the boundary $|y| =D$ at $t =T^*$. In other words, $t = T^*$ is the first time when $X(t;T,x)$ enters in the region $\{(t,x) : t\ge 1, |x|\le D\}$, to never leave it again for function $t\mapsto |X(t;T,x)|$ is strictly decreasing. A simple computation shows that $$\label{T* in terms of T} T^* = \frac{D^2}{D^2 - 1}(1-|x|^{-2})T <T\,.$$ Integrating expression over $[T^*, T] $ and using , we find that $$\begin{gathered} \label{w(T,x)} w(T,x) = \Big(\frac{T^*}{T}\Big)^\frac{1}{2}\Big(\frac{T - T(1 -|x|^{-2})}{T^* - T(1-|x|^{-2})}\Big) w(T^*, X(T^*;T,x)) \\ + i \int_{T^*}^T \Big(\frac{t}{T}\Big)^\frac{1}{2}\Big(\frac{T - T(1 -|x|^{-2})}{t - T(1-|x|^{-2})}\Big) f\big(t, X(t;T,x)\big) dt\,.\end{gathered}$$ From $$T^* - T(1-|x|^{-2}) = \frac{1}{D^2-1}(1-|x|^{-2})T \quad\text{and}\quad \frac{T^*}{T} = \frac{D^2}{D^2-1}(1-|x|^{-2})$$ so since $|w(t,x)|\lesssim \varepsilon t^{\beta'}$ whenever $|x|\ge D$, for some $\beta'>0$ by the hypothesis, we find that the first term in right hand side of is bounded by $C\varepsilon (|x|^2-1)^{-\frac{1}{2}}(T^*)^{\beta'}$, for some constant $C>0$. Setting $c= \frac{1}{D^2-1}$, by the hypothesis on $f$ we derive that $$\begin{split} \Big|\int_{T^*}^T \Big(\frac{t}{T}\Big)^\frac{1}{2}\Big(\frac{T - T(1 -|x|^{-2})}{t - T(1-|x|^{-2})}\Big) f\big(t, X(t;T,x)\big) & dt\Big| \lesssim\varepsilon T^\frac{1}{2}\int_{T^*}^T \big(t - T (1-|x|^{-2})\big)^{-1} t^{-\frac{1}{2}+\beta} dt \\ & =\varepsilon T^\frac{1}{2}\int_{T^*}^T \big(t-T^* + c (1-|x|^{-2})T\big)^{-1} t^{-\frac{1}{2}+\beta} dt \\ & \le\varepsilon T^{\frac{1}{2}}\int_0^{T-T^*}\frac{dt}{\big(t + c (1-|x|^{-2})T\big) t^{\frac{1}{2}-\beta}} \\ & \lesssim\varepsilon T^\frac{1}{2}\big((1-|x|^{-2})T\big)^{\beta-\frac{1}{2}} =\varepsilon T^\beta(1-|x|^{-2})^{\beta - \frac{1}{2}}\,. \end{split}$$ ### Propagation of the uniform estimate on the wave component {#Subs: propagation of the unif est wave} \[Prop: Propagation uniform estimate U,RU\] Let us fix $K_1>0$. There exist two integers $n\gg\rho\gg 1$ sufficiently large,, two constants $A,B>1$ sufficiently large, some small $0<\delta\ll \delta_2\ll \delta_1\ll \delta_0$, and $\varepsilon_0\in ]0,1[$ sufficiently small, such that, for any $0<\varepsilon<\varepsilon_0$, if $(u,v)$ is solution to - in some interval $[1,T]$, for a fixed $T>1$, and $u_\pm, v_\pm$ defined in satisfy a-priori estimates , for every $t\in [1,T]$, then it also verify in the same interval $[1,T]$. We warn the reader that, throughout the proof, $C,\beta, \beta'$ will denote some positive constants that may change line after line, such that $\beta\rightarrow 0$ as $\sigma\rightarrow0$ (resp. $\beta'\rightarrow 0$ as $\delta_1,\sigma\rightarrow 0$). We also remind that $h=1/t$. In proposition \[Prop: NF on wave\] we introduced function $u^{NF}$, defined from $u_{-}$ through , and showed that its $H^{\rho+1,\infty}$ norm (resp. the $H^{\rho+1,\infty}$ norm of $\mathrm{R}u^{NF}$) differs from that of $u_{-}$ (resp. of $\mathrm{R}u_{-}$) by a quantity satisfying (resp. ). If $n$ is sufficiently large with respect to $\rho$ (at least $n\ge \rho+18$), a-priori estimates , give that, for every $t\in [1,T]$, $$\begin{gathered} \label{Hrhoinfty_norm_u- in terms of uNF} \|u_{-}(t,\cdot)\|_{H^{\rho+1,\infty}} + \sum_{j=1}^2\|\mathrm{R}_j u_{-}(t,\cdot)\|_{H^{\rho+1,\infty}}\\ \le \|u^{NF}(t,\cdot)\|_{H^{\rho+1,\infty}}+\sum_{j=1}^2 \|\mathrm{R}_j u^{NF}(t,\cdot)\|_{H^{\rho,\infty}} + 2AB\varepsilon^2 t^{-1+\frac{\delta}{2}}.\end{gathered}$$ We successively considered $\widetilde{u}(t,x):=t\widetilde{u}^{NF}(t,tx)$ and decomposed it as in , with $\Sigma_j$ given by , showing that it satisfies (resp. ) when restricted to small frequencies $|\xi|\lesssim t^{-1}$ (resp. large frequencies $|\xi|\gtrsim t^\sigma)$. We then focused on $\widetilde{u}^{\Sigma_j,k}$ defined in , which is localized for frequencies supported in an annulus of size $2^k$ with $k\in K=\{k\in\mathbb{Z}: h\lesssim 2^k\lesssim h^{-\sigma}\}$, and further split it into the sum of functions $\widetilde{u}^{\Sigma_j,k}_{\Lambda_w}$, $\widetilde{u}^{\Sigma_j,k}_{\Lambda^c_w}$ (see ). On the one hand, from inequality and lemma \[Lem: from energy to norms in sc coordinates-WAVE\] we have that there is a positive constant $C$ such that, for every $t\in [1,T]$, $$\|\widetilde{u}^{\Sigma_j,k}_{\Lambda^c_w}(t,\cdot)\|_{L^\infty}\le C\varepsilon t^{\beta'}.$$ On the other hand, we proved in proposition \[Prop: transport equation for uLambda\] that, for any $D>0$ and any $(t,x)$ in truncated cylinder $\mathcal{C}^T_D=\{(t,x): 1\le t\le T, |x|\le D\}$, $\widetilde{u}^{\Sigma_j}_{\Lw}(t,x)$ defined in is solution to inhomogeneous transport equation , with inhomogeneous term $F_w(t,x)$ satisfying . Observe that, by definition of $\mathcal{M}$, symbolic calculus, and proposition \[Prop : Continuity on H\^s\], we have that $$\|\widetilde{u}^{\Sigma_j}_{\Lw}(1,\cdot)\|_{L^2} + \|x \widetilde{u}^{\Sigma_j}_{\Lw}(1,\cdot)\|_{L^2} \lesssim \|\widetilde{u}(1,\cdot)\|_{L^2} + \|\oph(\chi(h^\sigma \xi))\mathcal{M}\widetilde{u}(1,\cdot)\|_{L^2} \le C\varepsilon,$$ which means that $\varepsilon^{-1}\langle x\rangle\widetilde{u}^{\Sigma_j}_{\Lw}(1,x)\in L^2$. That hence implies that $|\widetilde{u}^{\Sigma_j}_{\Lw}(1,x)|\lesssim \varepsilon \langle x\rangle^{-2}$ for every $x\in\mathbb{R}^2$ (if not, we would have $\|\langle\cdot\rangle^{-1}\|_{L^2}\le \varepsilon^{-1} \|\langle \cdot\rangle \widetilde{u}^{\Sigma_j}_{\Lw}(1,\cdot)\|_{L^2}$). Moreover, if $D\gg 1$ is sufficiently large, from lemma \[Lem: est utilde Sigma,k large x\] below and \[Lem: from energy to norms in sc coordinates-WAVE\] in appendix \[Appendix B\] we deduce that $$\begin{gathered} \label{est_utilde_xBig} |\mathds{1}_{|x|\ge D} \widetilde{u}^{\Sigma_j}_{\Lw}(t,x)|\le C\frac{\log{|x|}}{|x|} h^{-\beta}\big(\|\oph(\chi(h^\sigma\xi))\widetilde{u}(t,\cdot)\|_{L^2} + \|\oph(\chi(h^\sigma\xi))\mathcal{M}\widetilde{u}(t,\cdot)\|_{L^2}\big)\\ \le C\varepsilon \frac{\log |x|}{|x|} t^{\beta'}.\end{gathered}$$ Therefore, from proposition \[Prop : estimate of solution of Tr in a cylinder\] we obtain that $$|\widetilde{u}^{\Sigma_j}_{\Lw}(t,x)|\lesssim C\varepsilon t^{\beta'} (1+|x|)^{-\frac{1}{2}}\big(t^{-1}+ |1-|x||\big)^{-\frac{1}{2}+\beta'}, \quad \forall (t,x)\in \mathcal{C}^T_D.$$ Summing up, denoting by $\mathds{1}_{\mathcal{C}^T_D}$ the characteristic function of cylinder $\mathcal{C}^T_D$, $$|\widetilde{u}^\Sigma(t,x)|\le C\varepsilon \mathds{1}_{\mathcal{C}^T_D} t^{\beta'} (1+|x|)^{-\frac{1}{2}}\big(t^{-1}+ |1-|x||\big)^{-\frac{1}{2}+\beta'} + C\varepsilon t^{\beta'}, \quad \forall (t,x)\in [1,T]\times\mathbb{R}^2.$$ Returning back to function $u^{NF}$ via , this means that, for every $(t,x)\in [1,T]\times \mathbb{R}^2$, $$\begin{gathered} \label{Hrhoinfty_bound_uNF} \left|\langle D_x\rangle^\rho u^{NF}(t,x)\right| + \sum_{j=1}^2 \left|\langle D_x\rangle^\rho R_j u^{NF}(t,x)\right|\\ \le C\varepsilon \mathds{1}_{\{|x|\le Dt \}} (t+|x|)^{-\frac{1}{2}}(1+ |t-|x||)^{-\frac{1}{2}+\beta'}+ C\varepsilon t^{-1+\beta'}.\end{gathered}$$ Finally, reminding definition \[def Sobolev spaces-NEW\] $(iii)$ of space $H^{\rho,\infty}$, injecting the above inequality in , and choosing $A>1$ sufficiently large such that $C<\frac{A}{3K_1}$, $\varepsilon_0>0$ sufficiently small so that $CB\varepsilon_0<(3K_1)^{-1}$, we deduce enhanced estimate . Beside the propagation of estimate , by combining inequalities , , and , we also deduce the following inequality $$| \partial_t u (t,x)| + |\nabla_x u(t,x)|\le C\varepsilon \mathds{1}_{\{|x|\le Dt \}} (t+|x|)^{-\frac{1}{2}}(1+ |t-|x||)^{-\frac{1}{2}+\beta'}+ C\varepsilon t^{-1+\beta'},$$ with $\beta'>0$ small as long as $\sigma, \delta_1$ are small, which almost corresponds to the optimal decay in time and space enjoyed by the linear wave in space dimension two. \[Lem: est utilde Sigma,k large x\] Let $\chi\in C^\infty_0(\mathbb{R}^2)$ be equal to 1 in a neighbourhood of the origin and $\sigma>0$ be small. Let also $\varphi\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$. There exists a constant $C>0$ such that for every $h\in ]0,1[, R\gg 1$, and any function $w(t,x)$ with $w(t,\cdot), \oph(\chi(h^\sigma\xi))\mathcal{M}w(t,\cdot)\in L^2(\mathbb{R}^2)$, $$\label{ineq:utilde xbig} \Big\|\varphi\Big(\frac{\cdot}{R}\Big)\oph(\chi(h^\sigma\xi))w(t,\cdot)\Big\|_{L^\infty}\le C R^{-1}(\log{R}+|\log h|) \sum_{|\gamma|=0}^1\|\oph(\chi(h^\sigma\xi))\mathcal{M}^\gamma w(t,\cdot)\|_{L^2}\,.$$ Let us fix $R\gg 1$ and, for seek of compactness, denote $\oph(\chi(h^\sigma\xi))w$ by $w^\chi$. For a new smooth cut-off function $\chi_1$ equal to 1 on the support of $\chi$, we have that $$\varphi\Big(\frac{x}{R}\Big) \oph(\chi(h^\sigma\xi)) w= \oph(\chi_1(h^\sigma\xi)) \Big[\varphi\Big(\frac{x}{R}\Big) w^\chi\Big] + \left[\varphi\Big(\frac{x}{R}\Big), \oph(\chi_1(h^\sigma\xi))\right]w^\chi,$$ where the symbol associated to above commutator is given by $$r_R(x,\xi) = -\frac{h^{1+\sigma} R^{-1}}{i(\pi h)^2}\int e^{\frac{2i}{h}\eta\cdot z}\left[\int_0^1 (\partial \varphi)\Big(\frac{x+tz}{R}\Big) dt\right] (\partial \chi_1)(h^\sigma(\xi+\eta)) dz d\eta,$$ as follows from and integration in $dy,d\zeta$. Since $(\partial \chi_1)(h^\sigma\xi)$ is supported for frequencies $|\xi|\le h^{-\sigma}$, and $R^{-1}, h^{1+\sigma}\le 1$, by making a change of coordinates $\eta/h\mapsto \eta$ and using that $e^{2i \eta\cdot z} = \big(\frac{1-2i\eta\cdot\partial_z}{1+4|\eta|^2}\big)\big(\frac{1-2i z\cdot\partial_\eta}{1+4|z|^2}\big)e^{2i \eta\cdot z}$, together with some integration by parts, one can check that $$\left\|\partial^\alpha_y \partial^\beta_\xi \big[r_R(\frac{x+y}{2}, h\xi)\big]\right\|_{L^2(d\xi)}\lesssim R^{-1}$$ for any $\alpha,\beta\in\mathbb{N}^2$, and hence obtain from lemma \[Lemma on inequalities for Op(A)\] that $$\|\oph(r^k_R(x,\xi))w^\chi(t,\cdot)\|_{L^\infty}\lesssim R^{-1}\|w^\chi(t,\cdot)\|_{L^2}.$$ Successively, taking a Littlewood-Paley decomposition such that $$\chi_1(h^\sigma\xi) \equiv \left[\phi\Big(\frac{R}{h}\xi\Big) + \sum_{hR^{-1}\le 2^j\le h^{-\sigma}}(1-\phi)\Big(\frac{R}{h}\xi\Big)\psi(2^{-j}\xi)\right] \chi_1(h^\sigma\xi),$$ with $\phi\in C^\infty_0(\mathbb{R}^2)$, equal to 1 close to the origin and $\psi\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$, we derive that $$\begin{gathered} \label{Linfty_wchi_split1} \left\|\oph(\chi_1(h^\sigma\xi))\Big[ \varphi\Big(\frac{x}{R}\Big) w^\chi\Big](t,\cdot) \right\|_{L^\infty} \lesssim \left\| \oph\Big(\phi\Big(\frac{R}{h}\xi\Big)\chi_1(h^\sigma\xi)\Big)\Big[ \varphi\Big(\frac{x}{R}\Big) w^\chi\Big](t,\cdot) \right\|_{L^\infty}\\ + \sum_{hR^{-1}\le 2^j\le h^{-\sigma}} \left\|\oph\Big((1-\phi)\Big(\frac{R}{h}\xi\Big)\psi(2^{-j}\xi)\chi_1(h^\sigma\xi)\Big)\Big[ \varphi\Big(\frac{x}{R}\Big) w^\chi \Big](t,\cdot)\right\|_{L^\infty},\end{gathered}$$ and immediately notice that $$\begin{gathered} \label{Linfty_wchi_phi(Rh-1xi)} \left\| \oph\Big(\phi\Big(\frac{R}{h}\xi\Big)\chi_1(h^\sigma\xi)\Big)\Big[ \varphi\Big(\frac{x}{R}\Big) w^\chi\Big](t,\cdot) \right\|_{L^\infty} \\ =\left\|\phi(RD_x) \oph(\chi_1(h^\sigma\xi))\Big[ \varphi\Big(\frac{x}{R}\Big) w^\chi\Big](t,\cdot) \right\|_{L^\infty} \lesssim R^{-1}\|w^\chi(t,\cdot)\|_{L^2},\end{gathered}$$ just by the classical Sobolev injection and the uniform continuity of $\oph(\chi_1(h^\sigma\xi))\varphi\Big(\frac{x}{R}\Big)$ on $L^2$. Introducing operators $\Theta_R, \Theta^{-1}_R$, where $\Theta_R u(x) := u(Rx)$, $\Theta^{-1}_Ru (x) := u\big(\frac{x}{R}\big)$, we have the following equality $$\begin{gathered} \label{operators Theta R} \oph\Big((1-\phi)\Big(\frac{R}{h}\xi\Big)\psi(2^{-j}\xi)\chi_1(h^\sigma\xi)\Big)\Big[\varphi\Big(\frac{x}{R}\Big)w^\chi\Big]\\ = \Big[\Theta^{-1}_R Op^w_{h_{Rj}}\Big( (1-\phi)\Big(\frac{\xi}{h_{Rj}}\Big)\psi(\xi)\chi_1(h^\sigma 2^j\xi)\Big)\varphi(x) \Theta_R\Big]w^\chi\end{gathered}$$ with $h_{Rj}:=\frac{h}{R2^j}\le 1$, and by $h_{Rj}$-symbolic calculus (that is proposition \[Prop: a sharp b\] with $h$ replaced by $h_{Rj}$), $$\begin{gathered} Op^w_{h_{Rj}}\Big( (1-\phi)\Big(\frac{\xi}{h_{Rj}}\Big)\psi(\xi)\chi_1(h^\sigma 2^j\xi)\Big)\varphi(x) =\\ Op^w_{h_{Rj}}\Big( (1-\phi)\Big(\frac{\xi}{h_{Rj}}\Big)\psi(\xi)\chi_1(h^\sigma 2^j\xi)\varphi(x)\Big) + Op^w_{h_{Rj}}(r(x,\xi))\end{gathered}$$ with $$r(x,\xi) = \frac{h_{Rj}}{2i (\pi h_{Rj})^2}\int e^{-\frac{2i}{h_{Rj}}y\cdot\zeta} \left[\int_0^1\partial_\xi\Big[(1-\phi)\Big(\frac{\xi}{h_{Rj}}\Big)\psi(\xi)\chi_1(h^\sigma 2^j\xi)\Big]\big|_{(\xi + t\zeta)} dt\right] (\partial\varphi)(x+y) dyd\zeta.$$ Similarly as before, one can prove that $$\left\|\partial^\alpha_x \partial^\beta_\xi \big[r(\frac{x+y}{2}, h\xi)\big]\right\|_{L^2(d\xi)}\lesssim 1$$ for any $\alpha,\beta\in\mathbb{N}^2$, observing that no negative power of $h_{Rj}$ appears in the right hand side of this inequality for the product of $\psi(\xi)$ with any derivative of $(1-\phi)(\frac{\xi}{h_{Rj}})$ is supported for $h_{Rj} \sim |\xi| \sim 1$. Hence lemma \[Lemma on inequalities for Op(A)\] gives that operator $Op^w_{h_{Rj}}(r(x,\xi))$ is uniformly bounded from $L^2$ to $L^\infty$ and $$\big\|Op^w_{h_{Rj}}(r(x,\xi))\Theta_R w^\chi(t,\cdot)\big\|_{L^\infty}\lesssim \|\Theta_Rw^\chi(t,\cdot)\|_{L^2}\lesssim R^{-1}\|w^\chi(t,\cdot)\|_{L^2}\,.$$ Since symbol $(1-\phi)\big(\frac{\xi}{h_{Rj}}\big)\psi(\xi)\chi_1(h^\sigma 2^j\xi)\varphi(x)$ is supported for $|x|\sim|\xi|\sim 1$, $$\begin{gathered} (1-\phi)\Big(\frac{\xi}{h_{Rj}}\Big)\psi(\xi)\chi_1(h^\sigma 2^j\xi)\varphi(x) \\ = \sum_{l=1}^2\underbrace{\frac{(1-\phi)\big(\frac{\xi}{h_{Rj}}\big)\psi(\xi)\chi_1(h^\sigma 2^j\xi)\varphi(x)(Rx_l|2^j\xi| - 2^j\xi_l)}{|Rx|2^j\xi|-2^j\xi|^2}}_{a_l(x,\xi)}\big(Rx_l|2^j\xi| - 2^j\xi_l\big),\end{gathered}$$ with $a_l(x,\xi) \in R^{-1}2^{-j}S_{0,0}(1)$ as long as $R\gg 1$, and by $h_{Rj}$-symbolic calculus $$(1-\phi)\Big(\frac{\xi}{h_{Rj}}\Big)\psi(\xi)\chi_1(h^\sigma 2^j\xi)\varphi(x) = \sum_{l=1}^2 a_l(x,\xi)\sharp \Big[(Rx_l|2^j\xi| - 2^j\xi_l)\widetilde{\psi}(\xi)\Big] + r_{Rj}(x,\xi),$$ with $\widetilde{\psi}\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$ such that $\widetilde{\psi}\psi \equiv \psi$, and $r_{Rj}\in h_{Rj}S_{0,0}(1)$. From semi-classical Sobolev injection $$\|Op^w_{h_{Rj}}(r_{Rj}(x,\xi))\Theta_Rw^\chi(t,\cdot)\|_{L^\infty}\lesssim \|\Theta_Rw^\chi(t,\cdot)\|_{L^2}\le R^{-1}\|w^\chi(t,\cdot)\|_{L^2}$$ while $$\label{last formula Op(al)} \begin{split} & Op^w_{h_{Rj}}(a_l(x,\xi))Op_{h_{Rj}}^w\big((Rx_l|2^j\xi| - 2^j\xi)\widetilde{\psi}(\xi)\big)\Theta_Rw^\chi \\ &= Op_{h_{Rj}}^w(a_l(x,\xi))\Theta_R \Big[\oph\big((x_l|\xi| - \xi)\widetilde{\psi}(2^{-j}\xi)\big)w^\chi\Big] \\ &= Op_{h_{Rj}}^w(a_l(x,\xi))\Theta_R \Big[ \oph(\widetilde{\psi}(2^{-j}\xi))\oph(x_l|\xi| - \xi)w^\chi -\frac{h}{2i} \oph((2^{-j}\xi)\cdot(\partial\widetilde{\psi})(2^{-j}\xi))w^\chi\Big]. \end{split}$$ The last thing to do to conclude the proof of the statement is to study continuity of operator $Op_{h_{Rj}}^w(a_l(x,\xi))$. \[Lemma: continuity of Op(al)\] We have that $Op^w_{h_{Rj}}(a_l(x,\xi)) :L^2 \rightarrow L^\infty$ is bounded with norm $$\left\|Op^w_{h_{Rj}}(a_l(x,\xi))\right\|_{\mathcal{L}(L^2;L^\infty)} \lesssim h^{-1}.$$ The result comes straightly from lemma \[Lemma on inequalities for Op(A)\]. Indeed, since symbol $a_l(x,\xi)$ is compactly supported in $x$ there is a smooth cut-off function $\varphi_1 \in C^\infty_0(\mathbb{R}^2\setminus\{0\})$, with $\varphi_1 \varphi \equiv \varphi$, such that $$\left|Op^w_{h_{Rj}}(a_l(x,\xi))w\right| \lesssim\|w\|_{L^2(dx)}\int \Big|\varphi_1\Big(\frac{x+y}{2}\Big)\Big| \sum_{|\alpha|\le 3}\left\|\partial^\alpha_y \left[a_l\Big(\frac{x+y}{2},h_{R_j}\xi\Big)\right]\right\|_{L^2(d\xi)}dy,$$ and for $|\alpha|\le 3$ $$\begin{aligned} &\left\|\partial^\alpha_y \left[a_l\Big(\frac{x+y}{2},h_{R_j}\xi\Big)\right]\right\|_{L^2(d\xi)} \\ &\hspace{20pt}\lesssim\frac{R}{h}\left\|\partial^\alpha_y \left[\frac{(1-\phi)(\xi)\psi(h_{Rj}\xi)\chi_1( h_{Rj} h^\sigma2^j\xi)\varphi_1(\frac{x+y}{2})}{|R(\frac{x+y}{2})|\xi| -\xi|^2}\Big(R\Big(\frac{x_l+y_l}{2}\Big)|\xi| -\xi_l\Big)\right]\right\|_{L^2(d\xi)}\\ &\hspace{20pt} \lesssim \frac{|\widetilde{\varphi}(\frac{x+y}{2})|}{h} \left(\int \frac{|\psi(h_{Rj}\xi)|^2}{|\xi|^2} d\xi\right)^{\frac{1}{2}} \lesssim \frac{|\widetilde{\varphi}(\frac{x+y}{2})|}{h},\end{aligned}$$ where $\widetilde{\varphi}\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$. Finally, summing up all formulas from to and using lemma \[Lemma: continuity of Op(al)\], we obtain that $$\Big\|\oph\Big((1-\phi)\Big(\frac{R}{h}\xi\Big)\psi(2^{-j}\xi)\chi_1(h^\sigma\xi)\Big)\Big[\varphi\Big(\frac{x}{R}\Big)w^\chi(t,\cdot)\Big]\Big\|_{L^\infty}\lesssim R^{-1}(\|w^\chi(t,\cdot)\|_{L^2} + \|\mathcal{M}w^\chi(t,\cdot)\|_{L^2}),$$ for any index $j\in \mathbb{Z}$ such that $hR^{-1}\le 2^j\le h^{-\sigma}$. Injecting and the above inequality in , and using that $[\mathcal{M}, \oph(\chi(h^\sigma\xi))]=i \oph((\partial \chi)(h^\sigma\xi)(h^\sigma |\xi|))$ is uniformly continuous on $L^2$, we deduce (the loss in $\log{R}+|\log h|$ arising from the fact that we are considering a sum over indices $j$, with $\log h-\log{R}\lesssim j \lesssim \log (h^{-1})$). ### Proof of the main theorems Straightforward after propositions \[Prop: Propagation of the energy estimate\], \[Prop:propagation\_unif\_est\_V\], \[Prop: Propagation uniform estimate U,RU\]. Let us prove that, for small enough data satisfying , Cauchy problem - has a unique global solution. This result follows by a local existence argument, after having proved that there exist two integers $n\gg \rho\gg 1$, two constants $A', B'>1$ sufficiently large, $\varepsilon_0>0$ sufficiently small, and $0<\delta\ll \delta_2\ll \delta_1\ll \delta_0$ small, such that, for any $0<\varepsilon<\varepsilon_0$, if $(u,v)$ is solution to - in $[1,T]\times\mathbb{R}^2$, for some $T>1$, with $\partial_{t,x}u\in C^0([1,T]; H^n(\mathbb{R}^2))$, $v\in C^0([1,T]; H^{n+1}(\mathbb{R}^2))\cap C^1([1,T];H^n(\mathbb{R}^2))$, and satisfies \[a-priori\_estimate\_u,v\] $$\begin{gathered} \|\partial_tu(t,\cdot)\|_{H^{\rho+1,\infty}} + \|\nabla_x u(t,\cdot)\|_{H^{\rho+1,\infty}} + \||D_x|u(t,\cdot)\|_{H^{\rho+1,\infty}}+ \sum_{j=1}^2\|\mathrm{R}_j\partial_tu(t,\cdot)\|_{H^{\rho+1,\infty}}\le A'\varepsilon t^{-\frac{1}{2}},\\ \|\partial_tv(t,\cdot)\|_{H^{\rho,\infty}} + \|v(t,\cdot)\|_{H^{\rho+1,\infty}}\le A'\varepsilon t^{-1}, \\ \|\partial_tu(t,\cdot)\|_{H^n} + \|\nabla_x u(t,\cdot)\|_{H^n}+ \|\partial_tv(t,\cdot)\|_{H^n}+\|\nabla_xv(t,\cdot)\|_{H^n}+\|v(t,\cdot)\|_{H^n}\le B'\varepsilon t^\frac{\delta}{2}, \end{gathered}$$ $$\begin{gathered} \sum_{|I|=k} \left[\|\partial_t \Gamma^I u(t,\cdot)\|_{L^2} + \|\nabla_x \Gamma^I u(t,\cdot)\|_{L^2}+ \|\partial_t \Gamma^I v(t,\cdot)\|_{L^2}+\|\nabla_x \Gamma^Iv(t,\cdot)\|_{L^2}\right.\\ \left.+\|\Gamma^I v(t,\cdot)\|_{L^2}\right]\le B'\varepsilon t^\frac{\delta_{3-k}}{2}, \quad 1\le k \le 3,\end{gathered}$$ for every $t\in [1,T]$, then in the same interval it satisfies \[a-priori\_enhance\_estimate\_u,v\] $$\begin{gathered} \|\partial_tu(t,\cdot)\|_{H^{\rho+1,\infty}} + \|\nabla_x u(t,\cdot)\|_{H^{\rho+1,\infty}} + \||D_x|u(t,\cdot)\|_{H^{\rho+1,\infty}}+ \sum_{j=1}^2\|\mathrm{R}_j\partial_tu(t,\cdot)\|_{H^{\rho+1,\infty}}\le \frac{A'}{2}\varepsilon t^{-\frac{1}{2}},\\ \|\partial_tv(t,\cdot)\|_{H^{\rho,\infty}} + \|v(t,\cdot)\|_{H^{\rho+1,\infty}}\le \frac{A'}{2}\varepsilon t^{-1}, \\ \|\partial_tu(t,\cdot)\|_{H^n} + \|\nabla_x u(t,\cdot)\|_{H^n}+ \|\partial_tv(t,\cdot)\|_{H^n}+\|\nabla_xv(t,\cdot)\|_{H^n}+\|v(t,\cdot)\|_{H^n}\le \frac{B'}{2}\varepsilon t^\frac{\delta}{2}, \\\end{gathered}$$ $$\begin{gathered} \sum_{|I|=k} \left[\|\partial_t \Gamma^I u(t,\cdot)\|_{L^2} + \|\nabla_x \Gamma^I u(t,\cdot)\|_{L^2}+ \|\partial_t \Gamma^I v(t,\cdot)\|_{L^2}+\|\nabla_x \Gamma^Iv(t,\cdot)\|_{L^2}\right.\\ \left. +\|\Gamma^I v(t,\cdot)\|_{L^2}\right]\le \frac{B'}{2}\varepsilon t^\frac{\delta_{3-k}}{2}, \quad 1\le k\le 3.\end{gathered}$$ We remind that, if $I=(i_1,\dots, i_n)$ is a multi-index of length $|I|=n$, with $i_j\in \{1,\dots,5\}$, $\Gamma^I = \Gamma_{i_1}\cdots \Gamma_{i_n}$ is a product of vector fields in family $\mathcal{Z}=\{\Omega, Z_j, \partial_j |j=1,2\}$. We can immediately observe that the above bounds are verified at time $t=1$ after and Sobolev injection. By definition we also notice that \[dependence\_unif\_norm\_u,v\_upm,vpm\] $$\begin{gathered} \|u_\pm (t,\cdot)\|_{H^{\rho+1,\infty}}+ \sum_{j=1}^2\|\mathrm{R}_ju_\pm (t,\cdot)\|_{H^{\rho+1,\infty}}\le 2 \|\partial_tu(t,\cdot)\|_{H^{\rho+1,\infty}}+ 2\||D_x|u(t,\cdot)\|_{H^{\rho+1,\infty}}\\ + 2 \sum_{j=1}^2\left( \|\partial_j u(t,\cdot)\|_{H^{\rho+1,\infty}}+\|\mathrm{R}_j\partial_tu(t,\cdot)\|_{H^{\rho+1,\infty}}\right),\end{gathered}$$ $$\|v_\pm (t,\cdot)\|_{H^{\rho,\infty}}\le 2\|\partial_tv(t,\cdot)\|_{H^{\rho,\infty}}+ 2\|v(t,\cdot)\|_{H^{\rho+1,\infty}},$$ and, conversely, \[dependence u,v in terms of upm vpm\] $$\begin{gathered} \|\partial_tu(t,\cdot)\|_{H^{\rho+1,\infty}} + \||D_x|u(t,\cdot)\|_{H^{\rho+1,\infty}}+ \sum_{j=1}^2\left(\|\partial_ju(t,\cdot)\|_{H^{\rho+1,\infty}}+ \|\mathrm{R}_j\partial_tu(t,\cdot)\|_{H^{\rho+1,\infty}}\right) \\ \le \|u_+(t,\cdot)\|_{H^{\rho+1,\infty}}+ \|u_{-}(t,\cdot)\|_{H^{\rho+1,\infty}} + \sum_{j=1}^2 \left(\|\mathrm{R}_j u_+(t,\cdot)\|_{H^{\rho+1,\infty}}+ \|\mathrm{R}_j u_{-}(t,\cdot)\|_{H^{\rho+1,\infty}}\right),\end{gathered}$$ $$\|\partial_tv(t,\cdot)\|_{H^{\rho,\infty}}+ \|v(t,\cdot)\|_{H^{\rho+1,\infty}}\le \|v_+(t,\cdot)\|_{H^{\rho,\infty}}+\|v_{-}(t,\cdot)\|_{H^{\rho,\infty}}.$$ Moreover, reminding definition of generalized energies $E_n(t;u_\pm, v_\pm)$, $E^k_3(t;u_\pm, v_\pm)$, for $n\ge 3$ and $0\le k \le 2$, and of set $\mathcal{I}^k_3$ in , there is a constant $C>0$ such that \[dependence\_energy\_u,v\_energy upm vpm\] $$\begin{gathered} C^{-1}E_n(t;u_\pm, v_\pm) \le \left[\|\partial_tu(t,\cdot)\|^2_{H^n} + \|\nabla_x u(t,\cdot)\|^2_{H^n}\right.\\ \left.+ \|\partial_tv(t,\cdot)\|^2_{H^n}+\|\nabla_xv(t,\cdot)\|^2_{H^n}+\|v(t,\cdot)\|^2_{H^n}\right] \le C E_n(t;u_\pm, v_\pm),\end{gathered}$$ and for any $0\le k\le 2$, $$\begin{gathered} C^{-1}E^k_3(t;u_\pm, v_\pm) \le \sum_{I\in\mathcal{I}^k_3} \left[\|\partial_t \Gamma^I u(t,\cdot)\|^2_{L^2} + \|\nabla_x \Gamma^I u(t,\cdot)\|^2_{L^2}\right. \\ \left. + \|\partial_t \Gamma^I v(t,\cdot)\|^2_{L^2}+\|\nabla_x \Gamma^Iv(t,\cdot)\|^2_{L^2}+\|\Gamma^I v(t,\cdot)\|^2_{L^2}\right] \le C E^k_3(t;u_\pm, v_\pm).\end{gathered}$$ Therefore, after , , and , we deduce that estimates are satisfied with $A=2A'$, $B=C_1B'$, for some new $C_1>0$, so choosing for instance $K_1=4$ and $K_2$ sufficiently large, theorem \[Thm: bootstrap argument\] and inequalities , imply . {#Appendix A} The aim of this appendix is to prove the continuity of some trilinear integral operators (see lemmas \[Lem\_appendix: est integrals Bj u v\] and \[Lem\_appendix: integral sigma\_tilde\_N\]) that arise in subsection \[sub: second normal form\] when performing a normal form argument at the energy level, and of some bilinear integral operators (see lemma \[Lem\_Appendix: est on Dj1j2\]) that instead appear in subsection \[Subsection: Section : Normal Forms for the Wave Equation\] when we perform a normal form the wave equation (see proposition \[Prop: NF on wave\]). All the other results of this chapter are stated and proved in view of the above mentioned lemmas. \[Lem\_appendix: Kernel with 1 function\] Let $\check{a}(x)$ denote the inverse transform of a function $a(\xi)$. $(i)$ If $a:\mathbb{R}^2\rightarrow \mathbb{C}$ is such that, for any $\alpha\in \mathbb{N}^2$ with $1\le |\alpha|\le 4$, $$| a(\xi)|\lesssim \langle\xi\rangle^{-3} \quad \text{and}\quad |\partial^\alpha a(\xi)| \lesssim_{\alpha} (|\xi|\langle\xi\rangle^{-1})^{1-|\alpha|}\langle \xi\rangle^{-3}\quad \forall \xi\in\mathbb{R}^2$$ then $$| \check{a}(x)|\lesssim |x|^{-1}\langle x \rangle^{-2}, \quad\forall x\in\mathbb{R}^2.$$ $(ii)$ If $a$ is such that, for any $\alpha\in\mathbb{N}^2$ with $|\alpha|\le 3$, $$|\partial^\alpha a(\xi)|\lesssim (|\xi|\langle\xi\rangle^{-1})^{-|\alpha|}\langle\xi\rangle^{-3},\quad \forall\xi\in\mathbb{R}^2$$ then $$|\check{a}(x)|\lesssim\langle x \rangle^{-2}, \quad \forall x\in\mathbb{R}^2;$$ $(iii)$ Let $N\in\mathbb{N}$. If for any $\alpha\in\mathbb{N}^2$ with $|\alpha|\le N$ there exists $f_\alpha\in L^1(\mathbb{R}^2)$ such that $|\partial^\alpha a(\xi)|\lesssim_\alpha |f_\alpha(\xi)|$ then $$|\check{a}(x)|\lesssim \langle x\rangle^{-N}, \quad \forall x\in\mathbb{R}^2.$$ *(i)* We consider a cut-off function $\phi\in C^\infty_0(\mathbb{R}^2)$ equal to 1 in the unit ball and write $$\label{splitting_check_a} \begin{gathered} \check{a}(x) = K_0(x) + K_1(x) \\ \text{with}\quad K_0(x) :=\frac{1}{(2\pi)^2} \int e^{ix\cdot\xi} a(\xi)\phi(\xi) d\xi, \quad K_1(x) :=\frac{1}{(2\pi)^2} \int e^{ix\cdot\xi} a(\xi)(1-\phi)(\xi) d\xi. \end{gathered}$$ On the one hand, since $|\partial^\alpha a(\xi)|\lesssim_{\alpha} \langle\xi\rangle^{-3}$ on the support of $(1-\phi)(\xi)$ for any $|\alpha|\le 4$, we immediately deduce by integration by parts that $| K_1(x)|\lesssim \langle x \rangle^{-4}$ for any $x\in \mathbb{R}^2$. On the other hand, again an integration by parts gives that $$x K_0(x) = \int e^{ix\cdot\xi} a_1(\xi) d\xi$$ with $a_1(\xi)$ supported for $|\xi|\lesssim 1$ and such that $|\partial^\alpha a_1(\xi)|\lesssim_\alpha |\xi|^{-|\alpha|}$ for any $\xi\in\mathbb{R}^2$, any $|\alpha|\le 3$. This implies that $|x K_0(x)|\lesssim 1$ for any $x\in\mathbb{R}^2$. Moreover, $|x^\alpha x\ K_0(x)|\lesssim_\alpha 1$ for any $|\alpha|\le 3$. This is obvious in the unit ball. Out of the unit ball we consider a Littlewood-Paley decomposition in frequencies so that $$\phi(\xi) = \phi(\xi) \left[\varphi_0(2^{-L_0}\xi) + \sum_{k=L_0+1}^0\varphi(2^{-k}\xi)\right],$$ with $supp\varphi_0\subset B_1(0)$, $\varphi\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$ and $L_0<0$ such that $2^{L_0}\sim |x|^{-1}$, and write $$\begin{gathered} xK_0(x) = K_0^0(x) + \sum_{k=L_0+1}^0 K_0^k(x) \\ \text{with}\quad K^0_0(x) := \int e^{ix\cdot\xi} a_1(\xi) \varphi_0(2^{-L_0}\xi) d\xi, \quad K^k_0(x) := \int e^{ix\cdot\xi} a_1(\xi) \varphi_k(2^{-k}\xi) d\xi. \end{gathered}$$ Performing a change of coordinates and making some integrations by parts we deduce that $$|K^0_0(x)|\lesssim 2^{2L_0} \quad \text{and} \quad |K^k_0(x)|\lesssim 2^{2k}\langle 2^k x\rangle^{-3}, \quad L_0+1\le k\le 0$$ for any $x\in\mathbb{R}^2$, which finally implies $|x K_0(x)|\lesssim 2^{2L_0}\sim |x|^{-2} $. *(ii)* The result follows splitting $\check{a}$ as in and applying to $K_0(x)$ the same argument previously used for $xK_0(x)$. $(iii)$ The result follows straightly from integration by parts and the fact that $f_\alpha\in L^1(\mathbb{R}^2)$ for any $|\alpha|\le N$. \[Cor\_appendix: decay of integral operators\] Let $d\in\mathbb{N}^*$, $N\in\mathbb{N}$ and $g_\beta \in L^1(\mathbb{R}^d)$ for every $|\beta|\le N$. $(i)$ If $a(\xi,\eta):\mathbb{R}^2\times\mathbb{R}^d\rightarrow \mathbb{C}$ is such that, for any $\beta\in\mathbb{N}^d$ with $|\beta|\le N$, $$\label{ineq_a_1} \begin{gathered} |\partial^\beta_\eta a(\xi,\eta)|\lesssim_\beta \langle\xi\rangle^{-3}|g_\beta(\eta)|, \\ |\partial^\alpha_\xi \partial^\beta_\eta a(\xi,\eta)|\lesssim_{\alpha,\beta} (|\xi|\langle\xi\rangle^{-1})^{1-|\alpha|} \langle\xi\rangle^{-3}|g_\beta(\eta)|, \quad 1\le |\alpha|\le 4. \end{gathered}$$ for any $(\xi,\eta)\in\mathbb{R}^2\times\mathbb{R}^d$, then $$\label{eq_cor_1} \left|\int e^{ix\cdot\xi + iy\cdot\eta} a(\xi,\eta) d\xi d\eta\right| \lesssim |x|^{-1}\langle x \rangle^{-2}\langle y \rangle^{-N}, \quad \forall (x,y)\in\mathbb{R}^2\times \mathbb{R}^d.$$ Moreover, if $d=2$ and $N=3$, for any $u,v\in L^2(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$ \[est:coroll\_app\_L\_norm\] $$\label{est: corollary_app L2 norm} \left\|\int e^{ix\cdot\xi}a(\xi,\eta)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{L^2(dx)} \lesssim \|u\|_{L^2}\|v\|_{L^\infty} \, (\text{or } \lesssim \|u\|_{L^\infty}\|v\|_{L^2})$$ and $$\left\|\int e^{ix\cdot\xi}a(\xi,\eta)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{L^\infty(dx)} \lesssim \|u\|_{L^\infty}\|v\|_{L^\infty}.$$ $(ii)$ If $a(\xi,\eta)$ is such that, for any $\alpha\in\mathbb{N}^2$ with $|\alpha|\le 3$, $\beta\in\mathbb{N}^d$ with $|\beta|\le N$, $$\label{ineq_a_2} |\partial^\alpha_\xi \partial^\beta_\eta a(\xi,\eta)|\lesssim_{\alpha,\beta} (|\xi|\langle\xi\rangle^{-1})^{-|\alpha|} \langle\xi\rangle^{-3}|g_\beta(\eta)|,$$ for any $(\xi,\eta)\in\mathbb{R}^2\times\mathbb{R}^d$, then $$\label{eq_cor_2} \left|\int e^{ix\cdot\xi + iy\cdot\eta} a(\xi,\eta) d\xi d\eta\right| \lesssim\langle x \rangle^{-2}\langle y \rangle^{-N}, \quad \forall (x,y)\in\mathbb{R}^2\times \mathbb{R}^d.$$ Moreover, if $d=2,N=3$, for any $u,v\in L^2(\mathbb{R}^2)$ \[ineq\_corA2\] $$\left\|\int e^{ix\cdot\xi}a(\xi,\eta)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{L^2(dx)} \lesssim \|u\|_{L^2}\|v\|_{L^2}$$ while if $u\in L^2(\mathbb{R}^2), v\in L^\infty(\mathbb{R}^2)$, $$\left\|\int e^{ix\cdot\xi}a(\xi,\eta)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{L^\infty(dx)} \lesssim \|u\|_{L^2}\|v\|_{L^\infty}.$$ Let $$K(x,\eta):=\int e^{ix\cdot\xi}a(\xi,\eta)d\xi \quad\text{and}\quad \widetilde{K}(x,y) := \int e^{ix\cdot\xi }K(x,\eta) d\eta.$$ By the hypothesis on $a(\xi,\eta)$ and lemma \[Lem\_appendix: Kernel with 1 function\] $(i)$ (resp. $(ii)$) we derive that, for any $\beta\in\mathbb{N}^d$ with $|\beta|\le N$, $$|\partial^\beta_\eta K(x,\eta)|\lesssim |x|^{-1}\langle x\rangle^{-2}|g_\beta(\eta)| \quad \left(\text{resp. } |\partial^\beta_\eta K(x,\eta)|\lesssim \langle x\rangle^{-2}|g_\beta(\eta)|\right) \quad \forall (x,\eta)\in\mathbb{R}^2\times\mathbb{R}^d.$$ Hence (resp. ) follows applying lemma \[Lem\_appendix: Kernel with 1 function\] $(iii)$ to $\widetilde{K}(x,y)$. $(i)$ If $d=2, N=3$, inequality from the fact that $$\int e^{ix\cdot\xi}a(\xi, \eta)\hat{u}(\xi-\eta) \hat{v}(\eta) d\eta = \int \widetilde{K}(x-y, y-z) u(y)v(z) dydx,$$ and by , for $L=L^2$ or $L=L^\infty$, $$\label{ineq: norm L(dx) kernel} \begin{split} \left\| \int \widetilde{K}(x-y,y-z) \widetilde{u}(y) \widetilde{v}(z) dydz\right\|_{L(dx)}& \lesssim \left\| \int |x-y|^{-1}\langle x-y\rangle^{-2}\langle y-z\rangle^{-3} \widetilde{u}(y) \widetilde{v}(z) dydz\right\|_{L(dx)} \\ & \lesssim \int |y|^{-1}\langle y \rangle^{-2}\langle z \rangle^{-3} \|\widetilde{u}(\cdot -y) \widetilde{v}(\cdot - y -z)\|_{L(dx)} dydz\\ & \lesssim \|\widetilde{u}\|_{L^\infty}\|\widetilde{v}\|_{L} \ (\text{or } \lesssim \|\widetilde{u}\|_{L}\|\widetilde{v}\|_{L^\infty} ). \end{split}$$ $(ii)$ By inequality $$\begin{split} &\left\| \int \widetilde{K}(x-y,y-z)u(y) v(z) dydz\right\|_{L^2(dx)}\lesssim \left\| \int\langle x-y\rangle^{-2}\langle y-z\rangle^{-3}| u(y) ||v(z)| dydz\right\|_{L^2(dx)}\\ & \lesssim \int \langle y- z\rangle^{-3}|u(y)| |v(z)| dydz \lesssim \int |v(z)| \left(\int \langle y-z\rangle^{-3}dy\right)^\frac{1}{2} \left(\int \langle y-z\rangle^{-3}|u(y)|^2 dy\right)^\frac{1}{2} dz\\ & \lesssim \|v\|_{L^2} \left(\int \langle y-z\rangle^{-3}|u(y)|^2 dydz\right)^\frac{1}{2}\lesssim \|u\|_{L^2}\|v\|_{L^2} \end{split}$$ and $$\begin{gathered} \left\| \int \widetilde{K}(x-y,y-z)u(y) v(z) dydz\right\|_{L^\infty(dx)}\lesssim \left\| \int\langle x-y\rangle^{-2}\langle y-z\rangle^{-3}| u(y) ||v(z)| dydz\right\|_{L^\infty(dx)}\\ \lesssim \|v\|_{L^\infty} \left\|\int \langle x-y\rangle^{-2}|u(y)| dy \right\|_{L^\infty(dx)} \lesssim \|u\|_{L^2}\|v\|_{L^\infty}.\end{gathered}$$ \[Lem:Sobolev norm of products\] Let $s\in\mathbb{N}^*$. For any $u,v\in H^{s}(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$, $$\label{Hs_norm_product} \|uv\|_{H^s}\lesssim \|u\|_{H^{s}}\|v\|_{L^\infty}+ \|u\|_{L^\infty}\|v\|_{H^{s}};$$ for any $u,v\in H^{s,\infty}(\mathbb{R}^2)\cap H^{s+2}(\mathbb{R}^2)$, any $\theta\in ]0,1[$, $$\label{Hsinfty_norm_product} \|uv\|_{H^{s,\infty}}\lesssim \|u\|^{1-\theta}_{H^{s,\infty}}\|u\|^\theta_{H^{s+2}}\|v\|_{L^\infty}+\|u\|_{L^\infty}\|v\|^{1-\theta}_{H^{s,\infty}}\|v\|^\theta_{H^{s+2}}.$$ Inequality is a classical result (see, for instance, [@Alinhac-Gerard]). In order to deduce we decompose product $uv$ as follows: $$\label{dec_para-products} uv= T_uv+ T_vu +R(u,v),$$ where $T_uv$ is the para-product of $u$ times $v$ defined by $$T_u v:= S_{-3}u S_0 v +\sum_{k\ge 1}S_{k-3}u \Delta_kv,$$ with $S_k = \chi(2^{-k}D_x)$, $\chi\in C^\infty_0(\mathbb{R}^2)$ such that $\chi(\xi)=1$ for $|\xi|\le 1/2$, $\chi(\xi)=0$ for $|\xi|\ge 1$, $\Delta_0= S_0$ and $\Delta_k = S_k- S_{k-1}$ for $k\ge 1$, and $R(u,v)=\sum_{k}\Delta_k u\widetilde{\Delta}_k v,$ with $\widetilde{\Delta}_k=\Delta_{k-1}+\Delta_k +\Delta_{k+1}$. Since $$T_uv =\sum_{j\ge 0}\Delta_j (T_uv) =\sum_{\substack{j,k\\ |j-k|\le N_0}} \Delta_j [S_{k-3}u \Delta_k v]$$ for a certain $N_0\in\mathbb{N}$, by definition \[def Sobolev spaces-NEW\] $(iii)$ of the $H^{s,\infty}$ norm and the fact that $\|\Delta_k v\|_{L^\infty}\lesssim 2^k\|\Delta_kv\|_{L^2}$ we deduce that, for any fixed $\theta\in ]0,1[$, $$\label{Hsinfty_norm_Tuv} \begin{split} &\|T_uv\|_{H^{s,\infty}} = \|\langle D_x\rangle^s T_uv\|_{L^\infty} \le \sum_{\substack{j,k\\ |j-k|\le N_0}}2^{js}\|\Delta_j [S_{k-3}u \Delta_k v]\|_{L^\infty}\\ & \le \sum_{\substack{j,k\\ |j-k|\le N_0}}2^{js} \|S_{k-3}u\|_{L^\infty} \|\Delta_kv\|_{L^\infty} \le \sum_{\substack{j,k\\ |j-k|\le N_0}}2^{js} \|u\|_{L^\infty} (2^{-ks}\|\Delta_k \langle D_x\rangle^s v\|_{L^\infty})^{1-\theta}(2^k\|\Delta_k v\|_{L^2})^\theta \\ &\lesssim \sum_{\substack{j,k\\ |j-k|\le N_0}}2^{(j-k)s} \|u\|_{L^\infty}\|\Delta_k \langle D_x\rangle^s v\|_{L^\infty}^{1-\theta}\left(2^{-k}\|\Delta_k \langle D_x\rangle^{s+2}v\|_{L^2}\right)^\theta\\ &\lesssim \|u\|_{L^\infty}\|v\|^{1-\theta}_{H^{s,\infty}}\|v\|^\theta_{H^{s+2}}. \end{split}$$ Similarly, $$\|T_vu\|_{H^{s,\infty}} + \|R(u,v)\|_{H^{s,\infty}}\lesssim \|u\|^{1-\theta}_{H^{s,\infty}}\|u\|^\theta_{H^{s+2}}\|v\|_{L^\infty}.$$ \[Cor\_appendixA:Hs-Hsinfty norm of bilinear expressions\] Let $s\in \mathbb{N}^*$, $a_1(\xi)\in S^{m_1}_0(\mathbb{R}^2)$, $a_2(\xi)\in S^{m_2}_0(\mathbb{R}^2)$, for some $m_1,m_2\ge 0$. For any $u\in H^{s+m_1}(\mathbb{R}^2)\cap H^{m_1,\infty}(\mathbb{R}^2)$, $v\in H^{s+m_2}(\mathbb{R}^2)\cap H^{m_2,\infty}(\mathbb{R}^2)$, $$\label{Hs_norm_bilinear-expressions} \left\|[a_1(D_x)u]\, [a_2(D_x)v] \right\|_{H^s}\lesssim \|u\|_{H^{s+m_1}}\|v\|_{H^{m_2,\infty}}+ \|u\|_{H^{m_1,\infty}}\|v\|_{H^{s+m_2}};$$ for any $u\in H^{s+m_1,\infty}(\mathbb{R}^2)\cap H^{s+m_1+2}(\mathbb{R}^2)$, $v\in H^{s+m_2,\infty}(\mathbb{R}^2)\cap H^{s+m_2+2}(\mathbb{R}^2)$, any $\theta\in ]0,1[$, $$\begin{gathered} \label{Hsinfty_norm_bilinear-expressions} \left\|[a_1(D_x)u]\, [a_2(D_x)v] \right\|_{H^{s,\infty}}\\ \lesssim \|u\|^{1-\theta}_{H^{s+m_1,\infty}} \|u\|^\theta_{H^{s+m_1+2}}\|v\|_{H^{m_2,\infty}}+ \|u\|_{H^{m_1,\infty}}\|v\|^{1-\theta}_{H^{s+m_2,\infty}}\|v\|^\theta_{H^{s+m_2+2}}.\end{gathered}$$ The result of the statement follows writing $[a_1(D_x)u]\, [a_2(D_x)v]$ in terms of para-products as in , and using that $T_{a_1(D)u}(a_2(D)v)$, $T_{a_2(D)v}(a_1(D)u)$ and remainder $R\big(a_1(D)u, a_2(D)v\big)$ can be written from $\widetilde{u}=\langle D_x\rangle^{m_1}u$, $\widetilde{v}=\langle D_x\rangle^{m_2}v$, as done below for the former of these terms: $$\begin{split} T_{a_1(D)u}(a_2(D)v) &= [S_{-3}a_1(D)\langle D_x\rangle^{-m_1}\widetilde{u}] [S_0a_2(D)\langle D_x\rangle^{-m_2}\widetilde{v}] \\ & + \sum_{k}[S_{k-3}a_1(D)\langle D_x\rangle^{-m_1}\widetilde{u}] [\Delta_k a_2(D)\langle D_x\rangle^{-m_2}\widetilde{v}]. \end{split}$$ Since $a_j(\xi)\langle\xi\rangle^{-m_j}\in S^0_0(\mathbb{R}^2)$, $j=1,2$, operators $S_k a_j(D)\langle D_x\rangle^{-m_j}$, $\Delta_k a_j(D)\langle D_x\rangle^{-m_j}$ have the same spectrum (i.e. the support of the Fourier transform) up to a negligible constant of $S_k$ and $\Delta_k$ respectively. In the following lemma we prove a result of continuity for a trilinear integral operator defined from multiplier $B^k_{(j_1,j_2,j_3)}(\xi,\eta)$ given by (resp. by ) for $k=1,2$ (resp. $k=3$), any $j_1,j_2,j_3\in \{+,-\}$. It is useful to observe that, since $$B^k_{(j_1,j_2,j_3)}(\xi,\eta) = \frac{j_1\langle\xi-\eta\rangle + j_2|\eta| - j_3\langle\xi\rangle}{2j_1 j_2\langle\xi-\eta\rangle|\eta|}\eta_k, \quad k=1,2$$ from , while $$B^3_{(j_1,j_2,j_3)}(\xi,\eta) = j_1\, \frac{j_1\langle\xi-\eta\rangle + j_2|\eta| - j_3\langle\xi\rangle}{2\langle\xi-\eta\rangle}$$ , we have that $$\begin{gathered} \label{explicit integral B} \frac{1}{(2\pi)^2}\int e^{ix\cdot\xi}B^k_{(j_1,j_2,j_3)}(\xi,\eta)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta = \frac{j_2}{2} (u\mathrm{R}_kv)(x) - \frac{j_1}{2} \Big[\Big(\frac{D_1}{\langle D_x\rangle}u\Big)v\Big](x) \\ + \frac{j_1}{2}D_1 \left[(\langle D_x\rangle^{-1}u)v\right](x) - \frac{j_3}{2j_1j_2}\langle D_x\rangle [(\langle D_x\rangle^{-1}u)(\mathrm{R}_kv)](x)\end{gathered}$$ for $k=1,2$, while for $k=3$ $$\begin{gathered} \label{explicit_integral_B3} \frac{1}{(2\pi)^2}\int e^{ix\cdot\xi}B^3_{(j_1,j_2,j_3)}(\xi,\eta)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta = \frac{1}{2}(uv)(x) + \frac{j_1j_2}{2}[ (\langle D_x\rangle^{-1}u) |D_x|v](x) \\ - \frac{j_1j_3}{2}\langle D_x\rangle\left[(\langle D_x\rangle^{-1} u) v\right](x).\end{gathered}$$ \[Lem\_appendix: est integrals Bj u v\] Let $B^k_{(j_1,j_2,j_3)}(\xi,\eta)$ be given by when $k=1,2$, and by when $k=3$, for any $j_1,j_2,j_3\in \{+,-\}$. Let also $\delta_k=1$ if $k\in \{1,2\}$, $\delta_k=0$ if $k=3$. For any $u,w\in L^2(\mathbb{R}^2), v\in H^{2,\infty}(\mathbb{R}^2)$ such that $\delta_k\mathrm{R}_kv\in H^{2,\infty}(\mathbb{R}^2)$, $$\label{estimate integral B uvw-new} \left|\int B^k_{(j_1,j_2,j_3)}(\xi,\eta)\hat{u}(\xi-\eta)\hat{v}(\eta)\hat{w}(-\xi) d\xi d\eta\right| \lesssim \|u\|_{L^2}\left(\|v\|_{H^{7,\infty}}+\delta_k \|\mathrm{R}_kv\|_{H^{7,\infty}}\right) \|w\|_{L^2}.$$ First of all we observe that for $k\in\{1,2\}$ $$\begin{gathered} \label{int_Bk_app} \int B^k_{(j_1,j_2,j_3)}(\xi,\eta)\hat{u}(\xi-\eta)\hat{v}(\eta)\hat{w}(-\xi) d\xi d\eta = \frac{j_2}{2} \int {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{u(\mathrm{R}_kv)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{u(\mathrm{R}_kv)}{\tmpbox}}(\xi) \hat{w}(-\xi)d\xi - \frac{j_1}{2} \int {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\big[\big(\frac{D_x}{\langle D_x\rangle}u\big)v\big]}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\big[\big(\frac{D_x}{\langle D_x\rangle}u\big)v\big]}{\tmpbox}}(\xi) \hat{w}(-\xi) d\xi \\ + \frac{j_1}{2}\int \frac{\xi_1}{\langle\xi-\eta\rangle} \hat{u}(\xi-\eta)\hat{v}(\eta)\hat{w}(-\xi) d\xi d\eta -\frac{j_3}{2j_1j_2}\int \frac{\langle\xi\rangle}{\langle\xi-\eta\rangle} \hat{u}(\xi-\eta)\widehat{\mathrm{R}_k v}(\eta)\hat{w}(-\xi) d\xi d\eta,\end{gathered}$$ while for $k=3$, $$\begin{gathered} \label{int_B3_app} \int B^3_{(j_1,j_2,j_3)}(\xi,\eta)\hat{u}(\xi-\eta)\hat{v}(\eta)\hat{w}(-\xi) d\xi d\eta = \frac{1}{2}\int {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{uv}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{uv}{\tmpbox}}(\xi)\hat{w}(-\xi)d\xi \\+ \frac{j_1j_2}{2} \int {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\left[(\langle D_x\rangle^{-1}u)|D_x|v\right]}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\left[(\langle D_x\rangle^{-1}u)|D_x|v\right]}{\tmpbox}}(\xi)\hat{w}(-\xi) d\xi -\frac{j_1j_3}{2} \int \frac{\langle\xi\rangle}{\langle\xi-\eta\rangle} \hat{u}(\xi-\eta)\hat{v}(\eta)\hat{w}(-\xi)d\xi.\end{gathered}$$ Hölder’s inequality shows immediately that the first two addends in both above right hand sides are bounded by the right hand side of . Then the result of the statement follows by proving that inequality is satisfied by integrals such as $$\int a(\xi,\eta) \hat{u}_1(\xi-\eta) \hat{u}_2(\eta)\hat{u}_3(-\xi) d\xi d\eta$$ with $a(\xi,\eta) = \xi_1\langle\xi-\eta\rangle^{-1}$ or $a(\xi,\eta) = \langle\xi\rangle \langle\xi-\eta\rangle^{-1}$, and some general functions $u_1,u_3\in L^2(\mathbb{R}^2), u_2\in L^\infty(\mathbb{R}^2)$. By taking a Littlewood-Paley decomposition we can split the above integral as $$\label{decomposition u1u2u3} \sum_{k,l\ge 0}\int a(\xi,\eta)\varphi_k(\xi)\varphi_l(\eta) \hat{u}_1(\xi-\eta) \hat{u}_2(\eta)\hat{u}_3(-\xi) d\xi d\eta,$$ with $\varphi_0\in C^\infty_0(\mathbb{R}^2)$, $\varphi \in C^\infty_0(\mathbb{R}^2\setminus\{0\})$ and $\varphi_k(\zeta) = \varphi(2^{-k}\zeta)$ for any $k\in\mathbb{N}^*$. Since frequencies $\xi,\eta$ are bounded on the support of $\varphi_0(\xi)\varphi_0(\eta)$, kernel $$K_0(x,y):=\int e^{ix\cdot\xi+iy\cdot\eta}a(\xi,\eta)\varphi_0(\xi)\varphi_0(\eta) d\xi d\eta$$ is such that $|K_0(x,y)|\lesssim \langle x \rangle^{-3}\langle y\rangle^{-3}$ for any $(x,y)$, after the first part of corollary \[Cor\_appendix: decay of integral operators\] $(i)$. Therefore $$\begin{split} & \left| \int a(\xi,\eta)\varphi_0(\xi)\varphi_0(\eta) \hat{u}_1(\xi-\eta) \hat{u}_2(\eta)\hat{u}_3(-\xi) d\xi d\eta\right| \\ &= \left| \int K_0(z-x, x-y) u_1(x) u_2(y) u_3(z) dxdydz\right| \\ &\lesssim \int \langle z-x\rangle^{-3}\langle x-y\rangle^{-3} |u_1(x)| |u_2(y)| |u_3(z)| dx dy dz \\ &\lesssim \|u_2\|_{L^\infty}\int \langle x\rangle^{-3}|u_1(z-x)||u_3(z)| dxdz \lesssim \|u_1\|_{L^2}\|u_2\|_{L^\infty}\|u_3\|_{L^2}, \end{split}$$ where last inequality obtained by Hölder inequality. For positive indices $l,k$ such that $l>k+N_0\ge 0$ (resp. $|l-k|\le N_0$), for a suitably large integer $N_0>1$, we have that $|\xi|<|\eta|\sim|\xi-\eta|$ (resp. $|\xi|\sim |\eta|$) on the support of $\varphi_k(\xi)\varphi_l(\eta)$. If we define $a_{l>k+N_0}(\xi,\eta):=a(\xi,\eta)\langle\eta\rangle^{-1}$ and $a_{|l-k|\le N_0}(\xi,\eta):=a(\xi,\eta)\langle\eta\rangle^{-7}$, it is a computation to check that, for any $\alpha,\beta\in\mathbb{N}^2$ with $|\alpha|, |\beta|\le 3$, $$|\partial^\alpha_\xi \partial^\beta_\eta [a_{l>k+N_0}(2^k\xi, 2^l\eta)]| + |\partial^\alpha_\xi \partial^\beta_\eta a_{|l-k|\le N_0}(\xi,\eta)|\lesssim 2^{-l}.$$ Hence, their associated kernels $K_{l>k+N_0}(x,y)$ and $K_{|l-k|\le N_0}(x,y)$ are such that $$|K_{l>k+N_0}(x,y)| + |K_{|l-k|\le N_0}(x,y)|\lesssim 2^{2k}2^l\langle 2^kx\rangle^{-3}\langle 2^l y\rangle^{-3}, \quad \forall (x,y)\in\mathbb{R}^2\times\mathbb{R}^2$$ as follows after a change of coordinates and some integrations by parts, and for any $l>k+N_0$ $$\label{eq_app_1} \begin{split} & \left| \int a(\xi,\eta)\varphi_k(\xi)\varphi_l(\eta) \hat{u}_1(\xi-\eta) \hat{u}_2(\eta)\hat{u}_3(-\xi) d\xi d\eta\right| \\ =& \left| \int K_{l>k+N_0}(z-x, x-y) u_1(x) [\langle D_x\rangle u_2](y) u_3(z) dxdydz\right| \\ \lesssim& 2^{2k}2^l \left|\int \langle 2^k(z-x)\rangle^{-3}\langle 2^l(x-y)\rangle^{-3} |u_1(x)||\langle D_x\rangle u_2(y)| |u_3(z)| dxdydz \right| \\ \lesssim & 2^{-\frac{k}{2}}2^{-\frac{l}{2}}\|u_1\|_{L^2}\|u_2\|_{H^{1,\infty}}\|u_3\|_{L^2}, \end{split}$$ while for $|l-k|\le N_0$ $$\label{eq_app_2} \begin{split} & \left| \int a(\xi,\eta)\varphi_l(\xi)\varphi_l(\eta) \hat{u}_1(\xi-\eta) \hat{u}_2(\eta)\hat{u}_3(-\xi) d\xi d\eta\right| \\ =& \left| \int K_{|l-k|\le N_0}(z-x, x-y) u_1(x) [\langle D_x\rangle^7 u_2](y) u_3(z) dxdydz\right| \\ \lesssim &2^{3l}\left|\int \langle 2^l(z-x)\rangle^{-3}\langle 2^l(x-y)\rangle^{-3} |u_1(x)||\langle D_x\rangle^7 u_2(y)| |u_3(z)| dxdydz \right| \\ \lesssim & 2^{-l}\|u_1\|_{L^2}\|u_2\|_{H^{7,\infty}}\|u_3\|_{L^2}. \end{split}$$ Finally, when positive indices $l,k$ are such that $k>l-N_0$ we observe that frequencies $\xi$ and $\xi-\eta$ are equivalent and of size $2^k$ on the support of $\varphi_k(\xi)\varphi_l(\eta)$. If we take $a_{k>l-N_0}(\xi,\eta)$ equal to $a_{l>k+N_0}(\xi,\eta)$, denote by $K_{k>l-N_0}(x,y)$ its associated kernel (which is hence equal to $K_{l>k+N_0}(x,y)$), and introduce two new smooth cut-off function $\varphi^1, \varphi^2\in C^\infty_0(\mathbb{R}^2)$ equal to 1 on the support of $\varphi$, together with operators $\Delta^1_k:=\varphi^1(2^{-k}D_x), \Delta^2_k:=\varphi^2(2^{-k}D_x)$, we deduce that $$\begin{split} & \left| \int a(\xi,\eta)\varphi_k(\xi)\varphi_l(\eta) \hat{u}_1(\xi-\eta) \hat{u}_2(\eta)\hat{u}_3(-\xi) d\xi d\eta\right| \\ =& \left| \int K_{k>l-N_0}(z-x, x-y) [\Delta^1_k u_1](x) [\langle D_x\rangle u_2](y) [\Delta^2_k u_3](z) dxdydz\right| \\ \lesssim& 2^{2k}2^l \left|\int \langle 2^k(z-x)\rangle^{-3}\langle 2^l(x-y)\rangle^{-3} |[\Delta^1_k u_1](x)||\langle D_x\rangle u_2(y)| |[\Delta^2_k u_3](z)| dxdydz \right| \\ \lesssim &2^{-l}\|\Delta^1_k u_1\|_{L^2}\|u_2\|_{H^{1,\infty}}\|\Delta^2_k u_3\|_{L^2}. \end{split}$$ Combining decomposition together with , and Cauchy-Schwarz inequality we finally obtain that $$\left| \int a(\xi,\eta) \hat{u}_1(\xi-\eta) \hat{u}_2(\eta)\hat{u}_3(-\xi) d\xi d\eta \right| \lesssim \|u_1\|_{L^2}\|u_2\|_{H^{7,\infty}}\|u_3\|_{L^2},$$ \[Lem\_appendix: integral sigma\_tilde\_N\] Let $\varepsilon>0$ be small, $N\in\mathbb{N}^*$, and $\sigma^N(\xi,\eta):\mathbb{R}^2\times\mathbb{R}^2\rightarrow \mathbb{C}$ be supported for $|\xi|\le \varepsilon \langle\eta\rangle$ and such that, for any $\alpha,\beta\in\mathbb{N}^2$, $$|\partial^\alpha_\xi \partial^\beta_\eta \sigma^N(\xi,\eta)|\lesssim |\xi|^{N+1-|\alpha|}\langle\eta\rangle^{-N-|\beta|}, \quad \forall (\xi,\eta)\in\mathbb{R}^2\times\mathbb{R}^2.$$ For any $(j_1,j_2,j_3)\in \{+,-\}^3$ let also $$\label{def_app_sigmatN} \widetilde{\sigma}^N_{(j_1,j_2,j_3)}(\xi,\eta):=\frac{\sigma^N(\eta,\xi-\eta)}{j_1\langle\xi-\eta\rangle + j_2|\eta| -j_3\langle\xi\rangle}.$$ Then for any $\alpha,\beta\in \mathbb{N}^2$ $$\label{derivatives sigma tilde N} \left|\partial^\alpha_\xi\partial^\beta_\eta \widetilde{\sigma}^N_{(j_1,j_2,j_3)}(\xi,\eta)\right|\lesssim_{\alpha,\beta} \langle \xi-\eta\rangle^{2-N+|\alpha|+2|\beta|}|\eta|^{N-|\beta|}, \quad \forall (\xi,\eta)\in\mathbb{R}^2\times\mathbb{R}^2$$ and if $N\ge 15$, for any $u,w\in L^2(\mathbb{R}^2)$, $v\in H^{N+3,\infty}(\mathbb{R}^2)$, $$\label{estimate_integral_sigmatildeN} \left|\int \widetilde{\sigma}^N_{(j_1,j_2,j_3)}(\xi,\eta)\hat{u}(\xi-\eta)\hat{v}(\eta)\hat{w}(-\xi) d\xi d\eta\right|\lesssim \|u\|_{L^2}\|v\|_{H^{N+3,\infty}}\|w\|_{L^2}.$$ From definition function $\widetilde{\sigma}^N_{(j_1,j_2,j_3)}$ can be written as follows $$\widetilde{\sigma}^N_{(j_1,j_2,j_3)}(\xi,\eta)=\frac{j_1\langle\xi-\eta\rangle +j_2|\eta| + j_3\langle\xi\rangle}{2j_1j_2 \langle\xi-\eta\rangle |\eta| - 2(\xi-\eta)\cdot\eta}\sigma^N_{(j_1,j_2,j_3)}(\eta,\xi-\eta).$$ We observe that $$[j_1j_2 \langle\xi-\eta\rangle|\eta| - (\xi-\eta)\cdot\eta]^{-1}\lesssim \langle\xi-\eta\rangle|\eta|^{-1}, \quad \forall (\xi,\eta)\in\mathbb{R}^2\times\mathbb{R}^2$$ and that for any multi-indices $\alpha,\beta\in\mathbb{N}^2$ of positive length $$\begin{gathered} \left|\partial^\alpha_\xi \left[(j_1j_2 \langle\xi-\eta\rangle|\eta| - (\xi-\eta)\cdot\eta)^{-1} \right] \right|\\ \lesssim \sum_{1\le |\alpha_1|\le |\alpha|} \left| j_1j_2 \langle\xi-\eta\rangle|\eta| - (\xi-\eta)\cdot\eta\right|^{-1-|\alpha_1|}|\eta|^{|\alpha_1|}\langle\xi-\eta\rangle^{-(|\alpha| - |\alpha_1|)}, \end{gathered}$$ $$\begin{gathered} \left|\partial^\beta_\eta \left[(j_1j_2 \langle\xi-\eta\rangle|\eta| - (\xi-\eta)\cdot\eta)^{-1} \right]\right| \\ \lesssim \sum_{0\le |\beta_1|<|\beta|}\left| j_1j_2 \langle\xi-\eta\rangle|\eta| - (\xi-\eta)\cdot\eta\right|^{-1-(|\beta|-|\beta_1|)}\sum_{\substack{i+j=|\beta| -2|\beta_1|\\ i,j\le |\beta|-|\beta_1|}}\langle\xi-\eta\rangle^i |\eta|^j.\end{gathered}$$ From above inequalities we hence deduce that on the support of $\sigma^N_{(j_1,j_2,j_3)}(\eta,\xi-\eta)$ (i.e. for $|\eta|\le \varepsilon |\xi-\eta|$), for any $\alpha,\beta\in\mathbb{N}^2$, $$\left| \partial^\alpha_\xi \partial^\beta_\eta \left[(j_1j_2 \langle\xi-\eta\rangle|\eta| - (\xi-\eta)\cdot\eta)^{-1} \right]\right|\lesssim_{\alpha,\beta} \langle \xi-\eta\rangle^{1+|\alpha| +2|\beta|}|\eta|^{-1-|\beta|},$$ and therefore that $$\left|\partial^\alpha_\xi \partial^\beta_\eta \left[\frac{j_1\langle\xi-\eta\rangle +j_2|\eta| + j_3\langle\xi\rangle}{2j_1j_2 \langle\xi-\eta\rangle |\eta| - 2(\xi-\eta)\cdot\eta}\right] \right| \lesssim_{\alpha,\beta} \langle\xi-\eta\rangle^{2+|\alpha|+2|\beta|}|\eta|^{-1-|\beta|} + \langle\xi\rangle\langle\xi-\eta\rangle^{1+|\alpha|+2|\beta|}|\eta|^{-1-|\beta|}.$$ The above estimates, summed up with the fact $$|\partial^\alpha_\xi\partial^\beta_\eta [\sigma^N_{(j_1,j_2,j_3)}(\eta,\xi-\eta)]|\lesssim_{\alpha,\beta} \langle\xi-\eta\rangle^{-N-|\alpha|}|\eta|^{N+1-|\beta|},$$ gives the first part of the statement. Let us now suppose that $N\ge 15$ and take $\chi\in C^\infty_0(\mathbb{R}^2)$ equal to 1 in a neighbourhood of the origin. We have that $$\begin{gathered} \int \widetilde{\sigma}^N_{(j_1,j_2,j_3)}(\xi,\eta)\hat{u}(\xi-\eta)\hat{v}(\eta)\hat{w}(-\xi) d\xi d\eta = \int K^N_0(z-x,x-y) u(x) v(y)w(z) dxdydz \\ + \int K_1^N(z-x,x-y) u(x) [\langle D_x\rangle^{N+3}v](y) w(z) dxdydz,\end{gathered}$$ with $$\begin{gathered} K^N_k(x,y):= \int e^{ix\cdot\xi + iy\cdot\eta}\widetilde{\sigma}^{N,k}_{(j_1,j_2,j_3)}(\xi,\eta) d\xi d\eta,\\ \widetilde{\sigma}^{N,0}_{(j_1,j_2,j_3)}(\xi,\eta)=\widetilde{\sigma}^N_{(j_1,j_2,j_3)}(\xi,\eta)\chi(\eta) \quad\text{and} \quad \widetilde{\sigma}^{N,1}_{(j_1,j_2,j_3)}(\xi,\eta)=\widetilde{\sigma}^N_{(j_1,j_2,j_3)}(\xi,\eta)\langle \eta\rangle^{-N-3}(1-\chi)(\eta). \end{gathered}$$ Then inequality is obtained using the fact that, for any $\widetilde{u},\widetilde{w}\in L^2, \widetilde{v}\in L^\infty$, $$\begin{split} \int \langle z-x\rangle^{-3}\langle x-y\rangle^{-3} |\widetilde{u}(x)| |\widetilde{v}(y)| |\widetilde{w}(z)| dxdydz &\lesssim \|v\|_{L^\infty}\int \langle z \rangle^{-3} |\widetilde{u}(x)| |\widetilde{w}(z-x)| dx dz \\ & \lesssim \|u\|_{L^2}\|v\|_{L^\infty}\|w\|_{L^2}. \end{split}$$ In the following lemma we derive some results on the Sobolev continuity of the bilinear integral operator $$(u,v)\mapsto \int e^{ix\cdot\xi}D_{(j_1,j_2)}(\xi,\eta)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta ,$$ with $D_{(j_1,j_2)}$ defined in . We warn the reader that we are not going to take advantage of factor $(1-\frac{\xi-\eta}{\langle\xi-\eta\rangle}\cdot\frac{\eta}{\langle\eta\rangle})$ in $D_{(j_1,j_2)}(\xi,\eta)$ when deriving the estimates mentioned below, since the Sobolev continuity of the above integral operator does not depend on the null structure $Q_0(v,\partial_1v)$ we chose for the Klein-Gordon self-interaction in the wave equation in system . \[Lem\_appendix: L2 Linfty inequalities integral D Dtilde\] Let $\rho\in \mathbb{N}$ and $D(\xi,\eta)$ a function satisfying, for any multi-indices $\alpha,\beta\in\mathbb{N}^2$, the following: $(i)$ if $|\xi|\lesssim 1$, $$\begin{gathered} |\partial^\beta_\eta D(\xi,\eta)|\lesssim_\beta \langle\eta\rangle^{\rho+|\beta|}, \\ |\partial^\alpha_\xi \partial^\beta_\eta D(\xi,\eta)|\lesssim_{\alpha,\beta} \langle\eta\rangle^{\rho+|\alpha|+|\beta|} + \sum_{|\alpha_1|+|\alpha_2|=|\alpha|}|\xi|^{1-|\alpha_1|}\langle\eta\rangle^{\rho+|\alpha_2|+|\beta|}, \quad |\alpha|\ge 1;\end{gathered}$$ $(ii)$ for $|\xi| \gtrsim 1$, $|\eta|\lesssim \langle\xi - \eta\rangle$, $$|\partial^\alpha_\xi \partial^\beta_\eta D(\xi,\eta)|\lesssim_{\alpha,\beta} \langle\xi-\eta\rangle^{\rho+|\alpha|+|\beta|};$$ $(iii)$ for $|\xi|\gtrsim 1$, $|\eta| \gtrsim \langle\xi-\eta\rangle$: $$|\partial^\alpha_\xi \partial^\beta_\eta D(\xi,\eta)|\lesssim_{\alpha,\beta} \langle\eta\rangle^{\rho+|\alpha|+|\beta|}.$$ Then for any $s\ge0$, any $u,v\in H^{s+\rho+13}(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$ (resp. $u,v\in H^{s+\rho+13,\infty}(\mathbb{R}^2)\cap L^2(\mathbb{R}^2)$) \[est: L2 Linfty integral D(xi,eta)\] $$\label{L2 est integral D(xi,eta)} \begin{split} \left\|\int e^{ix\cdot\xi} D(\xi,\eta) \hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{H^s(dx)} & \lesssim \|u\|_{H^{s+\rho+13}}\|v\|_{L^\infty} + \|u\|_{L^\infty}\|v\|_{H^{s+\rho+13}}\\ (\text{or } &\lesssim \|u\|_{H^{s+\rho+13,\infty}}\|v\|_{L^2} + \|u\|_{L^2}\|v\|_{H^{s+\rho+13,\infty}}), \end{split}$$ and for any $u,v\in H^{s+\rho+13,\infty}(\mathbb{R}^2)$ $$\label{Linfty est integral D(xi,eta)} \left\|\int e^{ix\cdot\xi} D(\xi,\eta) \hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{H^{s,\infty}(dx)} \lesssim \|u\|_{H^{s+\rho+13,\infty}}\|v\|_{L^\infty} + \|u\|_{L^\infty}\|v\|_{H^{s+\rho+13,\infty}} .$$ Furthermore, if $\phi \in C^\infty_0(\mathbb{R}^2)$, $t\ge 1$, $\sigma>0$ small, there exists $\delta>0$ depending linearly on $\sigma$, such that \[est:L2 Linfty integral D with cut-off\] $$\label{est:L2 integral D with cut-off} \begin{split} \left\|\phi(t^{-\sigma} D_x)\int e^{ix\cdot\xi} D(\xi,\eta) \hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{H^s(dx)} &\lesssim t^\delta \|u\|_{H^{\rho+13}}\|v\|_{L^\infty} \\ (\text{or } &\lesssim t^\delta \|u\|_{H^{\rho+13,\infty}}\|v\|_{L^2})\\ (\text{or } &\lesssim t^\delta \|u\|_{L^\infty}\|v\|_{H^{\rho+13}}),\\ (\text{or } &\lesssim t^\delta \|u\|_{L^2}\|v\|_{H^{\rho+13,\infty}}), \end{split}$$ $$\label{est: Linfty integral D with cut-off} \begin{split} \left\|\phi(t^{-\sigma} D_x)\int e^{ix\cdot\xi} D(\xi,\eta) \hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{H^{s,\infty}} &\lesssim t^\delta \|u\|_{H^{\rho+13,\infty}}\|v\|_{L^\infty} \\ (\text{or } &\lesssim t^\delta \|u\|_{L^\infty}\|v\|_{H^{\rho+13,\infty}}). \end{split}$$ Finally, if for any $\alpha,\beta\in\mathbb{N}^2$ $D(\xi,\eta)$ satisfies $(ii)$, $(iii)$ when $|\xi|\gtrsim 1$, together with: $(\widetilde{i})$ if $|\xi|\lesssim 1$ $$|\partial^\alpha_\xi\partial^\beta_\eta \widetilde{D}(\xi,\eta)|\lesssim_{\alpha,\beta} \langle\eta\rangle^{\rho+|\alpha|+|\beta|} + \sum_{|\alpha_1|+|\alpha_2|=|\alpha|}|\xi|^{-|\alpha_1|+1}\langle\eta\rangle^{\rho+|\alpha_2|+|\beta|},$$ then, for any $u,v\in H^{s+\rho+13}(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$, \[est: L2 Linfty integral D-tilde(xi,eta)\]$$\begin{gathered} \label{L2 est integral D-tilde(xi,eta)} \left\|\int e^{ix\cdot\xi} D(\xi,\eta) \hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{H^s(dx)} \lesssim \|u\|_{H^{\rho+10}}\|v\|_{L^2}+ \|u\|_{H^{s+\rho+13}}\|v\|_{L^\infty} + \|u\|_{L^\infty}\|v\|_{H^{s+\rho+13}}\\ (\text{or } \lesssim\|u\|_{L^2}\|v\|_{H^{\rho+10}}+ \|u\|_{H^{s+\rho+13,\infty}}\|v\|_{L^2} + \|u\|_{L^2}\|v\|_{H^{s+\rho+13,\infty}}),\end{gathered}$$and for any $u,v\in H^{s+\rho+13,\infty}(\mathbb{R}^2)$, with $u\in H^{\rho+10}(\mathbb{R}^2)$ (or $u\in L^2(\mathbb{R}^2)$), $$\begin{gathered} \label{Linfty est integral D-tilde(xi,eta)} \left\|\int e^{ix\cdot\xi} D(\xi,\eta) \hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{H^{s,\infty}(dx)} \lesssim\\ \|u\|_{H^{\rho+10}}\|v\|_{L^\infty}+ \|u\|_{H^{s+\rho+13,\infty}}\|v\|_{L^\infty} + \|u\|_{L^\infty}\|v\|_{H^{s+\rho+13,\infty}}\\ (\text{or } \lesssim \|u\|_{L^2}\|v\|_{H^{\rho+10,\infty}}+ \|u\|_{H^{s+\rho+13,\infty}}\|v\|_{L^\infty} + \|u\|_{L^\infty}\|v\|_{H^{s+\rho+13,\infty}}).\end{gathered}$$ Let $L(\mathbb{R}^2)$ denote either the $L^2(\mathbb{R}^2)$ space or the $L^\infty(\mathbb{R}^2)$ one. After definition \[def Sobolev spaces-NEW\] $(i)$ of space $H^s$ (resp. $(iii)$ of $H^{s,\infty}$), we should prove that the $L^2$ norm (resp. the $L^\infty$) norm of $$\label{integral Ds} \int e^{ix\cdot\xi} D^s(\xi,\eta)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta,$$ with $D^s(\xi,\eta):=D(\xi,\eta)\langle\xi\rangle^s$, is bounded by the right hand side of and (resp. and ). Let us first take $\chi\in C^\infty_0(\mathbb{R}^2)$ equal to 1 in a neighbourhood of the origin and split the above integral, distinguishing between bounded and unbounded frequencies $\xi$, as $$\label{split integral Ds} \int e^{ix\cdot\xi} D^s(\xi,\eta)\chi(\xi)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta +\int e^{ix\cdot\xi} D^s(\xi,\eta)(1-\chi)(\xi)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta.$$ On the support of $\chi(\xi)$ frequencies $\xi-\eta, \eta$ are either bounded or equivalent, thus if $$a^s_0(\xi,\eta):= \begin{cases} & D^s(\xi,\eta)\chi(\xi) \langle \xi-\eta\rangle^{-\rho-10}\\ &\qquad \text{or}\\ & D^s(\xi,\eta)\chi(\xi) \langle \eta\rangle^{-\rho-10} \end{cases}$$ $a^s_0(\xi,\eta)$ satisfies with $g_\beta(\eta)=\langle\eta\rangle^{-3}$ for any $|\beta|\le 3$, after hypothesis $(i)$ on $D(\xi,\eta)$. Then by and depending on the choice of $a^s_0(\xi,\eta)$, we have that \[est:L integral D chi(xi)\] $$\begin{gathered} \label{est:L integral D with derivatives on u} \left\|\int e^{ix\cdot\xi} D^s(\xi,\eta)\chi(\xi)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{L(dx)} = \left\|\int e^{ix\cdot\xi} a^s_0(\xi,\eta){\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\langle D_x\rangle^{\rho+10}u}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\langle D_x\rangle^{\rho+10}u}{\tmpbox}}(\xi-\eta)\hat{v}(\eta)d\xi d\eta \right\|_{L(dx)}\\ \lesssim \|\langle D_x\rangle^{\rho+10}u\|_{L}\|v\|_{L^\infty} (\text{or } \|\langle D_x\rangle^{\rho+10}u\|_{L^\infty}\|v\|_{L}),\end{gathered}$$or$$\begin{gathered} \label{est:L integral D with derivatives on v} \left\|\int e^{ix\cdot\xi} D^s(\xi,\eta)\chi(\xi)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{L(dx)} = \left\| \int e^{ix\cdot\xi} a^s_0(\xi,\eta) \hat{u}(\xi-\eta){\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\langle D_x\rangle^{\rho+10}v}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\langle D_x\rangle^{\rho+10}v}{\tmpbox}}(\eta)d\xi d\eta \right\|_{L(dx)}\\ \lesssim \|u\|_{L^\infty}\|\langle D_x\rangle^{\rho+10} v\|_{L} (\text{or } \|u\|_{L}\|\langle D_x\rangle^{\rho+10} v\|_{L^\infty}).\end{gathered}$$ Successively, we consider a Littlewood-Paley decomposition in order to write $$\begin{gathered} \label{LP decomposition integral D} \int e^{ix\cdot\xi}D^s(\xi,\eta)(1-\chi)(\xi) \hat{u}(\xi-\eta)\hat{v}(\eta)d\xi d\eta \\ = \sum_{k\ge 1,l\ge 0} \int e^{ix\cdot\xi}D^s(\xi,\eta)(1-\chi)(\xi)\varphi_k(\xi)\varphi_l(\eta) \hat{u}(\xi-\eta)\hat{v}(\eta)d\xi d\eta,\end{gathered}$$ where $\varphi_0 \in C^\infty_0(\mathbb{R}^2)$, $\varphi_k(\zeta)=\varphi(2^{-k}\zeta)$ with $\varphi\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$ for any $k\in\mathbb{N}^*$. When positive indices $l,k$ are such that $k>l+N_0$ for a certain large $N_0\in\mathbb{N}^*$, we have that $|\eta|< |\xi-\eta|$ and $|\xi-\eta|\sim |\xi|\sim 2^k$ on the support of $\varphi_k(\xi)\varphi_l(\eta)$. If $$a^s_{k>l+N_0}(\xi,\eta):=D^s(\xi,\eta) \varphi_k(\xi)\varphi_l(\eta)\langle\xi-\eta\rangle^{-s-\rho-13},$$ by hypothesis $(ii)$ we deduce that, for any $\alpha,\beta\in\mathbb{N}^2$ of length less or equal than 3, $$|\partial^\alpha_\xi\partial^\beta_\eta[ a^s_{k>l+N_0}(2^k\xi,2^l\eta)]|\lesssim 2^{-k}, \quad\forall (\xi,\eta)\in\mathbb{R}^2\times\mathbb{R}^2$$ and its associated kernel $$K^s_{k>l+N_0}(x,y):=\int e^{ix\cdot\xi + iy\cdot\eta} a^s_{k>l+N_0}(\xi,\eta)d\xi d\eta$$ verifies that $$|K^s_{k>l+N_0}(x,y)|\lesssim 2^k 2^{2l}\langle 2^kx\rangle^{-3}\langle 2^ly\rangle^{-3}, \quad \forall (x,y)\in\mathbb{R}^2\times\mathbb{R}^2$$ as one can check doing some integration by parts. Therefore $$\label{est:L akl k>l} \begin{split} &\left\| \int e^{ix\cdot\xi} D^s(\xi,\eta)\varphi_k(\xi)\varphi_l(\eta)\hat{u}(\xi-\eta)\hat{v}(\eta)d\xi d\eta \right\|_{L(dx)}\\ & = \left\|\int K^s_{k>l+N_0}(x-y,y-z) [\langle D_x\rangle^{s+\rho+13}u](y) v(z) dydz\right\|_{L(dx)} \\ &\lesssim \ 2^k 2^{2l} \left\| \int \langle 2^k(x-y)\rangle^{-3}\langle 2^l (y-z)\rangle^{-3} |\langle D_x\rangle^{s+\rho +13}u(y)| |v(z)| dydz \right\|_{L(dx)} \\ &\lesssim \ 2^k 2^{2l}\int \langle 2^ky\rangle^{-3} \langle 2^l z\rangle^{-3} \| [\langle D_x\rangle^{s+\rho+13} u](\cdot -y)v(\cdot -y-z)\|_{L(dx)} dydz \\ & \lesssim 2^{-\frac{k}{2}}2^{-\frac{l}{2}} \|\langle D_x\rangle^{s+\rho+13}u\|_{L}\|v\|_{L^\infty} \ (\text{ or } 2^{-\frac{k}{2}}2^{-\frac{l}{2}}\|\langle D_x\rangle^{s+\rho+13} u\|_{L^\infty}\|v\|_L). \end{split}$$ For indices $l,k$ such that $1\le k\le l+N_0$ we have that $|\xi-\eta|\lesssim |\eta|$ on the support of $\varphi_k(\xi)\varphi_l(\eta)$. If $$a^s_{k\le l+N_0}(\xi,\eta):=D^s(\xi,\eta)\varphi_k(\xi)\varphi_l(\eta)\langle\eta\rangle^{-s-\rho-13}$$ by hypothesis $(iii)$ for any multi-indices $\alpha,\beta$ of length less or equal than 3, $$|\partial^\alpha_\xi\partial^\beta_\eta [a^s_{k\le l+N_0}(2^k\xi,2^l\eta)]|\lesssim_{\alpha,\beta}2^{-l},$$ and its associated kernel $K^s_{k\le l+N_0}(x,y)$ is such that $$K^s_{k\le l+N_0}(x,y)|\lesssim 2^{2k}2^l \langle 2^kx\rangle^{-3}\langle 2^ly\rangle^{-3}, \quad \forall (x,y)\in\mathbb{R}^2\times\mathbb{R}^2.$$ Consequently $$\label{est:L akl k<l} \begin{split} & \left\| \int e^{ix\cdot\xi}D^s(\xi,\eta)\varphi_k(\xi)\varphi_l(\eta)\hat{u}(\xi-\eta)\hat{v}(\eta)d\xi d\eta \right\|_{L(dx)}\\ & \lesssim \ 2^{-\frac{k}{2}}2^{-\frac{l}{2}} \|u\|_{L^\infty}\|\langle D_x\rangle^{s+\rho+13}v\|_{L} \ (\text{or } 2^{-\frac{k}{2}}2^{-\frac{l}{2}} \|u\|_{L}\|\langle D_x\rangle^{s+\rho+13}v\|_{L^\infty}), \end{split}$$ and inequality (resp. ) is hence obtained by combining inequalities , , with $L=L^2$ (resp. $L=L^\infty$), and taking the sum over $k\ge 1, l\ge 0$. In order to derive inequalities , we first observe that we can reduce to study the $L^2$ and $L^\infty$ norm of with $s=0$ and $D(\xi,\eta)$ multiplied by $\phi(t^{-\sigma}\xi)$, up to a factor $t^{s\sigma}$. Here we use again decompositions , , and only need to modify some of the multipliers defined above, depending on if we want derivatives falling entirely on $u$ or rather on $v$. In fact, in order to prove the first two inequalities in and the first one in we introduce $$\begin{split} &a^\phi_{l\le k+N_0}(\xi,\eta):=D(\xi,\eta)\chi(t^{-\sigma}\xi)\varphi_k(\xi)\varphi_l(\eta)\\ &a^\phi_{l>k+N_0}(\xi,\eta):=D(\xi,\eta)\chi(t^{-\sigma}\xi)\varphi_k(\xi)\varphi_l(\eta)\langle\xi-\eta\rangle^{-\rho-13} \end{split}$$ and deduce from hypothesis $(ii)-(iii)$ on $D(\xi,\eta)$ and the fact that $|\xi|\lesssim t^\sigma$ on the support of $\phi(t^{-\sigma}\xi)$ that, for any $\alpha,\beta\in \mathbb{N}^2$ of length less or equal than 3, $$|\partial^\alpha_\xi \partial^\beta_\eta [a^\phi_{l\le k+N_0}(2^k\xi, 2^l\eta)]|\lesssim t^\delta 2^{-k} \quad \text{and} \quad |\partial^\alpha_\xi \partial^\beta_\eta [a^\phi_{ l>k+N_0}(2^k\xi, 2^l\eta)]|\lesssim 2^{-l}$$ with $\delta>0$, $\delta\rightarrow 0$ as $\sigma\rightarrow 0$. On the one hand, kernel $K^\phi_{l\le k+N_0}(x,y)$ associated to $a^\phi_{l\le k+N_0}(\xi,\eta)$ verifies $$|K^\phi_{l\le k+N_0}(x,y)|\lesssim t^\delta 2^{k}2^{2l}\langle2^k x\rangle^{-3}\langle 2^l y\rangle^{-3}, \quad \forall(x, y)\in\mathbb{R}^2\times\mathbb{R}^2$$ and then for any $l,k$ such that $l\le k+N_0$ $$\label{integral D with cut-off k bigger l} \begin{split} & \left\|\int e^{ix\cdot\xi} D(\xi,\eta)\phi(t^\sigma\xi)\varphi_k(\xi)\varphi_l(\eta)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{L(dx)} \\ = & \left\| \int K^\phi_{l\le k+N_0}(x-y,y-z) u(y) v(z) dydz \right\|_{L(dx)} \lesssim t^\delta 2^{-\frac{k}{2}}2^{-\frac{l}{2}} \|u\|_{L}\|v\|_{L^\infty}\\ & \hspace{7cm} (\text{or } \lesssim t^\delta 2^{-\frac{k}{2}}2^{-\frac{l}{2}}\|u\|_{L^\infty}\|v\|_{L}). \end{split}$$ On the other hand, kernel $K^\phi_{l>k+N_0}(x,y)$ associated to $a^\phi_{l>k+N_0}(\xi,\eta)$ satisfies $$|K^\phi_{l>k+N_0}(x,y)|\lesssim 2^{2k}2^l\langle2^k x\rangle^{-3}\langle 2^l y\rangle^{-3}, \quad \forall(x,y)\in\mathbb{R}^2\times\mathbb{R}^2$$ so for indices $l,k$ such that $l>k+N_0$ $$\begin{split} & \left\|\int e^{ix\cdot\xi} D(\xi,\eta)\phi(t^\sigma\xi)\varphi_l(\xi)\varphi_l(\eta)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{L(dx)} \\ &= \left\| \int K^\phi_{l>k+N_0}(x-y,y-z) [\langle D_x\rangle^{\rho+13} u](y) v(z) dydz \right\|_{L(dx)} \lesssim 2^{-\frac{k}{2}}2^{-\frac{l}{2}} \|\langle D_x\rangle^{\rho+13} u\|_{L}\|v\|_{L^\infty}\\ &\hspace{8.7cm} (\text{or } \lesssim 2^{-\frac{k}{2}}2^{-\frac{l}{2}}\|\langle D_x\rangle^{\rho+13} u\|_{L^\infty}\|v\|_{L}) \Big). \end{split}$$ Combining these two inequalities with and taking the sum over $k\ge 1, l\ge 0$ we obtain the wished estimates. Last two inequalities in and last one in are instead obtained combining with (that evidently holds for $D^s(\xi,\eta)$ replaced with $D(\xi,\eta)\phi(t^\sigma\xi)$) and . Finally, last part of the statement follows from the same argument of above, with the only difference that, after hypothesis $(\widetilde{i})$, multiplier $\widetilde{a}^s_0(\xi,\eta):=\widetilde{D}(\xi,\eta)\chi(\xi)\langle \eta\rangle^{-\rho-10}$ satisfies with $|g_\beta(\eta)|\lesssim \langle \eta\rangle^{-3}$ for any $|\beta|\le 3$, then by we have that $$\begin{gathered} \left\|\int e^{ix\cdot\xi} \widetilde{D}^s(\xi,\eta)\chi(\xi)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{L(dx)} \\ = \left\|\int e^{ix\cdot\xi} \widetilde{a}^s_0(\xi,\eta){\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\langle D_x\rangle^{\rho+10}u}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\langle D_x\rangle^{\rho+10}u}{\tmpbox}}(\xi-\eta)\hat{v}(\eta)d\xi d\eta \right\|_{L(dx)} \lesssim \|\langle D_x\rangle^{\rho+10}u\|_{L^2}\|v\|_{L},\end{gathered}$$or$$\begin{gathered} \left\|\int e^{ix\cdot\xi} \widetilde{D}^s(\xi,\eta)\chi(\xi)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{L(dx)} \\ =\left\| \int e^{ix\cdot\xi} \widetilde{a}^s_0(\xi,\eta) \hat{u}(\xi-\eta){\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\langle D_x\rangle^{\rho+10}v}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\langle D_x\rangle^{\rho+10}v}{\tmpbox}}(\eta)d\xi d\eta \right\|_{L(dx)} \lesssim \|u\|_{L^2}\|\langle D_x\rangle^{\rho+10} v\|_{L} .\end{gathered}$$ \[Lem\_Appendix: est on Dj1j2\] Let $j\in \{+,-\}$, $\phi\in C^\infty_0(\mathbb{R}^2)$, $t\ge 1, \sigma>0$, and $D_j(\xi,\eta)$ be the multiplier introduced in . For any $s\ge 0$, $i=1,2$, $D_j(\xi,\eta)$ and $\frac{\xi_i}{|\xi|}D_j(\xi,\eta)$ satisfy inequalities , with $\rho=2$, and \[est: L2 Linfty integral partial D\] $$\begin{gathered} \left\|\int e^{ix\cdot\xi} \partial_{\xi}D_j(\xi,\eta) \hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{H^s(dx)} \lesssim \|u\|_{H^{13}}\|v\|_{L^2}+ \|u\|_{H^{s+16}}\|v\|_{L^\infty} + \|u\|_{L^\infty}\|v\|_{H^{s+16}}\\ (\text{resp. }\lesssim \|u\|_{H^{13}}\|v\|_{L^2}+ \|u\|_{H^{s+16,\infty}}\|v\|_{L^2} + \|u\|_{L^2}\|v\|_{H^{s+16,\infty}}),\end{gathered}$$ $$\begin{gathered} \left\|\int e^{ix\cdot\xi} \partial_{\xi}D_j(\xi,\eta) \hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{H^{s,\infty}(dx)} \\ \lesssim \|u\|_{H^{13}}\|v\|_{L^\infty}+ \|u\|_{H^{s+16,\infty}}\|v\|_{L^\infty} + \|u\|_{L^\infty}\|v\|_{H^{s+16,\infty}},\end{gathered}$$ together with \[est:app\_partialD\_cutoff\] $$\begin{gathered} \label{est:L2 integral partialD with cut-off} \left\|\phi(t^{-\sigma}D_x)\int e^{ix\cdot\xi} \partial_{\xi}D_j(\xi,\eta) \hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{H^s(dx)} \lesssim t^\delta \|u\|_{H^{13}}\left(\|v\|_{L^2}+ \|v\|_{L^\infty}\right)\\ (\text{or }\lesssim t^\delta \|u\|_{L^2}\left(\|v\|_{H^{10}}+ \|v\|_{H^{13,\infty}}\right)),\end{gathered}$$ $$\begin{gathered} \left\|\phi(t^{-\sigma}D_x) \int e^{ix\cdot\xi} \partial_{\xi}D_j(\xi,\eta) \hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \right\|_{H^{s,\infty}(dx)} \lesssim t^\delta\left( \|u\|_{H^{13}}+ \|u\|_{H^{16,\infty}}\right)\|v\|_{L^\infty}\\ (\text{or } \lesssim t^\delta \left(\|u\|_{L^2}+\|u\|_{L^\infty}\right) \|v\|_{H^{16,\infty}}).\end{gathered}$$ Moreover, if $\Omega= x_1 \partial_2 - x_2\partial_1$ and $Z_n=x_n\partial_t +t\partial_n$, $n=1,2,$ \[est: L2 integral Omega and Zn cut-off\] $$\begin{gathered} \label{est: L2 Omega integral D with cut-off} \left\|\phi(t^{-\sigma}D_x)\Omega \int e^{ix\cdot\xi}D_j(\xi,\eta) \hat{u}(\xi-\eta)\hat{v}(\eta)d\xi d\eta\right\|_{L^2(dx)}\\ \lesssim t^\delta \left[ \left(\|u\|_{L^2}+\|\Omega u\|_{L^2}\right)\|v\|_{H^{17,\infty}}+ \|u\|_{H^{15,\infty}}\|\Omega v\|_{L^2}\right],\end{gathered}$$ $$\begin{gathered} \label{est:L2 Z integral D with cut-off} \left\|\phi(t^{-\sigma}D_x)Z_n \int e^{ix\cdot\xi}D_j(\xi,\eta) \hat{u}(\xi-\eta)\hat{v}(\eta)d\xi d\eta\right\|_{L^2(dx)}\\ \lesssim t^\delta \left[\|\partial_tu\|_{L^2}\|v\|_{H^{13}}+\|u\|_{H^{13}}\|\partial_tv\|_{L^2}+ \|Z_nu\|_{L^2}\|v\|_{H^{15,\infty}}+\|u\|_{H^{15,\infty}}\| Z_nv\|_{L^2} \right],\end{gathered}$$ $$\begin{gathered} \label{est: L2 DjZ integral D with cut-off} \left\|\phi(t^{-\sigma}D_x)D_j Z_n \int e^{ix\cdot\xi}D_j(\xi,\eta) \hat{u}(\xi-\eta)\hat{v}(\eta)d\xi d\eta\right\|_{L^2(dx)}\\ \lesssim t^\delta \left[\|\partial_tu\|_{L^2}\|v\|_{H^{14,\infty}}+\|u\|_{H^{14,\infty}}\|\partial_tv\|_{L^2}+ \|Z_nu\|_{L^2}\|v\|_{H^{17,\infty}}+\|u\|_{H^{17,\infty}}\| Z_nv\|_{L^2} \right],\end{gathered}$$ with $\delta>0$, $\delta\rightarrow 0$ as $\sigma\rightarrow 0$. The statement follows essentially from the observation that, for $j\in \{+,-\}$, functions $D_j(\xi,\eta)$ and $[(\xi_i\partial_{\xi_j})^{k_1}(\eta_i\partial_{\eta_j})^{k_2}D_j](\xi,\eta)$ satisfy hypothesis $(i)-(iii)$ of lemma \[Lem\_appendix: L2 Linfty inequalities integral D Dtilde\] with $\rho=2$ and $\rho=2+2(k_1+k_2)$ respectively, while $\partial_\xi D_j(\xi,\eta)$ satisfies $(\widetilde{i}),(ii),(iii)$ with $\rho=3$. In fact, we first remark that, for every $\xi, \eta$, denominator $1+\langle\xi-\eta\rangle\langle\eta\rangle - (\xi-\eta)\cdot\eta$ is bounded from below by a positive constant; secondly, the derivation of that denominator gives rise to losses in $\langle\xi-\eta\rangle, \langle\eta\rangle$, as $$\begin{aligned} \partial_{\xi_k}(1+\langle\xi-\eta\rangle\langle\eta\rangle - (\xi-\eta)\cdot\eta) &= \frac{\xi_k-\eta_k}{\langle\xi-\eta\rangle}\langle\eta\rangle + \eta_k, \\ \partial_{\eta_k}(1+\langle\xi-\eta\rangle\langle\eta\rangle - (\xi-\eta)\cdot\eta) & = \frac{\xi_k-\eta_k}{\langle\xi-\eta\rangle}\langle\eta\rangle + \langle\xi-\eta\rangle\frac{\eta_k}{\langle\eta\rangle} + \eta_k - (\xi_k-\eta_k).\end{aligned}$$ For $|\xi|\lesssim 1$ we have that $\langle\xi-\eta\rangle \lesssim \langle\eta\rangle$, so for any $\alpha,\beta\in\mathbb{N}^2$ $$\left|\partial^\alpha_\xi \partial^\beta_\eta \left[\frac{j\langle\xi-\eta\rangle + j\langle\eta\rangle }{1+\langle\xi-\eta\rangle\langle\eta\rangle - (\xi-\eta)\cdot\eta} \eta_1 \right]\right| \lesssim_{\alpha,\beta} \langle\eta\rangle^{2+|\alpha|+|\beta|},$$ while $$\begin{aligned} \left|\partial^\beta_\eta \left[\frac{|\xi|}{1+\langle\xi-\eta\rangle\langle\eta\rangle - (\xi-\eta)\cdot\eta} \eta_1 \right]\right| &\lesssim_\beta \langle\eta\rangle^{1+ |\beta|}, \\ \left|\partial^\alpha_\xi\partial^\beta_\eta \left[\frac{|\xi|}{1+\langle\xi-\eta\rangle\langle\eta\rangle - (\xi-\eta)\cdot\eta} \eta_1\right]\right| &\lesssim_{\alpha,\beta} \sum_{|\alpha_1|+|\alpha_2|=|\alpha|} |\xi|^{1-|\alpha_1|}\langle\eta\rangle^{1+ |\alpha_2|+|\beta|}, \quad |\alpha|\ge 1.\end{aligned}$$ For $|\xi|\gtrsim1$ and $|\eta|\lesssim \langle \xi-\eta\rangle$ (resp. $|\eta|\gtrsim \langle\xi -\eta\rangle$) we have that $|\xi|\lesssim |\xi-\eta|$ (resp.$|\xi|\lesssim |\eta|$), so each time a derivative hits the denominator of $D_j(\xi,\eta)$ we lose a factor $\langle\xi-\eta\rangle$ (resp. $\langle\eta\rangle$). Hence lemma \[Lem\_appendix: L2 Linfty inequalities integral D Dtilde\] immediately implies inequalities , with $D= D_j$ and $\rho=2$, together with , , while inequalities follow from and the fact that, after some integration by parts, $$\begin{split} \Omega & \int e^{ix\cdot\xi}D_j(\xi,\eta)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \\ &=\sum_{k_1+k_2+k_3+k_4=1}\int e^{ix\cdot\xi} [(\xi_1 \partial_{\xi_2} - \xi_2 \partial_{\xi_1})^{k_1}(\eta_1 \partial_{\eta_2} - \eta_2 \partial_{\eta_1})^{k_2}D_j](\xi,\eta)\widehat{\Omega^{k_3}u}(\xi-\eta){\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\Omega^{k_4}v}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\Omega^{k_4}v}{\tmpbox}}(\eta) d\xi d\eta ,\\ Z_n & \int e^{ix\cdot\xi}D_j(\xi,\eta)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \\ &= \int e^{ix\cdot\xi}[\partial_{\xi_n}D_j](\xi,\eta)D_t\big[\hat{u}(\xi-\eta)\hat{v}(\eta)\big] d\xi d\eta + \int e^{ix\cdot\xi}[\partial_{\eta_n}D_j](\xi,\eta)\hat{u}(\xi-\eta)\widehat{D_t v}(\eta) d\xi d\eta \\ & + \int e^{ix\cdot\xi}D_j(\xi,\eta)\widehat{Z_n u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta + \int e^{ix\cdot\xi}D_j(\xi,\eta)\hat{u}(\xi-\eta)\widehat{Z_nv}(\eta) d\xi d\eta, \end{split}$$ and, if $\delta_{jn}$ denotes the Kronecker delta, $$\begin{split} D_j & Z_n \int e^{ix\cdot\xi}D_j(\xi,\eta)\hat{u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta \\ & = \delta_{jn} \int e^{ix\cdot\xi}D_j(\xi,\eta) D_t\left[\hat{u}(\xi-\eta)\hat{v}(\eta)\right] d\xi d\eta \\ & + \int e^{ix\cdot\xi}\partial_{\xi_n} [\xi_j D_j](\xi,\eta)D_t\big[\hat{u}(\xi-\eta)\hat{v}(\eta)\big] d\xi d\eta + \int e^{ix\cdot\xi}\partial_{\eta_n}[\xi_j D_j](\xi,\eta)\hat{u}(\xi-\eta)\widehat{D_t v}(\eta) d\xi d\eta \\ & + \int e^{ix\cdot\xi} \xi_j D_j(\xi,\eta)\widehat{Z_n u}(\xi-\eta)\hat{v}(\eta) d\xi d\eta + \int e^{ix\cdot\xi}\xi_j D_j(\xi,\eta)\hat{u}(\xi-\eta)\widehat{Z_nv}(\eta) d\xi d\eta. \end{split}$$. {#Appendix B} The aim of this chapter is to show how, from the bootstrap assumptions , it is possible to derive a moderate growth in time for the $L^2$ norm of $\mathcal{L}^\mu\widetilde{v}$, with $0\le |\mu|\le 2$, and of $\Omega^\mu_h\mathcal{M}^\nu\widetilde{u}^{\Sigma,k}$, with $\mu, |\nu|=0,1$. These estimates are fundamentally used in propositions \[Prop:propagation\_unif\_est\_V\] and \[Prop: Propagation uniform estimate U,RU\]. Moreover, we also prove in lemma \[Lem\_appendix: sharp\_est\_VJ\] a sharp decay estimate for the uniform norm of the Klein-Gordon solution when one Klainerman vector field is acting on it (and when considered for frequencies less or equal than $t^\sigma$, with $\sigma>0$ small). We are hence going to assume for the rest of this chapter that a-priori estimates are satisfied in interval $[1,T]$, for some fixed $T>1$, and that $\varepsilon_0<(2A+B)^{-1}$. We remind that $\Gamma$ generally denotes one of the admissible vector fields belonging to $\mathcal{Z}$ (see ) and that, for a multi-index $I=(i_1,\dots,i_n)$ with $i_j\in \{1,\dots,5\}$ for $j=1,\dots, n$, $\Gamma^I = \Gamma_{i_1}\cdots \Gamma_{i_n}$. Also, we warn the reader that any norm $X$ ($X=L^\infty, H^s, H^s_h...$) of $w=w(t,x)$ is here considered with respect to spatial variable $x$. We will often write $\|\cdot\|_{X}$ in place of $\|\cdot\|_{X(dx)}$. Some preliminary lemmas {#Sub: App_B1} ----------------------- In the current section we list, on the one hand, some inequalities concerning the $H^s$ and $H^{s,\infty}$ norm of the quadratic non-linearities $Q^\mathrm{w}_0(v_\pm, D_1v_\pm), Q^\mathrm{kg}_0(v_\pm, D_1u_\pm)$ (see lemmas \[Lem: estimates NLkg NLw\], \[Lem: est DtU DtV\]), as they are very frequently recalled in the second part of the paper. On the other hand, we introduce some preliminary small results that will be useful in sections \[sec\_appB: first range of estimates\] and \[sec\_appB: second range of estimates\]. For seek of compactness, we denote $Q^\mathrm{w}_0(v_\pm, D_1v_\pm)$ and $Q^\mathrm{kg}_0(v_\pm, D_1 u_\pm)$ by $\textit{NL}_w$ and $\textit{NL}_{kg}$ respectively, i.e. $$\begin{aligned} \Nlw & := \frac{i}{4}\left[(v_+ + v_{-})D_1(v_+ + v_{-}) - \frac{D_x}{\langle D_x \rangle}(v_+ - v_{-})\cdot\frac{D_x D_1}{\langle D_x \rangle}(v_+ - v_{-})\right] ,\\ \Nlkg & := \frac{i}{4} \left[(v_+ + v_{-})D_1(u_+ + u_{-}) - \frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\cdot\frac{D_x D_1}{|D_x|}(u_+ - u_{-})\right]. \label{def_app_Nlkg}\end{aligned}$$ We recall the result of lemma \[Lem : new estimate 1-chi\], that can be also stated in the classical setting and says that, for any real positive $s>s'$ and $w\in H^s(\mathbb{R}^2)$, $$\label{ineq:1-chi} \|(1-\chi)(t^{-\sigma}D_x)w\|_{H^{s'}}\le C t^{-\sigma(s-s')}\|w\|_{H^s}, \quad \forall s>s'.$$ It is also useful to remind, in view of upcoming lemmas, that the $L^2$ norm of $(\Gamma^Iu)_\pm$ and $(\Gamma^Iv)_\pm$ is estimated with: $$\begin{split} E_n(t;W)^\frac{1}{2}, &\ \text{whenever }|I|\le n \text{ and }\Gamma^I \text{ is a product of spatial derivatives;}\\ E^k_3(t;W)^\frac{1}{2}, &\ \text{whenever }|I|\le 3 \text{ and at most }3-k \text{ vector fields in }\Gamma^I \text{, with } 0\le k\le 2 \\ &\text{ belong to }\{\Omega, Z_m, m=1,2\}. \end{split}$$ As assumed in , , such energies have a moderate growth in time and a hierarchy is established among them in the sense that $$0<\delta\ll \delta_2\ll \delta_1\ll \delta_0\ll 1.$$ We warn the reader that this hierarchy is often implicitly used throughout this chapter. \[Lem: estimates NLkg NLw\] For any $s\ge 0$, any $\theta\in ]0,1[$, $\textit{NL}_w$ satisfies the following inequalities: \[est Hs Hsinfty Nlw-new\] $$\begin{gathered} \|\textit{NL}_w(t,\cdot)\|_{L^2} \lesssim \|V(t,\cdot)\|_{H^{1,\infty}}\|V(t,\cdot)\|_{H^1}, \label{est L2 NLw} \\ \|\textit{NL}_w(t,\cdot)\|_{L^\infty} \lesssim \|V(t,\cdot)\|^2_{H^{2,\infty}},\label{est Linfty NLw} \\ \|\textit{NL}_w(t,\cdot)\|_{H^s} \lesssim \|V(t,\cdot)\|_{H^{s+1}}\|V(t,\cdot)\|_{H^{1,\infty}}, \label{est Hs NLw-New}\\ \|\textit{NL}_w(t,\cdot)\|_{H^{s,\infty}} \lesssim \|V(t,\cdot)\|_{H^{s+1,\infty}}^{2-\theta}\|V(t,\cdot)\|^\theta_{H^{s+3}}, \label{est Hsinfty for NLw-New}\\ \|\Omega \textit{NL}_w(t,\cdot)\|_{L^2}\lesssim \|V(t,\cdot)\|_{H^{2,\infty}}\left(\|V(t,\cdot)\|_{L^2}+ \|\Omega V(t,\cdot)\|_{H^1}\right), \label{est_Omega_NLw}\end{gathered}$$ while for $\textit{NL}_{kg}$ we have that: \[est on NLkg-new\] $$\begin{gathered} \|\textit{NL}_{kg}(t,\cdot)\|_{L^2} \lesssim \|V(t,\cdot)\|_{H^{1,\infty}}\|U(t,\cdot)\|_{H^1}, \label{est L2 NLkg} \\ \|\textit{NL}_{kg}(t,\cdot)\|_{L^\infty} \lesssim \|V(t,\cdot)\|_{H^{1,\infty}}\left(\|U(t,\cdot)\|_{H^{2,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}\right), \label{est Linfty NLkg}\\ \|\textit{NL}_{kg}(t,\cdot)\|_{H^s} \lesssim \|V(t,\cdot)\|_{H^{s}}\left(\|U(t,\cdot)\|_{H^{1,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|_{H^{1,\infty}}\right) + \|V(t,\cdot)\|_{L^\infty}\|U(t,\cdot)\|_{H^{s+1}}, \label{est Hs NLkg-New}\\\end{gathered}$$ $$\label{est Hsinfty NLkg-new} \begin{split} \|\textit{NL}_{kg}(t,\cdot)\|_{H^{s,\infty}}&\lesssim \|V(t,\cdot)\|^{1-\theta}_{H^{s,\infty}} \|V(t,\cdot)\|^\theta_{H^{s+2}}\left(\| U(t,\cdot)\|_{H^{1,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|_{H^{1,\infty}}\right)\\ &+ \|V(t,\cdot)\|_{L^\infty}\left(\| U(t,\cdot)\|^{1-\theta}_{H^{s+1,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|^{1-\theta}_{H^{s+1,\infty}}\right)\|U(t,\cdot)\|^\theta_{H^{s+3}}, \end{split}$$ $$\label{est L2 Omega NLkg} \begin{split} \|\Omega\textit{NL}_{kg}(t,\cdot)\|_{L^2}&\lesssim \left(\|V(t,\cdot)\|_{L^2}+\|\Omega V(t,\cdot)\|_{L^2}\right)\left(\|U(t,\cdot)\|_{H^{2,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}\right) \\ &+ \|V(t,\cdot)\|_{H^{1,\infty}}\|\Omega U(t,\cdot)\|_{H^1}. \end{split}$$ Inequalities , , , and are straightforward. The same is for and after commutation of $\Omega$ with the operators appearing in . All other inequalities in the statement are rather derived using corollary \[Cor\_appendixA:Hs-Hsinfty norm of bilinear expressions\]. \[Lem: est DtU DtV\] For any $s\ge 0$, any $\theta\in ]0,1[$, $$\begin{gathered} \|D_tU(t,\cdot)\|_{H^s}\lesssim \|U(t,\cdot)\|_{H^{s+1}} + \|V(t,\cdot)\|_{H^{s+1}}\|V(t,\cdot)\|_{H^{1,\infty}}, \label{Hs_norm_DtU}\\ \|D_tU(t,\cdot)\|_{H^{s,\infty}}\lesssim \|U(t,\cdot)\|_{H^{s+2,\infty}} + \|V(t,\cdot)\|^{2-\theta}_{H^{s+1,\infty}}\|V(t,\cdot)\|^\theta_{H^{s+3}}, \label{Hsinfty_norm_DtU} \\ \|D_t \mathrm{R}_1U(t,\cdot)\|_{H^{s,\infty}}\lesssim \|\mathrm{R}_1 U(t,\cdot)\|_{H^{s+1,\infty}} + \|V(t,\cdot)\|_{H^{s+3}}\|V(t,\cdot)\|_{H^{1,\infty}}, \label{Hsinfty norm Dt R1U-new}\\ \|D_t\Omega U(t,\cdot)\|_{L^2}\le \|\Omega U(t,\cdot)\|_{H^1}+ \|V(t,\cdot)\|_{H^{2,\infty}}\left(\|V(t,\cdot)\|_{L^2}+ \|\Omega V(t,\cdot)\|_{H^1}\right), \label{L2_norm_DtOmegaU}\end{gathered}$$ while $$\label{Hs norm DtV} \begin{split} \|D_tV(t,\cdot)\|_{H^s} &\lesssim \|V(t,\cdot)\|_{H^{s+1}} + \|V(t,\cdot)\|_{H^{s}}\left(\|U(t,\cdot)\|_{H^{1,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|_{H^{1,\infty}}\right)\\ & + \|V(t,\cdot)\|_{L^\infty}\|U(t,\cdot)\|_{H^{s+1}}, \end{split}$$ $$\label{est: Hsinfty Dt V} \begin{split} \|D_tV(t,\cdot)\|_{H^{s,\infty}}& \lesssim \|V(t,\cdot)\|_{H^{s+1,\infty}} + \|V(t,\cdot)\|^{1-\theta}_{H^{s,\infty}} \|V(t,\cdot)\|^\theta_{H^{s+1}}\left(\| U(t,\cdot)\|_{H^{1,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|_{H^{1,\infty}}\right)\\ &+ \|V(t,\cdot)\|_{L^\infty}\left(\| U(t,\cdot)\|^{1-\theta}_{H^{s+1,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|^{1-\theta}_{H^{s+1,\infty}}\right)\|U(t,\cdot)\|^\theta_{H^{s+3}}, \end{split}$$$$\label{L2_norm_DtOmegaV} \begin{split} \| D_t\Omega V(t,\cdot)\|_{L^2}& \le \|\Omega V(t,\cdot)\|_{H^1}+ \left(\|V(t,\cdot)\|_{L^2}+\|\Omega V(t,\cdot)\|_{L^2}\right)\left(\|U(t,\cdot)\|_{H^{2,\infty}}+ \|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}\right) \\ &+ \|V(t,\cdot)\|_{H^{1,\infty}}\|\Omega U(t,\cdot)\|_{H^1}. \end{split}$$ Straight consequence of the previous lemma and the fact that $(u_+, v_+, u_{-},v_{-})$ is solution to system . Observe that inequality is derived using that $$\|\mathrm{R}_1 \textit{NL}_w(t,\cdot)\|_{H^{s,\infty}}\lesssim \|\textit{NL}_w(t,\cdot)\|_{H^{s+2}}$$ after classical Sobolev injection and continuity of $\mathrm{R}_1: H^s\rightarrow H^s$, for any $s\ge 0$. Let $|I|=1$ be such that $\Gamma^I\in\{\Omega, Z_m, m=1,2\}$. Then $$\begin{gathered} \label{DtUI} \| D_t U^I(t,\cdot)\|_{L^2}\lesssim \|U^I(t,\cdot)\|_{H^1} + \|V(t,\cdot)\|_{H^{2,\infty}}\Big[\|V^I(t,\cdot)\|_{H^1} \\ + \|V(t,\cdot)\|_{H^1}\Big(1+\sum_{\mu=0}^1\|\mathrm{R}_1^\mu U(t,\cdot))\|_{H^{1,\infty}}\Big) +\|V(t,\cdot)\|_{L^\infty} \|U(t,\cdot)\|_{H^1}\Big],\end{gathered}$$ $$\begin{gathered} \label{DtVI} \| D_t V^I(t,\cdot)\|_{L^2}\lesssim \|V^I(t,\cdot)\|_{H^1} + \sum_{\mu=0}^1\|\mathrm{R}_1^\mu U(t,\cdot)\|_{H^{2,\infty}}\|V^I(t,\cdot)\|_{L^2} \\ + \|V(t,\cdot)\|_{H^{1,\infty}}\left(\|U^I(t,\cdot)\|_{H^1}+\|U(t,\cdot)\|_{H^1}+\|V(t,\cdot)\|_{H^{1,\infty}}\|V(t,\cdot)\|_{H^1}\right).\end{gathered}$$ The result of the statement follows using the equation satisfied, respectively, by $u^I_\pm$ and $v^I_\pm$, together with , with $s=0$. In fact, by with $|I|=1$, $$\begin{gathered} D_t u^I_\pm =\pm |D_x| u^I_\pm + Q^\mathrm{w}_0(v^I_\pm, D_1 v_\pm) + Q^\mathrm{w}_0(v_\pm, D_1 v^I_\pm) + G^\mathrm{w}_1(v_\pm,D v_\pm), \\ D_t v^I_\pm =\pm \langle D_x\rangle v^I_\pm + Q^\mathrm{kg}_0(v^I_\pm, D_1 u_\pm) + Q^\mathrm{kg}_0(v_\pm, D_1 u^I_\pm) + G^\mathrm{kg}_1(v_\pm,D u_\pm),\end{gathered}$$ with $G^\mathrm{w}_1(v_\pm,\partial v_\pm)=G_1(v, \partial v)$, $G^\mathrm{kg}_1(v_\pm,D u_\pm)= G_1(v, \partial u)$ and $G_1$ given by . Hence one can estimate the $L^2$ norm of the first two quadratic terms in above equalities with the $L^2$ norm of factors indexed in $I$ times the $L^\infty$ norm of the remaining one, while the $L^2$ norm of the latter quadratic terms can be instead bounded by taking the $L^2$ norm of one of the two factors times the $L^\infty$ norm of the remaining one, indifferently. We choose here to consider the $L^2$ norm of factors $Du_\pm, Dv_\pm$, and use , if the derivative $D$ is a time derivative. It is useful to remind that, if $w(t,x)$ is solution to inhomogeneous half wave equation from we have that for any $j,k\in\{1,2\}$ and $|\mu|\le 1$ \[equalities\_app\_xw\] $$\label{equality_xDkw} \begin{split} x_j D_k\Big(\frac{D_x}{|D_x|}\Big)^\mu w=& \frac{D_k}{|D_x|}\Big(\frac{D_x}{|D_x|}\Big)^\mu\left[x_j|D_x| -tD_j +\frac{1}{2i}\frac{D_j}{|D_x|}\right]w+t\frac{D_jD_k}{|D_x|}\Big(\frac{D_x}{|D_x|}\Big)^\mu w\\ & -\frac{1}{2i}\frac{D_jD_k}{|D_x|^2}\Big(\frac{D_x}{|D_x|}\Big)^\mu w + i Op\Big(\partial_j\Big(\frac{\xi_k}{|\xi|}\Big(\frac{\xi}{|\xi|}\Big)^\mu\Big)|\xi|\Big)w\\ =&\, i\frac{D_k}{|D_x|}\Big(\frac{D_x}{|D_x|}\Big)^\mu Z_jw + \frac{D_k}{|D_x|}\Big(\frac{D_x}{|D_x|}\Big)^\mu [x_jf(t,x)]\\ &+ t\frac{D_jD_k}{|D_x|}\Big(\frac{D_x}{|D_x|}\Big)^\mu w+ iOp\Big(\partial_j\Big(\frac{\xi_k}{|\xi|}\Big(\frac{\xi}{|\xi|}\Big)^\mu\Big)|\xi|\Big)w. \end{split}$$ Analogously, if $w(t,x)$ is solution to inhomogeneous half Klein-Gordon , from we have that $$\label{xjw_Zjw} \begin{split} x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu w &= \frac{1}{\langle D_x\rangle}\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu\left[\langle D_x\rangle x_j -tD_j\right] w+ t \frac{D_j}{\langle D_x\rangle}\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu w + i\oph\Big(\partial_j\Big(\frac{\xi}{\langle\xi\rangle}\Big)^\mu\Big)w \\ &= i\frac{1}{\langle D_x\rangle}\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu Z_jw -i \frac{D_j}{\langle D_x\rangle^2}\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu w + \frac{1}{\langle D_x\rangle}\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu[x_j f(t,x)]\\ & + t \frac{D_j}{\langle D_x\rangle}\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu w+ i\oph\Big(\partial_j\Big(\frac{\xi}{\langle\xi\rangle}\Big)^\mu\Big)w. \end{split}$$ We also remind the reader about equivalence , so we won’t particularly care if we are dealing with $\Gamma^I u_\pm, \Gamma^I v_\pm$ instead of $(\Gamma^I u)_\pm, (\Gamma^I v)_\pm$, when we bound the $L^2$ norm of those terms with the energy defined in . \[Lem\_appendix: Technical\_estimates\] There exists a positive constant $C>0$ such that, for every $j=1,2$, $t\in [1,T]$, \[norms\_H1\_Linfty\_xv-\] $$\begin{gathered} \sum_{|\mu|=0}^1 \left\| x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{H^1}\le CB\varepsilon t^{1+{\frac{\delta}{2}}}, \label{norm_xv-} \\ \sum_{|\mu|=0}^1 \left\| x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{H^{1,\infty}}\le C(A+B)\varepsilon t^{\frac{\delta_2}{2}}, \label{norm_Linfty_xv-}\end{gathered}$$ and $$\label{norm_xu-} \sum_{|\mu|=0}^1\left\|x_jD_x\Big(\frac{D_x}{|D_x|}\Big)^\mu u_\pm(t,\cdot)\right\|_{L^2}\le CB\varepsilon t^{1+\frac{\delta}{2}}.$$ We warn the reader that, throughout the proof, $C$ will denote a positive constant that may change line after line. As $v_+=-\overline{v_{-}}$ (resp. $u_+=-\overline{u_{-}}$), it is enough to prove the statement for $v_{-}$ (resp. for $u_{-}$). Since $v_{-}$ is solution to equation with $f=\Nlkg$, from it immediately follows that, for any $|\mu|\le 1$, $$\label{xjv_H1} \left\|x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_{-}(t,\cdot)\right\|_{H^1}\lesssim \|Z_jv_{-}(t,\cdot)\|_{L^2}+ t\|v_{-}(t,\cdot)\|_{H^1}+ \|x_j\textit{NL}_{kg}(t,\cdot)\|_{L^2(dx)}$$ along with $$\label{xjv_Linfty} \left\|x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_{-}(t,\cdot)\right\|_{H^{1,\infty}}\le \|Z_jv_{-}(t,\cdot)\|_{H^2}+ t\|v_{-}(t,\cdot)\|_{H^{2,\infty}}+ \|x_j\textit{NL}_{kg}(t,\cdot)\|_{L^\infty(dx)},$$ derived by using the classical Sobolev injection. Observe that $$\label{norm_Linfty_xNLkg} \left\| x_j\textit{NL}_{kg}(t,\cdot)\right\|_{L^\infty} \lesssim \left(\|x_jv_{-}(t,\cdot)\|_{L^\infty}+ \left\|x_j\frac{D_x}{\langle D_x\rangle}v_{-}(t,\cdot) \right\|_{L^\infty} \right) \sum_{\mu=0}^1\|\mathrm{R}^\mu_1U(t,\cdot)\|_{H^{2,\infty}},$$ but also $$\label{norm_L2_xNLkg} \|x_j\textit{NL}_{kg}(t,\cdot)\|_{L^2}\lesssim \sum_{|\mu|=0}^1 \left\| x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^2}\left(\|U(t,\cdot)\|_{H^{2,\infty}}+\|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}\right).$$ Thus, if $\varepsilon_0>0$ is assumed sufficiently small to verify $\varepsilon_0<(2A)^{-1}$, by injecting (resp. ) into (resp. in ), and using a-priori estimates , we obtain that $$\begin{split} \sum_{|\mu|=0}^1 \left\| x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{H^1}&\le C\left[E^2_3(t;W)^\frac{1}{2}+ tE_3(t;W)^\frac{1}{2}\right] + \|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}E_0(t;W)^\frac{1}{2}\\ &\le CB\varepsilon t^{1+\frac{\delta}{2}} \end{split}$$ $$\Big(\text{resp. }\sum_{|\mu|=0}^1 \left\| x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{H^{1,\infty}} \le C E^2_3(t;W)^\frac{1}{2}+t\|V(t,\cdot)\|_{H^{2,\infty}} \le C(A+B)\varepsilon t^{\frac{\delta_2}{2}}\Big),$$and the conclusion of the proof of . Analogously, from with $w=u_{-}$ and $f=\Nlw$, $$\begin{aligned} \sum_{|\mu|=0}^1\left\|x_j D_k \Big(\frac{D_x}{|D_x|}\Big)^\mu u_{-}(t,\cdot)\right\|_{L^2}&\lesssim \|Z_ju_\pm(t,\cdot)\|_{L^2}+ t\|u_\pm(t,\cdot)\|_{L^2}+\|x_j\Nlw(t,\cdot)\|_{L^2(dx)}\\ &\le CB\varepsilon t^{1+\frac{\delta}{2}},\end{aligned}$$ as follows , , and the fact that $$\label{xj_Nlw_Linfty} \|x_j \Nlw(t,\cdot)\|_{L^2}\lesssim \sum_{|\mu|=0}^1\left\|x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty}\|v_\pm(t,\cdot)\|_{H^1}.$$ There exists a constant $C>0$ such that, for every $j=1,2$, $t\in [1,T]$, \[est:L2,Linfty xNLkg\] $$\begin{aligned} \|x_j \textit{NL}_{kg}(t,\cdot)\|_{L^2}&\le C(A+B)B\varepsilon^2 t^\frac{\delta+\delta_2}{2}, \label{xj_NLkg}\\ \|x_j \textit{NL}_{kg}(t,\cdot)\|_{L^\infty}&\le C(A+B)B\varepsilon^2 t^{-\frac{1}{2}+\frac{\delta_2}{2}},\end{aligned}$$ and $$\begin{aligned} \|x_j \textit{NL}_w(t,\cdot)\|_{L^2}&\le C(A+B)B\varepsilon^2 t^\frac{\delta+\delta_2}{2}, \label{est_L2_xNLw}\\ \|x_j \textit{NL}_w(t,\cdot)\|_{L^\infty}&\le C(A+B)B\varepsilon^2 t^{-1+\frac{\delta_2}{2}}.\end{aligned}$$ From $$\|x_j \textit{NL}_{kg}(t,\cdot)\|_{L^2}\lesssim \sum_{\mu=0}^1 \left\| x_j (D_x\langle D_x\rangle^{-1})^\mu v_\pm(t,\cdot)\right\|_{L^\infty}\|u_\pm(t,\cdot)\|_{H^1},$$ and , together with and $$\begin{gathered} \|x_j\textit{NL}_w(t,\cdot)\|_{L^\infty}\lesssim \sum_{\mu=0}^1 \left\| x_j (D_x\langle D_x\rangle^{-1})^\mu v_\pm(t,\cdot)\right\|_{L^\infty}\|v_\pm(t,\cdot)\|_{H^{2,\infty}},\end{gathered}$$ we immediately derive the estimates of the statement using and a-priori estimates. There exists a positive constant $C>0$ such that, for any multi-index $I$ of length $k$, with $1\le k\le 2$, any $j=1,2$, $t\in [1,T]$, $$\label{norm_L2_xj-GammaIv-} \sum_{|\mu|=0}^1\left\|x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu (\Gamma^I v)_\pm(t,\cdot)\right\|_{H^1}+ \left\| x_j D_x\Big(\frac{D_x}{|D_x|}\Big)^\mu (\Gamma^I u)_\pm(t,\cdot)\right\|_{L^2} \le CB\varepsilon t^{1+\frac{\delta_{3-k}}{2}}.$$ We warn the reader that, throughout the proof, $C$ will denote a positive constant that may change line after line. As $\Gamma^I w_+=-\overline{\Gamma^I w_{-}}$, for any $I$ and $w\in\{v,u\}$, it is enough to prove the statement for $\Gamma^I v_{-}$, $\Gamma^I u_{-}$. From equalities together with the fact that, for any multi-index $I$, $(\Gamma^I v)_{-}$, $(\Gamma^I u)_{-}$ are solution to $$\label{half KG Gammav} [D_t + \langle D_x\rangle] (\Gamma^I v)_{-}(t,x) = \Gamma^I\textit{NL}_{kg}$$ and $$[D_t + \langle D_x\rangle] (\Gamma^I u)_{-}(t,x) = \Gamma^I\textit{NL}_{w}$$ respectively, we derive that, for any $j,k\in \{1,2\}$, $$\label{preliminary_norm_xGammav-} \sum_{|\mu|=0}^1\left\|x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu (\Gamma^I v)_\pm(t,\cdot)\right\|_{H^1}\le \|Z_j(\Gamma^I v)_{-}(t,\cdot)\|_{L^2}+t\|(\Gamma^I v)_{-}(t,\cdot)\|_{L^2}+\|x_j\Gamma^I\textit{NL}_{kg}(t,\cdot)\|_{L^2}$$ together with $$\label{preliminary_norm_xGammau} \sum_{|\mu|=0}^1 \left\| x_j D_x\Big(\frac{D_x}{|D_x|}\Big)^\mu (\Gamma^I u)_\pm(t,\cdot)\right\|_{L^2}\le \|Z_j(\Gamma^I u)_{-}(t,\cdot)\|_{L^2}+t\|(\Gamma^I u)_{-}(t,\cdot)\|_{L^2}+\|x_j\Gamma^I\textit{NL}_w(t,\cdot)\|_{L^2}.$$ The first two quantities in above right hand sides are bounded by $CB\varepsilon t^{1+\delta_{3-k}/2}$ after , so the quantities that need to be estimated in order to prove the statement are the $L^2$ norms of $x_j\Gamma^I\textit{NL}_{kg}$, $x_j\Gamma^I\textit{NL}_w$, for $1\le |I|\le 2$. We first prove for $|I|=1$ and $\Gamma^I=\Gamma$, reminding that from , $$\label{notation: NLGamma} \Gamma\textit{NL}_{kg} = Q^\mathrm{kg}_0\big((\Gamma v)_\pm, D_1u_\pm\big) + Q^\mathrm{kg}_0\big(v_\pm, D_1(\Gamma u)_\pm\big) + G^\mathrm{kg}_1\big(v_\pm, Du_\pm\big)$$ and $$\label{Gamma_Nlw} \Gamma\textit{NL}_w = Q^\mathrm{w}_0\big((\Gamma v)_\pm, D_1v_\pm\big) + Q^\mathrm{w}_0\big(v_\pm, D_1(\Gamma v)_\pm\big) + G^\mathrm{w}_1\big(v_\pm, Dv_\pm\big),$$ with $G^\mathrm{kg}_1\big(v_\pm, Du_\pm\big) = G_1(v, \partial u)$, $G^\mathrm{w}_1\big(v_\pm, Dv_\pm\big) = G_1(v, \partial v)$, and $G_1$ given by . By multiplying $x_j$ against the Klein-Gordon component in each product of $\Gamma\textit{NL}_{kg}$ we find that $$\begin{gathered} \label{norm_xmZn_NLkg} \|x_j\Gamma\textit{NL}_{kg}(t,\cdot)\|_{L^2}\lesssim \sum_{|\mu|=0}^1 \left\|x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu(\Gamma v)_{-}(t,\cdot)\right\|_{L^2}\left(\|U(t,\cdot)\|_{H^{2,\infty}}+\|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}\right) \\ +\sum_{|\mu|=0}^1\left\| x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty}\left(\|(\Gamma u)_\pm(t,\cdot)\|_{H^1}+ \|u_\pm(t,\cdot)\|_{H^1}+ \|D_t u_\pm(t,\cdot)\|_{L^2}\right),\end{gathered}$$ which injected into with $\Gamma^I=\Gamma$, together with with $s=0$, , and a-priori estimates , gives that $$\sum_{|\mu|=0}^1\left\|x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu (\Gamma^I v)_\pm(t,\cdot)\right\|_{H^1}\le CB\varepsilon t^{1+\frac{\delta_2}{2}}.$$ Similarly, using the above estimate together with with $s=0$, and a-priori estimates, we derive that $$\label{est_xGammaNLw_proof} \begin{split} &\|x_j\Gamma\textit{NL}_w(t,\cdot)\|_{L^2}\lesssim \sum_{|\mu|=0}^1 \left\|x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu(\Gamma v)_{-}(t,\cdot)\right\|_{L^2}\|v_\pm(t,\cdot)\|_{H^{2,\infty}}\\ &+ \sum_{ |\mu|=0}^1 \left\| x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty} \left(\| (\Gamma v)_\pm(t,\cdot)\|_{H^1}+\|v_\pm(t,\cdot)\|_{H^1}+\|D_t v_\pm(t,\cdot)\|_{L^2}\right) \\ &\le C(A+B)B\varepsilon^2 t^{\delta_2}. \end{split}$$ Plugging the above inequality in for $\Gamma^I=\Gamma$ and using again a-priori estimates we deduce that $$\sum_{|\mu|=0}^1\left\|x_j D_k\Big(\frac{D_x}{|D_x|}\Big)^\mu(\Gamma u)_{-}(t,\cdot)\right\|_{L^2}\le CB\varepsilon t^{1+\frac{\delta_2}{2}},$$ and conclude the proof of when $|I|=1$. When $|I|=2$ we observe that, from , $$\begin{gathered} \label{GammaI_NLkg} \Gamma^I \textit{NL}_{kg} = Q^\mathrm{kg}_0(v^I_\pm, D_1u_\pm)+Q^\mathrm{kg}_0(v_\pm, D_1u^I_\pm)+ \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|=|I_2|=1}}Q^\mathrm{kg}_0(v^{I_1}_\pm, D_1u^{I_2}_\pm)\\ + \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|+|I_2|\le 1}}c_{I_1,I_2}Q^\mathrm{kg}_0(v^{I_1}_\pm, Du^{I_2}_\pm),\end{gathered}$$ with $c_{I_1,I_2}\in \{-1,0,1\}$. Since the $L^2$ norm of terms indexed in $I_1,I_2$ with $|I_1|=|I_2|=1$ can be estimated using the Sobolev injection as follows: $$\label{xj_Q(vI1,uI2)_appendix} \left\|x_j Q^\mathrm{kg}_0(v^{I_1}_\pm, D_1u^{I_2}_\pm)\right\|_{L^2} \lesssim \sum_{ |\mu|=0}^1\|v^{I_1}_\pm(t,\cdot)\|_{H^2}\left\|x_jD_1\Big(\frac{D_x}{|D_x|}\Big)^\mu u^{I_2}_\pm(t,\cdot)\right\|_{L^2},$$ from we derive that $$\begin{aligned} &\|x_j\Gamma^I \textit{NL}_{kg}\|_{L^2}\lesssim \sum_{\substack{|J|\le 2\\ |\mu|,\nu=0}}^1 \left\| x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu(\Gamma^J v)_{-}(t,\cdot)\right\|_{L^2}\Big(\|\mathrm{R}_1^\nu u_\pm(t,\cdot)\|_{H^{2,\infty}}+ \|D_t\mathrm{R}_1^\nu u_\pm(t,\cdot)\|_{H^{1,\infty}}\Big) \\ & + \sum_{|\mu|=0}^1\left\|x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty}\Big[\|u^I_\pm(t,\cdot)\|_{H^1} + \sum_{|J|\le 1}\big(\|u^J_\pm(t,\cdot)\|_{H^1}+\|D_tu^J_\pm(t,\cdot)\|_{L^2}\big) \Big]\\ & + \sum_{\substack{|I_1|=|I_2|=1\\ |\mu|=0,1}}\|v^{I_1}_\pm(t,\cdot)\|_{H^2}\left\|x_jD_1\Big(\frac{D_x}{|D_x|}\Big)^\mu u^{I_2}_\pm(t,\cdot)\right\|_{L^2}.\end{aligned}$$ As before, injecting the above inequality into , using a-priori estimates and the fact that $\varepsilon_0<(2A)^{-1}$, together with with $s=0$, , with $s=1$, , , and with $k=1$, we obtain that $$\label{xGammaIv_I=2} \sum_{|\mu|=0}^1\left\|x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu(\Gamma^I v)_{-}(t,\cdot)\right\|_{H^1}\le CB\varepsilon t^{1+\frac{\delta_1}{2}}.$$ Analogously, since $$\begin{gathered} \Gamma^I \textit{NL}_{w} = Q^\mathrm{w}_0(v^I_\pm, D_1v_\pm)+Q^\mathrm{w}_0(v_\pm, D_1v^I_\pm)+ \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|=|I_2|=1}}Q^\mathrm{w}_0(v^{I_1}_\pm, D_1v^{I_2}_\pm)\\ + \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|+|I_2|<2}}c_{I_1,I_2}Q^\mathrm{w}_0(v^{I_1}_\pm, Dv^{I_2}_\pm),\end{gathered}$$ we have that $$\begin{split} &\|x_j\Gamma^I\textit{NL}_w\|_{L^2} \lesssim \sum_{\substack{|J|\le 2\\|\mu|=0,1}}\left\|x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu (\Gamma^J v)_\pm(t,\cdot)\right\|_{L^2}\Big(\|v_\pm(t,\cdot)\|_{H^{2,\infty}} + \|D_tv_\pm(t,\cdot)\|_{H^{1,\infty}}\Big)\\ &+ \sum_{ |\mu|=0}^1\left\| x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{H^{1,\infty}}\left(\sum_{|J|\le 2}\|(\Gamma^J v)_\pm(t,\cdot)\|_{H^1}+\sum_{|J|\le 1}\|D_tv^J_\pm(t,\cdot)\|_{L^2}\right)\\ & + \sum_{\substack{|I_1|=|I_2|=1\\|\mu|=0,1 }}\|(\Gamma^{I_1}v)_\pm(t,\cdot)\|_{H^2}\left\|x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu (\Gamma^{I_2} v)_\pm(t,\cdot)\right\|_{L^2}, \end{split}$$ so from with $s=0$, with $s=1$, , , with $|I|=1$, and a-priori estimates , we deduce $$\sum_{|\mu|=0}^1\left\|x_j D_k\Big(\frac{D_x}{|D_x|}\Big)^\mu (\Gamma^I u)_{-}(t,\cdot)\right\|_{L^2}\lesssim CB\varepsilon t^{1+\frac{\delta_1}{2}},$$ and hence conclude the proof of inequality also for the case $|I|=2$. There exists a positive constant $C>0$ such that, for any $\Gamma\in\mathcal{Z}$, $j=1,2$, and every $t\in [1,T]$, $$\begin{aligned} \|x_j\Gamma \textit{NL}_{kg}(t,\cdot)\|_{L^2} &\le C(A+B)B\varepsilon^2 t^{\frac{1}{2}+\frac{\delta_2}{2}}, \label{xjGamma_Nlkg}\\ \|x_j\Gamma \textit{NL}_w(t,\cdot)\|_{L^2} &\le C(A+B)B\varepsilon^2 t^{\delta_2}. \label{xjGamma_NLw}\end{aligned}$$ Estimate follows straightly from , with $s=0$, and estimates , , and with $k=1$, while has already been proved in . There exists a constant $C>0$ such that, for every $i,j=1,2$, every $t\in [1,T]$, \[est\_x2v-\] $$\begin{gathered} \sum_{|\mu|=0}^1\left\| x_jx_k \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^2}\le CB\varepsilon t^{2+\frac{\delta_2}{2}}, \label{norm_L2_xixjv-}\\ \sum_{|\mu|=0}^1\left\| x_jx_k \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty}\le C(A+B)\varepsilon t^{1+\frac{\delta_2}{2}}. \label{norm_Linfty_xixjv-}\end{gathered}$$ Moreover, for any $\Gamma\in \mathcal{Z}$, $$\label{est:xixjGamma v-} \sum_{|\mu|=0}^1\left\|x_i x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu (\Gamma v)_\pm(t,\cdot)\right\|_{L^2}\le CB\varepsilon t^{2+\frac{\delta_2}{2}}.$$ The proof of the statement follows from the fact that, by multiplying by $x_i$ and using that $$\|x_i x_j\textit{NL}_{kg}(t,\cdot)\|_{L^2}\lesssim \sum_{|\mu|=0}^1 \left\|x_i x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_{-}(t,\cdot)\right\|_{L^2} \left(\|u_\pm(t,\cdot)\|_{H^{2,\infty}}+\|\mathrm{R}_1u_\pm(t,\cdot)\|_{H^{2,\infty}}\right)$$ together with $$\|x_i x_j \textit{NL}_{kg}(t,\cdot)\|_{L^\infty} \lesssim\sum_{|\mu|=0}^1\left\| x_jx_k \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_{-}(t,\cdot)\right\|_{L^\infty}\left(\|u_\pm (t,\cdot)\|_{H^{2,\infty}}+\|\mathrm{R}_1u_\pm(t,\cdot)\|_{H^{2,\infty}}\right),$$ we derive that $$\begin{gathered} \sum_{|\mu|=0}^1\left\| x_jx_k \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^2} \lesssim \sum_{|\mu|=0}^1\left(\|x^\mu_i (Z_jv)_{-}(t,\cdot)\|_{L^2}+ t\|x_i^\mu v_{-}(t,\cdot)\|_{L^2}\right)\\ + \sum_{\mu=0}^1 \left\|x_i x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_{-}(t,\cdot)\right\|_{L^2} \left(\|u_\pm(t,\cdot)\|_{H^{2,\infty}}+\|\mathrm{R}_1u_\pm(t,\cdot)\|_{H^{2,\infty}}\right),\end{gathered}$$ and using that operator $\langle D_x\rangle^{-1}$ is bounded from $H^1$ to $L^\infty$ $$\begin{gathered} \sum_{|\mu|=0}^1\left\|x_i x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_{-}(t,\cdot)\right\|_{L^\infty} \lesssim \sum_{k=0}^1\left(\|x_i^k (Z_j v)_{-}(t,\cdot)\|_{H^1} + t \|x^k_i v_{-}(t,\cdot)\|_{H^{1,\infty}}\right)\\ +\sum_{k, |\mu|=0}^1 \left\| x^k_i x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_{-}(t,\cdot)\right\|_{L^\infty}\left(\|u_\pm(t,\cdot)\|_{H^{2,\infty}}+\|\mathrm{R}_1u_\pm(t,\cdot)\|_{H^{2,\infty}}\right).\end{gathered}$$ As $\varepsilon_0>0$ verifies that $\varepsilon_0<(2A)^{-1}$, inequality , with $k=1$, and a-priori estimates imply that $$\sum_{ |\mu|=0}^1 \left\|x_i x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_{-}(t,\cdot)\right\|_{L^2}\lesssim CB\varepsilon t^{2+\frac{\delta_2}{2}},$$ while from , with $k=1$ and a-priori estimates, $$\sum_{ |\mu|=0}^1 \left\|x_i x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_{-}(t,\cdot)\right\|_{L^\infty}\le C(A+B)\varepsilon t^{1+\frac{\delta_2}{2}}.$$ As $v_+=-\overline{v_{-}}$, that implies the first part of the statement. Analogously, using with $w= (\Gamma v)_{-}$ and multiplying that relation by $x_i$ we find that $$\begin{gathered} \label{xixjGammav_preliminary_1} \sum_{|\mu|=0}^1\left\| x_i x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu (\Gamma v)_{-}(t,\cdot)\right\|_{L^2}\\ \lesssim \sum_{\mu=0}^1 \big( \|x_i^\mu Z_j (\Gamma v)_{-}(t,\cdot)\|_{L^2} + t\|x^\mu_i (\Gamma v)_{-}(t,\cdot)\|_{L^2} + \|x_i^\mu x_j\Gamma\textit{NL}_{kg}(t,\cdot)\|_{L^2}\big),\end{gathered}$$ and after , and a-priori estimates, $$\label{xixjGammav_preliminary_2} \sum_{\mu=0}^1\big( \|x_i^\mu Z_j (\Gamma v)_{-}(t,\cdot)\|_{L^2} + t\|x^\mu_i (\Gamma v)_{-}(t,\cdot)\|_{L^2}\big) + \| x_j\Gamma\textit{NL}_{kg}(t,\cdot)\|_{L^2} \le CB\varepsilon t^{2+\frac{\delta_2}{2}}.$$ By multiplying both $x_i, x_j$ against each Klein-Gordon factor in $\Gamma\textit{NL}_{kg}$ (see equality ) we derive that $$\begin{gathered} \|x_ix_j \Gamma\textit{NL}_{kg}(t,\cdot)\|_{L^2}\lesssim \sum_{|\mu|\nu=0}^1 \left\|x_i x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu (\Gamma v)_{-}(t,\cdot)\right\|_{L^2}\|\mathrm{R}_1^\nu u_\pm(t,\cdot)\|_{H^{2,\infty}}\\ + \sum_{|\mu|=0}^1 \left\|x_i x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty}\left(\|(\Gamma u)_\pm(t,\cdot)\|_{H^1}+\|u_\pm(t,\cdot)\|_{H^1}+\|D_tu_\pm(t,\cdot)\|_{L^2}\right),\end{gathered}$$ so by with $s=0$, , a-priori estimates and the fact that $\varepsilon_0<(2A)^{-1}$, $$\|x_ix_j \Gamma\textit{NL}_{kg}(t,\cdot)\|_{L^2} \le \frac{1}{2} \|x_i x_j(\Gamma v)_{-}(t,\cdot)\|_{L^2}+ C(A+B)B\varepsilon^2 t^{1+\delta_2},$$ which injected in , together with , implies . There exists a constant $C>0$ such that, for every $i,j=1,2$, every $t\in [1,T]$, \[Lem\_appendix: Technical\_estimates\], $$\label{est_L2:xixj_NL} \|x_i x_j\textit{NL}_{kg}(t,\cdot)\|_{L^2} + \|x_i x_j\textit{NL}_w(t,\cdot)\|_{L^2}\le C(A+B)B\varepsilon^2 t^{1+\frac{\delta+\delta_2}{2}}.$$ Straightforward after , and the following inequality $$\begin{gathered} \|x_i x_j\textit{NL}_{kg}(t,\cdot)\|_{L^2} + \|x_i x_j\textit{NL}_w(t,\cdot)\|_{L^2} \\ \lesssim \sum_{|\mu|=0}^1\left\|x_ix_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty}\left(\|u_\pm(t,\cdot)\|_{H^1}+\|v_\pm(t,\cdot)\|_{H^1}\right).\end{gathered}$$ There exists a constant $C>0$ such that, for any $i,j,k=1,2$, every $t\in [1,T]$, \[Lem\_appendix: Technical\_estimates\], $$\label{est:x3_v-} \sum_{|\mu|=0}^1\left\|x_i x_j x_k \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^2}\le CB\varepsilon t^{3+\frac{\delta_2}{2}}.$$ Using equality we derive that $$\begin{gathered} \|x_i x_j x_k v_{-}(t,\cdot)\|_{L^2} \lesssim \sum_{\mu_1, \mu_2=0}^1\left(\|x^{\mu_1}_i x_j^{\mu_2} (Z_kv)_{-}(t,\cdot)\|_{L^2}+ t\|x^{\mu_1}_i x_j^{\mu_2} v_{-}(t,\cdot)\|_{L^2}\right)\\ + \sum_{\mu_1, \mu_2, |\mu|=0}^1 \left\|x_i^{\mu_1} x_j^{\mu_2}x_k\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_{-}(t,\cdot)\right\|_{L^2} \left(\|u_\pm(t,\cdot)\|_{H^{2,\infty}}+\|\mathrm{R}_1u_\pm(t,\cdot)\|_{H^{2,\infty}}\right),\end{gathered}$$ so the result of the statement is a straight consequence of , , , , a-priori estimates, and the fact that $\varepsilon_0$ is smaller than $(2A)^{-1}$. First range of estimates {#sec_appB: first range of estimates} ------------------------ The aim of this section is to show that, if a-priori estimates are satisfied for every $t\in [1,T]$, for some fixed $T>1$, then in the same interval the semi-classical Sobolev norms of the semi-classical functions $\ut, \vt$ introduced in grow in time at a moderate rate $t^\beta$, for some small $\beta>0$. More precisely, in lemma \[Lem: from energy to norms in sc coordinates-WAVE\] we prove that this is the case for the $H^s_h(\mathbb{R}^2)$ norm of $\ut$, $\ut^{\Sigma_j,k}$ (see definition ) for any $s\le n-15$, and for the $L^2(\mathbb{R}^2)$ norm of those functions when operators $\Omega_h$ and $\Mcal$, introduced in and respectively, are acting on them and frequencies are less or equal than $h^{-\sigma}$, for some small $\sigma>0$. Lemma \[Lem: from energy to norms in sc coordinates-KG\] shows that this moderate growth is also enjoyed by the $H^s_h(\mathbb{R}^2)$ norm of $\vt$, again for $s\le n=15$, and by the $L^2(\mathbb{R}^2)$ norm of $\Lcal\vt$ (see ) when restricted to frequencies $|\xi|\lesssim h^{-\sigma}$. The proof of this latter lemma will require some intermediate results, among which lemma \[Lem\_appendix: preliminary est VJ\] that provides us with a first non-sharp estimate of the $L^\infty(\mathbb{R}^2)$ norm of Klein-Gordon functions $v_\pm$ when one Klainerman vector field is acting on them (and again frequencies are localized for $|\xi|\lesssim t^\sigma$). This estimate will successively improved to the sharpest one in lemma \[Lem\_appendix: sharp\_est\_VJ\] of section \[Sec\_App\_4\]. As said at the beginning of this chapter, we prove the below results under the hypothesis that a-priori estimates are satisfied in some fixed $[1,T]$, with $\varepsilon_0<(2A+B)^{-1}$. We remind here that, if $\chi\in C^\infty_0(\mathbb{R}^2)$ and $\sigma>0$, $\chi(t^{-\sigma}D_x)$ is a bounded operator from $H^s$ to $L^2$ with norm $O(t^{\sigma s})$, and on $L^\infty$ uniformly in time. \[Lem: from energy to norms in sc coordinates-WAVE\] Let $\widetilde{u}, \widetilde{u}^{\Sigma_j,k}$ be defined, respectively, in and , and $s\le n-15$. There exists a constant $C>0$ such that, for any $\theta_0, \chi \in C^\infty_0(\mathbb{R}^2)$ and every $t\in [1,T]$, \[inequality: from\_energy\_to\_norm\_insc\] $$\begin{gathered} \|\widetilde{u}(t,\cdot)\|_{H^s_h}+ \|\widetilde{u}^{\Sigma_j,k}(t,\cdot)\|_{H^s_h} \le CB\varepsilon t^{\frac{\delta}{2}+\kappa}, \label{est:utilde-Hs}\\ \|\Omega_h\widetilde{u}^{\Sigma_j,k}(t,\cdot)\|_{L^2}\le CB\varepsilon t^{\frac{\delta_2}{2}+\kappa},\label{est:Omega-utilde}\\ \sum_{|\mu|=1}\left(\|\oph(\chi(h^\sigma\xi))\mathcal{M}^\mu \widetilde{u}(t,\cdot)\|_{L^2}+ \|\mathcal{M}^\mu\widetilde{u}^{\Sigma_j,k}(t,\cdot)\|_{L^2}\right) \le C(A+B)\varepsilon t^{\frac{\delta_2}{2}+\kappa},\label{est:Mutilde}\\ \sum_{|\mu|=1}\|\theta_0(x)\Omega_h\mathcal{M}^\mu\widetilde{u}^{\Sigma_j,k}(t,\cdot)\|_{L^2} \le CB\varepsilon t^{\frac{\delta_1}{2}+\kappa},\label{est:Omega-M-utilde}\end{gathered}$$ with $\kappa=\sigma\rho$ if $\rho\ge 0$, 0 otherwise. We warn the reader that, throughout the proof, $C$ and $\beta$ will denote positive constants that may change line after line, with $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. We will also use the following concise notation $$\begin{gathered} \phi^j_k(\xi):= \Sigma(\xi)(1-\chi_0)(h^{-1}\xi)\varphi(2^{-k}\xi)\chi_0(h^\sigma\xi), \end{gathered}$$ reminding that $$\label{Op(Sigma varphi)} \left\| \oph(\phi^j_k(\xi))\right\|_{\mathcal{L}(L^2)} =O(h^{-\kappa}),$$ with $\kappa=\sigma\rho$ if $\rho\ge 0$, 0 otherwise. Inequality is straightforward after , definitions and , inequality , and a-priori estimate . By commutating $\oph(\phi^j_k(\xi))$ with $\mathcal{M}$ (the commutator with $\Omega_h$ being zero if $\varphi, \chi_0$ are supposed to be radial) and using we observe that there is some $\chi\in C^\infty_0(\mathbb{R}^2)$ such that $$\begin{gathered} \|\Omega_h \widetilde{u}^{\Sigma_j,k}(t,\cdot)\|_{L^2}\lesssim h^{-\kappa}\|\oph(\chi_0(h^\sigma\xi))\Omega_h \widetilde{u}(t,\cdot)\|_{L^2},\\ \|\mathcal{M} \widetilde{u}^{\Sigma_j,k}(t,\cdot)\|_{L^2}\lesssim h^{-\kappa}\sum_{|\nu|=0}^1\|\oph(\chi(h^\sigma\xi))\mathcal{M}^\nu \widetilde{u}(t,\cdot)\|_{L^2}, \end{gathered}$$ $$\|\theta_0(x)\Omega_h \mathcal{M}\widetilde{u}^{\Sigma_j,k}(t,\cdot)\|_{L^2} \lesssim \|\theta_0(x)\oph(\phi^j_k(\xi))\Omega_h \mathcal{M}\widetilde{u}(t,\cdot)\|_{L^2} + h^{-\kappa}\sum_{\mu=0}^1\|\oph(\chi(h^\sigma\xi))\Omega^\mu_h\widetilde{u}(t,\cdot)\|_{L^2}.$$Therefore, as $h=t^{-1}$, in order to prove - it is enough to show that, for any $\chi\in C^\infty_0(\mathbb{R}^2)$, $$\begin{gathered} \|\oph(\chi(h^\sigma\xi))\Omega_h \widetilde{u}(t,\cdot)\|_{L^2}\le CB\varepsilon t^{\frac{\delta_2}{2}},\label{est_Omega-utilde_proof} \\ \|\oph(\chi(h^\sigma\xi))\mathcal{M} \widetilde{u}(t,\cdot)\|_{L^2} \le C(A+B)\varepsilon t^{\frac{\delta_2}{2}},\label{est_Mutilde-proof} \\ \|\theta_0(x)\oph(\phi^j_k(\xi))\Omega_h \mathcal{M}\widetilde{u}(t,\cdot)\|_{L^2} \le CB\varepsilon t^{\frac{\delta_1}{2}}. \label{est_Omega-M-utilde-proof}\end{gathered}$$ Estimate follows from definitions and , inequality with $u=v=v_\pm$, and a-priori estimates , as $$\begin{aligned} \|\oph(\chi(h^\sigma\xi))\Omega_h\widetilde{u}(t,\cdot)\|_{L^2} &\lesssim \|(\Omega u)_{-}(t,\cdot)\|_{L^2}+ \|\chi(t^{-\sigma}D_x)\Omega(u^{NF}-u_{-})(t,\cdot)\|_{L^2}\\ &\lesssim \|\Omega U(t,\cdot)\|_{L^2}+t^\beta \left(\|V(t,\cdot)\|_{L^2}+\|\Omega V(t,\cdot)\|_{L^2}\right)\|V(t,\cdot)\|_{H^{17,\infty}}\\ & \le C(1+A\varepsilon t^{-1+\beta})E^2_3(t;W)^\frac{1}{2}\le CB\varepsilon t^{\frac{\delta_2}{2}}.\end{aligned}$$ From equality and definition of $u^{NF}$ we deduce that $$\label{L2_norm_Mutilde} \begin{split} &\|\oph(\chi(h^\sigma\xi))\mathcal{M}_n\widetilde{u}(t,\cdot) \|_{L^2}\lesssim \|Z_nU(t,\cdot)\|_{L^2}+ \|\chi(t^{-\sigma}D_x)Z_n(u^{NF}-u_{-})(t,\cdot)\|_{L^2}\\ & + \|\widetilde{u}(t,\cdot)\|_{L^2}+ \|\oph(\chi(h^\sigma\xi))[t(tx_j) [q_w+c_w](t,tx)]\|_{L^2(dx)}+ \|\chi(t^{-\sigma}D_x)(x_nr^{NF}_w)(t,\cdot)\|_{L^2}, \end{split}$$ with $q_w$, $c_w$ and $r^{NF}_w$ given by , and respectively. We first notice that, after inequality with $u=v=v_\pm$, with $s=0$, a-priori estimates, and the fact that $A\varepsilon_0\le 1$, $$\begin{gathered} \label{norm_Zn(uNf-u-)} \|\chi(t^{-\sigma}D_x)Z_n(u^{NF}-u_{-})(t,\cdot)\|_{L^2}\\ \lesssim t^\beta \left(\|D_tV(t,\cdot)\|_{L^2}\|V(t,\cdot)\|_{H^{13}}+ \|V(t,\cdot)\|_{H^{15,\infty}}\|Z_nV(t,\cdot)\|_{L^2}\right)\le CB\varepsilon t^{\beta+\delta}.\end{gathered}$$ Let us also observe that from , we have that $$\label{qw+cw} \begin{split} q_w(t,x) + c_w(t,x) &= \frac{1}{2}\Im\left[\overline{v_{-}}\, D_1 v_{-} - \overline{\frac{D_x}{\langle D_x\rangle}v_{-}}\cdot\frac{D_xD_1}{\langle D_x\rangle}v_{-}\right] (t,x)\\ & = \frac{h^2}{2} \Im \left[\overline{\widetilde{V}}\, \oph(\xi_1)\widetilde{V} - \overline{\oph\Big(\frac{\xi_1}{\langle \xi\rangle}\Big)\widetilde{V}}\cdot \oph\Big(\frac{\xi\xi_1}{\langle \xi\rangle}\Big)\widetilde{V}\right]\Big(t,\frac{x}{t}\Big), \end{split}$$ where $\widetilde{V}(t,x):=tv_{-}(t,tx)$ is such that, for every $s, \rho\ge 0$, $$\|\widetilde{V}(t,\cdot)\|_{H^s_h}=\| v_{-}(t,\cdot)\|_{H^s}, \quad \|\widetilde{V}(t,\cdot)\|_{H^{\rho,\infty}_h} = t\|v_{-}(t,\cdot)\|_{H^{\rho,\infty}}.$$ Moreover, by with $w=v_{-}$ and $f=\textit{NL}_{kg}$ $$\label{Lwidetilde(V)} \begin{split} \|\mathcal{L}_j\widetilde{V}(t,\cdot)\|_{H^1_h}&\lesssim \|Z_j v_{-}(t,\cdot)\|_{L^2}+\|v_{-}(t,\cdot)\|_{L^2}\\ &+ \left(\|x_jv_\pm(t,\cdot)\|_{L^\infty}+\left\|x_j\frac{D_x}{\langle D_x\rangle}v_\pm(t,\cdot)\right\|_{L^\infty}\right)\|U(t,\cdot)\|_{H^1}. \end{split}$$ Using along with the definition of $\mathcal{L}_j$ in we derive that $$\label{x_jqw} \begin{split} & t(tx_j)[q_w+ c_w](t,tx) = \frac{1}{2}\Im\left[\overline{\widetilde{V}} \oph(\xi_1)(h\mathcal{L}_j\widetilde{V}) +\overline{\widetilde{V}}\oph\Big(\frac{\xi_1\xi_j}{\langle\xi\rangle}\Big)\widetilde{V} + \overline{\widetilde{V}}[x_j, \oph(\xi_1)]\widetilde{V} \right. \\ &- \overline{\oph\Big(\frac{\xi}{\langle\xi\rangle}\Big)\widetilde{V}}\cdot \oph\Big(\frac{\xi\xi_1}{\langle\xi\rangle}\Big)(h\mathcal{L}_j\widetilde{V}) - \overline{\oph\Big(\frac{\xi}{\langle\xi\rangle}\Big)\widetilde{V}}\cdot \oph\Big(\frac{\xi\xi_1\xi_j}{\langle\xi\rangle^2}\Big)\widetilde{V}\\ &\left. - \overline{\oph\Big(\frac{\xi}{\langle\xi\rangle}\Big)\widetilde{V}}\cdot \Big[x_j,\oph\Big(\frac{\xi\xi_1}{\langle\xi\rangle}\Big)\Big]\widetilde{V}\right](t,x), \end{split}$$ so after estimates and $$\label{est_xj_qw+cw} \begin{split} \|t(tx_j)[q_w+c_w](t,t\cdot)\|_{L^2(dx)}& \lesssim \left[\|\widetilde{V}(t,\cdot)\|_{H^1_h}+ h\|\mathcal{L}_j\widetilde{V}(t,\cdot)\|_{H^1_h}\right]\|\widetilde{V}(t,\cdot)\|_{H^{1,\infty}_h} \\ &\le CA(A+B)\varepsilon^2 t^{\frac{\delta}{2}}. \end{split}$$ Moreover, from , the fact that $x_je^{ix\cdot\xi}=D_{\xi_j}e^{ix\cdot\xi}$, integration by parts, and inequalities with $\rho=2$ (after the first part of lemma \[Lem\_Appendix: est on Dj1j2\]), , we get that $$\label{norm_xnrNF} \begin{split} \|\chi(t^{-\sigma}D_x)&(x_nr^{NF}_w)(t,\cdot)\|_{L^2}\\ &\lesssim t^\beta \left[\|x_nv_{-}(t,\cdot)\|_{L^\infty}\|\textit{NL}_{kg}(t,\cdot)\|_{H^{15}}+\|V(t,\cdot)\|_{H^{15}}\|x_n\textit{NL}_{kg}(t,\cdot)\|_{L^\infty} \right.\\ &\left. + \|\textit{NL}_{kg}(t,\cdot)\|_{L^2}\left(\|V(t,\cdot)\|_{H^{13}}+ \|V(t,\cdot)\|_{H^{13,\infty}}\right) + \|V(t,\cdot)\|_{H^{13}}\|\textit{NL}_{kg}(t,\cdot)\|_{L^\infty}\right]\\ &\le CB\varepsilon t^\frac{\delta_2}{2}, \end{split}$$ where last estimate follows from , , inequalities , , with $s=15$, and a-priori estimates . Consequently, from , , , , and a-priori estimate with $k=2$, we obtain . Let us now apply $\theta_0\big(\frac{x}{t}\big)\phi^j_k(D_x)\Omega$ to both sides of to deduce that $$\label{L2_norm_thetaM-utilde} \begin{split} &\left\| \theta_0(x) \oph(\phi^j_k(\xi))\Omega_h\mathcal{M}_n\widetilde{u}(t,\cdot)\right\|_{L^2} \lesssim \|\Omega Z_n U(t,\cdot)\|_{L^2}\\ &+ \left\|\theta_0\Big(\frac{x}{t}\Big)\phi^j_k(D_x)\Omega Z_n(u^{NF}-u_{-})(t,\cdot)\right\|_{L^2}+\sum_{\mu=0}^1 \|\oph(\chi_0(h^\sigma\xi))\Omega^\mu_h\widetilde{u}(t,\cdot)\|_{L^2}\\ &+ \left\|\theta_0(x)\oph(\phi^j_k(\xi))\Omega_h[t(tx_j)(q_w+c_w)(t,tx)]\right\|_{L^2(dx)}+ \left\|\theta_0(x)\oph(\phi^j_k(\xi))\Omega_h [t (tx_n)r^{NF}_w](t,tx)]\right\|_{L^2(dx)}. \end{split}$$In order to estimate the second addend in the above right hand side we first commute $Z_n$ to $\Omega$, reminding that $$[\Omega, Z_1]=-Z_2 \quad \text{and}\quad [\Omega, Z_2]=Z_1,$$ and use that $$\theta_0\Big(\frac{x}{t}\Big)\phi^j_k(D_x) Z_j = \Big[t\theta^j_0\Big(\frac{x}{t}\Big)\phi^j_k(D_x) + \theta_0\Big(\frac{x}{t}\Big)[\phi^j_k(D_x),x_j]\Big]\partial_t + t \theta_0\Big(\frac{x}{t}\Big)\phi^j_k(D_x)\partial_j,$$ with $\theta^j_0(z):=\theta_0(z)z_j$. Observe that commutator $[\phi^j_k(D_x),x_j]$ is bounded on $L^2$ with norm $O(t)$, and that its symbol is still supported for moderate frequencies $|\xi|\lesssim t^{-\sigma}$. Therefore, for some new $\chi\in C^\infty_0(\mathbb{R}^2)$ we have that $$\begin{split} \left\|\theta_0\Big(\frac{x}{t}\Big)\phi^j_k(D_x)\Omega Z_n(u^{NF}-u_{-})(t,\cdot)\right\|_{L^2} &\lesssim t \left\|\chi(t^{-\sigma}D_x)\partial_{t,x}(u^{NF}-u_{-})(t,\cdot)\right\|_{L^2}\\ &+ t\left\| \chi(t^{-\sigma}D_x) \partial_{t,x}\Omega(u^{NF}-u_{-})(t,\cdot)\right\|_{L^2}, \end{split}$$ so using with $\rho=2$ (because of first part of lemma \[Lem\_Appendix: est on Dj1j2\]) and , both considered with $u=\partial_{t,x}v_\pm, v=v_\pm$, and $u=v_\pm, v=\partial_{t,x}v_\pm$, we obtain that the above right hand side is estimated by $$t^{1+\beta}\left[\left(\|\partial_{t,x}V(t,\cdot)\|_{L^2}+\|\Omega \partial_{t,x}V(t,\cdot)\|_{L^2}\right)\|V(t,\cdot)\|_{H^{17,\infty}} + \left(\|V(t,\cdot)\|_{L^2}+\|\Omega V(t,\cdot)\|_{L^2} \right)\|\partial_{t,x}V(t,\cdot)\|_{H^{17,\infty}}\right]$$ From and with $s=0$, and a-priori estimates, we hence deduce that $$\label{OmegaZ(uNF-u)} \left\|\theta_0\Big(\frac{x}{t}\Big)\phi^j_k(D_x)\Omega Z_n(u^{NF}-u_{-})(t,\cdot)\right\|_{L^2} \le CB\varepsilon t^{\beta+\frac{\delta_2}{2}}.$$ As concerns, instead, the estimate of the fourth $L^2$ norm in the right hand side of , we observe that from equality , Leibniz rule and $$\begin{gathered} \label{Omegah(xjqw)} \left\|\theta_0(x)\oph(\phi^j_k(\xi))\Omega_h[t(tx_j)[q_w+c_w](t,tx)]\right\|_{L^2} \lesssim \sum_{\mu=0}^1h^{-\kappa}\|\widetilde{V}(t,\cdot)\|_{H^{2,\infty}_h}\|\Omega^\mu_h\widetilde{V}(t,\cdot)\|_{H^1_h} \\ +\sum_{\mu=0}^1 h^{1-\kappa}\|\widetilde{V}(t,\cdot)\|_{H^{1,\infty}_h}\|\Omega^\mu_h\mathcal{L}_j\widetilde{V}(t,\cdot)\|_{H^1}+ h^{1-\kappa}\|\Omega_h \widetilde{V}(t,\cdot)\|_{L^\infty}\|\mathcal{L}_j \widetilde{V}(t,\cdot)\|_{H^1_h},\end{gathered}$$ with $\kappa=\sigma\rho$ if $\rho\ge 0$, 0 otherwise. Using the semi-classical Sobolev injection, and the fact that $\|\Omega_h\widetilde{V}(t,\cdot)\|_{H^s_h}=\|\Omega v_{-}(t,\cdot)\|_{H^s}$ for any $s\ge 0$, together with and a-priori estimates, we see that $$\label{hOmegaV LjV} h\|\Omega_h \widetilde{V}(t,\cdot)\|_{L^\infty}\|\mathcal{L}_j \widetilde{V}(t,\cdot)\|_{H^1_h} \lesssim \|\Omega \widetilde{V}(t,\cdot)\|_{H^2_h}\|\mathcal{L}_j \widetilde{V}(t,\cdot)\|_{H^1_h} \le CB\varepsilon t^{\frac{3\delta_2}{2}}.$$ Also, from with $w=v_{-}$ and $f=\textit{NL}_{kg}$$$\|\Omega_h\mathcal{L}_j\widetilde{V}(t,\cdot)\|_{L^2} \lesssim \|\Omega Z_jv_{-}(t,\cdot)\|_{L^2}+\sum_{\mu=0}^1\|\Omega^\mu v_{-}(t,\cdot)\|_{L^2} + \left\|\Omega \left(x_j\textit{NL}_{kg}\right)(t,\cdot)\right\|_{L^2}\le C(A+B)B\varepsilon^2 t^{\frac{1}{2}+\frac{\delta_2}{2}},$$where last inequality is obtained using , and estimates , . Therefore $$h\|\widetilde{V}(t,\cdot)\|_{H^{1,\infty}_h}\|\Omega_h\mathcal{L}_j\widetilde{V}(t,\cdot)\|_{L^2} \le CAB(A+B)\varepsilon^3 t^{-\frac{1}{2}+\frac{\delta_2}{2}},$$ which combined with , and a-priori estimates gives that $$\label{est_Omegah(xqw)} \left\|\theta_0(x)\oph(\phi^j_k(\xi))\Omega_h[t(tx_j)[q_w+c_w](t,tx)]\right\|_{L^2} \le CB\varepsilon t^{\frac{3\delta_2}{2}}.$$ We estimate the latter $L^2$ norm in recalling definition of $r^{NF}_w$, commutating $\Omega$ and $x_n$, and using that $$\theta_0(x) \oph(\phi^j_k(\xi))x_n = \theta^n_0(x)\oph(\phi^j_k(\xi)) + \theta_0(x)[\oph(\phi^j_k(\xi)), x_n],$$ where $$[\oph(\phi^j_k(\xi)), x_n] = -i h\oph(\partial_n \phi^j_k(\xi))$$ is uniformly bounded on $L^2$. After , with $\theta\ll 1$ small, and a-priori estimates we derive that, for some $\chi\in C^\infty_0(\mathbb{R}^2)$, $$\begin{split} & \left\|\theta_0(x)\oph(\phi^j_k(\xi))\Omega_h [t (tx_n)r^{NF}_w](t,tx)]\right\|_{L^2(dx)} \lesssim \sum_{\mu=0}^1 t\|\chi(t^{-\sigma}D_x)\Omega^\mu r^{NF}_w(t,\cdot)\|_{L^2}\le CB\varepsilon. \end{split}$$ Combining , , and above estimate together with , , , and assuming $3\delta_2\le \delta_1$, we finally obtain and the conclusion of the proof. In the following lemma we explain how we estimate the $L^2$ or the $L^\infty$ norm of products supported for moderate frequencies $|\xi|\lesssim t^\sigma$, when we have a control on high Sobolev norms of, at least, all factors but one. This type of estimate will be frequently used in most of the results that follow. \[Lem\_appendix:L\_estimate of products\] Let $n\in\mathbb{N}$, $n\ge 2$, and $w_1,\dots, w_n$ such that $w_1\in L^2(\mathbb{R}^2)$, $w_2,\dots,w_n\in L^\infty(\mathbb{R}^2)\cap H^s(\mathbb{R}^2)$, for some large positive $s$. Let also $\chi\in C^\infty_0(\mathbb{R}^2)$ and $\sigma>0$. There exists some $\chi_1\in C^\infty_0(\mathbb{R}^2)$, equal to 1 on the support of $\chi$, such that for $L=L^2$ or $L=L^\infty$ $$\begin{gathered} \left\|\chi(t^{-\sigma}D_x)\left[w_1\dots w_n\right]\right\|_{L} \lesssim \left\|\left[\chi_1(t^{-\sigma}D_x)w_1\right] \prod_{j=2}^n \chi(t^{-\sigma}D_x)w_j\right\|_{L(dx)} \\ + t^{-N(s)}\|w_1\|_{L^2(dx)}\sum_{j=2}^n\prod_{k\ne j}\|w_k\|_{L^\infty}\|w_j\|_{H^s(dx)},\end{gathered}$$ with $N(s)$ as large as we want as long as $s>0$ is large. The idea of the proof is to decompose each factor $w_j$, for $j=2,\dots,n$ into $$\label{dec_wj} \chi(t^{-\sigma}D_x)w_j + (1-\chi)(t^{-\sigma}D_x)w_j,$$ and to estimate the $L^2$ norm of product $$\label{prod_trunc} \chi(t^{-\sigma}D_x)\left[w_1\prod_{\substack{k=2,\dots,n\\ k\ne j}}\widetilde{w}_k \left[ (1-\chi)(t^{-\sigma}D_x)w_j \right]\right],$$ where $\widetilde{w}_k$ is either $w_k$ or $\chi(t^{-\sigma}D_x)w_k$, with the $L^2$ norm of $w_1$ times the $L^\infty$ norm of all remaining factors, reminding that $\chi(t^{-\sigma}D_x)$ is uniformly bounded on $L^\infty$ and that by Sobolev injection and , $$\left\|(1-\chi)(t^{-\sigma}D_x)w_j\right\|_{L^\infty(dx)}\lesssim t^{-N(s)}\|w_j\|_{H^s(dx)},$$ with $N(s)$ as large as we want as long as $s>0$ is large. The $L^\infty$ norm of is estimated in the same way, using firstly the $L^2-L^\infty$ continuity of operator $\chi(t^{-\sigma}D_x)$ acting on the entire product. The end of the statement follows from the observation that, if $\text{supp}\chi\subset B_C(0)$ for some $C>0$, then $$\label{property_support} \text{supp}\hat{w}_1\subset\{\xi : |\xi|\ge C_1>nC\} \quad \Rightarrow \quad\chi(t^{-\sigma}D_x)\Big[w_1\prod_{j=2}^n\chi(t^{-\sigma}D_x)w_j\Big]\equiv 0.$$ Property is more general, meaning that if $\chi,\chi_j\in C^\infty_0(\mathbb{R}^2)$ with $\text{supp}\chi\subset B_C(0)$, $\text{supp}\chi_j\subset B_{C_j}(0)$ for some $C,C_j>0$, for every $j=2,\dots,n$, then $$\text{supp}\hat{w}_1\subset \Big\{\xi : |\xi|\ge C_1>C+\sum_{j=2}^n C_j\Big\} \quad \Rightarrow \quad\chi(t^{-\sigma}D_x)\Big[w_1\prod_{j=2}^n\chi_j(t^{-\sigma}D_x)w_j\Big]\equiv 0.$$ We have seen at the beginning of section \[Sub: App\_B1\], and already used in the previous lemma’s proof, that, if $w\in H^s(\mathbb{R}^2)$ for some large $s>0$, the $L^2$ norm (resp. $L^\infty$ norm) of this function when restricted to large frequencies $|\xi|\gtrsim t^\sigma$ decays fast in time as $t^{-\sigma s}$ (resp. $t^{-\sigma(s-1)-1}$ after the semi-classical Sobolev injection). The aim of the following lemma is to show that, even if we don’t have a control on the $H^s(\mathbb{R}^2)$ norm of $(\Gamma u)_\pm$, $(\Gamma v)_\pm$, for $\Gamma\in\{\Omega,Z_m, m=1,2\}$ and $s$ larger than 2, the $L^2$ norm (resp. $L^\infty$ norm) of products as in still have a good decay in time. \[Lem\_app:products\_Gamma\] Let $w\in \{u,v\}$ and for any $\Gamma\in \{\Omega, Z_m , m=1,2\}$ $$(\Gamma w)_\pm = \begin{cases} (D_t\pm |D_x|)(\Gamma u), \quad & \text{if } w=u,\\ (D_t\pm \langle D_x\rangle)(\Gamma v), \quad & \text{if } w=v. \end{cases}$$ Let also $n\in\mathbb{N}^*$, $w_1,\dots, w_n$ be such that $w_1, xw_1\in L^2(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$, $w_j\in L^\infty(\mathbb{R}^2)$ for $j=2,\dots,n$, $\chi\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$, and $a(D_x)= D_x^\alpha\big(\frac{D_x}{\langle D_x\rangle}\big)^\beta \big(\frac{D_x}{|D_x|}\big)^\gamma$ for any $\alpha,\beta,\gamma\in\mathbb{N}^2$ with $|\alpha|, |\beta|, |\gamma|\le 1$. Then for $L=L^2$ or $L=L^\infty$ we have that \[lem\_prod\_Omega\_Z\] $$\label{lem_prod_Omegaw} \begin{split} \left\| a(D_x)(\Omega w)_\pm w_1\dots w_n \right\|_{L(dx)} &\lesssim \left\|\left[\chi(t^{-\sigma}D_x)a(D_x) (\Omega w)_\pm\right] \prod_{j=1}^n w_j\right\|_{L(dx)}\\ &+t^{-N(s)} \|w_\pm(t,\cdot)\|_{H^s}\Big(\sum_{|\mu|=0}^1\|x^\mu w_1\|_{L(dx)}\Big)\prod_{j=2}^n \|w_j\|_{L^\infty(dx)} \end{split}$$ and, for $m=1,2,$ $$\begin{gathered} \label{lem_prod_Zmw} \left\| a(D_x)(Z_m w)_\pm w_1\dots w_n \right\|_{L(dx)} \lesssim \left\|\left[\chi(t^{-\sigma}D_x)a(D_x) (Z_m w)_\pm\right] \prod_{j=1}^n w_j\right\|_{L(dx)}\\ +t^{-N(s)}\left( \|w_\pm(t,\cdot)\|_{H^s}+\|D_tw_\pm(t,\cdot)\|_{H^s}\right)\Big(\sum_{\mu=0}^1\|x^\mu_m w_1\|_{L(dx)}+ t\|w_1\|_{L(dx)}\Big)\prod_{j=2}^n \|w_j\|_{L^\infty(dx)},\end{gathered}$$ with $N(s)$ as large as we want as long as $s>0$ is large. Let us remind definition of Klainerman vector fields $\Omega, Z_m$, for $m=1,2$, and decompose factor $a(D_x)(\Gamma w)_\pm$ in frequencies by means of operator $\chi(t^{-\sigma}D_x)$. When dealing with product $$\label{prod_app_w1wn} \big[(1-\chi)(t^{-\sigma}D_x)a(D_x)(\Gamma w)_\pm\big]w_1\cdots w_n$$ the idea is to discharge on $w_1$ factors $x$ and/or $t$ defining $\Gamma$, after a previous commutation between $D_t\pm |D_x|$ if $w=u$ (resp. $D_t\pm \langle D_x\rangle$ if $w=v$) and $\Gamma$, and between $(1-\chi)(t^{-\sigma}D_x)a(D_x)$ and the mentioned factors $x, t$. For instance, if $w=u$ and $\Gamma=Z_1$ $$\begin{gathered} \big[(1-\chi)(t^{-\sigma}D_x)a(D_x)(Z_1 u)_\pm\big] w_1 = \big[ (1-\chi)(t^{-\sigma}D_x)a(D_x)(\partial_t u)_\pm\big](x_1 w_1)\\ + \big[(1-\chi)(t^{-\sigma}D_x)a(D_x)(\partial_1 u)_\pm\big] (tw_1)+ \Big[(1-\chi)(t^{-\sigma}D_x)a(D_x)\frac{D_1}{|D_x|} u_\pm\Big] w_1 \\ + \Big[\big[(1-\chi)(t^{-\sigma}D_x)a(D_x), x_1\big]D_t u_\pm\Big] w_1,\end{gathered}$$from which we deduce, using the Sobolev injection together with , that $$\left\|\big[ (1-\chi)(t^{-\sigma}D_x)a(D_x)(Z_1 u)_\pm\big] w_1\right\|_{L} \lesssim t^{-N(s)}\left(\|u_\pm(t,\cdot)\|_{H^s}+\|D_t u(t,\cdot)\|_{H^s}\right)\Big(\sum_{\mu=0}^1 \|x_1^\mu w_1\|_{L} + t\|w_1\|_{L}\Big),$$with $N(s)$ large as long as $s$ is large. Analogous inequalities can be obtained for $\Gamma=\Omega, Z_2$ and/or $w=v$. This concludes the proof of the statement since the $L$ norm of is bounded by the $L$ norm of $\big[(1-\chi)(t^{-\sigma}D_x)a(D_x)(Z_1 u)_\pm\big] w_1$ times the $L^\infty$ norm of the remaining factors. If the hypothesis of lemma \[Lem\_app:products\_Gamma\] are satisfied and in addition $w_1,\dots,w_n \in H^s(\mathbb{R}^2)$, we have that \[cor\_app\_est\_1\]$$\begin{split} \left\| a(D_x)(\Omega w)_\pm w_1\cdots w_n \right\|_{L} &\lesssim \left\|\left[\chi(t^{-\sigma}D_x)a(D_x) (\Omega w)_\pm\right] \prod_{j=1}^n \chi(t^{-\sigma}D_x)w_j\right\|_{L}\\ &+t^{-N(s)} \|w_\pm(t,\cdot)\|_{H^s(dx)}\Big(\sum_{|\mu|=0}^1\|x^\mu w_1\|_{L}\Big)\prod_{j=2}^n \|w_j\|_{L^\infty}\\ &+ t^{-N(s)} \|(\Omega w)_\pm(t,\cdot)\|_{L^2}\sum_{j=1}^n \prod_{k\ne j}\|w_k\|_{L^\infty}\|w_j\|_{H^s} \end{split}$$and, for $m=1,2$, $$\begin{split} &\left\| a(D_x)(Z_m w)_\pm w_1\cdots w_n \right\|_{L} \lesssim \left\|\left[\chi(t^{-\sigma}D_x)a(D_x) (Z_m w)_\pm\right] \prod_{j=1}^n \chi(t^{-\sigma}D_x)w_j\right\|_{L}\\ & +t^{-N(s)}\left( \|w_\pm(t,\cdot)\|_{H^s}+\|D_tw_\pm(t,\cdot)\|_{H^s}\right)\Big(\sum_{\mu=0}^1\|x^\mu_m w_1\|_{L}+ t\|w_1\|_{L}\Big)\prod_{j=2}^n \|w_j\|_{L^\infty}\\ &+ t^{-N(s)} \|(Z_m w)_\pm(t,\cdot)\|_{L^2}\sum_{j=1}^n \prod_{k\ne j}\|w_k\|_{L^\infty}\|w_j\|_{H^s}, \end{split}$$ with $N(s)$ as large as we want as long as $s>0$ is large. Moreover, there exists $\chi_1\in C^\infty_0(\mathbb{R}^2)$ such that, for any fixed $j_0\in \{1,\dots,n\}$, \[cor\_app\_est\_2\] $$\label{cor_estOmega_2} \begin{split} &\left\|\chi(t^{-\sigma}D_x)\big[a(D_x)(\Omega w)_\pm w_1\cdots w_n\big] \right\|_{L} \\ &\lesssim \left\|\big[\chi(t^{-\sigma}D_x)a(D_x)(\Omega w)_\pm \big]\big[\chi_1(t^{-\sigma}D_x)w_{j_0}\big]\prod_{\substack{j=1,\dots,n\\j\ne j_0}}\chi(t^{-\sigma}D_x)w_j \right\|_{L} \\ &+t^{-N(s)} \|w_\pm(t,\cdot)\|_{H^s}\Big(\sum_{|\mu|=0}^1\|x^\mu w_1\|_{L}\Big)\prod_{j=2}^n \|w_j\|_{L^\infty}\\ &+ t^{-N(s)} \|(\Omega w)_\pm(t,\cdot)\|_{L^2}\sum_{\substack{j=1,\dots,n \\ j\ne j_0}} \prod_{k\ne j}\|w_k\|_{L^\infty}\|w_j\|_{H^s} \end{split}$$ and, for $m=1,2$, $$\label{cor_estZm_2} \begin{split} &\left\|\chi(t^{-\sigma}D_x)\big[a(D_x)(Z_m w)_\pm w_1\cdots w_n\big] \right\|_{L}\\ & \lesssim \left\|\big[\chi(t^{-\sigma}D_x)a(D_x) (Z_m w)_\pm \big]\big[\chi_1(t^{-\sigma}D_x)w_{j_0}\big]\prod_{\substack{j=1,\dots,n\\j\ne j_0}}\chi(t^{-\sigma}D_x)w_j \right\|_{L} \\ & +t^{-N(s)}\left( \|w_\pm(t,\cdot)\|_{H^s}+\|D_tw_\pm(t,\cdot)\|_{H^s}\right)\Big(\sum_{\mu=0}^1\|x^\mu_m w_1\|_{L}+ t\|w_1\|_{L}\Big)\prod_{j=2}^n \|w_j\|_{L^\infty}\\ &+ t^{-N(s)} \|(Z_m w)_\pm(t,\cdot)\|_{L^2}\sum_{\substack{j=1,\dots,n \\ j\ne j_0}} \prod_{k\ne j}\|w_k\|_{L^\infty}\|w_j\|_{H^s}. \end{split}$$ The inequalities of the statement mainly follows from . In fact, by decomposing each factor $w_j$ appearing in the first norm in the right hand sides of as in , and then using the following inequality, for $\Gamma\in \{\Omega, Z_m, m=1,2\}$ and $\widetilde{w}_k$ either equal to $w_k$ or to $\chi(t^{-\sigma}D_x)w_k$, $$\begin{gathered} \left\| [\chi(t^{-\sigma}D_x)a(D_x)(\Gamma w)_\pm] \prod_{\substack{k=1,\dots,n\\ k\ne j}}\widetilde{w}_k \left[ (1-\chi)(t^{-\sigma}D_x)w_j \right]\right\|_{L} \\ \lesssim t^{-N(s)}\|(\Gamma w)_\pm(t,\cdot)\|_{L^2}\prod_{\substack{k=1,\dots,n\\ k\ne j}}\|w_k\|_{L^\infty}\|w_j\|_{H^s},\end{gathered}$$ with $N(s)$ as large as we want as long as $s>0$, which is obtained from together with the $L^2-L^\infty$ and $L^\infty-L^\infty$ continuity of operator $\chi(t^{-\sigma}D_x)$, we obtain . On the other hand, if the product in the left hand side of is localized in frequencies by means of operator $\chi(t^{-\sigma}D_x)$, so it is for the product in the first norm of the same inequalities. Inequalities are then derived by bounding these $L$ norms by means lemma \[Lem\_appendix:L\_estimate of products\], where the role of $w_1$ is here played by $w_{j_0}$, for some fixed $j_0\in \{1,\dots,n\}$. The following two lemmas are stated and proved in view of lemma \[Lem\_appendix: preliminary est VJ\], in which we recover a first non-sharp estimate on the $L^\infty$ norm of the Klein-Gordon component when one Klainerman vector field is acting on it and its frequencies are less or equal than $t^\sigma$, for some small $\sigma>0$. This estimate will be successively refined in lemma \[Lem\_appendix: sharp\_est\_VJ\]. \[Lem\_appendix:Linfty\_bound\_chi\_w\] Let $\chi\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$ small, and $w=w(t,x)$ such that, if $\widetilde{w}(t,x):=tw(t,tx)$, $\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu \widetilde{w}(t,\cdot)\in L^2(\mathbb{R}^2)$ for any $|\mu|\le 1$. Then $$\label{ineq:norm_Linfty_chi-w} \left\| \chi(t^{-\sigma}D_x)w(t,\cdot)\right\|_{L^\infty} \lesssim t^{-1+\beta} \sum_{|\mu|=0}^1\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu \widetilde{w}(t,\cdot) \right\|_{L^2},$$ with $\beta>0$ small, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. Since $$\chi(t^{-\sigma}D_x)w(t,y) = t^{-1}\oph(\chi(h^\sigma\xi))\widetilde{w}(t,x)|_{x=\frac{y}{t}},$$ the goal is to prove that $$\label{norm_Linfty_vtildeGamma} \left\|\oph(\chi(h^\sigma\xi))\widetilde{w}(t,\cdot) \right\|_{L^\infty}\lesssim h^{-\beta}\sum_{|\mu|=0}^1\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu \widetilde{w}(t,\cdot) \right\|_{L^2},$$ for a small $\beta>0$, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. So let $w^\chi:=\oph(\chi(h^{\sigma}\xi)) \widetilde{w}$ and take $\chi_1\in C^\infty_0(\mathbb{R}^2)$ equal to 1 on the support of $\chi$, so that $$\oph(\chi(h^{\sigma}\xi)) \widetilde{w} = \oph(\chi_1(h^{\sigma}\xi)) \widetilde{w}^\chi.$$ For a $\gamma\in C^\infty_0(\mathbb{R}^2)$, equal to 1 in a neighbourhood of the origin and with sufficiently small support, we consider the following decomposition $$\oph\Big(\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi_1(h^\sigma\xi)\Big)\widetilde{w}^\chi + \oph\Big((1-\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi_1(h^\sigma\xi)\Big)\widetilde{w}^\chi$$ and immediately observe that, from inequality , $$\left\| \oph\Big((1-\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi_1(h^\sigma\xi)\Big)\widetilde{w}^\chi(t,\cdot) \right\|_{L^\infty}\lesssim h^{-\beta}\sum_{|\mu|=0}^1\| \oph(\chi_1(h^\sigma\xi))\mathcal{L}^\mu \widetilde{w}^\chi(t,\cdot)\|_{L^2}.$$ After lemma \[Lem:family\_thetah\] there exists a family of smooth cut-off functions $\theta_h(x)$ such that equality holds. Then, if we also consider a new cut-off function $\chi_2$ equal to 1 on the support of $\chi_1$ and a small $\sigma_1>\sigma$, by symbolic calculus and remark \[Remark:symbols\_with\_null\_support\_intersection\] we derive that for any $N\in\mathbb{N}$ $$\begin{split} \oph\Big(\gamma\left(\frac{x-p'(\xi)}{\sqrt{h}}\right)\chi_1(h^\sigma\xi)\Big)\widetilde{w}^\chi & = \theta_h(x)\oph(\chi_2(h^\sigma\xi)) \oph\Big(\gamma\left(\frac{x-p'(\xi)}{\sqrt{h}}\right)\chi_1(h^\sigma\xi)\Big)\widetilde{w}^\chi\\ & + \oph(r_\infty(x,\xi))\widetilde{w}^\chi + \theta_h(x) \oph(r^1_\infty(x,\xi))\widetilde{w}^\chi \end{split}$$ with $r_\infty, r^1_\infty\in h^N S_{\frac{1}{2},\sigma}(\langle \frac{x-p'(\xi)}{\sqrt{h}}\rangle^{-1})$. It is enough to take $N=1$ to have, by proposition \[Prop : Continuity from $L^2$ to L\^inf\], that $$\|\oph(r_\infty)\widetilde{w}^\chi(t,\cdot)\|_{L^\infty} + \|\theta_h(x)\oph(r^1_\infty)\widetilde{w}^\chi(t,\cdot)\|_{L^\infty}\le h^{-\beta}\|\widetilde{w}^\chi(t,\cdot)\|_{L^2}.$$ As function $\phi(x):=\sqrt{1-|x|^2}$ is well defined on the support of $\theta_h$ we are allowed to to write the following: $$\begin{aligned} & \left\| \theta_h(x) \oph(\chi_2(h^{\sigma_1}\xi))\oph\Big(\gamma\left(\frac{x-p'(\xi)}{\sqrt{h}}\right)\chi_1(h^\sigma\xi)\Big)\widetilde{w}^\chi(t,\cdot)\right\|_{L^\infty} \\ & = \left\| e^{\frac{i}{h}\phi}\theta_h(x) \oph(\chi_2(h^{\sigma_1}\xi)) \oph\Big(\gamma\left(\frac{x-p'(\xi)}{\sqrt{h}}\right)\chi_1(h^\sigma\xi)\Big)\widetilde{w}^\chi(t,\cdot)\right\|_{L^\infty} \\ & \lesssim \left\| \oph(\chi_2(h^{\sigma_1}\xi))\left[ e^{\frac{i}{h}\phi}\theta_h(x) \oph\Big(\gamma\left(\frac{x-p'(\xi)}{\sqrt{h}}\right)\chi_1(h^\sigma\xi)\Big)\widetilde{w}^\chi\right]\right\|_{L^\infty} + \|\oph(r_\infty)\widetilde{w}^\chi(t,\cdot)\|_{L^\infty},\end{aligned}$$ for a new $r_\infty\in h^NS_{\frac{1}{2},\sigma}\big(\langle\frac{x-p'(\xi)}{\sqrt{h}}\rangle^{-1}\big)$. This latter $r_\infty$ comes out from the commutation between $e^{\frac{i}{h}\phi}\theta_h(x)$ and $\oph(\chi_2(h^{\sigma_1}\xi))$, whose symbol is computed using until a large enough order $M$. We notice that we gain a factor $h^{|\alpha|(\sigma_1-\sigma)}$ at each order of the mentioned asymptotic development as $\sigma_1>\sigma$. Moreover, those terms write in terms of the derivatives of $\chi_2$ and hence vanish on the support of $\chi_1$. By proposition \[Prop: a sharp b\] and remark \[Remark:symbols\_with\_null\_support\_intersection\] we then deduce that the composition of the mentioned commutator with $\oph\big(\gamma\big(\frac{x-p'(\xi)}{\sqrt{h}}\big)\chi_1(h^\sigma\xi)\big)$ is an operator of symbol $r_\infty$, with $N$ as large as we want. Using the classical Sobolev injection, symbolic calculus and lemma \[Lem: (xi+dphi)Op(gamma)\] we find that$$\begin{aligned} & \left\| \oph(\chi_2(h^{\sigma_1}\xi))\left[ e^{\frac{i}{h}\phi}\theta_h(x) \oph\Big(\gamma\left(\frac{x-p'(\xi)}{\sqrt{h}}\right)\chi_1(h^\sigma\xi)\Big)\widetilde{w}^\chi\right]\right\|_{L^\infty}\\ & \lesssim |\log h| \left[ \|\widetilde{w}^\chi(t,\cdot)\|_{L^2} + \sum_{j=1}^2\left\| D_j\left[ e^{\frac{i}{h}\phi}\theta_h(x) \oph\Big(\gamma\left(\frac{x-p'(\xi)}{\sqrt{h}}\right)\chi_1(h^\sigma\xi)\Big)\widetilde{w}^\chi\right]\right\|_{L^2} \right] \\ &\lesssim |\log h| \left[ \|\widetilde{w}^\chi(t,\cdot)\|_{L^2} + \sum_{j=1}^2 h^{-1}\left\| \oph\big((\xi_j+d_j\phi(x))\theta_h(x)\big)\oph\Big(\gamma\left(\frac{x-p'(\xi)}{\sqrt{h}}\right)\chi_1(h^\sigma\xi)\Big)\widetilde{w}^\chi\right\|_{L^2}\right] \\ &\lesssim |\log h| \left[ \|\widetilde{w}^\chi(t,\cdot)\|_{L^2} + h^{-\beta}\sum_{|\mu|=0}^1 \left\|\oph(\chi_1(h^\sigma\xi))\mathcal{L}^\mu \widetilde{w}^\chi(t,\cdot)\right\|_{L^2}\right].\end{aligned}$$Finally, commutating $\mathcal{L}$ with $\oph(\chi(h^\sigma\xi))$ defining $\widetilde{w}^\chi$, and reminding that $\chi_1\equiv 1$ on the support of $\chi$, we obtain $$\|\oph(\chi(h^\sigma\xi))\widetilde{w}^\chi(t,\cdot)\|_{L^\infty}\lesssim h^{-\beta}\sum_{|\mu|=0}^1\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu \widetilde{w}(t,\cdot)\|_{L^2},$$ for every $t\in[1,T]$, and hence . \[Lem\_appendix:Intro\_of\_vNFGamma\] Let $I$ be a multi-index of length $j$, with $j=1,2$, and $$\label{def_vNF-Gamma} \vNFGamma(t,x): = (\Gamma^I v)_{-}(t,x)-\frac{i}{4(2\pi)^2}\sum_{j_1,j_2\in \{+,-\}}\int e^{ix\cdot\xi}B^1_{(j_1,j_2,+)}(\xi,\eta) \widehat{v^I_{j_1}}(\xi-\eta) \hat{u}_{j_2}(\eta) d\xi d\eta,$$ with $B^1_{(j_1,j_2,+)}$ given by with $j_3=+$ and $k=1$. Then there exists a constant $C>0$ such that, for any $\chi\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$ small, and every $t\in [1,T]$, $$\label{ineq:Linfty_ vINF - vI} \left\|\chi(t^{-\sigma}D_x)\left(\vNFGamma - (\Gamma^I v)_{-}\right)(t,\cdot) \right\|_{L^\infty}\le \frac{1}{2} \left\|\chi(t^{-\sigma}D_x)(\Gamma^I v)_{-}(t,\cdot) \right\|_{L^\infty} + CB\varepsilon t^{-1}.$$ Moreover, for every $m=1,2,$ $t\in[1,T]$, $$\label{ineq:L2 Zm vINF-vI} \left\|\chi(t^{-\sigma}D_x)Z_m\left(\vNFGamma - (\Gamma^I v)_{-}\right)(t,\cdot) \right\|_{L^2} \le C(A+B)B\varepsilon^2 t^{2\sigma+\frac{\delta_{3-j}+\delta_2}{2}}.$$ We first notice that, after definition and inequalities , , we have the following explicit expression: $$\label{explicit vNfGamma-vJ-} \vNFGamma-(\Gamma^I v)_{-} = -\frac{i}{2}\left[(D_t\Gamma^I v)(D_1u) - (D_1\Gamma^I v)(D_tu) + D_1[(\Gamma^I v) D_tu] - \langle D_x\rangle [(\Gamma^I v) D_1u] \right].$$ From the above equality together with lemma \[Lem\_appendix:L\_estimate of products\] with $L=L^\infty$ and $w_1=(\Gamma^I v)_\pm$, and , , we deduce that there exists some $\chi_1\in C^\infty_0(\mathbb{R}^2)$, equal to 1 on the support of $\chi$, and $s>0$ sufficiently large such that $$\label{est_1_vNFGamma-vJ} \begin{split} &\left\|\chi(t^{-\sigma}D_x)(\vNFGamma - v^I_{-})(t,\cdot) \right\|_{L^\infty}\\ &\lesssim t^\sigma \sum_{\mu=0}^1\left\|[\chi_1(t^{-\sigma}D_x)(\Gamma^I v)_\pm(t,\cdot)] [\chi(t^{-\sigma}D_x)\mathrm{R}^\mu_1u_\pm](t,\cdot)\right\|_{L^\infty}\+ t^{-2}\|(\Gamma^I v)_\pm(t,\cdot)\|_{L^2}\|u_\pm(t,\cdot)\|_{H^s}\\ &\lesssim t^\sigma \sum_{\mu=0}^1\left\|[\chi_1(t^{-\sigma}D_x)(\Gamma^I v)_\pm(t,\cdot)] [\chi(t^{-\sigma}D_x)\mathrm{R}^\mu_1u_\pm](t,\cdot)\right\|_{L^\infty} + B^2\varepsilon^2 t^{-3/2}, \end{split}$$ where the latter inequality follows from after a-priori energy estimates , . Our aim is to truncate factor $(\Gamma^I v)_\pm$ in the above right hand side rather with the same operator $\chi(t^{-\sigma}D_x)$ appearing on the left hand side. We hence proceed by picking some $\kappa\ge 1$ and decomposing $\chi(t^{-\sigma}D_x)\mathrm{R}^\mu_1 u_\pm$ as $$\label{dec_small_frequencies_Ru} \chi(t^{-\sigma}D_x)\mathrm{R}^\mu_1 u_\pm = \chi(t^\kappa D_x) \mathrm{R}^\mu_1 u_\pm + (1-\chi)(t^\kappa D_x)\chi(t^{-\sigma}D_x)\mathrm{R}^\mu_1 u_\pm,$$ noticing that, as $\chi(t^\kappa \xi)$ is supported for very small frequencies $|\xi|\lesssim t^{-\kappa}$, by Sobolev injection we have that $$\left\| \chi(t^\kappa D_x)\mathrm{R}^\mu_1 u_\pm(t,\cdot)\right\|_{L^\infty}\lesssim t^{-\kappa}\|u_\pm(t,\cdot)\|_{L^2}.$$ Consequently, using the $L^2-L^\infty$ continuity of $\chi_1(t^{-\sigma}D_x)$ along with a-priori estimates , , we have that for any for $\mu=0,1$ $$\begin{aligned} \left\|[\chi_1(t^{-\sigma}D_x)(\Gamma^I v)_\pm(t,\cdot)] \chi(t^\kappa D_x)\mathrm{R}^\mu_1u_\pm(t,\cdot)\right\|_{L^\infty}&\lesssim t^{\sigma-\kappa} \|(\Gamma^I v)_\pm (t,\cdot)\|_{L^2}\|u_\pm(t,\cdot)\|_{L^2}\\ &\le CB\varepsilon t^{-\kappa+\sigma + \frac{\delta_{3-j}+\delta}{2}}.\end{aligned}$$ Choosing $\kappa=1+\sigma +\frac{\delta+\delta_1}{2}$ we deduce from and the above inequality that $$\begin{gathered} \label{cut_2} \left\|[\chi_1(t^{-\sigma}D_x)(\Gamma^I v)_\pm(t,\cdot)] \chi(t^{-\sigma}D_x)\mathrm{R}^\mu_1u_\pm(t,\cdot)\right\|_{L^\infty} \\ \lesssim \left\|[\chi_1(t^{-\sigma}D_x)(\Gamma^I v)_\pm(t,\cdot)] (1-\chi)(t^\kappa D_x)\chi(t^{-\sigma}D_x)\mathrm{R}^\mu_1u_\pm(t,\cdot)\right\|_{L^\infty} +CB\varepsilon t^{-1}.\end{gathered}$$ We then decompose $(\Gamma^Iv)_\pm$ in frequencies using the wished operator $\chi(t^{-\sigma}D_x)$. In order to estimate the $L^\infty$ norm of $$[(1-\chi)(t^{-\sigma} D_x)\chi_1(t^{-\sigma}D_x)(\Gamma^I v)_\pm] (1-\chi)(t^\kappa D_x)\chi(t^{-\sigma}D_x)\mathrm{R}^\mu_1u_\pm$$ we first commute $\Gamma^I$ to operator $D_t\pm \langle D_x\rangle$ (see ) and successively look at it as a linear combination of derivations of the form $x^\alpha t^a\partial^\alpha_x\partial^b_t$, with $1\le |\alpha| + a\le 2$, $1\le |\beta| + b\le 2$. By commutating $x^\alpha$ to $(1-\chi)(t^{-\sigma} D_x)\chi_1(t^{-\sigma}D_x)$, multiplying it against the wave factor, and successively combining the classical Sobolev injection with inequality , we find that $$\begin{gathered} \label{cut_1} \left\|[(1-\chi)(t^{-\sigma} D_x)\chi_1(t^{-\sigma}D_x)(\Gamma^I v)_\pm(t,\cdot)] (1-\chi)(t^\kappa D_x)\chi(t^{-\sigma}D_x)\mathrm{R}^\mu_1u_\pm(t,\cdot)\right\|_{L^\infty} \\ \lesssim t^{-N(s)}\left(\|v_\pm(t,\cdot)\|_{H^s}+\|D_tv_\pm(t,\cdot)\|_{H^s} + \|D^2_tv_\pm(t,\cdot)\|_{H^s}\right) \\ \times \sum_{\substack{1\le |\alpha|+ a \le 2\\ |\mu|=0,1}}\left\|x^\alpha t^a (1-\chi)(t^\kappa D_x)\chi(t^{-\sigma}D_x)\mathrm{R}^\mu_1u_\pm\right\|_{L^\infty}.\end{gathered}$$ Using system with $|I|=0$, with $s=1$, and a-priori estimates, it is straightforward to check that $$\label{est:A} \|v_\pm(t,\cdot)\|_{H^s}+\|D_tv_\pm(t,\cdot)\|_{H^s} + \|D^2_tv_\pm(t,\cdot)\|_{H^s} \le CB\varepsilon t^\frac{\delta}{2}.$$ Also, $$\label{est:B} t^a\left\| (1-\chi)(t^\kappa D_x)\chi(t^{-\sigma}D_x)\mathrm{R}^\mu_1u_\pm\right\|_{L^\infty}\lesssim t^{a+\sigma}\|u_\pm(t,\cdot)\|_{L^2}\le CB\varepsilon t^{a+\sigma+\frac{\delta}{2}},$$ and for $|\alpha|\in \{1,2\}$ we have that $$\label{ineq:x_alpha_R1u} \left\|x^\alpha (1-\chi)(t^\kappa D_x)\chi(t^{-\sigma}D_x)\mathrm{R}^\mu_1u_\pm\right\|_{L^\infty}\le CB\varepsilon t^{|\alpha|+|\alpha|\kappa+\frac{\delta}{2}}.$$ In fact, when $|\alpha|=1$ this can be proved by commutating $x^\alpha$ with $(1-\chi)(t^\kappa D_x)\chi(t^{-\sigma}D_x)$, using that $$[x_n, (1-\chi)(t^\kappa D_x)\chi(t^{-\sigma}D_x)] =-it^\kappa (\partial_n\chi)(t^\kappa D_x) +i t^{-\sigma}(\partial_n\chi)(t^{-\sigma}D_x), \quad n=1,2,$$ is bounded from $L^2$ to $L^\infty$ uniformly in $t$, and together with estimates , , and the following inequality $$\begin{gathered} \left\| (1-\chi)(t^\kappa D_x)\chi(t^{-\sigma}D_x)\big[x^\alpha \mathrm{R}^\mu_1u_\pm\big](t,\cdot)\right\|_{L^\infty} \\ \lesssim t^\kappa \left[\sum_{|\mu|=1}\|Z^\mu u_\pm(t,\cdot)\|_{L^2} + t\|u_\pm(t,\cdot)\|_{H^1} + \|x\textit{NL}_w(t,\cdot)\|_{L^2}\right],\end{gathered}$$ which is obtained by writing $$\begin{aligned} &(1-\chi)(t^\kappa D_x)\chi(t^{-\sigma}D_x) x_n \mathrm{R}^\mu_1\\ &= t^\kappa \widetilde{\chi}_1(t^\kappa D_x)\chi(t^{-\sigma}D_x) x_n|D_x|\mathrm{R}^\mu_1 + t^\kappa \widetilde{\chi}_1(t^\kappa D_x)\chi(t^{-\sigma}D_x)[|D_x|,x]\mathrm{R}^\mu_1 \\ &= t^\kappa\widetilde{\chi}_1(t^\kappa D_x)\chi(t^{-\sigma}D_x)\mathrm{R}^\mu_1 \left[x_n|D_x| -tD_n + \frac{1}{2i}\frac{D_n}{|D_x|}\right] \\ &+ t^\kappa\widetilde{\chi}_1(t^\kappa D_x)\chi(t^{-\sigma}D_x)\mathrm{R}^\mu_1\left[tD_n - \frac{1}{2i}\frac{D_n}{|D_x|}\right]+\delta_{\mu=1} it^\kappa\widetilde{\chi}_1(t^\kappa D_x)\chi(t^{-\sigma}D_x)Op(|\xi|\partial_n(\xi_1|\xi|^{-1}))\\ &-it^\kappa \widetilde{\chi}_1(t^\kappa D_x)\chi(t^{-\sigma}D_x)\mathrm{R}_n\mathrm{R}^\mu_1\end{aligned}$$ with $\widetilde{\chi}(\xi):=(1-\chi)(\xi)|\xi|^{-1}$ and using relation with $w=u_\pm$. The proof for $|\alpha|=2$ is analogous. It is based on the commutation of $x^\alpha$ with $(1-\chi)(t^\kappa D_x)\chi(t^{-\sigma}D_x)$ (the commutator is here a $L^2-L^\infty$ bounded operator with norm $O(t^{\kappa})$), on the fact that we can rewrite $(1-\chi)(t^\kappa D_x)\chi(t^{-\sigma}D_x) x^\alpha \mathrm{R}^\mu_1$ making appear $(x|D_x| -tD_x + \frac{1}{2i}\frac{D_x}{|D_x|})^\alpha$ by considering $\widetilde{\chi}_2(\xi) := (1-\chi)(\xi)|\xi|^{-2}$ instead of previous $\widetilde{\chi}_1$, and use relation . Doing so we derive the following inequality $$\begin{gathered} \left\| (1-\chi)(t^\kappa D_x)\chi(t^{-\sigma}D_x)\big[x^\alpha \mathrm{R}^\mu_1u_\pm\big](t,\cdot)\right\|_{L^\infty} \\ \lesssim t^{2\kappa} \left[\sum_{|\mu|=2}\|Z^\mu u_\pm(t,\cdot)\|_{L^2} + \sum_{|\mu|\le 1} t^{2-|\mu|}\|Z^\mu u_\pm(t,\cdot)\|_{H^1} + \sum_{|\mu|=1}^2\|x^\mu\textit{NL}_w(t,\cdot)\|_{L^2}\right] \le CB\varepsilon t^{2+2\kappa +\frac{\delta}{2}},\end{gathered}$$last estimate following from a-priori estimates, and . Summing up , , , together with the previous choice of $\kappa$ and the fact that in $N(s)\ge 6$ if $s>0$ is sufficiently large, we deduce that $$\label{est_3_vNFGamma-vJ} \left\|[(1-\chi)(t^{-\sigma} D_x)\chi_1(t^{-\sigma}D_x)(\Gamma^I v)_\pm(t,\cdot)] (1-\chi)(t^\kappa D_x)\chi(t^{-\sigma}D_x)\mathrm{R}^\mu_1u_\pm(t,\cdot)\right\|_{L^\infty}\le CB\varepsilon t^{-\frac{3}{2}}.$$ Therefore, from , ,, and the uniform continuity on $L^\infty$ of $\chi(t^{-\sigma} D_x)$, we find that $$\left\| \chi(t^{-\sigma}D_x)\left(\vNFGamma-v^I_{-} \right)(t,\cdot)\right\|_{L^\infty} \lesssim \sum_{\mu=0}^1 t^\sigma \|\chi(t^{-\sigma}D_x)(\Gamma^I v)_\pm (t,\cdot)\|_{L^\infty}\|\mathrm{R}^\mu_1u_\pm(t,\cdot)\|_{L^\infty} + CB\varepsilon t^{-1},$$ and as $\sigma$ is small and $\varepsilon_0<(2A)^{-1}$, from we obtain . In order to prove we apply $Z_m$ to equality and apply the Leibniz rule. As $$\label{commutator_Z_DtD1} \begin{split} [Z_m, D_t]= - D_m, \quad [Z_m, D_1]=-\delta_{m1} D_t, \quad [Z_m, \langle D_x\rangle]= - D_m\langle D_x\rangle^{-1} D_t, \end{split}$$ with $\delta_{m1}$ the Kronecker delta, we find that $$\label{Zm(vgamma-vJ)} \begin{split} &2i\chi(t^{-\sigma}D_x) Z_m (\vNFGamma - v^I_{-}) \\ &=\chi(t^{-\sigma}D_x) \Big[ (D_t Z_m\Gamma^I v)(D_1u) - (D_1Z_m\Gamma^I v)(D_tu)+ D_1[(Z_m\Gamma^I v)(D_tu)] - \langle D_x\rangle [(Z_m \Gamma^I v)(D_1u)]\\ &\hspace{15pt}+ (D_t \Gamma^I v)(D_1Z_m u) - (D_1\Gamma^I v)(D_t Z_mu)+ D_1[(\Gamma^I v)(D_t Z_mu)] - \langle D_x\rangle [ (\Gamma^I v)(D_1Z_m u)]\\ &\hspace{15pt}- (D_m\Gamma^I v)(D_1u) + \delta_{m1}(D_t\Gamma^I v)(D_tu) - \delta_{m1} D_t[(\Gamma^I v)(D_t u)] + \frac{D_m}{\langle D_x\rangle} D_t[(\Gamma^I v)(D_1 u)] \\ & \hspace{15pt}-\delta_{m1} (D_t\Gamma^I v)(D_tu) + (D_1\Gamma^I v)(D_m u)-\delta_{m1} D_1[(\Gamma^I v)(D_t u)] + \delta_{m1}\langle D_x\rangle [(\Gamma^I v)(D_tu)]\Big]. \end{split}$$The $L^2$ norm of all products in the above second, fourth and fifth line, i.e. those in which $Z_m$ is not acting on the wave component $u$, is estimated by $$\begin{gathered} \label{first_terms_Zm(vgamma-vj)} \sum_{\mu=0}^1 t^\sigma \left(\|(Z_m\Gamma^I v)_\pm (t,\cdot)\|_{L^2}+ \|(\Gamma^I v)_\pm (t,\cdot)\|_{L^2}\right)\left(\|\mathrm{R}^\mu_1 u_\pm (t,\cdot)\|_{L^\infty}+\|D_tu_\pm (t,\cdot)\|_{L^\infty}\right)\\ \le CAB\varepsilon^2 t^{-\frac{1}{2} + \frac{\delta_0}{2}+\sigma},\end{gathered}$$ after inequality with $s=0$ and a-priori estimates. The $L^2$ norm of products appearing in the second line are, instead, estimated by using and with $L=L^2$, $\Gamma w= Z_mu$, $s>0$ sufficiently large so that $N(s)\ge 2$. It is hence bounded by $$\begin{split} & t^\sigma \left\|\chi(t^{-\sigma}D_x)(\Gamma^I v)_\pm(t,\cdot)\right\|_{L^\infty}\| (Z_mu)_\pm(t,\cdot)\|_{L^2}\\ &+ t^{-2}\Big(\sum_{|\mu|=0}^1 \|x^\mu (\Gamma^Iv)_\pm(t,\cdot)\|_{L^2} + t\|(\Gamma^I v)_\pm(t,\cdot)\|_{L^2}\Big)\left(\|u_\pm(t,\cdot)\|_{H^s}+ \|D_tu_\pm(t,\cdot)\|_{H^s}\right) \\ & \le CB^2\varepsilon^2 t^{2\sigma+ \frac{\delta_{3-j}+\delta_2}{2}}, \end{split}$$ where the latter estimate is obtained using the fact that $\chi(t^{-\sigma}D_x)$ is a bounded operator from $L^2$ to $L^\infty$ with norm $O(t^\sigma)$, together with , and a-priori estimates. That concludes, together with , the proof of and of the statement. \[Lem\_appendix: preliminary est VJ\] There exists a constant $C>0$ such that, for any $\rho\in\mathbb{N}$, $\chi\in C^\infty_0(\mathbb{R}^2)$, equal to 1 in a neighbourhood of the origin, $\sigma>0$ small, and every $t\in [1,T]$, $$\label{Linfty_est_VJ} \sum_{|I|=1}\|\chi(t^{-\sigma}D_x)V^I(t,\cdot)\|_{H^{\rho,\infty}} \le CB\varepsilon t^{-1+\beta+\frac{\delta_1}{2}},$$ with $\beta>0$ small such that $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. Since $\chi(t^{-\sigma}D_x)$ is a bounded operator from $L^\infty$ to $H^{\rho,\infty}$ with norm $O(t^{\sigma\rho})$, for any $\rho\in\mathbb{N}$, it is enough to prove that the $L^\infty$ norm of $\chi(t^{-\sigma}D_x) V^I(t,\cdot)$ is bounded by the right hand side of . Moreover, as this latter inequality is automatically satisfied when $\Gamma$ is a spatial derivative after a-priori estimate and the fact that operator $\chi(t^{-\sigma}D_x)$ is uniformly bounded on $L^\infty$, for the rest of the proof we will assume that $\Gamma\in \{\Omega, Z_j, j=1,2\}$ is a Klainerman vector field. We also warn the reader that, throughout the proof, $C$ and $\beta$ will denote some positive constants that may change line after line, with $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. Instead of proving the result of the statement directly on $\chi(t^{-\sigma}D_x)v^I_\pm$ we do it for $\chi(t^{-\sigma}D_x)\vNFGamma$, where $\vNFGamma$ has been introduced in and is considered here for $|I|=1$ and $\Gamma^I=\Gamma$. In fact, by $$\label{vI-_bounded_by_vNFGamma} \left\| \chi(t^{-\sigma}D_x)v^I_{-}(t,\cdot)\right\|_{L^\infty}\le 2 \left\| \chi(t^{-\sigma}D_x)\vNFGamma(t,\cdot)\right\|_{L^\infty}+CB\varepsilon t^{-1}.$$ The advantage of dealing with this new function is related to the fact that it is solution to a half Klein-Gordon equation with a more suitable non-linearity (see ) than the equation satisfied by $v^I_{-}$. In fact, it is a computation to show that from definition $$\label{KG_vNF-Gamma} [D_t+\langle D_x\rangle]\vNFGamma(t,x) = \NLNF,$$ where $$\label{def_NLNF} \NLNF= r^{I,\textit{NF}}_{kg}(t,x) + Q^\mathrm{kg}_0(v_\pm, D_1 u^I_\pm) + G^\mathrm{kg}_1(v_\pm, Du_\pm),$$ $G^\mathrm{kg}_1(v_\pm, Du_\pm) = G_1(v,\partial u)$ with $G_1$ given by , and $$\begin{gathered} \label{def:rNF-Gamma-kg} r^{I,\textit{NF}}_{kg}(t,x) = -\frac{i}{4(2\pi)^2}\\ \times \sum_{j_1,j_2\in \{+,-\}}\int e^{ix\cdot\xi} B^1_{(j_1,j_2,+)}(\xi,\eta) \left[{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\Gamma^I\textit{NL}_{kg}}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\Gamma^I\textit{NL}_{kg}}{\tmpbox}}(\xi-\eta)\hat{u}_{j_2}(\eta) - \widehat{v^I_{j_1}}(\xi-\eta) \widehat{\textit{NL}_w}(\eta)\right] d\xi d\eta,\end{gathered}$$ with $B^1_{(j_1,j_2,+)}$ given by when $j_3=+$ and $k=1$. After and it appears that $r^{I,\textit{NF}}_{kg}$ has the following nice explicit expression $$\label{explicit rNFkg-Gamma} r^{I,\textit{NF}}_{kg} =-\frac{i}{2}\left[(\Gamma^I\textit{NL}_{kg}) D_1u - (D_1\Gamma^I v)\textit{NL}_w + D_1[(\Gamma^I v)\textit{NL}_w]\right].$$ Using lemma \[Lem\_appendix:Linfty\_bound\_chi\_w\] and relation with $w=\vNFGamma$, and reminding that $\|t w(t,t\cdot)\|_{L^2}=\|w(t,\cdot)\|_{L^2}$, we find the following $$\begin{gathered} \label{prelimary_ineq_vNFGamma} \left\| \chi(t^{-\sigma}D_x)\vNFGamma(t,\cdot)\right\|_{L^\infty} \\ \lesssim t^{-1+\beta} \sum_{|\mu|=0}^1\left\|\chi(t^{-\sigma}D_x)Z^\mu \vNFGamma(t,\cdot)\right\|_{L^2}+ \sum_{j=1}^2 t^{-1+\beta}\left\|\chi(t^{-\sigma}D_x) \big[x_j\NLNF\big](t,\cdot)\right\|_{L^2}.\end{gathered}$$ From equality , along with , , and a-priori estimates , , we immediately see that $$\label{est_preliminary_(vNFGamma - vJ-)} \begin{split} \left\| \chi(t^{-\sigma}D_x)(\vNFGamma -v^I_{-})(t,\cdot)\right\|_{L^2}&\lesssim t^\sigma\|v^I_\pm(t,\cdot)\|_{L^2}\left(\|u_\pm(t,\cdot)\|_{L^\infty}+\|\mathrm{R}_1u_\pm(t,\cdot)\|_{L^\infty}\right)\\ & \le CAB\varepsilon^2 t^{-\frac{1}{2}+\frac{\delta_2}{2}+\sigma}, \end{split}$$ and as $\sigma, \delta_2\ll 1$ are small $$\label{est:vNFGamma} \begin{split} \left\|\chi(t^{-\sigma}D_x) \vNFGamma(t,\cdot)\right\|_{L^2} &\le \left\|\chi(t^{-\sigma}D_x) v^I_{-}(t,\cdot)\right\|_{L^2}+ \left\| \chi(t^{-\sigma}D_x)(\vNFGamma -v^I_{-})(t,\cdot)\right\|_{L^2} \\ &\le CB\varepsilon t^{\frac{\delta_2}{2}}. \end{split}$$ Moreover, from and a-priori estimate we have that, for every $m=1,2$, $t\in [1,T]$, $$\label{est_L2_ZmvNFGamma} \left\|\chi(t^{-\sigma}D_x)Z_m \vNFGamma(t,\cdot) \right\|_{L^2}\le CB\varepsilon t^\frac{\delta_1}{2}.$$ Finally, from , , , , and a-priori estimates, we derive that $$\label{est:L2_xj_rINF_kg} \begin{split} &\left\|\chi(t^{-\sigma}D_x) \left[x_j r^{I,\textit{NF}}_{kg}\right](t,\cdot)\right\|_{L^2} \lesssim \|x_j\Gamma^I\textit{NL}_{kg}(t,\cdot)\|_{L^2}\left(\|u_\pm(t,\cdot)\|_{L^\infty}+ \|\mathrm{R}_1u_\pm(t,\cdot)\|_{L^\infty}\right)\\ & + \sum_{\mu=0}^1 t^\sigma\left(\|x_j^\mu v_\pm(t,\cdot)\|_{L^\infty}+\left\| x_j^\mu \frac{D_x}{\langle D_x\rangle}v_\pm(t,\cdot)\right\|_{L^\infty}\right)\|v^I_\pm(t,\cdot)\|_{L^2}\|v_\pm(t,\cdot)\|_{H^{2,\infty}}\\ & \le C(A+B)B\varepsilon^2 t^\frac{\delta_2}{2}, \end{split}$$ while from with $s=0$, and a-priori estimates $$\begin{split} &\left\| \chi(t^{-\sigma}D_x)\left[x_j Q^\mathrm{kg}_0\left(v_\pm, D_1 u^I_\pm\right)\right](t,\cdot)\right\|_{L^2}+ \left\| \chi(t^{-\sigma}D_x)\left[x_j G^\mathrm{kg}_1\left(v_\pm, D u_\pm\right)\right](t,\cdot)\right\|_{L^2}\\ &\lesssim \left(\|x_j v_\pm(t,\cdot)\|_{L^\infty}+\left\| x_j \frac{D_x}{\langle D_x\rangle}v_\pm(t,\cdot)\right\|_{L^\infty}\right)\left(\|u^I_\pm(t,\cdot)\|_{H^1}+ \|D_tu_\pm(t,\cdot)\|_{L^2}\right)\\ & \le C(A+B)B\varepsilon t^{\delta_2}. \end{split}$$ Therefore, from we deduce that $$\label{est_xj_NL-kg-NF-Gamma} \|\chi(t^{-\sigma}D_x)\big[ x_j\NLNF\big](t,\cdot)\|_{L^2} \le C(A+B)B\varepsilon^2 t^{\delta_2},$$ so injecting , , into , and summing it up with , we obtain the result of the statement. As done for the Klein-Gordon component in the above lemma, we also derive an estimate for the uniform norm of the wave component when a Klainerman vector field acts on it and its frequencies less or equal than $t^\sigma$ (see lemma \[Lem\_appendix: est UJ\]). We first need the following result. \[Lem\_appendix: L\^2 estimates uJ\] Let $\Gamma\in \mathcal{Z}$, index $J$ be such that $\Gamma^J=\Gamma$, and $\widetilde{u}^J (t,x):= t(\Gamma u)_{-}(t,tx)$. There exists a constant $C>0$ such that, for any $\theta_0, \chi\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$, and every $t\in [1,T]$, $$\begin{gathered} \|\widetilde{u}^J(t,\cdot)\|_{L^2}\le CB\varepsilon t^\frac{\delta_2}{2}, \label{utildeJ_L2}\\ \|\Omega_h \widetilde{u}^J(t,\cdot)\|_{L^2}\le CB\varepsilon t^{\frac{\delta_1}{2}}\label{Omega_utildeJ_L2} ,\\ \left\|\mathcal{M} \widetilde{u}^J(t,\cdot)\right\|_{L^2} \le CB\varepsilon t^{\frac{\delta_1}{2}}, \label{MuJ}\\ \left\| \theta_0(x) \oph(\chi(h^\sigma\xi))\Omega_h\mathcal{M}\widetilde{u}^J(t,\cdot)\right\|_{L^2}\le CB\varepsilon t^\frac{\delta_0}{2}.\end{gathered}$$ We warn the reader that, throughout the proof, $C$ will denote a positive constant that may change line after line. We also recall that $$[D_t + \langle D_x\rangle] (\Gamma u)_{-}(t,x)=\Gamma \textit{NL}_w(t,x).$$ Estimates and are straightforward after and the fact that $$\|\widetilde{u}^J(t,\cdot)\|_{L^2}=\|(\Gamma u)_{-}(t,\cdot)\|_{L^2}, \quad \|\Omega_h\widetilde{u}^J(t,\cdot)\|_{L^2}=\|(\Omega\Gamma u)_{-}(t,\cdot)\|_{L^2}.$$ From with $w=(\Gamma u)_{-}$ and $f=\Gamma \textit{NL}_w$, estimates , , along with the fact that $\delta_2\ll \delta_1$ (e.g. $2\delta_2\le\delta_1$), and $(A+B)\varepsilon_0<1$, we obtain . By we also derive that, for any $n=1,2,$ $$\label{preliminary_Omega-M-uJ} \begin{split} \left\| \theta_0(x) \oph(\chi(h^\sigma\xi)) \Omega_h\mathcal{M}_n\widetilde{u}^J(t,\cdot)\right\|_{L^2} &\lesssim \|\Omega Z_n (\Gamma u)_{-}(t,\cdot)\|_{L^2} + \sum_{\mu=0}^1\|\Omega^\mu (\Gamma u)_{-}(t,\cdot)\|_{L^2} \\ & + \left\|\theta_0\Big(\frac{x}{t}\Big) \chi(t^{-\sigma}D_x)\Omega [x_n \Gamma \textit{NL}_w](t,\cdot) \right\|_{L^2}. \end{split}$$ The first two norms in the above right hand side are controlled by $E^0_3(t;W)^{1/2}$ and are hence bounded by $CB\varepsilon t^\frac{\delta_0}{2}$. By commutating $x_n$ with $\chi(t^{-\sigma}D_x)\Omega$, and using that $\theta_0\big(\frac{x}{t}\big)x_n=t \theta^n_0\big(\frac{x}{t}\big)$, with $\theta^n_0(z):=\theta_0(z)z_n$, we deduce that $$\begin{split} \left\|\theta_0\Big(\frac{x}{t}\Big)\chi(t^{-\sigma}D_x) \Omega [x_n \Gamma\textit{NL}_w](t,\cdot) \right\|_{L^2}\lesssim t \sum_{\mu=0}^1 \|\chi_1(t^{-\sigma}D_x)\Omega^\mu \Gamma \textit{NL}_w\|_{L^2}, \end{split}$$ for some new $\chi_1\in C^\infty_0(\mathbb{R}^2)$. On the one hand, using , with $s=0$ and a-priori estimates we derive that $$\label{est_t_GammaNLw} \begin{split} t\|\Gamma \textit{NL}_w\|_{L^2} &\lesssim t \|v_\pm(t,\cdot)\|_{H^{2,\infty}}\left(\|(\Gamma v)_\pm(t,\cdot)\|_{H^1}+ \|v_\pm(t,\cdot)\|_{H^1}+\|D_tv_\pm(t,\cdot)\|_{L^2}\right)\lesssim CB\varepsilon t^\frac{\delta_2}{2}. \end{split}$$ On the other hand, when we compute $\Omega \Gamma \textit{NL}_w$ we find among the out-coming quadratic terms the following ones $$Q^\mathrm{w}_0((\Omega v)_\pm, D_1(\Gamma v)_\pm) \quad \text{and} \quad Q^\mathrm{w}_0((\Gamma v)_\pm, D_1( \Omega v)_\pm),$$ which we estimate in the $L^2$ norm (when truncated for frequencies less or equal than $t^\sigma$) by means of with $L=L^2$, $\Gamma w= \Omega v$, and $s>0$ large enough to have $N(s)\ge 3$. From , and a-priori estimates, we obtain that $$\begin{split} &\left\| \chi(t^{-\sigma}D_x)Q^\mathrm{w}_0((\Omega v)_\pm, D_1(\Gamma v)_\pm)\right\|_{L^2}+ \left\|\chi(t^{-\sigma}D_x)Q^\mathrm{w}_0((\Gamma v)_\pm, D_1( \Omega v)_\pm)\right\|_{L^2}\\ & \lesssim t^\sigma \|\chi(t^{-\sigma}D_x)(\Omega v)_\pm (t,\cdot)\|_{L^\infty}\|(\Gamma v)_\pm(t,\cdot)\|_{H^1} +\sum_{|\mu|=0}^1 t^{-3}\|v_\pm(t,\cdot)\|_{H^s}\|x^\mu (\Gamma v)_\pm (t,\cdot)\|_{H^1} \\ & \le CB^2\varepsilon^2 t^{-1+\beta+\frac{\delta_1+\delta_2}{2}}, \end{split}$$ with $\beta>0$ small such that $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. All remaining quadratic contributions to $\Omega \Gamma \textit{NL}_w$ are estimated with $$\begin{gathered} \|(\Omega \Gamma v)_\pm (t,\cdot)\|_{H^1}\|v_\pm(t,\cdot)\|_{H^{2,\infty}} + \|(\Omega v)_\pm(t,\cdot)\|_{L^2}\left(\|v_\pm(t,\cdot)\|_{H^{1,\infty}}+\|D_tv_\pm(t,\cdot)\|_{L^\infty}\right) \\ + \|v_\pm(t,\cdot)\|_{H^{1,\infty}}\left(\|(\Omega v)_\pm(t,\cdot)\|_{H^1} + \|D_t(\Omega v)_\pm (t,\cdot)\|_{L^2}\right),\end{gathered}$$ and are hence bounded by $C(A+B)B\varepsilon^2 t^{-1+\frac{\delta_1}{2}}$ after , and the a-priori estimates. This finally implies that $$t\left\|\chi(t^{-\sigma}D_x)\Omega \Gamma \textit{NL}_w(t,\cdot) \right\|_{L^2} \le C(A+B)B\varepsilon^2 t^{\beta+\frac{\delta_1+\delta_2}{2}},$$ which, together with and the fact that $\beta+\frac{\delta_1+\delta_2}{2}\le \frac{\delta_0}{2}$, as $\delta_2\ll \delta_1 \ll \delta_0$ and $\beta>0$ is as small as we want provided that $\sigma$ is small, gives $$\left\|\theta_0\Big(\frac{x}{t}\Big)\chi(t^{-\sigma}D_x) \Omega [x_n \Gamma\textit{NL}_w](t,\cdot) \right\|_{L^2} \le CB\varepsilon t^\frac{\delta_0}{2}.$$ \[Lem\_appendix: est UJ\] There exists a constant $C>0$ such that, for any $\rho\in\mathbb{N}$, $\chi\in C^\infty_0(\mathbb{R}^2)$ equal to 1 in a neighbourhood of the origin, $\sigma>0$ small, and every $t\in [1,T]$, $$\label{Linfty_est_UJ} \sum_{|J|=1}\sum_{|\mu|=0}^1\|\chi(t^{-\sigma}D_x)\mathrm{R}^\mu U^J(t,\cdot)\|_{H^{\rho,\infty}} \le C(A+B)\varepsilon t^{-\frac{1}{2}+\beta+\frac{\delta_1}{2}},$$ for a small $\beta>0$, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. We warn the reader that, throughout the proof, $C$ and $\beta$ will denote two positive constants that may change line after line, with $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. Moreover, since $\chi(t^{-\sigma}D_x)$ is a bounded operator from $L^\infty$ to $H^{\rho,\infty}$ with norm $O(t^{\sigma\rho})$, for any $\rho\in\mathbb{N}$, we can reduce to prove that the $L^\infty$ norm of $\chi(t^{-\sigma}D_x)\mathrm{R}^\mu U^J(t,\cdot)$ is bounded by the right hand side of . We observe that this estimate is automatically satisfied when $J$ is such that $\Gamma^J$ is a spatial derivative, as a consequence of a-priori estimate . We therefore assume that $\Gamma^J$ is one of the Klainerman vector fields $\Omega, Z_m$, for $m\in\{1,2\}$. Introducing $\widetilde{u}^J(t,x):=t u^J_{-}(t,tx)$, passing to the semiclassical setting ($t\mapsto t$, $x\mapsto \frac{x}{t}$, and $h:=1/t$), and reminding that $u^J_+ = -\overline{u^J_{-}}$, inequality becomes $$\label{est_Linfty_uJ_semiclassical} \sum_{|\mu|=0}^1\left\|\oph\Big(\chi(h^\sigma\xi)(\xi|\xi|^{-1})^\mu\Big)\widetilde{u}^J_{-}(t,\cdot) \right\|_{L^\infty}\le C(A+B)\varepsilon h^{-\frac{1}{2}-\beta-\frac{\delta_1}{2}}.$$ We consider a Littlewood-Paley decomposition such that $$\label{dec_UJ} \chi(h^\sigma\xi)= \widetilde{\chi}(h^{-1}\xi)+\sum_k(1-\widetilde{\chi})(h^{-1}\xi)\psi(2^{-k}\xi)\chi(h^\sigma\xi),$$ for some suitably supported $\widetilde{\chi}\in C^\infty_0(\mathbb{R}^2)$, $\psi\in C^\infty_0(\mathbb{R}^2\setminus \{0\})$, and immediately observe that the above sum is restricted to indices $k$ such that $h\lesssim 2^k\lesssim h^{-\sigma}$. By the classical Sobolev injection, the uniform continuity of $\oph(\xi|\xi|^{-1})$ on $L^2$, and a-priori estimate , we derive that for any $|\mu|\le 1$, every $t\in [1,T]$, $$\label{est_utildeJ_xi<h} \begin{split} \left\|\oph\big(\widetilde{\chi}(h^{-1}\xi)(\xi|\xi|^{-1})^\mu\big)\widetilde{u}^J(t,\cdot)\right\|_{L^\infty} &= \|\chi( D_x) \oph((\xi|\xi|^{-1})^\mu)\widetilde{u}^J(t,\cdot)\|_{L^\infty}\\ &\lesssim \| u^J_{-}(t,\cdot)\|_{L^2} \le CB\varepsilon t^{\frac{\delta_2}{2}}. \end{split}$$ If we concisely denote by $\phi_k(\xi)$ the $k$-th addend in decomposition and introduce two smooth cut-off functions $\chi_0$, $\gamma$, with $\chi_0$ radial and equal to 1 on the support of $\phi_k$, $\gamma$ with sufficiently small support, we can write $$\begin{aligned} \oph\left(\phi_k(\xi)(\xi|\xi|^{-1})^\mu\right)\widetilde{u}^J &= \oph\Big(\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\phi_k(\xi) (\xi|\xi|^{-1})^\mu\Big) \oph(\chi_0(h^\sigma\xi))\widetilde{u}^J \\ &+ \oph\Big((1-\gamma)\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\phi_k(\xi) (\xi|\xi|^{-1})^\mu\Big)\oph(\chi_0(h^\sigma\xi))\widetilde{u}^J.\end{aligned}$$ On the one hand, after proposition \[Prop : continuity of Op(gamma1):X to L2\], the fact that $2^k\lesssim h^{-\sigma}$, a-priori estimate , and the uniform $L^2$ continuity of $\oph(\chi_0(h^\sigma\xi))$, we have that for any $|\mu|\le 1$ $$\begin{gathered} \label{gamma_uJ} \left\| \oph\Big(\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\phi_k(\xi) (\xi|\xi|^{-1})^\mu\Big)\oph(\chi_0(h^\sigma\xi))\widetilde{u}^J(t,\cdot)\right\|_{L^\infty} \\ \lesssim h^{-\frac{1}{2}-\beta}\left(\|\oph(\chi_0(h^\sigma\xi))\widetilde{u}^J(t,\cdot)\|_{L^2}+ \|\theta_0(x)\Omega_h\oph(\chi_0(h^\sigma\xi))\widetilde{u}^J(t,\cdot)\|_{L^2}\right)\\ \lesssim h^{-\frac{1}{2}-\beta}\left(\|u^J_{-}(t,\cdot)\|_{L^2}+ \|\Omega u^J_{-}(t,\cdot)\|_{L^2}\right) \le CB\varepsilon h^{-\frac{1}{2}-\beta-\frac{\delta_1}{2}}.\end{gathered}$$ On the other hand, using that $(1-\gamma)(z)=\gamma_1^j(z)z_j$, where $\gamma_1^j(z):=(1-\gamma)(z)z_j|z|^{-2}$ is such that $|\partial^\alpha_z\gamma_1^j(z)|\le \langle z\rangle^{-1-|\alpha|}$, we derive from , the commutation between $\mathcal{M}$ with $\oph(\chi_0(h^\sigma\xi))$, and lemma \[Lem\_appendix: L\^2 estimates uJ\], that $$\begin{gathered} \left\| \oph\Big((1-\gamma)\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\phi_k(\xi) (\xi|\xi|^{-1})^\mu\Big)\oph(\chi_0(h^\sigma\xi))\widetilde{u}^J\right\|_{L^\infty}\\ \lesssim h^{-\beta}\sum_{\gamma,|\nu|=0}^1\|(\theta_0(x)\Omega_h)^\gamma \mathcal{M}^\nu \oph(\chi_0(h^\sigma\xi)) \widetilde{u}^J(t,\cdot)\|_{L^2} \le CB\varepsilon t^{\beta+\frac{\delta_0}{2}}.\end{gathered}$$ Combining this estimate with we deduce that $$\|\oph\left(\phi_k(\xi)(\xi|\xi|^{-1})^\mu\right)\widetilde{u}^J(t,\cdot)\|_{L^\infty}\le C(A+B)\varepsilon h^{-\frac{1}{2}-\beta-\frac{\delta_1}{2}},$$ for any $|\mu|\le 1$, and hence after , , up to a further loss $|\log h|$, as a consequence of the fact that the sum in is finite and taken over indices $k$ such that $\log h \lesssim k\lesssim \log h^{-1}$. \[Lem\_appendix:xGammav\_Linfty\] There exists a positive constant $C>0$ such that, for any $\chi\in C^\infty_0(\mathbb{R}^2)$ equal to 1 in a neighbourhood of the origin, $\sigma>0$, and every $t\in [1,T]$, $$\label{norm_Linfty_xjGammav-} \sum_{|\mu|=0}^1\left\| \chi(t^{-\sigma}D_x)\left[x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu (\Gamma v)_\pm(t,\cdot)\right]\right\|_{L^\infty}\le CB\varepsilon t^{\beta+\frac{\delta_1}{2}},$$ with $\beta>0$ small, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. We warn the reader that, throughout the proof, $C$ and $\beta$ will denote two positive constants that may change line after line, with $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. As $\Gamma v_+=-\overline{\Gamma v_{-}}$, it is enough to prove the statement for $\Gamma v_{-}$. If $\Gamma$ is a spatial derivative, estimate is just consequence of the uniform continuity of $\chi(t^{-\sigma}D_x)$ on $L^\infty$ and of . We then assume that $\Gamma\in \{\Omega, Z_m, m=1,2\}$ is a Klainerman vector field. First of all, we observe that by with $w= (\Gamma v)_{-}$ and $f=\Gamma\textit{NL}_{kg}$, along with the classical Sobolev injection, $$\label{Linfty_xj_Gammav_preliminary} \sum_{|\mu|=0}^1\left\| x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu (\Gamma v)_{-}(t,\cdot)\right\|_{L^\infty}\lesssim \|Z_j(\Gamma v)_{-}(t,\cdot)\|_{H^1}+ t\|(\Gamma v)_{-}(t,\cdot)\|_{H^2}+ \sum_{\mu=0}^1\|x^\mu_j\Gamma\textit{NL}_{kg}(t,\cdot)\|_{L^\infty}.$$ From equality and lemma \[Lem\_app:products\_Gamma\] with $L=L^\infty$ and $s>0$ large enough so that $N(s)\ge3$, together with estimates , , , , , , , , and , we get that $$\label{Gamma_NLkg_Linfty} \begin{split} &\|\Gamma\Nlkg(t,\cdot)\|_{L^\infty}\\ &\lesssim \sum_{\mu=0}^1\left(\|\chi(t^{-\sigma}D_x)(\Gamma v)_\pm(t,\cdot)\|_{H^{1,\infty}}\|\mathrm{R}^\mu_1u_\pm(t,\cdot)\|_{H^{2,\infty}}\right) + \|v_\pm(t,\cdot)\|_{H^{1,\infty}}\left\|\chi(t^{-\sigma}D_x)(\Gamma u)_\pm(t,\cdot)\right\|_{H^{2,\infty}} \\ &+ \|v_\pm(t,\cdot)\|_{H^{1,\infty}}\times \sum_{|\mu|=0}^1\left(\|\mathrm{R}^\mu u_\pm(t,\cdot)\|_{H^{2,\infty}}+ \|D_t\mathrm{R}^\mu u_\pm(t,\cdot)\|_{H^{1,\infty}}\right)\\ & + t^{-3}\left(\|v_\pm(t,\cdot)\|_{H^s}+\|D_tv_\pm(t,\cdot)\|_{H^s}\right)\Big(\sum_{|\mu|, |\nu|=0}^1\left\|x^\mu D_1\Big(\frac{D_x}{|D_x|}\Big)^\nu u_\pm(t,\cdot)\right\|_{L^2}+ t \|u_\pm(t,\cdot)\|_{L^2}\Big) \\ & + t^{-3}\Big(\sum_{|\mu|=0}^1 \|x^\mu v_\pm(t,\cdot)\|_{L^2}+t\|v_\pm(t,\cdot)\|_{L^2}\Big)\left(\|u_\pm(t,\cdot)\|_{H^s}+\|D_tu_\pm(t,\cdot)\|_{H^s}\right) \\ &\le CAB\varepsilon^2 t^{-\frac{3}{2}+\beta + \frac{\delta_1}{2}}. \end{split}$$Moreover, as $$\label{xj_QGammav_1} \left\|x_j Q^\mathrm{kg}_0\big((\Gamma v)_\pm, D_1u_\pm\big)\right\|_{L^\infty} \lesssim \sum_{|\mu|, \nu=0}^1\left\|x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu(\Gamma v)_{-}(t,\cdot)\right\|_{L^\infty}\|\mathrm{R}^\nu_1u_\pm(t,\cdot)\|_{H^{2,\infty}},$$ $$\label{xjG1} \left\| x_j G^\mathrm{kg}_1\big(v_\pm, D u_\pm\big)\right\|_{L^\infty}\lesssim \sum_{|\mu|=0}^1\left\|x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty} \left(\|u_\pm(t,\cdot)\|_{H^{2,\infty}}+\|D_tu_\pm(t,\cdot)\|_{H^{1,\infty}}\right),$$ and by lemma \[Lem\_app:products\_Gamma\] with $L=L^\infty$, $w= u$, and $s>0$ large enough so that $N(s)\ge 3$, $$\label{xj_QGammu} \begin{split} &\left\|x_j Q^\mathrm{kg}_0(v_\pm, D_1 (\Gamma u)_\pm)\right\|_{L^\infty}\lesssim \sum_{|\mu|=0}^1\left\|x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty} \|\chi(t^{-\sigma}D_x)(\Gamma u)_\pm(t,\cdot)\|_{H^{2,\infty}}\\ & + t^{-3}\sum_{|\mu|,\nu=0}^1\left( \left\| x^\mu x_j^\nu v_\pm(t,\cdot)\right\|_{L^2} + t \|x^\nu_j v_\pm(t,\cdot)\|_{L^2}\right)\left(\|u_\pm(t,\cdot)\|_{H^s}+\|D_tu_\pm(t,\cdot)\|_{H^s}\right), \end{split}$$ we derive that $$\label{xjGamma_NL_preliminary} \left\|x_j\Gamma\Nlkg(t,\cdot)\right\|_{L^\infty}\le CA\varepsilon t^{-\frac{1}{2}} \sum_{|\mu|, \nu=0}^1\left\|x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu(\Gamma v)_{-}(t,\cdot)\right\|_{L^\infty} + C(A+B)B\varepsilon^2 t^{-\frac{1}{2}+\beta+\frac{\delta_1+\delta_2}{2}},$$ as follows using , with $s=1$, , ,, and a-priori estimates. By injecting the above inequality into and using the fact that $\varepsilon_0<(2CA)^{-1}$, we initially obtain that $$\label{est_xj Gammav_preliminary} \| x_j (\Gamma v)_{-}(t,\cdot)\|_{L^\infty} + \left\|x_j\frac{D_x}{\langle D_x\rangle} (\Gamma v)_{-}(t,\cdot) \right\|_{L^\infty}\le CB\varepsilon t^{1+\frac{\delta_2}{2}}.$$ If we take any smooth cut-off function $\chi$ and use equality , instead of we find that $$\begin{gathered} \label{ineq_chi_xGammav-} \sum_{|\mu|=0}^1\left\|\chi(t^{-\sigma} D_x)\Big[ x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu (\Gamma v)_{-}(t,\cdot)\Big]\right\|_{L^\infty}\lesssim \|Z_j(\Gamma v)_{-}(t,\cdot)\|_{H^1}+ t\|\chi(t^{-\sigma}D_x)(\Gamma v)_{-}(t,\cdot)\|_{L^\infty}\\ + \sum_{\mu=0}^1\left\|\chi(t^{-\sigma}D_x)\big[ x^\mu_j\Gamma\textit{NL}_{kg}(t,\cdot)\big]\right\|_{L^\infty},\end{gathered}$$ where now $$\left\|\chi(t^{-\sigma}D_x)\big[x_j\Gamma\textit{NL}_{kg}(t,\cdot)\big]\right\|_{L^\infty}\lesssim \left\|x_j\Gamma\textit{NL}_{kg}(t,\cdot)\right\|_{L^\infty} \le C(A+B)B\varepsilon^2 t^{\frac{1}{2}+\frac{\delta_2}{2}},$$ as follows injecting into . Therefore, from , lemma \[Lem\_appendix: preliminary est VJ\] and a-priori estimate with $k=2$, we find that $$\label{xj_Gamma v-_preliminary2} \left\|\chi(t^{-\sigma} D_x) \big[x_j (\Gamma v)_{-}(t,\cdot)\big]\right\|_{L^\infty} + \left\|\chi(t^{-\sigma} D_x) \left[x_j \frac{D_x}{\langle D_x\rangle} (\Gamma v)_{-}(t,\cdot)\right]\right\|_{L^\infty} \le CB\varepsilon t^{\frac{1}{2}+\frac{\delta_2}{2}}.$$ Finally, by means of lemma \[Lem\_appendix:L\_estimate of products\] with $L=L^\infty$, $w_1 = x(\Gamma v)_\pm$, and $s>0$ such that $N(s)\ge 2$, we derive that for any $\chi\in C^\infty_0(\mathbb{R}^2)$ there is some $\chi_1\in C^\infty_0(\mathbb{R}^2)$ such that $$\begin{gathered} \left\|\chi(t^{-\sigma}D_x) x_jQ^\mathrm{kg}_0((\Gamma v)_\pm, D_1u_\pm) \right\|_{L^\infty} \\ \lesssim \sum_{|\mu|, \nu=0}^1\left\| \chi_1(t^{-\sigma}D_x)\left[x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu(\Gamma v)_{-}(t,\cdot)\right] \right\|_{L^\infty} \|\chi(t^{-\sigma}D_x)\mathrm{R}^\nu_1u_\pm(t,\cdot)\|_{H^{2,\infty}} \\ + \sum_{\mu=0}^1 t^{-2}\left\|x^\mu_j (\Gamma v)_\pm(t,\cdot)\right\|_{L^2} \|u_\pm(t,\cdot)\|_{H^s}.\end{gathered}$$ Then, combining such inequality with , , together with , , and all the other inequalities to which we already referred before, from we find that $$\left\| \chi(t^{-\sigma}D_x) \big[x_j \Gamma\textit{NL}_{kg}(t,\cdot)\big]\right\|_{L^\infty}\le C(A+B)\varepsilon^2 t^{\delta_2},$$ which injected into finally implies, together with with $k=2$, lemma \[Lem\_appendix: preliminary est VJ\], and , the wished estimate . Making use of lemmas \[Lem\_appendix: preliminary est VJ\] and \[Lem\_appendix:xGammav\_Linfty\], estimate can be improved of a factor $t^{-\frac{1}{2}}$. This improvement, that will be useful to derive , is showed in the following lemma. Let $I$ be a multi-index of length 1 and $r^{I,\textit{NF}}_{kg}$ be given by . There exists a constant $C>0$ such that, for any $\rho\in\mathbb{N}$, $\chi\in C^\infty_0(\mathbb{R}^2)$, equal to 1 in a neighbourhood of the origin, $\sigma>0$ small, $j=1,2$, and every $t\in [1,T]$, $$\label{enhanced_xjrINF_kg} \left\|\chi(t^{-\sigma}D_x)\left[x_j r^{I,\textit{NF}}_{kg}\right](t,\cdot)\right\|_{L^2}\le C(A+B)AB\varepsilon^3 t^{-\frac{1}{2}+\beta+\frac{\delta+\delta_1}{2}},$$ with $\beta>0$ small, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. Let us remind the explicit expression of $r^{I,\textit{NF}}_{kg}$ and consider the cubic term $x_j \Gamma^I\textit{NL}_{kg}(D_1u)$. Reminding and applying lemma \[Lem\_appendix:L\_estimate of products\] with $L=L^2$ and $s>0$ sufficiently large so that $N(s)\ge 2$, together with and a-priori estimates, we derive that there is some $\chi_1\in C^\infty_0(\mathbb{R}^2)$ such that $$\label{est_xNLI_D1u_prel} \begin{split} & \left\|\chi(t^{-\sigma}D_x)\left[x_j \Gamma^I\textit{NL}_{kg}(D_1u)\right](t,\cdot)\right\|_{L^2}\\ &\lesssim \left\|\chi_1(t^{-\sigma}D_x)\left[x_j \Gamma^I\textit{NL}_{kg}\right](t,\cdot)\right\|_{L^2}\|\mathrm{R}_1u_\pm(t,\cdot)\|_{L^\infty} + t^{-2}\|x_j\Gamma^I\textit{NL}_{kg}(t,\cdot)\|_{L^2}\|u_\pm(t,\cdot)\|_{H^s}\\ & \le CA\varepsilon t^{-\frac{1}{2}} \left\|\chi_1(t^{-\sigma}D_x)\left[x_j \textit{NL}^I_{kg}\right](t,\cdot)\right\|_{L^2} + C(A+B)B\varepsilon^2 t^{-1}. \end{split}$$ Then, recalling and using again lemma \[Lem\_appendix:L\_estimate of products\] with $L=L^2$, $w_1 = (\Gamma v)_\pm$, and $s$ as before, in order to estimate the contribution coming from the first quadratic term in the right hand side of , we find that there is a new $\chi_2\in C^\infty_0(\mathbb{R}^2)$ such that $$\begin{split} & \left\|\chi_1(t^{-\sigma}D_x)\left[x_j \textit{NL}^I_{kg}\right](t,\cdot)\right\|_{L^2} \\ & \lesssim \sum_{|\mu|=0}^1\left\|\chi_2(t^{-\sigma}D_x)\Big[x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu (\Gamma v)_\pm\Big] (t,\cdot)\right\|_{L^\infty}\|u_\pm(t,\cdot)\|_{H^1}+ t^{-2}\|x_j(\Gamma v)_\pm(t,\cdot)\|_{L^2}\|u_\pm(t,\cdot)\|_{H^s} \\ & +\sum_{|\mu|=0}^1 \left\| x_j \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty}\left(\|(\Gamma u)_\pm(t,\cdot)\|_{H^1}+\|u_\pm(t,\cdot)\|_{H^1}+ \|D_t u_\pm(t,\cdot)\|_{L^2}\right)\\ &\le C(A+B)B\varepsilon^2 t^{\beta+\frac{\delta+\delta_1}{2}}, \end{split}$$ where the latter estimate is obtained from with $s=0$, , with $k=1$, and a-priori estimates. This implies, combined with , that $$\left\|\chi(t^{-\sigma}D_x)\left[x_j \textit{NL}^I_{kg}(D_1u)\right](t,\cdot)\right\|_{L^2} \le C(A+B)AB\varepsilon^3 t^{-\frac{1}{2}+\beta+\frac{\delta+\delta_1}{2}},$$ and from , and a-priori estimates, $$\begin{split} &\left\|\chi(t^{-\sigma}D_x) \left[x_j r^{I,\textit{NF}}_{kg}\right](t,\cdot)\right\|_{L^2} \lesssim \left\|\chi(t^{-\sigma}D_x)\left[x_j \textit{NL}^I_{kg}(D_1u)\right](t,\cdot)\right\|_{L^2}\\ & + \sum_{\mu=0}^1 t^\sigma\left(\|x_j^\mu v_\pm(t,\cdot)\|_{L^\infty}+\left\| x_j^\mu \frac{D_x}{\langle D_x\rangle}v_\pm(t,\cdot)\right\|_{L^\infty}\right)\|v^I_\pm(t,\cdot)\|_{L^2}\|v_\pm(t,\cdot)\|_{H^{2,\infty}}\\ & \le C(A+B)AB\varepsilon^3 t^{-\frac{1}{2}+\beta+\frac{\delta+\delta_1}{2}}, \end{split}$$ which concludes the proof of the statement. Let $I$ be a multi-index of length 2. There exists a constant $C>0$ such that, for every $j=1,2$, $t\in [1,T]$, $$\label{xj_GammaI_NLkg_I=2} \left\| x_j \Gamma^I\textit{NL}_{kg}(t,\cdot)\right\|_{L^2}\le C(A+B)B\varepsilon^2 t^{\frac{1}{2}+\beta + \frac{\delta_1+\delta_2}{2}},$$ with $\beta>0$ small, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. We remind the reader about . Instead of using , which was obtained by Sobolev injection, we apply lemma \[Lem\_app:products\_Gamma\] with $L=L^2$, $\Gamma w =\Gamma^{I_2}u$, $s>0$ sufficiently large so that $N(s)\ge 3$, and exploit the fact that we have an estimate of the $H^{\rho,\infty}$ norm of $D_1u^{I_2}$ when truncated for frequencies less or equal than $t^\sigma$ (see lemma \[Lem\_appendix: est UJ\]). Therefore, for $(I_1,I_2)\in \mathcal{I}(I)$ such $|I_1|=|I_2|=1$ we obtain that $$\begin{split} & \left\| x_j Q^\mathrm{kg}_0\left(v^{I_1}_\pm, D_1 u^{I_2}_\pm\right)(t,\cdot)\right\|_{L^2}\lesssim \sum_{\mu=0}^1\left\| x^\mu_j v^{I_1}_\pm(t,\cdot)\right\|_{L^2}\left\|\chi(t^{-\sigma}D_x) u^{I_2}_\pm (t,\cdot)\right\|_{H^{2,\infty}} \\ & + t^{-3}\left(\|u_\pm(t,\cdot)\|_{H^s}+\|D_tu_\pm(t,\cdot)\|_{H^s}\right) \left[\sum_{|\mu|=0}^2 \|x^\mu v^{I_1}_\pm(t,\cdot)\|_{L^2} + \sum_{|\mu|=0}^1 t\|x^\mu v^{I_1}_\pm(t,\cdot)\|_{L^2}\right]\\ & \le C(A+B)B\varepsilon^2 t^{\frac{1}{2}+\beta+\frac{\delta_1+\delta_2}{2}}, \end{split}$$ last estimate following from lemma \[Lem\_appendix: est UJ\] together with , with $k=1$, , a-priori estimates, and the fact that $\delta_1,\delta_2\ll 1$ are small. Consequently, from the following inequality $$\begin{aligned} &\|x_j\Gamma^I \textit{NL}_{kg}(t,\cdot)\|_{L^2}\lesssim \sum_{\mu=0}^1\|\mathrm{R}_1^\mu u_\pm(t,\cdot)\|_{H^{2,\infty}}\sum_{\substack{|J|\le 2\\ \mu=0,1}}\| x^\mu_j(\Gamma^J v)_{-}(t,\cdot)\|_{L^2} \\ & + \sum_{|\mu|=0}^1\left\|x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty}\Big[\|u^I_\pm(t,\cdot)\|_{H^1} + \sum_{|J|<2}\big(\|u^J_\pm(t,\cdot)\|_{H^1}+\|D_tu^J_\pm(t,\cdot)\|_{L^2}\big) \Big]\\ & + \sum_{|I_1|=|I_2|=1}\left\|x_j Q^\mathrm{kg}_0\left(v^{I_1}_\pm, D_1 u^{I_2}_\pm\right)(t,\cdot)\right\|_{L^2},\end{aligned}$$ together with , with $s=0$, , and with $k=1$, we finally derive . \[Lem: from energy to norms in sc coordinates-KG\] Let us fix $s\in\mathbb{N}$. There exists a constant $C>0$ such that, if we assume that a-priori estimates are satisfied in some interval $[1,T]$, for a fixed $T>1$, with $n\ge s+2$, then we have, for any $\chi \in C^\infty_0(\mathbb{R}^2)$ and $\sigma>0$ small, $$\begin{gathered} \|\widetilde{v}(t,\cdot)\|_{H^s_h} \le CB\varepsilon t^\frac{\delta}{2},\label{est:Hs_vtilde}\\ \sum_{|\mu|=1}\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu\widetilde{v}(t,\cdot)\|_{L^2} \le CB\varepsilon t^\frac{\delta_2}{2}, \label{est:Lvtilde}\end{gathered}$$ for every $t\in [1,T]$. We warn the reader that, throughout the proof, $C$ and $\beta$ will denote two positive constants that may change line after line, with $\beta>0$ is small as long as $\sigma$ is small. It is straightforward to check that the $H^s_h$ norm of $\widetilde{v}$ is bounded by energy $E_n(t;W)^\frac{1}{2}$ whenever $n\ge s+2$, after definitions , , inequality , and a-priori estimates , . In order to prove we first use relation and definition to derive that $$\label{norm_L2_Lnwidetilde(v)} \begin{split} \|\oph(\chi(h^\sigma\xi))\mathcal{L}_m\widetilde{v}(t,\cdot)\|_{L^2}&\lesssim \|Z_mV(t,\cdot)\|_{L^2}+\|\chi(t^{-\sigma}D_x)Z_m(v^{NF}-v_{-})(t,\cdot)\|_{L^2}\\ & + \|\widetilde{v}(t,\cdot)\|_{L^2}+ \|\chi(t^{-\sigma}D_x)[x_m r^{NF}_{kg}](t,\cdot)\|_{L^2}, \end{split}$$ with $r^{NF}_{kg}$ given by . Using we can rewrite and similarly to , , as: $$\label{explicit vNf-v-} v^{NF}-v_{-} = -\frac{i}{2}\left[(D_tv)(D_1u) - (D_1v)(D_tu) + D_1[v D_tu] - \langle D_x\rangle [v D_1u] \right]$$ and $$\label{explicit rNFkg} r^{NF}_{kg} =-\frac{i}{2}\left[\textit{NL}_{kg} D_1u - (D_1v)\textit{NL}_w + D_1(v\textit{NL}_w)\right].$$ From and , together with estimates and , $$\label{chi_xm_rNFkg} \begin{split} \|\chi(t^{-\sigma}D_x)&(x_m r^{NF}_{kg})(t,\cdot) \|_{L^2} \lesssim t^{\sigma}\left( \|x_n v_{-}(t,\cdot)\|_{L^\infty}+ \left\| x_n\frac{D_x}{\langle D_x\rangle} v_{-}(t,\cdot)\right\|_{L^\infty}\right) \\ &\times \Big[\left(\|U(t,\cdot)\|_{H^{2,\infty}} + \|\mathrm{R}_1U(t,\cdot)\|_{H^{2,\infty}}\right) \|U(t,\cdot)\|_{L^2} + \|V(t,\cdot)\|_{H^{2,\infty}}\|V(t,\cdot)\|_{L^2}\Big]\\ & + \|V(t,\cdot)\|^2_{H^{1,\infty}}\|V(t,\cdot)\|_{H^1} \le C(A+B)AB\varepsilon^3 t^{-\frac{1}{2}+\sigma+\frac{(\delta+\delta_2)}{2}}. \end{split}$$ Similarly to , $$\label{Zm(vNF-v-)} \begin{split} &2i\chi(t^{-\sigma}D_x) Z_m (v^{NF}-v_{-}) \\ & = \chi(t^{-\sigma}D_x) \Big[(D_t Z_m v)(D_1u) - (D_1Z_m v)(D_tu)+ D_1[(Z_m v)(D_tu)] - \langle D_x\rangle [(Z_m v)(D_1u)]\\ &\hspace{15pt}+ (D_t v)(D_1Z_m u) - (D_1 v)(D_t Z_mu)+ D_1[v (D_t Z_mu)] - \langle D_x\rangle [ v(D_1Z_m u)]\\ &\hspace{15pt} - (D_m v)(D_1u) + \delta_{m1}(D_t v)(D_tu) - \delta_{m1} D_t[v (D_t u)] + \frac{D_m}{\langle D_x\rangle} D_t[v (D_1 u)] \\ & \hspace{15pt} -\delta_{m1} (D_t v)(D_tu) + (D_1 v)(D_m u)-\delta_{m1} D_1[v(D_t u)] + \delta_{m1}\langle D_x\rangle [v(D_tu)]\Big]. \end{split}$$ We bound the $L^2$ norm of all products in the first line of the above equality by means of lemma \[Lem\_appendix:L\_estimate of products\], and all the others by the $L^\infty$ norm of the Klein-Gordon factor times the $L^2$ norm of the wave one. In this way we get that, for some new $\chi_1\in C^\infty_0(\mathbb{R}^2)$ and $s>0$ sufficiently large, we derive that $$\begin{gathered} \left\|\chi(t^{-\sigma}D_x)Z_m (v^{NF}-v_{-})(t,\cdot) \right\|_{L^2}\\ \lesssim t^\sigma \left\|\chi_1(t^{-\sigma}D_x)(Z_m v)_\pm(t,\cdot)\right\|_{L^\infty}\|u_\pm(t,\cdot)\|_{L^2} + t^{-1}\|(Z_mv)_\pm(t,\cdot)\|_{L^2}\|u_\pm(t,\cdot)\|_{H^s}\\ + t^\sigma \|v_\pm(t,\cdot)\|_{H^{1,\infty}}\left(\|(Z_mu)_\pm(t,\cdot)\|_{L^2}+\|u_\pm(t,\cdot)\|_{L^2}+\|D_tu_\pm(t,\cdot)\|_{L^2}\right).\end{gathered}$$ Consequently, using estimates , with $s=0$, and , we obtain that $$\label{est_Zm(vNF-v-)} \|\chi(t^{-\sigma}D_x)Z_m(v^{NF}-v_{-})(t,\cdot)\|_{L^2}\le C(A+B)B\varepsilon^2 t^{-1+\beta+\frac{\delta+\delta_1}{2}},$$ which plugged into , along with , and , gives . Last range of estimates {#sec_appB: second range of estimates} ----------------------- The aim of this section is to show that a-priori estimates also infer a moderate growth in time of the $L^2(\mathbb{R}^2)$ norm of $\mathcal{L}^\mu\vt$, for $|\mu|=2$, when this function is restricted to frequencies less or equal than $h^{-\sigma}$, for $\sigma>0$ small. This is proved in lemma \[Lem\_appendix: estimate L2vtilde\]. Lemmas from \[Lem\_appendix:Lm(Zv - vNF-Z)\] to \[Lem\_appendix: L xnrNFkg\] are intermediate technical results. \[Lem\_appendix:Lm(Zv - vNF-Z)\] Let us consider $v^{NF}$ introduced in and $\vNFGamma$ as in with $|I|=1$ and $\Gamma^I=Z_n$, for $n\in \{1,2\}$. There exists a constant $C>0$ such that, for any $\chi \in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$ small, and every $t\in [1,T]$, $$\label{est:t(Zmv - vNFZn)} \left\| \chi(t^{-\sigma}D_x)\big[ (Z_n v)_{-} - \vNFGamma\big](t,\cdot)\right\|_{L^2}\le C(A+B)B\varepsilon^2 t^{-1+\beta + \frac{\delta+\delta_1}{2}},$$ $$\begin{gathered} \label{est:xm Zn(vNF -v-)} \left\| \chi(t^{-\sigma}D_x)\big[x_m Z_n(v^{NF}-v_{-})(t,\cdot)\big] \right\|_{L^2} + \left\| \chi(t^{-\sigma}D_x)\big[x_m \big((Z_n v)_{-} - \vNFGamma \big)\big](t,\cdot)\right\|_{L^2}\\ \le C(A+B)B \varepsilon^2 t^{\beta+ \frac{\delta_1+\delta_2}{2}}.\end{gathered}$$ The same estimates hold true when $Z_n$ is replaced with $\Omega$. By comparing equality , with $|I|=1$ and $\Gamma^I=Z_n$, with we see that $\chi(t^{-\sigma}D_x)(\vNFGamma - (Z_nv)_{-})$ corresponds to the first line in the right hand side of . Therefore, inequality is automatically satisfied after , which was obtained by estimating the right hand side of term by term. In order to prove , let us consider equality but with $\chi(t^{-\sigma}D_x)$ replaced with $\chi(t^{-\sigma}D_x)x_m$. The $L^2$ norm of each product in the second to fourth line is bounded by $$t^\sigma\sum_{\mu,\nu=0}^1 \left\|x^\mu_m \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\nu v_\pm(t,\cdot)\right\|_{L^\infty}\left(\|(Z_m u)_\pm(t,\cdot)\|_{L^2}+\|u_\pm(t,\cdot)\|_{L^2}+\|D_tu_\pm(t,\cdot)\|_{L^2}\right),$$ and then by the right hand side of after , with $s=0$, and . Using lemma \[Lem\_appendix:L\_estimate of products\] with $L=L^2$ and $s>0$ large enough to have $N(s)\ge 2$, we obtain that the $L^2$ norm of products in the first line of (the modified) is bounded by $$\begin{gathered} \sum_{\mu,\nu=0}^1 \left\|\chi_1(t^{-\sigma}D_x)\Big[x^\mu_m \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\nu (Z_m v)_\pm(t,\cdot)\Big]\right\|_{L^\infty}\|u_\pm(t,\cdot)\|_{L^2} \\ + \sum_{\mu=0}^1 t^{-N(s)}\|x^\mu_m (Z_m v)_\pm(t,\cdot)\|_{L^2}\|u_\pm(t,\cdot)\|_{H^s},\end{gathered}$$ for some smooth cut-off $\chi_1$, and hence by the right hand side of after , and with $\Gamma=Z_m$. This concludes the proof of . When $Z_n$ is replaced with $\Omega$, instead of referring to one uses that $$\begin{split} 2i\Omega \left(v^{NF}-v_{-}\right) & = (D_t \Omega v)(D_1u) - (D_1\Omega v)(D_tu)+ D_1[(\Omega v)(D_tu)] - \langle D_x\rangle [(\Omega v)(D_1u)]\\ &+ (D_t v)(D_1\Omega u) - (D_1 v)(D_t \Omega u)+ D_1[v (D_t \Omega u)] - \langle D_x\rangle [ v(D_1\Omega u)]\\ & - (D_t v)(D_2 u) + (D_2 v)(D_t u)- D_2[v (D_t u)] + \langle D_x\rangle [ v(D_2 u)] \end{split}$$ and applies the same argument as above to recover the wished estimates. \[Lem\_appendix: ZnvNF\_LmZnvNF\] Let $v^{\textit{NF}}$ be defined as in . There exists a constant $C>0$ such that, for any $\chi\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$ small, $m=1,2$, $$\label{est:ZnvNF} \left\|\oph(\chi(h^\sigma\xi)) [tZ_nv^{NF}(t,tx)]\right\|_{L^2(dx)}\le CB\varepsilon t^{\frac{\delta_2}{2}},$$ $$\label{est:Lm_ZnvNF} \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m [tZ_nv^{NF}(t,tx)]\right\|_{L^2(dx)}\le CB\varepsilon t^{\frac{\delta_1}{2}},$$ for every $t\in [1, T]$. Let us write $Z_nv^{NF}$ as follows: $$\label{Zn_vNF} Z_nv^{NF} = Z_n(v^{NF}-v_{-}) + \left[(Z_n v)_{-} - \vNFGamma\right] + \vNFGamma + \frac{D_n}{\langle D_x\rangle}v^{NF}+\frac{D_n}{\langle D_x\rangle}(v_{-}-v^{NF}),$$ with $\vNFGamma$ given by with $|I|=1$ and $\Gamma^I=Z_n$. From the fact that $\|tw(t,t\cdot)\|_{L^2}= \|w(t,\cdot)\|_{L^2}$ and estimates , , ,, , along with the following one $$\begin{gathered} \|\chi(t^{-\sigma}D_x)D_n\langle D_x\rangle^{-1} v^{NF}(t,\cdot)\|_{L^2}\\\le \|\chi(t^{-\sigma}D_x)D_n\langle D_x\rangle^{-1} v_{-}(t,\cdot)\|_{L^2} + \|\chi(t^{-\sigma}D_x)D_n\langle D_x\rangle^{-1}(v_{-} - v^{NF})(t,\cdot)\|_{L^2} \le CB\varepsilon t^\frac{\delta}{2},\end{gathered}$$ we immediately obtain . From we also derive that $$\label{ineq: L-Zn-vNF} \begin{split} &\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m [t Z_nv^{NF}(t,tx)]\right\|_{L^2(dx)} \lesssim \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m \big[tZ_n(v^{NF}-v_{-})(t,tx)\big] \right\|_{L^2(dx)}\\ & + \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m \big[t\big((Z_n v)_{-}-\vNFGamma\big)(t,tx)\big]\right\|_{L^2(dx)}+ \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m \big[t\vNFGamma(t,tx)\big]\right\|_{L^2(dx)}\\ &+ \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m \big[tD_n\langle D_x\rangle^{-1}v^{NF}(t,tx)\big]\right\|_{L^2(dx)}\\ & + \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m \big[tD_n\langle D_x\rangle^{-1}(v_{-} - v^{NF})(t,tx)\big]\right\|_{L^2(dx)}. \end{split}$$ By relation with $w=\vNFGamma$ and estimates , , , it follows that $$\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m \big[t\vNFGamma(t,tx)\big]\right\|_{L^2} \le CB\varepsilon t^\frac{\delta_1}{2},$$ while from and we have that $$\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m \big[tD_n\langle D_x\rangle^{-1}v^{NF}(t,tx)\big]\right\|_{L^2}\le CB\varepsilon t^\frac{\delta_2}{2}.$$ The remaining $L^2$ norms in the right hand side of are estimated reminding definition of $\mathcal{L}_m$ and using the fact that $$\label{dec_norm_Lm} \|\oph(\chi(h^\sigma\xi))\mathcal{L}_m [tw(t,tx)]\|_{L^2(dx)}\lesssim \|\chi(t^{-\sigma}D_x)[x_m w(t,\cdot)]\|_{L^2}+ t\|\chi(t^{-\sigma}D_x) w(t,\cdot)\|_{L^2}.$$ Therefore, by and lemma \[Lem\_appendix:Lm(Zv - vNF-Z)\] we derive that $$\begin{gathered} \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m \big[tZ_n(v^{NF}-v_{-})(t,tx)\big] \right\|_{L^2(dx)}\\+ \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m \big[t\big((Z_n v)_{-}-\vNFGamma\big)(t,tx)\big]\right\|_{L^2(dx)} \le C(A+B)B\varepsilon^2 t^{\beta + \frac{\delta+\delta_1}{2}},\end{gathered}$$ while from , a-priori estimates, together with the following inequality $$\begin{gathered} \|\chi(t^{-\sigma}D_x)[x_m (v_{-} - v^{NF})](t,\cdot)\|_{L^2} \lesssim \sum_{\mu,\nu=0}^1t^\sigma \left\|x_m^\mu \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\nu v_{-}(t,\cdot) \right\|_{L^\infty}\|u_\pm(t,\cdot)\|_{L^2}\\ \le C(A+B)B\varepsilon^2 t^{\sigma+ \frac{\delta+\delta_2}{2}},\end{gathered}$$ which follows by , , , , , we derive $$\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m \big[tD_n\langle D_x\rangle^{-1}(v_{-} - v^{NF})(t,tx)\big]\right\|_{L^2} \le C(A+B)B\varepsilon^2 t^{\sigma + \frac{(\delta+\delta_2)}{2}}.$$ In the following lemma we are going to prove that the product of the semiclassical wave function $\ut$ with the Klein-Gordon one $\vt$ enjoys a better $L^2$ (resp. $L^\infty$) estimate than the one roughly obtained by taking the $L^2$ (resp. $L^\infty$) norm of the former times the $L^\infty$ norm of the latter. Estimates $$\begin{aligned} \left\| \vt\ut (t,\cdot)\right\|_{L^2}& \lesssim \|\vt(t,\cdot)\|_{L^\infty}\|\ut(t,\cdot)\|_{L^2}\le CAB\varepsilon^2 h^{-\frac{\delta}{2}},\\ \left\| \vt\ut (t,\cdot)\right\|_{L^\infty}& \lesssim \|\vt(t,\cdot)\|_{L^\infty}\|\ut(t,\cdot)\|_{L^\infty}\le CA^2\varepsilon^2 h^{-\frac{1}{2}-\frac{\delta}{2}},\end{aligned}$$ which follows from , , , can be in fact improved of a factor $h^{1/2}$ (see ). This comes from the fact that the main contribution to $\ut$ is localized around manifold $\Lw$ introduced in , whereas $\vt$ concentrates around $\Lkg$ defined in , and these two manifolds are disjoint. (0,-1.7) – (0,1.7); (-2,0) – (2,0); at (1.9,0) [$x$]{}; at (0,1.6) [$\xi$]{}; (-1,-1.7) – (-1,1.7); (1,-1.7) – (1,1.7); at (-1,0) [$-1$]{}; at (1,0) [$1$]{}; at (1,1) [$\Lw$]{}; plot(, [/((1-\^2)\^(1/2))]{}); plot(-, [-/((1-\^2)\^(1/2))]{}); at (-0.8,-1.6) [$\Lkg$]{}; \[Lem\_appendix:product\_Vtilde\_Utilde\] Let $h=t^{-1}$, $\widetilde{u}, \widetilde{v}$ be defined in , $a_0(\xi)\in S_{0,0}(1)$, and $b_1(\xi)=\xi_j$ or $b_1(\xi)=\xi_j\xi_k|\xi|^{-1}$, with $j,k\in \{1,2\}$. There exists a constant $C>0$ such that, for any $\chi, \chi_1\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$, and every $t\in [1,T]$, we have that \[est\_L2Linfty\_Vtilde\_Utilde\] $$\label{est_L2_Vtilde_Utilde} \big\|[\oph(\chi(h^\sigma\xi)a_0(\xi))\widetilde{v}(t,\cdot)][\oph(\chi_1(h^\sigma\xi)b_1(\xi))\widetilde{u}(t,\cdot)]\big\|_{L^2} \le C(A+B)B\varepsilon^2 h^{\frac{1}{2}-\beta-\frac{\delta+\delta_1}{2}},$$ $$\label{est_Linfty_Vtilde Utilde} \big\|[\oph(\chi(h^\sigma\xi)a_0(\xi))\widetilde{v}(t,\cdot)][\oph(\chi_1(h^\sigma\xi)b_1(\xi))\widetilde{u}(t,\cdot)]\big\|_{L^\infty} \le C(A+B)B\varepsilon^2 h^{-\beta-\frac{\delta+\delta_1}{2}},$$ with $\beta>0$ small, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. Before entering in the details of the proof, we warn the reader that $C$ and $\beta$ denote two positive constants that may change line after line, with $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. Also, we will denote by $R(t,x)$ any contribution, in what follows, that satisfies inequalities , and by $\chi_2$ a smooth cut-off function, identically equal to 1 on the support of $\chi_1$, so that $$\oph(\chi_1(h^\sigma\xi))\widetilde{u} = \oph(\chi_1(h^\sigma\xi))\oph(\chi_2(h^\sigma\xi))\widetilde{u},$$ assuming that at any time $\widetilde{u}$ can be replaced with $\oph(\chi_2(h^\sigma\xi))\widetilde{u}$. Finally, it is useful to remind that from , , , , and a-priori estimates, $$\label{Hrho_infty_utilde_appendix} \|\widetilde{u}(t,\cdot)\|_{H^{\rho+1,\infty}_h}+\sum_{|\mu|=1}\| \oph\big((\xi|\xi|^{-1})^\mu\big)\widetilde{u}(t,\cdot)\|_{H^{\rho+1,\infty}_h} \le CA\varepsilon h^{-\frac{1}{2}},$$ while by , (with $\theta \ll 1$ small enough) and a-priori estimates, $$\label{Hrho_infty_vtilde_appendix} \|\widetilde{v}(t,\cdot)\|_{H^{\rho,\infty}_h}\le CA\varepsilon,$$ for every $t\in [1,T]$. First of all, we take $\gamma\in C^\infty_0(\mathbb{R}^2)$ equal to 1 in a neighbourhood of the origin and with sufficiently small support, and define $$\begin{gathered} \widetilde{v}_{\Lambda_{kg}}(t,x):= \oph\Big(\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)a_0(\xi)\Big)\widetilde{v}(t,x), \\ \widetilde{v}_{\Lambda^c_{kg}}(t,x):=\oph\Big((1-\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)a_0(\xi)\Big)\widetilde{v}(t,x),\end{gathered}$$ with $p(\xi):=\langle \xi\rangle$, so that $$\label{dec_vt_app} \oph(\chi(h^\sigma\xi)a_0(\xi))\widetilde{v}=\widetilde{v}_{\Lambda_{kg}}+\widetilde{v}_{\Lambda^c_{kg}}.$$ The following estimates hold: $$\begin{gathered} \|\widetilde{v}_{\Lambda_{kg}}(t,\cdot)\|_{L^\infty}\le CA\varepsilon h^{-\beta}, \label{est:vLambda_appendix} \\ \|\widetilde{v}_{\Lambda^c_{kg}}(t,\cdot)\|_{L^\infty} \le CB\varepsilon h^{\frac{1}{2}-\beta-\frac{\delta_1}{2}}. \label{est:vLambdac_appendix}\end{gathered}$$ The former one is a straight consequence of proposition \[Prop:Continuity Lp-Lp\] with $p=+\infty$ and . On the other hand, if we write $$(1-\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)a_0(\xi) =\sum_{j=1}^2 \gamma_1^j\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)a_0(\xi) \Big(\frac{x_j-p'_j(\xi)}{\sqrt{h}}\Big)$$ with $\gamma_1^j(z):=(1-\gamma)(z)z_j|z|^{-2}$ such that $|\partial^\alpha_z\gamma_1^j(z)|\lesssim \langle z\rangle^{-1-|\alpha|}$, and use with $c(x,\xi)=\chi(h^\sigma\xi)a_0(\xi)$, we obtain that $$\label{est:vLmabdakgc_appendix_preliminary} \begin{split} \|\widetilde{v}_{\Lambda^c_{kg}}(t,\cdot)\|_{L^\infty}&\lesssim \sum_{j=1}^2 \sqrt{h} \left\|\oph\Big(\gamma_1^j\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)a_0(\xi)\Big)\mathcal{L}_j\widetilde{v}(t,\cdot) \right\|_{L^\infty}\\ & + \sum_{j=1}^2 \sqrt{h} \left\|\oph\Big(\gamma_1^j\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\partial_j\big(\chi(h^\sigma\xi)a_0(\xi)\big)\Big)\widetilde{v}(t,\cdot) \right\|_{L^\infty} \\ & + \sum_{j=1}^2 \sum_{|\alpha|=2}\sqrt{h} \left\|\oph\Big((\partial^\alpha\gamma_1^j)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)a_0(\xi)(\partial^\alpha_\xi p')(\xi)\Big)\widetilde{v}(t,\cdot) \right\|_{L^\infty} \\ &+ \|\oph(r(x,\xi))\widetilde{v}(t,\cdot)\|_{L^\infty}, \end{split}$$ with $r\in h^{1-\beta}S_{\frac{1}{2},\sigma}(\langle\frac{x-p'(\xi)}{\sqrt{h}}\rangle^{-1})$. Since $\gamma_1^j$ vanishes in a neighbourhood of the origin, we derive from inequality , equation and relation with $w=v^{NF}$, lemmas \[Lem: from energy to norms in sc coordinates-KG\], \[Lem\_appendix: ZnvNF\_LmZnvNF\], and estimate , that the first sum in the above right hand side is bounded by the right hand side of . The same is true for the above second and third sums after and lemma \[Lem: from energy to norms in sc coordinates-KG\], and for the above latter $L^\infty$ norm because of proposition \[Prop : Continuity from $L^2$ to L\^inf\] and estimate . After decomposition and estimates , , and , we see that $$\oph(\chi(h^\sigma\xi)a_0(\xi))\widetilde{v} \, \oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u} = \widetilde{v}_{\Lambda_{kg}}\oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u} + R(t,x).$$ For some suitably supported $\chi_0\in C^\infty_0(\mathbb{R}^2)$, $\varphi\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$, we also consider the following decomposition $$\oph(\chi_1(h^\sigma\xi)b_1(\xi))\widetilde{u} = \oph(\chi_0(h^{-1}\xi)b_1(\xi))\widetilde{u} + \sum_k \oph\big((1-\chi_0)(h^{-1}\xi)\varphi(2^{-k}\xi)\chi_1(h^\sigma\xi)b_1(\xi)\big)\widetilde{u},$$ and observe that, from proposition \[Prop : Continuity on H\^s\] and the classical Sobolev injection, $$\begin{gathered} \left\|\oph(\chi_0(h^{-1}\xi)b_1(\xi))\widetilde{u}(t,\cdot)\right\|_{L^2}+ \left\|\oph(\chi_0(h^{-1}\xi)b_1(\xi))\widetilde{u}(t,\cdot)\right\|_{L^\infty}\lesssim h\|\widetilde{u}(t,\cdot)\|_{L^2}.\end{gathered}$$ Combining the above decomposition and estimate with and we derive that $$\label{dec_k_app} \widetilde{v}_{\Lambda_{kg}}\oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u} = \sum_k \widetilde{v}_{\Lambda_{kg}}\oph(\phi_k(\xi)b_1(\xi))\widetilde{u} + R(t,x),$$ where $\phi_k(\xi):=(1-\chi_0)(h^{-1}\xi)\varphi(2^{-k}\xi)\chi(h^\sigma\xi)$. We can further decompose $\oph(\phi_k(\xi)b_1(\xi))\widetilde{u}$ by defining $$\begin{gathered} \widetilde{u}^k_{\Lambda_w}(t,x):=\oph\Big(\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\phi_k(\xi)b_1(\xi)\Big)\widetilde{u}(t,x),\\ \widetilde{u}^k_{\Lambda^c_w}(t,x):=\oph\Big((1-\gamma)\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\phi_k(\xi)b_1(\xi)\Big)\widetilde{u}(t,x), \end{gathered}$$ and observe that $$\begin{gathered} \left\|\widetilde{u}^k_{\Lambda^c_w}(t,\cdot)\right\|_{L^2} \lesssim h^{\frac{1}{2}-\beta}\left[\|\widetilde{u}(t,\cdot)\|_{L^2} +\sum_{\mu, |\nu|=0}^1\|(\theta_0(x) \Omega_h)^\mu \mathcal{M}^\nu \oph(\chi_2(h^\sigma\xi))\widetilde{u}(t,\cdot)\|_{L^2}\right]\\\le CB\varepsilon h^{\frac{1}{2}-\beta-\frac{\delta_1}{2}},\end{gathered}$$ and $$\begin{gathered} \left\|\widetilde{u}^k_{\Lambda^c_w}(t,\cdot)\right\|_{L^\infty} \lesssim h^{-\beta}\left[\|\widetilde{u}(t,\cdot)\|_{L^2} +\sum_{\mu, |\nu|=0}^1\|(\theta_0(x) \Omega_h)^\mu \mathcal{M}^\nu \oph(\chi_2(h^\sigma\xi))\widetilde{u}(t,\cdot)\|_{L^2}\right]\\\le CB\varepsilon h^{-\beta-\frac{\delta_1}{2}},\end{gathered}$$ as follows by using the following equality $$(1-\gamma)\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\phi_k(\xi)b_1(\xi) = \sum_{j=1}^2\gamma_1^j\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\Big(\frac{x_j|\xi|-\xi_j}{h^{1/2-\sigma}}\Big)\phi_k(\xi)b_1(\xi),$$ with $\gamma_1^j(z):=(1-\gamma)(z)z_j|z|^{-2}$, together with with $a\equiv 1$, $p=1$, and lemma \[Lem: from energy to norms in sc coordinates-WAVE\]. Then, as the sum over $k$ in the right hand side of is actually restricted to indices $k$ such that $h\lesssim 2^k\lesssim h^{-\sigma}$, the above estimates and imply that $$\sum_k \widetilde{v}_{\Lambda_{kg}}\oph(\phi_k(\xi)b_1(\xi))\widetilde{u} = \sum_k \widetilde{v}_{\Lambda_{kg}}\ut^k_{\Lw} + R(t,x).$$ Moreover, using lemma \[Lem:family\_thetah\], symbolic calculus and remark \[Remark:symbols\_with\_null\_support\_intersection\], each $\vt_{\Lkg}\ut^k_{\Lw}$ in the above right hand side can be replaced with $$\frac{\theta_h(x)}{|x|^2-1}\vt_{\Lkg} \ (|x|^2-1)\ut^k_{\Lw}$$ up to a new remainder $R(t,x)$. Since $|\theta_h(x) (|x|^2-1)^{-1}|\lesssim h^{-2\sigma}$ on the support of $\theta_h(x)$, from proposition \[Prop : Continuity on H\^s\] and estimates , , we get that \[est\_uLambda\_vLambda\] $$\begin{gathered} \left\|\theta_h(x)\widetilde{v}_{\Lambda_{kg}}(t,\cdot) \widetilde{u}^k_{\Lambda_w} (t,\cdot)\right\|_{L^2}\le CB\varepsilon h^{-\frac{\delta}{2}-\beta}\|\theta_h(x)(|x|^2-1)\widetilde{u}^k_{\Lambda_w}(t,\cdot)\|_{L^\infty},\\ \left\|\theta_h(x)\widetilde{v}_{\Lambda_{kg}} (t,\cdot)\widetilde{u}^k_{\Lambda_w} (t,\cdot)\right\|_{L^\infty}\le CA\varepsilon h^{-\beta}\|\theta_h(x)(|x|^2-1)\widetilde{u}^k_{\Lambda_w}(t,\cdot)\|_{L^\infty}.\end{gathered}$$ Then the end of the proof relies on the fact that $\theta_h(x)(|x|^2-1)\widetilde{u}^k_{\Lambda_w}$ can be expressed in terms of $h\Mcal \ut$. In fact, for a fixed $N\in\mathbb{N}$ and up to some negligible multiplicative constants, we have from proposition \[Prop: a sharp b\] that $$\label{symb_prod_(x2-1)} \begin{split} \left[\theta_h(x)(|x|^2-1)\right]\sharp &\left[\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\phi_k(\xi)b_1(\xi)\right] = \gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\phi_k(\xi)b_1(\xi)\theta_h(x) (|x|^2-1)\\ &+ h\left\{\theta_h(x) (|x|^2-1), \gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\phi_k(\xi)b_1(\xi) \right\}\\ & + \sum_{ |\alpha|=2}^{N-1} h^{|\alpha|} \partial^{\alpha}_x\left[\theta_h(x)(|x|^2-1)\right] \partial^\alpha_\xi\left[\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\phi_k(\xi)b_1(\xi)\right] +r_N(x,\xi), \end{split}$$ with $$\begin{gathered} \label{rN_lemma_appendixB} r_N(x,\xi) = \frac{h^N}{(\pi h)^4}\sum_{|\alpha|=N}\int e^{\frac{2i}{h}(\eta\cdot z-y\cdot\zeta)} \int_0^1 \partial^\alpha_x[\theta_h(x)(|x|^2-1)]|_{(x+tz)}(1-t)^{N-1}dt \\ \times \partial^\alpha_\xi\left[\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\phi_k(\xi)b_1(\xi)\right]|_{(x+y, \xi+\eta)} dy dz d\eta d\zeta.\end{gathered}$$ As $$|x|^2-1 = x\cdot x - \frac{\xi\cdot\xi}{|\xi|^2} = (x|\xi|-\xi)\cdot \frac{x}{|\xi|} + (x|\xi|-\xi)\cdot\frac{\xi}{|\xi|^2},$$ the first term in the right hand side of appears to be linear combination of products of the form $\gamma\big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\big)\phi_k(\xi)a(x)b_0(\xi)(x_j|\xi|-\xi_j)$, for some smooth compactly supported function $a(x)$, and $b_0(\xi)$ such that $|\partial^\alpha b_0(\xi)|\lesssim |\xi|^{-|\alpha|}$. From and lemma \[Lem: from energy to norms in sc coordinates-WAVE\], we hence deduce that $$\begin{gathered} \label{est_first_term} \left\| \oph\Big(\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\phi_k(\xi)b_1(\xi)\theta_h(x) (|x|^2-1)\Big)\widetilde{u}(t,\cdot)\right\|_{L^\infty} \le CB\varepsilon h^{\frac{1}{2}-\beta-\frac{\delta_1}{2}}.\end{gathered}$$ An explicit computation shows that $$\begin{gathered} h \left\{\theta_h(x) (|x|^2-1), \gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\phi_k(\xi)b_1(\xi) \right\}\\ =\sum_i h^{\frac{1}{2}+\sigma}\partial_{x_i}[\theta_h(x)(|x|^2-1)]\sum_j(\partial_j\gamma)\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\Big(x_j\frac{\xi_i}{|\xi|}-\delta_{ij}\Big)\phi_k(\xi)b_1(\xi) \\ +h \gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big) \partial_x[\theta_h(x)(|x|^2-1)]\partial_\xi[\phi_k(\xi)b_1(\xi)],\end{gathered}$$ with $\delta_{ij}=1$ being the Kronecker delta. One the one hand, since the first contribution to the above right hand side is still supported for $|x|<1-ch^{2\sigma}$, we can multiply and divide it by $|x|^2-1$ so that it writes as linear combination of terms of the form $h^{\frac{1}{2}-\sigma}\gamma_1\big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\big)\phi_k(\xi)a(x)b_0(\xi)(x_j|\xi|-\xi_j)$, for a new $\gamma_1\in C^\infty_0(\mathbb{R}^2)$, and some new $a(x), b_0(\xi)$ with the same properties as the ones we considered before. On the other hand, as $\partial_\xi[\phi_k(\xi)b_1(\xi)]$ is uniformly bounded and supported for frequencies of size $2^k$, the second term in the above right hand side writes as linear combination of products of the form $h\gamma\big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\big)\phi^1_k(\xi)a(x)b_0(\xi)$, for some new $\phi^1_k\in C^\infty_0(\mathbb{R}^2\setminus\{0\})$. Therefore, inequality , proposition \[Prop : continuity of Op(gamma1):X to L2\], and lemma \[Lem: from energy to norms in sc coordinates-WAVE\], give that $$\begin{gathered} \label{est_second_term} \left\| h \oph\Big(\Big\{\theta_h(x) (|x|^2-1), \gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\phi_k(\xi)b_1(\xi) \Big\}\Big)\widetilde{u}(t,\cdot)\right\|_{L^\infty} \le CB\varepsilon h^{\frac{1}{2}-\beta-\frac{\delta_1}{2}}.\end{gathered}$$ As concerns $|\alpha|$-order terms, for each fixed $2\le |\alpha|\le N-1$, we find using that they are given by $$\begin{gathered} h^{|\alpha|}\gamma\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big) \partial^\alpha_x[\theta_h(x)(|x|^2-1)]\partial^\alpha_\xi (\phi_k(\xi)b_1(\xi))\\ + \sum_{\substack{|\beta_1|+|\beta_2|=|\alpha|\\|\beta_1|\ge 1}}\sum_{j=1}^{|\beta_1|}h^{|\alpha| - j(\frac{1}{2}-\sigma)} \gamma_j\Big(\frac{x|\xi|-\xi}{h^{1/2-\sigma}}\Big)\widetilde{\theta}_j(x)b_{j-|\beta_1|}(\xi)\partial^{\beta_2}_\xi(\phi_k(\xi)b_1(\xi)),\end{gathered}$$ for some $\gamma_j, \widetilde{\theta}_j\in C^\infty_0(\mathbb{R}^2)$. Since $|\alpha|\ge 2$ and $|\partial^\mu_\xi(\phi_k(\xi)b_1(\xi))|\lesssim 2^{-k(|\mu|-1)}$, for any $\mu\in\mathbb{N}^2$, by proposition \[Prop : continuity of Op(gamma1):X to L2\] and lemma \[Lem: from energy to norms in sc coordinates-WAVE\] we obtain that the action of their quantization on $\widetilde{u}$ is estimated in the uniform norm by $$\begin{gathered} \label{est_third_term} \left[h^{|\alpha|-\frac{1}{2}-\beta}2^{-k(|\alpha|-1)} + \sum_{1\le j\le |\alpha|} h^{|\alpha|-j(\frac{1}{2}-\sigma)}2^{k(j+1-|\alpha|)}h^{-\frac{1}{2}-\beta}\right]\\ \times \left[\|\widetilde{u}(t,\cdot)\|_{L^2} +\sum_{\mu, |\nu|=0}^1\|(\theta \Omega_h)^\mu \oph(\chi_1(h^\sigma\xi)\widetilde{u}(t,\cdot)\|_{L^2}\right]\le CB\varepsilon h^{\frac{1}{2}-\beta-\frac{\delta_1}{2}}.\end{gathered}$$ Finally, by integrating in $dyd\zeta$ and using in we find that $r_N(x,\xi)$ can be written as $$\begin{gathered} \sum_{j\le N} h^{N-j(\frac{1}{2}-\sigma)}\frac{1}{(\pi h)^2}\int e^{\frac{2i}{h}\eta\cdot z}\int_0^1 \theta_N(x+tz)(1-t)^{N-1}dt \\ \times \gamma_j\Big(\frac{x|\xi+\eta|-(\xi+\eta)}{h^{1/2-\sigma}}\Big)\phi_k^j(\xi+\eta)b_{j+1-N}(\xi+\eta) dzd\eta,\end{gathered}$$ for some new smooth compactly supported $\theta_N, \gamma_j, \phi_k^j$.From the last part of proposition \[Prop : Linfty est of integral remainders\] then follows that the quantization of the above integral is a bounded operator from $L^2$ to $L^\infty$, with norm controlled by $$\sum_{\substack{j\le N\\ i\le 6}}h^{N-j(\frac{1}{2}-\sigma)}2^{k(1+j-N)}(h^{-\frac{1}{2}+\sigma}2^k)^i (h^{-1}2^k)\lesssim h$$ if $N$ is sufficiently large (e.g. $N\ge 10$), and consequently that $$\|\oph(r_N(x,\xi))\widetilde{u}(t,\cdot)\|_{L^\infty}\lesssim h\|\widetilde{u}^k(t,\cdot)\|_{L^2} \le CB\varepsilon h^{1-\frac{\delta}{2}}.$$ Finally, summing up the above estimates with formulas from to we obtain that $$\begin{gathered} \|\theta_h(x)(|x|^2-1)\widetilde{u}_{\Lambda_w}(t,\cdot)\|_{L^\infty}\lesssim CB\varepsilon h^{\frac{1}{2}-\beta-\frac{\delta_1}{2}},\end{gathered}$$ which injected in gives that $\theta_h(x)\widetilde{v}_{\Lambda_{kg}}\widetilde{u}^k_{\Lambda_w}$ is a remainder $R(t,x)$. That concludes the proof of the statement. A similar result to the one proved in lemma \[Lem\_appendix:product\_Vtilde\_Utilde\] holds true when $\widetilde{u}$ in the left hand side of is replaced with $$\label{def_utildeJ} \widetilde{u}^J(t,x):= t (\Gamma u)_{-}(t,tx)$$ with $\Gamma \in \{\Omega, Z_m, m=1,2\}$ being a Klainerman vector field, as briefly shown in the following: \[Lem\_appendix: product vtilde\_utildeJ\] Let $h=t^{-1}$, $\widetilde{v}$ be defined in , $\widetilde{u}^J$ as in , $a_0(\xi)\in S_{0,0}(1)$, and $b_1(\xi)=\xi_j$ or $b_1(\xi)=\xi_j\xi_k|\xi|^{-1}$, with $j,k\in \{1,2\}$. There exists a constant $C>0$ such that, for any $\chi, \chi_1\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$, and every $t\in [1,T]$, we have that $$\label{est_vtilde_utildeJ_L2} \big\|[\oph(\chi(h^\sigma\xi)a_0(\xi))\widetilde{v}(t,\cdot)][\oph(\chi_1(h^\sigma\xi)b_1(\xi))\widetilde{u}^J(t,\cdot)]\big\|_{L^2} \le C(A+B)B\varepsilon^2 h^{\frac{1}{2}-\beta'},$$ $$\label{est_Linfty_vtilde_utildeJ} \big\|[\oph(\chi(h^\sigma\xi)a_0(\xi))\widetilde{v}(t,\cdot)][\oph(\chi_1(h^\sigma\xi)b_1(\xi))\widetilde{u}^J(t,\cdot)]\big\|_{L^\infty} \le C(A+B)B\varepsilon^2 h^{-\beta'},$$ with $\beta'>0$ small, $\beta\rightarrow 0$ as $\sigma,\delta_0\rightarrow 0$. The proof of this result is analogous to that of lemma \[Lem\_appendix:product\_Vtilde\_Utilde\] except that, instead of referring to , we should use that $$\label{est:Hrho_utildeJ} \|\oph(\chi(h^\sigma\xi))\widetilde{u}^J(t,\cdot)\|_{H^{\rho+1,\infty}_h}+\sum_{|\mu|=1}\| \oph\big(\chi(h^\sigma\xi))(\xi|\xi|^{-1})^\mu\big)\widetilde{u}^J(t,\cdot)\|_{H^{\rho+1,\infty}_j} \le CA\varepsilon h^{-\frac{1}{2}-\beta-\frac{\delta_1}{2}},$$ which is the semiclassical translation of , and to lemma \[Lem\_appendix: L\^2 estimates uJ\] instead of lemma \[Lem: from energy to norms in sc coordinates-WAVE\]. \[Lem\_appendix:Linfty\_est\_rNFkg\]Let $a_0(\xi)\in S_{0,0}(1)$, $b_1(\xi)\in \{\xi_j, \xi_j\xi_k|\xi|^{-1}, |\xi|, j,k=1,2\}$, $b_0(\xi)\in \{1,\xi_j|\xi|^{-1}, j=1,2\}$. There exists a constant $C>0$ such that, for any $\chi\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$ small, and every $t\in [1,T]$, $$\label{est_Linfty_prod_a0v_b1u_R1u} \left\| \chi(t^{-\sigma}D_x) \big[[a_0(D_x)v_{-}] [b_1(D_x)u_{-}] b_0(D_x)u_{-}\big](t,\cdot)\right\|_{L^\infty} \le C(A+B)AB\varepsilon^3 t^{-\frac{5}{2}+\beta + \frac{\delta+\delta_1}{2}},$$ with $\beta>0$ small, $\beta\rightarrow 0$ as $\sigma\rightarrow0$. Consequently $$\label{est_Linfty_chi_rnfkg} \left\|\chi(t^{-\sigma}D_x)r^{NF}_{kg}(t,\cdot) \right\|_{L^\infty}\le C(A+B)AB\varepsilon^3 t^{-\frac{5}{2}+\beta + \frac{\delta+\delta_1}{2}},$$ where $r^{NF}_{kg}$ is given by . We warn the reader that we denote by $C$ and $\beta$ two positive constants that may change line after line during this proof, with $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. Moreover, we are going to denote generically by $R(t,x)$ each term satisfying $$\|R(t,\cdot)\|_{L^\infty}\le C(A+B)AB\varepsilon^3 t^{-\frac{5}{2}+\beta + \frac{\delta+\delta_1}{2}}.$$ From lemma \[Lem\_appendix:L\_estimate of products\] with $L=L^\infty$ and $s>0$ large enough to have $N(s)\ge 3$, and a-priori estimates , we can reduce ourselves to estimate the $L^\infty$ norm of the product in the left hand side of when all its factors are supported for moderate frequencies less or equal than $t^\sigma$, up to remainders $R(t,x)$. Moreover, since \[est:vNF-v-\_uBF-u-\_appendix\] $$\label{a0_vNF-v-} \left\|\chi(t^{-\sigma}D_x)a_0(D_x)[v^{NF}-v_{-}](t,\cdot) \right\|_{L^\infty}\le CA^2\varepsilon^2 t^{-\frac{3}{2}+\sigma}$$ and $$\label{b1_uNF-u-} \left\|\chi(t^{-\sigma}D_x)b_1(D_x)[u^{NF}-u_{-}](t,\cdot) \right\|_{L^\infty}\le CA^2\varepsilon^2 t^{-2+\beta},$$ as follows by and , with $\rho=2$ (as consequence of lemma \[Lem\_Appendix: est on Dj1j2\]), together with a-priori estimates, we can also suppose $v_{-}$ (resp. $u_{-}$) be replaced with $v^{NF}$ (resp. $u^{NF}$), up to some new $R(t,x)$. This reduces us to prove that $$\begin{gathered} \left\|[\chi(t^{-\sigma}D_x)a_0(D_x)v^{NF}][\chi(t^{-\sigma}D_x)b_1(D_x)u^{NF}][\chi(t^{-\sigma}D_x)b_0(D_x) u_{-}](t,\cdot)\right\|_{L^\infty}\\ \le C(A+B)AB\varepsilon^3 t^{-\frac{5}{2}+\beta+\frac{\delta+\delta_1}{2}},\end{gathered}$$ or rather, reminding , to show that $$\left\|[\chi(t^{-\sigma}D_x)a_0(D_x)v^{NF}][\chi(t^{-\sigma}D_x)b_1(D_x)u^{NF}](t,\cdot)\right\|_{L^\infty}\le C(A+B)B\varepsilon^2 t^{-2+\beta+\frac{\delta+\delta_1}{2}}.$$ But after writing the above product in the semi-classical setting and reminding definition , one can immediately check that this estimate is satisfied thanks to , which concludes the proof of . The last part of the statement follows from , the fact that $$\left\|\chi(t^{-\sigma}D_x)\left[ - \frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\, \Nlw + D_1\big[\langle D_x\rangle^{-1}(v_+-v_{-})\, \Nlw\right](t,\cdot) \right\|_{L^\infty}\le CA^3\varepsilon^3 t^{-3+\sigma}$$ for every $t\in [1,T]$, which is consequence of and a-priori estimate , and from the observation that the remaining contributions to $\rnfkg$ are products of the form $$[a_0(D_x)v_{-}] [b_1(D_x)u_{-}] \mathrm{R}_1u_{-},$$ with $a_0(\xi)$ equal to 1 or to $\xi_j\langle\xi\rangle^{-1}$, and $b_1(\xi)$ equal to $\xi_1$ or to $\xi_j \xi_1|\xi|^{-1}$, for $j=1,2$. \[Lem\_appendix: L xnrNFkg\] Under the same assumptions as in lemma \[Lem\_appendix:Linfty\_est\_rNFkg\], \[est\_xx\_prod\_a0v\_b1u\_R1u\] $$\begin{aligned} \left\|\chi(t^{-\sigma}D_x) \big[x_n [a_0(D_x)v_{-}] [b_1(D_x)u_{-}] b_0(D_x) u_{-}\big](t,\cdot)\right\|_{L^2(dx)}&\le C(A+B)^2B\varepsilon^3 t^{-1+\beta+\frac{\delta_1}{2}}, \\ \left\|\chi(t^{-\sigma}D_x) \big[x_m x_n [a_0(D_x)v_{-}][ b_1(D_x)u_{-}]b_0(D_x)u_{-}\big](t,\cdot)\right\|_{L^2(dx)}&\le C(A+B)^2B\varepsilon^3 t^{\beta+\frac{\delta_1}{2}},\end{aligned}$$ for every $t\in [1,T]$, $m,n=1,2$, with $\beta>0$ small, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. Moreover, \[est\_xxrnfkg\] $$\begin{aligned} \left\|\chi(t^{-\sigma}D_x) \big[x_n r^{NF}_{kg}(t,\cdot)\big]\right\|_{L^2(dx)} &\le C(A+B)^2B\varepsilon^3 t^{-1+\beta+\frac{\delta+\delta_1}{2}}, \label{est:t_xn_rNFkg}\\ \left\|\chi(t^{-\sigma}D_x) \big[x_m x_n r^{NF}_{kg}(t,\cdot)\big]\right\|_{L^2(dx)} &\le C(A+B)^2B\varepsilon^3 t^{\beta+\frac{\delta+\delta_1}{2}}. \label{est:xm_xn_rNFkg}\end{aligned}$$ We warn the reader that we will denote by $C$ and $\beta$ two positive constants that may change line after line, with $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. We also denote by $R(t,x)$ any contribution verifying \[est\_R\_lemma\_rnfkg\] $$\begin{aligned} \left\|\chi(t^{-\sigma}D_x) \big[x_nR(t,\cdot)\big]\right\|_{L^2(dx)} &\le C(A+B)^2B\varepsilon^3 t^{-1+\beta+\frac{\delta+\delta_1}{2}}, \label{xn_R} \\ \left\|\chi(t^{-\sigma}D_x) \big[x_m x_n R(t,\cdot)\big]\right\|_{L^2(dx)} &\le C(A+B)^2B\varepsilon^3 t^{\beta+\frac{\delta+\delta_1}{2}}.\label{xnxm_R}\end{aligned}$$ Let us first notice that, after , , and a-priori estimates, we have that $$\begin{gathered} \left\|\chi(t^{-\sigma}D_x)\left[ - x_n\frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\, \Nlw +x_n D_1\big[\langle D_x\rangle^{-1}(v_+-v_{-})\, \Nlw\right](t,\cdot) \right\|_{L^2} \\ \lesssim t^\sigma \sum_{\mu=0}^1\left\|x_n^\mu v_\pm(t,\cdot) \right\|_{L^2}\|\textit{NL}_w(t,\cdot)\|_{L^\infty}\le CA^2B\varepsilon^3 t^{-1+\sigma+\frac{\delta}{2}}\end{gathered}$$ and $$\begin{gathered} \left\|\chi(t^{-\sigma}D_x)\left[ - x_mx_n\frac{D_x}{\langle D_x\rangle}(v_+ - v_{-})\, \Nlw + x_mx_nD_1\big[\langle D_x\rangle^{-1}(v_+-v_{-})\, \Nlw\right](t,\cdot) \right\|_{L^2} \\ \lesssim t^\sigma \sum_{\mu_1, \mu_2=0}^1\left\| x_m^{\mu_1}x_n^{\mu_2} v_\pm(t,\cdot) \right\|_{L^2}\|\textit{NL}_w(t,\cdot)\|_{L^\infty}\le C(A+B)AB\varepsilon^3 t^{\sigma+\frac{\delta}{2}}.\end{gathered}$$ Therefore, since from and the remaining contributions to $\rnfkg$ are of the form $$[a_0(D_x)v_{-}] [b_1(D_x)u_{-}] \mathrm{R}_1u_{-}$$ with $a_0(\xi)$ equal to 1 or to $\xi_j\langle\xi\rangle^{-1}$, and $b_1(\xi)$ equal to $\xi_1$ or to $\xi_j \xi_1|\xi|^{-1}$, for $j=1,2$, estimates will follow from . Our aim is hence to prove that the above product is a remainder $R(t,x)$. Applying lemma \[Lem\_appendix:L\_estimate of products\] with $L=L^2$, $w_1=x_n a_0(D_x) v_{-}$ (resp. $w_1=x_m x_n a_0(D_x) v_{-}$), $s>0$ sufficiently large so that $N(s)> 2$, and using estimates (resp. ), , , we can suppose all above factors truncated for moderate frequencies less or equal than $t^\sigma$, up to remainders $R(t,x)$. Let us also observe that, from , and ,$$\begin{aligned} \left\|\chi(t^{-\sigma}D_x) \Big[ [\chi_1(t^{-\sigma}D_x)[x_n a_0(D_x)v_{-}]][ \chi(t^{-\sigma}D_x) b_1(D_x)(u^{NF}- u_{-})] [\chi(t^{-\sigma}D_x)b_0(D_x)u_{-}]\Big](t,\cdot)(t,\cdot) \right\|_{L^2} \\ \lesssim \sum_{|\mu|=0}^1 \left\|x_n\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty}\| \chi(t^{-\sigma}D_x) b_1(D_x)(u^{NF}- u_{-})\|_{L^\infty}\| u_\pm(t,\cdot)\|_{L^2}\\ \le C(A+B)A^2B\varepsilon^4 t^{-2+\beta+\frac{\delta+\delta_2}{2}},\end{aligned}$$and that, using additionally estimate , $$\begin{aligned} \left\|\chi(t^{-\sigma}D_x) \Big[ [\chi_1(t^{-\sigma}D_x)[x_m x_n a_0(D_x)v_{-}][\chi(t^{-\sigma}D_x) b_1(D_x)(u^{NF}- u_{-})] [\chi(t^{-\sigma}D_x)b_0(D_x)u_{-}]\Big](t,\cdot)(t,\cdot) \right\|_{L^2} \\ \lesssim \sum_{|\mu|, |\nu| =0}^1 \left\|x_m x_n\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty}\| \chi(t^{-\sigma}D_x) b_1(D_x)(u^{NF}- u_{-})\|_{L^2}\|\mathrm{R}^\nu u_\pm(t,\cdot)\|_{L^\infty}\\ \le C(A+B)A^2B\varepsilon^4 t^{-1+\beta+\frac{\delta+\delta_2}{2}}.\end{aligned}$$This means that we can actually replace $u_{-}$ by $u^{NF}$ up to some new $R(t,x)$. Furthermore, we can also substitute $\chi_1(t^{-\sigma}D_x)[x^k_m x_na_0(D_x)v_{-}]$ with $\chi(t^{-\sigma}D_x)[x^k_m x_n a_0(D_x)v^{NF}_{-}]$, for any $k\in \{0,1\}$, up to a new remainder $R(t,x)$ in consequence of a-priori estimate , the fact that $$\label{est_L2_uNF} \|u^{NF}(t,\cdot)\|_{L^2}\le CB\varepsilon t^\frac{\delta}{2},$$ (see in semi-classical coordinates), and the following inequalities \[xx\_(vnf-v-)\] $$\begin{gathered} \label{x_(vnf-v)} \left\|\chi_1(t^{-\sigma}D_x)\left[x_na_0(D_x) (v^{NF}-v_{-})\right](t,\cdot)\right\|_{L^\infty}\\ \lesssim \sum_{\mu,\nu,\kappa=0}^1t^\sigma\left\|x^\mu_n \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\nu v_\pm(t,\cdot) \right\|_{L^\infty} \|\mathrm{R}^\kappa_1u_\pm(t,\cdot)\|_{L^\infty} \le C(A+B)A\varepsilon^2 t^{-\frac{1}{2}+\sigma+\frac{\delta_2}{2}}\end{gathered}$$ and $$\begin{gathered} \left\| \chi_1(t^{-\sigma}D_x)\left[x_m x_n a_0(D_x) (v^{NF}-v_{-})\right](t,\cdot)\right\|_{L^\infty}\\ \lesssim \sum_{\mu_1,\mu_2,\nu,\kappa=0}^1\left\|x^{\mu_1}_m x^{\mu_2}_n \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\nu v_\pm(t,\cdot) \right\|_{L^\infty} \|\mathrm{R}^\kappa_1u_\pm(t,\cdot)\|_{L^\infty}\le C(A+B)A\varepsilon^2 t^{\frac{1}{2}+\frac{\delta_2}{2}},\end{gathered}$$ derived from , , , and . This reduces us to prove that, for $k=0,1$, $$\begin{gathered} \left\|\big[\chi_1(t^{-\sigma}D_x)[x_m^k x_n a_0(D_x)v^{NF}_{-}]\big] \big[\chi(t^{-\sigma}D_x)b_1(D_x)u^{NF}\big]\big[\chi(t^{-\sigma}D_x)b_0(D_x)u_{-}\big](t,\cdot)\right\|_{L^2(dx)}\\ \le C(A+B)^2B\varepsilon^3 t^{-1+k+\beta+\frac{\delta_1}{2}},\end{gathered}$$ or rather, after , that $$\left\| \left[\chi_1(t^{-\sigma}D_x)[x_m^k x_n a_0(D_x) v^{NF}_{-}]\right] [\chi(t^{-\sigma}D_x)b_1(D_x) u^{NF}](t,\cdot) \right\|_{L^2(dx)}\le C(A+B)B\varepsilon^2 t^{-\frac{1}{2}+k +\beta+\frac{\delta+\delta_1}{2}}.$$Passing to the semi-classical setting, this corresponds to prove that$$\label{est_sc_setting_appendix} \sum_{k=0}^1\Big\| \left[\oph(\chi_1(h^\sigma\xi))[x_m^k x_n \oph(a_0(\xi))\widetilde{v}\right] [\oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u}] (t,\cdot)\Big\|_{L^2(dx)}\le C(A+B)B\varepsilon^2 h^{\frac{1}{2}-\beta-\frac{\delta+ \delta_1}{2}}.$$First of all let us notice that, from the commutation of $x_n$ with $\oph(a_0(\xi))$ and definition of $\mathcal{L}_n$, $$\label{eq:xn_a0_vtilde} \begin{split} x_n \oph(a_0(\xi))\widetilde{v} & = h \oph(a_0(\xi))\mathcal{L}_n\widetilde{v} + \oph\Big(a_0(\xi)\frac{\xi_n}{\langle\xi\rangle}\Big)\widetilde{v} - \frac{h}{2i}\oph\big(\partial_{\xi_n}a_0(\xi)\big)\widetilde{v}, \end{split}$$ while from the commutation of $x_m$ with $\oph(\chi(h^\sigma\xi)b_1(\xi))$, definition of $\mathcal{M}_m$, and symbolic calculus, $$\label{eq:xm_b1_utilde} \begin{split} x_m \oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u}& = h \oph(\chi(h^\sigma\xi)b_1(\xi)|\xi|^{-1})\mathcal{M}_m\widetilde{u} - \frac{h}{2i}\oph\big(\partial_{\xi_m}(\chi(h^\sigma\xi)b_1(\xi)|\xi|^{-1})|\xi|\big)\widetilde{u}\\ & + \oph(\chi(h^\sigma\xi)b_1(\xi)\xi_m|\xi|^{-1})\widetilde{u} -\frac{h}{2i}\oph\big(\partial_{\xi_m}(\chi(h^\sigma\xi)b_1(\xi))\big)\widetilde{u}. \end{split}$$ On the one hand, using equality , lemma \[Lem: from energy to norms in sc coordinates-KG\], and estimates , , we deduce that $$\label{ineq:xn_utilde_vtilde} \begin{split} &\Big\|\left[\oph(\chi_1(h^\sigma\xi)[x_n \oph(a_0(\xi))\widetilde{v}]\right] [\oph(\chi(h^\sigma\xi) b_1(\xi))\widetilde{u}](t,\cdot) \Big\|_{L^2} \\ &\le \left\| \Big[\oph\Big(\chi_1(h^\sigma\xi)a_0(\xi)\frac{\xi_n}{\langle\xi\rangle}\Big)\widetilde{v}(t,\cdot)\Big] [\oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u}(t,\cdot)] \right\|_{L^2} + CAB\varepsilon^2 h^{\frac{1}{2}-\beta-\frac{\delta_2}{2}}\\ & \le C(A+B)B\varepsilon^2 h^{\frac{1}{2}-\beta-\frac{\delta+\delta_1}{2}}. \end{split}$$ On the other hand, when we deal with the $L^2$ norm in the left hand side of corresponding to $k=1$ we first commute $x_m$ with $\oph(\chi_1(h^\sigma\xi))$ and see, using symbolic calculus, that $$\begin{aligned} &\Big\| \left[\oph(\chi_1(h^\sigma\xi))[x_m x_n \oph(a_0(\xi))\widetilde{v}\right] [\oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u}] (t,\cdot)\Big\|_{L^2(dx)} \\ &\le \Big\| \left[h^\sigma \oph((\partial\chi_1)(h^\sigma\xi))[ x_n \oph(a_0(\xi))\widetilde{v}\right] [\oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u}] (t,\cdot)\Big\|_{L^2(dx)} \\ & + \Big\| \left[\oph(\chi_1(h^\sigma\xi))[ x_n \oph(a_0(\xi))\widetilde{v}\right] [x_m\oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u}](t,\cdot) \Big\|_{L^2(dx)}.\end{aligned}$$ The first norm in the above right hand side satisfies an inequality analogous to . In order to derive an estimate for the latter one, we first use equality and observe the following: from the semi-classical Sobolev injection and estimates , , we have that $$\begin{gathered} \label{est_Lvt_Mut} \left\| h^2\big[\oph(\chi_1(h^\sigma\xi)a_0(\xi))\mathcal{L}_n\widetilde{v}\big]\big[ \oph(\chi(h^\sigma\xi)b_1(\xi)\xi_m|\xi|^{-1})\mathcal{M}_m\widetilde{u}\big](t,\cdot)\right\|_{L^2}\\ \lesssim h \left\| \oph(\chi_1(h^\sigma\xi)a_0(\xi))\mathcal{L}_n\widetilde{v}(t,\cdot)\right\|_{L^2}\left\| \oph(\chi(h^\sigma\xi)b_1(\xi)\xi_m|\xi|^{-1})\mathcal{M}_m\widetilde{u}(t,\cdot)\right\|_{L^2}\\ \le C(A+B)B\varepsilon^2 h^{1-\delta_2-\beta};\end{gathered}$$ a similar chain of inequalities as in , together with , and , gives that for any $\theta\in ]0,1[$ $$\begin{gathered} \label{RRutillde} \left\| \oph(b_1(\xi)\xi_m|\xi|^{-1})\widetilde{u}(t,\cdot)\right\|_{L^\infty}=t \left\|b_1(D_x)D_m|D_x|^{-1}u^{NF}(t,\cdot)\right\|_{L^\infty}\\ \lesssim t \|u^{NF}(t,\cdot)\|^{1-\theta}_{H^{3,\infty}}\|u^{NF}(t,\cdot)\|_{H^2} \le CA^{1-\theta}B^\theta\varepsilon t^{\frac{1}{2}+\frac{(1+\delta)}{2}\theta}.\end{gathered}$$ Therefore, from equality and estimates , , , , we find that $$\label{est_Lv_Mu} h\left\|\oph(\chi_1(h^\sigma\xi)a_0(\xi))\mathcal{L}_n\widetilde{v}\, \big[x_m \oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u}\big](t,\cdot)\right\|_{L^2}\le C(A+B)B\varepsilon^2 h^{\frac{1}{2}-\frac{\delta_2}{2} -\frac{(1+\delta)\theta}{2}}.$$ Moreover, using again along with, , and , $$\begin{aligned} &\left\|\Big[\oph\Big(\chi_1(h^\sigma\xi)a_0(\xi)\frac{\xi_n}{\langle\xi\rangle}\Big)\vt + h\oph\big(\chi_1(h^\sigma\xi) \partial_{\xi_n}a_0(\xi)\big)\vt\Big] \, \big[x_m \oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u}\big](t,\cdot)\right\|_{L^2(dx)}\\ & \le \left\|\Big[\oph\Big(\chi_1(h^\sigma\xi)a_0(\xi)\frac{\xi_n}{\langle\xi\rangle}\Big)\vt\Big] \Big[\oph\Big(\chi(h^\sigma\xi)b_1(\xi)\frac{\xi_m}{|\xi|}\Big)\ut\Big](t,\cdot)\right\|_{L^2(dx)} + C(A+B)B\varepsilon^2 h^{1-\beta-\frac{\delta_2}{2}}\\ & \le C(A+B)B \varepsilon^2 h^{\frac{1}{2}-\beta-\frac{\delta+\delta_1}{2}}.\end{aligned}$$ Choosing $\theta\ll 1$ small enough, this concludes that $$\label{ineq:xmxn_vtilde_utilde} \Big\| \left[\oph(\chi_1(h^\sigma\xi))[x_m x_n \oph(a_0(\xi))\widetilde{v}\right] [\oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u}] (t,\cdot)\Big\|_{L^2(dx)} \le C(A+B)B\varepsilon^2 h^{\frac{1}{2}-\beta-\frac{\delta+\delta_1}{2}}$$ and, together with , the proof of . We can finally prove the following: \[Lem\_appendix: estimate L2vtilde\] There exists a constant $C>0$ such that, for any $\chi \in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$ small, and every $t\in [1,T]$, $$\label{est:L2vtilde} \sum_{|\mu|=2}\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu\widetilde{v}(t,\cdot)\|_{L^2} \le CB\varepsilon t^{\beta+\frac{\delta+\delta_1}{2}},$$ with $\beta>0$ small, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. From relation and the commutation between $\mathcal{L}_m$ and $\oph(\langle\xi\rangle)$ we deduce that $$\label{ineq_L2vtilde} \begin{split} &\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m\mathcal{L}_n \widetilde{v}(t,\cdot)\right\|_{H^1_h}\lesssim \sum_{\mu=0}^1\Big[\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu_m \big[tZ_n v^{NF}(t,tx)\big] \right\|_{L^2(dx)}\\ & + \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu_m \oph\Big(\frac{\xi_n}{\langle\xi\rangle}\Big)\widetilde{v}(t,\cdot)\big]\right\|_{L^2(dx)}+ \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu_m \big[t(tx_n)r^{NF}_{kg}(t,tx)\big]\right\|_{L^2(dx)}\Big], \end{split}$$ so the result of the statement follows from lemmas \[Lem: from energy to norms in sc coordinates-KG\], \[Lem\_appendix: ZnvNF\_LmZnvNF\], and inequalities , . The sharp decay estimate of the Klein-Gordon solution with a Klainerman vector field {#Sec_App_4} ------------------------------------------------------------------------------------ This last section is devoted to prove that, for any admissible vector field $\Gamma$, the $L^\infty(\mathbb{R}^2)$ norm of functions $(\Gamma v)_\pm$, when restricted to moderate frequencies less or equal than $t^\sigma$, for some small $\sigma>0$, decays in time at the same sharp rate $t^{-1}$ of the two-dimensional linear Klein-Gordon solution. This result is proved in lemma \[Lem\_appendix: sharp\_est\_VJ\] under the hypothesis that a-priori estimates are satisfied in some fixed interval $[1,T]$, with $\varepsilon_0<(2A+B)^{-1}$ and $0<\delta\ll \delta_2\ll\delta_1\ll \delta_0\ll 1$ sufficiently small, and is fundamental when proving lemmas \[Lem: L2 est nonlinearities\] and \[Lem:L2 est nonlinearity Dt\]. All the other lemmas of this section are to be meant as preparatory intermediate results. With the convention that $D=D_1$ whenever $|I_1|+|I_2|=2$, $D\in \{D_j, D_t, j=1,2\}$ otherwise, there exists a positive constant $C>0$ such that, for any $\chi\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$ small, $n=1,2$, and every $t\in [1,T]$, $$\label{xn_Qkg0_vI1uI2} \sum_{\substack{|I_1|+|I_2|\le 2 \\ |I_1|<2}}\left\| \chi(t^{-\sigma}D_x)\big[x_n Q^\mathrm{kg}_0(v^{I_1}_\pm, D u^{I_2}_\pm)\big](t,\cdot)\right\|_{L^2(dx)}\le C(A+B)B\varepsilon^2 t^{\beta+\frac{\delta+\delta_2}{2}},$$ with $\beta>0$ small such that $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. We estimate the $L^2$ norms in the left hand side of separately. $\bullet$ When $|I_1|=0$, $|I_2|=2$, we derive from and that $$\begin{split} \left\| \chi(t^{-\sigma}D_x)\big[x_n Q^\mathrm{kg}_0(v_\pm, D_1 u^{I_2}_\pm)\big](t,\cdot)\right\|_{L^2(dx)} &\lesssim \sum_{|\mu|=0}^1\left\| x_n \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty}\|u^{I_2}_\pm(t,\cdot)\|_{H^1}\\ &\le C(A+B)B\varepsilon t^\frac{\delta_1+\delta_2}{2}; \end{split}$$ $\bullet$ When $|I_1|=|I_2|=1$ and $\Gamma^{I_2}\in \{\Omega,Z_m, m=1,2\}$ is a Klainerman vector field we use inequalities with $L=L^2$, $w_{j_0} =x_n(D_x\langle D_x\rangle^{-1})^\mu v^{I_1}_\pm$ with $|\mu|=0,1,$ and $s$ large enough so that $N(s)\ge 2$, to derive that $$\begin{split} & \left\|\chi(t^{-\sigma}D_x)\left[x_n Q^\mathrm{kg}_0(v^{I_1}_\pm, D_1 u^{I_2}_\pm)\right](t,\cdot)\right\|_{L^2(dx)}\\ & \lesssim \sum_{|\mu|=0}^1\left\|\chi_1(t^{-\sigma}D_x)\Big[ x_n \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v^{I_1}_\pm\Big](t,\cdot)\right\|_{L^\infty}\|u^{I_2}_\pm(t,\cdot)\|_{H^1}\\ & +\sum_{\substack{|\mu|=0,1,2 \\|\nu|=0,1 }}t^{-2}\left(\|x^\mu v^{I_1}_\pm(t,\cdot)\|_{L^2} + t\|x^\nu v^{I_1}_\pm(t,\cdot)\|_{L^2}\right) \left(\|u_\pm(t,\cdot)\|_{H^s} + \|D_tu_\pm(t,\cdot)\|_{H^s}\right) \\ &\le CB^2\varepsilon^2 t^\frac{\delta_1+\delta_2}{2}, \end{split}$$ where last estimate is deduced using , , , and ; $\bullet$ When $|I_1|=|I_2|=1$ and $\Gamma^{I_2}$ is a spatial derivative we use lemma \[Lem\_appendix:L\_estimate of products\] with $L=L^2$, $w_1 =x_n(D_x\langle D_x\rangle^{-1})^\mu v^{I_1}_\pm$ with $|\mu|=0,1$, $s$ large enough so that $N(s)\ge 1$, and again estimates , and . We obtain that $$\begin{split} & \left\|\chi(t^{-\sigma}D_x)\left[x_n Q^\mathrm{kg}_0(v^{I_1}_\pm, D_1 u^{I_2}_\pm)\right](t,\cdot)\right\|_{L^2(dx)}\\ & \lesssim \sum_{|\mu|=0}^1\left\|\chi_1(t^{-\sigma}D_x)\Big[ x_n \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v^{I_1}_\pm\Big](t,\cdot)\right\|_{L^\infty}\|u_\pm(t,\cdot)\|_{H^2}\\ & +\sum_{|\mu|=0}^1t^{-1}\|x^\mu v^{I_1}_\pm(t,\cdot)\|_{L^2} \|u_\pm(t,\cdot)\|_{H^s} \le CB^2\varepsilon^2 t^\frac{\delta+\delta_1}{2}; \end{split}$$ $\bullet$ When $|I_1|+|I_2|\le 1$, by the assumption derivative $D$ can be equal to $D_x$ or to $D_t$. Then - If $|I_1|=0$, after , , and $$\begin{split} &\left\|\chi(t^{-\sigma}D_x)\big[x_n Q^\mathrm{kg}_0(v_\pm, D u^{I_2}_\pm)\big](t,\cdot) \right\|_{L^2(dx)}\\ & \lesssim \sum_{|\mu|=0}^1 \left\| x_n\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty}\left(\|u^{I_2}_\pm(t,\cdot)\|_{H^1} + \|D_tu^{I_2}_\pm(t,\cdot)\|_{L^2}\right) \le C(A+B)B\varepsilon^2 t^{\delta_2}; \end{split}$$ - If $|I_1|=1$, $|I_2|=0$, using lemma \[Lem\_appendix:L\_estimate of products\] as done above, together with , , , and a-priori estimates, we derive that $$\begin{split} &\left\|\chi(t^{-\sigma}D_x)\big[x_n Q^\mathrm{kg}_0(v^{I_1}_\pm, D u_\pm)\big] (t,\cdot)\right\|_{L^2(dx)} \\ &\lesssim \sum_{|\mu|=0}^1 \left\|\chi_1(t^{-\sigma}D_x)\Big[ x_n\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v^{I_1}_\pm(t,\cdot)\Big]\right\|_{L^\infty}\left(\|u_\pm(t,\cdot)\|_{H^1} + \|D_tu_\pm(t,\cdot)\|_{L^2}\right)\\ & + \sum_{|\mu|=0}^1t^{-1} \left\| x_n\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v^{I_1}_\pm(t,\cdot)\right\|_{L^2}\left(\|u_\pm(t,\cdot)\|_{H^s} + \|D_tu_\pm(t,\cdot)\|_{H^s}\right) \\ & \le CB^2\varepsilon^2 t^{\beta+\frac{\delta+\delta_1}{2}}. \end{split}$$ \[Lem\_appendix:est vI I=2\] There exists a positive constant $C>0$ such that, for any $\chi\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$ small, $\rho\in\mathbb{N}$, and every $t\in [1,T]$, $$\label{est:Linfty_vI_I=2-new} \sum_{|I|=2}\left\|\chi(t^{-\sigma}D_x)V^I(t,\cdot)\right\|_{H^{\rho,\infty}}\le C B\varepsilon t^{-1+\beta+\frac{\delta_0}{2}},$$ with $\beta>0$ small such that $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. Estimate is evidently satisfied when $\Gamma^I$ contains at least one spatial derivative thanks to lemma \[Lem\_appendix: preliminary est VJ\]. We then focus on the case when $\Gamma^I$ is the product of two Klainerman vector fields. As $v^I_+ = -\overline{v^I_{-}}$, we prove the statement for $\chi(t^{-\sigma}D_x)v^I_{-}$. Moreover, from the $L^\infty-H^{\rho,\infty}$ continuity of $\chi(t^{-\sigma}D_x)$ with norm $O(t^{\sigma\rho})$, we can assume the $H^{\rho,\infty}$ norm in replaced with the $L^\infty$ one, up to a loss $t^{\sigma\rho}$. As done in lemma \[Lem\_appendix: preliminary est VJ\], instead of proving the statement directly on $\chi(t^{-\sigma}D_x)v^I_{-}$ we do it for $\chi(t^{-\sigma}D_x)\vNFGamma$, with $\vNFGamma$ introduced in and considered here with $|I|=2$. This is justified by inequality . From definition of $\vNFGamma$, equation and equality one can check that $$\label{def_NLNF_I=2} \begin{gathered}[] [D_t + \langle D_x\rangle] \vNFGamma = \NLNF\\ \text{where } \NLNF = r^{I,\textit{NF}}_{kg}(t,x) + \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|<2}} c_{I_1,I_2}Q^\mathrm{kg}_0(v^{I_1}_\pm, D u^{I_2}_\pm), \end{gathered}$$ with $r^{I,\textit{NF}}_{kg}$ given by the same integral expression as in but with $|I|=2$ (and hence having the explicit expression ), and $c_{I_1,I_2}\in \{-1,0,1\}$, $c_{I_1,I_2}=1$ when $|I_1|+|I_2|=2$ (in which case derivative $D$ corresponds to $D_1$). It is straightforward to show that inequalities , , hold even when $|I|=2$, up to replacing $\delta_2$ with $\delta_1$. Therefore, using those latter ones together with $$\sum_{j=1}^2 \left\|\chi(t^{-\sigma}D_x)Z_j\vNFGamma(t,\cdot)\right\|_{L^2}\le CB\varepsilon t^{\frac{\delta_0}{2}},$$ which is consequence of with $k=0$ and of with $j=2$, we derive that $$\left\|\chi(t^{-\sigma}D_x)\vNFGamma(t,\cdot)\right\|_{L^\infty}\le CB\varepsilon t^\frac{\delta_0}{2} + \sum_{j=1}^2 Ct^{-1+\beta}\left\|\chi(t^{-\sigma}D_x)\left[ x_j \NLNF\right](t,\cdot)\right\|_{L^2(dx)}.$$ The only thing we need to show in order to prove the statement is hence that $$\label{xj_NLNF_I=2} \left\|\chi(t^{-\sigma}D_x)\left[ x_j \NLNF\right](t,\cdot)\right\|_{L^2(dx)}\le C(A+B)B\varepsilon^2 t^{\beta+\frac{\delta_1+\delta_2}{2}}.$$ But from and with $|I|=2$ we have that $$\begin{split} &\left\|\chi(t^{-\sigma}D_x)\left[ x_j \NLNF\right](t,\cdot)\right\|_{L^2(dx)} \lesssim \|x_j\textit{NL}^I_{kg}(t,\cdot)\|_{L^2}\left(\|u_\pm(t,\cdot)\|_{L^\infty}+ \|\mathrm{R}_1u_\pm(t,\cdot)\|_{L^\infty}\right)\\ & + \sum_{\mu=0}^1 t^\sigma \left(\|x_j^\mu v_\pm(t,\cdot)\|_{L^\infty}+\left\| x_j^\mu \frac{D_x}{\langle D_x\rangle}v_\pm(t,\cdot)\right\|_{L^\infty}\right)\|v^I_\pm(t,\cdot)\|_{L^2}\|v_\pm(t,\cdot)\|_{H^{2,\infty}} \\ & + \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|<2}}\left\| \chi(t^{-\sigma}D_x)\left[x_j Q^\mathrm{kg}_0(v^{I_1}_\pm, D u^{I_2}_\pm)\right](t,\cdot)\right\|_{L^2} \end{split}$$ so follows from a-priori estimates, , and . As $\delta_2\ll \delta_1\ll \delta_0$, that concludes that $$\label{est_Linfty_vINF_2} \left\|\chi(t^{-\sigma}D_x)\vNFGamma(t,\cdot)\right\|_{L^\infty} \le C B\varepsilon t^{-1+\beta+\frac{\delta_0}{2}}.$$ There exists a positive constant $C>0$ such that, for any multi-index $I$ of length 2, any $\chi\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$ small, $j=1,2$, and every $t\in [1,T]$ $$\label{est:Linfty_xjGammaI_v} \left\|\chi(t^{-\sigma}D_x)\left[x_j (\Gamma^Iv)_\pm\right](t,\cdot)\right\|_{L^\infty}\le CB\varepsilon t^{\beta + \frac{\delta_0}{2}},$$ with $\beta>0$ small, $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. If $\Gamma^I$ contains at least one spatial derivative is satisfied after , and . Let us then assume that $\Gamma^I$ is product of two Klainerman vector fields. From equation , equality with $w=(\Gamma^I v)_{-}$, the $L^2-L^\infty$ continuity of operator $\chi(t^{-\sigma}D_x)\langle D_x\rangle^{-1}$ with norm $O(t^\sigma)$, and the $L^\infty$ continuity of $\chi(t^{-\sigma}D_x)D_x\langle D_x\rangle^{-1}$ with norm $O(t^{\sigma})$, we derive that $$\begin{gathered} \label{xj_GammaIv_preliminary} \left\|\chi(t^{-\sigma}D_x)\left[x_j (\Gamma^Iv)_{-}\right](t,\cdot)\right\|_{L^\infty(dx)} \lesssim t^\sigma \|Z_j (\Gamma^I v)_{-}(t,\cdot)\|_{L^2}+ t\left\|\chi(t^{-\sigma}D_x)(\Gamma^I v)_{-}(t,\cdot)\right\|_{L^\infty}\\+t^{\sigma}\left\|\chi(t^{-\sigma}D_x)\left[x_j\Gamma^I \textit{NL}_{kg}\right](t,\cdot)\right\|_{L^\infty}.\end{gathered}$$ Reminding and applying lemma \[Lem\_appendix:L\_estimate of products\] with $L=L^\infty$ and $w_1 = (D_x\langle D_x\rangle^{-1})^\mu v^I_\pm$, for $|\mu|=0,1$, to the contribution coming from the first quadratic term in the right hand side of , we find that there is some $\chi_1\in C^\infty_0(\mathbb{R}^2)$ such that $$\label{xj_GammaI_NLkg_Linfty} \begin{split} \left\|\chi(t^{-\sigma}D_x)\left[x_j\Gamma^I \textit{NL}_{kg}\right](t,\cdot)\right\|_{L^\infty} &\lesssim \sum_{\mu, \nu=0}^1 \left\|\chi_1(t^{-\sigma}D_x)\left[x^\mu_j (\Gamma^I v)_\pm\right](t,\cdot)\right\|_{L^\infty}\|\mathrm{R}_1^\nu u_\pm(t,\cdot)\|_{H^{2,\infty}} \\ &+ t^{-N(s)}\sum_{\mu=0}^1 \left\|x^\mu_j(\Gamma^I v)_\pm(t,\cdot)\right\|_{L^2}\|u_\pm(t,\cdot)\|_{H^s}\\& + \sum_{\substack{(I_1,I_2)\in\mathcal{I}(I)\\ |I_1|<2}}\left\|\chi(t^{-\sigma}D_x)\left[x_jQ^\mathrm{kg}_0\left(v^{I_1}_\pm, D u^{I_2}_\pm\right)\right](t,\cdot)\right\|_{L^\infty}. \end{split}$$ Therefore, picking $s>0$ large so that $N(s)>1$ and using the $L^2-L^\infty$ continuity of $\chi_1(t^{-\sigma}D_x)$ with norm $O(t^{\sigma})$, together with the estimates , with $k=2$, along with , we find at first that $$\left\|\chi(t^{-\sigma}D_x)\left[x_j\Gamma^I \textit{NL}_{kg}\right](t,\cdot)\right\|_{L^\infty} \le CAB\varepsilon^2 t^{\frac{1}{2}+\sigma+\frac{\delta_1}{2}}.$$ Injecting the above estimate, together with and , into we derive that $$\left\|\chi(t^{-\sigma}D_x)\left[x_j (\Gamma^Iv)_{-}\right](t,\cdot)\right\|_{L^\infty} \le CB\varepsilon t^{\frac{1}{2}+\sigma+\frac{\delta_1}{2}}.$$ The above inequality holds for any $\chi\in C^\infty_0(\mathbb{R}^2)$, so injecting it into and using again a-priori estimates, , , together with the fact that $\beta+ (\delta + \delta_2)/2\le \delta_1/2$ as $\beta$ is as small as we want as long as $\sigma$ is small and $\delta, \delta_2\ll \delta_1$, we find the following enhanced estimate $$\left\|\chi(t^{-\sigma}D_x)\left[x_j\Gamma^I \textit{NL}_{kg}\right](t,\cdot)\right\|_{L^\infty}\le C(A+B)B\varepsilon^2 t^{\sigma +\frac{\delta_1}{2}}.$$ Consequently, summing up this latter one with and , we end up with . Let $\Gamma\in \mathcal{Z}$ be an admissible vector field. There exists a positive constant $C$ such that, for any $\chi\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$ small, $i,j=1,2$, and every $t\in [1,T]$, $$\label{est:Linfty_xixjGammav-new} \left\|\chi(t^{-\sigma}D_x)\left[x_i x_j (\Gamma v)_\pm(t,\cdot)\right]\right\|_{L^\infty(dx)}\le CB\varepsilon t^{1+\beta+\frac{\delta_1}{2}},$$ with $\beta>0$ such that $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. Since $(\Gamma v)_+ = -\overline{(\Gamma v)_+}$ we reduce to prove that inequality holds true for $(\Gamma v)_{-}$. Moreover, we only focus on the case where $\Gamma\in \{\Omega, Z_m, m=1,2\}$ is a Klainerman vector field, as with $\Gamma$ being a spatial derivative is simply a consequence of . We remind that $(\Gamma v)_{-}$ is solution to non-linear Klein-Gordon equation with $\Gamma^I=\Gamma$, and that the non-linearity $\Gamma\Nlkg$ is given by . Hence, multiplying $x_i$ to relation with $w=(\Gamma v)_{-}$ and making some commutations we find that $$\label{est_1_xixjGammav} \begin{split} & \left\|\chi(t^{-\sigma}D_x)\left[x_i x_j (\Gamma v)_\pm(t,\cdot)\right]\right\|_{L^\infty(dx)} \\ &\lesssim \sum_{\mu=0}^1 \left[\left\| \chi(t^{-\sigma}D_x)\left[x_i^\mu Z_j(\Gamma v)_{-}\right](t,\cdot)\right\|_{L^\infty(dx)} + t \left\| \chi(t^{-\sigma}D_x)\left[x_i^\mu (\Gamma v)_{-}\right](t,\cdot)\right\|_{L^\infty(dx)}\right] \\ & + \sum_{\mu=0}^1\left\|\chi(t^{-\sigma}D_x)\left[x_i^{\mu} x_j \Gamma \textit{NL}_{kg}\right](t,\cdot)\right\|_{L^\infty(dx)}. \end{split}$$ At first, we estimate the latter contribution in the above right hand side using that $\chi(t^{-\sigma}D_x)$ is a continuous $L^2-L^\infty$ operator with norm $O(t^{\sigma})$ together with estimates , with $s=0$, , , , : $$\label{first_estimate_xixjGammaNL} \begin{split} & \sum_{\mu=0}^1\left\|\chi(t^{-\sigma}D_x)\left[x_i^{\mu} x_j \Gamma \textit{NL}_{kg}\right](t,\cdot)\right\|_{L^\infty(dx)} \lesssim \sum_{\mu=0}^1t^{\sigma}\left\|\chi(t^{-\sigma}D_x)\left[x_i^{\mu} x_j \Gamma \textit{NL}_{kg}\right](t,\cdot)\right\|_{L^2(dx)}\\ & \lesssim \sum_{\mu_1, \mu_2, \nu=0}^1 t^{\sigma} \|x_i^{\mu_1} x_j^{\mu_2} (\Gamma v)_\pm(t,\cdot)\|_{L^2(dx)}\|\mathrm{R}^\nu_1 u_\pm(t,\cdot)\|_{H^{2,\infty}}\\ & + \sum_{\mu, |\nu|=0}^1 t^{\sigma} \left\|x_i^{\mu} x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\nu v_\pm(t,\cdot)\right\|_{L^\infty(dx)}\left(\| (\Gamma u)_\pm(t,\cdot)\|_{H^1}+\|u_\pm(t,\cdot)\|_{H^1}+\|D_tu_\pm(t,\cdot)\|_{L^2}\right)\\ & \le C(A+B)B \varepsilon^2 t^{\frac{3}{2}+\sigma+\frac{\delta_2}{2}}. \end{split}$$Injecting this estimate, along with , , and , into we deduce that for any smooth cut-off function $\chi$ $$\label{first_est_xixjGammav} \left\|\chi(t^{-\sigma}D_x)\left[x_i x_j (\Gamma v)_\pm(t,\cdot)\right]\right\|_{L^\infty(dx)} \le CB\varepsilon t^{\frac{3}{2}+\sigma+\frac{\delta_2}{2}}.$$ Now, if we change the approach of bounding the $L^\infty(dx)$ norm of $x_i^{\mu}x_j Q^\mathrm{kg}_0((\Gamma v)_{-}, D_1u_\pm)$, which is one of the contributions to $x_i^{\mu}x_j\Gamma\textit{NL}_{kg}$ after , and make use of lemma \[Lem\_appendix:L\_estimate of products\] with $L=L^\infty$ instead of , we see that $$\begin{split} &\sum_{\mu=0}^1 \left\|\chi(t^{-\sigma}D_x)\left[x_i^{\mu} x_j \Gamma \textit{NL}_{kg}\right](t,\cdot)\right\|_{L^\infty(dx)}\\ & \lesssim \sum_{\mu_1, \mu_2, \nu=0}^1\left\| \chi_1(t^{-\sigma}D_x)\left[x_i^{\mu_1}x_j^{\mu_2} (\Gamma v)_\pm\right](t,\cdot)\right\|_{L^\infty(dx)}\|\mathrm{R}^\nu_1u_\pm(t,\cdot)\|_{H^{2,\infty}}\\ & + \sum_{\mu_1, \mu_2=0}^1 t^{-N(s)}\|x_i^{\mu_1}x_j^{\mu_2} (\Gamma v)_\pm(t,\cdot)\|_{L^2(dx)}\|u_\pm(t,\cdot)\|_{H^s}\\ & + \sum_{\mu, |\nu|=0}^1 t^{\sigma} \left\|x_i^{\mu} x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\nu v_\pm(t,\cdot)\right\|_{L^\infty(dx)}\left(\| (\Gamma u)_\pm(t,\cdot)\|_{H^1}+\|u_\pm(t,\cdot)\|_{H^1}+\|D_tu_\pm(t,\cdot)\|_{L^2}\right). \end{split}$$ Then, choosing $s>0$ sufficiently large so that $N(s)\ge 3$ and using again , , with $k=1$, , , , together with , we obtain that $$\sum_{\mu=0}^1 \left\|\chi(t^{-\sigma}D_x)\left[x_i^{\mu} x_j \Gamma \textit{NL}_{kg}\right](t,\cdot)\right\|_{L^\infty(dx)} \le C(A+B)B\varepsilon^2 t^{1+\sigma+\frac{\delta_2}{2}},$$ which enhances of a factor $t^{1/2}$. Combining the above estimate with , , and , we finally end up with the result of the statement. Let $\Gamma\in \{\Omega, Z_m, m=1,2\}$ be a Klainerman vector field, $\vNFGamma$ the function defined in with $|I|=1$ and $\Gamma^I=\Gamma$, and $B^k_{(j_1,j_2,j_3)}(\xi,\eta)$ the multiplier introduced in (resp. in ) for any $k=1,2$ (resp. $k=3$), any $j_i\in \{+,-\}$ for $i=1,2,3$. Let us define $$\label{def_VNF} \begin{split} \VNF(t,x):= \vNFGamma(t,x) &-\frac{i}{4(2\pi)^2}\sum_{j_1,j_2\in \{+,-\}}\int e^{ix\cdot\xi}B^1_{(j_1,j_2,+)}(\xi,\eta) \hat{v}_{j_1}(\xi-\eta) \widehat{(\Gamma u)_{j_2}}(\eta) d\xi d\eta\\ & +\delta_{\Omega}\frac{i}{4(2\pi)^2}\sum_{j_1,j_2\in \{+,-\}}\int e^{ix\cdot\xi}B^2_{(j_1,j_2,+)}(\xi,\eta) \hat{v}_{j_1}(\xi-\eta) \hat{u}_{j_2}(\eta) d\xi d\eta \\ & + \delta_{Z_1}\frac{i}{4(2\pi)^2}\sum_{j_1,j_2\in \{+,-\}}\int e^{ix\cdot\xi}B^3_{(j_1,j_2,+)}(\xi,\eta) \hat{v}_{j_1}(\xi-\eta) \hat{u}_{j_2}(\eta) d\xi d\eta, \end{split}$$ where $\delta_\Omega$ (resp. $\delta_{Z_1}$) is equal to 1 if $\Gamma=\Omega$ (resp. if $\Gamma=Z_1$), 0 otherwise. There exists a constant $C>0$ such that, for any $\chi\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$, and every $t\in [1,T]$, $$\label{est:VNF-(Gammav)} \left\|\chi(t^{-\sigma}D_x)\big(\VNF - (\Gamma v)_{-}\big)(t,\cdot) \right\|_{L^\infty} \le C(A+B)A\varepsilon^2 t^{-\frac{5}{4}}.$$ Moreover, for every $m=1,2$ and $t\in [1,T]$ $$\label{est:L2_Zm(VNF-Gammav)} \left\||\chi(t^{-\sigma}D_x)Z_m\big(\VNF - (\Gamma v)_{-}\big)(t,\cdot) \right\|_{L^2} \le C(A+B)B\varepsilon^2 t^{3\sigma +\delta_2}.$$ From definition of $\VNF$ and equalities , , we find that $$\label{explicit VNF - VJ} \begin{split} \VNF - (\Gamma v)_{-} &= \vNFGamma - (\Gamma v)_{-}\\ & -\frac{i}{2}\left[(D_t v)(D_1 \Gamma u) - (D_1 v)(D_t \Gamma u) + D_1[v (D_t \Gamma u)] - \langle D_x\rangle [v (D_1 \Gamma u)] \right] \\ & +\delta_{\Omega} \frac{i}{2}\left[(D_tv)(D_2u) - (D_2v)(D_tu) + D_2[v D_tu] - \langle D_x\rangle [v D_2u] \right] \\ & +\delta_{Z_1} \frac{i}{2}\left[(D_t v)(D_t u) + v( |D_x|^2u )- \langle D_x\rangle[v (D_tu)]\right], \end{split}$$ where $\vNFGamma - (\Gamma v)_{-}$ has the explicit expression . We use , and lemma \[Lem\_appendix:L\_estimate of products\] with $L=L^\infty$, $w_1=\mathrm{R}_1^\mu u_\pm$ (resp. $w_1=\mathrm{R}_1^\mu(\Gamma u)_\pm$) for $\mu=0,1$, and $s>0$ large enough to have $N(s)\ge 2$, in order to estimate the $L^\infty$ norm of products appearing in (resp. in the second line in the above right hand side). For some new $\chi_1\in C^\infty_0(\mathbb{R}^2)$ we have that $$\begin{split} &\left\|\chi(t^{-\sigma}D_x)\big(\VNF - (\Gamma v)_{-}\big)(t,\cdot) \right\|_{L^\infty} \lesssim \left\|\chi(t^{-\sigma}D_x)\big(\vNFGamma - (\Gamma v)_{-}\big)(t,\cdot) \right\|_{L^\infty} \\ & +\sum_{|\mu|=0}^1 t^\sigma \|v_\pm(t,\cdot)\|_{H^{1,\infty}}\left\|\chi_1(t^{-\sigma}D_x) \mathrm{R}^\mu_1 (\Gamma u)_\pm(t,\cdot)\right\|_{L^\infty} + t^{-2}\|v_\pm(t,\cdot)\|_{H^s} \|(\Gamma u)_\pm(t,\cdot)\|_{L^2}\\ & + \sum_{|\mu|=0}^1 t^\sigma \|v_\pm(t,\cdot)\|_{H^{1,\infty}} \|\mathrm{R}^\mu_1 u_\pm(t,\cdot)\|_{H^{2,\infty}}, \end{split}$$ with $$\begin{split} \left\|\chi(t^{-\sigma}D_x)\big(\vNFGamma - (\Gamma v)_{-}\big)(t,\cdot) \right\|_{L^\infty} &\lesssim \sum_{\mu=0}^1 t^\sigma \left\|\chi_1(t^{-\sigma}D_x)(\Gamma v)_\pm(t,\cdot)\right\|_{L^\infty}\|\mathrm{R}^\mu_1 u_\pm(t,\cdot)\|_{L^\infty}\\ & + t^{-2}\|(\Gamma v)_\pm(t,\cdot)\|_{L^2}\|u_\pm(t,\cdot)\|_{H^s}. \end{split}$$ Estimate follows then from , and . In order to derive we apply $Z_m$ to and use the Leibniz rule, reminding formulas . Among the quadratic terms coming out from the action of $Z_m$ on the second line in we see appear products where $Z_m$ is acting on $v$ and $\Gamma$ on $u$. We estimate the $L^2$ norm of those ones, when truncated by operator $\chi(t^{-\sigma}D_x)$, using inequalities with $L=L^2$, $w=u$, $w_{j_0}=(D_x\langle D_x\rangle^{-1})^\mu Z_m v$ for $|\mu|=0,1$, and $s>0$ large enough to have $N(s)>1$. We bound instead the $L^2$ norm of all other remaining products with the $L^\infty$ norm of factor that does not contain any vector field times the $L^2$ norm of the remaining one. Hence $$\begin{split} &\left\|\chi(t^{-\sigma}D_x)Z_m\big(\VNF - (\Gamma v)_{-}\big)(t,\cdot)\right\|_{L^2} \lesssim \left\|\chi(t^{-\sigma}D_x)Z_m\big(\vNFGamma - (\Gamma v)_{-}\big)(t,\cdot)\right\|_{L^2} \\ & + t^\sigma \left\|\chi_1(t^{-\sigma}D_x)(Z_m v)_\pm(t,\cdot)\right\|_{L^\infty}\|(\Gamma u)_\pm(t,\cdot)\|_{L^2}\\ & + t^{-N(s)}\Big(\sum_{|\mu|=0}^1\|x^\mu (Z_mv)_\pm(t,\cdot)\|_{L^2}+t\|(Z_m v)_\pm(t,\cdot)\|_{L^2}\Big)\left(\|u_\pm(t,\cdot)\|_{H^s}+\|D_tu_\pm(t,\cdot)\|_{H^s}\right)\\ &+t^\sigma \| v_\pm(t,\cdot)\|_{H^{1,\infty}}\left(\|(Z_m \Gamma u)_\pm(t,\cdot)\|_{L^2} + \|(\Gamma u)_\pm(t,\cdot)\|_{L^2}+ \|D_t(\Gamma u)_\pm(t,\cdot)\|_{L^2}\right)\\ & + \sum_{|\mu|=0}^1t^\sigma\|(\Gamma v)_\pm(t,\cdot)\|_{L^2}\|\mathrm{R}^\mu u_\pm(t,\cdot)\|_{H^{2,\infty}} \\ &+ t^\sigma\|v_\pm(t,\cdot)\|_{H^{1,\infty}}\left(\|(Z_m u)_\pm(t,\cdot)\|_{H^1}+ \|u_\pm(t,\cdot)\|_{H^1}+ \|D_tu_\pm(t,\cdot)\|_{L^2}\right), \end{split}$$ and estimate is obtained from , , , , with $j=1$, and . \[Lem\_appendix: LvtildeGamma\] Let $\Gamma\in \{\Omega, Z_m, m=1,2\}$ be a Klainerman vector field, $\VNF$ the function defined in and $$\label{def_VtildeGamma} \widetilde{V}^\Gamma(t,x) :=t\VNF(t,tx).$$ There exists a positive constant $C>0$ such that, for any $\chi \in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$ small, and every $t\in [1,T]$, $$\begin{gathered} \left\| \widetilde{V}^\Gamma(t,\cdot)\right\|_{L^2}\le CB\varepsilon t^\frac{\delta_2}{2}, \label{est:VtildeGamma}\\ \sum_{|\mu|=1}\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu \widetilde{V}^\Gamma(t,\cdot)\right\|_{L^2}\le CB\varepsilon t^\frac{\delta_1}{2}.\label{est:Lcal_VtildeGamma}\end{gathered}$$ Let us recall equalities with $\Gamma^I=\Gamma$ and . From a-priori estimates we immediately derive that, for every $t\in [1,T]$, $$\|[\VNF -(\Gamma v)_{-}](t,\cdot)\|_{L^2} \le CAB\varepsilon t^{-\frac{1}{2}+\frac{\delta_2}{2}+\sigma},$$ and consequently that $$\label{est_L2_VNF} \|\widetilde{V}^\Gamma(t,\cdot)\|_{L^2}=\|\VNF(t,\cdot)\|_{L^2}\le CB\varepsilon t^{\frac{\delta_2}{2}}.$$ Using definition one can check that $\VNF$ is solution to $$\label{half KG VNF} [D_t + \langle D_x\rangle]\VNF(t,x) = \NLT(t,x) - \delta_{Z_1} Q^\mathrm{kg}_0\big(v_\pm, Q^\mathrm{w}_0(v_\pm, D_1 v_\pm)\big)$$ with $$\label{def_NLT} \begin{split} \NLT(t,x)& = r^{I,\textit{NF}}_{kg}(t,x)\\ &- \frac{i}{4(2\pi)^2}\int e^{ix\cdot\xi}B^1_{(j_1,j_2,+)}(\xi,\eta) \left[\widehat{\textit{NL}_{kg}}(\xi-\eta) \widehat{(\Gamma u)_{j_2}}(\eta) - \hat{v}_{j_1}(\xi-\eta){\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\Gamma\textit{NL}_{w}}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\Gamma\textit{NL}_{w}}{\tmpbox}}(\eta)\right] d\xi d\eta \\ & + \delta_{\Omega} \frac{i}{4(2\pi)^2}\int e^{ix\cdot\xi}B^2_{(j_1,j_2,+)}(\xi,\eta) \left[\widehat{\textit{NL}_{kg}}(\xi-\eta) \hat{u}_{j_2}(\eta) - \hat{v}_{j_1}(\xi-\eta){\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\textit{NL}_{w}}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\textit{NL}_{w}}{\tmpbox}}(\eta)\right] d\xi d\eta\\ &+ \delta_{Z_1} \frac{i}{4(2\pi)^2}\int e^{ix\cdot\xi}B^3_{(j_1,j_2,+)}(\xi,\eta) \left[\widehat{\textit{NL}_{kg}}(\xi-\eta) \hat{u}_{j_2}(\eta) - \hat{v}_{j_1}(\xi-\eta){\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\textit{NL}_{w}}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\textit{NL}_{w}}{\tmpbox}}(\eta)\right] d\xi d\eta, \end{split}$$ and $r^{I,\textit{NF}}_{kg}$ given by (or, explicitly, by ) with $|I|=1$. Superscript *c* in $\NLT$ stands for *cubic* and wants to stress out the fact that, passing from function $(\Gamma v)_{-}$ to $\VNF$, we have replaced all quadratic terms in the right hand side of (when $|I|=1$ and $\Gamma^I=\Gamma$) with cubic ones. Hence, from relation with $w=\VNF$ and equation we get that $$\label{ineq:Lcal_VTildeGamma_proof} \begin{split} &\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m \widetilde{V}^\Gamma(t,\cdot)\right\|_{H^1} \lesssim \|\chi(t^{-\sigma}D_x)Z_m\VNF(t,\cdot)\|_{L^2} + \left\|\oph(\chi(h^\sigma\xi)\xi_m\langle\xi\rangle^{-1})\widetilde{V}^\Gamma(t,\cdot)\right\|_{L^2}\\ & + \left\|\chi(t^{-\sigma}D_x)\left[x_m\NLT\right](t,\cdot) \right\|_{L^2(dx)} + \delta_{Z_1}\left\|\chi(t^{-\sigma}D_x)\left[x_mQ^\mathrm{kg}_0\big(v_\pm, Q^\mathrm{w}_0(v_\pm, D_1 v_\pm)\big)\right](t,\cdot) \right\|_{L^2(dx)} \end{split}$$ After with $k=1$,, and the fact that $\sigma$ can be chosen sufficiently small so that $3\sigma+\delta_2\le \delta_1/2$, as $\delta_2\ll \delta_1$, it is straightforward to see that $$\label{est:L2_ZmVNF} \|\chi(t^{-\sigma}D_x)Z_m\VNF(t,\cdot)\|_{L^2} \le CB\varepsilon t^\frac{\delta_1}{2}.$$ Moreover, from , and a-priori estimates, $$\begin{gathered} \label{est:L2_xj_cubic} \left\|\chi(t^{-\sigma}D_x)\left[x_mQ^\mathrm{kg}_0\big(v_\pm, Q^\mathrm{w}_0(v_\pm, D_1 v_\pm)\big)\right](t,\cdot) \right\|_{L^2(dx)} \\\lesssim \sum_{|\mu|=0}^1 \left\|x_n\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot) \right\|_{L^\infty(dx)} \|\textit{NL}_w(t,\cdot)\|_{L^2} \le C(A+B)AB \varepsilon^3 t^{-1+\frac{\delta+\delta_2}{2}}.\end{gathered}$$ Using instead equalities and we derive the following explicit expression for $\NLT$: $$\label{explicit NLT} \begin{split} \NLT(t,x)= r^{I,\textit{NF}}_{kg}(t,x) &- \frac{i}{2}\left[\textit{NL}_{kg}(D_1 \Gamma u) - (D_1v)\Gamma\textit{NL}_w + D_1[v \Gamma\textit{NL}_w]\right] \\ & + \delta_\Omega \frac{i}{2}\left[\textit{NL}_{kg}(D_2 u) - (D_2v)\textit{NL}_w+ D_2[v \textit{NL}_w]\right] \\ &+\delta_{Z_1}\left[\textit{NL}_{kg}(D_tu) + (D_tv)\textit{NL}_w - \langle D_x\rangle[v\textit{NL}_w]\right]. \end{split}$$ Hence, reminding estimates , , with $s=0$, , , and equality from which follows that $$\label{L2_norm_NLIw} \|\Gamma\textit{NL}_w(t,\cdot)\|_{L^2}\lesssim \|v_\pm(t,\cdot)\|_{H^{1,\infty}}\left(\|v^I_\pm(t,\cdot)\|_{H^1}+\|v_\pm(t,\cdot)\|_{H^1}+\|D_tv_\pm(t,\cdot)\|_{L^2}\right),$$ we find that $$\label{first_estimate_xNLT} \begin{split} &\left\|\chi(t^{-\sigma}D_x)\left[x_j \NLT\right](t,\cdot)\right\|_{L^2(dx)} \lesssim \left\|\chi(t^{-\sigma}D_x)\left[x_j r^{I,\textit{NF}}_{kg}(t,\cdot)\right](t,\cdot)\right\|_{L^2(dx)} \\& + \sum_{|\mu|, \nu=0}^1 \left\|x_j\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot) \right\|_{L^\infty(dx)} \|\mathrm{R}^\nu u_\pm(t,\cdot)\|_{H^{2,\infty}}\left(\|(\Gamma u)_\pm(t,\cdot)\|_{L^2}+ \|u_\pm(t,\cdot)\|_{L^2}\right) \\ & + \sum_{k,|\mu|=0}^1 \left\|x_j^k\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot) \right\|_{L^\infty(dx)} \left(\|\Gamma\textit{NL}_w(t,\cdot)\|_{L^2}+\|\textit{NL}_w(t,\cdot)\|_{L^2}\right)\\ & \le C(A+B)AB \varepsilon^2 t^{-\frac{1}{2}+\beta+\frac{\delta+\delta_1}{2}}. \end{split}$$ By injecting the above estimate, together with , , , into we finally deduce and conclude the proof of the statement. \[Lem\_appendix:est\_Zm(vINF-Gammav)\] Let $\Gamma\in \{\Omega, Z_m, m=1,2\}$ be a Klainerman vector field and $I_1, I_2$ two multi-indices such that $\Gamma^{I_1}=\Gamma$ and $\Gamma^{I_2}=Z_m\Gamma$, with $m\in\{1,2\}$. Let also $v^{I,\textit{NF}}$ be the function defined in for a generic multi-index $I$ of length equal to 1 or 2. There exists a constant $C>0$ such that, for any $\chi\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$ small, $m,n=1,2$, every $t\in [1,T]$, \[est:L2\_Zn(vINF-Gammav)\_xmZN(vINF-Gammav)\] $$\begin{gathered} \label{est:L2_Zn(vINF-Gammav)} \left\|\chi(t^{-\sigma}D_x)\left[Z_m \left(v^{I_1,\textit{NF}} - (\Gamma v)_{-}\right)\right](t,\cdot) \right\|_{L^2} + \left\|\chi(t^{-\sigma}D_x)\left(v^{I_2,\textit{NF}} - (Z_m\Gamma v)_{-}\right)(t,\cdot) \right\|_{L^2}\\ \le C(A+B)B\varepsilon^2 t^{-1+\beta + \frac{\delta+\delta_1+\delta_2}{2}}\end{gathered}$$ and $$\begin{gathered} \label{est:L2_xmZn(vINF-Gammav)} \left\|\chi(t^{-\sigma}D_x)\left[x_nZ_m \left(v^{I_1,\textit{NF}} - (\Gamma v)_{-}\right)\right](t,\cdot) \right\|_{L^2}+\left\|\chi(t^{-\sigma}D_x)\left[x_n\left(v^{I_2,\textit{NF}} - (Z_m\Gamma v)_{-}\right)\right](t,\cdot) \right\|_{L^2} \\ \le C(A+B)B\varepsilon^2 t^{\beta + \frac{\delta+\delta_1+\delta_2}{2}},\end{gathered}$$ with $\beta>0$ small such that $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. Moreover, if $\VNF$ is the function defined in , then for every $t\in [1,T]$ \[est:L2\_Zm(VNF-Gammav)\_xnZm(VNF-Gammav)\] $$\begin{aligned} \left\|\chi(t^{-\sigma}D_x)\left[Z_m \left(\VNF - (\Gamma v)_{-}\right)\right](t,\cdot) \right\|_{L^2} &\le C(A+B)B\varepsilon^2 t^{-1+\beta + \frac{\delta+\delta_1+\delta_2}{2}}, \label{est:L2_Zm(VNF-Gammav)_enhanced} \\ \left\|\chi(t^{-\sigma}D_x)\left[x_nZ_m \left(\VNF - (\Gamma v)_{-}\right)\right](t,\cdot) \right\|_{L^2} &\le C(A+B)B\varepsilon^2 t^{\beta + \frac{\delta+\delta_1+\delta_2}{2}}. \label{est:L2_xnZm(VNF-Gammav)}\end{aligned}$$ We warn the reader that throughout the proof we denote by $C$ and $\beta$ two positive constants that may change line after line, with $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. We refer to equality with $I=I_1$ and bound the $L^2$ norm of each product in the first, third and fifth line of its right hand side by means of lemma \[Lem\_appendix:L\_estimate of products\] with $L=L^2$. The $L^2$ norm of the remaining products in the second line of the mentioned equality is instead estimated using inequalities with $L=L^2$ and $w_{j_0}=(D_x\langle D_x\rangle^{-1})^\mu (\Gamma^{I_1}v)_\pm$. In this way we obtain that there is some $\chi_1\in C^\infty_0(\mathbb{R}^2)$ such that $$\begin{split} &\left\|\chi(t^{-\sigma}D_x)\left[Z_m \left(v^{I_1,\textit{NF}} - (\Gamma v)_{-}\right)\right](t,\cdot) \right\|_{L^2} \lesssim t^\sigma \|\chi_1(t^{-\sigma}D_x)(Z_m \Gamma v)_\pm(t,\cdot)\|_{L^\infty}\|u_\pm(t,\cdot)\|_{L^2} \\ &+ t^{-N(s)}\|(Z_m\Gamma v)_\pm(t,\cdot)\|_{L^2}\|u_\pm(t,\cdot)\|_{H^s} + t^\sigma \|\chi_1(t^{-\sigma}D_x)(\Gamma v)_\pm(t,\cdot)\|_{L^\infty} \|(Z_mu)_\pm(t,\cdot)\|_{L^2}\\ & +t^{-N(s)} \left(\sum_{\mu=0}^1\|x_m^\mu (\Gamma v)_\pm(t,\cdot)\|_{L^2} + t\|(\Gamma v)_\pm(t,\cdot)\|_{L^2}\right) \left(\|u_\pm(t,\cdot)\|_{H^s}+ \|D_tu_\pm(t,\cdot)\|_{H^s}\right)\\ &+ t^\sigma\|\chi_1(t^{-\sigma}D_x)(\Gamma v)_\pm(t,\cdot)\|_{L^\infty}\left(\|u_\pm(t,\cdot)\|_{L^2} + \|D_t u_\pm(t,\cdot)\|_{L^2}\right) \\ & +t^{-N(s)}\|(\Gamma v)_\pm(t,\cdot)\|_{L^2}\left(\|u_\pm(t,\cdot)\|_{H^s}+ \|D_t u_\pm(t,\cdot)\|_{H^s}\right). \end{split}$$ Choosing $s>0$ large so that $N(s)>1$ and using estimates , , , together with lemmas \[Lem\_appendix: preliminary est VJ\] and \[Lem\_appendix:est vI I=2\], we hence find that $$\left\|\chi(t^{-\sigma}D_x)\left[Z_m \left(v^{I_1,\textit{NF}} - (\Gamma v)_{-}\right)\right](t,\cdot) \right\|_{L^2} \le C(A+B)B\varepsilon^2 t^{-1+\beta + \frac{\delta+\delta_1+\delta_2}{2}}.$$ Analogously,$$\begin{split} &\left\|\chi(t^{-\sigma}D_x)\left[x_n Z_m \left(v^{I_1,\textit{NF}} - (\Gamma v)_{-}\right)\right](t,\cdot) \right\|_{L^2} \lesssim t^\sigma \|\chi_1(t^{-\sigma}D_x)\left[x_n(Z_m \Gamma v)_\pm\right](t,\cdot)\|_{L^\infty} \|u_\pm(t,\cdot)\|_{L^2}\\ &+ t^{-N(s)}\|x_n(Z_m\Gamma v)_\pm(t,\cdot)\|_{L^2}\|u_\pm(t,\cdot)\|_{H^s} + t^\sigma \|\chi_1(t^{-\sigma}D_x)\left[x_n(\Gamma v)_\pm\right](t,\cdot)\|_{L^\infty} \|(Z_mu)_\pm(t,\cdot)\|_{L^2}\\ & +t^{-N(s)} \left(\sum_{\mu=0}^1\|x_m^\mu x_n (\Gamma v)_\pm(t,\cdot)\|_{L^2} + t\|x_n(\Gamma v)_\pm(t,\cdot)\|_{L^2}\right) \left(\|u_\pm(t,\cdot)\|_{H^s}+ \|D_tu_\pm(t,\cdot)\|_{H^s}\right)\\ & + t^\sigma \|\chi_1(t^{-\sigma}D_x)\left[x_n(\Gamma v)_\pm\right](t,\cdot)\|_{L^\infty}\left(\|u_\pm(t,\cdot)\|_{L^2} + \|D_t u_\pm(t,\cdot)\|_{L^2}\right)\\ &+ t^{-N(s)}\|x_n(\Gamma v)_\pm(t,\cdot)\|_{L^2}\left(\|u_\pm(t,\cdot)\|_{H^s}+ \|D_t u_\pm(t,\cdot)\|_{H^s}\right),\\ \end{split}$$ so from , , , , and , we derive that $$\left\|\chi(t^{-\sigma}D_x)\left[x_n Z_m \left(v^{I_1,\textit{NF}} - (\Gamma v)_{-}\right)\right](t,\cdot) \right\|_{L^2} \le C(A+B)B\varepsilon^2 t^{\beta+\frac{\delta+\delta_1+\delta_2}{2}}.$$ Inequalities follows then just by the observation that, after the hypothesis on multi-indices $I_1,I_2$ and the comparison between with $I=I_2$ and with $I=I_1$, $2i\chi(t^{-\sigma}D_x)\left(v^{I_2,\textit{NF}} - (Z_m\Gamma v)_{-}\right)$ corresponds to the first line in the right hand side of . In order to derive estimate we apply $Z_m$ to both sides of equality , use , formulas , and successively proceed as follows: products in which $Z_m$ acts on $v$ and $\Gamma$ on $u$, that arise from the action of $Z_m$ on the second line of , are estimated using inequalities with $L=L^2$ and $w=u$; products in which $Z_m$ is acting on $v$ and there are no Klainerman vector fields acting on $u$ are estimated applying lemma \[Lem\_appendix:L\_estimate of products\] with $L=L^2$; the $L^2$ norm of the remaining ones are controlled by the $L^\infty$ norm of the Klein-Gordon factor times the $L^2$ norm of the wave one. In this way we get that $$\begin{split} & \left\|\chi(t^{-\sigma}D_x)\left[Z_m \left(\VNF - (\Gamma v)_{-}\right)\right](t,\cdot) \right\|_{L^2} \lesssim \left\|\chi(t^{-\sigma}D_x)\left[Z_m \left(v^{I_1,\textit{NF}} - (\Gamma v)_{-}\right)\right](t,\cdot) \right\|_{L^2} \\ & +t^\sigma \left\|\chi_1(t^{-\sigma}D_x)(Z_m v)_\pm(t,\cdot)\right\|_{L^\infty}\left( \|(\Gamma u)_\pm(t,\cdot)\|_{L^2} + \|u_\pm(t,\cdot)\|_{L^2}\right)\\ & + t^{-N(s)}\left(\sum_{|\mu|=0}^1 \|x^\mu (Z_m v)_\pm(t,\cdot)\|_{L^2}+ t\|(Z_mv)_\pm(t,\cdot)\|_{L^2}\right)\left(\|u_\pm(t,\cdot)\|_{H^s}+\|D_tu_\pm(t,\cdot)\|_{H^s}\right) \\ & + \|v_\pm(t,\cdot)\|_{H^{1,\infty}}\left(\|(Z_m\Gamma u)_\pm(t,\cdot)\|_{L^2}+ \|(\Gamma u)_\pm(t,\cdot)\|_{L^2} + \|D_t(\Gamma u)_\pm(t,\cdot)\|_{L^2}\right.\\ &\hspace{9cm}\left. + \|u_\pm(t,\cdot)\|_{L^2} +\|D_tu_\pm(t,\cdot)\|_{L^2} \right). \end{split}$$ Choosing $s>0$ large so that $N(s)>2$ and using , , with $k=1$, , , we hence recover . An analogous procedure leads us to the following inequality $$\begin{split} & \left\|\chi(t^{-\sigma}D_x)\left[x_n Z_m \left(\VNF - (\Gamma v)_{-}\right)\right](t,\cdot) \right\|_{L^2} \lesssim \left\|\chi(t^{-\sigma}D_x)\left[x_n Z_m \left(v^{I_1,\textit{NF}} - (\Gamma v)_{-}\right)\right](t,\cdot) \right\|_{L^2} \\ & +t^\sigma \left\|\chi_1(t^{-\sigma}D_x)\left[x_n(Z_m v)_\pm\right](t,\cdot)\right\|_{L^\infty}\left( \|(\Gamma u)_\pm(t,\cdot)\|_{L^2} + \|u_\pm(t,\cdot)\|_{L^2}\right)\\ & + t^{-N(s)}\left(\sum_{|\mu|=0}^1 \|x^\mu x_n(Z_m v)_\pm(t,\cdot)\|_{L^2}+ t\|x_n (Z_mv)_\pm(t,\cdot)\|_{L^2}\right)\left(\|u_\pm(t,\cdot)\|_{H^s}+\|D_tu_\pm(t,\cdot)\|_{H^s}\right) \\ & +\sum_{|\mu|=0}^1 \left\|x_n \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty}\left(\|(Z_m\Gamma u)_\pm(t,\cdot)\|_{L^2}+ \|(\Gamma u)_\pm(t,\cdot)\|_{L^2} + \|D_t(\Gamma u)_\pm(t,\cdot)\|_{L^2}\right.\\ &\hspace{10cm}\left. + \|u_\pm(t,\cdot)\|_{L^2} +\|D_tu_\pm(t,\cdot)\|_{L^2} \right) , \end{split}$$ and estimate is obtained by choosing $s>0$ large so that $N(s)>1$ and using , , , with $k=1$, , , and a-priori estimates. \[Lem\_appendix:Lm\_ZnVNF\] Let $\Gamma\in \{\Omega, Z_m, m=1,2\}$ be a Klainerman vector field and $\VNF$ be the function defined in . There exists a constant $C>0$ such that, for any $\chi\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$ small, $m,n=1,2$, and every $t\in [1,T]$, $$\label{est_Lm_ZnVNF} \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m\left[tZ_n\VNF(t,tx)\right]\right\|_{L^2(dx)}\le CB\varepsilon t^\frac{\delta_0}{2}.$$ We warn the reader that, throughout the proof, we denote by $C$ and $\beta$ two positive constants that may change line after line, with $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. Let $\vNFGamma$ be the function defined in for a generic multi-index $I$ of length 1 or 2, and $I_1,I_2$ two multi-indices such that $\Gamma^{I_1}=\Gamma$, $\Gamma^{I_2}=Z_n\Gamma$. Using we rewrite $Z_n\VNF$ as follows: $$Z_n\VNF = Z_n\left(\VNF - (\Gamma v)_{-}\right) + \left[(Z_n\Gamma v)_{-} - v^{I_2,\textit{NF}}\right] + v^{I_2,\textit{NF}} + \frac{D_n}{\langle D_x\rangle}v^{I_1,\textit{NF}} + \frac{D_n}{\langle D_x\rangle}\left[(\Gamma v)_{-} - v^{I_1,\textit{NF}}\right]$$ so that $$\label{ineq:Lm_ZnVNF} \begin{split} &\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m\left[tZ_n\VNF(t,tx)\right]\right\|_{L^2(dx)} \\ &\lesssim \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m\left[tZ_n\left(\VNF - (\Gamma v)_{-}\right)(t,tx)\right]\right\|_{L^2(dx)} \\ & + \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m\left[t\left[(Z_n\Gamma v)_{-} - v^{I_2,\textit{NF}}\right](t,tx)\right]\right\|_{L^2(dx)}\\ & + \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m\left[t v^{I_2,\textit{NF}} (t,tx)\right]\right\|_{L^2(dx)}+ \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m\left[t\frac{D_n}{\langle D_x\rangle}v^{I_1,\textit{NF}}(t,tx)\right]\right\|_{L^2(dx)}\\ & + \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m\left[t\frac{D_n}{\langle D_x\rangle}\left[(\Gamma v)_{-} - v^{I_1,\textit{NF}}\right](t,tx)\right]\right\|_{L^2(dx)}. \end{split}$$ Since $v^{I_2,\textit{NF}}$ satisfies with $I=I_2$, we derive from relation with $w=v^{I_2,\textit{NF}}$ that $$\begin{split} &\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m\left[t v^{I_2,\textit{NF}} (t,tx)\right]\right\|_{L^2(dx)} \lesssim \left\|\chi(t^{-\sigma}D_x)Z_m(\Gamma^{I_2}v)_{-}(t,\cdot) \right\|_{L^2} \\ &+\left\|\chi(t^{-\sigma}D_x)Z_m\left[v^{I_2, \textit{NF}}- (\Gamma^{I_2}v)_{-}\right](t,\cdot) \right\|_{L^2} + \left\|\chi(t^{-\sigma}D_x)v^{I_2,\textit{NF}}(t,\cdot) \right\|_{L^2} \\ &+ \left\| \chi(t^{-\sigma}D_x)\left[x_m \textit{NL}^{I_2,\textit{NF}}_{kg}\right](t,\cdot)\right\|_{L^2}. \end{split}$$ A-priori estimate with $k=0$, with $I=I_2$, , , the fact that $\delta\ll \delta_2\ll \delta_1\ll \delta_0$ and that $\beta$ is small as long as $\sigma$ is small, imply $$\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m\left[t v^{I_2,\textit{NF}} (t,tx)\right]\right\|_{L^2(dx)} \le CB\varepsilon t^\frac{\delta_0}{2}.$$ Analogously, commutating $\mathcal{L}_m$ with $\oph(\xi_n \langle \xi\rangle^{-1})$, using with $w=v^{I_1,\textit{NF}}$ and the fact that $v^{I_1,\textit{NF}}$ is solution to with non-linear term given by , together with inequalities , , , we derive that $$\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m\left[t\frac{D_n}{\langle D_x\rangle}v^{I_1,\textit{NF}}(t,tx)\right]\right\|_{L^2(dx)} \le CB\varepsilon t^\frac{\delta_1}{2}.$$ Finally, the remaining norms in the right hand side of are estimated by the right hand side of after and lemma \[Lem\_appendix:est\_Zm(vINF-Gammav)\]. Lemmas \[Lem\_appendix: preliminary est VJ\], \[Lem\_appendix: LvtildeGamma\] and \[Lem\_appendix:Lm\_ZnVNF\] allow us to prove an analogous result to that of lemma \[Lem\_appendix:product\_Vtilde\_Utilde\], where $\widetilde{v}$ is replaced with $\widetilde{V}^\Gamma$ introduced in . \[Lem\_appendix: product\_VtildeGamma\_utilde\] Let $h=t^{-1}$, $\ut, \widetilde{V}^\Gamma$ be respectively defined in and, $a_0(\xi)\in S_{0,0}(1)$, and $b_1(\xi)=\xi_j$ or $b_1(\xi)=\xi_j\xi_k|\xi|^{-1}$, with $j,k\in \{1,2\}$. There exists a constant $C>0$ such that, for any $\chi, \chi_1\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$, and every $t\in [1,T]$, we have that $$\label{est_VtildeGamma_utilde} \big\|[\oph(\chi(h^\sigma\xi)a_0(\xi))\widetilde{V}^\Gamma(t,\cdot)][\oph(\chi_1(h^\sigma\xi)b_1(\xi))\widetilde{u}(t,\cdot)]\big\|_{L^2} \le C(A+B)B\varepsilon^2 h^{\frac{1}{2}-\beta'},$$ $$\label{est_Linfty_VtildeGamma_utilde} \big\|[\oph(\chi(h^\sigma\xi)a_0(\xi))\widetilde{V}^\Gamma(t,\cdot)][\oph(\chi_1(h^\sigma\xi)b_1(\xi))\widetilde{u}(t,\cdot)]\big\|_{L^\infty} \le C(A+B)B\varepsilon^2 h^{-\beta'},$$ with $\beta'>0$ small, $\beta\rightarrow 0$ as $\sigma,\delta_0\rightarrow 0$. The proof of this result has the same structure as that of lemma \[Lem\_appendix:product\_Vtilde\_Utilde\]. Only few differences occur due to to the fact that we are replacing $\widetilde{v}$ with $\widetilde{V}^\Gamma$. We limit here to indicate them. Instead of referring to estimate we use the fact that, after in classical coordinates, there exists a constant $C>0$ such that for any $\rho\in\N$ $$\label{est:Hrho_VtildeGamma} \left\|\oph(\chi(h^\sigma\xi))\widetilde{V}^\Gamma(t,\cdot)\right\|_{H^{\rho,\infty}}\le CB\varepsilon h^{-\beta-\frac{\delta_1}{2}},$$ with $\beta>0$ small such that $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. We successively decompose $\widetilde{V}^\Gamma$ into $\widetilde{V}^\Gamma_{\Lambda_{kg}}+\widetilde{V}^\Gamma_{\Lambda^c_{kg}}$, with $$\begin{gathered} \widetilde{V}^\Gamma_{\Lambda_{kg}}(t,x):= \oph\Big(\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)a_0(\xi)\Big)\widetilde{V}^\Gamma(t,x), \\ \widetilde{V}^\Gamma_{\Lambda^c_{kg}}(t,x):=\oph\Big((1-\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi(h^\sigma\xi)a_0(\xi)\Big)\widetilde{V}^\Gamma(t,x).\end{gathered}$$ On the one hand, from the fact that above operators are supported for frequencies $|\xi|\lesssim h^\sigma$, together with proposition \[Prop:Continuity Lp-Lp\] with $p=+\infty$ and , we have that $$\left\|\widetilde{V}^\Gamma_{\Lambda_{kg}}(t,\cdot)\right\|_{L^\infty}\le CB\varepsilon h^{-\beta-\frac{\delta_1}{2}}.$$ On the other hand, combining the analogous of with lemma \[Lem\_appendix: LvtildeGamma\] (instead of \[Lem: from energy to norms in sc coordinates-KG\]), estimates , (instead of lemma \[Lem\_appendix: ZnvNF\_LmZnvNF\]) and (instead of ), $$\left\|\widetilde{V}^\Gamma_{\Lambda^c_{kg}}(t,\cdot)\right\|_{L^\infty}\le CB\varepsilon h^{\frac{1}{2}-\beta-\frac{\delta_1}{2}}.$$ \[Lem\_appendix:xx(VNF-Gammav)\] Let $\Gamma\in \{\Omega, Z_m, m=1,2\}$ be a Klainerman vector field and $\VNF$ be the function defined in . There exists a constant $C>0$ such that, for any $\chi\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$ small, $m,n=1,2$, and every $t\in [1,T]$, \[est:x\_x\_(VNF-Gammav)\] $$\begin{aligned} \left\|\chi(t^{-\sigma}D_x)\left[x_m\left(\VNF - (\Gamma v)_{-}\right)\right](t,\cdot)\right\|_{L^\infty}& \le C(A+B)^2\varepsilon^2 t^{-\frac{1}{2}+\beta +\frac{\delta_1+\delta_2}{2}},\label{est:xn(VNF-Gammav)} \\ \left\|\chi(t^{-\sigma}D_x)\left[x_n x_m\left(\VNF - (\Gamma v)_{-}\right)\right](t,\cdot)\right\|_{L^\infty} &\le C(A+B)^2\varepsilon^2 t^{\frac{1}{2}+\beta +\frac{\delta_1+\delta_2}{2}}, \label{est:xmxn(VNF-Gammav)}\end{aligned}$$ with $\beta>0$ small such that $\beta\rightarrow 0$ as $\sigma\rightarrow 0$. We remind the reader about explicit expression of the difference $\VNF - (\Gamma v)_{-}$, and , here considered with $|I|=1$ such that $\Gamma^I=\Gamma$. We first use equalities , , and, after some commutations, multiply $x_m$ (together with $x_n$ when proving ) against each Klein-Gordon factor. Successively, we estimate the contribution coming from $\vNFGamma - (\Gamma v)_{-}$ using lemma \[Lem\_appendix:L\_estimate of products\] with $L=L^\infty$, and all products coming from the second line of by means of inequalities with $L=L^\infty$, $w=u$ and $w_{j_0}=(D_x\langle D_x\rangle^{-1})^\mu Z_m v$ for $|\mu|=0,1$. On the one hand, we obtain that $$\begin{split} & \left\|\chi(t^{-\sigma}D_x)\left[x_m\left(\VNF - (\Gamma v)_{-}\right)\right](t,\cdot)\right\|_{L^\infty} \\ &\lesssim \sum_{\mu=0}^1 t^\sigma \left\|\chi_1(t^{-\sigma}D_x)\left[x_m (\Gamma v)_\pm(t,\cdot)\right]\right\|_{L^\infty}\|\mathrm{R}^\mu_1 u_\pm(t,\cdot)\|_{L^\infty} + t^{-N(s)}\|x_m (\Gamma v)_\pm(t,\cdot)\|_{L^2}\|u_\pm(t,\cdot)\|_{H^s}\\ & + \sum_{|\mu|=0}^1 t^\sigma\left\|x_m \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm\right\|_{L^\infty}\|\chi_1(t^{-\sigma}D_x)(\Gamma u)_{-}(t,\cdot)\|_{L^\infty} \\ &+ \sum_{|\mu|=0}^2 t^{-N(s)}\|x^\mu v_\pm(t,\cdot)\|_{L^2}\left(\|u_\pm(t,\cdot)\|_{H^s}+\|D_tu_\pm(t,\cdot)\|_{H^s}\right) \\ & + \sum_{|\mu|, |\nu|=0}^1 t^\sigma\left\|x_m \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm\right\|_{L^\infty} \|\mathrm{R}^\mu u_\pm(t,\cdot)\|_{H^{2,\infty}} \end{split}$$ and estimate follows choosing $s>0$ large enough to have $N(s)\ge 2$ and using , , , with $k=1$, , . On the other hand, $$\begin{split} & \left\|\chi(t^{-\sigma}D_x)\left[x_n x_m\left(\VNF - (\Gamma v)_{-}\right)\right](t,\cdot)\right\|_{L^\infty} \\ &\lesssim \sum_{\mu=0}^1 t^\sigma \left\|\chi_1(t^{-\sigma}D_x)\left[x_n x_m (\Gamma v)_\pm(t,\cdot)\right]\right\|_{L^\infty}\|\mathrm{R}^\mu_1 u_\pm(t,\cdot)\|_{L^\infty} \\ &+ t^{-N(s)}\|x_nx_m (\Gamma v)_\pm(t,\cdot)\|_{L^2}\|u_\pm(t,\cdot)\|_{H^s}\\ & + \sum_{|\mu|=0}^1 t^\sigma\left\|x_nx_m \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm\right\|_{L^\infty}\|\chi_1(t^{-\sigma}D_x)(\Gamma u)_{-}(t,\cdot)\|_{L^\infty} \\ &+ \sum_{|\mu|=0}^3 t^{-N(s)}\|x^\mu v_\pm(t,\cdot)\|_{L^2}\left(\|u_\pm(t,\cdot)\|_{H^s}+\|D_tu_\pm(t,\cdot)\|_{H^s}\right) \\ & + \sum_{|\mu|, |\nu|=0}^1 t^\sigma\left\|x_n x_m \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm\right\|_{L^\infty} \|\mathrm{R}^\mu u_\pm(t,\cdot)\|_{H^{2,\infty}} \end{split}$$ so picking the same $s$ as before and using , , , , , and , together with a-priori estimates, we derive . \[Lem\_appendix:xxNLT\] Let $\Gamma\in \{\Omega, Z_m, m=1,2\}$ be a Klainerman vector field and $\NLT$ be given by . There exists a constant $C>0$ such that for any $\chi\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$ small, $m,n=1,2$, and every $t\in [1,T]$, $$\begin{aligned} \left\|\chi(t^{-\sigma}D_x)\left[x_n \NLT\right](t,\cdot)\right\|_{L^2} &\le C(A+B)^2B\varepsilon^3 t^{-1+\beta'}, \\ \left\|\chi(t^{-\sigma}D_x)\left[x_m x_n \NLT\right](t,\cdot)\right\|_{L^2} &\le C(A+B)^2B\varepsilon^3 t^{\beta'},\end{aligned}$$ with $\beta'>0$ small such that $\beta'\rightarrow 0$ as $\sigma, \delta_0\rightarrow 0$. Moreover, in the same time interval $$\label{est_Linfty_NLT} \left\|\chi(t^{-\sigma}D_x) \NLT(t,\cdot)\right\|_{L^\infty} \le C(A+B)^2B\varepsilon^3 t^{-\frac{5}{2}+\beta'}.$$ We warn the reader that we will denote by $C,\beta, \beta'$ some positive constants that may change line after line, with $\beta\rightarrow 0$ (resp. $\beta'\rightarrow 0$) as $\sigma\rightarrow 0$ (resp. as $\sigma,\delta_0\rightarrow 0$). For a seek of compactness we also denote by $R(t,x)$ any contribution verifying $$\label{est_L2_R} \begin{split} \left\|\chi(t^{-\sigma}D_x)\left[x_nR(t,\cdot)\right]\right\|_{L^2}& \le C(A+B)^2B\varepsilon^3 t^{-1+\beta'}, \\ \left\|\chi(t^{-\sigma}D_x)\left[x_m x_n R(t,\cdot)\right]\right\|_{L^2}& \le C(A+B)^2B\varepsilon^3 t^{\beta'}, \end{split}$$ together with $$\label{est_Linfty_R} \left\|\chi(t^{-\sigma}D_x)R(t,\cdot)\right\|_{L^\infty}\le C(A+B)^2B\varepsilon^3 t^{-\frac{5}{2}+\beta'}.$$ Let us introduce $\textit{NL}^{\emph{cub}}_v$ as follows $$\label{NL_cub_v} \begin{split} \textit{NL}^{\emph{cub}}_v :=& -\frac{i}{2}\left[-(D_1\Gamma v)\Nlw +D_1\left[(\Gamma v)\Nlw\right]\right] \\ & -\frac{i}{2}\left[-(D_1v)\Gamma\textit{NL}_w + D_1[v \Gamma\textit{NL}_w]\right] + \frac{i}{2}\delta_\Omega \left[-(D_2v)\textit{NL}_w + D_2[v \textit{NL}_w]\right] \\ & + \delta_{Z_1}\left[(D_tv)\textit{NL}^I_w - \langle D_x\rangle[v \textit{NL}_w]\right], \end{split}$$ so that from $$\label{dec_NLT} \begin{split} \NLT = \frac{i}{2}\left[(\Gamma\Nlkg) (D_1u) + \Nlkg(D_1\Gamma u)\right] + \delta_\Omega \frac{i}{2} \Nlkg (D_2u) + \delta_{Z_1}\Nlkg (D_tu) + \textit{NL}^\emph{cub}_v, \end{split}$$ with $\delta_\Omega$ (resp. $\delta_{Z_1}$) equal to 1 when $\Gamma = \Omega$ (resp. $\Gamma =Z_1$), 0 otherwise. After , , and estimates , , with $s=0$, , , $\textit{NL}^\emph{cub}_v$ verifies the following: $$\begin{split} & \left\| \chi(t^{-\sigma}D_x)\left[x_n \textit{NL}^\emph{cub}_v\right](t,\cdot)\right\|_{L^2}\\ &\lesssim \sum_{\mu, |\nu|=0}^1 t^\sigma \left\|x^\mu_n\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\nu v_\pm(t,\cdot)\right\|_{L^\infty} \left[\|\Gamma\textit{NL}_w(t,\cdot)\|_{L^2}+\|\textit{NL}_w(t,\cdot)\|_{L^2} + \|v_\pm(t,\cdot)\|_{H^{2,\infty}}\|(\Gamma^I v)_\pm(t,\cdot)\|_{L^2}\right] \\ & \le C(A+B)AB\varepsilon^3 t^{-1+\sigma+\delta_2}. \end{split}$$From the mentioned inequalities and the additional , it also satisfies $$\begin{split} & \left\| \chi(t^{-\sigma}D_x)\left[x_m x_n \textit{NL}^\emph{cub}_v\right](t,\cdot)\right\|_{L^2}\\ &\lesssim \sum_{\mu_1,\mu_2,|\nu|=0}^1 t^\sigma \left\|x^{\mu_1}_mx^{\mu_2}_n\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\nu v_\pm(t,\cdot)\right\|_{L^\infty} \left[\|\Gamma\textit{NL}_w(t,\cdot)\|_{L^2}+\|\textit{NL}_w(t,\cdot)\|_{L^2}\right.\\ &\hspace{7.5cm}\left. + \|v_\pm(t,\cdot)\|_{H^{2,\infty}}\|(\Gamma^I v)_\pm(t,\cdot)\|_{L^2}\right]\\ & \le C(A+B)AB\varepsilon^3 t^{\sigma+\delta_2}. \end{split}$$ Moreover, applying twice lemma \[Lem\_appendix:L\_estimate of products\] with $L=L^\infty$ and $s>0$ large enough to have $N(s)\ge 2$, the first time to estimate products involving $\Gamma v$ and $\textit{NL}_w$ in , the second one to estimate the first two quadratic contributions to $\Gamma\textit{NL}_w$ (see ), we derive that there are two smooth cut-off functions $\chi_1, \chi_2$ such that $$\begin{split} &\left\| \chi(t^{-\sigma}D_x)\textit{NL}^\emph{cub}_v(t,\cdot)\right\|_{L^\infty} \lesssim t^\sigma\left\|\chi_1(t^{-\sigma}D_x)(\Gamma v)_\pm(t,\cdot)\right\|_{L^\infty}\|\Nlw(t,\cdot)\|_{L^\infty} \\ &+ t^{-2}\|(\Gamma v)_\pm(t,\cdot)\|_{L^2}\|\Nlw(t,\cdot)\|_{H^s}+ t^\sigma \|\chi_1(t^{-\sigma}D_x)\Gamma\textit{NL}_w(t,\cdot)\|_{L^\infty}\|v_\pm(t,\cdot)\|_{H^{1,\infty}} \\ & + t^{-2}\|\textit{NL}^I_w(t,\cdot)\|_{L^2}\|v_\pm(t,\cdot)\|_{H^s} + t^\sigma\|v_\pm(t,\cdot)\|_{H^{1,\infty}}\|\textit{NL}_w(t,\cdot)\|_{L^\infty} \end{split}$$ and $$\begin{split} & \|\chi_1(t^{-\sigma}D_x)\Gamma\textit{NL}_w(t,\cdot)\|_{L^\infty} \lesssim \|\chi_2(t^{-\sigma}D_x)(\Gamma v)_\pm(t,\cdot)\|_{H^{2,\infty}}\|v_\pm(t,\cdot)\|_{H^{2,\infty}} \\ &+ t^{-2}\|(\Gamma v)_\pm(t,\cdot)\|_{H^1}\|v_\pm(t,\cdot)\|_{H^s} + \|v_\pm(t,\cdot)\|_{H^{1,\infty}}\left(\|v_\pm(t,\cdot)\|_{H^{2,\infty}}+ \|D_tv_\pm(t,\cdot)\|_{H^{1,\infty}}\right). \end{split}$$ From a-priori estimates, , , with $s=0$, with $s=1$ and $\theta\ll 1$ small, , , we then recover $$\left\| \chi(t^{-\sigma}D_x)\textit{NL}^\emph{cub}_v(t,\cdot)\right\|_{L^\infty} \le CA^2B\varepsilon^3 t^{-3+\beta'}.$$ Those inequalities make $\textit{NL}^\emph{cub}_v$ a contribution of the form $R(t,x)$, so from we are left to prove that the same is true for $\Gamma\textit{NL}_{kg}(D_1u)$, $\textit{NL}_{kg}(D_1\Gamma u)$, $\textit{NL}_{kg}(D_2 u)$ and $\textit{NL}_{kg}(D_tu)$. We immediately observe, from and , that the cubic contributions to $\textit{NL}_{kg}(D_2u)$ and $\textit{NL}_{kg} (D_t u)$ are of the form $$\label{prod_a0v_b1u_b0u} [a_0(D_x)v_{-}] [b_1(D_x)u_{-}] b_0(D_x)u_{-},$$ with $a_0(\xi)\in \{1,\xi_j\langle \xi\rangle^{-1}, j=1,2\}$, $b_1(\xi)\in \{\xi_1, \xi_j\xi_1|\xi|^{-1}, j=1,2\}$, $b_0(\xi)\in \{1,\xi_2|\xi|^{-1}\}$. Therefore, lemmas \[Lem\_appendix:Linfty\_est\_rNFkg\], \[Lem\_appendix: L xnrNFkg\] imply that $\textit{NL}_{kg}(D_2 u)$ and $\textit{NL}_{kg}(D_tu)$ are remainders $R(t,x)$. Furthermore, from , and the equation satisfied by $u_\pm$ in with $|I|=0$, $$\begin{gathered} \Gamma\textit{NL}_{kg}= Q^\mathrm{kg}_0((\Gamma v)_\pm, D_1 u_\pm) + Q^\mathrm{kg}_0(v_\pm, D_1 (\Gamma u)_\pm) \\ -\delta_{\Omega}Q^\mathrm{kg}_0(v_\pm, D_2 u_\pm) - \delta_{Z_1} \Big[Q^\mathrm{kg}_0(v_\pm, |D_x| u_\pm) + Q^\mathrm{kg}_0\big(v_\pm, Q^\mathrm{w}_0(v_\pm, D_1 v_\pm)\big)\Big],\end{gathered}$$ with $\delta_\Omega$ (resp. $\delta_{Z_1}$) equal to 1 if $\Gamma =\Omega$ (resp. $\Gamma =Z_1$), 0 otherwise. Estimates and imply that $$\left\|\chi(t^{-\sigma}D_x)\left[x_n Q^\mathrm{kg}_0\big(v_\pm, Q^\mathrm{w}_0(v_\pm, D_1 v_\pm)\big) (D_1u) \right](t,\cdot)\right\|_{L^2(dx)} \le C(A+B)A^2B\varepsilon^4 t^{-\frac{3}{2}+\frac{\delta+\delta_2}{2}},$$ while after , , , $$\begin{gathered} \left\|\chi(t^{-\sigma}D_x)\left[x_mx_n Q^\mathrm{kg}_0\big(v_\pm, Q^\mathrm{w}_0(v_\pm, D_1 v_\pm)\big) (D_1u) \right](t,\cdot)\right\|_{L^2(dx)}\\ \lesssim \sum_{|\mu|=0}^1\left\| x_mx_n\Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty(dx)}\|\textit{NL}_w(t,\cdot)\|_{L^2(dx)}\|\mathrm{R}_1 u_\pm(t,\cdot)\|_{L^\infty}\\ \le C(A+B)A^2B\varepsilon t^{-\frac{1}{2}+\frac{\delta+\delta_2}{2}}\end{gathered}$$ Also, for any $\theta\in ]0,1[$, $$\begin{gathered} \left\|\chi(t^{-\sigma}D_x)\left[ Q^\mathrm{kg}_0\big(v_\pm, Q^\mathrm{w}_0(v_\pm, D_1 v_\pm)\big) (D_1u) \right](t,\cdot)\right\|_{L^\infty(dx)}\\ \lesssim \|v_\pm(t,\cdot)\|_{H^{1,\infty}}\|\textit{NL}_w(t,\cdot)\|_{H^{1,\infty}}\|\mathrm{R}_1u_\pm(t,\cdot)\|_{L^\infty} \le CA^{4-\theta}B^\theta \varepsilon^4 t^{-\frac{7}{2}+\theta(1+\frac{\delta}{2})},\end{gathered}$$ as follows from with $s=1$ and a-priori estimates. Thus $Q^\mathrm{kg}_0\big(v_\pm, Q^\mathrm{w}_0(v_\pm, D_1 v_\pm)\big) (D_1u)$ is a remainder $R(t,x)$. The same holds true for $$\left[-\delta_{\Omega}Q^\mathrm{kg}_0(v_\pm, D_2 u_\pm) -\delta_{Z_1}Q^\mathrm{kg}_0(v_\pm, |D_x| u_\pm)\right](D_1u)$$ thanks to lemmas \[Lem\_appendix:Linfty\_est\_rNFkg\] and \[Lem\_appendix: L xnrNFkg\], since the above term is linear combination of products of the form $$[a_0(D_x)v_{-}]\, [b_1(D_x)u_{-}] \, \mathrm{R}_1 u_{-},$$ with the same $a_0(\xi)$ as before and $b_1(\xi)\in \{\xi_2, \xi_2\xi_j|\xi|^{-1}, |\xi|, j=1,2\}$, as one can check using and . Summing up, the very contributions for which we have to prove estimates and are the following: \[fst\_products\] $$\begin{gathered} [a_0(D_x)(\Gamma v)_{-}]\, [b_1(D_x)u_{-}] \, \mathrm{R}_1 u_{-} \label{first_product}\\ [a_0(D_x)v_{-}]\, [b_1(D_x)(\Gamma u)_{-}] \, \mathrm{R}_1 u_{-},\label{second_product} \end{gathered}$$ which are the remaining types of products in $(\Gamma\textit{NL}_{kg})(D_1u)$, and $$[a_0(D_x)v_{-}]\, [b_1(D_x)u_{-}] \, \mathrm{R}_1 (\Gamma u)_{-}, \label{third_product}$$ which are the products appearing in $\Nlkg (D_1\Gamma u$), with $a_0$ being the same as above and $b_1(\xi)$ equal to $\xi_1$ or to $\xi_j\xi_1|\xi|^{-1}$, with $j=1,2$. All the manipulations we are going to make in what follows are aimed at showing that these estimates follow from lemmas \[Lem\_appendix:product\_Vtilde\_Utilde\], \[Lem\_appendix: product vtilde\_utildeJ\] and \[Lem\_appendix: product\_VtildeGamma\_utilde\]. Firstly, we can assume that all factors in are truncated for moderate frequencies less or equal than $t^\sigma$, up to $R(t,x)$ contributions. As regards , this comes out from the application of lemma \[Lem\_appendix:L\_estimate of products\]. In fact, taking $L=L^2$, $w_1=x_m^k x_n a_0(D_x) (\Gamma v)_{-}$for $k\in \{0,1\}$, $s>0$ large enough to have $N(s)>2$, and using a-priori estimates and , , we find that there is some $\chi_1\in C^\infty_0(\mathbb{R}^2)$ such that, for $k=0,1$, $$\begin{split} &\left\|\chi(t^{-\sigma}D_x)\big[x_m^k x_n [a_0(D_x)(\Gamma v)_{-}]\, [b_1(D_x)u_{-}] \, \mathrm{R}_1 u_{-} \big]\right\|_{L^2(dx)} \\ &\lesssim \left\|\left[\chi_1(t^{-\sigma}D_x)[x_m^k x_na_0(D_x)(\Gamma v)_{-}]\right] \left[\chi(t^{-\sigma}D_x)b_1(D_x)u_{-}\right] \left[\chi(t^{-\sigma}D_x)\mathrm{R}_1 u_{-}\right]\right\|_{L^2(dx)} \\ &+ t^{-2}\sum_{\mu_1,\mu_2,|\nu|=0}^1 \|x_m^{\delta_{k1}\mu_1}x^{\mu_2}_n(\Gamma v)_{-}(t,\cdot)\|_{L^2(dx)} \|\mathrm{R}^\nu u_{-}(t,\cdot)\|_{H^{2,\infty}}\|u_{-}(t,\cdot)\|_{H^s}\\ &\lesssim \left\|\left[\chi_1(t^{-\sigma}D_x)[x_m^kx_na_0(D_x)(\Gamma v)_{-}]\right] \left[\chi(t^{-\sigma}D_x)b_1(D_x)u_{-}\right] \left[\chi(t^{-\sigma}D_x)\mathrm{R}_1 u_{-}\right]\right\|_{L^2(dx)} \\ &+ CAB^2\varepsilon^3 t^{-\frac{3}{2}(1-k)+\frac{\delta+\delta_2}{2}}, \end{split}$$ where $\delta_{k1}$ is the Kronecker delta. Taking instead $L=L^\infty$, from a-priori estimates we derive that $$\begin{split} &\left\|\chi(t^{-\sigma}D_x)\big[ [a_0(D_x)(\Gamma v)_{-}]\, [b_1(D_x)u_{-}] \, \mathrm{R}_1 u_{-} \big]\right\|_{L^\infty(dx)} \\ & \lesssim \left\|\left[\chi_1(t^{-\sigma}D_x)[a_0(D_x)(\Gamma v)_{-}]\right] \left[\chi(t^{-\sigma}D_x)b_1(D_x)u_{-}\right] \left[\chi(t^{-\sigma}D_x)\mathrm{R}_1 u_{-}\right]\right\|_{L^\infty(dx)}\\ & + t^{-2}\sum_{|\mu|=0}^1 \|(\Gamma v)_{-}(t,\cdot)\|_{L^2} \|\mathrm{R}^\mu u_{-}(t,\cdot)\|_{H^{2,\infty}}\|u_{-}(t,\cdot)\|_{H^s}\\ & \lesssim \left\|\left[\chi_1(t^{-\sigma}D_x)[a_0(D_x)(\Gamma v)_{-}]\right] \left[\chi(t^{-\sigma}D_x)b_1(D_x)u_{-}\right] \left[\chi(t^{-\sigma}D_x)\mathrm{R}_1 u_{-}\right]\right\|_{L^\infty(dx)}\\ & + CAB^2\varepsilon^3 t^{-\frac{5}{2}+\frac{\delta+\delta_2}{2}}. \end{split}$$ As concerns instead products and , this follows applying inequalities with $w=u$, $w_{j_0}=x_m^kx_na_0(D_x)v_{-}$ for $k=0,1$, and $s>0$ such that $N(s)\ge$. In fact, for $L=L^2$ we use estimates , , , together with , to derive that for $k\in \{0,1\}$ $$\begin{split} & \left\|\chi(t^{-\sigma}D_x)\left[x_m^kx_n [a_0(D_x)v_{-}]\, [b_1(D_x)(\Gamma u)_{-}] \, \mathrm{R}_1 u_{-} \right]\right\|_{L^2(dx)}\\ & \lesssim \left\|\left[\chi_1(t^{-\sigma}D_x)[x_m^kx_n a_0(D_x) v_{-}]\right] \left[\chi(t^{-\sigma}D_x)b_1(D_x)(\Gamma u)_{-}\right] \chi(t^{-\sigma}D_x)\mathrm{R}_1 u_{-} \right\|_{L^2(dx)} \\ & + t^{-3}\sum_{|\mu_1|, \mu_2, \mu_3=0}^1 \left(\left\|x^{\mu_1}x_m^{\delta_{k1}\mu_2}x^{\mu_3}_n v_{-}(t,\cdot) \right\|_{L^2(dx)} + t \|x^{\mu_2}_n v_{-}(t,\cdot)\|_{L^2}\right) \|u_\pm(t,\cdot)\|_{H^s}\|\mathrm{R}_1u_{-}(t,\cdot)\|_{L^\infty} \\ &+ t^{-3}\left\|x^k_mx_na_0(D_x)v_{-}(t,\cdot)\right\|_{L^\infty(dx)}\|(\Gamma u)_{-}(t,\cdot)\|_{L^2}\|u_{-}(t,\cdot)\|_{H^s}\\ & \lesssim \left\| \left[\chi_1(t^{-\sigma}D_x)[x_m^kx_n a_0(D_x) v_{-}]\right] \left[\chi(t^{-\sigma}D_x)b_1(D_x)(\Gamma u)_{-}\right] \chi(t^{-\sigma}D_x)\mathrm{R}_1 u_{-} \right\|_{L^2(dx)}\\& + CAB^2\varepsilon^2 t^{-(1-k) + \frac{\delta+\delta_2}{2}}. \end{split}$$ Using instead with $L=L^\infty$ along with and , $$\begin{split} & \left\|\chi(t^{-\sigma}D_x)\left[ [a_0(D_x)v_{-}]\, [b_1(D_x)(\Gamma u)_{-}] \, \mathrm{R}_1 u_{-} \right]\right\|_{L^\infty} \\ & \lesssim \left\| \left[\chi_1(t^{-\sigma}D_x)a_0(D_x) v_{-}\right] \left[\chi(t^{-\sigma}D_x)b_1(D_x)(\Gamma u)_{-}\right] \chi(t^{-\sigma}D_x)\mathrm{R}_1 u_{-} \right\|_{L^\infty} + CAB^2\varepsilon^2 t^{-\frac{5}{2}+ \frac{\delta+\delta_2}{2}}. \end{split}$$ Secondly, we can assume that in (resp. in ) $b_1(D_x) u_{-}$ is replaced with $b_1(D_x)\unf$ (with $\unf$ introduced in ). This is justified up to some $R(t,x)$ terms that satisfy as consequence of , , (resp. , ), , and also because of , (resp. ) and . Hence we are led to estimate the $L^2$ norm of $$\begin{aligned} & \left[\chi_1(t^{-\sigma}D_x)[x_m^kx_n^l a_0(D_x) (\Gamma v)_{-}]\right] \left[\chi(t^{-\sigma}D_x)b_1(D_x) \unf\right] \chi(t^{-\sigma}D_x)\mathrm{R}_1 u_{-} \label{first}\\ & \left[\chi_1(t^{-\sigma}D_x)[x_m^kx_n^l a_0(D_x) v_{-}]\right] \left[\chi(t^{-\sigma}D_x)b_1(D_x)(\Gamma u)_{-}\right] \chi(t^{-\sigma}D_x)\mathrm{R}_1 u_{-} \label{second}\\ & \left[\chi_1(t^{-\sigma}D_x)[x_m^kx_n^l a_0(D_x) v_{-}\right] \left[\chi(t^{-\sigma}D_x)b_1(D_x) \unf\right] \chi(t^{-\sigma}D_x)\mathrm{R}_1 (\Gamma u)_{-}\label{third}\end{aligned}$$ for $k=0,1$, $l=1$, and the $L^\infty$ norm of above products when $k=l=0$. Thirdly, we can think of $a_0(D_x)(\Gamma v)_{-}$ in and of $a_0(D_x)v_{-}$ in , as replaced with $a_0(D_x)\VNF$ and $a_0(D_x)\vnf$ respectively, where $\VNF$ has been introduced in and $\vnf$ in . For (resp. ) this substitution is justified up to some $R(t,x)$ terms that satisfy and , the former because of a-priori estimate , and (resp. , and ), the latter after , (resp. , ) and the classical translation of the semi-classical $$\|\unf(t,\cdot)\|_{H^{\rho,\infty}}+ \|\mathrm{R}\unf(t,\cdot)\|_{H^{\rho,\infty}}\le CB\varepsilon t^{-\frac{1}{2}}.$$ Therefore, in order to conclude the proof we must prove that, for some $\chi, \chi_1\in C^\infty_0(\R^2)$ and $k\in \{0,1\}$, $$\begin{aligned} & \left\|\left[\chi_1(t^{-\sigma}D_x)[x_m^kx_n a_0(D_x) (\Gamma v)_{-}]\right] \left[\chi(t^{-\sigma}D_x)b_1(D_x) \unf\right] \chi(t^{-\sigma}D_x)\mathrm{R}_1 u_{-} \right\|_{L^2(dx)} \\ & + \left\|\left[\chi_1(t^{-\sigma}D_x)[x_m^kx_n a_0(D_x) v_{-}]\right] \left[\chi(t^{-\sigma}D_x)b_1(D_x)(\Gamma u)_{-}\right] \chi(t^{-\sigma}D_x)\mathrm{R}_1 u_{-} \right\|_{L^2(dx)} \\ & +\left\| \left[\chi_1(t^{-\sigma}D_x) [x_m^k x_n^la_0(D_x) v_{-}]\right] \left[\chi(t^{-\sigma}D_x)b_1(D_x) \unf\right] \chi(t^{-\sigma}D_x)\mathrm{R}_1 (\Gamma u)_{-}\right\|_{L^2(dx)} \\ &\le C(A+B)^2B\varepsilon^3 t^{-1+k+\beta'}\end{aligned}$$ and $$\begin{aligned} & \left\|\left[\chi_1(t^{-\sigma}D_x) a_0(D_x) (\Gamma v)_{-}\right] \left[\chi(t^{-\sigma}D_x)b_1(D_x) \unf\right] \chi(t^{-\sigma}D_x)\mathrm{R}_1 u_{-} \right\|_{L^\infty(dx)} \\ & + \left\|\left[\chi_1(t^{-\sigma}D_x) a_0(D_x) v_{-}\right] \left[\chi(t^{-\sigma}D_x)b_1(D_x)(\Gamma u)_{-}\right] \chi(t^{-\sigma}D_x)\mathrm{R}_1 u_{-} \right\|_{L^\infty(dx)} \\ & + \left\| \left[\chi_1(t^{-\sigma}D_x) a_0(D_x) v_{-}\right] \left[\chi(t^{-\sigma}D_x)b_1(D_x) \unf\right] \chi(t^{-\sigma}D_x)\mathrm{R}_1 (\Gamma u)_{-}\right\|_{L^\infty(dx)}\\ & \le C(A+B)^2B\varepsilon^3 t^{-\frac{5}{2}+\beta'}.\end{aligned}$$ Actually, using , , and passing to the semi-classical framework and unknowns with $\widetilde{V}^\Gamma$ introduced in , $\ut, \vt$ in , and $\ut^I(t,x)=t^{-1}(\Gamma u)_{-}(t, t^{-1}x)$, above inequalities will follow respectively from $$\label{est_L2_prod_sem} \begin{split} \sum_{k=0}^1&\Big[\left\| \left[\oph(\chi_1(h^\sigma\xi))[x^k_m x_n\oph(a_0(\xi))\widetilde{V}^\Gamma]\right]\big[ \oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u}\big](t,\cdot) \right\|_{L^2(dx)}\\ &+\left\| \left[\oph(\chi_1(h^\sigma\xi))[x^k_m x_n\oph(a_0(\xi))\vt]\right]\big[ \oph(\chi(h^\sigma\xi)b_1(\xi))\ut^I\big](t,\cdot) \right\|_{L^2(dx)}\\ &+\left\| \left[\oph(\chi_1(h^\sigma\xi))[x^k_m x_n\oph(a_0(\xi))\vt]\right]\big[ \oph(\chi(h^\sigma\xi)b_1(\xi))\ut\big](t,\cdot) \right\|_{L^2(dx)}\Big]\\ & \le C(A+B)B\varepsilon^3 h^{-\frac{1}{2}-\beta'} \end{split}$$ and $$\label{est_Linfty_prod_sem} \begin{split} &\left\| \big[\oph(\chi_1(h^\sigma\xi)a_0(\xi))\widetilde{V}^\Gamma\big]\big[ \oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u}\big](t,\cdot) \right\|_{L^\infty(dx)}\\ &+\left\| \big[\oph(\chi_1(h^\sigma\xi)a_0(\xi))\vt\big]\big[ \oph(\chi(h^\sigma\xi)b_1(\xi))\ut^I\big](t,\cdot) \right\|_{L^\infty(dx)} \\ &+\left\| \big[\oph(\chi_1(h^\sigma\xi)a_0(\xi))\vt\big]\big[ \oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u}\big](t,\cdot) \right\|_{L^\infty(dx)}\le C(A+B)B\varepsilon^3 h^{-\beta'}. \end{split}$$ We immediately obtain from inequalities and that$$\sum_{k=0}^1\left\| \left[\oph(\chi_1(h^\sigma\xi))[x^k_m x_n\oph(a_0(\xi))\vt]\right]\big[ \oph(\chi(h^\sigma\xi)b_1(\xi))\ut\big](t,\cdot) \right\|_{L^2(dx)}\le C(A+B)B\varepsilon^3 h^{-\frac{1}{2}-\beta'}.$$Moreover, one can check that $$\begin{gathered} \left\| \left[\oph(\chi_1(h^\sigma\xi))[ x_n\oph(a_0(\xi))\widetilde{V}^\Gamma]\right]\big[ \oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u}\big](t,\cdot) \right\|_{L^2(dx)} \\ \le \left\| \Big[\oph\Big(\chi_1(h^\sigma\xi)a_0(\xi)\frac{\xi_n}{\langle\xi\rangle}\Big)\widetilde{V}^\Gamma\Big] [\oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u}](t,\cdot) \right\|_{L^2} + CAB\varepsilon^2 h^{\frac{1}{2}-\beta'},\end{gathered}$$ $$\begin{split} &\left\| \left[\oph(\chi_1(h^\sigma\xi))[ x_m x_n\oph(a_0(\xi))\widetilde{V}^\Gamma]\right]\big[ \oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u}\big](t,\cdot) \right\|_{L^2(dx)} \\ &\lesssim \left\| \Big[\oph\Big(\chi_1(h^\sigma\xi)a_0(\xi)\frac{\xi_n}{\langle\xi\rangle}\Big)\widetilde{V}^\Gamma\Big]\Big[\oph\Big(\chi(h^\sigma\xi)b_1(\xi)\frac{\xi_m}{|\xi|}\Big)\widetilde{u}\Big](t,\cdot) \right\|_{L^2(dx)}+ C(A+B)B\varepsilon^2 h^{\frac{1}{2}-\beta'}, \end{split}$$ and $$\begin{gathered} \left\| \left[\oph(\chi_1(h^\sigma\xi))\big[ x_n \oph(a_0(\xi))\vt\big]\right]\big[ \oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u}^I\big](t,\cdot) \right\|_{L^2(dx} \\ \le \left\| \Big[\oph\Big(\chi_1(h^\sigma\xi)a_0(\xi)\frac{\xi_n}{\langle\xi\rangle}\Big)\vt\Big] [\oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u}^I](t,\cdot) \right\|_{L^2} + CAB\varepsilon^2 h^{\frac{1}{2}-\beta'},\end{gathered}$$ $$\begin{split} & \left\| \left[\oph(\chi_1(h^\sigma\xi))\big[x_m x_n \oph(a_0(\xi))\vt\big]\right]\big[ \oph(\chi(h^\sigma\xi)b_1(\xi))\widetilde{u}^J\big](t,\cdot) \right\|_{L^2} \\ & \lesssim \left\| \Big[\oph\Big(\chi_1(h^\sigma\xi)a_0(\xi)\frac{\xi_n}{\langle\xi\rangle}\Big)\vt\Big]\Big[\oph\Big(\chi(h^\sigma\xi)b_1(\xi)\frac{\xi_m}{|\xi|}\Big)\widetilde{u}^I\Big](t,\cdot) \right\|_{L^2(dx)} + C(A+B)B\varepsilon^2 h^{\frac{1}{2}-\beta'}. \end{split}$$ This can be done using a similar argument to the one that led us to and , up to replacing $\widetilde{v}$ with $\widetilde{V}^\Gamma$ in , referring to lemma \[Lem\_appendix: LvtildeGamma\] instead of \[Lem: from energy to norms in sc coordinates-KG\], and to estimate instead of , in order to derive the former two inequalities; up to replacing $\widetilde{u}$ with $\widetilde{u}^I$ in , using lemma \[Lem\_appendix: L\^2 estimates uJ\] instead of , , estimate instead of , and the fact that for any $\theta\in]0,1[$ $$\begin{gathered} \left\| \oph(\chi(h^\sigma\xi)b_1(\xi)\xi_m|\xi|^{-1})\widetilde{u}^I(t,\cdot)\right\|_{L^\infty}=t \left\|\chi(t^{-\sigma}D_x)b_1(D_x)D_m|D_x|^{-1}(\Gamma u)_{-}(t,\cdot)\right\|_{L^\infty}\\ \lesssim t \|\chi(t^{-\sigma}D_x) (\Gamma u)_{-}(t,\cdot)\|^{1-\theta}_{H^{3,\infty}}\|(\Gamma u)_{-}(t,\cdot)\|_{H^2}^\theta \le C(A+B)^{1-\theta}B^\theta\varepsilon t^{\frac{1}{2}+\beta+\frac{\delta_1}{2}+\frac{(1+\delta_1+\delta_2)}{2}\theta},\end{gathered}$$ which is the analogous of (last estimate deduced using and with $k=1$), to demonstrate the latter two ones. Therefore, above inequalities and , imply . Finally, is consequence of , and . That concludes the proof of the statement. \[Lem\_appendix:Lm xn NLT\] Let $\NLT$ be given by . There exists a constant $C>0$ such that, for any $\chi\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$ small, $m,n=1,2$, and every $t\in [1,T]$, $$\begin{gathered} \left\| \oph(\chi(h^\sigma\xi))\mathcal{L}_m \left[t(tx_n)\NLT(t,tx)\right]\right\|_{L^2(dx)}\le C(A+B)^2B\varepsilon^3 t^{\beta'}, \\ \left\| \oph(\chi(h^\sigma\xi))\mathcal{L}_m \left[t(tx_n) Q^\mathrm{kg}_0\big(v_\pm, Q^\mathrm{w}_0(v_\pm, D_1 v_\pm)\big)(t,tx)\right]\right\|_{L^2(dx)}\le C(A+B)AB\varepsilon^3 t^{\frac{\delta+\delta_2}{2}},\end{gathered}$$ with $\beta'>0$ such that $\beta'\rightarrow 0$ as $\sigma,\delta_0\rightarrow 0$. Straightforward after , lemma \[Lem\_appendix:xxNLT\], estimate and the following inequality $$\begin{gathered} \left\|\chi(t^{-\sigma}D_x)\left[x_mx_n Q^\mathrm{kg}_0\big(v_\pm, Q^\mathrm{w}_0(v_\pm, D_1 v_\pm)\big)\right](t,\cdot)\right\|_{L^2} \\\lesssim \sum_{|\mu|=0}^1 \left\|x_mx_n \Big(\frac{D_x}{\langle D_x\rangle}\Big)^\mu v_\pm(t,\cdot)\right\|_{L^\infty} \|\textit{NL}_w(t,\cdot)\|_{L^2} \le C(A+B)AB\varepsilon^3 t^{\frac{\delta+\delta_2}{2}},\end{gathered}$$ deduced from , and . \[Lem\_appendix:Lcal2\_VtildeGamma\] Let $\widetilde{V}^\Gamma$ be the function defined in . There exists some positive constant $C$ such that, for any $\chi\in C^\infty_0(\mathbb{R}^2)$, $\sigma>0$ small, and every $t\in [1,T]$, $$\sum_{|\mu|=2}\left\|\oph(\chi(h^\sigma\xi)\mathcal{L}^\mu\widetilde{V}^\Gamma(t,\cdot)\right\|_{L^2}\le CB\varepsilon t^{\beta'},$$ with $\beta'>0$ small, $\beta'\rightarrow 0$ as $\sigma,\delta_0\rightarrow 0$. First of all we remind that $\VNF$ is solution to . From relation and the commutation between $\mathcal{L}_m$ and $\oph(\langle\xi\rangle)$ we deduce that, for any $m,n=1,2$, $$\begin{split} &\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}_m\mathcal{L}_n \widetilde{V}^\Gamma(t,\cdot)\right\|_{H^1_h}\lesssim \sum_{\mu=0}^1\Big[\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu_m \big[tZ_n \VNF(t,tx)\big] \right\|_{L^2(dx)}\\ & + \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu_m \oph\Big(\frac{\xi_n}{\langle\xi\rangle}\Big)\widetilde{V}^\Gamma(t,\cdot)\big]\right\|_{L^2}+ \left\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu_m \big[t(tx_n)\NLT(t,tx)\big]\right\|_{L^2(dx)}\\ &+ \left\| \oph(\chi(h^\sigma\xi))\mathcal{L}^\mu_m \left[t(tx_n) Q^\mathrm{kg}_0\big(v_\pm, Q^\mathrm{w}_0(v_\pm, D_1 v_\pm)\big)(t,tx)\right]\right\|_{L^2(dx)}\Big]. \end{split}$$ The result of the statement follows then from , , , , and lemmas \[Lem\_appendix: LvtildeGamma\], \[Lem\_appendix:Lm xn NLT\]. \[Lem\_appendix: sharp\_est\_VJ\] There exists a constant $C>0$ such that, for any $\chi\in C^\infty_0(\mathbb{R}^2)$ equal to 1 in a neighbourhood of the origin, $\sigma>0$ small, and every $t\in [1,T]$, $$\label{sharp_est_VI} \sum_{|I|=1}\|\chi(t^{-\sigma}D_x)V^I(t,\cdot)\|_{L^\infty} \le CB\varepsilon t^{-1}.$$ As this estimate is evidently satisfied when $I$ is such that $\Gamma^I$ is a spatial derivative after a-priori estimate , we focus on proving the statement for $\Gamma^I\in \{\Omega, Z_m, m=1,2\}$ being a Klainerman vector field. For simplicity, we refer to $\Gamma^I$ simply by $\Gamma$. Instead of proving the result of the statement directly on $(\Gamma v)_\pm$ we show that $$\label{sharp_est_VNF} \left\|\VNF(t,\cdot)\right\|_{L^\infty}\le CB\varepsilon t^{-1},$$ where $\VNF$ has been introduced in . After , the above inequality evidently implies the statement. The main idea to derive the sharp decay estimate in is to use the same argument that, in subsection \[Subsection : The Derivation of the ODE Equation\], led us to the propagation of a-priori estimate , i.e. to move to the semi-classical setting and deduce an ODE from equation satisfied by $\VNF$. The most important feature that will provide us with is that the uniform norm of all involved non-linear terms is integrable in time. Before going into the details, we also remind the reader our choice to denote by $C, \beta$ and $\beta'$ some positive constants that may change line after line, with $\beta\rightarrow 0$ (resp. $\beta'\rightarrow 0$) as $\sigma\rightarrow 0$ (resp. as $\sigma,\delta_0\rightarrow 0$). Let us consider $\widetilde{V}^\Gamma(t,x):=t \VNF(t,tx)$, operator $\Gamma^{kg}$ as follows $$\Gamma^{kg}:=\oph\Big(\gamma\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi_1(h^\sigma\xi)\Big),$$ with $\gamma, \chi_1\in C^\infty_0(\mathbb{R}^2)$ such that $\gamma\equiv 1$ close to the origin, $\chi_1\equiv 1$ on the support of $\chi$, $p(\xi):=\langle \xi\rangle$, and $$\begin{aligned} \widetilde{V}^\Gamma_{\Lambda_{kg}}(t,x)&:= \Gamma^{kg}\oph(\chi(h^\sigma\xi))\widetilde{V}^\Gamma(t,x), \\ \widetilde{V}^\Gamma_{\Lambda^c_{kg}}(t,x)&:=\oph\Big((1-\gamma)\Big(\frac{x-p'(\xi)}{\sqrt{h}}\Big)\chi_1(h^\sigma\xi)\Big)\oph(\chi(h^\sigma\xi))\widetilde{V}^\Gamma(t,x),\end{aligned}$$ so that $$\oph(\chi(h^\sigma\xi))\widetilde{V}^\Gamma(t,\cdot) =\widetilde{V}^\Gamma_{\Lambda_{kg}}+ \widetilde{V}^\Gamma_{\Lambda^c_{kg}}.$$ It immediately follows from inequality and lemmas \[Lem\_appendix: LvtildeGamma\], \[Lem\_appendix:Lcal2\_VtildeGamma\], that $$\label{est_widetildeV_Lambda-kg-c} \left\|\widetilde{V}^\Gamma_{\Lambda^c_{kg}}(t,\cdot)\right\|_{L^\infty}\lesssim \sum_{|\mu|=0}^2 h^{\frac{1}{2}-\beta}\left\|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu \widetilde{V}^\Gamma(t,\cdot) \right\|_{L^2}\le CB\varepsilon t^{-\frac{1}{2}+\beta'}.$$ On the other hand, as $\VNF$ is solution to an explicit computation shows that $\widetilde{V}^\Gamma$ satisfies the following semi-classical pseudo-differential equation: $$\left[D_t - \oph(x\cdot\xi - \langle\xi\rangle)\right]\widetilde{V}^\Gamma(t,x)=h^{-1}\NLT(t,tx) - \delta_{Z_1}h^{-1} Q^\mathrm{kg}_0\big(v_\pm, Q^\mathrm{w}_0(v_\pm, D_1 v_\pm)\big)(t,tx),$$ with $\NLT$ given explicitly by . Applying successively operators $\oph(\chi(h^\sigma\xi))$ and $\Gamma^{kg}$ to the above equation we find, from symbolic calculus and the first part of lemma \[Lem: Commutator Gamma-kg\], that $\widetilde{V}^\Gamma_{\Lambda_{kg}}$ satisfies $$\begin{gathered} \label{KG_VtildeGamma_Lambda} \left[D_t - \oph(x\cdot\xi - \langle\xi\rangle)\right]\widetilde{V}^\Gamma_{\Lambda_{kg}}(t,x) = h^{-1}\Gamma^{kg}\oph(\chi(h^\sigma\xi))\left[\NLT(t,tx)\right]\\ - \delta_{Z_1} h^{-1}\Gamma^{kg}\oph(\chi(h^\sigma\xi))\big[ Q^\mathrm{kg}_0\big(v_\pm, Q^\mathrm{w}_0(v_\pm, D_1 v_\pm)\big)(t,tx)\big] - \oph(b(x,\xi))\oph(\chi(h^\sigma\xi))\widetilde{V}^\Gamma(t,x)\\ + i\sigma h^{1+\sigma}\Gamma^{kg}\oph\big((\partial\chi)(h^\sigma\xi)\cdot(h^\sigma\xi)\big)\widetilde{V}^\Gamma,\end{gathered}$$ with symbol $b$ given by . Since $\gamma$’s derivatives vanish in a neighbourhood of the origin and $\partial\chi_1\equiv 0$ on the support of $\chi$, from symbolic calculus of lemma \[Lem : a sharp b\] and remark \[Remark:symbols\_with\_null\_support\_intersection\], \[Lem : composition gamma-1 and its argument\], together with inequalities , , that $$\begin{gathered} \left\| \oph(b(x,\xi))\oph(\chi(h^\sigma\xi))\widetilde{V}^\Gamma(t,\cdot)\right\|_{L^\infty}\\ \lesssim h^{\frac{3}{2}-\beta} \sum_{|\mu|=0}^2 \|\oph(\chi(h^\sigma\xi))\mathcal{L}^\mu \widetilde{V}^\Gamma(t,\cdot)\|_{L^2} + h^2\left\|\widetilde{V}^\Gamma(t,\cdot)\right\|_{L^2} \le CB\varepsilon t^{-\frac{3}{2}+\beta'},\end{gathered}$$ where last estimate is obtained using lemmas \[Lem\_appendix: LvtildeGamma\], \[Lem\_appendix:Lcal2\_VtildeGamma\]. Moreover, reminding lemma \[Lem:family\_thetah\] and using symbolic calculus we see that, for any $N\in \N$ as large as we want, $$\begin{gathered} \label{Gamma_Op(partialchi)_VtildeGamma} h^{1+\sigma} \left\| \Gamma^{kg}\oph((\partial\chi(h^\sigma\xi)\cdot(h^\sigma\xi))\widetilde{V}^\Gamma(t,\cdot)\right\|_{L^\infty}\\ \le h^{1+\sigma} \left\|\Gamma^{kg}\theta_h(x)\oph((\partial\chi(h^\sigma\xi)\cdot(h^\sigma\xi))\widetilde{V}^\Gamma(t,\cdot) \right\|_{L^\infty} + h^N\|\widetilde{V}^\Gamma(t,\cdot)\|_{L^2},\end{gathered}$$ where $\theta_h(x)$ is a smooth cut-off function supported in closed ball $\overline{B_{1-ch^{2\sigma}}(0)}$, with $c>0$ small. Denoting $(\partial\chi)(\xi)\cdot \xi$ concisely by $\widetilde{\chi}(\xi)$, we observe from proposition \[Prop:Continuity Lp-Lp\] with $p=+\infty$, together with the uniform continuity on $L^\infty$ of operator $\widetilde{\chi}(t^{-\sigma}D_x)$, the definition of $\widetilde{V}^\Gamma$ in terms of $\VNF$, and , that $$\begin{gathered} h^{1+\sigma}\left\| \Gamma^{kg} \theta_h(x)\oph(\widetilde{\chi}(h^\sigma\xi)) \widetilde{V}^\Gamma(t,\cdot)\right\|_{L^\infty} \lesssim h^{1-\beta}\left\|\theta_h(x)\oph(\widetilde{\chi}(h^\sigma\xi)) \widetilde{V}^\Gamma(t,\cdot)\right\|_{L^\infty} \\ \le t^{\beta} \left\|\theta_h\Big(\frac{\cdot}{t}\Big)\widetilde{\chi}(t^{-\sigma}D_x) (\Gamma v)_{-}(t,\cdot) \right\|_{L^\infty}+C(A+B)B\varepsilon^2 t^{-\frac{5}{4}+\beta}.\end{gathered}$$ Using the fact that, for $\theta_h^j(z):=\theta_h(z)z_j$, $$\theta_h\Big(\frac{x}{t}\Big) (\Omega v)_{-} = t\Big[\theta_h^1\Big(\frac{x}{t}\Big) \partial_2v_{-} - \theta_h^2\Big(\frac{x}{t}\Big) \partial_1 v_{-}\Big]$$ and $$\theta_h\Big(\frac{x}{t}\Big) (Z_m v)_{-} = t\Big[\theta_h^m\Big(\frac{x}{t}\Big) \partial_t v_{-} + \theta_h\Big(\frac{x}{t}\Big)\partial_m v_{-}\Big] + \theta_h\Big(\frac{x}{t}\Big) \frac{D_m}{\langle D_x\rangle}v_{-}, \quad m=1,2,$$ and making some commutations, we can express $(\Gamma v)_{-}$ in terms of $v_{-}$ and its derivatives up to a loss in $t$. Thus, from the classical Sobolev injection combined inequality , we obtain that $$\begin{split} t^{-\beta}\left\|\widetilde{\chi}(t^{-\sigma}D_x)\theta_h\Big(\frac{\cdot}{t}\Big)(\Gamma v)_{-}(t,\cdot) \right\|_{L^\infty}& \lesssim t^{-N(s)+1+\beta}\left(\|D_tv_\pm(t,\cdot)\|_{H^s}+\|v_\pm(t,\cdot)\|_{H^s}\right)\\ & \le CB\varepsilon t^{-\frac{3}{2}}, \end{split}$$ last estimate following by taking $s>0$ large enough to have $N(s)\ge 3$ and using a-priori estimates along with with $s=0$. From and we hence derive that $$h^{1+\sigma} \left\| \Gamma^{kg}\oph((\partial\chi(h^\sigma\xi)\cdot(h^\sigma\xi))\widetilde{V}^\Gamma(t,\cdot)\right\|_{L^\infty} \le CB\varepsilon t^{-\frac{3}{2}},$$ so the last two terms in the right hand side of equation are remainders $R(t,x)$ such that $$\label{remainder R(V)} \|R(t,\cdot)\|_{L^\infty}\le CB\varepsilon t^{-\frac{5}{4}},$$ for every $t\in [1,T]$. After proposition \[Prop:Continuity Lp-Lp\] with $p=+\infty$, estimate , and the fact that for any $\theta\in ]0,1[$, $$\left\|Q^{\mathrm{kg}}_0\left(v_\pm, Q^\mathrm{w}_0(v_\pm , D_1v_\pm)\right) (t,\cdot)\right\|_{L^\infty(dx)}\le CA^{3-\theta}B^\theta\varepsilon^3 t^{-3+\theta(1+\frac{\delta}{2})},$$ as follows by with $s=1$ and a-priori estimates, we deduce (up to taking $\theta\ll 1$ small in the above inequality) that also the first two non-linear terms in the right hand side of satisfy and can be included into $R(t,x)$. Therefore, $\widetilde{V}^\Gamma_{\Lambda_{kg}}$ satisfies $$\left[D_t - \oph(x\cdot\xi - \langle\xi\rangle)\right]\widetilde{V}^\Gamma_{\Lambda_{kg}}(t,x) = R(t,x),$$ and using along with inequality , together with lemmas \[Lem\_appendix: LvtildeGamma\], \[Lem\_appendix:Lcal2\_VtildeGamma\], we deduce that, for the same family of cut-off functions $\theta_h$ introduced above, $\widetilde{V}^\Gamma_{\Lambda_{kg}}$ is solution to the following ODE: $$\label{ODE_Vtilde} D_t \widetilde{V}^\Gamma_{\Lambda_{kg}}(t,x) = -\theta_h(x)\phi(x)\widetilde{V}^\Gamma_{\Lambda_{kg}}(t,x)+ R(t,x),$$ with $\phi(x)=\sqrt{1-|x|^2}$. Since the inhomogeneous term $R(t,x)$ decays, in the uniform norm, at a rate which is integrable in time, we get that $$\|\widetilde{V}^\Gamma_{\Lambda_{kg}}(t,\cdot)\|_{L^\infty}\lesssim \|\widetilde{V}^\Gamma_{\Lambda_{kg}}(1,\cdot)\|_{L^\infty}+CB\varepsilon \le CB\varepsilon,$$ which summed up with implies , and hence the conclusion of the proof. [^1]: The author is supported by a PhD fellowship funded by the FSMP and the Labex MME-DII, and by For Women in Science fellowship funded by Fondation L’Oréal-UNESCO. Keywords: Global solution of coupled wave-Klein-Gordon systems, Klainerman vector fields, Semiclassical Analysis.
--- abstract: | [Scharlemann and Schultens have shown that for any pair of knots $K_1$ and $K_2$, $w(K_1 \# K_2) \geq max\{w(K_1), w(K_2)\}$. Scharlemann and Thompson have given a scheme for possible examples where equality holds. Using results of Scharlemann–Schultens, Rieck–Sedgwick and Thompson, it is shown that for $K= \#_{i=1}^n K_i$ a connected sum of mp-small knots and $K^{\prime}$ any non-trivial knot, $w(K\#K^{\prime})>w(K)$. ]{} address: 'Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, USA' author: - Jacob Hendricks title: 'Mp-small summands increase knot width' --- Introductory remarks ==================== Thin position, introduced by David Gabai [@g], has applications in solving difficult problems, yet there remains much to understand about the width of knots. Current understanding of the behavior of width under connect sum is incomplete, but has some form. For example, given two knots $K_1$ and $K_2$, it is easily seen that $w(K_1 \# K_2) \leq w(K_1) + w(K_2) -2$; simply stack the knots in thin position, perform the connect sum, and calculate the upperbound. Also, Rieck and Sedgwick [@rs] have shown that if $K_1$ and $K_2$ are mp-small (see definition \[11\]) then $w(K_1 \# K_2) = w(K_1) + w(K_2) - 2$. By means of a fascinating method (see [@ss]), it has also been shown that for any two knots $K_1$ and $K_2$, $w(K_1 \# K_2) \geq \mbox{max}\{w(K_1), w(K_2)\}$. It is natural to wonder if there exists a pair of knots such that $w(K_1 \# K_2) = w(K_1)$, for this would be a peculiar property of width. In fact, a class of possible examples of equality is presented in [@st], where it is argued, though not proven, that for any knot $K_2$ there is a (typically quite complicated) knot $K_1$, apparently determined by the bridge number of $K_2$, so that $w(K_1 \# K_2) = w(K_1)$. In contrast, we show here that if $K_1$ is the connect sum of mp-small knots, then for any knot $K_2$, $w(K_1 \# K_2) > w(K_1)$. I would like to thank the referees for many helpful remarks and Yo’av Rieck for many helpful conversations. Preliminaries ============= For the sake of brevity, familiarity with *width, thin/thick levels, thin position, bridge position, swallow-follow torus, satellite knot* as well as the following definitions and theorems will be assumed. One can find definitions of the italicized words in many sources, for example [@hk] and [@ss]. A knot $K \subset S^3$ is called meridionally planar small (mp-small) if the only incompressible meridional surface in its complement is a boundary-parallel annulus. \[11\] In [@t], Thompson proved: If a knot $K$ in thin position countains a thin level, then it is not mp-small. \[1\] Thus, for an mp-small knot $K$, thin position must equal bridge position. In [@rs] (Theorem 4.1), a converse of Theorem \[1\] is given in the case that $K$ is the connect sum of two non-trivial knots. Let $K$ be a connected sum of non-trivial knots, $K = K_1 \# K_2$. Then any thin position for $K$ is not bridge position for K. \[3\] The following was also shown in [@rs]: Let $K= \#_{i=1}^n K_i$ be a connected sum of mp-small knots. If K is in thin position, then there is an ordering of the summands $K_{i_1}, K_{i_2},\ldots ,K_{i_n}$ and a collection of leveled decomposing annuli $A_{i_1}, A_{i_2},\ldots ,A_{i_n-1}$ so that the thin levels of the presentation are precisely the annuli $\{ A_{i_j} \}$ occurring in order, where the annulus $A_{i_j}$ separates the connected sum $K_{i_1} \# K_{i_2} \#\ldots \# K_{i_j}$ from the connected sum $K_{i_j+1} \#\ldots \# K_{i_n}$. \[rs2\] Scharlemann–Schultens [@ss] describe a method for reimbedding a knot $K$ so that a height function on $K$ would be preserved. Suppose $p: S^3 \rightarrow \mathbb{R}$ is the standard height function. Let $K \subset S^3$ be a knot that lies inside a standard unknotted torus $H \subset S^3$. Let $f: H \rightarrow S^3$ be a possibly knotted embedding of $H$ in $S^3$. Then there is a reimbedding $f^{\prime}: H \rightarrow S^3$ so that 1. $pf^{\prime} = pf$, i.e. the reimbedding preserves height 2. $S^3 - f^{\prime}(H)$ is a solid torus (so $f^{\prime}(H)$ is an unknotted solid torus) and 3. $f(K^{\prime})$ is isotopic to $K$ in $S^3$. \[2\] This is a specific case of what is shown in Corollary 5.4 of [@ss]. In this paper will apply the theorem in this setting: We will be given a connected sum $K\#L$ embedded in $S^3$ via a knotted embedding $f: H \rightarrow S^3$. $K$ has here been placed in $H$ as a wrapping number one knot and the embedding $f$ is known to take the core of $H$ to the knot $L$. (The boundary of $f(H)$ is commonly called a “swallow-follow” torus for $K\#L$, for it swallows $K$ and follows $L$.) When summands are mp-small ========================== In [@ss], it is shown that $w(K_1\#K_2)\geq w(K_i)$ for $i=1,2$. One may ask if there is a pair of knots $K_1$ and $K_2$ such that equality holds (eg. [@st]). It is shown in [@rs] that for $K_1$ and $K_2$ both mp-small knots, $w(K_1\#K_2) = w(K_1)+w(K_2)-2$. In particular, when both are non-trivial knots, $w(K_1\#K_2)$ $>w(K_i)$, $i=1,2$. A natural question is whether this inequality remains true if only one of the knots is mp-small. If $K_1$ is an mp-small knot, then for any non-trivial knot $K_2$, $w(K_1\#K_2)>w(K_1)$. \[1case\] Since $K_1$ is mp-small, Theorem \[1\] implies that thin position of $K_1$ equals bridge position. Then, take $K_1\#K_2$ to be a satellite knot with companion knot $K_2$ via the *swallow-follow torus*. Put $K_1\#K_2$ in thin position; notice by Theorem \[3\], $K_1\#K_2$ cannot be in bridge position, so there exists a thin level. By Theorem \[2\], there exists a height preserving reimbedding that changes the knot $K_1\#K_2$ to the knot $K_1$. Since $K_1\#K_2$ has a thin level, so must a height preserving reimbedding of $K_1\#K_2$. So, this reimbedding yields an embedding of $K_1$ that has a thin level. (i.e. an embedding of $K_1$ that is not in bridge position.) It follows that this embedding of $K_1$ cannot be in thin position, so its width as embedded, and hence the width of $K_1\#K_2$, is greater than the minimal width of $K_1$. One can use the method for showing proposition \[1case\] to prove a more general statement; that is, $K_1$ can be taken to be the connect sum of mp-small knots. Let $K= \#_{i=1}^n K_i$ be a connected sum of mp-small knots and let $K^{\prime}$ be any non-trivial knot. Then, $w(K\#K^{\prime})>w(K)$. Suppose $w(K\#K^{\prime})=w(K)$; we will show that $K^{\prime}$ must be the unknot. Now, take $K^{\prime}$ to be the companion knot of the satellite knot given by $K\#K^{\prime}$ with $H$ a torus that swallows $K$ and follows $K^{\prime}$. Put $K\#K^{\prime}$ in thin position. By Theorem \[2\], there exists a reimbedding $f$ such that $f$ preserves a height function on $K\#K^{\prime}$ and $f(K\#K^{\prime})$ is isotopic to $K$ in $S^3$. We are assuming $w(K\#K^{\prime})=w(K)$, thus $f(K\#K^{\prime})$ is a thin presentation of $K$; therefore, by Theorem \[rs2\], every thin level of $K$ is one of the $n-1$ decomposing annuli, which decompose the connect sum $\#_{i=1}^n K_i$ into mp-small knots. Since each $K_i$ is mp-small, by Theorem \[1\], thin position of each $K_i$ is bridge position of each $K_i$. Now, since $f$ is a height preserving reimbedding, $K\#K^{\prime}$ must have $n-1$ decomposing annuli that constitute all of the thin levels; hence, $K\#K^{\prime} = \#_{i=1}^n K^{\prime}_i$. Note that each of the $n$ components $K^{\prime}_i$ is in bridge position. This must also be thin position; for if not, one could thin $K\#K^{\prime}$ by thinning the summand for which bridge does not equal thin, but we have put $K\#K^{\prime}$ in thin position. It follows from Theorem \[3\] that each $K^{\prime}_i$ must be prime. Hence, we have $K\#K^{\prime} = \#_{i=1}^n K^{\prime}_i$ where each $K^{\prime}_i$ is prime, and $K=\#_{i=1}^n K_i$ where each $K_i$ is prime. Since knot factorizations are unique, $K^{\prime}$ must be the unknot. plus 1pt **D Gabai**, [*Foliations and the topology of 3-manifolds, III*]{}, J. Differential Geom. 26 (1987) 479-536 **DJ Heath**, **T Kobayashi**, [*[Essential Tangle Decomposition from Thin Position of a Link]{}*]{}, Pacific J. Math. 179 (1997) 101–117 **Y Rieck**, **E Sedgwick**, [*[Thin position for a connected sum of small knots]{}*]{}, 2[2002]{}[14]{}[297]{}[309]{} **M Scharlemann**, [**J Schultens**]{}, [*3-manifolds with planar presentations and the width of satellite knots*]{}, **M Scharlemann**, [**A Thompson**]{}, [*On the additivity of knot width*]{}, from: “Proceedings of the Casson Fest”, (Cameron Gordon and Yoav Rieck, editors), 7[2004]{}5[135]{}[144]{} **A Thompson**, [*Thin position and bridge number for knots in the 3-sphere*]{}, Topology 36 (1997) 505–507
--- abstract: 'We present H$\alpha$ velocity fields for a sample of nearly face–on spiral galaxies observed with DensePak on the WIYN telescope. We combine kinematic inclinations and position angles measured from these data with photometric inclinations and position angles measured from $I$-band images to show that spiral disks are intrinsically non-circular. -0.25in' author: - 'David R. Andersen$^1$, M. A. Bershady$^2$, L. S. Sparke$^2$, J. S. Gallagher III$^2$, E. M. Wilcots$^2$, W. van Driel$^3$, D. Monnier-Ragaigne$^3$' title: 'H$\alpha$ Velocity Fields of Normal Spiral Disks' --- DensePak H$\alpha$ Velocity Fields ================================== DensePak is a 30$\times$45 arcsec fiber-optic integral-field unit feeding a bench spectrograph on the WIYN 3.5m telescope (Barden et al. 1998). By using multiple pointings of DensePak in echelle mode ($\lambda/\Delta\lambda=13,000$), we have constructed H$\alpha$ velocity fields for 8 nearby, normal, apparently face–on spiral disks. The velocity fields, typically complete to $\sim$3 disk scale lengths, reach the peak of the rotation curve (Figure 1), and are modeled successfully by a single, inclined disk with a hyperbolic tangent function for V(r) (Figure 2). Residuals from this simple model are small (typically 5 km/s). Our first discovery was that this sample, while selected to appear photometrically face-on, had projected velocities indicative of significant inclinations (up to 35$^\circ$). Clearly these disks are not circular. To quantify this observation, we developed an efficient method to estimate disk elongation of nearly face-on galaxies by combining measurements from H$\alpha$ velocity fields and $I$-band images. In the context of a simple geometric model, we interpret differences between kinematic and photometric inclinations and position angles in terms of intrinsic disk ellipticity. Five galaxies in the sample are non-circular at greater than 99% CL; the range of ellipticities estimated within the context of our model is 0.02 to 0.20 (Andersen et al. 2000). Even such modest disk ellipticity can account for much of the scatter in the Tully-Fisher (TF) relation (Franx & de Zeeuw 1992), as discussed by Bershady & Andersen (these proceedings). Support for this research comes from NSF/AST-9970780. -0.3in -0.1in Andersen, D.R., Bershady, M.A., Sparke, L.S., Gallagher, J.S., & Wilcots, E.M. 2000, [*submitted to ApJ Letters*]{} -0.025in Barden, S.C., Sawyer, D.G., & Honeycutt, R.K. 1998, SPIE, 3355, 892 -0.025in Bershady, M.A. & Andersen, D. R. 2000, ASPCS, 197, 175 -0.025in Franx, M. & de Zeeuw, T. 1992, ApJ Letters, 392, L47
--- author: - 'J. R. Walsh' - 'G. H. Jacoby' - 'R. F. Peletier' - 'N. A. Walton' date: 'Received ; accepted' title: 'The Light Element Abundance Distribution in NGC 5128 from Planetary Nebulae[^1] ' --- [Planetary nebulae in the nearest large elliptical galaxy provide light element abundances difficult or impossible to measure by other means in a stellar system very different from the galaxies in the Local Group.]{} [The light element abundance pattern from many planetary nebulae (PNe) at a range of radial distances was measured from optical spectroscopy in the elliptical galaxy NGC 5128, which hosts the radio source Centaurus A. The PN abundances, in particular for oxygen, and the PN progenitor properties are related to the galaxy stellar properties.]{} PNe in NGC 5128 covering the upper 4 mag. of the luminosity function were selected from a catalogue. VLT FORS1 multi-slit spectra in blue and red ranges were obtained over three fields at 3, 9 and 15$'$ projected radii (4, 8 and 17 kpc, for an adopted distance of 3.8 Mpc) and spectra were extracted for 51 PNe. Accurate electron temperature and density diagnostics are usually required for abundance determination, but were not available for most of the PNe. Cloudy photoionization models were run to match the spectra by a spherical, constant density nebula ionized by a black body central star. He, N, O and Ne abundances with respect to H were determined and, for brighter PN, S and Ar; central star luminosities and temperatures are also derived. [ Emission line ratios for the 51 PNe are entirely typical of PN such as in the Milky Way. The temperature sensitive \[O III\]4363Å line was weakly detected in 10 PNe, both \[O II\] and \[O III\] lines were detected in 30 PNe, and only the bright \[O III\]5007Å line was detected in 7 PN. For 40 PNe with Cloudy models, from the upper 2 mag. of the \[O III\] luminosity function, the most reliably estimated element, oxygen, has a mean 12+Log(O/H) of 8.52 with a narrow distribution. No obvious radial gradient is apparent in O/H over a range 2-20 kpc. Comparison of the PN abundances with the stellar population, from the spectra of the integrated stellar light on the multi-slits and existing photometric studies, suggests an average metallicity of \[Fe/H\]=-0.4 and \[O/Fe\]=0.25. ]{} [The masses of the PN central stars in NGC 5128 deduced from model tracks imply an epoch of formation even more recent than found for the minority young population from colour magnitude studies. The PN may belong to the young tail of a recent, minor, star formation episode or derive from other evolutionary channels, perhaps involving binary stars.]{} Introduction ============ Planetary nebulae provide a multi-facetted probe of the galactic environment. As a short-lived product of the evolution of low mass stars, they sample the bulk, by both number and mass, of the stellar population of galaxies of all Hubble types. Their size makes them unresolved from the ground at distances greater than $\sim$1 Mpc but their strong emission lines makes them easy targets for observational work, even when observed against the high stellar background of a galaxy bulge. These advantages have been exploited in a number of distinct areas. Statistics of PN in galaxies of various types show a late time indicator of the star formation rate through the luminosity specific PN frequency $\phi$ ( Ciardullo et al. [@Ciardullo89]). The PN luminosity function (PNLF), despite theoretical difficulties, still proves to be a powerful distance method (Jacoby et al. [@Jacoby92]; Ciardullo et al [@Ciardullo02]). The radial velocity, simply measured from one emission line, provides a kinematic test particle for study of the galaxy potential and in intra-cluster gas of the cluster potential and turbulence (Arnaboldi et al. [@Arnaboldi]). The abundances of a number of light elements can be measured from emission line ratios in PN spectra, such as He and N and, in particular, the $\alpha$-elements O, Ne, S and Ar, which are difficult to study in all but high resolution and high signal-to-noise spectra of individual stars. As the closest example of a large early-type galaxy, NGC 5128 (Hubble type S0p) occupies a central place in studies of resolved stellar populations in a galaxy different to the Milky Way and M31. This large elliptical galaxy in the Centaurus group shows signs of major activity with an active nucleus, Faranoff-Riley Class I radio lobes, presence of dust and young stars in its inner region (Graham [@Graham]), a young ($\sim$0.3 Gyr) tidal stream (Peng et al. [@Peng02]) and stellar shells in its outer regions (Malin et al. [@Malin]); see Israel ([@Israel]) for an earlier review. This activity can be attributed to a minor merger which has had little influence on the bulk of the older passively evolving population (Woodley [@Woodley06]), making NGC 5128 a nearby exemplar of more distant massive early-type galaxies; Harris ([@Harris10]) provides a recent review of the underlying galaxy properties. The spectroscopy of globular clusters by Beasley et al. ([@Beasley]) indicates typical ages of 7-8 Gyr from stellar population fitting, reinforcing the presence of a large-scale intermediate age star formation episode. Fits to deep colour-magnitude diagrams also provide evidence for a minority, much younger, population of age 2-4 Gyr (Rejkuba et al. [@Rejkuba11]). As representative of the low mass stars, planetary nebulae in NGC 5128 have been catalogued over a period of two decades, beginning with the catalogue of Hui et al. ([@Hui93a]). 785 PN were discovered by \[O III\]5007Å  emission line and off-band imaging; from the PN \[O III\] magnitudes Hui et al. ([@Hui93b]) determined a PNLF distance of 3.5 Mpc. Independent measurements from the Mira period-luminosity relation and the luminosity of the tip of the red giant branch (Rejkuba et al. [@Rejkuba05]), surface brightness fluctuations (Tonry et al. [@Tonry]), globular cluster luminosity function (Harris et al. [@Harris88]) and 42 classical Cepheid variables (Ferrarese et al. [@Ferrarese]) result in distance estimate in the range 3.4 to 4.1 Mpc. Harris et al. ([@HRH10]) present a comprehensive review of distance estimates to NGC 5128, and recommend a best-estimate of 3.8 Mpc, which is adopted here (distance modulus 27.90mag.). Peng et al. ([@Peng04]) extended the original catalogue of PN and found a further 356 PN by filter imaging; 780 out of a total of 1141 were spectroscopically confirmed. Further emission line mapping and follow-up intermediate dispersion spectroscopy has extended this list to over 1200 confirmed PN (Rejkuba & Walsh [@RejWal]). This number of PN makes NGC 5128 a rich source for statistical extra-galactic PN studies, rivalled only by the Milky Way and M 31 (Merrett et al. [@Merrett06]). Since there are both a large number of PN and they have been catalogued to large galactocentric distances (to 80 kpc along the major axis by Peng et al. ([@Peng04])), the radial velocities allow the dynamics of the halo mass distribution to be studied. Hui et al. ([@Hui95]) measured \[O III\] radial velocities for 431 out of their 785 catalogued PN. The offset of the rotation axis from the minor axis was attributed to evidence of triaxiality; fitting the rotation curve and velocity dispersion revealed a radially increasing mass-to-light ratio and hence the presence of a dark matter halo. Peng et al. ([@Peng04]) extended the PN kinematic work to 780 PN and showed the large rotation along the major axis but with a pronounced zero-velocity twist produced by the triaxial-prolate mass distribution. The kinematics of the large population of globular clusters (with 563 available radial velocities – Woodley et al. [@Woodley10], Woodley et al. [@Woodley07], Beasley et al. [@Beasley]) show similar kinematics to the PN but with small differences, such as lower rotation amplitude (Woodley et al. [@Woodley07]). The NGC 5128 globular clusters (GCs) also provide fundamental evidence of the star formation history. Of the 605 confirmed GCs in NGC 5128 (Woodley et al. [@Woodley07], Woodley et al. [@Woodley10]), more than half have metallicity measurements and they divide roughly half and half into metal rich and metal poor above and below \[Fe/H\]$=-1.0$. The metal poor GCs (\[Z/H\]$\sim-1.3$) are older with ages similar to Milky Way GCs and the metal rich GCs (Peng et al. [@Peng04]; Beasley et al. [@Beasley]) have intermediate ages (4-8 Gyr) and metallicity (\[Z/H\]$\sim-0.5$). There are many more GC candidates (e.g. Harris et al. [@Harris04]) and the total number of GCs has been estimated at around 1300 (Harris [@Harris10]). NGC 5128 is close enough that individual stars can be resolved with HST imaging (and from the ground with adaptive optics in the near-IR) and several studies have been obtained outside of the bright bulge where crowding is lower. Soria et al. ([@Soria]) detected the Red Giant Branch (RGB) and the Asymptotic Giant Branch (AGB) from intermediate age stars. The same field was followed up with NICMOS photometry and the IR colour magnitude diagram shows stars well above the tip of the red giant branch for old stars, confirming the presence of intermediate age stars (Marleau et al. [@Marleau]). Colour magnitude diagrams from HST WFPC2 were constructed for two halo fields at projected radii of 19 and 29 kpc (Harris et al. [@Harris99]; Harris et al. [@Harris00]). The colour magnitude diagrams are dominated by old stars. Under the assumption that the ages of the stars in NGC 5128 are similar to those of old globular clusters in our Galaxy, the mean metallicity of the two fields was found to be similar at \[Fe/H\]$\sim-0.4$ with about one third of the stars in a metal poor component and the rest metal rich. The metallicities in the halo fields were compared by Harris et al. ([@Harris02]) to the metallity distribution function (MDF) in a field at 7.4kpc, containing a mixture of outer bulge and inner halo. A broad MDF was found as in the halo fields, peaking at \[Fe/H\]$\sim-0.4$; but subtracting the MDF of the halo fields reveals a peak at slightly higher metallicity ($\sim-0.2$). Much deeper ACS imaging by Rejkuba et al. ([@Rejkuba05]) in a halo field at 36 kpc reached to the horizontal branch (core helium burning population) as well as showing the red giant branch, red clump and AGB bump. Again a broad MDF was found with an average metallicity of $-0.64$, but with a broad tail to higher metallicity ($>0$). The age sensitive indicators imply an average age for the halo of 8$^{+3}_{-4}$ Gyr. Comparison of the MDF in the four fields shows the peak shifting to lower metallicity with increasing projected radius but the distribution is similarly wide at all radii. In order to determine light element abundances of PN in NGC 5128 fairly high signal-to-noise spectroscopy is required. Close to the inner bulge the stellar continuum surface brightness is too high and only the strongest few lines can be measured. However in the outer regions by selecting bright PN, diagnostic lines of He, Ne, Ar and S as well as the brighter lines of O and N can be measured and abundances derived. Walsh et al. ([@Walsh99]) performed deep ESO 3.6m long slit spectra of a few selected PN with a long slit spectrograph and measured lines in addition to the brightest line (\[O III\]5007Å) in five PN. O/H could be determined in two PN and values of 12 + Log(O/H) $\sim$8.5 (i.e. \[O/H\] - 0.2, adopting the Solar oxygen abundance of 8.69 from Scott et al. ([@Scott])) were derived. This initial work has been expanded to deeper spectra of many more PNe with the ESO VLT and FORS1 instrument in multi-slit mode. In section 2 the observations are described and the reduction of the spectra and derivation of the abundances are presented in section 3. The results for the PN line fluxes and derived abundances both for the individual PN and for summations of many spectra are presented in section 4. Since the diagnostic lines for electron temperature and density, important for abundance determination, are weak or undetected, photoionization modelling of all well-detected lines was undertaken using Cloudy and is described in section 5. Consideration of the lack of a gradient in the PN O abundances, comparison of the results with the stellar, photometrically-derived metallicity, and relation of the PN to the stellar populations in NGC 5128 are discussed in section 6. Conclusions are collected in section 7. Observations ============ FORS imaging and spectroscopy ----------------------------- The European Southern Observatory Very Large Telescope (VLT) FORS1 instrument mounted on VLT Unit Telescope 1 was used for the observations described. FORS1 has a number of modes for imaging; for spectroscopy, long slit, multi-slit, or multi-slit mask are available (see Appenzeller et al. [@Appenzeller] for details). The multi-object mode (MOS) allows 19 slitlets of height varying from 20 to 22 $\arcsec$ to be placed over a 6.8$\times$6.8$\arcmin$ field of view. In the regions of NGC 5128 where PNe were catalogued by Hui et al ([@Hui93a]) and ([@Hui93b]), the surface density is such that there are usually enough PNe to be able to fill the slitlets, thus ensuring an optimal match between target density and instrument multiplex. This match was realised in practice for regions over the bright disk of NGC 5128, whilst in the outer regions, around 50% of the slitlets could be utilised. Three regions were selected for MOS spectroscopy to fulfill the following criteria: sample the inner and outer regions at a range of effective radii to detect any abundance gradient from the PNe; sample the major and minor axes; ensure that some of the brightest PNe were observed to maximize the probability of detecting the faintest line species; collect as many PN spectra as possible. Not all of these criteria are mutually consistent, but the three regions selected at radial offset distances of 3.4, 8.7 and 15.4$\arcmin$ provided spectra of 51 PNe at a range of m$_{5007A}$ ( m$_{5007A}$ = -2.5 log F$_\lambda$ (erg cm$^{-2}$ s$^{-1}$) -13.74; Jacoby [@Jacoby89]) from 23.5 to $>$27.0 mag. ### Pre-imaging Direct images with FORS1 and a narrow band \[O III\] filter (called OIII+50, centred at 5005Å and FWHM 57Å) were obtained in December 1999-January 2000[^2] in service mode. Tab. \[PreImaObs\] lists details of the imaging observations whose primary purpose was to provide images for the MOS slit assignment using the FIMS tool. The standard resolution collimator was used giving a field of 6.8$\times$6.8$\arcmin$ with a pixel size of 0.2$\arcsec$ per pixel. The field names are taken from those of Hui et al. ([@Hui93a]). A companion filter (actually a redshifted \[O III\] filter OIII/6000, centred at 5109 Åwith FWHM 61Å) was also used to obtain an \[O III\] off-band subtraction to confirm the reality of the PNe; all the previously catalogued PNe in the fields were confirmed as \[O III\] emission line objects. Seeing at the time of observations was between 0.5 and 1.2$\arcsec$. [l l l l c]{} Position &        $\alpha$              $\delta$ & \[O III\] Exp & Cont. & Date\ (radius, kpc) &  h   m   s       $^\circ$    $\arcmin$    $\arcsec$ & (sec) & (sec) &\ \ F56 (9.6) & 13 25 50.6 -43 06 09.2 & 2$\times$480 & 2$\times$220 & 1999 12 27\ F42 (3.8) & 13 25 09.0 -43 02 29.3 & 2$\times$480 & 2$\times$220 & 2000 01 13\ F34 (17.0) & 13 24 47.2 -43 13 32.2 & 2$\times$480 & 2$\times$220 & 2000 01 13\ ### MOS Spectroscopy FORS1 MOS observations in service mode were conducted in three sessions[^3]. Tab. \[PNSpeObs\] lists the salient details in chronological order. To cover the wavelength range with useful diagnostic emission lines (essentially from \[O II\]3726,3729Å to beyond \[O II\]7320,7330Å), two grisms were employed: grism 600B (called GRIS\_600B+12) has a dispersion of 1.2Å/pixel and, for a centred MOS slitlet, a wavelength coverage of 3450 to 5900Å; grism 300V (named GRIS\_300V+10 and used with a GG435 blocking filter) has a lower dispersion of 2.7Å/pix and a coverage of 4450 to 8650Å. The overlap region between both spectra is 4500-5900Å thus allowing at least the strong \[O III\]4959,5007Å lines to be used to tie the two spectra to a common flux scale. The slit width for the campaigns in 2000 and 2001 was 0.8$\arcsec$ whilst the observations in 2003 had a slit width of 1.0$\arcsec$. The resulting spectral resolutions are 4.8 and 6.0Å for 600B and 10.7 and 13.4Å for 300V observations respectively. The supporting observations of the spectrophotometric standard stars for flux calibration, which were taken with a broad slit of 5$''$ width, are listed in Tab. \[SpecSpeObs\]. [l l l l l l]{} Position &         $\alpha$       $\delta$ & Grism &   Exp. &   Date & Seeing\ &  h   m   s       $^\circ$    $\arcmin$    $\arcsec$ & &   (sec) & & ($\arcsec$)\ \ F56 & 13 25 50.6 -43 06 09.2 & 600B & 4$\times$2400 & 2000 05 02 & 0.8-1.2\ F56 & 13 25 50.6 -43 06 09.2 & 300V & 2$\times$2400 & 2000 05 02 & 0.8-1.2\ & & & & &\ F42 & 13 25 09.0 -43 02 29.3 & 600B & 2$\times$1500 & 2001 03 22 & 0.8\ F34 & 13 24 47.2 -43 13 32.2 & 600B & 2$\times$1500 & 2001 03 24 & 0.9\ F34 & 13 24 47.2 -43 13 32.2 & 600B & 2$\times$1500 & 2001 03 27 & 0.8\ F42 & 13 25 09.0 -43 02 29.3 & 600B & 2$\times$1500 & 2001 03 27 & 0.7\ & & & & &\ F34 & 13 24 47.2 -43 13 32.2 & 300V & 2$\times$1320 & 2003 03 25 & 0.7\ F34 & 13 24 47.2 -43 13 32.2 & 600B & 8$\times$1320 & 2003 04 08 & 0.5\ F42 & 13 25 09.0 -43 02 29.3 & 300V & 2$\times$1320 & 2003 04 24 & 1.0\ F42 & 13 25 09.0 -43 02 29.3 & 600B & 3$\times$1320 & 2003 04 24 & 1.0\ F42 & 13 25 09.0 -43 02 29.3 & 600B & 3$\times$1320 & 2003 04 30 & 0.6\ F42 & 13 25 09.0 -43 02 29.3 & 300V & 2$\times$1320 & 2003 04 30 & 0.5\ F56 & 13 25 50.6 -43 06 09.2 & 300V & 2$\times$1320 & 2003 05 05 & 0.5\ F56 & 13 25 50.6 -43 06 09.2 & 600B & 4$\times$1320 & 2003 05 05 & 0.6\ F34 & 13 24 47.2 -43 13 32.2 & 300V & 2$\times$1320 & 2003 06 02 & 1.1\ F56 & 13 25 50.6 -43 06 09.2 & 600B & 2$\times$1320 & 2003 07 21 & 1.0\ F56 & 13 25 50.6 -43 06 09.2 & 300V & 2$\times$1340 & 2003 07 22 & 0.4\ [l l r l]{} Star & Grism & Exp. & Date\ & & (sec) &\ \ LTT 7379 & 600B & 25 & 2000 05 02\ LTT 7379 & 300V & 7 & 2000 05 02\ & & &\ LTT 7379 & 600B & 25 & 2001 03 24\ LTT 7379 & 600B & 25 & 2001 03 27\ & & &\ EG 274 & 600B & 30 & 2003 04 08\ LTT 7379 & 600B & 25 & 2003 04 24\ LTT 7379 & 300V & 7 & 2003 04 24\ LTT 7379 & 600B & 25 & 2003 04 30\ LTT 7987 & 600B & 100 & 2003 05 05\ LTT 7987 & 300V & 30 & 2003 05 05\ LTT 7987 & 300V & 30 & 2003 06 02\ LTT 7379 & 300V & 7 & 2003 07 22\ Reduction and analysis ====================== All the data frames (flats, science frames on PN and standard stars) were bias subtracted using master bias frames provided by the ESO reduction pipeline contemporaneous with the observing data. Standard IRAF[^4] routines were used for the reduction. Pixel-to-pixel flat fields were constructed from dome flats by extracting the 2D area of each slitlet, collapsing in the cross-dispersion direction, box car smoothing in the dispersion direction and dividing the dome flat by the smoothed version. Identical reductions were performed for the bluer (600B grism) and redder (300V grism) spectra. Wavelength calibration was achieved by fitting 4th order Chebyshev polynomials to the dependence of pixel position on wavelength for the arc lamp lines on each slitlet separately. The separate slitlet images, rebinned to an identical wavelength scale, were then recombined into a single image which was corrected for atmospheric extinction. Individual exposures were combined with clipping to remove discrepant pixels caused by detector bad pixels and cosmic ray events. Flux calibration was achieved by observations of one or more spectrophotometric standard stars, the ones applied being listed in Tab. \[SpecSpeObs\]. The standard star spectra were analysed in an identical way to the PN, except that the one slitlet pertaining to the standard star was reduced. In the few cases where a standard star was not available in the same configuration as the PN on the night of observation, a standard star from another night had to be used. In general conditions were good or photometric, and no large flux calibration discrepancies were found between spectra of the same field observed in different runs (see Tab. \[PNSpeObs\]). Narrow band magnitudes for the spectrophotometric standards were taken from Hamuy et al. ([@Hamuy]) and the PN spectra were flux calibrated using iraf.noao.onedspec routines. Fig. \[MOSspec\] shows an example of the reduced multi-slit spectra for field F56. The unequal wavelength coverage of the spectra is dependent on the relative position of the target centred slitlet within the field. Thus a target to the right edge of the field is truncated to the red but extends to lower wavelengths than a target in the centre of the FORS1 MOS field. Fig. \[MOSspec\] demonstrates that there can be more than one PN per slitlet; in some cases this was planned, by placing the slitlet such that two PN from the Hui et al. catalogue lay on the slit, but in some cases in the two inner fields, PN were spectroscopically detected that were not present in the Hui et al. catalogue, on account of their faintness or proximity to another catalogued PN. These previously uncatalogued PN typically had m$_{5007A} > 27$mag. On the basis of the strongest line - \[O III\]5007Å - the number of PN detectable in the three fields were: 21 in F56; 21 in F42; and 9 in F34. On each slitlet, regions distinct from the PN spectra were located and used to subtract the background, with a 1st or 2nd order fit in the cross-dispersion direction. In determining the spectra of the PN, no attempt was made to perform a separate sky background subtraction since there is galaxy background at all slitlet positions, except perhaps for the slitlets at largest galactocentric distance in the outermost field (F34). 1D spectra of each PN were formed by summing the background subtracted signal along the slit; 7-9 pixels (1.4-1.8$\arcsec$) were used to collect around 60 to 70% of the flux for 0.9$\arcsec$ seeing (assuming a Gaussian PSF). In addition to producing 1D spectra of flux, the same extraction was performed for non-flux calibrated spectra in order to determine the random noise errors on the extracted spectra. No attempt was made to propagate other sources of error, such as flat fielding, into the resultant error vectors As is evident from Tab. \[PNSpeObs\] all fields were observed on more than one occasion. In order not to lose any information, all the flux calibrated and extracted 1D spectra were employed to form averaged spectra per PN. It was expected that the average should be formed using the inverse of the exposure time as weight; however the sky transmission, airmass, seeing and moon phase could all affect the resulting noise in a spectrum, so a simple unweighted mean was finally employed. However the F56 600B spectra obtained in 2003 were found to have lower fluxes but similar signal-to-noise (S/N) as the 2000 observations. These later observations were rescaled in forming the average. For each PN, the various spectra were intercompared before averaging; if a line was found to be significantly discrepant between spectra, a careful examination of the raw 2D spectra was made to determine if a processing step, such as incorrect CR removal in the PN spectrum or the local background, had affected the emission line flux. If a line was deemed to be badly affected, the final spectrum was substituted using only the clean spectrum. The 300V spectra overlap with the 600B spectra for the range 4600 - 5800Å allowing the red lines (He I 5876Å, H$\alpha$+\[N II\], \[S II\], etc) to be placed on the same relative flux scale as the blue lines, primarily using the strong 5007Å line to scale the spectra. The errors on the final spectra were combined using Gaussian error propagation though these combination steps. The resultant spectra thus have a range of line S/N and the magnitude of the propagated errors were checked in two ways: regions of line-free continua were chosen and the root-mean square on the mean was compared with the mean of the statistical errors for the same regions; the rms on the mean value of the fixed \[O III\]5007/4959Å ratio was compared with the mean of the errors on this observed ratio from the Gaussian line fits (see Section 4.1). It was found that the naively propagated errors over-estimated the real errors on the data values as demonstrated by these two tests; the factor varied slightly between data sets but was around 1.7 for each spectrum. The statistical errors in each spectrum were amended by this amount and are those that are listed as the propagated errors on the measured line fluxes (see Tab. \[ObsFlux\]). In order to examine the stellar spectra at the positions of the PN, an attempt was made to subtract the sky background for the bluer (600B) spectra. The F34 field contains the highest proportion of sky over galaxy background, so was used to remove sky from the exposures of the other fields when F34 and other fields were observed on the same night (observing runs in 2001 and 2003 - see Tab. \[PNSpeObs\]). The flux calibrated sky spectrum per pixel for the F34 field, excluding the PN spectra, was formed and subtracted from the F56 and F42 data. Some mismatches in terms of the sky lines were noted, particularly \[N I\], and some of the sky spectra produced poor subtraction in the sense of over-subtraction to short wavelengths. These problematic spectra were not used in forming a mean galactic background spectrum in the two fields F42 and F56. Results ======= Individual PN spectra --------------------- The 1D spectra of the 51 PN in NGC 5128 were analysed by interactively fitting Gaussians to the emission lines with a linear interpolation to the underlying galaxy continuum over the line extent. Errors on the line fits were propagated from the flux errors. The extinction correction was determined by comparison with the case B values for 12000K and 5000cm$^{-3}$ and the Seaton ([@Seaton]) Galactic reddening law with R=3.2 (in the absence of other information on the appropriate reddening law to adopt for NGC 5128), in all cases where at least the H$\alpha$ and H$\beta$ lines were detected. The observed line fluxes (normalised to H$\beta$=100) and errors are presented in Tab. \[ObsFlux\]. The field number and slitlet numbering of the target is provided, the number from the catalogue of Hui et al ([@Hui93b]) together with the $\Delta \alpha$, $\Delta \delta$ offsets (in arcsec) from the position of the nucleus (taken as the SIMBAD coordinate 13$^{h}$ 25$^{m}$ 27.6$^{s}$ -43$^\circ$ 01$'$ 08.8$''$ (J2000)), the observed logarithmic H$\beta$ line flux and error, and the m$_{5007A}$ determined from the observed \[O III\]5007Å line flux and error. The extinction correction c, and $E_{B-V}$ values are listed in Tab. \[DeredFlux\]. The errors on the extinction were not propagated to the dereddened line errors. Seven PNe with weak emission lines, whose large errors on the H$\beta$ line precluded reliable determination of line ratios, were also analysed but are not included in Tab. \[ObsFlux\]; these are listed separately in Tab. \[FaintPN\]. Except for the PN F56\#7 (5509 in Hui et al. [@Hui93b]), the absolute coordinates of these faint PN were determined from the relative positions of the PN on the slitlet with respect to the PN coordinates from Hui et al.([@Hui93b]). The \[O III\]/H$\beta$ ratios are listed in Tab. \[FaintPN\] where H$\beta$ was detected; however given the large uncertainty on the H$\beta$ line flux, these ratios are poorly determined. The PN not detected from the narrow band imaging of Hui et al. ([@Hui93b]) are all faint (m$_{5007A}$ $>$ 26.5) and are near the centre of the galaxy where the stellar continuum is high. Slit spectroscopy allows lower equivalent width emission lines to be detected and can thus probe deeper than filter imaging. ------------- ------------ ----------------- --------------------- -------------------------------------- ------------- -------------------- MOS Slitlet Hui et al.        $\alpha$      $\delta$      Offsets m$_{5007A}$ \[O III\]/H$\beta$ No.   ($h$ $m$ $s$) ($^\circ$ $'$ $''$)      $\Delta \alpha$ $\Delta \delta$ mag.$^\ast$ approx.     (arcsec) F56\#7 5509 13 25 56.41 $-$43 06 23.2 ($+$293.6,$-$314.4) 26.2$^\dag$ 29 F56\#13a 13 25 46.50 $-$43 04 57.3 ($+$207.3,$-$228.5) 27.6 4 F56\#13c 13 25 45.78 $-$43 04 53.3 ($+$199.4,$-$224.5) 27.5 13 F42\#12a 13 25 15.21 $-$43 03 12.5 ($-$135.9,$-$123.7) 26.5 23 F42\#14a 13 25 18.20 $-$43 02 23.1 ($-$103.1,$-$74.3) 26.8 22 F42\#14c 13 25 19.24 $-$43 07 19.9 ($-$91.7,$-$71.1) 27.1 9 F42\#16b 13 25 23.13 $-$43 02 49.6 ($-$49.0,$-$100.8) 27.2 9 ------------- ------------ ----------------- --------------------- -------------------------------------- ------------- -------------------- The comparison between the m$_{5007A}$ magnitudes from this work, converting the measured 5007Å flux to m$_{5007A}$, and those from Hui et al. ([@Hui93b]) is shown in Fig. \[O3mags\]. For Field F56, the observed m$_{5007A}$ magnitudes are on average 0.33$\pm$0.12 mag. brighter than the Hui et al. measurements, excluding the two faintest PN. The absolute H$\beta$ flux calibration was adopted from the May 2000 600B observations; presumably the spectrophotometric standard was taken under slightly poorer conditions, leading to an apparent higher absolute flux for the PN observations. For the F42 field, the m$_{5007A}$ magnitudes are 0.67$\pm$0.21 mag. fainter than the Hui et al. ones, which can be attributed to an offset of the slits from the PN positions, since all the F42 observations display lower m$_{5007A}$ than the imaging observations. For the F34 field, the m$_{5007A}$ magnitudes are on average 0.51$\pm$0.21 mag. fainter than the imaging magnitudes, which are consistent with a $\approx$0.2$''$ slit offset for 0.8$''$ wide slits in 1.0$''$ seeing (assuming a Gaussian seeing profile). Region-averaged PN spectra -------------------------- Single spectra consisting of the sum of all the observed brighter PN (i.e. those contained in Tab. \[ObsFlux\]) in each of the three fields were constructed. The primary motivation was to enable the detection of fainter diagnostic lines (such as He I, \[S II\], \[Ar IV\], etc.) which were not detected, or were marginally detected, on the individual spectra. This was also an explorative study to measure how different the derived abundances would be from the summed spectra in comparison with the ensemble of the individual PN abundances. This aspect is of interest for studies of more distant PN populations where all line detections from individual PN spectra have low S/N and summed spectra are mandatory for measurement of abundances. The radial velocity of each PN was measured from the strong lines in the spectrum (principally \[O III\] 4959 and 5007Åand H$\alpha$) and employed to shift all the spectra in a single region to zero radial velocity. The spectra were then summed and the resulting emission line spectra fitted in the same way as the individual PN spectra (Section 4.1). Figure 2 of Walsh et al. ([@Walsh05]) shows the resulting three spectra. Tab. \[RegObsFlux\] lists the resulting observed fluxes. The extinction was calculated from the H line ratios with the same assumptions as for the individual PN spectra (Sect. 4.1) and the dereddened line fluxes are listed in Tab. \[RegDerFlux\]. The ionic abundances of the elements corresponding to the detected species can be determined from the dereddened line ratios with respect to H$\beta$ (Tab. \[RegDerFlux\]). The electron temperature of the O$^{++}$ emitting region can be directly measured from the 5007/4363Å, but also required for abundances is an electron density measure. The only directly determined N$_e$ value is from the \[S II\]6716/6731Å ratio, which has two disadvantages in that it samples the minority low ionization nebular volume and has a relatively low critical density for collisional de-excitation. Coppeti & Writzel ([@CoWR02]) show that \[S II\] N$_e$ measured in PN is similar to N$_e$ measured for higher ionization species, such as \[Cl III\] and \[Ar IV\]. Adopting a value of N$_e$ of 5000 cm$^{-3}$ for the NGC 5128 PN appears to be fair and will not lead to bias on the abundance estimates, unless the density is very high ($^{>}_\sim$20000 cm$^{-3}$). In the high density case, the \[O III\]5007/4363Å line ratio is decreased by collisional de-excitation, leading to too high a value of T$_e$. Tab. \[Regabunds\] lists the derived ionic abundances for the summed region spectra. The sources for the atomic data for the collisionally excited lines were taken from the compilation in Liu et al. ([@Liu2000]) and the routines from Storey & Hummer ([@StorHum]) were used for the recombination line emissivities of H and He. The corrections for unseen stages of ionization (ionization correction factors, ICFs) were taken from Kingsburgh & Barlow ([@KiBa]). The errors in the total abundances do not take into account the errors in the electron temperature; if these are propagated the errors rise to $\pm$0.10 for the O abundance. The He/H abundance is only listed if the lines at 4471 and or 5876Åwere detected; if He II 4686Å was detected in addition, it was added to derive the total He/H. If only He II was detected no He/H is listed. --------------------------- --------------- ----------- ------- ----------- ------- ----------- ------- Species $\lambda$ (Å) F$_{Obs}$ $\pm$ F$_{Obs}$ $\pm$ F$_{Obs}$ $\pm$  \[O II\] 3727 28.6 1.7 23.9 2.2 54.5 2.9  \[Ne III\] 3868 84.0 1.7 62.3 1.5 74.0 1.9  H I 3889 17.3 4.4 13.3 1.6 14.3 1.2  \[Ne III\] + H$\epsilon$ 3970 33.5 1.0 26.9 1.5 34.4 1.9  H$\delta$ 4101 21.1 1.2 10.7 1.2 21.3 1.1  H$\gamma$ 4340 46.7 1.9 41.5 2.0 40.8 1.5  \[O III\] 4363 8.7 0.7 6.1 1.3 8.5 0.6  He I 4471 8.0 1.0 6.5 1.0 5.0 3.3  He II 4686 10.7 1.2 9.3 1.9 13.1 1.1  H$\beta$ 4861 100.0 0.0 100.0 0.0 100.0 0.0  \[O III\] 4959 391.3 5.7 396.4 7.1 417.1 7.1  \[O III\] 5007 1159.6 16.0 1213.6 20.2 1247.8 20.6  He I 5016 3.7 0.4  \[N I\] 5199 3.2 0.2  He I 5876 14.5 1.9 29.9 5.6  \[N II\] 6548 50.3 1.2 62.7 4.4 62.7 4.6  H$\alpha$ 6562 390.6 3.8 422.7 9.8 412.8 14.7  \[N II\] 6583 152.2 1.8 149.2 4.3 115.7 5.5  He I 6678 4.6 0.8 7.1 4.6  \[S II\] 6716 4.5 0.4 7.3 2.8  \[S II\] 6730 10.9 1.1 27.9 3.0  \[Ar III\] 7133 28.5 1.1 22.9 3.1  \[O II\] 7325 27.2 2.4 19.6 4.6 log F(H$\beta$) -15.46 0.01 -15.42 0.01 -15.78 0.01 m$_{5007A}$ 22.24 0.03 22.10 0.04 22.98 0.04 --------------------------- --------------- ----------- ------- ----------- ------- ----------- ------- --------------------------- --------------- ----------- ------- ----------- ------- ----------- ------- Species $\lambda$ (Å) F$_{Obs}$ $\pm$ F$_{Obs}$ $\pm$ F$_{Obs}$ $\pm$  \[O II\] 3727 37.0 2.3 33.0 3.0 73.8 3.9  \[Ne III\] 3868 105.9 2.1 83.3 2.1 97.2 2.5  H I 3889 72.0 5.5 17.7 2.1 18.7 1.5  \[Ne III\] + H$\epsilon$ 3970 41.4 1.2 35.0 1.9 44.2 2.4  H$\delta$ 4101 25.4 1.4 13.5 1.4 26.4 1.4  H$\gamma$ 4340 53.1 1.8 48.7 2.4 47.4 1.7  \[O III\] 4363 9.8 0.8 6.6 1.3 9.9 0.6  He I 4471 8.9 1.0 7.4 1.2 5.6 3.6  He II 4686 11.2 1.3 9.8 2.0 13.8 1.1  H$\beta$ 4861 100.0 0.0 100.0 0.0 100.0 0.0  \[O III\] 4959 381.8 5.6 384.6 6.9 405.5 6.9  \[O III\] 5007 1118.3 15.4 1160.0 20.0 1196.3 19.7  He I 5016 3.6 0.4  \[N I\] 5199 3.0 0.2  He I 5876 11.6 1.5 22.8 4.3  \[N II\] 6548 36.5 0.9 42.1 2.9 43.0 3.1  H$\alpha$ 6562 282.9 2.8 282.8 6.6 282.8 10.1  \[N II\] 6583 109.9 1.3 99.4 2.9 79.0 3.7  He I 6678 3.3 0.5 4.7 3.0  \[S II\] 6716 3.2 0.3 4.8 1.8  \[S II\] 6730 7.7 0.8 18.1 1.9  \[Ar III\] 7133 19.1 0.8 13.9 1.9  \[O II\] 7325 17.9 1.6 11.6 2.7 c 0.44 0.03 0.55 0.07 0.51 0.12 log F(H$\beta$) -15.03 0.01 -14.88 0.01 -15.27 0.01 m$_{5007A}$ 21.18 0.03 20.74 0.04 21.74 0.04 --------------------------- --------------- ----------- ------- ----------- ------- ----------- ------- Species ------------------------------ ---------- -------- ---------- -------- ---------- -------- Value $\pm$ Value $\pm$ Value $\pm$  \[O III\] (5007+4959)/4363Å 153.1 12.6 234.0 46.2 161.8 10.0  \[O III\] T$_e$ (K) 10970 300 9620 560 10780 220  O$^{+}$ 1.76E-5 1.1E-6 2.69E-5 2.4E-6 3.52e-5 1.8E-6  Ne$^{++}$ 7.93E-5 1.6E-6 1.03E-4 2.5E-6 7.28E-5 1.9E-6  He$^{+}$ 0.173 0.019 0.142 0.023 0.109 0.070  He$^{++}$ (4471Å) 0.0094 0.0011 0.0081 0.0016 0.0115 0.0009  O$^{++}$ 3.091E-4 4.3E-6 4.997E-4 8.6E-6 3.307E-4 5.4E-6  He$^{+}$ (5876Å) 0.084 0.011 0.160 0.031  N$^{+}$ 1.83E-5 2.2E-7 2.28E-5 6.6E-7 1.32E-5 6.2E-7  S$^{+}$ 4.40E-7 4.6E-8 1.39E-6 1.5E-7  Ar$^{++}$ 1.50E-6 6.3E-8 1.46E-6 2.0E-7 12 + Log(O/H) 8.53 0.01 8.74 0.01 8.59 0.02 He/H 0.182 0.019 0.150 0.023 0.122 0.0070 Ne/O 0.257 0.006 0.206 0.006 0.220 0.007 N/O 1.04 0.07 0.85 0.08 0.38 0.03 12 + log(S/H) 6.55 0.05 7.06 0.04 12 + log (Ar/H) 6.49 0.04 6.44 0.06 Stellar absorption lines ------------------------ The slitlets sample a spectroscopic background consisting of sky with a substantial contribution from the galaxy continuum, except for the outermost field, F34. The regions of the slitlets not occupied by planetary nebula spectra can provide the spectrum of the stellar continuum of NGC 5128. The inner fields, F56 and F42, contain no slitlets sampling the sky free of stellar continuum, and so no simultaneous sky subtraction can be performed. For the outer field F34 however, the stellar continuum is very weak and was not detected, as shown by the absence of a gradient in the background with radial offset from the galaxy centre. The 600B F34 spectrum taken on 2003-04-08 was thus used to produce a candidate mean sky spectrum for subtraction from the spectra for the two inner fields (F56 and F42). This sky was interactively subtracted from the F42 and F56 spectra for each slitlet. Good results were found only for the F42 spectra taken in April 2001 and for the F56 field spectra taken in May 2003. These results are attributable to very differing sky contributions to the galaxy spectra at the different observation epochs. For the F56 field, the sky-subtracted data show very low continuum in the five slitlets (1-5) furthest from the galaxy centre; these slitlets were not considered for analysis of the galaxian stellar continuum. Fig. \[Galcont\] shows the resulting stellar continuum from fields F56 (blue line) and F42 (red line), for all the slitlets summed. Strong CaII H & K, H$\gamma$, H$\beta$ and Mg II lines typical of an intermediate-old stellar population are clearly visible and are indicated on the Figure. The absorption lines in the individual galaxy continuum spectra for each slitlet were analysed by computing Lick indices (Worthey et al. [@Worthey]). The spectra were smoothed with a Gaussian to simulate the $\sim$8Å resolution of the Lick/IDS stellar spectra and shifted to zero velocity before the equivalent widths were computed. Errors on the indices were computed by propagating the flux errors in the equivalent width determination. A single radial velocity of 580 kms$^{-1}$ was used to shift the spectra to rest wavelength. Errors of $\pm$50 kms$^{-1}$ in this radial velocity cause an offset in the bounds of the Lick indices and can introduce errors up to about 0.2Å in EW. Three of the Lick indices (H$\beta$, Mgb and the combined Fe index $<$Fe$>$, defined as 0.5\*\[EW(Fe$_{5270A}$) + EW(Fe$_{5335A}$)\], are plotted in Fig. \[Licks\] as a function of the projected position of the slitlet centres from the centre of the galaxy. The $<$Fe$>$ index shows no radial gradient nor does the H$\beta$ index, although the latter shows higher scatter to larger projected distance from the nucleus. The Mg b index shows a weak negative radial gradient, typical of early type galaxies (c.f. Davies et al. [@Davies93]). Chemical abundance determination ================================ Extragalactic PN abundances --------------------------- The electron temperature (T$_e$) is required In order to determine reliable chemical abundances for extra-galactic PNe from the collisionally excited lines on account of the exponential dependence of line emissivity with temperature. The electron density (N$_e$) is also required since these lines have collision cross sections dependent on density. In order to avoid large corrections for unseen ionization species a range of lines of different ionization from neutral and singly ionized up to high ionization is also desirable; however the well-established empirical technique using ICFs can compensate for the lack of some ionization species from the optical wavelength range. As is typical for faint and distant PNe such as observed here, these criteria are not fully met. In the absence of measurements of weak T$_e$ and N$_e$ diagnostic line ratios (such as \[O III\] 5007/4363Å and \[S II\]6716/6731Å or \[Ar IV\]4711/4740Å), the practice adopted by Stasińska et al. ([@SRM98]) was to use the upper limits to the strength of these faint lines to constrain T$_e$ and N$_e$ within reasonable limits, taking typical values for PN from studies in the Milky Way and Magellanic Clouds (e.g. T$_e$ $\sim$12000K, N$_e$ $\sim$5000 cm$^{-3}$). Jacoby & Ciardullo ([@JC99]) introduced a different approach to abundance determination for extra-galactic PNe (in this case for M31): that of photoionization modelling using the very well-established Cloudy (Ferland et al. [@Ferland]) code. The gains of this latter method are that both strong lines with low errors, weak lines with large errors and upper limits can all be used in arriving at a satisfactory model, [*and*]{} the parameters of the central stars (luminosity, $\cal L$ and effective temperature, $T_{eff}$) are derived as part of the photoionization modelling. Extra-galactic PNe are spatially unresolved but have known distances and absolute fluxes available though m$_{5007\AA}$ photometry. An input data set for photoionization modelling can be assembled, given some simplifying assumptions in the absence of more detailed information; in particular, the nebular geometry and information on the stellar atmosphere are not available. A black body has been shown to be an acceptable assumption for PN central stars (Howard et al. [@Howard97]), which leaves the nebular geometry. Taking a spherical shell of constant density as a baseline, Jacoby & Ciardullo ([@JC99]), hereafter JC99, compared nebular abundances using directly measured T$_e$ and N$_e$ values and ICFs to check that photoionization modelling was indeed a valid approach. The approach was shown to work well. Additional complexity, such as two zone density structure and the presence of dust within the nebula, was found to be required to satisfactorily match the observed spectra in some cases. Magrini et al. ([@Magrini]) followed the same procedure as JC99 in determining abundances of PNe in M33. They also tested the method by applying it to the integrated spectra of well-observed Milky Way PNe and compared the model abundances to those from the ICF method, finding agreement within 0.15 dex for O/H and 0.3 dex for N/H. In addition Magrini et al. ([@Magrini]) compared abundances derived from Cloudy models with those from ICFs for three PNe of their M33 sample. They also considered comparisons employing model atmospheres and an $r^{-2}$ density in the nebular shell, and found no major discrepancies. The technique, involving the development of a large number of models in working to a satisfactory match of observed and model spectrum, is labour-intensive and therefore not suitable for large samples. It also requires simplifying assumptions. The technique, however, is well suited to the present sample of PN spectra in NGC 5128 where the range of abundances is of particular interest and the data set is limited; it also serves as a probe of the star formation history. Only 10 of the PN spectra presented in Tables \[ObsFlux\] and \[DeredFlux\] have detectable \[O III\]4363Å, enabling direct measurement of the electron temperature, and only 7 have an electron density sensitive ratio (\[S II\]6716/6731Å) measured. The typical S/N on the weak 4363Å line ($\leq 3$) implies that T$_e$ errors of 1250K at 11000K affect abundance determination, leading to errors of $^{+0.17}_{-0.13}$ on log (O$^{++}$/H$^{+}$). For the PNe without detectable diagnostic line ratios, two approaches are possible: employ the region-averaged spectra to assign T$_e$ and N$_e$ values to the single PN, although these values may be substantially in error for particular nebulae; or, employ photoionization modelling of the strong lines to find the best-matching model that does not violate the upper limit constraints for the weak lines. Modelling procedure ------------------- The simplest possible photoionization model was employed: a black body central star characterized by its temperature and luminosity, the latter given by the H$\beta$ luminosity derived from the m$_{5007\AA}$ magnitude (Hui et al. [@Hui93b]), the reddening and the dereddened 5007Å/H$\beta$; a spherical shell with the inner radius set to a small value (0.005 pc) to ensure high ionization emission close to the central star, but not so close that instabilities arise in the model; the outer radius set to a large value (0.5 pc) since the nebulae are expected to be optically thick, as given the high luminosity of the PNe observed in NGC 5128 (i.e. within 2 mag. of the peak of the PNLF); and an initial set of abundances. For the abundances of He, N, O, Ne, S and Ar, the initial values were taken from the ICF analysis for the integrated spectra listed in Tab. \[Cloumods\] and the C/O ratio taken as 0.5. However these values were immediately allowed to vary to fit the individual spectra (with C varying in lock-step with O) and should not be seen as prejudicing the individual determinations from the higher S/N integrated spectra. The initial assumption of an optically thick nebula could also have been relaxed in the modelling process but was not found to be demanded by the fits to the PN spectra. The philosophy was to aim to match the dereddened lines fluxes within the listed errors for all the spectra listed in Tab. \[DeredFlux\] (except F56\#9, F56\#17, F42\#15b and F42\#17 without detected \[O II\] or He lines), thus 40 PN spectra. These PNe represent the upper 2.1 mag. of the PNLF. The modelling procedure followed closely that outlined by JC99 and also Marigo et al. [@Marigo]. An initial default density of 5000cm$^{-3}$ was used. The initial estimate of $T_{eff}$ was taken from He II 4686Å/H$\beta$; in the absence of 4686Å, its upper limit and the 5007Å/H$\beta$ ratio was used. The presence of 4686Å enabled $T_{eff}$ to be refined within a few runs of Cloudy; if 4686Åwas not present, both $T_{eff}$ and O/H had to be adjusted. Then He was adjusted to match the He I lines, with minor adjustments to T$_e$ for the concomitant changes in He II flux. O/H and the density were adjusted to match \[O II\]3727Å and, if present, the \[O II\]7325Å quadruplet. For large changes in O abundance, Ne and C were initially changed in proportion. Once the basic physical parameters ($T_{eff}$, density, He, O) were roughly determined, Ne, N, Ar and S were altered to match the observed lines. In the final phase all lines and $T_{eff}$ and density were subject to small changes to improve the fit, where possible. A guide to the goodness of fit of the observed and predicted line strengths was formed, taking the sum of the line flux differences normalised by the S/N: thus larger discrepancies for weak lines could be weighted lower. A discussion of the goodness of the matches and the estimation of errors in presented in section 5.4. In general around 10 iterations per PN was enough to reach a satisfactory match to the spectrum but in more difficult cases (F56\#12b, F42\#10 and F34\#7 for example) up to 30 iterations were required. In total, $\sim$600 Cloudy models were run to complete this analysis. ---------- -------- ----------- --------- -------- --------------- ------------ ------ --------------- ------------- --- PN \# T$_{eff}$ Log $L$ Ne    Quality$^{3}$ (He/H) (N/H) (O/H) (Ne/H) (S/H)$^1$ (Ar/H)$^2$ (kK) ($L_{\odot}$) (cm$^{-3}$) F56\#1 11.06 8.20 8.40 7.70 6.55 6.00 60 3.68 5000 b F56\#2 11.15 7.85 8.43 7.72 6.00 155 4.20 10000 a F56\#3 11.06 8.30 8.48 7.78 6.55 6.00 64 3.66 2000 b F56\#4 11.06 8.30 8.38 7.70 6.55 6.50 78 3.75 5000 b F56\#5 11.06 7.40 8.55 7.85 6.35 150 3.74 10000 c F56\#6 11.20 8.30 8.64 7.95 6.60 75 4.15 6000 b F56\#8 11.20 8.62 8.60 7.80 6.80 6.50 63 3.77 9000 b F56\#10 11.10 8.20 8.38 7.60 6.55 6.60 88 4.03 15000 b F56\#11 10.90 8.42 8.54 7.70 7.00 6.40 90 3.67 22500 a F56\#12a 11.10 8.53 8.80 8.10 6.55 6.40 200 3.86 10000 a F56\#12b 11.00 8.80 8.55 7.85 6.50 45 3.93 35000 b F56\#13b 11.00 7.60 8.52 7.70 6.20 105 3.76 20000 a F56\#14 11.06 8.40 8.46 7.70 6.55 6.30 95 3.86 15000 b F56\#15 11.06 7.90 8.47 7.70 6.55 6.20 70 3.86 10000 c F56\#16 11.06 8.20 8.47 7.60 6.55 6.20 70 3.91 10000 b F56\#18 11.06 8.63 8.51 7.85 6.90 90 3.87 15000 b F42\#1 11.12 7.85 8.45 7.65 6.90 6.20 80 3.83 10000 c F42\#2 11.06 8.19 8.68 8.03 155 4.05 10000 a F42\#3 11.06 8.26 8.44 7.50 6.40 50 3.99 10000 c F42\#4 11.01 8.12 8.03 7.10 6.40 6.20 86 4.00 12000 a F42\#6 11.15 7.30 8.44 7.60 7.10 63 3.63 13000 b F42\#7 11.06 8.00 8.51 7.70 127 3.64 10000 b F42\#8 11.06 8.20 8.46 7.70 93 3.86 15000 b F42\#9 11.12 8.20 8.45 7.70 6.50 74 3.78 30000 b F42\#10 11.28 8.18 8.10 7.40 6.85 6.10 120 4.29 2500 a F42\#11 11.06 8.25 8.28 7.35 6.85 6.20 100 3.72 10000 c F42\#12b 11.06 8.22 8.69 8.07 108 4.09 10000 b F42\#13 11.06 8.28 8.90 8.25 190 3.84 16000 a F42\#14b 11.06 8.20 9.00 8.40 6.70 220 4.06 20000 a F42\#16a 11.06 8.28 8.72 7.92 100 3.78 14000 a F42-18 11.06 7.68 8.49 7.45 120 4.35 10000 c F34\#1 11.06 8.25 8.73 8.00 5.65 71 3.89 8500 a F34\#2 11.23 7.62 8.09 7.38 90 4.11 1500 a F34\#4 11.06 8.10 8.46 7.70 6.30 67 3.90 8000 b F34\#7 11.06 8.30 8.91 8.30 6.50 82 3.99 13000 a F34\#11 11.16 7.40 8.53 7.78 180 3.59 30000 a F34\#12 11.16 7.35 8.48 7.30 167 3.13 30000 a F34\#14 11.06 7.90 8.62 7.82 70 3.61 20000 b F34\#15 11.06 7.40 8.43 7.76 123 3.90 10000 b F34\#16 11.06 8.28 8.63 7.95 6.55 180 3.70 5000 a ---------- -------- ----------- --------- -------- --------------- ------------ ------ --------------- ------------- --- Results and scrutiny -------------------- Tab. \[Cloumods\] lists the derived abundances for He, N, O, Ne, Ar, and S with respect to H where lines of these elements were detected, together with the stellar luminosity, black body temperature and the constant shell density. The quality of the derived parameters are broadly classified into three categories based on: He II (and or \[O III\]4363Å) and \[O II\] both detected (15 PN: grade a); \[O II\] but not He II detected (18 PN: grade b); neither He II nor \[O II\], but at least He I, \[O III\]5007Å, \[Ne III\]3868Å and \[N II\]6583Å, detected (6 PN: grade c). Ten of the quality “a" spectra had 4363Å detected and the models tailored to fit the strong lines (i.e. a model was not tweaked to exactly fit the 4363Å line) were found to match the 4363Åline within the $1\sigma$ errors in all cases except one (F42\#16a; 4207 in Hui et al. [@Hui93b]). The results presented are not claimed to be unique, as the adopted geometry of the nebulae is the simplest possible and the use of a black body atmosphere does not produce the most satisfactory fits to the level of ionization in detailed photoionization modelling of Galactic PNe (e.g. Wright et al. [@Wright]). However these simple models are capable of matching the observed variety of spectra with a very plausible range of abundances, central star temperatures and nebular densities. Carbon is the most important coolant after O, but no lines of C were detected so there are no direct constraints on the C/H abundance. In the absence of other evidence, the abundance of C was assumed to be similar to O (C/O=0.5) and to vary in lock-step with O as the abundance of O was altered to match the spectrum. Once an adequate model had been found, tests were performed on four of the spectra, altering the C abundance by a factor 2.5 and refitting for the other species. A maximum difference in 12+log(O/H) of -0.10 was found for 12+log(C/H) higher by +0.40; for 12+log(C/H) lower by 0.40, a maximum difference of +0.02 in 12+log(O/H) was found. Smaller changes in N and Ne abundances were required to match the spectra with these altered C abundances. Thus the abundances are not very sensitive to modest changes in the assumed C/O ratio. For the best observed PN spectrum (viz. the target with the brightest m$_{5007\AA}$) PN, (F56\#2; 5601 in Hui et al. [@Hui93b]), a set of Cloudy models were run with NLTE model atmospheres from the Tübingen compilation (Rauch [@Rauch03]), and also with the presence of dust inside the ionized region. H, He and PG1159 model atmospheres were employed. While the H model atmosphere showed a much lower $T_{eff}$, an He or PG1159 atmosphere at 170-180 kK produced satisfactory fits, with the resulting O/H lower by up to 0.16 dex. Including dust with an atmosphere model required that O/H be reduced by up to 0.05 for a gas-to-dust mass ratio of 1000; as the primary coolant of the nebula, the lower O/H is compensated by the enhanced cooling from dust. Stasińska ([@Stas2002]) questioned the uniqueness of abundance determinations even in the case that a direct measure of T$_e$ from e.g. \[O III\]4363Å is available. Depending on the temperature of the central star, a strong gradient in the electron temperature can exist within the nebula and a single value of T$_e$ may not be a good representation for the bulk of the emission. The T$_e$ gradient is far less steep for PNe with hot central stars ($\sim$100000K) than for cooler central stars and the use of a single T$_e$ should yield reliable abundances. Stasińska ([@Stas2002]) presented photoionization models for a Milky Way Bulge PN with a central star temperature of 39000K that can be equally well fit by a lower abundance (\[O/H\]=-0.31) and by a higher abundance model (\[O/H\]=+0.39)[^5]. 95% of the PNe observed in NGC 5128 appear to have $T_{eff} \geq$ 60kK, so the likelihood of a double-valued abundance is very low. However since the lowest $T_{eff}$ values derived in the Cloudy modelling (Tab. 7) are $\sim$50kK, then uniqueness may be a more significant concern for these few objects. A concerted attempt was made to fit the spectra of the PNe with the lowest derived T$_{eff}$ by low and high metallicity (Z) models. Here there is the liberty to change the stellar temperature since the He II 4686/H$\beta$ ratio can be taken as an upper limit based on the errors on local lines. Two PN were found that could not be distinguished in terms of lower and higher Z Cloudy models. In the case of F56\#12b (a class b spectrum with the \[O II\] line detected but not He II), the higher Z solution with 12+log(O/H) of 8.55 for a BB $T_{eff}$ of 49 kK could not be distinguished from a model with $T_{eff}$ of 73 kK and 12+log(O/H) of 8.06 if the density was decreased from $4.0\times10^{4}$ to $1.8\times10^{4}$ cm$^{-3}$. For F42\#3 (class c spectrum without \[O II\] and He II lines detected), a higher Z solution with 12+log(O/H) of 8.44 for $T_{eff}$ of 50 kK was indistinguishable from that with 12+log(O/H) of 8.01 but $T_{eff}$ of 73kK, for the same density in the shell. For the class c PN, an attempt was made to match each spectrum by a forced lower Z model but no convincing matches could be found; nevertheless, one may exist given the rather weak constraints implied by these lower quality spectra. Conservatively, the higher abundance solutions were adopted (in Tab. \[Cloumods\]) on the grounds that they do not differ strongly from the values for other PNe at similar radii. The two PNe with alternate low O/H values are not improbable and are comparable to the abundances of PNe in dwarf galaxies (e.g. Dopita & Meatheringham, [@Dopita91] and Leisy & Dennefeld, [@Leisy06]); F42\#4 also shows a similarly low value of O/H but has a well-detected spectrum. In the context of a giant elliptical galaxy, such low abundance values are surprizing because they imply low metallicity progenitor stars,. Radial velocities of these PNe (available for F56\#12b and F42\#4 from Peng et al. [@Peng04]) do not show ’peculiar’ velocities, being within 1$\sigma$ of the mean velocity of NGC 5128. There is no indication of these PNe being obvious ’halo’ objects or of resulting from a recent low-mass galaxy interloper not yet in dynamical equilibrium with the parent galaxy. The abundances of the individual PN from Tab. \[Cloumods\] were combined and compared to the abundances derived from the integrated region spectra with empirically determined temperatures and ICFs presented in Tab. \[Regabunds\]. The means were determined by weighting the individual abundances by m$_{5007A}$. For the combined region F56, the weighted mean abundances from Tab. \[Cloumods\] for O agreed to within 0.01 dex and for F34 to within 0.06 dex (weighted means of the log and the fractional O/H abundances were calculated with similar results). For field F42, the discrepancy on the weighted mean of the log O abundance was larger, with a value up to 0.22 dex lower than the value in Tab. \[Regabunds\]. The difference for F42 is quite striking in this comparison; the electron temperature was measured to be $\sim$1000K lower than for the other two regions. It is suggested that the value of T$_e$ was underestimated in this merged spectrum due to a poorly determined \[O III\]4363Å flux. A further check on the integrated region spectra was performed by running photoionization models for the region spectra presented in Tab. \[RegDerFlux\]. The abundances were compared to the empirical abundances derived using ICFs (Tab. \[Regabunds\]). Satisfactory Cloudy models could not be fitted without departing from the simplifying assumptions used for the individual PN models. In particular, when the stellar temperature was chosen from the He II/H$\beta$ ratio, this could not be made compatible with the electron temperature derived from the \[O III\]5007/4363Å ratio. If He II4686Å matched the observations within the errors, then \[O III\]4363Å was predicted much too strong (higher T$_e$); when \[O III\]4363Å matched the model, He II was modelled too low (lower $T_{eff}$). The addition of modest amounts of dust inside the ionized region of the photoionization model to provide extra heating through photoelectric emission from the grains did not resolve this discrepancy. However allowing for these modelling issues, the range of O/H was close (within 0.1 dex) to the empirical values listed in Tab. \[Regabunds\] for the sum of PNe in fields F56 and F34, but only within 0.2 dex for F42. These demonstrations show that summing PN spectra provides a useful indicator of the mean O abundance in cases when the line detections have low S/N, such as for PNe in more distant galaxies. This conclusion is subject to the condition that the PN abundances do not differ strongly, as in this case. Méndez et al. ([@Mendez05]) have presented lower limits on O and Ne abundances based on just such a summed spectrum for 14 PNe in NGC 4697 (an elliptical at 11 Mpc). Towards a comparative study of models ------------------------------------- A number of Cloudy photoionization models were run in order to arrive at an adopted model for each PN matched in Tab. \[Cloumods\]. The models presented cannot be claimed to be unique but represent a feasible match to the spectra in terms of the abundances and physical conditions within the restricted set of parameters (black body atmosphere for the stars, linear density law, etc). Magrini et al. ([@Magrini]) and JC99 discussed the accuracy of photoionization models to their PN spectra. In general the “a" quality spectra of the NGC 5128 PN approach those of the fainter sample of the much closer PNe observed in M33 and M31 by these authors. Simply adopting the range of models within the statistical errors on the line ratios underestimates the real uncertainties on the abundance determinations. For a selection of models, independent Cloudy runs were performed by two of the authors and the results compared (c.f. Jacoby et al. [@Jacoby97]) and with the ICF determinations in the cases where \[O III\]4363Å was detected (using an adopted N$_e$ value of 5000 cm$^{-3}$). While a full error analysis is outside the scope of this work, a limited investigation was attempted. A range of parameter values around the adopted ones were explored to determine the sensitivity of the models, given the errors on the spectra. It would be very time consuming to perform an investigation of the likely parameter range for all PN in Tab. \[Cloumods\], so two PN were selected. F56\#2 was chosen as a high S/N case with both \[O III\] 4363Å and He II 4686Å well detected (a class “a" spectrum in Table \[Cloumods\]) and a high stellar temperature. F34\#1 was chosen as a PN with lower S/N and only a marginal detection of He II 4686Å (class “b" in Table \[Cloumods\]) and a lower stellar temperature and representing the typical average quality of spectra. Cloudy models were run for these two exemplars varying a number of parameters about the adopted solution in Tab. \[Cloumods\]: O abundance varied by $\pm$0.2 dex; stellar black body temperature by 10-15% (20000K for F56\#2, with $T_{BB}$ 155000K and 10000 K for F34\#1, with $T_{BB}$ 71000 K); the density by a factor of two; and a model atmosphere rather than a black body. For each modified parameter set, Cloudy models were run to match the spectra within errors as far as possible by freely varying all the other, non-displaced, parameters. Even this limited analysis involved running more than 180 separate Cloudy models. The results of the comparison of Cloudy models are summarised in Tables \[VarF56\] and \[VarF34\]. The third columns list the dereddened spectrum from Tab. \[DeredFlux\] and the 4th column the spectrum resulting from the adopted model given in Tab. \[Cloumods\]. The successive columns list the resulting spectra from the incremented parameters. The lower half of each table lists the stellar temperature, density and abundances corresponding to the spectrum in the upper part of the same column; values in column 4 being identical to those in Tab. \[Cloumods\]. An overall merit factor, $FoM$, of the match of the model spectrum with the observed spectrum, taking account of the S/N of the fluxes, is defined by: $$FoM = \sum_{i=1}^{n} |((F_{obs}-F_{mod})/F_{obs})|\times (F_{obs}/Err_{obs}))$$ where $F_{obs}$ and $Err_{obs}$ are respectively the dereddened flux and error and $F_{mod}$ the model flux; the summation is taken over the number of emission lines. Since some lines are in fixed ratio, such as H lines, a subset of the lines was adopted to dispense with some redundant information (except for the more important lines such as H$\alpha$ and H$\beta$, the weak He I lines and the \[O III\] 4959Å  line). Table \[VarF56\] presents the comparative models for the bright PN F56\#2 (5601 in Hui et al. [@Hui93a]). The comparison of the observed and model spectra makes it clear that the He I 5876Å line is strongly under-estimated, most probably because of the poor removal of the strong Na I telluric emission. In general satisfactory models could be found, except for the cases of O - 0.2 abundance (column 6 of Table \[VarF56\]) and $T_{BB}$ - 20000 K (column 8). What constituted a satisfactory match to the spectrum was not simply given by the FoM value, but certain line ratios, which act as important diagnostics of nebular conditions, e.g. He II/H$\beta$, \[O III\] 5007Å/H$\beta$ and \[O III\] 5007/4363Å), were given higher weight in assessing the quality of the match. It is clear that a range of O $\pm$0.2 produces model spectra in which the important \[O III\] lines are not well-matched, and thus representative errors on O in the range 0.1 to 0.15 are suggested. Similarly the range of stellar temperature of $\pm$20000 K and density varying by a factor 2 also over-estimate the allowed range of some of the line ratios. However it is clear from these parameter ranges that the error bars are not necessarily symmetric. H, He and PG1159 atmospheres from Rauch et al. [@Rauch03] were input to Cloudy and although the choice of temperatures is not very extensive, a satisfactory fit with the PG1159 atmosphere for 170000 K was found. However this should not be considered as a best estimate since a comprehensive range of temperature and gravity was not tested. The comparative models for the fainter PN F34\#1 (2906 in Hui et al. [@Hui93a], $m_{5007A}$=24.96) are given in Tab. \[VarF34\]. Here the number of lines is less than F56\#2, since \[O III\]4363Å and no He I lines were detected and He II is a limit. The \[O II\] 7325Åline is clearly under-estimated in the measured spectrum by comparison with the model spectra. Satisfactory matches could generally be obtained with the ranges of parameters listed. The models with a range of O of $\pm$0.2 is clearly accommodated by the data. Only the lowering of the stellar temperature by 10000 K (column 8 of \[VarF34\]) produced an unsatisfactory fit. A PG1159 model atmosphere (Rauch [@Rauch03]) of 70000K produced a good fit (column 11), but the 70000 K He model produced a similar fit with an O abundance lower by 0.40 dex than the black body. From this comparison of observed and model spectra with a higher and a lower S/N, we conclude that the typical dependence of errors on the choice of Cloudy models are generally in the range of 0.15 - 0.20 for the O abundance, around 10-15% for He and around 0.2 for Ne and N. These values are similar to those found by Magrini et al. [@Magrini] comparing abundances from Cloudy models with those of the ICF method. These representative abundance errors are shown in plots, such as Figs. \[abund\_rels\] and \[oh\_grad\]. -------------------- --------------- --------------- ------- --------- --------- ------------------ ------------------ -------------------- -------------------- ------------ Species $\lambda$ (Å) Flux & Model Model Model Model Model Model Model Model     error O + 0.2 O - 0.2 T$_{BB}$ + 20 kK T$_{BB}$ - 20 kK N$_e$ $\times$ 2.0 N$_e$ $\times$ 0.5 Mod.atmos.  \[O II\] 3727 56 $\pm$ 13 59 58 47 53 59 32 53 57  \[Ne III\] 3868 124 $\pm$ 13 124 127 128 122 123 126 125 125  \[O III\] 4363 28 $\pm$  8 24 15 27 26 22 22 22 27  He I 4471 6 $\pm$  3 6 6 6 3 11 6 9 6  He II 4686 39 $\pm$  8 39 38 34 46 30 37 30 38  H$\beta$ 4861 100 $\pm$  0 100 100 100 100 100 100 100 100  \[O III\] 4959 457 $\pm$ 31 449 448 398 451 471 454 458 456  \[O III\] 5007 1367 $\pm$ 90 1350 1348 1199 1358 1417 1367 1377 1371  He I 5876 6 $\pm$  3 18 18 19 11 33 19 28 20  H$\alpha$ 6562 281 $\pm$ 19 286 285 290 286 286 284 285 288  \[N II\] 6583 55 $\pm$  5 54 57 58 53 53 54 53 57  He I 6678 7 $\pm$  4 4 5 4 2 8 4 7 5  \[Ar III\] 7133 8 $\pm$  3 9 6 10 8 9 7 9 8  \[O II\] 7325 14 $\pm$  4 16 14 13 19 14 15 9 16 Model input  T$_{BB}$(kK) 155 152.5 150 175 135 150 132.5 170$^{1}$  N$_H$ (cm$^{-3}$) 10000 10000 10000 12000 8000 20000 5000 10000  12+log(He) 11.15 11.15 11.20 11.00 11.35 11.15 11.30 11.18  12+log(C) 8.40 9.20 6.60 8.40 8.40 8.70 8.40 8.20  12+log(N) 7.85 8.02 7.84 7.85 7.88 8.01 8.05 7.85  12+log(O) 8.43 8.63 8.23 8.48 8.43 8.49 8.40 8.39  12+log(Ne) 7.72 7.96 7.57 7.75 7.70 7.78 7.70 7.67  12+log(Ar) 6.00 6.00 6.15 6.00 6.00 6.00 6.00 5.95  FoM 7.53 8.30 13.46 7.34 14.79 9.06 12.90 8.56 -------------------- --------------- --------------- ------- --------- --------- ------------------ ------------------ -------------------- -------------------- ------------ -------------------- --------------- --------------- ------- --------- --------- ------------------ ------------------ -------------------- -------------------- ------------ Species $\lambda$ (Å) Flux & Model Model Model Model Model Model Model Model     error O + 0.2 O - 0.2 T$_{BB}$ + 10 kK T$_{BB}$ - 10 kK N$_e$ $\times$ 2.0 N$_e$ $\times$ 0.5 Mod.atmos.  \[O II\] 3727 86 $\pm$  9 89 85 88 87 60 49 144 85  \[Ne III\] 3868 93 $\pm$  6 91 92 93 92 95 92 94 96  He II 4686 2 $\pm$  2 2 2 4 4 1 2 4 0  H$\beta$ 4861 100 $\pm$  0 100 100 100 100 100 100 100 100  \[O III\] 4959 393 $\pm$ 15 384 391 384 393 248 385 387 390  \[O III\] 5007 1164 $\pm$ 41 1156 1177 1157 1183 747 1157 1166 1174  H$\alpha$ 6562 283 $\pm$ 12 288 290 287 286 286 287 288 287  \[N II\] 6583 104 $\pm$  6 104 104 104 104 103 104 105 103  \[Ar III\] 7133 4 $\pm$  3 3 4 4 4 3 3 4 4  \[O II\] 7325 4 $\pm$  3 17 16 17 17 8 16 17 14 Model input  T$_{BB}$(kK) 71 68 78 81 61 67.5 80 70$^{1}$  N$_H$ (cm$^{-3}$) 8500 10000 7500 8500 8500 17000 4250 7000  12+log(He/H) 11.06 11.06 11.06 10.90 11.30 11.06 10.90 11.20  12+log(C) 8.60 8.50 8.40 8.80 8.50 8.60 8.30 8.20  12+log(N) 8.25 8.38 8.13 8.16 8.50 8.40 8.03 8.18  12+log(O) 8.73 8.93 8.53 8.60 9.00 8.78 8.54 8.60  12+log(Ne) 8.00 8.23 7.78 7.84 8.60 8.07 7.77 7.88  12+log(Ar) 5.65 5.85 5.55 5.60 5.90 6.65 5.55 5.60  FoM 6.44 5.79 6.48 6.74 26.62 10.20 12.90 5.30 -------------------- --------------- --------------- ------- --------- --------- ------------------ ------------------ -------------------- -------------------- ------------ Discussion ========== Individual PN spectra and abundances ------------------------------------ A sample of 51 PNe in NGC 5128 have been observed over a range of projected galactocentric distances from 1.7 to 18.9 kpc and covering the PN luminosity function from the brightest known PN in NGC 5128 (PN 5601, $m_{5007A} = 23.51$) to 4.1 mags fainter. The mean 5007Å/H$\beta$ ratio is 11.00 $\pm$ 0.55 (r.m.s on the mean) for the 42 spectra with the highest S/N. The extrema of the 5007Å/H$\beta$ values for this subset are 6.0 ($\pm$ 0.67) to 18.8 ($\pm$3.7); the value of this ratio is very sensitive to the subtraction of the underlying continuum and the variation of the H$\beta$ absorption line along the slitlet. The mean logarithmic extinction correction at H$\beta$ is 0.56 $\pm$ 0.20. The overall Galactic extinction to NGC 5128 is E$_{B-V}$=0.123 (Burstein & Heiles [@BurHei]) or E$_{B-V}$=0.115 (Schlegel et al [@Schlegel]), the latter is equivalent to c=0.17 for a Galactic extinction law with R=3.2 (Seaton [@Seaton]). The lowest values of extinction measured in the PNe (Tab. \[DeredFlux\]) are compatible with no local extinction in NGC 5128. The histogram of c values peaks at 0.45 ($E_{B-V}$ = 0.30) and shows no obvious trend with radial offset beyond 200$''$ from the nucleus; the five highest values occur within a radius of 200$''$ (3.7 kpc). The frequency of higher c values at lower radii is probably due to the high line of sight extinction in the vicinity of the dust lane, rather than any intrinsic PN property (e.g. high intrinsic dust content associated with young PNe, such as NGC 7027; Zhang et al. [@Zhang]). The ratio of 5007Å/H$\beta$ does not show any significant overall gradient with projected radial offset from the galaxy centre; the highest values lie in field F42 where the stellar continuum is strongest and removal of the underlying H$\beta$ is the most critical. There is no evidence for a gradient in \[Ne III\]/\[O III\] implying a constant O$^{++}$/Ne$^{++}$ ratio. If there is any O enrichment, e.g. during the third dredge-up (Péquignot et al. [@Pequ]), or depletion (through the ON cycle, Leisy & Dennefeld [@Leisy06]) occurring in the nebular envelopes of the PNe, then this constant value of the ionic ratio implies that any alteration in O abundance must be accompanied by a corresponding change in Ne. Since O-Ne correlation is not predicted by AGB evolution (Karakas & Lattanzio [@KarLat]), the rather constant O/Ne ratio implies that O enrichment/depletion is not an important effect. The mean 12 + log(O/H) is 8.52 $\pm$ 0.03 (median value 8.48) and the range of \[O/H\] is $-$0.66 to $+$0.31. The mean O/H abundances inside and outside 10 kpc are identical. Of the three PN with the lowest O/H, two occur close to the nucleus (F42\#4 and F42\#10) and the other is at 19.5 kpc (F34\#2) and all have low \[O III\]/H$\beta$, without notably strong \[O II\]; F34\#2 has lower N/H. The PN with the highest O/H occur in fields 42 and 56 and are distinguished by very high \[O III\]5007/H$\beta$ but with large errors. The problems in removing the underlying stellar continuum from slit measurements can contribute to large uncertainty on the line fluxes for fainter nebulae in the central regions (see the discussion in Roth ([@Roth06])), and very high \[O III\]/H$\beta$ of $>$ 15 must be viewed with caution since the H$\beta$ may be considerably underestimated. The mean N/O ratio is 0.51 $\pm$ 0.06. Four PNe were found with high N/O ratio (F56\#8, F56\#12b, F56\#18 and F42\#4) but only F56\#12b and F56\#18 have He/H and N/O which could be classified as characteristic of Type I nebulae (Peimbert [@Peimbert]). Since F56\#12b was modelled by a lower temperature central star (the lowest of all the PN modelled, Tab. \[Cloumods\]), it may be that optical depth effects and the assumption of a black body may yield misleading results. F56\#18 is perhaps a better candidate for a Type I nebula, having a relatively high stellar temperature and a higher reddening, but this PN does not possess the highest luminosity central star. The brightest PN, and incidentally the nebula with the highest luminosity star (Tab. \[Cloumods\]), has N/O of 0.12 and shows no elevation of this ratio, compared for example to the mean Milky Way value of 0.47 (Kingsburgh & Barlow [@KiBa]). The shorter timescales for PN evolution of Type I PN make them minor contributors to a PN population luminosity function, but they contribute about $\sim$10% of Milky Way PN by number. The mean oxygen abundances in NGC 5128 PNe are intermediate between values for the LMC (12 + log(O/H) mean 8.4 (e.g. Leisy & Dennefeld[@Leisy06]), and that of the Milky Way (mean 8.68, Kingsburgh & Barlow [@KiBa]). A plot of the dependence of the \[O III\]5007Å luminosity on 12+log(O/H) (Fig. \[o3lum\]) shows a tendency to increase with the oxygen abundance, in line with the theoretical relation of of Dopita et al. ([@DopJac]) as observationally transformed by Richer ([@Rich93]; see Figure 2 and equation 1). A second order fit to the observed points demonstrates a comparable dependence of \[O III\] luminosity on (O/H) to the theoretical relation and is shown in Fig. \[o3lum\] by a dashed line. The region summed spectra provide good detections of a single ionization species of S and Ar (Tab. \[RegDerFlux\]) and lines of these species were detected in a number of the brightest PNe (Tab. \[ObsFlux\] and \[DeredFlux\]) allowing useful indications of S/H and Ar/H (Tab. \[Regabunds\] and Tab. \[Cloumods\]). S and Ar abundances serve an important purpose in that they are not considered to be affected by the AGB nucleosynthesis (e.g. Herwig [@Herwig]). Whilst O is the element which is most easily measured in PN since it is the dominant coolant, it can be destroyed by CNO cycling or synthesized during helium burning which impacts its usefulness as a metallicity indicator. Péquignot et al. ([@Pequ]) have found evidence for third dredge-up O enhancement in a PN in the Sagittarius dwarf galaxy (12+log(O/H) = 8.36), but only at the level of $<$0.03 dex. By comparing the lock-step dependence of Ne and O in PN samples, Richer & McCall ([@Rich08]) conclude, in agreement with Karakas & Lattanzio ([@KarLat]), that the majority of PNe do not show any changes in the O or Ne abundances across the AGB and PN transition. The Ne/H abundances are well correlated with the O abundances for the NGC 5128 PNe as shown by Fig. \[abund\_rels\] (slope 1.22$\pm$0.09, or 1.17$\pm$0.04 excluding the 4 points furthest from the linear relation) This result is similar to the value found by Leisy & Dennefeld ([@Leisy06]) for the LMC (slope 1.13). The relation between N/O and O/H is also shown in Fig. \[abund\_rels\]. There is considerable scatter with a weak but insignificant trend for an anti-correlation, as also found by Magrini et al. [@Magrini] for PNe in M33. The mean Ar/H abundance (6.3 for 21 PNe) is indistinguishable from the mean for 70 PN in the Milky Way (Kingsburgh & Barlow [@KiBa]). The mean S abundance (6.8 for 9 PNe) also appears to be similar to the mean value from the Kingsburgh & Barlow ([@KiBa]) sample. One is led to the surprising conclusion that most of the PN abundances in NGC 5128 are not significantly different from the mean values for Milky Way PNe. PN 5601 ------- PN 5601 (F56\#2) is the brightest known PN and the best-studied individual PN in NGC 5128, and indeed of all PN known beyond the Local Group. Walsh et al. ([@Walsh99]) measured long slit observations of this PN amongst a few others and the major difference between the spectrum presented in Tab. \[ObsFlux\] is the H$\alpha$/H$\beta$ ratio and the higher He II/H$\beta$ ratio. The earlier observations were taken with a fixed slit position angle over a considerable range of parallactic angle and suffer from wavelength dependent flux loss from the slit. This is demonstrated by the very low value of extinction derived. The value of extinction derived for this nebula is now seen to be quite large, but only slightly higher than the mean value for all the observed PN (E$_{B-V}$=0.32), giving no evidence that it is intrinsically dustier than the fainter PNe. There is some expectation from examples of Milky Way PNe, that young, high mass, optically thick PN (c.f. NGC 7027, Zhang et al. [@Zhang]; NGC 6302 Wright et al. [@Wright]) have higher intrinsic dust extinction. Abundance gradient from PNe --------------------------- One of the primary aims of this study was to investigate any abundance gradient of the $\alpha$-elements, as revealed by the PN spectra, and to compare it to gradients in the stellar properties such as the metallicity distribution function. A plot of the variation of O/H v. projected radial offset (Fig. \[oh\_grad\]), for the PNe with O/H values from the Cloudy models (Tab. \[Cloumods\]), shows two particular features. First, the range of O/H values is similar in the inner regions (projected radius, $\cal R$ $<$4 kpc) to the range in the outer regions ($>$13 kpc). Second, there is no evidence of a gradient in O/H. A least squares fit confirms this; the linear correlation coefficient is -0.07 for 40 points. Rather, there is a common mean value of 8.52 for the inner and outer points (considered as less than and greater than $\cal R$ of 10 kpc). However, the highest O/H values do occur at $\cal R$ $<$ 4 kpc, in the inner field F42. Although our sample is not large, the result that there is no significant O/H gradient in NGC 5128 appears to be robust. The distribution of \[$\alpha$/Fe\] v. \[Fe\] has an important role in determining the enrichment history of a galaxy (e.g. Matteucci & Recchi [@MatRec]). O, as representing the $\alpha$ elements, is primarily a core collapse SN product released early in star formation activity, whilst Fe, as the chief product of Type Ia SNe, is released over many Gyr. Figure \[O\_Fe\_rel\] shows the Fe v. Mg b EW plot for the integrated spectra of regions F42 and F56 with several tracks from Thomas et al. ([@Thomas]) models for selected metallicity and \[$\alpha$/Fe\] ratio. To be compatible with Harris et al. ([@Harris00]), who calibrated their metallicities by assuming that their stars were as old as those in old Galactic globular clusters, then the models must be compared at an age of 15 Gyr. In this case the datapoints for fields F42 and F56 lie close to the tracks of \[$\alpha$/Fe\] = 0.3, giving an \[$\alpha$/Fe\] ratio of 0.25, and show metallicities \[Z\] around -0.6 (there are no tracks at metallicity intermediate between -0.33 and -1.35 in Thomas et al. [@Thomas]). If, however, the age of the stars were younger (as indicated e.g. by Rejkuba et al. ([@Rejkuba05], [@Rejkuba11])), the derived metallicities would be about 0.2 higher, giving an average value of -0.4. This would imply an oxygen abundance relative to solar of -0.15, a value that is consistent with the mean O abundance of -0.17 with respect to the solar value (Scott et al. [@Scott]). These abundances can also be compared with the MDF from Harris & Harris ([@Harris02]) in three fields at 7.6 kpc SW of the centre, at 20 kpc S and 29 kpc S (distances rescaled to 3.8 Mpc). Fig. \[O\_Z\] shows the histogram of all the PN O/H abundances with respect to Solar overplotted against the completeness corrected MDFs in the inner field (7.6 kpc) and two combined outer fields (Table 1 of Harris & Harris [@Harris02]). The PN sample covers the range around the inner and outer fields of Harris et al. ([@Harris02]). A shift of -0.25 is required to align the peak of the PN O/H distribution with the stellar MDF in the inner field (magenta histogram). However it is strongly apparent that the distribution of (O/H) for the PNe is much narrower than for the stellar metallicity, so matching the distribution peaks may not be a justifiable simplification. Splitting the PN into two samples with positions greater and less than 10 kpc shows a shift in the peak value of (O/H) of $\sim$-0.2 for the inner field and $\sim$-0.3 for the outer field, although the number of PNe with abundances is lower in the outer field (11). For the full PN sample, with an average O abundance of -0.17 compared to Solar, the total metallicity is then -0.4 assuming a mean \[$\alpha$/Fe\]=0.25. Harris et al. ([@Harris02]) find an average metallicity of -0.6 in their outer fields, which becomes -0.4 assuming an age of 8 Gyr; thus taken at face value, assuming the metallicity of the field stars is that of the PN, the age of these stars in NGC 5128 is around 10 Gyr. Both methods – matching observed spectral indices to stellar population models and comparison with photometrically derived stellar metallicity distributions – provide an average \[O/Fe\] ratio of 0.25 at a stellar metallicity (viz. \[Fe/H\]) of around -0.4, and an average age of 8 Gyr. This single value is obviously a simplification for the behaviour in a whole galaxy, but demonstrates that the \[O/Fe\] v. \[Fe/H\] value for NGC 5128 is within the range of $\alpha$-enhanced stellar metallicities for early type galaxies (c.f. Thomas et al. [@Thomas]). The modelling of the colour-magnitude diagram by Rejkuba et al. ([@Rejkuba11]) also independently favours $\alpha$-enhanced stellar abundances in NGC 5128. However it can be argued that while all the stars are represented in the integrated stellar spectra and photometric surveys, only a subset of these stars become PNe. So, the mean values of the two samples of the PN \[O/H\] and the stellar \[Fe/H\] in Fig. \[O\_Z\] are not necessarily strictly comparable. In particular the PNe may derive from a younger population, on average. Only the higher Z stars produce observable PN, while the stellar continuum is comprised of stars of all metallicities. Overall the occurrence of PN is highly delimited relative to all the stars in a galaxy and in addition the PN central star masses also have a very restricted range relative to all white dwarfs. Thus the PNe may only come from a young population; that would explain the tendency toward both higher Z (Fig. \[O\_Z\]) and higher central star masses (see Fig. \[HRdiag\]). The old stars, which have lower Z on average, fail to produce PNe because their masses are too low. Chiappini et al. ([@Chiap]) present a careful comparison of the spectroscopically determined abundances of Milky Way Bulge stars with Bulge planetary nebulae. Their sample of 166 PN in the Bulge with well-determined abundances shows a broad distribution of 12 + log(O/H) from about 8.2 to 8.9, with a mean of 8.57, rather similar to the mean value and range determined here in NGC 5128. However the O/H abundance collated from various spectroscopic studies of Bulge giant stars shows a lower value by $\sim$0.3 dex. Various suggestions are offered by Chiappini et al. ([@Chiap]) for this discrepancy, but none appears to be conclusive. Comparing O from the Bulge PN with \[O/Fe\] from the stars, implies \[O$_{PN}$/Fe$_{Stars}$\] $\sim$ -0.1. This value is not strictly comparable to that in NGC 5128 since \[Fe/H\] is derived from photometry; spectroscopy of individual stars in NGC 5128 is ruled out until larger telescopes are available. Taken at face value the difference in \[O/Fe\] could imply that the formation history of NGC 5128 as a classical giant elliptical, is dissimilar to that of the Bulge; however the discrepancy in O abundances of Bulge PN and giants found by Chiappini et al. ([@Chiap]) weakens this conclusion as does the large spread in stellar Fe abundances (in both NGC 5128 and the Bulge). PN and the star formation history --------------------------------- One of the advantages of running Cloudy models to determine the PN abundances is that the central star luminosities and temperatures are derived. Since the PNe are optically thick, the match to the spectrum effectively provides the temperature and the $m_{5007\AA}$ photometry, combined with the reddening, provides the luminosity. Fig. \[HRdiag\] shows the Log $L$, Log T$_{eff}$ points for the 40 PNe modelled (Tab. \[Cloumods\]). The log $L$ values span a range of a factor 6, but are, not surprizingly, at the high end for PNe central stars, since only the top few magnitudes of the PNLF is explored by these data. No radial trend in the central star luminosities is apparent, paralleling the lack of an O abundance gradient (Fig. \[oh\_grad\]). A few evolutionary tracks from Vassiliadis & Wood ([@VasWood]) for higher mass progenitor stars are shown on Fig. \[HRdiag\], although there is currently no information on whether the central stars are on the H or He burning track. For the H burning tracks the bulk of the points match stars with masses above 2 M$_\odot$. There are multiple routes in terms of stellar mass, metallicity and mode of turn-off from the AGB (H or He burning track) to reach the same PN luminosity, so relating a given PN in a galaxy to its progenitor star formation episode is non-trivial. Marigo et al. ([@Marigo]) have modelled effects of stellar populations on the \[O III\] cut-off of a population of PN, as measured for the PNLF. They find that the peak \[O III\] luminosity is emitted by stars with initial (main sequence) masses of about 2.5 M$_\odot$ with a very strong dependence with age. PN from higher mass progenitors, although being intrinsically luminous, evolve very rapidly and so make a minor contribution to the bright end of the PN population. In addition, Ciardullo & Jacoby ([@Ciardullo99]) showed that the intrinsically massive and luminous progenitors are prolific dust producers, and therefore self-extinct. Thus the most massive progenitors will generally appear as rare and faint PNe in a galactic sample. Ciardullo et al. ([@Ciardullo05]) suggest that the progenitors of the most luminous PNe in galaxies, if they are not formed from stars younger than $\sim$1 Gyr, may occur predominantly in binary systems. In principle the calibration of PN luminosity v. stellar progenitor mass can be used to identify the age of the starburst giving rise to an observed PN population. If indeed the bulk of the PN observed in this study arise in progenitors of about 2.5 M$_\odot$, then from the grid of evolutionary tracks of low mass stars of Girardi et al. ([@Girardi]) one can infer a progenitor age of 0.7 Gyr (see Fig. \[AGBage\] derived from the Girardi et al. tracks for $[Z/Z_\odot]$ 0.0, -0.4 and -0.7 and the stellar age to AGB termination). Such an age would place the PN in a very recent star formation episode. This seems rather unlikely since there is little evidence for an extensive stellar population so young. A young (2-4 Gyr) minority (20-30%) component was inferred from the HST stellar photometric studies of Rejkuba et al. ([@Rejkuba05] and [@Rejkuba11]), but not with stars younger than 2 Gyr as appears to be required to match the PN. Younger stars are brighter and so should have been well-detected in optical photometry, ruling out their presence in NGC 5128. If the PNe studied here belong to this minority younger population of age 2-4 Gyr, then the deduced stellar masses, as indicated by the HR diagram of the PN central stars in Fig. \[HRdiag\], are still unexpectedly high. However since these PN belong to the bright end of the luminosity function, there is a selection effect in favour of the more luminous (viz. highest mass) PN progenitor stars. If however the observed PNe arose from an intermediate age population (5-8 Gyr), the progenitor stars would only have a mass of $\sim$1.2 M$_\odot$ and the discrepancy between their deduced masses and age is more difficult to reconcile with evolutionary tracks of single low mass stars (e.g. Vassiliadis & Wood [@VasWood]). It would be profitable to study spectroscopically a group of lower luminosity PNe in NGC 5128 to determine if their properties differ from the high $L$ PNe presented in this study. Conclusions =========== Low resolution spectroscopy of 51 PN in NGC 5128 at a range of galactocentric distance of 2-20 kpc have been obtained with the ESO VLT and FORS1 instrument in multi-slit mode (MOS) in three fields. The PN are drawn from the upper 4 magnitudes of the PN \[O III\] luminosity function. The emission line spectra have been analysed and lines typical of ionization by hot PN central stars have been measured. The weak \[O III\]4363Å line was just detected in 20% of the PNe. In order to determine element abundances for a larger fraction of the observed PNe, photoionization modelling was conducted with Cloudy for 40 PN with the highest quality spectra (representing the upper 2 mag. of the PNLF). He, N, O and Ne abundances were determined for all the PNe, and S and Ar for about half of the sample. For the most reliably estimated element, oxygen, no radial gradient is seen and the slope of O/H v. projected radius is identical inside and outside of 10 kpc. The range of \[O/H\] in the sample spanned -0.66 to +0.31 dex with the mean 12 + log O/H of 8.52 (median 8.48). The PN O abundances were compared to the stellar abundance measured from Lick indices from the continuum spectroscopy (i.e. along the MOS slitlets not occupied by PN emission) and from resolved star photometry from the literature. If the stars in NGC 5128 have an average age of 8 Gyr, the stellar and PN metallicities agree in the outer parts of NGC 5128 with average values around -0.4 Solar. 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C., et al. 2002, , 124, 3144 Peng, E. W., Ford, H. C., Freeman, K. C. 2004, , 602, 705 Péquignot, D., Walsh, J. R., Zijlstra, A. A., et al. 2000, , 361, L1 Phillips, M. M. 1981, , 197, 659 Quillen, A. C., Brookes, M. H., Keene, J., et al. 2006, ApJ, 645, 1092 Quillen, A. Bland-Hawthorn, J., Green, J. D., et al. 2008, , 384, 1469 Rauch, T. 2003, , 403, 709 Rejkuba, M., Greggio, L., Harris, W. E., et al. 2005, , 631, 262 Rejkuba, M., Walsh, J. R. 2006, in [*Planetary Nebulae Beyond the Milky Way*]{}. Eds. L. Stanghellini, J. R. Walsh & N. Douglas, Springer, p. 292 Rejkuba, M., Harris, W. E., Greggio, L., et al. 2011, , 526, 123 Richer, M. G. 1993, , 389, 27 Richer, M. G., McCall, M. L. 2008, , 684, 1190 Roth, M. M. 2006, in [*Science Perspectives for 3D Spectroscopy*]{}, eds. M. Kissler-Patig, J. R. Walsh, M. M. Roth. ESO Astrophysics Symposia, Springer, Berlin. p. 291 Schlegel, D. J., Finkbeiner, D. P., Davis, M. 1998, , 500, 525 Scott, P., Asplund, M., Grevesse, N. et al. 2009, , 691, L119 Seaton, M. J. 1979, , 187, 73P Soria. R., Mould, J. R., Watson, A. M., et al. 1996, , 465, 79 Stasińska, G., Richer, M., McCall, M. L. 1998, , 336, 667 Stasińska, G. 2002, , 12, 62 Storey, P. J., Hummer, D. G. 1995, , 272, 41 Thomas, D., Maraston, C., Bender, R., 2003, , 339, 897 Tonry, J., L., Dressler, A., Blakeslee, J. P., et al. 2001, , 546, 681 Vassiliadis, E., Wood, P. R. 1994, , 92, 125 Walsh, J. R., Walton, N. J., Jacoby, G. H., Peletier, R. 1999, A&A, 346 753 Walsh, J. R., Jacoby, G. H., Peletier, R., et al. 2005, in Planetary Nebulae Beyond the Milky Way, eds. L. Stanghellini, J. R. Walsh, N. G. Douglas, Springer, 262 Woodley, K. A. 2006, , 132, 2424 Woodley, K. A., Harris, W. E., Beasley, M. A., et al. 2007, , 134, 494 Woodley, K. A., Gomez, M., Harris, W. E., et al. 2010, , 139, 1871 Worthey, G., Faber, S. M., Gonzalez, J. J., Burstein, D., 1994, , 94, 687 Wright, N. J., Barlow, M. J., Ercolano, B., et al. 2011, ,in press Zhang, Y., Liu, X.-W., Luo, S.-G., et al. 2005, , 442, 249 Zoccali, M., Hill, V., Lecureur, A., et al. 2008, , 486, 177 [l l | r r | r r | r r | r r | r r]{} & & & & & &\ & & & & & &\ & & & & & &\ Species & $\lambda$ (Å) & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$\  \[O II\] & 3727 & 38 & 32 & 43 & 10 & 84 & 20 & & & &\  \[Ne III\] & 3868 & 38 & 14 & 97 & 10 & 55 & 16 & 72 & 19 & 112 & 60\  \[Ne III\] + H$\epsilon$ & 3970 & & & 29 & 6 & & & & & &\  H$\delta$ & 4101 & & & 22 & 8 & & & & & &\  H$\gamma$ & 4340 & 36 & 13 & 48 & 9 & 43 & 14 & 31 & 13 & 46 & 48\  \[O III\] & 4363 & & & 24 & 7 & & & & & &\  He I & 4471 & & & 5 & 3 & & & & & &\  He II & 4686 & & & 37 & 7 & & & & & &\  H$\beta$ & 4861 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 298 & 27 & 469 & 32 & 276 & 35 & 332 & 49 & 549 & 64\  \[O III\] & 5007 & 867 & 69 & 1421 & 93 & 812 & 92 & 965 & 135 & 1606 & 174\  He I & 5876 & & & 8 & 4 & & & 24 & 7 & &\  \[N II\] & 6548 & 44 & 7 & 23 & 6 & 72 & 15 & 74 & 14 & &\  H$\alpha$ & 6562 & 392 & 32 & 396 & 26 & 440 & 51 & 412 & 58 & 420 & 47\  \[N II\] & 6583 & 147 & 14 & 78 & 7 & 284 & 34 & 246 & 35 & 25 & 10\  He I & 6678 & & & 10 & 6 & & & & & &\  \[S II\] & 6716 & & & & & 39 & 30 & 21 & 8 & &\  \[S II\] & 6730 & 13 & 7 & & & 20 & 11 & 11 & 6 & &\  \[Ar III\] & 7133 & 17 & 10 & 12 & 4 & 21 & 11 & 59 & 13 & 39 & 34\  \[O II\] & 7325 & & & 22 & 6 & 104 & 35 & 28 & 12 & &\ log F(H$\beta$) & & -16.37 & 0.03 & -16.06 & 0.03 & -16.50 & 0.05 & -16.39 & 0.06 & -16.59 & 0.05\ m$_{5007A}$ & & 24.84 & 0.09 & 23.52 & 0.07 & 25.22 & 0.12 & 24.76 & 0.15 & 24.72 & 0.12\ & & & & & & & & & & &\ & & & & & & & & & & &\ & & & & & &\ & & & & & &\ & & & & & &\ Species & $\lambda$ (Å) & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$\  \[O II\] & 3727 & 42 & 9 & 38 & 10 & & & 24 & 15 & 29 & 5\  \[Ne III\] & 3868 & 75 & 12 & 49 & 13 & & & 69 & 11 & 96 & 9\  \[Ne III\] + H$\epsilon$ & 3970 & 52 & 8 & 31 & 16 & & & 53 & 31 & 47 & 23\  H$\delta$ & 4101 & 13 & 10 & 27 & 18 & & & 27 & 7 & 27 & 10\  H$\gamma$ & 4340 & 43 & 7 & 42 & 10 & & & 43 & 8 & 48 & 9\  \[O III\] & 4363 & & & & & & & & & 15 & 4\  He I & 4471 & & & & & & & & & &\  He II & 4686 & & & & & & & & & &\  H$\beta$ & 4861 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 432 & 35 & 292 & 27 & 263 & 108 & 382 & 29 & 476 & 37\  \[O III\] & 5007 & 1273 & 100 & 865 & 78 & 794 & 311 & 1179 & 87 & 1370 & 103\  He I & 5876 & 14 & 3 & 33 & 8 & & & 26 & 8 & 11 & 7\  \[N II\] & 6548 & 39 & 5 & 76 & 9 & & & 29 & 5 & 59 & 11\  H$\alpha$ & 6562 & 468 & 37 & 362 & 33 & 301 & 123 & 388 & 29 & 336 & 77\  \[N II\] & 6583 & 149 & 13 & 233 & 22 & 51 & 36 & 113 & 10 & 161 & 35\  He I & 6678 & 16 & 5 & 18 & 11 & & & & & &\  \[S II\] & 6716 & & & 13 & 4 & & & & & &\  \[S II\] & 6730 & & & 19 & 5 & & & 12 & 6 & 10 & 3\  \[Ar III\] & 7133 & 56 & 9 & 39 & 8 & & & 51 & 8 & 31 & 7\  \[O II\] & 7325 & 29 & 12 & 46 & 16 & & & 25 & 19 & &\ log F(H$\beta$) & & -16.08 & 0.03 & -16.13 & 0.04 & -17.37 & 0.19 & -16.14 & 0.03 & -16.12 & 0.03\ m$_{5007A}$ & & 23.70 & 0.09 & 24.23 & 0.10 & 27.43 & 0.48 & 23.93 & 0.08 & 23.71 & 0.08\ & & & & & & & & & & &\ & & & & & & & & & & &\ & & & & & &\ & & & & & &\ & & & & & &\ Species & $\lambda$ (Å) & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$\  \[O II\] & 3727 & 77 & 52 & 15 & 11 & 31 & 12 & 28 & 11 & &\  \[Ne III\] & 3868 & 129 & 27 & 32 & 11 & 82 & 14 & 84 & 11 & 55 & 12\  \[Ne III\] + H$\epsilon$ & 3970 & 66 & 49 & 27 & 14 & 40 & 13 & 50 & 40 & &\  H$\delta$ & 4101 & & & 15 & 8 & & & 26 & 9 & 26 & 13\  H$\gamma$ & 4340 & 41 & 13 & 36 & 9 & 40 & 14 & 30 & 11 & 44 & 13\  \[O III\] & 4363 & & & & & & & & & &\  He I & 4471 & & & & & & & & & &\  He II & 4686 & 57 & 13 & & & 13 & 11 & & & &\  H$\beta$ & 4861 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 684 & 97 & 180 & 20 & 529 & 50 & 445 & 41 & 366 & 45\  \[O III\] & 5007 & 1956 & 271 & 516 & 51 & 1540 & 140 & 1336 & 121 & 1083 & 128\  He I & 5876 & 20 & 11 & 16 & 8 & 15 & 6 & 23 & 6 & &\  \[N II\] & 6548 & 111 & 19 & 77 & 9 & & & 67 & 9 & 24 & 8\  H$\alpha$ & 6562 & 420 & 59 & 365 & 36 & 383 & 36 & 382 & 35 & 415 & 49\  \[N II\] & 6583 & 335 & 48 & 266 & 26 & 34 & 7 & 194 & 19 & 63 & 10\  He I & 6678 & & & & & 11 & 4 & & & &\  \[S II\] & 6716 & & & & & & & & & &\  \[S II\] & 6730 & 12 & 9 & & & & & 11 & 5 & 20 & 8\  \[Ar III\] & 7133 & 31 & 8 & 32 & 7 & 19 & 8 & 33 & 7 & 21 & 8\  \[O II\] & 7325 & 69 & 20 & 20 & 12 & & & 58 & 15 & &\ log F(H$\beta$) & & -16.37 & 0.06 & -16.17 & 0.04 & -16.22 & 0.04 & -16.11 & 0.04 & -16.21 & 0.05\ m$_{5007A}$ & & 23.97 & 0.15 & 24.91 & 0.11 & 23.83 & 0.10 & 23.71 & 0.10 & 24.19 & 0.13\ & & & & & & & & & & &\ & & & & & & & & & & &\ & & & &\ & & & &\ & & & &\ Species & $\lambda$ (Å) & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$\  \[O II\] & 3727 & 41 & 16 & & & 26 & 18\  \[Ne III\] & 3868 & 48 & 12 & 26 & 10 & 88 & 21\  \[Ne III\] + H$\epsilon$ & 3970 & 16 & 7 & 18 & 10 & 29 & 15\  H$\delta$ & 4101 & & & & & &\  H$\gamma$ & 4340 & 37 & 7 & 30 & 8 & &\  \[O III\] & 4363 & & & & & &\  He I & 4471 & 20 & 7 & & & &\  He II & 4686 & & & & & &\  H$\beta$ & 4861 & 100 & 0 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 369 & 31 & 275 & 34 & 416 & 74\  \[O III\] & 5007 & 1063 & 84 & 861 & 101 & 1315 & 230\  He I & 5876 & 24 & 9 & 30 & 12 & 28 & 13\  \[N II\] & 6548 & 36 & 7 & & & 102 & 23\  H$\alpha$ & 6562 & 415 & 34 & 356 & 42 & 447 & 79\  \[N II\] & 6583 & 147 & 14 & & & 323 & 58\  He I & 6678 & & & & & &\  \[S II\] & 6716 & & & & & &\  \[S II\] & 6730 & 11 & 6 & & & 30 & 11\  \[Ar III\] & 7133 & 24 & 8 & & & &\  \[O II\] & 7325 & & & & & &\ log F(H$\beta$) & & -16.15 & 0.03 & -16.19 & 0.05 & -16.35 & 0.08\ m$_{5007A}$ & & 24.07 & 0.09 & 24.41 & 0.13 & 24.34 & 0.19\ & & & & & & &\ & & & & & & & & & & &\ & & & & & &\ & & & & & &\ & & & & & &\ Species & $\lambda$ (Å) & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$\  \[O II\] & 3727 & & & 75 & 37 & & & 21 & 6 & 44 & 34\  \[Ne III\] & 3868 & 61 & 8 & 134 & 17 & 19 & 4 & 28 & 4 & 42 & 7\  \[Ne III\] + H$\epsilon$ & 3970 & 33 & 6 & 51 & 8 & & & 20 & 3 & &\  H$\delta$ & 4101 & 17 & 5 & & & 19 & 10 & 17 & 4 & &\  H$\gamma$ & 4340 & 37 & 6 & 45 & 29 & 43 & 7 & 42 & 5 & 37 & 7\  \[O III\] & 4363 & & & & & & & & & &\  He I & 4471 & & & & & & & 5 & 2 & &\  He II & 4686 & & & 36 & 8 & & & & & &\  H$\beta$ & 4861 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 350 & 24 & 625 & 60 & 182 & 13 & 222 & 12 & 289 & 25\  \[O III\] & 5007 & 1078 & 70 & 1872 & 177 & 582 & 38 & 663 & 33 & 778 & 64\  He I & 5876 & 28 & 8 & & & & & 7 & 2 & 32 & 11\  \[N II\] & 6548 & & & & & 65 & 9 & & & &\  H$\alpha$ & 6562 & 367 & 25 & 462 & 46 & 437 & 29 & 361 & 18 & 349 & 30\  \[N II\] & 6583 & 78 & 7 & 231 & 26 & 203 & 15 & 147 & 8 & 18 & 7\  \[S II\] & 6716 & 20 & 5 & & & & & 6 & 2 & &\  \[S II\] & 6730 & 27 & 7 & & & & & 12 & 4 & 26 & 11\  \[Ar III\] & 7133 & 23 & 6 & & & 37 & 8 & 30 & 5 & &\  \[O II\] & 7325 & & & & & & & & & 37 & 6\ log F(H$\beta$) & & -16.56 & 0.03 & -16.78 & 0.04 & -16.62 & 0.03 & -16.43 & 0.02 & -16.62 & 0.04\ m$_{5007A}$ & & 25.09 & 0.07 & 25.04 & 0.10 & 25.89 & 0.07 & 25.27 & 0.05 & 25.59 & 0.09\ & & & & & & & & & & &\ & & & & & & & & & & &\ & & & & & &\ & & & & & &\ & & & & & &\ Species & $\lambda$ (Å) & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$\  \[O II\] & 3727 & & & 40 & 7 & 12 & 8 & 48 & 4 & &\  \[Ne III\] & 3868 & 80 & 9 & 78 & 8 & 66 & 10 & 54 & 5 & 45 & 7\  \[Ne III\] + H$\epsilon$ & 3970 & 36 & 7 & 31 & 7 & 27 & 10 & 18 & 3 & &\  H$\delta$ & 4101 & 20 & 6 & 18 & 6 & & & 22 & 5 & 19 & 5\  H$\gamma$ & 4340 & 40 & 7 & 39 & 7 & 50 & 33 & 44 & 7 & 43 & 10\  \[O III\] & 4363 & & & & & & & 10 & 6 & &\  He I & 4471 & & & & & & & & & &\  He II & 4686 & 23 & 8 & & & & & 20 & 4 & &\  H$\beta$ & 4861 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 512 & 42 & 431 & 31 & 339 & 30 & 271 & 14 & 313 & 26\  \[O III\] & 5007 & 1522 & 121 & 1315 & 90 & 1081 & 90 & 827 & 41 & 973 & 78\  He I & 5876 & & & & & 29 & 12 & & & &\  \[N II\] & 6548 & & & & & & & 58 & 20 & 117 & 24\  H$\alpha$ & 6562 & 410 & 34 & 390 & 28 & 412 & 37 & 447 & 23 & 372 & 33\  \[N II\] & 6583 & 142 & 16 & 161 & 14 & 118 & 14 & 278 & 15 & 226 & 21\  He I & 6678 & & & & & & & 15 & 4 & &\  \[S II\] & 6716 & & & & & & & 22 & 5 & &\  \[S II\] & 6730 & & & & & & & 37 & 7 & 16 & 7\  \[Ar III\] & 7133 & & & & & 48 & 10 & 29 & 6 & 84 & 25\  \[O II\] & 7325 & & & & & 72 & 31 & & & &\ log F(H$\beta$) & & -16.63 & 0.03 & -16.60 & 0.03 & -16.70 & 0.04 & -16.38 & 0.02 & -16.59 & 0.04\ m$_{5007A}$ & & 24.87 & 0.09 & 24.97 & 0.08 & 25.43 & 0.09 & 24.90 & 0.05 & 25.25 & 0.09\ & & & & & & & & & & &\ & & & & & & & & & & &\ & & & & & &\ & & & & & &\ & & & & & &\ Species & $\lambda$ (Å) & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$\  \[O II\] & 3727 & & & 56 & 32 & 50 & 38 & & & 65 & 14\  \[Ne III\] & 3868 & 114 & 15 & 151 & 80 & 160 & 32 & 43 & 9 & 100 & 16\  \[Ne III\] + H$\epsilon$ & 3970 & 64 & 10 & 53 & 32 & & & & & 34 & 10\  H$\delta$ & 4101 & & & & & & & & & &\  H$\gamma$ & 4340 & 40 & 12 & & & & & 31 & 17 & 44 & 15\  \[O III\] & 4363 & & & & & & & & & 19 & 8\  He I & 4471 & & & & & & & & & &\  He II & 4686 & 14 & 12 & 31 & 24 & 58 & 29 & & & &\  H$\beta$ & 4861 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 569 & 57 & 690 & 356 & 796 & 140 & 270 & 33 & 574 & 74\  \[O III\] & 5007 & 1747 & 172 & 2185 & 1125 & 2438 & 426 & 872 & 101 & 1749 & 224\  He I & 5876 & & & & & & & & & &\  \[N II\] & 6548 & & & & & & & & & 64 & 16\  H$\alpha$ & 6562 & 504 & 51 & 507 & 263 & 493 & 89 & 432 & 51 & 385 & 52\  \[N II\] & 6583 & 197 & 23 & 246 & 130 & 171 & 36 & & & 192 & 30\  He I & 6678 & & & & & & & & & &\  \[S II\] & 6716 & & & & & & & & & &\  \[S II\] & 6730 & & & & & & & 35 & 10 & &\  \[Ar III\] & 7133 & & & & & & & & & &\  \[O II\] & 7325 & & & & & & & & & 44 & 19\ log F(H$\beta$) & & -16.73 & 0.04 & -17.10 & 0.22 & -16.88 & 0.08 & -16.64 & 0.05 & -16.65 & 0.06\ m$_{5007A}$ & & 24.97 & 0.11 & 25.66 & 0.56 & 24.99 & 0.19 & 25.51 & 0.13 & 24.77 & 0.14\ & & & & & & & & & & &\ & & & & & & & & & & &\ & & &\ & & &\ & & &\ Species & $\lambda$ (Å) & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$\  \[O II\] & 3727 & & & &\  \[Ne III\] & 3868 & & & 40 & 9\  \[Ne III\] + H$\epsilon$ & 3970 & & & &\  H$\delta$ & 4101 & & & &\  H$\gamma$ & 4340 & 34 & 8 & 46 & 11\  \[O III\] & 4363 & & & &\  He I & 4471 & & & &\  He II & 4686 & & & 17 & 14\  H$\beta$ & 4861 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 210 & 16 & 553 & 61\  \[O III\] & 5007 & 608 & 42 & 1616 & 175\  He I & 5876 & & & &\  \[N II\] & 6548 & 51 & 11 & &\  H$\alpha$ & 6562 & 495 & 36 & 583 & 65\  \[N II\] & 6583 & 191 & 18 & 69 & 16\  He I & 6678 & & & &\  \[S II\] & 6716 & & & &\  \[S II\] & 6730 & & & &\  \[Ar III\] & 7133 & & & &\  \[O II\] & 7325 & & & &\ log F(H$\beta$) & & -16.61 & 0.03 & -16.65 & 0.05\ m$_{5007A}$ & & 25.81 & 0.07 & 24.85 & 0.12\ & & & & & & & & & & &\ & & & & & & & & & & &\ & & & & & &\ & & & & & &\ & & & & & &\ Species & $\lambda$ (Å) & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$\  \[O II\] & 3727 & 64 & 7 & 64 & 7 & 48 & 9 & 55 & 28 & 30 & 11\  \[Ne III\] & 3868 & 72 & 5 & 45 & 5 & 49 & 5 & 98 & 6 & 95 & 11\  \[Ne III\] + H$\epsilon$ & 3970 & 25 & 4 & & & 35 & 6 & 39 & 5 & 28 & 7\  H$\delta$ & 4101 & 23 & 4 & 16 & 4 & 24 & 4 & 25 & 5 & 24 & 6\  H$\gamma$ & 4340 & 35 & 3 & 44 & 5 & 46 & 5 & 43 & 6 & 49 & 8\  \[O III\] & 4363 & & & 9 & 4 & & & 8 & 2 & 18 & 9\  He I & 4471 & & & 17 & 4 & & & 9 & 3 & 18 & 7\  He II & 4686 & 2 & 2 & 8 & 4 & & & t & & 48 & 9\  H$\beta$ & 4861 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 404 & 15 & 261 & 13 & 319 & 16 & 535 & 21 & 530 & 45\  \[O III\] & 5007 & 1214 & 43 & 782 & 37 & 972 & 45 & 1591 & 62 & 1574 & 130\  He I & 5876 & & & 32 & 9 & & & & & &\  \[N II\] & 6548 & 33 & 9 & & & & & & & &\  H$\alpha$ & 6562 & 410 & 17 & 528 & 28 & 454 & 24 & 447 & 20 & 421 & 43\  \[N II\] & 6583 & 152 & 9 & 108 & 12 & 151 & 11 & 116 & 9 & 27 & 11\  He I & 6678 & & & 26 & 6 & & & & & 8 & 8\  \[S II\] & 6716 & & & & & & & & & &\  \[S II\] & 6730 & & & & & & & & & &\  \[Ar III\] & 7133 & 6 & 4 & & & 39 & 10 & 37 & 7 & &\  \[O II\] & 7325 & 6 & 5 & & & 33 & 16 & & & &\ log F(H$\beta$) & & -16.56 & 0.02 & -16.70 & 0.03 & -16.69 & 0.03 & -16.59 & 0.03 & -16.94 & 0.05\ m$_{5007A}$ & & 24.96 & 0.04 & 25.78 & 0.05 & 25.51 & 0.05 & 24.73 & 0.04 & 25.62 & 0.09\ & & & & & & & & & & &\ & & & & & & & & & & &\ & & & & &\ & & & & &\ & & & & &\ Species & $\lambda$ (Å) & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$ & F$_{Obs}$ & $\pm$\  \[O II\] & 3727 & 48 & 16 & 38 & 4 & & & 138 & 16\  \[Ne III\] & 3868 & 34 & 7 & 66 & 5 & 93 & 9 & 111 & 14\  \[Ne III\] + H$\epsilon$ & 3970 & & & 30 & 4 & & & 46 & 9\  H$\delta$ & 4101 & & & 15 & 10 & 20 & 6 & &\  H$\gamma$ & 4340 & 35 & 8 & 47 & 5 & 45 & 6 & 46 & 12\  \[O III\] & 4363 & & & 7 & 3 & 13 & 5 & 25 & 8\  He I & 4471 & & & 5 & 3 & & & &\  He II & 4686 & 45 & 9 & & & 21 & 15 & 47 & 12\  H$\beta$ & 4861 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 477 & 45 & 403 & 18 & 455 & 30 & 532 & 50\  \[O III\] & 5007 & 1426 & 132 & 1193 & 53 & 1349 & 87 & 1647 & 151\  He I & 5876 & 49 & 25 & & & & & &\  \[N II\] & 6548 & & & & & & & 88 & 13\  H$\alpha$ & 6562 & 321 & 30 & 380 & 19 & 475 & 34 & 458 & 48\  \[N II\] & 6583 & 21 & 8 & 59 & 7 & 43 & 11 & 418 & 44\  He I & 6678 & & & & & & & &\  \[S II\] & 6716 & & & & & & & &\  \[S II\] & 6730 & & & & & & & 36 & 22\  \[Ar III\] & 7133 & & & & & & & &\  \[O II\] & 7325 & 57 & 26 & & & & & &\ log F(H$\beta$) & & -16.98 & 0.05 & -16.63 & 0.03 & -16.83 & 0.04 & -17.03 & 0.04\ m$_{5007A}$ & & 25.84 & 0.10 & 25.16 & 0.05 & 25.50 & 0.07 & 25.81 & 0.10\ & & & & & & & & & & &\ [l l | r r | r r | r r | r r | r r]{} & & & & & &\ & & & & & &\ & & & & & &\ Species & $\lambda$ (Å) & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$\  \[O II\] & 3727 & 50 & 42 & 56 & 13 & 120 & 29 & & & &\  \[Ne III\] & 3868 & 49 & 18 & 124 & 13 & 76 & 22 & 95 & 25 & 150 & 81\  \[Ne III\] + H$\epsilon$ & 3970 & & & 37 & 8 & & & & & 72 & 42\  H$\delta$ & 4101 & & & 27 & 10 & & & & & &\  H$\gamma$ & 4340 & 41 & 15 & 58 & 10 & 51 & 17 & 35 & 14 & &\  \[O III\] & 4363 & & & 28 & 8 & & & & & &\  He I & 4471 & & & 6 & 3 & & & & & &\  He II & 4686 & & & 39 & 8 & & & & & &\  H$\beta$ & 4861 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 291 & 26 & 457 & 31 & 268 & 34 & 323 & 47 & 533 & 62\  \[O III\] & 5007 & 836 & 66 & 1367 & 90 & 772 & 87 & 925 & 129 & 1536 & 166\  He I & 5876 & & & 6 & 3 & & & 18 & 5 & &\  \[N II\] & 6548 & 31 & 5 & 18 & 4 & 46 & 10 & 51 & 10 & &\  H$\alpha$ & 6562 & 282 & 23 & 281 & 19 & 282 & 32 & 282 & 40 & 281 & 32\  \[N II\] & 6583 & 105 & 10 & 55 & 5 & 181 & 21 & 168 & 24 & 17 & 7\  He I & 6678 & & & 7 & 4 & & & & & &\  \[S II\] & 6716 & & & & & 24 & 19 & 14 & 5 & &\  \[S II\] & 6730 & 9 & 5 & & & 13 & 7 & 7 & 4 & &\  \[Ar III\] & 7133 & 11 & 6 & 8 & 3 & 12 & 6 & 37 & 8 & 24 & 21\  \[O II\] & 7325 & & & 14 & 4 & 58 & 20 & 17 & 7 & &\ c & & 0.46 & 0.11 & 0.48 & 0.09 & 0.62 & 0.16 & 0.53 & 0.20 & 0.56 & 0.16\ log F(H$\beta$) & & -15.91 & 0.03 & -15.58 & 0.03 & -15.88 & 0.05 & -15.86 & 0.06 & -16.03 & 0.05\ m$_{5007A}$ & & 23.73 & 0.09 & 22.37 & 0.07 & 23.73 & 0.12 & 23.49 & 0.15 & 23.37 & 0.12\ & & & & & & & & & & &\ & & & & & & & & & & &\ & & & & & &\ & & & & & &\ & & & & & &\ Species & $\lambda$ (Å) & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$\  \[O II\] & 3727 & 63 & 13 & 46 & 12 & & & 31 & 19 & 33 & 6\  \[Ne III\] & 3868 & 109 & 17 & 59 & 15 & & & 87 & 14 & 109 & 11\  \[Ne III\] + H$\epsilon$ & 3970 & 73 & 11 & 36 & 19 & & & 66 & 38 & 52 & 26\  H$\delta$ & 4101 & 18 & 13 & 32 & 20 & & & 33 & 9 & 30 & 11\  H$\gamma$ & 4340 & 53 & 8 & 47 & 11 & & & 49 & 10 & 51 & 10\  \[O III\] & 4363 & & & & & & & & & &\  He I & 4471 & & & & & & & & & &\  He II & 4686 & & & & & & & & & &\  H$\beta$ & 4861 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 416 & 34 & 286 & 27 & 262 & 107 & 373 & 29 & 470 & 36\  \[O III\] & 5007 & 1203 & 94 & 841 & 76 & 789 & 309 & 1137 & 84 & 1344 & 101\  He I & 5876 & 10 & 2 & 28 & 7 & & & 21 & 6 & 10 & 6\  \[N II\] & 6548 & 23 & 3 & 59 & 7 & & & 21 & 4 & 49 & 10\  H$\alpha$ & 6562 & 281 & 22 & 282 & 26 & 282 & 115 & 281 & 21 & 282 & 64\  \[N II\] & 6583 & 89 & 8 & 181 & 17 & 48 & 34 & 81 & 7 & 135 & 29\  He I & 6678 & 10 & 3 & 14 & 9 & & & & & &\  \[S II\] & 6716 & & & 10 & 3 & & & & & &\  \[S II\] & 6730 & & & 14 & 4 & & & 10 & 5 & 9 & 2\  \[Ar III\] & 7133 & 30 & 5 & 29 & 6 & & & 34 & 5 & 25 & 6\  \[O II\] & 7325 & 19 & 8 & 33 & 11 & & & 16 & 12 & &\ c & & 0.71 & 0.11 & 0.35 & 0.13 & 0.09 & 0.30 & 0.45 & 0.11 & 0.24 & 0.32\ log F(H$\beta$) & & -15.37 & 0.03 & -15.78 & 0.04 & -17.37 & 0.19 & -15.69 & 0.03 & -15.88 & 0.03\ m$_{5007A}$ & & 21.99 & 0.09 & 23.39 & 0.10 & 27.44 & 0.43 & 23.93 & 0.08 & 23.13 & 0.08\ & & & & & & & & & & &\ & & & & & & & & & & &\ & & & & & &\ & & & & & &\ & & & & & &\ Species & $\lambda$ (Å) & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$\  \[O II\] & 3727 & 106 & 72 & 19 & 14 & 40 & 16 & 36 & 14 & &\  \[Ne III\] & 3868 & 172 & 35 & 39 & 13 & 102 & 17 & 105 & 14 & 73 & 16\  \[Ne III\] + H$\epsilon$ & 3970 & 85 & 63 & 32 & 16 & 48 & 15 & 61 & 49 & &\  H$\delta$ & 4101 & & & 18 & 9 & & & 30 & 11 & 32 & 16\  H$\gamma$ & 4340 & 48 & 15 & 40 & 10 & 46 & 16 & 35 & 11 & 52 & 15\  \[O III\] & 4363 & & & & & & & & & &\  He I & 4471 & & & & & & & & & &\  He II & 4686 & 60 & 14 & & & 14 & 11 & & & &\  H$\beta$ & 4861 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 665 & 94 & 177 & 19 & 517 & 48 & 435 & 40 & 356 & 44\  \[O III\] & 5007 & 1872 & 259 & 502 & 49 & 1488 & 136 & 1292 & 117 & 1037 & 123\  He I & 5876 & 15 & 9 & 14 & 7 & 12 & 5 & 19 & 5 & &\  \[N II\] & 6548 & 75 & 13 & 60 & 7 & & & 50 & 7 & 17 & 6\  H$\alpha$ & 6562 & 283 & 40 & 282 & 28 & 281 & 26 & 283 & 26 & 282 & 34\  \[N II\] & 6583 & 225 & 32 & 205 & 20 & 25 & 5 & 143 & 14 & 42 & 7\  He I & 6678 & & & & & 8 & 3 & & & &\  \[S II\] & 6716 & & & & & & & & & &\  \[S II\] & 6730 & 8 & 6 & & & & & 8 & 4 & 13 & 6\  \[Ar III\] & 7133 & 19 & 5 & 23 & 5 & 13 & 6 & 23 & 5 & 13 & 5\  \[O II\] & 7325 & 42 & 13 & 13 & 8 & & & 37 & 10 & &\ c & & 0.55 & 0.20 & 0.36 & 0.14 & 0.43 & 0.13 & 0.42 & 0.13 & 0.54 & 0.17\ log F(H$\beta$) & & -15.83 & 0.06 & -15.81 & 0.04 & -15.79 & 0.04 & -15.69 & 0.04 & -15.67 & 0.05\ m$_{5007A}$ & & 22.64 & 0.15 & 24.04 & 0.11 & 22.79 & 0.10 & 22.70 & 0.10 & 24.19 & 0.13\ & & & & & & & & & & &\ & & & & & & & & & & &\ & & & &\ & & & &\ & & & &\ Species & $\lambda$ (Å) & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$\  \[O II\] & 3727 & 57 & 21 & & & 37 & 26\  \[Ne III\] & 3868 & 63 & 16 & 31 & 12 & 123 & 30\  \[Ne III\] + H$\epsilon$ & 3970 & 21 & 9 & 21 & 12 & 40 & 20\  H$\delta$ & 4101 & & & & & &\  H$\gamma$ & 4340 & 43 & 9 & 34 & 9 & &\  \[O III\] & 4363 & & & & & &\  He I & 4471 & 22 & 8 & & & &\  He II & 4686 & & & & & &\  H$\beta$ & 4861 & 100 & 0 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 359 & 30 & 270 & 33 & 402 & 72\  \[O III\] & 5007 & 1018 & 81 & 840 & 98 & 1249 & 218\  He I & 5876 & 19 & 7 & 26 & 11 & 20 & 10\  \[N II\] & 6548 & 25 & 5 & & & 64 & 15\  H$\alpha$ & 6562 & 282 & 23 & 283 & 33 & 282 & 50\  \[N II\] & 6583 & 100 & 10 & & & 203 & 36\  He I & 6678 & & & & & &\  \[S II\] & 6716 & & & & & &\  \[S II\] & 6730 & 7 & 4 & & & 18 & 7\  \[Ar III\] & 7133 & 15 & 5 & & & &\  \[O II\] & 7325 & & & & & &\ c & & 0.54 & 0.11 & 0.32 & 0.17 & 0.64 & 0.25\ log F(H$\beta$) & & -15.61 & 0.03 & -15.87 & 0.05 & -15.71 & 0.08\ m$_{5007A}$ & & 22.77 & 0.09 & 23.63 & 0.13 & 22.90 & 0.19\ & & & & & & &\ & & & & & & & & & & &\ & & & & & &\ & & & & & &\ & & & & & &\ Species & $\lambda$ (Å) & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$\  \[O II\] & 3727 & & & 111 & 56 & & & 26 & 7 & 52 & 40\  \[Ne III\] & 3868 & 74 & 10 & 192 & 24 & 26 & 5 & 34 & 5 & 50 & 9\  \[Ne III\] + H$\epsilon$ & 3970 & 40 & 8 & 71 & 11 & & & 24 & 4 & &\  H$\delta$ & 4101 & 20 & 6 & & & 24 & 12 & 20 & 4 & &\  H$\gamma$ & 4340 & 41 & 7 & 55 & 36 & 52 & 8 & 47 & 6 & 41 & 7\  \[O III\] & 4363 & & & & & & & & & &\  He I & 4471 & & & & & & & 5 & 3 & &\  He II & 4686 & & & 39 & 9 & & & & & &\  H$\beta$ & 4861 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 343 & 23 & 602 & 58 & 176 & 13 & 218 & 12 & 284 & 25\  \[O III\] & 5007 & 1046 & 68 & 1772 & 168 & 554 & 36 & 645 & 32 & 760 & 63\  He I & 5876 & 24 & 6 & & & & & 6 & 2 & 28 & 10\  \[N II\] & 6548 & & & & & 42 & 6 & & & &\  H$\alpha$ & 6562 & 281 & 19 & 281 & 28 & 282 & 19 & 281 & 14 & 281 & 24\  \[N II\] & 6583 & 60 & 6 & 140 & 16 & 131 & 10 & 114 & 6 & 15 & 6\  \[S II\] & 6716 & 15 & 4 & & & & & 5 & 2 & &\  \[S II\] & 6730 & 20 & 6 & & & & & 9 & 3 & 21 & 8\  \[Ar III\] & 7133 & 16 & 4 & & & 22 & 5 & 22 & 4 & &\  \[O II\] & 7325 & & & & & & & & & 28 & 5\ c & & 0.37 & 0.09 & 0.69 & 0.14 & 0.61 & 0.09 & 0.35 & 0.07 & 0.30 & 0.12\ log F(H$\beta$) & & -16.19 & 0.03 & -16.09 & 0.04 & -16.01 & 0.03 & -16.08 & 0.02 & -16.32 & 0.04\ m$_{5007A}$ & & 24.19 & 0.07 & 23.37 & 0.10 & 24.41 & 0.07 & 24.43 & 0.05 & 24.86 & 0.09\ & & & & & & & & & & &\ & & & & & & & & & & &\ & & & & & &\ & & & & & &\ & & & & & &\ Species & $\lambda$ (Å) & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$\  \[O II\] & 3727 & & & 52 & 9 & 16 & 11 & 69 & 6 & &\  \[Ne III\] & 3868 & 105 & 12 & 99 & 10 & 87 & 13 & 76 & 6 & 55 & 9\  \[Ne III\] + H$\epsilon$ & 3970 & 46 & 9 & 38 & 9 & 34 & 13 & 24 & 5 & &\  H$\delta$ & 4101 & 25 & 7 & 22 & 7 & & & 28 & 7 & 22 & 6\  H$\gamma$ & 4340 & 47 & 8 & 44 & 8 & 59 & 38 & 52 & 9 & 48 & 11\  \[O III\] & 4363 & & & & & & & 12 & 7 & &\  He I & 4471 & & & & & & & & & &\  He II & 4686 & 24 & 8 & & & & & 21 & 5 & &\  H$\beta$ & 4861 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 498 & 41 & 421 & 30 & 330 & 29 & 262 & 14 & 306 & 26\  \[O III\] & 5007 & 1460 & 116 & 1268 & 87 & 1036 & 86 & 786 & 39 & 944 & 76\  He I & 5876 & & & & & 23 & 9 & & & &\  \[N II\] & 6548 & & & & & & & 37 & 13 & 89 & 18\  H$\alpha$ & 6562 & 282 & 23 & 283 & 21 & 282 & 25 & 282 & 15 & 281 & 25\  \[N II\] & 6583 & 97 & 11 & 116 & 10 & 80 & 10 & 175 & 10 & 170 & 16\  He I & 6678 & & & & & & & 9 & 3 & &\  \[S II\] & 6716 & & & & & & & 14 & 3 & &\  \[S II\] & 6730 & & & & & & & 23 & 4 & 12 & 5\  \[Ar III\] & 7133 & & & & & 30 & 7 & 17 & 3 & 59 & 17\  \[O II\] & 7325 & & & & & 44 & 19 & & & &\ c & & 0.52 & 0.12 & 0.45 & 0.10 & 0.53 & 0.13 & 0.64 & 0.07 & 0.39 & 0.12\ log F(H$\beta$) & & -16.11 & 0.03 & -16.15 & 0.03 & -16.17 & 0.04 & -15.74 & 0.02 & -16.20 & 0.04\ m$_{5007A}$ & & 23.61 & 0.09 & 23.88 & 0.08 & 24.15 & 0.09 & 23.36 & 0.05 & 24.31 & 0.09\ & & & & & & & & & & &\ & & & & & & & & & & &\ & & & & & &\ & & & & & &\ & & & & & &\ Species & $\lambda$ (Å) & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$\  \[O II\] & 3727 & & & 91 & 52 & 78 & 60 & & & 83 & 18\  \[Ne III\] & 3868 & 174 & 22 & 230 & 122 & 240 & 48 & 59 & 13 & 125 & 20\  \[Ne III\] + H$\epsilon$ & 3970 & 94 & 14 & 78 & 46 & & & & & 42 & 12\  H$\delta$ & 4101 & & & & & & & & & &\  H$\gamma$ & 4340 & 51 & 15 & & & & & 37 & 20 & 50 & 17\  \[O III\] & 4363 & & & & & & & & & 21 & 9\  He I & 4471 & & & & & & & & & &\  He II & 4686 & 15 & 13 & 33 & 26 & 62 & 32 & & & &\  H$\beta$ & 4861 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 545 & 55 & 660 & 341 & 763 & 135 & 261 & 32 & 561 & 73\  \[O III\] & 5007 & 1637 & 161 & 2046 & 1053 & 2290 & 401 & 831 & 96 & 1688 & 216\  He I & 5876 & & & & & & & & & &\  \[N II\] & 6548 & & & & & & & & & 46 & 12\  H$\alpha$ & 6562 & 282 & 29 & 282 & 146 & 281 & 51 & 281 & 33 & 281 & 38\  \[N II\] & 6583 & 110 & 13 & 136 & 72 & 97 & 21 & & & 139 & 22\  He I & 6678 & & & & & & & & & &\  \[S II\] & 6716 & & & & & & & & & &\  \[S II\] & 6730 & & & & & & & 22 & 6 & &\  \[Ar III\] & 7133 & & & & & & & & & &\  \[O II\] & 7325 & & & & & & & & & 29 & 12\ c & & 0.81 & 0.14 & 0.82 & 0.72 & 0.78 & 0.25 & 0.60 & 0.17 & 0.44 & 0.19\ log F(H$\beta$) & & -15.92 & 0.04 & -16.28 & 0.22 & -16.10 & 0.08 & -16.04 & 0.05 & -16.21 & 0.06\ m$_{5007A}$ & & 23.01 & 0.11 & 23.69 & 0.56 & 23.11 & 0.19 & 24.06 & 0.13 & 23.71 & 0.14\ & & & & & & & & & & &\ & & & & & & & & & & &\ & & &\ & & &\ & & &\ Species & $\lambda$ (Å) & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$\  \[O II\] & 3727 & & & &\  \[Ne III\] & 3868 & & & 67 & 14\  \[Ne III\] + H$\epsilon$ & 3970 & & & &\  H$\delta$ & 4101 & & & &\  H$\gamma$ & 4340 & 43 & 10 & 61 & 14\  \[O III\] & 4363 & & & &\  He I & 4471 & & & &\  He II & 4686 & & & 19 & 15\  H$\beta$ & 4861 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 201 & 16 & 524 & 58\  \[O III\] & 5007 & 571 & 39 & 1490 & 161\  He I & 5876 & & & &\  \[N II\] & 6548 & 29 & 6 & &\  H$\alpha$ & 6562 & 281 & 20 & 283 & 32\  \[N II\] & 6583 & 108 & 10 & 33 & 8\  He I & 6678 & & & &\  \[S II\] & 6716 & & & &\  \[S II\] & 6730 & & & &\  \[Ar III\] & 7133 & & & &\  \[O II\] & 7325 & & & &\ c & & 0.79 & 0.10 & 1.01 & 0.16\ log F(H$\beta$) & & -15.82 & 0.03 & -15.64 & 0.05\ m$_{5007A}$ & & 23.91 & 0.07 & 22.41 & 0.12\ & & & & & & & & & & &\ & & & & & & & & & & &\ & & & & & &\ & & & & & &\ & & & & & &\ Species & $\lambda$ (Å) & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$\  \[O II\] & 3727 & 86 & 9 & 106 & 12 & 71 & 13 & 80 & 41 & 41 & 15\  \[Ne III\] & 3868 & 93 & 6 & 71 & 7 & 70 & 7 & 137 & 8 & 127 & 15\  \[Ne III\] + H$\epsilon$ & 3970 & 31 & 5 & & & 48 & 8 & 53 & 7 & 36 & 9\  H$\delta$ & 4101 & 28 & 5 & 23 & 6 & 31 & 6 & 33 & 7 & 31 & 7\  H$\gamma$ & 4340 & 41 & 4 & 56 & 7 & 56 & 7 & 52 & 7 & 57 & 10\  \[O III\] & 4363 & & & 11 & 5 & & & 9 & 3 & 21 & 11\  He I & 4471 & & & 20 & 5 & & & 10 & 3 & 20 & 8\  He II & 4686 & 2 & 2 & 8 & 4 & & & t & & 51 & 10\  H$\beta$ & 4861 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 393 & 15 & 249 & 13 & 307 & 15 & 517 & 21 & 514 & 43\  \[O III\] & 5007 & 1164 & 41 & 729 & 34 & 922 & 43 & 1512 & 59 & 1505 & 124\  He I & 5876 & & & 21 & 6 & & & & & &\  \[N II\] & 6548 & 23 & 6 & & & & & & & &\  H$\alpha$ & 6562 & 283 & 12 & 281 & 15 & 281 & 15 & 282 & 12 & 282 & 28\  \[N II\] & 6583 & 104 & 6 & 57 & 6 & 93 & 7 & 73 & 6 & 18 & 8\  He I & 6678 & & & 13 & 3 & & & & & 5 & 5\  \[S II\] & 6716 & & & & & & & & & &\  \[S II\] & 6730 & & & & & & & & & &\  \[Ar III\] & 7133 & 4 & 3 & & & 21 & 5 & 21 & 4 & &\  \[O II\] & 7325 & 4 & 3 & & & 17 & 9 & & & &\ c & & 0.52 & 0.06 & 0.88 & 0.07 & 0.67 & 0.08 & 0.64 & 0.06 & 0.56 & 0.14\ log F(H$\beta$) & & -16.04 & 0.02 & -15.82 & 0.03 & -16.02 & 0.03 & -15.95 & 0.03 & -16.38 & 0.05\ m$_{5007A}$ & & 23.70 & 0.04 & 23.66 & 0.05 & 23.89 & 0.05 & 23.18 & 0.04 & 24.27 & 0.09\ & & & & & & & & & & &\ & & & & & & & & & & &\ & & & & &\ & & & & &\ & & & & &\ Species & $\lambda$ (Å) & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$ & F$_{Dered}$ & $\pm$\  \[O II\] & 3727 & 53 & 17 & 48 & 6 & & & 203 & 24\  \[Ne III\] & 3868 & 37 & 7 & 82 & 6 & 135 & 14 & 157 & 19\  \[Ne III\] + H$\epsilon$ & 3970 & & & 36 & 5 & & & 64 & 13\  H$\delta$ & 4101 & & & 18 & 13 & 26 & 8 & &\  H$\gamma$ & 4340 & 37 & 8 & 53 & 6 & 55 & 8 & 55 & 15\  \[O III\] & 4363 & & & 8 & 3 & 16 & 6 & 30 & 10\  He I & 4471 & & & 5 & 3 & & & &\  He II & 4686 & 46 & 9 & & & 23 & 16 & 50 & 13\  H$\beta$ & 4861 & 100 & 0 & 100 & 0 & 100 & 0 & 100 & 0\  \[O III\] & 4959 & 473 & 45 & 394 & 18 & 438 & 29 & 513 & 48\  \[O III\] & 5007 & 1405 & 130 & 1153 & 51 & 1274 & 82 & 1560 & 143\  He I & 5876 & 45 & 23 & & & & & &\  \[N II\] & 6548 & & & & & & & 54 & 8\  H$\alpha$ & 6562 & 282 & 26 & 281 & 14 & 282 & 20 & 281 & 29\  \[N II\] & 6583 & 18 & 7 & 43 & 6 & 25 & 6 & 255 & 27\  He I & 6678 & & & & & & & &\  \[S II\] & 6716 & & & & & & & &\  \[S II\] & 6730 & & & & & & & 21 & 13\  \[Ar III\] & 7133 & & & & & & & &\  \[O II\] & 7325 & 48 & 22 & & & & & &\ c & & 0.18 & 0.13 & 0.42 & 0.07 & 0.73 & 0.10 & 0.68 & 0.15\ log F(H$\beta$) & & -16.81 & 0.05 & -16.22 & 0.03 & -16.10 & 0.04 & -16.36 & 0.04\ m$_{5007A}$ & & 25.41 & 0.10 & 24.15 & 0.05 & 23.73 & 0.07 & 24.17 & 0.10\ & & & & & & & & & & &\ [^1]: Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile in observing proposals 64.N-0219, 66.B-0134, 67.B-0111 and 71.B-0134. [^2]: ESO programme 64.N-0219 [^3]: ESO programmes 64.B-0219, 66.B-0134 and 71.B-0134 [^4]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. [^5]: Attempts to reproduce the photoionization fits of Stasińska ([@Stas2002]) using Cloudy 08.01 were not entirely successful. The lower metallicity model could be satisfactorily reproduced with a slightly higher $T_{eff}$ of 43 kK and lower O of \[O/H\]=-1.02; however the higher metallicity model required \[O/H\]=+0.86 and at such high abundance the Cloudy models were very sensitive to extremely small changes in abundances of O, Ne, N and of the nebular size and density. From this numerical experiment, it is tentatively concluded that such extreme conditions are rare.
--- abstract: 'Recently, the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets $A$ of the $N$-dimensional Euclidean space ${\mathbb R}^N$, for any integer $N\ge1$. It is defined by the Lebesgue integral $\zeta_A(s)=\int_{A_{\delta}}d(x,A)^{s-N}{\mathrm d}x$, for all $s\in{\mathbb{C}}$ with $\operatorname{Re}\,s$ sufficiently large, and we call it the [*distance zeta function*]{} of $A$. Here, $d(x,A)$\[d(x,A)\] denotes the Euclidean distance from $x$ to $A$ and $A_{\delta}$ is the $\delta$-neighborhood of $A$, where ${\delta}$ is a fixed positive real number. We prove that the abscissa of absolute convergence of $\zeta_A$ is equal to $\overline\dim_BA$, the upper box (or Minkowski) dimension of $A$. Particular attention is payed to the principal complex dimensions of $A$, defined as the set of poles of $\zeta_A$ located on the critical line $\{\mathop{\mathrm{Re}} s=\overline\dim_BA\}$, provided $\zeta_A$ possesses a meromorphic extension to a neighborhood of the critical line. We also introduce a new, closely related zeta function, $\tilde\zeta_A(s)=\int_0^{\delta}t^{s-N-1}|A_t|\,{\mathrm d}t$, called the [*tube zeta function*]{} of $A$. Assuming that $A$ is Minkowski measurable, we show that, under some mild conditions, the residue of $\tilde\zeta_A$ computed at $D=\dim_BA$ (the box dimension of $A$), is equal to the Minkowski content of $A$. More generally, without assuming that $A$ is Minkowski measurable, we show that the residue is squeezed between the lower and upper Minkowski contents of $A$. We also introduce [*transcendentally quasiperiodic sets*]{}, and construct a class of such sets, using generalized Cantor sets, along with Baker’s theorem from the theory of transcendental numbers.' address: - 'University of California, Department of Mathematics, 900 University Ave., Riverside, California 92521-0135, USA' - 'University of Zagreb, Faculty of Electrical Engineering and Computing, Department of Applied Mathematics, Unska 3, 10000 Zagreb, Croatia' author: - 'M. L. Lapidus' - 'G. Radunović' - 'D. Žubrinić' title: Distance and tube zeta functions of fractals and arbitrary compact sets --- zeta function, distance zeta function, tube zeta function, fractal set, fractal string, box dimension, complex dimensions, principal complex dimensions, Minkowski content, Minkowski measurable set, residue, Dirichlet-type integral, transcendentally quasiperiodic set, fractality and complex dimensions. Introduction {#intro} ============ In this article, we provide a far-reaching extension of the theory of zeta functions for fractal strings, to arbitrary fractal sets in Euclidean spaces of any dimension. Fractal strings have been introduced by the first author (M. L. Lapidus) in the early 1990s. The related theory of zeta functions of fractal strings and their complex dimensions, developed in the course of the last two decades of active research, is presented in an extensive monograph of the first author with M. van Frankenhuijsen [@lapidusfrank12]. The new zeta function $\zeta_A$, associated with any fractal set $A$ in ${\mathbb{R}}^N$, has been introduced in 2009 by the first author, and its definition can be found in Equation (\[z\]) below. We refer to it as the [*distance zeta function*]{} of $A$. Here, by a fractal set, we mean any bounded set $A$ of the Euclidean space ${\mathbb{R}}^N$, with $N\geq 1$. The reason is that, in this paper, the key role is played by a certain notion of fractal dimension, more specifically, by the upper box dimension of a bounded set (also called the upper Minkowski dimension, Bouligand dimension, or limit capacity, etc.). This new class of zeta functions enables us to obtain a nontrivial extension of the theory of [*complex dimensions of fractal strings*]{}, to arbitrary bounded fractal sets in Euclidean spaces of any dimension. A systematic study of the zeta functions associated with fractal strings and fractal sprays was motivated and undertaken, in particular, in the 1990s in papers of the first author, \[Lap1–3\], as well as in joint papers of the first author with C. Pomerance \[[LapPo1–2]{}\] and with H. Maier [@LapMa2]. In a series of papers, as well as in two monographs with M. van Frankenhuijsen \[[Lap-vFr1–2]{}\], and in the book [@lapz], it has grown into a well-established theory of fractal complex dimensions, and is still an active area of research, with applications to a variety of subjects, including spectral theory, harmonic analysis, number theory, dynamical systems, probability theory and mathematical physics. We also draw the reader’s attention to [@DubSep], \[Es1–2\], \[EsLi1–2\], [@fal2], [@hamlap], [@lapidushe], [@HerLa1], [@Kom], [@LaLeRo], [@LaLu], [@lappe2], [@lappewi1], \[LapRaŽu1–8\], [@lemen], [@MorSep], \[MorSepVi1–2\], \[Ol1–2\], [@winter], \[Tep1–2\], along with the many relevant references therein. Other, very different approaches to a higher-dimensional theory of some special classes of fractal sets, namely, fractal sprays and self-similar tilings, were developed by the first author and E. Pearse in [@lappe2], as well as by the first author, E. Pearse and S. Winter in [@lappewi1] via fractal tube formulas and the associated scaling and tubular zeta functions. (See also [@pe2] and [@pewi].) The definitions of the tubular zeta functions introduced in [@lappe2] and [@lappewi1] differ considerably from those studied in this article. The precise connection between these zeta functions and the fractal zeta functions introduced in this paper is provided in [@cras2]. We point out that by using the fractal zeta functions introduced in this paper, it is possible to generalize the fractal tube formulas and a Minkowski measurability criterion obtained for fractal strings in [@lapidusfrank12] to arbitrary compact sets in Euclidean spaces; see \[LapRaŽu4–5\]. We also refer to [@mm] for a key use of these fractal tube formulas to obtain a Minkowski measurability criterion, expressed in terms of the nonexistence of nonreal (principal) complex dimensions and generalizing to any dimension its counterpart established for fractal strings in [@lapidusfrank12 Chapter 8]. Contents -------- The rest of this paper is organized as follows: In Section \[ch\_distance\], the distance zeta function $\zeta_A$ of a bounded set $A{\subset}{\mathbb{R}}^N$ is introduced in Definition \[z\]. Then, the main result of Section \[ch\_distance\] is obtained in Theorem \[an\], in which it is shown (among other things) that the abscissa of (absolute) convergence of the distance zeta function $\zeta_A$ of any bounded subset $A$ of ${\mathbb{R}}^N$ is equal to ${\overline{\dim}}_BA$, the upper box dimension (or the upper Minkowski dimension) of $A$. (All of the subsets denoted by $A$ appearing in this paper are implicitly assumed to be nonempty.) As a useful technical tool in the study of fractal zeta functions, we introduce the notion of ‘equivalence’ between tamed Dirichlet-type integrals (see Definition \[equ\]). We also define the set of ‘principal complex dimensions’ of $A$, denoted by $\dim_{PC}A$ (see Definition \[dimc\]), as a refinement of the notion of the upper box dimension of $A$. Moreover, in the one-dimensional case (i.e., in the case of a bounded fractal string $\mathcal L$), we show that $\zeta_A$, the distance zeta function of $A$ (the boundary of the string $\mathcal L$), and $\zeta_{\mathcal L}$, the geometric zeta function of $\mathcal L$, contain essentially the same information. In particular, $\zeta_A$ and $\zeta_{\mathcal L}$ are equivalent in the above sense, and hence, have the same principal complex dimensions (see Subsections \[zeta\_s\] and \[eqzf\]); they also have the same (visible) complex dimensions (with the same multiplicities) in every domain of ${\mathbb{C}}\setminus\{0\}$ to which one (and hence, both) of these zeta functions can be meromorphically continued. Finally, we show that the distance zeta function has a nice and very useful scaling property; see Proposition \[scalingd\]. In Section \[residues\_m\], we introduce the so-called ‘tube zeta function’ $\tilde\zeta_A$ of the bounded set $A$ (which is closely related to the distance zeta function $\zeta_A$; see Theorem \[equr\] and the associated functional equation ), and study its properties; see, in particular, Definition \[zeta\_tilde\] in Subsection \[residues\_m\_tube\]. Under suitable natural conditions, we show that the residue of the tube zeta function $\tilde\zeta_A$, computed at $D=\dim_BA$ (assuming that the box dimensions exists), always lies between the lower and upper ($D$-dimensional) Minkowski contents of $A$; see Theorem \[pole1mink\_tilde\]. In particular, if $A$ is Minkowski measurable, then the residue of $\tilde\zeta_A$ at $D$ coincides with the Minkowski content of $A$. Similar results are obtained for the distance zeta function $\zeta_A$ of the fractal set $A$; see Theorem \[pole1\]. In fact, we also show that $\zeta_A$ and $\tilde\zeta_A$, the distance and tube zeta functions of $A$, contain essentially the same information. These results are illustrated by means of several examples, including a class of generalized Cantor sets (Examples \[res-cantor\], \[res-cantor2\] and \[Cmae\]), the $(N-1)$-dimensional sphere in ${\mathbb{R}}^N$ (see Example \[sphere\]), $a$-strings (Example \[a-string2\]), as well as ‘fractal grills’ introduced in Subsection \[dtx\]; see Theorem \[Axm\]. In Section \[quasi0\], we introduce a class of ‘$n$-quasiperiodic sets’ (Definition \[quasiperiodic\]). The main result is stated in Theorem \[quasi1\], which can be considered as a fractal set-theoretic interpretation of Baker’s theorem (Theorem \[baker0\]) from transcendental number theory and in which we construct a family of transcendentally $n$-quasiperiodic sets, for any integer $n\ge2$. An important role in the construction of quasiperiodic sets is played by the class of generalized Cantor sets $C^{(m,a)}$ depending on two parameters, introduced in Definition \[Cma\]. Moreover, in Subsection \[hyperfractal\], we close the main part of this paper by connecting the present work to future extensions (notably, the construction of transcendentally ${\infty}$-quasiperiodic sets), the notion of hyperfractal (and even, maximally hyperfractal) set, and more broadly, the notion of fractality within the context of this new general theory of complex dimensions. In short, much as in [@lapidusfrank12], we say that a bounded subset $A{\subset}{\mathbb{R}}^N$ is [*fractal*]{} if its associated zeta function (i.e., the distance or the tube zeta function, $\zeta_A$ or $\tilde\zeta_A$, of $A$ or when $N=1$, the geometric zeta function $\zeta_{\mathcal L}$, where $\mathcal L$ is the fractal string associated with $A$) has at least one nonreal complex dimension or else has a natural boundary beyond which it cannot be meromorphically continued (i.e., $A$ is “hyperfractal”). Observe that, unlike in the one-dimensional theory of complex dimensions developed in [@lapidusfrank12], we now have at our disposal precise definitions of the fractal zeta functions of arbitrary bounded subsets of ${\mathbb{R}}^N$ and hence, of the complex dimensions of those sets (i.e., of the poles of these fractal zeta functions); see Definition \[1.331/2\] and the beginning of Subsection \[residues\_m\_tube\]. The complex dimensions of a variety of classic and less well-known fractals will be computed in subsequent work, \[LapRaŽu1–8\]. The aim of Appendix A is to introduce the class of ‘extended Dirichlet-type integrals’ (or functions), i.e., of EDTIs, which contains all of the fractal zeta functions studied in the present paper; see Definition A.1. We study some of the key properties of EDTIs and introduce two closely related (but distinct) notions of equivalence; see Definitions A.2 and A.6. Notation -------- Throughout this paper, we shall use the following notation. By $|E|=|E|_N$, we denote the $N$-dimensional Lebesgue measure of a measurable subset $E$ of ${\mathbb{R}}^N$. Given $r\ge0$, the [*lower and upper $r$-dimensional Minkowski contents*]{} ${{\cal M}}^{*r}(A)$ and ${{\cal M}}_*^r(A)$ of a bounded subset $A$ of ${\mathbb{R}}^N$ are defined by $$\label{mink} {{\cal M}}_*^r(A)=\liminf_{t\to 0^+}\frac{|A_t|}{t^{N-r}}, {\quad}{{\cal M}}^{*r}(A)=\limsup_{t\to 0^+}\frac{|A_t|}{t^{N-r}}.$$ Here, $A_t:=\{x\in{\mathbb{R}}^N:d(x,A)<t\}$ denotes the $t$-[*neighborhood*]{} (or [*tubular neighborhood of radius*]{} $t$) of $A$, and $d(x,A)$ is the Euclidean distance from $x$ to $A$. The function $t\mapsto|A_t|$, defined for $t$ positive and close to $0$, is called the [*tube function*]{} associated with $A$. From our point of view, one of the basic tasks of fractal analysis is to understand the nature of the tube functions for various fractal sets. The above definition coincides with Federer’s definition (in [@federer]), up to a (positive) multiplicative constant depending only on $N$ and $r$, the value of which is not important for the purposes of this article. The [*upper box dimension*]{} of $A$ is defined by $$\label{dim} {\overline{\dim}}_BA=\inf\{r\ge0:{{\cal M}}^{*r}(A)=0\};$$ it is easy to see that we also have $$\label{Mty} {\overline{\dim}}_BA=\sup\{r\ge0:{{\cal M}}^{*r}(A)=+{\infty}\}.$$ The lower box dimension of $A$, denoted by $\underline \dim_BA$, is defined analogously, with ${{\cal M}}_*^r(A)$ instead of ${{\cal M}}^{*r}(A)$ on the right-hand side of (\[dim\]) and (\[Mty\]). Clearly, since $A$ is bounded, we always have $0\le\underline\dim_BA\le{\overline{\dim}}_BA\le N$. If both ${\overline{\dim}}_BA$ and $\underline\dim_BA$ coincide, their common value is denoted by $\dim_BA$ and is called the [*box dimension*]{} of $A$ (or [*Minkowski–Bouligand dimension*]{}, or else, [*Minkowski dimension*]{}\[MinkDim\]). Various properties of the box dimension can be found, e.g., in [@falc], [@mattila], [@tricot] and [@lapidusfrank12]. If there exists a nonnegative real number $D$ such that $$0<{{\cal M}}_*^D(A)\le{{\cal M}}^{*D}(A)<{\infty},$$ we say that $A$ is [*Minkowski nondegenerate*]{}. If $A$ is nondegenerate, it then follows that $\dim_BA$ exists and is equal to $D$. If ${{\cal M}}_*^D(A)={{\cal M}}^{*D}(A)$, their common value is denoted by ${{\cal M}}^D(A)$ and called the [*Minkowski content*]{} of $A$. If, in addition, $${{\cal M}}^D(A)\in(0,+{\infty}),$$ then $A$ is said to be [*Minkowski measurable*]{}.\[Minkowski\_measurable\] The notion of Minkowski measurability seems to have been introduced by Hadwiger in [@hadwiger] and was later used by Federer in [@federer], as well as by Stachó [@stacho] (inspired by [@federer]), and in many other works, including [@BroCar], [@Lap1] [@lapiduspom], [@fal2], [@tricot], [@mink], [@rae], [@Kom] and [@KomPeWi]. The notion of Minkowski nondegeneracy has been introduced in [@rae] (and was studied earlier in [@lapiduspom] and \[Lap-vFr1–2\] when $N=1$; see also [@LapPo2] when $N\ge3$). The notion of Minkowski (or box) dimension was introduced by Bouligand in [@bouligand]. Throughout this paper, we will assume implicitly that the bounded set $A{\subset}{\mathbb{R}}^N$ is nonempty. We note that since $|A_t|=|({\overline{A}})_t|$ for every $t>0$, the values of ${{\cal M}}_*^r(A)$, ${{\cal M}}^{*r}(A)$, $\underline\dim_BA$, ${\overline{\dim}}_BA$ (as well as of ${{\cal M}}^D(A)$ and $\dim_BA$, when they exist) do not change when we replace the bounded set $A{\subset}{\mathbb{R}}^N$ by its closure ${\overline{A}}$ in ${\mathbb{R}}^N$. Therefore, throughout this paper, we might as well assume a priori that $A$ is an arbitrary (nonempty) compact subset of ${\mathbb{R}}^N$. Observe that, as is well known, this is in sharp contrast with the Hausdorff dimension (and associated Hausdorff measure $\mathcal{H}_H$); see, e.g., [@falc]. For example, if $A=\{1/j:j\in{\mathbb{N}}\}$, then (since $A$ is countable), $\dim_HA=0$ and $\mathcal{H}_H(A)=0$, while $D:=\dim_BA=1/2$ and $\mathcal{M}^D(A)=2\sqrt2$; see [@Lap1 Example 5.1]. Finally, given an extended real number ${\alpha}\in{\mathbb{R}}\cup\{\pm{\infty}\}$, we denote by $\{{\mathop{\mathrm{Re}}}s>{\alpha}\}$ the open right half-plane $\{s\in{\mathbb{C}}:{\mathop{\mathrm{Re}}}s>{\alpha}\}$ (which coincides with ${\mathbb{C}}$ or $\emptyset$ if ${\alpha}=-{\infty}$ or $+{\infty}$, respectively). Furthermore, if ${\alpha}\in{\mathbb{R}}$, we denote by $\{{\mathop{\mathrm{Re}}}s={\alpha}\}$ the vertical line $\{s\in{\mathbb{C}}:{\mathop{\mathrm{Re}}}s={\alpha}\}$. Also, we let ${\mathbbm{i}}:=\sqrt{-1}$. Distance and tube zeta functions of fractal sets {#ch_distance} ================================================ In this section, we introduce and study a new fractal zeta function, namely, the distance zeta function attached to an arbitrary bounded subset of ${\mathbb{R}}^N$, for any $N\ge1$; see Subection \[properties\_definition\] and Subsection \[properties\_analyticity\]. In Subsection \[properties\_zeta\], we then consider the special case when $N=1$ and compare this new fractal zeta function with the known geometric zeta function of a fractal string. Finally, in Subsection \[eqzf\], we introduce a suitable equivalence relation which enables us to capture some of the main features of fractal zeta functions. Definition of the distance zeta functions of fractal sets {#properties_definition} --------------------------------------------------------- We study here some basic properties of the distance zeta function $\zeta_A=\zeta_A(s)$ associated with an arbitrary bounded subset $A$ of ${\mathbb{R}}^N$, and introduced by the first author in 2009. \[defn\] Let $\delta$ be any given positive number. The [*distance zeta function*]{} $\zeta_A$ of a bounded subset $A$ of ${\mathbb{R}}^N$ is defined by $$\label{z} \zeta_A(s):=\int_{A_\delta}d(x,A)^{s-N}{\mathrm d}x.$$ Here, the integral is taken in the sense of Lebesgue (hence, the complex-valued function $d(\,\cdot\,,A)^{s-N}$ is absolutely integrable on $A_{\delta}$) and we assume that $s\in{\mathbb{C}}$ is such that ${\mathop{\mathrm{Re}}}s$ is sufficiently large. As we shall see in Theorem \[an\], the Lebesgue integral in (\[z\]) is well defined if ${\mathop{\mathrm{Re}}}s$ is larger than ${\overline{\dim}}_BA$, the upper box dimension of $A$; furthermore, ${\overline{\dim}}_BA=D(\zeta_A)$, the [*abscissa of $($absolute$)$ convergence*]{} of $\zeta_A$. Moreover, under the additional hypotheses of Theorem \[an\](c), ${\overline{\dim}}_BA$ also coincides with $D_{\rm hol}(\zeta_A)$, the [*abscissa of holomorphic continuation*]{} of $\zeta_A$. Here, by definition, $$\label{DzetaA} D(\zeta_A):=\inf\left\{{\alpha}\in{\mathbb{R}}:\int_{A_{\delta}}d(x,A)^{{\alpha}-N}{\mathrm d}x<{\infty}\right\}$$ while $$\label{Dhol} D_{\rm hol}(\zeta_A):=\inf\big\{{\alpha}\in{\mathbb{R}}:\mbox{$\zeta_A$ is holomorphic on $\{{\mathop{\mathrm{Re}}}s>{\alpha}\}$}\,\big\}.$$ Hence, the [*half-plane of $($absolute$)$ convergence*]{} of $\zeta_A$, $\Pi(\zeta_A):=\{{\mathop{\mathrm{Re}}}s>D(\zeta_A)\}$ (resp., the [*half-plane of holomorphic continuation*]{} of $\zeta_A$, $\mathcal{H}(\zeta_A):=\{{\mathop{\mathrm{Re}}}s>D_{\rm hol}(\zeta_A)\}$) is the largest open half-plane of the form $\{{\mathop{\mathrm{Re}}}s>{\alpha}\}$, for some ${\alpha}\in{\mathbb{R}}\cup\{\pm{\infty}\}$, on which the Lebesgue integral $\int_{A_{\delta}}d(x,A)^{s-N}{\mathrm d}x$ is convergent or, equivalently, absolutely convergent (resp., to which $\zeta_A$ can be holomorphically continued). It will follow from our results that $D(\zeta_A)\in[0,N]$ while $D_{\rm hol}(\zeta_A)\in[-{\infty},D(\zeta_A)]$, and that both $D(\zeta_A)$ and $D_{\rm hol}(\zeta_A)$ are independent of the choice of ${\delta}>0$; see Proposition \[O\] along with Definition \[equ\]. Again, the same comment can be made about $D(\tilde\zeta_A)$ and $D_{\rm hol}(\tilde\zeta_A)$, given exactly as in and , respectively, except for $\zeta_A$ replaced by $\tilde\zeta_A$ (the tube zeta function of $A$, see Definition \[zeta\_tilde\]). Actually, if ${\overline{\dim}}_BA<N$, then $D(\zeta_A)=D(\tilde\zeta_A)$ and $D_{\rm hol}(\zeta_A)=D_{\rm hol}(\tilde\zeta_A)$; see Corollary \[equr\_c\]. Given any meromorphic function $f$, the [*abscissa of holomorphic continuation*]{} of $f$, denoted by $D_{\rm hol}(f)$, can be defined in exactly the same way as $D_{\rm hol}(\zeta_A)$, except with $\zeta_A$ replaced by $f$ in the counterpart of . The same comment is not true for $D(f)$, which may not make sense unless $f$ is given by a Dirichlet-type integral (DTI); see Subsection \[eqzf\] and Appendix A below. As will be shown in Proposition \[O\], the dependence of $\zeta_A$ on the choice of $\delta$ is inessential, since the difference of two distance zeta functions corresponding to the same set $A$ and different values of ${\delta}$ can be identified with an entire function. Note that without loss of generality (in fact, simply by replacing $A$ by its closure), we could assume that $A$ is an arbitrary (nonempty) compact subset of ${\mathbb R}^N$. Similar comments could be made about the tube zeta functions introduced in Definition \[zeta\_tilde\] below. Analyticity of the distance zeta functions {#properties_analyticity} ------------------------------------------ The main result of this section is stated in Theorem \[an\]. It shows that the zeta function $\zeta_A$ is analytic (i.e., holomorphic) in the half-plane $\{{\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA\}$, and that (under the mild hypotheses of part $(c)$ of Theorem \[an\]) the lower bound is optimal. In other words, the [*abscissa of absolute convergence*]{} $D(\zeta_A)$ of the Dirichlet-type integral defined by the right-hand side of (\[z\]) is always equal to the upper box dimension of $A$ and, under the additional hypotheses of Theorem \[an\](c), it also coincides with the abscissa of holomorphic continuation $D_{\rm hol}(\zeta_A)$. In order to prove Theorem \[an\], we shall need a result due to Harvey\[harvey\] and Polking\[polking\] (see [@acta p. 42]), obtained in order to study the singularities of the solutions of certain linear partial differential equations, and which we now formulate in a different, but equivalent way: $$\label{int} \mbox{If{\quad}$\gamma\in(-{\infty},N-{\overline{\dim}}_BA)$,{\quad}then{\quad}$\int_{A_\delta}d(x,A)^{-\gamma}{\mathrm d}x<{\infty}$,}$$ where $\delta$ is an arbitrary positive number. This result and its various extensions is discussed in [@rae Sections 3 and 4]. For the sake of completeness, we provide an extension of (\[int\]), which we shall need later on. We omit the proofs of the following two lemmas. They can be obtained by using, e.g., the identity $\int_{{\mathbb{R}}^N}f(x)^{\alpha}{\mathrm d}x={\alpha}\int_0^{+{\infty}} t^{{\alpha}-1}|\{f>t\}|\,{\mathrm d}t$, where $f:{\mathbb{R}}^N\to[0,+{\infty}]$ is a Lebesgue measurable function and ${\alpha}\in(0,+{\infty})$ (see [@folland p. 198]), and by using the definition of the upper box dimension ${\overline{\dim}}_BA$ given in and above. \[identity0\] Let $A$ be a bounded subset of ${\mathbb{R}}^N$, ${\delta}>0$ and $\gamma\in(-{\infty},N-{\overline{\dim}}_BA)$. Then $$\label{identity} \int_{A_\delta}d(x,A)^{-\gamma}\,{\mathrm d}x=\delta^{-\gamma}|A_\delta|+\gamma\int_0^\delta t^{-\gamma-1}|A_t|\,{\mathrm d}t.$$ Furthermore, both of the integrals appearing in  are finite; hence, they are convergent Lebesgue integrals. \[optimal\] Let $A$ be a bounded subset of ${\mathbb{R}}^N$, $\delta>0 $ and $\gamma>N-{\overline{\dim}}_BA$. Then $\int_{A_\delta}d(x,A)^{-\gamma}{\mathrm d}x=+{\infty}$. \[rm2.5\] If $\gamma:=N-{\overline{\dim}}_BA$, then the conclusion of Lemma \[optimal\] does not hold, in general. Indeed, a class of counterexamples is provided in [@rae Theorem 4.3]. In the sequel, we shall usually say more briefly that $D(\zeta_A)$ is the [*abscissa of convergence*]{} of $\zeta_A$, meaning the abscissa of Lebesgue (i.e., absolute) convergence of $\zeta_A$; see and the comment following it. \[an\] Let $A$ be an arbitrary bounded subset of ${\mathbb{R}}^N$ and let $\delta>0$. Then$:$ $(a)$ The zeta function $\zeta_A$ defined by is holomorphic in the half-plane $\{{\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA\}$, and for all complex numbers $s$ in that region, we have $$\label{zeta'} \zeta_A'(s)=\int_{A_\delta}d(x,A)^{s-N}\log d(x,A)\,{\mathrm d}x.$$ $(b)$ We have $$\overline\dim_BA=D(\zeta_A),$$ where $D(\zeta_A)$ is the abscissa of Lebesgue $($i.e., absolute$)$ convergence of $\zeta_A$. Furthermore, in light of part $(a)$, we always have $D_{\rm hol}(\zeta_A)\le D(\zeta_A)$. $(c)$ If the box $($or Minkowski$)$ dimension $D:=\dim_BA$ exists, $D<N$, and ${{\cal M}}_*^D(A)>0$, then $\zeta_A(s)\to+{\infty}$ as $s\to D^+$, $s\in{\mathbb{R}}$. In particular, in this case, we also have that $$\label{dimDDhol} \dim_BA=D(\zeta_A)=D_{\rm hol}(\zeta_A).$$ $(a)$ Denoting the right-hand side of (\[zeta’\]) by $I(s)$, and choosing any $s\in{\mathbb{C}}$ such that ${\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA$, it suffices to show that $$\begin{aligned} \label{der} R(h)&:=&\frac{\zeta_A(s+h)-\zeta_A(s)}h-I(s)\\ &=&\int_{A_\delta}\left( \frac{d(x,A)^h-1}h-\log d(x,A) \right)\,d(x,A)^{s-N}{\mathrm d}x\nonumber\end{aligned}$$ converges to zero as $h\to0$ in ${\mathbb{C}}$, with $h\ne0$. Let $d:=d(x,A)\in(0,\delta)$. Defining $$\label{fhdef} f(h):=\frac{d^h-1}h-\log d=\frac1h({\mathrm e}^{(\log d)h}-1)-\log d,$$ and using the MacLaurin series ${\mathrm e}^z=\sum_{j\ge0}\frac{z^j}{j!}$, we obtain that $$\label{fh} f(h)=h(\log d)^2\sum_{k=0}^{\infty}\frac1{(k+2)(k+1)}\cdot\frac{(\log d)^kh^k}{k!}.$$ Furthermore, assuming without loss of generality that $0<\delta\le1$, and hence $\log d\le0$, we have $$\begin{aligned} |f(h)|&\le&\frac12|h|\,(\log d)^2\sum_{k=0}^{\infty}\frac{(|\log d|\,|h|)^k}{k!}\nonumber\\ &=&\frac12|h|\,(\log d)^2{\mathrm e}^{-(\log d)|h|}=\frac12|h|\,(\log d)^2d^{-|h|}.\nonumber\end{aligned}$$ Therefore, $$|R(h)|\le\frac12|h|\int_{A_\delta}|\log d(x,A)|^2d(x,A)^{{\mathop{\mathrm{Re}}}s-N-|h|}{\mathrm d}x.$$ Let ${\varepsilon}>0$ be a sufficiently small number, to be specified below. Taking $h\in{\mathbb{C}}$ such that $|h|<{\varepsilon}$, since $\delta\le1$ and hence $d(x,A)\le1$ for all $x\in A_{\delta}$, we have $$|R(h)|\le\frac12|h|\int_{A_\delta}|\log d(x,A)|^2d(x,A)^{\varepsilon}d(x,A)^{{\mathop{\mathrm{Re}}}s-N-2{\varepsilon}}{\mathrm d}x.$$ Since there exists a positive constant $C=C(\delta,{\varepsilon})$ such that $|\log d|^2d^{\varepsilon}\le C$ for all $d\in(0,\delta)$, we see that $$\label{R} |R(h)|\le \frac12C|h|\int_{A_\delta}d(x,A)^{{\mathop{\mathrm{Re}}}s-N-2{\varepsilon}}{\mathrm d}x.$$ Letting $\gamma:=2{\varepsilon}+N-{\mathop{\mathrm{Re}}}s$, we see that the integrability condition $\gamma<N-{\overline{\dim}}_BA$ stated in (\[int\]) is equivalent to ${\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA+2{\varepsilon}$. Observe that this latter inequality holds for all positive ${\varepsilon}$ small enough, due to the assumption ${\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA$. Hence, $R(h)\to0$ as $h\to0$ in ${\mathbb{C}}$, with $h\ne0$. This proves part $(a)$. $(b)$ Lemma \[optimal\] implies that for any real number $\alpha<D={\overline{\dim}}_BA$, we have $\int_{A_\delta}d(x,A)^{\alpha-N}\, {\mathrm d}x=+\infty$. On the other hand, in light of estimate (\[int\]), we know that $\zeta_A(\alpha)=\allowbreak \int_{A_\delta}d(x,A)^{\alpha-N}\,{\mathrm d}x<\infty$ for any $\alpha>D$. We therefore deduce from the definition of $D(\zeta_A)$ that $D(\zeta_A)={\overline{\dim}}_BA$. This completes the proof of part $(b)$. $(c)$ Condition ${{\cal M}}_*^D(A)>0$ implies that for any fixed $\delta>0$ there exists $C>0$ such that for all $t\in(0,\delta)$, we have $|A_t|\ge Ct^{N-D}$. Using (\[int\]) and Lemma \[identity0\], we see that for any $\gamma\in(0,N-D)$, $$\begin{aligned} {\infty}&>&I(\gamma):=\int_{A_\delta}d(x,A)^{-\gamma}{\mathrm d}x=\delta^{-\gamma}|A_\delta|+\gamma\int_0^\delta t^{-\gamma-1}|A_t|\,dt\nonumber\\ &\ge&\gamma C\int_0^\delta t^{N-D-\gamma-1}{\mathrm d}t= \gamma C\frac{\delta^{N-D-\gamma}}{N-D-\gamma}.\nonumber\end{aligned}$$ Therefore, if $\gamma\to N-D$ from the left, then $I(\gamma)\to+{\infty}$. Equivalently, if $s\in{\mathbb{R}}$ is such that $s\to D^+$, then $\zeta_A(s)\to+{\infty}$. Hence, $\zeta_A$ has a singularity at $s=D$. Since, in light of part $(a)$, we know that $\zeta_A$ is holomorphic for ${\mathop{\mathrm{Re}}}s>D$, we deduce that $\{{\mathop{\mathrm{Re}}}s>D\}$ is the maximal right half-plane to which $\zeta_A$ can be holomorphically continued; i.e., $\mathcal{H}(\zeta_A)=\{{\mathop{\mathrm{Re}}}s>D\}$ and so $D_{\rm hol}(\zeta_A)=D$. Since, in light of part $(b)$ (and because $\dim_BA$ exists, according to the assumptions of part $(c)$), $D:=\dim_BA=D(\zeta_A)$, we conclude that holds and hence, the proof of part $(c)$ is complete. This concludes the proof of the theorem. An alternative proof of part $(a)$ of Theorem \[an\] can be given by using a well-known theorem concerning the holomorphicity of functions defined by integrals on $A_{\delta}$ depending holomorphically on a parameter. In applying this theorem (see \[LapRaŽu1\] and the text of Definition \[abscissa\_f\] below) one needs to use the (obvious) fact according to which the function $x\mapsto d(x,A)$ is bounded from above (by ${\delta}$); in other words, $\zeta_A$ (as defined by ) is a tamed DTI (in the sense of Definition \[abscissa\_f\] below). Next, we comment on some of the hypotheses and conclusions of Theorem \[an\]. $(i)$ The condition ${{\cal M}}_*^D(A)>0$ in the hypotheses of Theorem  \[an\](c) cannot be omitted. Indeed, for $N=1$, there exists a class of subsets $A{\subset}[0,1]$ such that $D=\dim_BA$ exists and ${{\cal M}}_*^D(A)=0$, while $\zeta_A(D)=\int_{A_\delta}d(x,A)^{D-N}{\mathrm d}x<{\infty}$; see [@rae Theorem 4.3]. This class of bounded subsets of ${\mathbb{R}}$ can be easily extended to ${\mathbb{R}}^N$ for any $N\ge2$ by letting $B:=A\times[0,1]^{N-1}{\subset}[0,1]^N$ and using the results of Subsection \[dtx\]. $(ii)$ The inequality $D_{\rm hol}(\zeta_A)\le D(\zeta_A)$ is sharp. Indeed, there exist compact subsets of ${\mathbb{R}}^N$ such that $D_{\rm hol}(\zeta_A)=D(\zeta_A)$. For example, $A=C\times[0,1]^{N-1}$, where $C$ is the ternary Cantor set or, more generally, $C={\partial}{\Omega}$ is the boundary of any (nontrivial) bounded fractal string ${\Omega}{\subset}{\mathbb{R}}$. (In that case, we have $D_{\rm hol}(\zeta_A)=D(\zeta_A)={\overline{\dim}}_BA=N-1+\dim_B C$.) This follows from Theorem \[1.2.31\] in Subsection \[zeta\_s\] below and the comment following it. $(iii)$ The assumptions of part $(c)$ of Theorem \[an\] are satisfied by most fractals of interest to us. (One notable exception is the boundary $A$ of the Mandelbrot set (viewed as as a subset of ${\mathbb{R}}^2\simeq{\mathbb{C}}$), for which $\dim_HA=2$ (and consequently, $\dim_BA=2$ since $\dim_HA\le\dim_BA$), according to Shishikura’s well-known theorem [@shishikura].) We note that, on the other hand, there exists a bounded subset of ${\mathbb{R}}^N$ not satisfying the hypotheses of part $(c)$ of Theorem \[an\] and such that $D_{\rm hol}(\zeta_A)<D(\zeta_A)$. Indeed, an easy computation shows that, for example, for $N=1$ and $A=[0,1]$, we have that $D_{\rm hol}(\zeta_A)=0$ and $D(\zeta_A)=1$. At present, however, we do not know whether there exist nontrivial subsets $A$ of ${\mathbb{R}}$ (or, more generally, of ${\mathbb{R}}^N$) for which $D_{\rm hol}(\zeta_A)<D(\zeta_A)$. Zeta functions of fractal strings and of associated fractal sets {#zeta_s} ---------------------------------------------------------------- \[properties\_zeta\] In Example \[L\] below, we show that Definition \[defn\] provides a natural extension of the zeta function associated with a (bounded) [*fractal string*]{} $\mathcal L=(\ell_j)_{j\ge1}$, where $(\ell_j)_{j\geq 1}$ is a nonincreasing sequence of positive numbers such that $\sum_{j=1}^\infty \ell_j<{\infty}$: $$\label{string} \zeta_{\mathcal L}(s)=\sum_{j=1}^\infty \ell_j^s,$$ for all $s\in{\mathbb{C}}$ with ${\mathop{\mathrm{Re}}}s$ sufficiently large. Note that the sequence $(\ell_j)_{j\ge1}$ of positive numbers is assumed to be infinite. The study of zeta functions of fractal strings arose naturally in the early 1990s in joint work of the first author\[lapidus2\] with Carl Pomerance \[[LapPo1–2]{}\] and with Helmut Maier\[maier\] [@LapMa2] while investigating direct and inverse spectral problems associated with the vibrations of a fractal string. Such a zeta function, $\zeta_{\mathcal L}$, called the [*geometric zeta function*]{}\[geometric\_zf\] of $\mathcal L$, has since then been studied in a number of references, including the monograph [@lapidusfrank12].. (See the broader list of references given in the introduction.) Recall that, geometrically, a (bounded) [*fractal string*]{}\[fr\_str\] is a bounded open set $\Omega\subseteq{\mathbb{R}}$. It can be uniquely written as a disjoint union of open intervals $I_j$ ($\Omega=\cup_{j=1}^\infty I_j$) with lengths $\ell_j$ (i.e., $\ell_j=|I_j|$ for all $j\ge1$). Without loss of generality, one may assume that $(\ell_j)_{j\geq 1}$ is written in nonincreasing order and that $\ell_j\to 0$ as $j\to\infty$: $\ell_1\geq \ell_2\geq\cdots$. In order to avoid trivial special cases, we will assume implicitly throughout this paper that $\mathcal L$ is nontrivial; i.e., that $\mathcal L$ consists of an infinite sequence of lengths (or ‘scales’) and hence, that ${\Omega}$ does not consist of a finite union of bounded open intervals. If $\mathcal L$ is trivial, then we must replace $D_{\rm hol}(\zeta_{\mathcal L})$ by $\max\{D_{\rm hol}(\zeta_{\mathcal{L}}),0\}$ in of Theorem \[2.181/2\] (since then, $D_{\rm hol}(\zeta_{\mathcal L})=-{\infty}$ and $D(\zeta_{\mathcal L})={\delta}_{{\partial}{\Omega}}\ge0$). From the point of view of fractal string theory, one may identify a fractal string with the sequence $\mathcal L$ of its lengths $($or [*scales*]{}$)$: $\mathcal L=(\ell_j)_{j\geq 1}$. The bounded open set ${\Omega}$ is then called a [*geometric realization of $\mathcal{L}$*]{}. Note that $|{\Omega}|=\sum_{j=1}^{\infty}\ell_j<{\infty}$, where $|{\Omega}|=|{\Omega}|_1$ denotes the $1$-dimensional Lebesgue measure (or length) of ${\Omega}$. We now recall a basic property of $\zeta_{\mathcal L}$, first observed in [@Lap2], using a key result of Besicovich\[besicovich\] and Taylor\[taylor\] [@BesTay]. (For a direct proof, see [@lapidusfrank12 Theorem 1.10].) \[2.181/2\] If $\mathcal L$ is a nontrivial bounded fractal string $($i.e., $\mathcal L=(\ell_j)_{j\geq 1}$ is an infinite sequence$)$, then the abscissa of convergence $D(\zeta_{\mathcal L})$ of $\zeta_{\mathcal L}$ coincides with the $($inner$)$ Minkowski dimension $\delta_{\partial\Omega}$ of $\partial\mathcal L=\partial\Omega:$ $$\label{2.201/2} D(\zeta_{\mathcal L})=D_{\rm hol}(\zeta_{\mathcal L})=\delta_{\partial\Omega}.$$ Recall that, by definition, $$\label{Ddef} D(\zeta_{\mathcal L}):=\inf\Big\{\alpha\in{\mathbb{R}}\ :\ \sum_{j=1}^{\infty}\ell_j^{\alpha}<\infty\Big\},$$ while $\delta_{\partial\Omega}$ is then defined in terms of the volume (i.e., length) of the inner epsilon (or tubular) neighborhoods of ${\partial}\Omega$, namely, $({\partial}\Omega)_{{\varepsilon}}\cap\Omega=\{x\in\Omega\ :\ d(x,\partial\Omega)<{\varepsilon}\}$; see [@lapidusfrank12 Chapter 1]. In order to establish the equality $D(\zeta_{\mathcal L})=D_{\rm hol}(\zeta_{\mathcal L})$ from Theorem \[2.181/2\], one first notes that $\zeta_{\mathcal L}$ is holomorphic for ${\mathop{\mathrm{Re}}}s>D(\zeta_{\mathcal L})$ and that $\{{\mathop{\mathrm{Re}}}s>D(\zeta_{\mathcal L})\}$ is the largest open right half-plane having this property; i.e., $D(\zeta_{\mathcal L})=D_{\rm hol}(\zeta_{\mathcal{L}})$. The latter property follows from the fact that (because $\zeta_{\mathcal L}(s)$ is initially given in by a Dirichlet series with positive coefficients), $\zeta_{\mathcal L}(s)\to+{\infty}$ as $s\to D^+$, $s\in{\mathbb{R}}$, where $D:=D(\zeta_{\mathcal L})={\delta}_{{\partial}{\Omega}}$; see, e.g., [@serre Section VI.2.3]. The proof of the equality $D(\zeta_{\mathcal L})={\delta}_{{\partial}{\Omega}}$ requires significantly more work; see the aforementioned references. Note that, more precisely, ${\overline{\dim}}_BA_{\mathcal L}={\delta}_{{\partial}{\Omega}}$ is equal to ${\overline{\dim}}_B({\partial}{\Omega},{\Omega})$, the Minkowski dimension of ${\partial}{\Omega}$ relative to ${\Omega}$ (also called the inner Minkowski dimension of ${\partial}{\Omega}$, or, equivalently, of $\mathcal L$) which is defined in terms of the volume (i.e., length) of the inner tubular neighborhoods of ${\Omega}$. More specifically, ${\delta}_{{\partial}{\Omega}}$ is given by or , except for $|A_t|$ replaced by $|A_t\cap{\Omega}|_1$, with $A:={\partial}{\Omega}$, in the counterpart of the second equality of . In fractal string theory, one is particularly interested in the meromorphic continuation of $\zeta_{\mathcal L}$ to a suitable region (when it exists), along with its poles, which are called the [*complex dimensions*]{}\[c\_dim\] of $\mathcal L$. In particular, in [@lapidusfrank12], explicit formulas are obtained that are applicable to various counting functions associated with the geometry and the spectra of fractal strings, as well as to $|({\partial}{\Omega})_t\cap{\Omega}|_1$, now defined as the volume of the inner tubular neighborhood of ${\partial}{\Omega}$ (i.e., of $\mathcal L$). These explicit formulas are expressed in terms of the complex dimensions (i.e., the poles of $\zeta_{\mathcal L}$) and the associated residues. Furthermore, they enable one to obtain a very precise understanding of the oscillations underlying the geometry and spectra of fractal strings (as well as of more general fractal-like objects). From the perspective of the theory developed in the present work, a convenient choice for the set $A_{\mathcal L}$ corresponding to the fractal string $\mathcal L=(\ell_j)_{j\ge1}$ is $$\label{A_L} A_{\mathcal L}:=\{a_k : k\geq 1\},{\quad}\mbox{where{\quad}$a_k:=\sum_{j\ge k}\ell_j${\quad}for each{\quad}$k\geq 1$.}$$ As follows easily from Theorem \[2.181/2\] and the definition of $A_{\mathcal L}$ (see Equations below) and , the function $\zeta_{\mathcal L}$ in (\[string\]) is holomorphic for all $s\in{\mathbb{C}}$ with ${\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA_{\mathcal L}$. Moreover, this bound is optimal. In other words, $\overline{\dim}_BA$ coincides both with the abscissa of holomorphic continuation $D_{\rm hol}(\zeta_{\mathcal L})$ and the abscissa of (absolute) convergence $D(\zeta_{\mathcal L})$ of $\mathcal L$. Furthermore, $\zeta_{\mathcal L}(s)\to+{\infty}$ as $s\in{\mathbb{R}}$ converges to ${\overline{\dim}}_BA_{\mathcal L}$ from the right; compare with Theorem \[an\] above. In light of Theorem $\ref{2.181/2}$, Theorem \[an\](b), Equation $(\ref{2.201/2})$ and Equations –, we then have the following equalities: $$\label{eq} {\overline{\dim}}_BA_{\mathcal L}=D(\zeta_{A_{\mathcal L}})=D_{\rm hol}(\zeta_{A_{\mathcal L}})=D(\zeta_{\mathcal L})=D_{\rm hol}(\zeta_{\mathcal L})={\delta}_{{\partial}{\Omega}}.$$ The following example shows that the study of the geometric zeta function $\zeta_{\mathcal L}$ of any (bounded) fractal string $\mathcal L$ can be reduced to the study of the distance zeta function $\zeta_{A_{\mathcal L}}$ of the associated bounded set $A_{\mathcal L}$ on the real line. (See also Remark \[entirely\] below for a more general statement.) \[L\] Let $(I_k)_{k\ge1}$ be a sequence of bounded intervals, $I_k=(a_{k+1},a_k)$, $k\ge1$, where the $a_k$’s are defined by , and let $s$ be a complex variable. Using (\[z\]), we see that the distance zeta function of $A=A_{\mathcal L}$ for ${\mathop{\mathrm{Re}}}s>D(\zeta_{\mathcal L})$ is given by $$\label{zetal} \zeta_A(s)=2\int_0^\delta x^{s-1}{\mathrm d}x+\sum_{k=1}^\infty\int_{I_k}d(x,{\partial}I_k)^{s-1}{\mathrm d}x\\ =2s^{-1}\delta^s+\sum_{k=1}^{\infty}J_k(s), $$ where the first term in this last expression corresponds to the boundary points of the interval $(0,a_1)$. Assuming that $\delta\ge \ell_1/2$, we have that for all $k\ge1$, $$\label{zetalk} J_k(s)=s^{-1}2^{1-s}\ell_k^s.$$ Note that we assume that $s\in{\mathbb{C}}$ is such that ${\mathop{\mathrm{Re}}}s>D(\zeta_{\mathcal L})$, so that the series $\sum_{k=1}^{\infty}J_k(s)$ appearing in is convergent. In light of (\[string\])–(\[A\_L\]) and , we then obtain the following relation: $$\label{cantor_string} \zeta_A(s)=s^{-1}2^{1-s}\zeta_{\mathcal L}(s)+2s^{-1}\delta^{s}.$$ The case when $0<\delta<\ell_1/2$ yields an analogous relation: $$\label{simeq} \zeta_A(s)=u(s)\zeta_{\mathcal L}(s)+v(s),$$ where again $u(s):=s^{-1}2^{1-s}$, with a simple pole at $s=0$. Note that here, $u(s)$ and $v(s)=v(s,\delta)$ are holomorphic functions in the right half-plane $\{{\mathop{\mathrm{Re}}}s>0\}$. Hence, by the principle of analytic continuation and since $\zeta_{\mathcal L}$ is holomorphic for ${\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA$, the same relation still holds for the meromorphic extensions of $\zeta_A$ and of $\zeta_{\mathcal L}$ (when they exist, see Theorem \[1.2.31\]) within the right half-plane $\{{\mathop{\mathrm{Re}}}s>0\}$. The following result is in accordance with Theorem \[2.181/2\]. \[1.2.31\] Let $\mathcal{L}=(\ell_j)_{j\ge1}$ be a $($nontrivial$)$ fractal string such that $\sum_{j\ge1}\ell_j<{\infty}$, and let $A_{\mathcal L}=\big\{a_k=\sum_{j\ge k}\ell_j:k\ge1\big\}$. Then $$\label{DAL} D(\zeta_{A_{\mathcal L}})=D_{\rm hol}(\zeta_{A_{\mathcal L}})=D(\zeta_{\mathcal L})=D_{\rm hol}(\zeta_{\mathcal L})={\overline{\dim}}_BA_{\mathcal L}.$$ Furthermore, given $c\ge0$, the sets of poles of the meromorphic extensions of $\zeta_{A_{\mathcal L}}$ and $\zeta_{\mathcal L}$ $($if one, and therefore both, of the extensions exist$)$ to the open right half-plane $\{{\mathop{\mathrm{Re}}}s>c\}$ coincide. Moreover, the poles of $\zeta_{A_{\mathcal{L}}}$ and $\zeta_{\mathcal{L}}$ $($in such a half-plane$)$ have the same multiplicities. More generally, given any subdomain $U$ of ${\mathbb{C}}\setminus\{0\}$ containing the critical line $\{{\mathop{\mathrm{Re}}}s=D(\zeta_{\mathcal L})\}$, $\zeta_{A_{\mathcal L}}$ has a meromorphic continuation to $U$ if and only $\zeta_{\mathcal L}$ does, and in that case, $\zeta_{A_{\mathcal L}}$ and $\zeta_{\mathcal L}$ have the same visible poles in $U$ and with the same multiplicities. The first claim follows from Theorem \[2.181/2\] combined with parts $(a)$ and $(b)$ of Theorem \[an\]. The second and the third claims are an immediate consequence of the identities  and (\[simeq\]) in Example \[L\]. \[entirely\] An entirely similar proof shows that, in Example \[L\] and Theorem \[1.2.31\], we can replace $A_{\mathcal L}$ with $A:={\partial}{\Omega}$, where the bounded open set ${\Omega}{\subset}{\mathbb{R}}$ is any geometric realization of the (nontrivial) fractal string $\mathcal L$, provided ${\overline{\dim}}_BA:={\delta}_{{\partial}{\Omega}}$, as defined in the comment following . Hence, with the notation used in , we also have the following counterpart of in this more general situation: $$D(\zeta_{\mathcal L})=D_{\rm hol}(\zeta_{\mathcal L})=D(\zeta_{{\partial}{\Omega}})=D_{\rm hol}(\zeta_{{\partial}{\Omega}})={\delta}_{{\partial}{\Omega}}:={\overline{\dim}}_B({\partial}{\Omega},{\Omega}).$$ Actually, a direct computation shows that, in that case, the relation between $\zeta_{\mathcal L}$ and $\zeta_{{\partial}{\Omega},{\Omega}}$ (the distance zeta function of the fractal string $\mathcal L$, viewed as a relative fractal drum, in the sense of [@memoir], is even more straightforward: $$\zeta_{{\partial}{\Omega},{\Omega}}(s)=\frac{2^{1-s}}{s}\zeta_{\mathcal L}(s),$$ for every $s\in{\mathbb{C}}$ such that ${\mathop{\mathrm{Re}}}s>{\delta}_{{\partial}{\Omega}}$ and, more generally, in every domain of ${\mathbb{C}}$ to which one (and hence both) of these two fractal zeta functions can be meromorphically continued. Equivalent zeta functions {#eqzf} ------------------------- In this subsection, we shall introduce an equivalence relation $\sim$ on the set of zeta functions (see Definition \[equ\]). Let us illustrate its purpose in the case of the distance zeta function $\zeta_A$ of a given nonincreasing infinite sequence $A=(a_k)_{k\ge1}$, converging to zero in ${\mathbb{R}}$. As we saw in Example \[L\], it makes sense to identify it with its simpler form $\zeta_{\mathcal L}$, where $\mathcal L=(\ell_j)_{j\ge1}$ is the associated bounded fractal string, defined by $\ell_j=a_j-a_{j+1}$. This is done by removing the inessential functions $u(s)$ and $v(s)$ appearing in Equation (\[simeq\]) above. Therefore, $\zeta_A\sim\zeta_{\mathcal L}$. Throughout this subsection (and Appendix A in which this topic is further developed), we will assume that $E$ is a locally compact, Hausdorff topological (and metrizable) space and that $\mu$ is a [*local*]{} (roughly speaking, locally bounded) positive or complex measure (in the sense of [@dolfr], [@johlap], or [@lapidusfrank12 Chapter 4]). In short, a [*local measure*]{} is a $[0,+{\infty}]$-valued or ${\mathbb{C}}$-valued set-function on $\mathcal{B}:=\mathcal{B}(E)$ (the Borel ${\sigma}$-algebra of $E$), whose restriction to $\mathcal{B}(K)$, where $K$ is an arbitrary compact subset of $E$, is a bounded positive measure or is a complex (and hence, bounded) measure, respectively. The [*total variation measure*]{} of $\mu$ (see, e.g., \[Coh\] or \[Ru\]) is denoted by $|\mu|$; it is a (local) positive measure and, if $\mu$ is itself positive, then $|\mu|=\mu$. We refer to \[Coh, Fol, Ru\] for the theory of standard positive or complex measures. We assume that the $\mu$-measurable function ${\varphi}:E\to{\mathbb{R}}\cup\{+{\infty}\}$ appearing in Definition \[abscissa\_f\] just below is [*tamed*]{}, in the following sense: there exists a positive constant $C=C({\varphi})$ such that $$\label{E1} |\mu|(\{{\varphi}>C\})=0;$$ i.e., ${\varphi}$ is essentially bounded from above with respect to $|\mu|$. We then say that $f$, defined by below, is a [*tamed*]{} DTI. \[abscissa\_f\] Given a [*tamed Dirichlet-type integral*]{} (tamed DTI, in short) function $f=f(s)$ of the form $$\label{Efi} f(s):=\int_E{\varphi}(x)^s\,{\mathrm d}\mu(x),$$ where $\mu$ is a suitable (positive or complex) local (i.e., locally bounded) measure on a given (measurable) space $E$ \[i.e., $\mu:\mathcal{B}\to[0,+{\infty}]$ or $\mu:\mathcal{B}\to{\mathbb{C}}$\], and ${\varphi}:E\to{\mathbb{R}}\cup\{+{\infty}\}$ is a $\mu$-measurable function such that ${\varphi}\ge0$ $\mu$-a.e. on $E$, we define the [*abscissa of $($absolute$)$ convergence*]{} $D(f)\in{\mathbb{R}}\cup\{\pm{\infty}\}$ by $$\label{D(f)} \begin{aligned} D(f)&:=\inf\left\{\alpha\in{\mathbb{R}}:\int_E{\varphi}(x)^{\alpha}{\mathrm d}|\mu|(x)<{\infty}\right\}\\ &\phantom{:}=\inf\big\{\alpha\in{\mathbb{R}}:\mbox{${\varphi}(x)^s$ is Lebesgue integrable for ${\mathop{\mathrm{Re}}}s>{\alpha}$}\big\}. \end{aligned}$$ It follows that the [*half-plane of $($absolute$)$ convergence of $f$*]{}, namely, $\Pi(f):=\{{\mathop{\mathrm{Re}}}s>D(f)\}$, is the [*maximal*]{} open right half-plane (of the form $\{{\mathop{\mathrm{Re}}}s>{\alpha}\}$, for some ${\alpha}\in{\mathbb{R}}\cup\{\pm{\infty}\}$) on which the function $x\mapsto{\varphi}(x)^s$ is absolutely (i.e., Lebesgue) integrable. (Note that $D(f)$ is well defined for any tamed Dirichlet-type integral $f$.) In , by definition, $\inf\emptyset:=+\infty$ and $\inf{\mathbb{R}}=-{\infty}$. Using a classic theorem about the holomorphicity of integrals depending analytically on a parameter, one can show that $f$ is holomorphic on $\{{\mathop{\mathrm{Re}}}s>D(f)\}$. Hence, it follows that $D_{\rm hol}(f)\le D(f)$. Here, $D_{\rm hol}(f)\in{\mathbb{R}}\cup\{\pm{\infty}\}$, the [*abscissa of holomorphic continuation of $f$*]{}, is defined exactly as $D_{\rm hol}(\zeta_A)$ in , except for $\zeta_A$ replaced by $f$. In , the integral is taken with respect to $|\mu|$, the total variation measure of $\mu$; recall that if $\mu$ is positive, then $|\mu|=\mu$. Note that we may clearly replace ${\varphi}(x)^s$ by ${\varphi}(x)^{{\mathop{\mathrm{Re}}}s}$ in the second equality of , since for a measurable function, Lebesgue integrability is equivalent to absolute integrability. There are many examples for which $D_{\rm hol}(f)=D(f)$ (see, e.g., Equation in Theorem \[1.2.31\] or Equation  in Theorem \[an\]) and other examples for which $D_{\rm hol}(f)<D(f)$ (this is so for Dirichlet $L$-functions with a nontrivial primitive character, in which case $D_{\rm hol}(f)=-{\infty}$ but $D(f)=1$; see, e.g., [@serre Section VI.3]). This is the case, for instance, if $f(s):=\sum_{n=1}^{\infty}(-1)^{n-1}/n^s$. All of the fractal zeta functions encountered in this work, namely, the distance and tube zeta functions (see Subsection \[properties\_definition\] above and Subsection \[residues\_m\_tube\] below), their counterparts for relative fractal drums, the geometric zeta function of (possibly generalized) fractal strings ([@lapidusfrank12 Chapters 1 and 4]), as well as the spectral zeta functions of (relative) fractal drums (see [@Lap3; @brezish]) are tamed DTIs; i.e., they are Dirichlet-type integrals (in the sense of , and for a suitable choice of set $E$, function ${\varphi}$ and measure $\mu$) satisfy condition . This justifies, in particular, the use of the expression “abscissa of (absolute) convergence” and “half-plane of (absolute) convergence” for all of these fractal zeta functions, including the tube and distance zeta functions which are key objects in the present paper. For example, for the distance zeta function $\zeta_A$ (as in Definition \[defn\] above), we can choose $E:=A_{\delta}$ (or else, $E:=A_{\delta}\setminus{\overline{A}}$), ${\varphi}(x):=d(x,A)$ for $x\in E$ and $\mu({\mathrm d}x):=d(x,A)^{-N}{\mathrm d}x$, while for the tube zeta function (as in Definition \[zeta\_tilde\] below), we can choose $E:=(0,{\delta})$, ${\varphi}(t):=t$ for $t\in E$ and $\mu({\mathrm d}x):=t^{-N-1}|A_t|{\mathrm d}t=t^{-N}|A_t|\,({\mathrm d}t/t)$. In both cases, it is easy to check that the tameness condition is satisfied, with $C:={\delta}$. In closing, we note that the class of tamed Dirichlet-type integrals also contains all arithmetic zeta functions (that is, all zeta functions occurring in number theory); see, e.g., \[ParSh1–2, Pos, Ser, Tit, Lap-vFr2, Lap4\]. Recall from part (b) of Theorem \[an\] that we have the following result, which is very useful for the computation of the upper box dimension of fractal sets. \[an1\] Let $A$ be any bounded subset of ${\mathbb{R}}^N$. Then $${\overline{\dim}}_BA=D(\zeta_A).$$ Hence, we have $0\le D(\zeta_A)\le N$. Following [@lapidusfrank12 Sections 1.2.1 and 5.1], assume that the set $A$ has the property that $\zeta_A$ can be extended to a meromorphic function defined on $G{\subseteq}{\mathbb{C}}$, where $G$ is an open and connected neighborhood of the [*window*]{}\[window\] $\bm W$ defined by $$\bm{W}:=\{s\in{\mathbb{C}}: {\mathop{\mathrm{Re}}}s\ge S({\mathop{\mathrm{Im}}}s)\}.$$ Here, the function $S:{\mathbb{R}}\to(-{\infty},D(\zeta_A)]$, called the [*screen*]{},\[screen\] is assumed to be Lipschitz continuous. Note that the closed set $\bm W$ contains the [*critical line*]{} (of convergence) $\{{\mathop{\mathrm{Re}}}s=D(\zeta_A)\}$.\[cr\_line\_w\] In other words, we assume that $A$ is such that its distance zeta function can be extended meromorphically to an open domain $G$ containing the closed right half-plane $\{{\mathop{\mathrm{Re}}}s\ge D(\zeta_A)\}$. (Following the usual conventions, we still denote by $\zeta_A$ the meromorphic continuation of $\zeta_A$ to $G$, which is necessarily unique due to the principle of analytic continuation. Furthermore, as in [@lapidusfrank12], we assume that the [*screen*]{} $$\bm{S}:={\partial}\bm{W}=\{S(\tau)+{\mathbbm{i}}\tau:\tau\in{\mathbb{R}}\}$$ does not contain any poles of $\zeta_A$.) A set $A$ satisfying this property is said to be [*admissible*]{}\[admissible\]. (There exist nonadmissible fractal sets; see Subsection \[hyperfractal\].) The notion of admissibility used here is weaker than the one used in [@cras2] and [@mm] because we do not establish fractal tube formulas in this paper. We will also need to consider [*the set of poles of $\zeta_A$ located on the critical line*]{} $\{{\mathop{\mathrm{Re}}}s=D(\zeta_A)\}$, where $D(\zeta_A)$ is assumed to be a real number (see Definition \[dimc\]): $$\label{po} {{\mathop{\mathcal P}}}_c(\zeta_A)=\{\omega\in \bm{W}:\mbox{$\omega$ is a pole of $\zeta_A$ and ${\mathop{\mathrm{Re}}}\omega=D(\zeta_A)$}\}.$$ It is a subset of [*the set of all poles of*]{} $\zeta_A$ in $\bm W$, that we denote by ${{\mathop{\mathcal P}}}(\zeta_A)$ or ${{\mathop{\mathcal P}}}(\zeta_A,\bm{W})$ (see Definition \[1.331/2\]). \[-ty\] We assume in the definition of ${{\mathop{\mathcal P}}}_c(\zeta_A)$ that $D(\zeta_A)\in{\mathbb{R}}$, which is the case for example if $A$ is bounded, according to Corollary \[an1\]. Note that clearly (and in contrast to ${{\mathop{\mathcal P}}}(\zeta_A)={{\mathop{\mathcal P}}}(\zeta_A,\bm{W})$, to be introduced in Definition \[1.331/2\]), ${{\mathop{\mathcal P}}}_c(\zeta_A)$ is independent of the choice of the window $\bm W$. The following definition is a slight modification of the notion of complex dimension for fractal strings. \[dimc\] Let $A$ be an admissible subset of ${\mathbb{R}}^N$ such that $D(\zeta_A)\in{\mathbb{R}}$. Then, the [*set of principal complex dimensions*]{} of $A$, denoted by $\dim_{PC} A$, is defined as the set of poles of $\zeta_A$ which are located on the critical line $\{{\mathop{\mathrm{Re}}}s=D(\zeta_A)\}$: $$\dim_{PC} A:={{\mathop{\mathcal P}}}_c(\zeta_A),$$ where ${{\mathop{\mathcal P}}}_c(\zeta_A)$ is given by (\[po\]). As we see, in Definition \[dimc\], if $A\subset{\mathbb{R}}^N$ is bounded, the singularities of $\zeta_A$ we are interested in are located on the vertical line $\{{\mathop{\mathrm{Re}}}s={\overline{\dim}}_BA\}$. \[1.331/2\] Let $A$ be an admissible subset of ${\mathbb{R}}^N$. Then, the [*set of visible complex dimensions*]{} of $A$ [*with respect to a given window $\bm W$*]{} (often called, in short, the [*set of complex dimensions of $A$ relative to $\bm W$*]{}, or simply the [*set of $($visible$)$ complex dimensions*]{} of $A$ if no ambiguity may arise or if $\bm{W}={\mathbb{C}}$), is defined as the set of all the poles of $\zeta_A$ which are located in the window $\bm W$: $$\label{1.401/2} \mathcal{P}(\zeta_A)=\{\omega\in \bm{W} : \omega\textrm{ is a pole of } \zeta_A\}.$$ Instead of $\mathcal{P}(\zeta_A)$, we can also write $\mathcal{P}(\zeta_A,\bm{W})$, in order to stress that this set depends on $\bm W$ as well. Furthermore, all the sets of complex dimensions appearing in this paper are interpreted as multisets, i.e., with the multiplicities of the poles taken into account Next, we would like to extend the class of zeta functions to which a slight modification of Definition \[dimc\] and Definition \[1.331/2\] can be applied. Given a meromorphic function $f$ on a domain $G\subseteq{\mathbb{C}}$ containing the vertical line $\{{\mathop{\mathrm{Re}}}s=D(f)\}$ (as in Remark \[-ty\] above, we assume here that $D(f)\in{\mathbb{R}}$), and which (for all $s\in {\mathbb{C}}$ with ${\mathop{\mathrm{Re}}}s$ sufficiently large) is given by a convergent Dirichlet-type integral of the form and satisfying condition , so that $D(f)<{\infty}$ is well defined by ), we define the set ${{\mathop{\mathcal P}}}_c(f)$ in much the same way as in (\[po\]): $$\label{pof} {{\mathop{\mathcal P}}}_c(f)=\{\omega\in G:\mbox{$\omega$ is a pole of $f$ and ${\mathop{\mathrm{Re}}}\omega=D(f)$}\}.$$ It is a subset of the set ${{\mathop{\mathcal P}}}(f)$ of all the poles of $f$ belonging to $G$. In other words, $$\label{1.411/2} \mathcal{P}(f)=\{\omega\in G:\omega\textrm{ is a pole of } f\}.$$ \[1.333/4\] If $f=\zeta_A$, where $A$ is an admissible set for a given window $\bm W$, then (with $G:=\mathring{\bm{W}}$, the interior of the window) ${{\mathop{\mathcal P}}}_c(f)={{\mathop{\mathcal P}}}_c(\zeta_A)$, the set of principal complex dimensions of $A$, while ${{\mathop{\mathcal P}}}(f,\mathring{\bm{W}})={{\mathop{\mathcal P}}}(f)={{\mathop{\mathcal P}}}(\zeta_A)={{\mathop{\mathcal P}}}(\zeta_A,\bm{W})$, the set of (visible) complex dimensions of $A$ (relative to $\bm W$). This follows from the fact that since $A$ is admissible, $\zeta_A$ does not have any poles along the screen $\bm S$. \[1.334/5\] Observe that ${{\mathop{\mathcal P}}}_c(f)$ is independent of the choice of the domain $G$ containing the vertical line $\{{\mathop{\mathrm{Re}}}s=D(f)\}$. Moreover, since as was noted earlier, the function $f$ is holomorphic for ${\mathop{\mathrm{Re}}}s>D(f)$, there are no poles of $f$ located in the open half-plane $\{{\mathop{\mathrm{Re}}}s>D(f)\}$; this is why we could equivalently require that the domain $G\subseteq{\mathbb{C}}$ contains the closed half-plane $\{{\mathop{\mathrm{Re}}}s\geq D(f)\}$ in order to define ${{\mathop{\mathcal P}}}_c(f)$ and ${{\mathop{\mathcal P}}}(f)$. Finally, we note that since ${{\mathop{\mathcal P}}}(f)$ is the set of poles of a meromorphic function, it is a discrete subset of ${\mathbb{C}}$; in particular, it is at most countable. Since ${{\mathop{\mathcal P}}}_c(f){\subseteq}{{\mathop{\mathcal P}}}(f)$, the same is true for ${{\mathop{\mathcal P}}}_c(f)$. (An entirely analogous comment can be made about ${{\mathop{\mathcal P}}}_c(\zeta_A)$ and ${{\mathop{\mathcal P}}}(\zeta_A)$ in Definition \[dimc\] and Definition \[1.331/2\], respectively.) We next define the equivalence of a given distance zeta function $f$ to a suitable meromorphic function $g$ (of a preferably simpler form), a notion which will be useful to us in the sequel. Note that the relation $\sim$ introduced in Definition \[equ\] is clearly an equivalence relation on the set of all tamed DTIs. \[equ\] Let $f$ and $g$ be tamed Dirichlet-type integrals, as in Definition \[abscissa\_f\], both admitting a (necessarily unique) meromorphic extension to an open connected subset $U$ of ${\mathbb{C}}$ which contains the closed right half-plane $\{{\mathop{\mathrm{Re}}}s\ge D(f)\}$. (As follows from the complete definition, this closed half-plane is actually the closure of the common half-plane of convergence of $f$ and $g$, given by $\Pi:=\Pi(f)=\Pi(g)$.) Then, the function $f$ is said to be [*equivalent*]{} to $g$, and we write $f\sim g$, if $D(f)=D(g)$ (and this common value is a real number) and furthermore, the sets of poles of $f$ and $g$, located on the common critical line $\{{\mathop{\mathrm{Re}}}s=D(f)\}$, coincide. Here, the multiplicities of the poles should be taken into account. In other words, we view the set of principal poles ${\mathcal{P}}_c(f)$ of $f$ as a multiset. More succinctly, $$\label{equ2} f\sim g\quad\overset{\mbox{\tiny def.}}\Longleftrightarrow\quad D(f)=D(g)\,\,(\in{\mathbb{R}}){\quad}\mathrm{and}{\quad}{{\mathop{\mathcal P}}}_c(f)={{\mathop{\mathcal P}}}_c(g).$$ If a tamed Dirichlet-type integral $f$ is given (for example, a distance zeta function $\zeta_A$ corresponding to a given fractal set $A$), the aim is to find an equivalent meromorphic function $g$, defined by a simpler expression. Satisfactory results can already be obtained with functions $g$ of the form $g(s)=u(s)f(s)+v(s)$, for a suitable choice of the holomorphic functions $u$ and $v$, with $u$ nowhere vanishing in the given domain, as we have seen in Example \[L\]. We refer to Definition A.2 in Appendix A to this paper for an extension of Definition \[equ\] to the broader class of extended Dirichlet-type integrals (extended DTIs, for short), as introduced in Definition A.1. We also refer to Definition A.6 (and the comments surrounding it) at the end of Appendix A for a closely related, but somewhat different (and perhaps more practical) definition, allowing the meromorphic function $g$ not to be a DTI (or more generally, an EDTI of type I, in the terminology of Appendix A). These new definitions (Definitions A.2 and A.6) can be applied to (essentially) all the examples of interest in this paper and in our general theory. Towards the end of Appendix A, the interested reader can find a large class of functions $g$ giving the “leading behavior” of fractal zeta functions $f$. (See Theorem A.3 in Appendix A, along with its consequences.) In the following proposition, we consider the dependence of the distance zeta function $\zeta_A$ on ${\delta}>0$. For this reason, we denote $\zeta_A$ by $\zeta_A(\,\cdot\,,A_{{\delta}})$. \[O\] Let $A$ be a bounded subset of ${\mathbb{R}}^N$. Then, for any two positive real numbers ${\delta}_1$ and ${\delta}_2$, we have $\zeta_A(\,\cdot\,,A_{{\delta}_1})\sim\zeta_A(\,\cdot\,,A_{{\delta}_2})$, in the sense of Definition \[equ\]. We assume without loss of generality that ${\delta}_1<{\delta}_2$, since for ${\delta}_1={\delta}_2$ there is nothing to prove. For ${\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA$, the difference of the functions $\zeta_A(s,A_{{\delta}_2})$ and $\zeta_A(s,A_{{\delta}_1})$ is equal to $$\label{int12} \int_{A_{{\delta}_2}\setminus A_{{\delta}_1}}d(x,A)^{s-N}{\mathrm d}x.$$ Note that ${\delta}_1\le d(x,A)<{\delta}_2$ for every $x\in A_{{\delta}_2}\setminus A_{{\delta}_1}$. Hence, the integral given by (\[int12\]) is an entire function of the variable $s$. The following result deals with the scaling property of the distance zeta function. \[scalingd\] For any bounded subset $A$ of ${\mathbb{R}}^N$, ${\delta}>0$ and ${\lambda}>0$, we have $D(\zeta_{{\lambda}A}(\,\cdot\,,{\lambda}(A_{\delta})))=D(\zeta_A(\,\cdot\,,A_{\delta}))={\overline{\dim}}_BA$ and $$\label{zetalA} \zeta_{{\lambda}A}(s,{\lambda}(A_{\delta}))={\lambda}^s\zeta_A(s,A_{\delta}),$$ for all $s\in{\mathbb{C}}$ with ${\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA$. Furthermore, if ${\omega}\in{\mathbb{C}}$ is a simple pole of the meromorphic extension of $\zeta_A(s,A_{\delta})$ to some open connected neighborhood of the critical line $\{{\mathop{\mathrm{Re}}}s={\overline{\dim}}_BA\}$ $($we use the same notation for the meromophically extended function$)$, then $$\label{reslA} {\operatorname{res}}(\zeta_{{\lambda}A}(\,\cdot\,,{\lambda}(A_{\delta})),{\omega})={\lambda}^{{\omega}}{\operatorname{res}}(\zeta_A,{\omega}).$$ Equation follows easily by noting that ${\lambda}(A_{\delta})=({\lambda}A)_{{\lambda}{\delta}}$; we leave the details to the interested reader. To prove Equation , note that by using , we obtain that $$\begin{aligned} {\operatorname{res}}(\zeta_{{\lambda}A}(\,\cdot\,,{\lambda}(A_{\delta})),{\omega})&=\lim_{s\to {\omega}}(s-{\omega})\zeta_{{\lambda}A}(s,{\lambda}A)\\ &=\lim_{s\to {\omega}}(s-{\omega}){\lambda}^s\zeta_A(s,A)={\lambda}^{{\omega}}{\operatorname{res}}(\zeta_A,{\omega}), \end{aligned}$$ which concludes the proof of the proposition. This scaling result is useful, in particular, in the study of fractal sprays and self-similar sets in Euclidean spaces; see \[LapRaŽu3,5\]. Residues of zeta functions and Minkowski contents {#residues_m} ================================================= In this section, we show that the residue of any suitable meromorphic extension of the distance zeta function $\zeta_A$ of a fractal set $A$ in ${\mathbb{R}}^N$ is closely related to the Minkowski content of the set; see Theorems \[pole1\] and \[pole1mink\_tilde\]. Therefore, the distance zeta functions, as well as the tube zeta functions that we introduce below (see Definition \[zeta\_tilde\]), can be considered as a useful tool in the study of the geometric properties of fractals. Distance zeta functions of fractal sets and their residues {#residues_m_distance} ---------------------------------------------------------- Here we use the notation $\zeta_A(s,A_\delta)$ for the distance zeta function instead of $\zeta_A(s)$, in order to stress the dependence of the zeta function on $\delta$. We start with an identity or functional equation, which will motivate us to introduce a new class of zeta functions, described by (\[zeta\_tilde\]). \[equr\] Let $A$ be a bounded subset of ${\mathbb{R}}^N$, and let $\delta$ be a fixed positive number. Then, for all $s\in{\mathbb{C}}$ such that ${\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA$, the following identity holds$:$ $$\label{equality} \int_{A_\delta}d(x,A)^{s-N}{\mathrm d}x=\delta^{s-N}|A_\delta|+(N-s)\int_0^\delta t^{s-N-1}|A_t|\,{\mathrm d}t.$$ Furthermore, the function $\tilde\zeta_A(s):=\int_0^\delta t^{s-N-1}|A_t|\,{\mathrm d}t$ is absolutely convergent $($and hence, holomorphic$)$ on $\{{\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA\}$. The function $\tilde\zeta_A$, which we have just introduced, is called the tube zeta function of $A$ $($see Definition \[zeta\_tilde\]$)$ and will be studied in Subsection \[residues\_m\_tube\]. Equality (\[equality\]) holds for all real numbers $s\in({\overline{D}},+{\infty})$, where ${\overline{D}}:={\overline{\dim}}_BA$. Indeed, it follows immedately from Lemma \[identity0\], if we take $\gamma:=N-s$ (note that then $\gamma<N-{\overline{D}}$). Let us denote the left-hand side of (\[equality\]) by $f(s)$, and the right-hand side by $g(s)$. Since $f(s)=g(s)$ on the subset $({\overline{D}},+{\infty}){\subset}{\mathbb{C}}$, to prove the theorem, it suffices to show that $f(s)$ and $g(s)$ are both holomorphic in the region $\{{\mathop{\mathrm{Re}}}s>{\overline{D}}\}$. Indeed, the fact that (\[equality\]) then holds for all $s\in{\mathbb{C}}$ with ${\mathop{\mathrm{Re}}}s>\overline{D}$ follows from the principle of analytic continuation; see, e.g., [@conway Corollary 3.8]. The holomorphicity of $f(s)$ in that region is precisely the content of Theorem \[an\]$(a)$. In order to prove the holomorphicity of $g(s)$ on $\{{\mathop{\mathrm{Re}}}s>{\overline{D}}\}$, it suffices to show that $\tilde\zeta_A(s)$ is absolutely convergent on $\{{\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA\}$. Note that $\tilde\zeta_A(s)$ is the Dirichlet-type integral, $\tilde\zeta_A(s)=\int_E{\varphi}(t)^s{\mathrm d}\mu(x)$, where $E:=(0,\delta)$, ${\varphi}(t):=t$, $d\mu(x):=t^{-N-1}|A_t|\,dt$, and the latter measure is positive. Therefore, it suffices to show that for any $s\in{\mathbb{C}}$ such that ${\mathop{\mathrm{Re}}}s>{\overline{D}}$, the Dirichlet-type integral $\tilde\zeta_A(s)$ is well defined. To see this, let ${\varepsilon}>0$ be small enough, so that ${\mathop{\mathrm{Re}}}s>{\overline{D}}+{\varepsilon}$. Since ${{\cal M}}^{*({\overline{D}}+{\varepsilon})}(A)=0$, there exists $C_\delta>0$ such that $|A_t|\le C_\delta t^{N-{\overline{D}}-{\varepsilon}}$ for all $t\in(0,\delta]$. Then $$\nonumber \begin{aligned} |\tilde\zeta_A(s)|&\le\int_0^\delta t^{{\mathop{\mathrm{Re}}}s-N-1}|A_t|\,{\mathrm d}t\\ &\le C_\delta\int_0^\delta t^{{\mathop{\mathrm{Re}}}s-{\overline{D}}-{\varepsilon}-1}{\mathrm d}t=C_\delta\frac{\delta^{{\mathop{\mathrm{Re}}}s-{\overline{D}}-{\varepsilon}}}{{\mathop{\mathrm{Re}}}s-{\overline{D}}-{\varepsilon}}<{\infty}, \end{aligned}$$ which concludes the proof of the theorem. \[equr\_c\] If ${\overline{\dim}}_BA<N$, then $$D(\zeta_A)=D(\tilde\zeta_A){\quad}\mbox{and}{\quad}D_{\rm hol}(\zeta_A)=D_{\rm hol}(\tilde\zeta_A).$$ This follows at once from Equation of Theorem \[equr\] and from the definition of $D(f)$ and $D_{\rm hol}(f)$, for $f=\zeta_A$ or $f=\tilde\zeta_A$. The following theorem is, in particular, a higher-dimensional generalization of [@lapidusfrank12 Theorem 1.17] and yields more information than the latter result, when $N=1$. (The problem of constructing meromorphic extensions of fractal zeta functions is studied in [@mezf].) \[pole1\] Assume that the bounded set $A{\subset}{\mathbb{R}}^N$ is Minkowski nondegenerate $($that is, $0<{{\cal M}}_*^D(A)\le{{\cal M}}^{*D}(A)<{\infty}$, and, in particular, $\dim_BA=D$$)$, and $D<N$. If, in addition, $\zeta_A(\,\cdot\,,A_\delta)$ can be extended meromorphically to a neighborhood of $s= D$, then $D$ is necessarily a simple pole of $\zeta_A(\,\cdot\,,A_\delta)$, and the value of the residue of $\zeta_A(\,\cdot\,,A_{\delta})$ at $D$, ${\operatorname{res}}(\zeta_A(\,\cdot\,,A_\delta), D)$, does not depend on $\delta>0$. Furthermore, $$\label{res} (N-D){{\cal M}}_*^D(A)\le{\operatorname{res}}(\zeta_A(\,\cdot\,,A_\delta),D)\le(N-D){{\cal M}}^{*D}(A),$$ and in particular, if $A$ is Minkowski measurable, then $$\label{pole1minkg1=} {\operatorname{res}}(\zeta_A(\,\cdot\,,A_\delta), D)=(N-D){{\cal M}}^D(A).$$ Since ${{\cal M}}_*^D(A)>0$, using Theorem \[an\](c) we conclude that $s=D$ is a pole of $\zeta_A=\zeta_A(\,\cdot\,,A_{\delta})$. Therefore, it suffices to show that the order of the pole at $s=D$ is not larger than $1$. Let us take any fixed $\delta>0$, and let $$\label{Cdelta} C_\delta:=\sup_{t\in(0,\delta]}\frac{|A_t|}{t^{N-D}}.$$ Note that $C_\delta<{\infty}$ because ${{\cal M}}^{*D}(A)<{\infty}$. Then, in light of , for all $s\in{\mathbb{R}}$ with $D<s<N$, we have $$\label{res0} \begin{aligned} \zeta_A(s,A_\delta)&=\int_{A_\delta}d(x,A)^{s-N}{\mathrm d}x=\delta^{s-N}|A_\delta|+(N-s)\int_0^\delta t^{s-N-1}|A_t|\,{\mathrm d}t\\ &\le C_\delta\delta^{s-D}+C_\delta(N-s)\frac{\delta^{s-D}}{s-D}=C_\delta(N-D)\delta^{s-D}\frac1{s-D}. \end{aligned}$$ Therefore, $0<\zeta_A(s,A_\delta)\leq C_1(s-D)^{-1}$ for all $s\in(D,N)$. This shows that $s=D$ is a pole of $\zeta_A(s,A_\delta)$ which is at most of order $1$, and the first claim is established. Namely, $D$ is a simple pole of $\zeta_A(s,A_{\delta})$. The fact that the residue of $\zeta_A(s,A_\delta)$ at $s=D$ is independent of the value of $\delta>0$ follows immediately from Proposition \[O\]. In order to prove the second inequality in (\[res\]), is suffices to multiply (\[res0\]) by $s-D$, with $s$ real, and take the limit as $s\to D^+$ along the real axis: $$\label{res-delta} {\operatorname{res}}(\zeta_A(\,\cdot\,,A_\delta),D)\le(N-D)\lim_{s\to D^+}C_\delta\delta^{s-D}=(N-D)C_\delta.$$ Since the residue of $\zeta_A(s,A_\delta)$ at $D$ does not depend on $\delta$, (\[res\]) follows from (\[res-delta\]) by recalling the definition of $C_\delta$ given in (\[Cdelta\]) and passing to the limit as $\delta\to0^+$ (note that the function ${\delta}\mapsto C_{\delta}$ is nondecreasing and that $C_{\delta}\to{{\cal M}}^{*D}(A)$ as $\delta\to0^+$) on the right-hand side of (\[res-delta\]). The first inequality in (\[res\]) is proved analogously by replacing the supremum by an infimum in the definition of $C_{\delta}$ given in . \[res-cantor\] Let $A=C^{(a)}$ be the generalized Cantor set defined by the parameter $a\in(0,1/2)$. Recall that $C^{(a)}$ is obtained by deleting the middle interval of length $1-2a$ from the interval $[0,1]$, and then continuing in the usual way, scaling by the factor $a$ at each step. (For $a=1/3$, we obtain the middle third Cantor set, which is studied in detail in [@lapiduspom] and, from the point of view of geometric zeta functions and the associated complex dimensions, in [@lapidusfrank12].) By a direct computation, we obtain the corresponding zeta function: $$\label{cantor_z} \zeta_A(s,A_\delta):=\frac{2^{1-s}(1-2a)^s}{s(1-2a^s)}+2\delta^ss^{-1}.$$ Its residue computed at $D=D(a):=\dim_BA=\log_{1/a}2$ is given by $$\label{cantor_res} {\operatorname{res}}(\zeta_A(\,\cdot\,,A_\delta),D)=\frac{2}{\log 2}\left(\frac12-a\right)^{D}.$$ On the other hand, the values of the lower and upper $D$-dimensional Minkowski contents are respectively equal to (see [@mink Equations (3.12) and (3.13) for $m=2$]): $$\label{cantorM} {{\cal M}}_*^D(A)=\frac 1D\left(\frac{2D}{1-D}\right)^{1-D},\quad {{\cal M}}^{*D}(A)=2(1-a)\left(\frac12-a\right)^{D-1},$$ and thus ${{\cal M}}_*^D(A)<{{\cal M}}^{*D}(A)$ (see also Remark \[&lt;\] below). It follows that $C^{(a)}$ is not Minkowski measurable. Therefore, for any generalized Cantor set $A=C^{(a)}$, with $a\in(0,1/2)$, we have that $$\label{cantorM1} (1-D){{\cal M}}_*^D(A)<{\operatorname{res}}(\zeta_A(\,\cdot\,,A_\delta),D)<(1-D){{\cal M}}^{*D}(A).$$ This is in agreement with (\[res\]) in Theorem \[pole1\]. In particular, since the functions $(0,1/2)\ni a\mapsto {{\cal M}}_*^D(A)$ and $a\mapsto {{\cal M}}^{*D}(A)$ are bounded, and $D=\log_{1/a}2\to 1^-$ as $a\to1/2^-$, we have that for any positive $\delta$, $$\lim_{a\to1/2^-}{\operatorname{res}}(\zeta_A(\,\cdot\,,A_\delta),D)=0.\nonumber$$ The residues of $\zeta_A(s,A_\delta)$ at the poles $s_k:=D+k\mathbf{p}{{\mathbbm{i}}}$, $k\in{\mathbb{Z}}$, on the critical line $\{{\mathop{\mathrm{Re}}}s=D\}$, expressed in terms of the residue at $D$ and the ‘oscillatory period’ (see [@lapidusfrank12]) $\mathbf{p}:=2\pi/\log(1/a)$, are the following: $${\operatorname{res}}(\zeta_A(\,\cdot\,,A_\delta),s_k)=\frac{D2^{-k\mathbf{p}{{\mathbbm{i}}}}(1-2a)^{k\mathbf{p}{{\mathbbm{i}}}}} {s_ka^{k\mathbf{p}{{\mathbbm{i}}}}}{\operatorname{res}}(\zeta_A(\,\cdot\,,A_\delta),D),\quad k\in{\mathbb{Z}}.$$ \[&lt;\] As we have already noted, the two inequalities in (\[cantorM1\]) are in agreement with (\[res\]) in Theorem \[pole1\]. In [@mezf], we prove that the strict inequalities in (\[res\]) are not just a coincidence: indeed, they hold for a large class of Minkowski nonmeasurable sets in Euclidean spaces. An analogous remark applies to the inequalities (\[zeta\_tilde\_M\]) in Theorem \[pole1mink\_tilde\] below, dealing with tube zeta functions. Tube zeta functions of fractal sets and their residues {#residues_m_tube} ------------------------------------------------------ Going back to Theorem \[equr\], we see that it is natural to introduce a new fractal zeta function of bounded subsets $A$ of ${\mathbb{R}}^N$. \[zeta\_tilde\] Let $\delta$ be a fixed positive number, and let $A$ be a bounded subset of ${\mathbb{R}}^N$. Then, the [*tube zeta function*]{} of $A$, denoted by $\tilde\zeta_A$, is defined by $$\label{tildz} \tilde\zeta_A(s)=\int_0^\delta t^{s-N-1}|A_t|\,{\mathrm d}t,$$ for all $s\in{\mathbb{C}}$ with ${\mathop{\mathrm{Re}}}s$ sufficiently large. As we know from Theorem \[equr\], the tube zeta function is (absolutely) convergent (and hence, holomorphic) on the open right half-plane $\{{\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA\}$. We call $\tilde\zeta_A$ the tube zeta function of $A$ since its definition involves the tube function $(0,\delta)\ni t\mapsto |A_t|$. Relation (\[equality\]) can be written as follows (with $\zeta_A(s)=\zeta_A(s,A_{\delta})$, as before, and $\tilde\zeta_A(s)=\tilde\zeta_A(s,A_{\delta})$, for emphasis): $$\label{equ_tilde} \zeta_A(s,A_\delta)=\delta^{s-N}|A_\delta|+(N-s)\tilde\zeta_A(s,A_{\delta}),$$ for any $\delta>0$ and for all $s\in{\mathbb{C}}$ such that ${\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA$. From the [*functional equation*]{} relating $\zeta_A$ and $\tilde\zeta_A$, it would seem that $\tilde\zeta_A$ has a singularity at $s=N$. However, from the second part of Theorem \[equr\] we see that for ${\overline{\dim}}_BA<N$, the value $s=N$ is regular (i.e., holomorphic) for $\tilde\zeta_A$. It then follows from that the two fractal zeta functions $\zeta_A$ and $\tilde\zeta_A$ contain essentially the same information. In particular, still assuming that ${\overline{\dim}}_BA<N$, $\tilde\zeta_A$ has a meromorphic continuation to a given domain $U{\subseteq}{\mathbb{C}}$ (containing the critical line $\{{\mathop{\mathrm{Re}}}s={\overline{\dim}}_BA\}$) if and only if $\zeta_A$ does, and in that case (according to the principle of analytic continuation), the unique meromorphic continuations to $U$ of $\zeta_A$ and $\tilde\zeta_A$ are still related by the functional equation . Also in that case, the residues (or, more generally, the principal parts) of $\zeta_A$ and $\tilde\zeta_A$ of a given simple (resp., multiple) pole of $s={\omega}\in U$ are related in a very simple manner; see, e.g., Equation below in the case of the simple pole $s={\overline{\dim}}_BA$. Furthermore, $\mathcal{P}(\zeta_A)=\mathcal{P}(\tilde\zeta_A)$ and (assuming that $U$ contains the critical line $\{{\mathop{\mathrm{Re}}}s={\overline{\dim}}_BA\}$), $\mathcal{P}_c(\zeta_A)=\mathcal{P}_c(\tilde\zeta_A)$. Moreover, we have that $D(\tilde\zeta_A)=D(\zeta_A)$, $D_{\rm hol}(\tilde\zeta_A)=D_{\rm hol}(\zeta_A)$ and $D_{\rm mer}(\tilde\zeta_A)=D_{\rm mer}(\zeta_A)$. (Here, $D_{\rm mer}(f)$, the [*abscissa of meromorphic continuation*]{} of a given meromorphic function $f$, is defined exactly as $D_{\rm hol}(f)$ in Equation and the surrounding text, except for “holomorphic” replaced by “meromorphic”; and similarly for the half-plane of meromorphic continuation of $f$.) Also, we have $\Pi(\tilde\zeta_A)=\Pi(\zeta_A)$ and $\mathcal{H}(\tilde\zeta_A)=\mathcal{H}(\zeta_A)$; similarly, the half-planes of meromorphic continuation of $\tilde\zeta_A$ and $\zeta_A$ coincide. Still in light of , it follows from Theorem \[equr\] that $\tilde\zeta_A$ is holomorphic on $\{{\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA\}$ and that (provided ${\overline{\dim}}_BA<N$), the lower bound ${\overline{\dim}}_BA$ is optimal from the point of view of the convergence of the Lebesgue integral defining $\zeta_A$ in ; i.e., $D(\tilde\zeta_A)\,(=D(\zeta_A))={\overline{\dim}}_BA$. More generally, the exact analog of Theorem \[an\] holds for $\tilde\zeta_A$ (instead of $\zeta_A$), except for the fact that in the counterpart of part $(c)$ of Theorem \[an\] we no longer need to assume that $D<N$ (where $D:=\dim_BA$). Assuming that there exists a meromorphic extension of $\zeta_A(s,A_\delta)$ to an open connected neighborhood of ${\overline{D}}:={\overline{\dim}}_BA$, and ${\overline{D}}$ is a simple pole, ${\overline{D}}<N$, then it easily follows from (\[equ\_tilde\]) that $$\label{1.3.18} {\operatorname{res}}(\tilde\zeta_A,{\overline{D}})=\frac1{N-{\overline{D}}}{\operatorname{res}}(\zeta_A(\,\cdot\,,A_\delta),{\overline{D}}).$$ Indeed, $$\begin{aligned} {\operatorname{res}}(\zeta_A(\,\cdot\,,A_\delta),{\overline{D}})&=&\lim_{s\to {\overline{D}}}(s-{\overline{D}})[\delta^{s-N}|A_\delta|+(N-s)\tilde\zeta_A(s)]\nonumber\\ &=&(N-{\overline{D}})\lim_{s\to{\overline{D}}}(s-{\overline{D}})\tilde\zeta_A(s)\nonumber\\ &=&(N-{\overline{D}}){\operatorname{res}}(\tilde\zeta_A,{\overline{D}}).\nonumber\end{aligned}$$ Hence, the following result, in the case when $D<N$, is an immediate consequence of Theorem \[pole1\] and relation (\[equality\]) (or, equivalently, ), while in the case when $D=N$, it can be shown directly. \[pole1mink\_tilde\] Assume that $A$ is a bounded subset of ${\mathbb{R}}^N$ such that $D:=\dim_BA$ exists, $0<{{\cal M}}_*^D(A)\le{{\cal M}}^{*D}(A)<{\infty}$, and there exists a meromorphic extension of $\tilde\zeta_A$ to an open neighborhood of $D$. Then $D$ is a simple pole, and for any positive $\delta$, the value of ${\operatorname{res}}(\tilde{\zeta}_A,D)$ is independent of $\delta$. Furthermore, we have $$\label{zeta_tilde_M} {{\cal M}}_*^D(A)\le{\operatorname{res}}(\tilde\zeta_A, D)\le {{\cal M}}^{*D}(A),$$ and, in particular, if $A$ is Minkowski measurable, then $$\label{zeta_tilde_Mm} {\operatorname{res}}(\tilde\zeta_A, D)={{\cal M}}^D(A).$$ In the following example, we compute the complex dimensions of the unit $(N-1)$-dimensional sphere in ${\mathbb{R}}^N$, using the tube zeta function of the sphere. \[sphere\] Let $A:={\partial}B_1(0)$ be the unit $(N-1)$-dimensional sphere in ${\mathbb{R}}^N$ centered at the origin. We would like to compute its complex dimensions. To this end, we first compute the corresponding tube zeta function $\tilde\zeta_A$. Let us fix any ${\delta}\in(0,1)$. Since $|A_t|={\omega}_N(1+t)^N-{\omega}_N(1-t)^N$, where $t\in(0,1)$ and ${\omega}_N$ is the $N$-dimensional Lebesgue measure of the unit ball in ${\mathbb{R}}^N$, we have that for any fixed ${\delta}\in(0,1)$, $$\begin{aligned} \tilde\zeta_A(s)&=\int_0^{\delta}t^{s-N-1}|A_t|\,{\mathrm d}t={\omega}_N\int_0^{\delta}t^{s-N-1}((1+t)^N-(1-t)^N)\,{\mathrm d}t\\ &={\omega}_N\int_0^{\delta}t^{s-N-1}\Bigg(\sum_{k=0}^N\binom Nk\big(1-(-1)^k\big) t^k\Bigg)\,{\mathrm d}t\\ &={\omega}_N\sum_{k=1}^N\big(1-(-1)^k\big) \binom Nk\frac{{\delta}^{s-N+k}}{s-(N-k)}, \end{aligned}$$ for all $s\in{\mathbb{C}}$ with ${\mathop{\mathrm{Re}}}s>N-1$. The last expression can be meromorphically extended to the whole complex plane, and we still denote it by $\tilde\zeta_A(s)$. Therefore, we have $$\label{SN-1} \tilde\zeta_A(s)={\omega}_N\sum_{k=0}^N\big(1-(-1)^k\big)\binom Nk\frac{{\delta}^{s-N+k}}{s-(N-k)},$$ for all $s\in{\mathbb{C}}$. It follows that $$\label{dimBAN-1} \begin{gathered} \dim_BA=D(\tilde\zeta_A)=D(\zeta_A)=N-1,\\ {{\mathop{\mathcal P}}}_c(\tilde\zeta_A)={{\mathop{\mathcal P}}}_c(\zeta_A)=\{N-1\}, \end{gathered}$$ as expected. (Note that $\dim_BA=N-1<N$, so that ${{\mathop{\mathcal P}}}_c(\tilde\zeta_A)={{\mathop{\mathcal P}}}_c(\zeta_A)$ and ${{\mathop{\mathcal P}}}(\tilde\zeta_A)={{\mathop{\mathcal P}}}(\zeta_A)$.) Moreover, still in light of , the set of complex dimensions of $A$ is given by (with $\lfloor x\rfloor$ denoting the integer part of $x\in{\mathbb{R}}$) $$\label{dimSN-1} \begin{aligned} {{\mathop{\mathcal P}}}(\tilde\zeta_A)={{\mathop{\mathcal P}}}(\zeta_A)&=\Big\{N-(2j+1):j=0,1,2,\dots,\Big\lfloor\frac{N-1}2\Big\rfloor\Big\}\\ &=\Big\{N-1,N-3,\dots,N-\Big(2\Big\lfloor\frac{N-1}2\Big\rfloor+1\Big)\Big\}. \end{aligned}$$ For odd $N$, the last number in this set is equal to $0$, while for even $N$, it is equal to $1$. Furthermore, the residue of the tube zeta function $\tilde\zeta_A$ at any of its poles $N-k\in{{\mathop{\mathcal P}}}(\tilde\zeta_A)$ is given by ${\operatorname{res}}(\tilde\zeta_A,N-k)=2{\omega}_N\binom Nk$; that is, $$\label{resSN-1b} {\operatorname{res}}(\tilde\zeta_A,d)=2{\omega}_N\binom Nd,{\quad}\mbox{for all{\quad}$d\in{{\mathop{\mathcal P}}}(\tilde\zeta_A)$}.$$ Note that in the case when $d=D:=N-1$, we obtain $$\label{resBR0} {\operatorname{res}}(\tilde\zeta_A,D)=2N{\omega}_N={{\cal M}}^D(A),$$ where the last equality is easily obtained from the definition of the Minkowski content, as follows: $${{\cal M}}^D(A)=\lim_{t\to0^+}\frac{|A_t|}{t^{N-D}}=\lim_{t\to0^+}\frac{{\omega}_N(1+t)^N-{\omega}_N(1-t)^N}{t}=2N{\omega}_N.$$ In other words, $A$ is Minkowski measurable and $$\label{MHD} {{\cal M}}^D(A)=2\, \mathcal{H}^D(A),$$ where $\mathcal{H}^D$ denotes the $D$-dimensional Hausdorff measure. (Equation is a special case of a much more general result proved by Federer in [@federer Theorem 3.2.39].) Equation is in agreement with Equation in Theorem \[pole1mink\_tilde\]. Residues of tube zeta functions of generalized Cantor sets and $a$-strings {#residues_m_zeta} -------------------------------------------------------------------------- We provide here two simple examples illustrating some of the main results of this section. \[res-cantor2\] As an illustration of inequality (\[zeta\_tilde\_M\]), we consider generalized Cantors sets, $A=C^{(a)}$, $a\in(0,1/2)$. We obtain $$\label{cantorM2} {{\cal M}}_*^D(A)<{\operatorname{res}}(\tilde\zeta_A(\,\cdot\,,A_\delta),D)<{{\cal M}}^{*D}(A),$$ where the values of the lower and upper Minkowski contents, ${{\cal M}}_*^D(A)$ and ${{\cal M}}^{*D}(A)$, are given by and $D=D(a)=\log_{1/a}2$. It is worth observing that $C^{(a)}$ becomes almost like a Minkowski measurable set for $a$ close to $1/2$, since both ${{\cal M}}^{*D}(A)$ and ${{\cal M}}_*^D(A)$ tend to the common limit $1$ as $a\to1/2^-$. On the other hand, in the limit where $a\to0^+$, $C^{(a)}$ remains Minkowski nonmeasurable since $$\lim_{a\to0^+}{{\cal M}}^{*D}(A)=4,\quad \lim_{a\to0^+}{{\cal M}}_*^D(A)=2.$$ \[a-string2\] Given $a>0$, the associated [*$a$-string*]{} is defined by $\mathcal L=(\ell_j)_{j\ge1}$, where $\ell_j=j^{-a}-(j+1)^{-a}$. Let $A=A_{\mathcal L}=\{j^{-a}:j\in{\mathbb{N}}\}$ be the associated set; see Example \[L\] and the discussion preceding it. This set is Minkowski measurable, $$\label{a-string} {{\cal M}}^D(A)=\frac{2^{1-D}}{D(1-D)}a^D,\quad D=D(a)=\frac1{1+a}.$$ This fractal string has been introduced in [@Lap1 Example 5.1]. Due to (\[pole1minkg1=\]) and (\[zeta\_tilde\_Mm\]), we know that $$\label{a-string-mink} {\operatorname{res}}(\zeta_A(\,\cdot\,,A_\delta),D)=(1-D){{\cal M}}^D(A),\quad {\operatorname{res}}(\tilde\zeta_A,D)={{\cal M}}^D(A).$$ Distance and tube zeta functions of fractal grills {#dtx} -------------------------------------------------- It is of interest to understand the behavior of the distance and tube zeta functions with respect to the Cartesian products of sets. In this subsection, we restrict our attention to Cartesian products of the form $A\times[0,1]^k{\subset}{\mathbb{R}}^{N+k}$, which we call [*fractal grills*]{}. Here, $A$ is a bounded subset of ${\mathbb{R}}^N$ and $k$ is any positive integer. Since the set $A$ can be naturally identified with $A\times\{0\}{\subset}{\mathbb{R}}^{N+1}$, it will be convenient to introduce the following notation for all $s\in{\mathbb{C}}$ with ${\mathop{\mathrm{Re}}}s$ sufficiently large: $$\label{[N]} \zeta_A^{[N]}(s):=\int_{A_{\delta}}d(x,A)^{s-N}\,{\mathrm d}x,{\quad}\tilde\zeta_A^{[N]}(s):=\int_0^{\delta}t^{s-N-1}|A_t|_N{\mathrm d}t,$$ where the index $[N]$ indicates that we view $A$ as a subset of ${\mathbb{R}}^N$ and $|A_t|_N$ is the $N$-dimensional Lebesgue measure of the $t$-neighborhood of $A$ in ${\mathbb{R}}^N$. Hence, $\tilde\zeta_A^{[N+1]}(s)=\int_0^{\delta}t^{s-N-2}|(A\times\{0\})_t|_{N+1}{\mathrm d}t$. Note that, by writing $|(A\times\{0\})_t|_{N+1}$, we interpret $(A\times\{0\})_t$ as the $t$-neighborhood of $A\times\{0\}$ in ${\mathbb{R}}^{N+1}$. Furthermore, observe that, in , $\zeta_A^{[N]}$ and $\tilde\zeta_A^{[N]}$, are, respectively, the usual distance and tube zeta functions of $A$ (viewed as a bounded subset of ${\mathbb{R}}^N$) whereas, for example, $\tilde\zeta_A^{[N+1]}$ is the tube zeta function of $A\times\{0\}$, but now viewed instead as a subset of ${\mathbb{R}}^{N+1}$. Moreover, in and of Lemma \[cartesian\] just below, $\zeta_{A\times[0,1]}^{[N+1]}$ and $\tilde\zeta_{A\times[0,1]}^{[N+1]}$ stand, respectively, for the usual distance and tube zeta functions of $A\times[0,1]$ (naturally viewed as a subset of ${\mathbb{R}}^{N+1}$). In the sequel, if $\Sigma$ is a given set of complex numbers and $\kappa\in{\mathbb{C}}$ a fixed complex number, we let $\Sigma+\kappa:=\{s+\kappa:s\in \Sigma\}$. We shall also need the following definition. \[simeq1\] Assume that $f(s)$ and $g(s)$ are two tamed Dirichlet-type integrals (DTIs, in short) which are (absolutely) convergent on an open right half-plane $\{{\mathop{\mathrm{Re}}}s>{\alpha}\}$, for some ${\alpha}\in{\mathbb{R}}$. Let their difference $h(s):=f(s)-g(s)$ be a tamed DTI such that $D(h)<D(g)$. (Or, equivalently, that there exists a real number ${\beta}$, with ${\beta}<D(g)$, such that the integral defining $h$ is absolutely convergent (and hence, holomorphic) on $\{{\mathop{\mathrm{Re}}}s>{\beta}\}$.) Then we say that $f$ and $g$ are [*weakly equivalent*]{} and write $f\simeq g$. It can be checked that if $f$ and $g$ are tamed DTIs, then $f-g$ (or, more generally, any linear combination of $f$ and $g$) is a tamed DTI (as is required in Definition \[simeq1\] just above) provided both the DTIs $f$ and $g$ are based on the same underlying pair $(E,{\varphi})$ in the notation of Definition \[abscissa\_f\]. Therefore, $D(h)$ and $\Pi(h)$ are well defined in that case. This situation arises, for example, for the tube zeta function discussed in the present subsection. We then have $E:=(0,{\delta})$ and ${\varphi}(t):=t$ for all $t\in E$. Note that in Definition \[simeq1\], we do not assume that $g$ possesses a meromorphic continuation to a neighborhood of any point on its critical line $\{{\mathop{\mathrm{Re}}}s=D(g)\}$. Case $(c)$ of Lemma \[simeq2\] below provides a simple and useful condition for the implication $f\simeq g$ $\implies$ $f\sim g$ to hold, where the equivalence $\sim$ is described in Definition \[equ\] above. \[simeq2\] Assume that $f$ and $g$ are two tamed Dirichlet-type integrals such that $f\simeq g$. Then, the following properties hold$:$ $(a)$ We have $D(f)=D(g)$. $(b)$ The relation $\simeq$ is reflexive and symmetric. $(c)$ If there exists a connected open set $U{\subseteq}\{{\mathop{\mathrm{Re}}}s>D(f-g)\}$ containing the critical line $\{{\mathop{\mathrm{Re}}}s = D(g)\}$ and such that $g$ can be meromorphically continued to $U$, then $f$ has the same property and ${{\mathop{\mathcal P}}}_c(f)={{\mathop{\mathcal P}}}_c(g)$. In particular, $f\sim g$ in the sense of Definition \[equ\]. $(a)$ Since, by Definition \[simeq1\], $f(s)=g(s)+h(s)$ and $D(h)<D(g)$, we conclude that $D(f)\le D(g)$. If we had $D(f)<D(g)$, then we would have $$\label{contr} \max\{D(f),D(h)\}<D(g).$$ On the other hand, the function (i.e., the DTI) $g(s)=f(s)-h(s)$ is absolutely convergent on $\{{\mathop{\mathrm{Re}}}s>\max\{D(f),D(h)\}\}$, which is impossible due to . This contradiction proves that $D(f)=D(g)$. Property $(b)$ follows at once from $(a)$ and Definition \[simeq1\]. Finally, property $(c)$ follows easily from the relation $f(s)=g(s)+h(s)$. \[cartesian\] Let $A$ be a bounded subset of ${\mathbb{R}}^N$. Then $$\label{1} \zeta_{A\times[0,1]}^{[N+1]}(s)=\zeta_A^{[N]}(s-1)+\zeta_A^{[N+1]}(s)$$ and $$\label{2} \tilde\zeta_{A\times[0,1]}^{[N+1]}(s)=\tilde\zeta_A^{[N]}(s-1)+\tilde\zeta_A^{[N+1]}(s)$$ for all $s\in{\mathbb{C}}$ with ${\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA+1$. In particular, if $A$ is such that $\zeta_A$ or $($equivalently, provided ${\overline{\dim}}_BA<N$$)$ $\tilde\zeta_A$ admits a $($necessarily unique$)$ meromorphic continuation to a connected open neighborhood of the critical line of Lebesgue $($absolute$)$ convergence $\{{\mathop{\mathrm{Re}}}s=D(\zeta_A)\}$ $($recall from Theorem \[an\] that $D(\zeta_A)={\overline{\dim}}_BA$$)$, then $$\zeta_{A\times[0,1]}^{[N+1]}(s)\simeq\zeta_A^{[N]}(s-1){\quad}\mbox{\rm and}{\quad}\tilde\zeta_{A\times[0,1]}^{[N+1]}(s)\simeq\tilde\zeta_A^{[N]}(s-1).$$ Hence, if $\zeta_A$ can be meromorphically continued to a connected, open set $U$ containing the critical line $\{{\mathop{\mathrm{Re}}}s=D(\zeta_A)\}$, then $\mathcal{P}_c(\zeta_{A\times[0,1]}^{[N+1]})=\mathcal{P}_c(\zeta_A^{[N]})+1$; that is, $$\dim_{PC}(A\times[0,1])=\dim_{PC} A+1.$$ In particular, if ${\overline{\dim}}_BA<N$, then $$\begin{aligned} D(\zeta_{A\times[0,1]}^{[N+1]})&=D(\zeta_A^{[N]})+1=D(\tilde\zeta_A^{[N]})+1=D(\tilde\zeta_{A\times[0,1]}^{[N+1]})\\ &={\overline{\dim}}_B(A\times[0,1])={\overline{\dim}}_BA+1. \end{aligned}$$ Let us first prove Equation . It is easy to see (cf. [@maja Remark 1]) that: $$\label{Aid} |(A\times[0,1])_t|_{N+1}=|A_t|_N\cdot 1+|(A\times\{0\})_t|_{N+1}.$$ Substituting into the second equality of , we conclude that $$\label{m=1} \begin{aligned} \tilde\zeta_{A\times[0,1]}^{[N+1]}(s)&=\int_0^{\delta}t^{s-N-2}(|A_t|_N+|(A\times\{0\})_t|_{N+1})\,{\mathrm d}t\\ &=\int_0^{\delta}t^{(s-1)-N-1}|A_t|_N{\mathrm d}t+ \int_0^{\delta}t^{s-(N+1)-1}|(A\times\{0\})_t|_{N+1}{\mathrm d}t\\ &=\tilde\zeta_A^{[N]}(s-1)+\tilde\zeta_A^{[N+1]}(s) \end{aligned}$$ for all $s\in{\mathbb{C}}$ with ${\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA+1$. (Here, we also use the fact that ${\overline{\dim}}_BA$ is the same in the case of $A\times\{0\}{\subset}{\mathbb{R}}^{N+1}$, as in the case of $A{\subset}{\mathbb{R}}^N$; that is, the upper box dimension of a bounded set, as well as the lower box dimension, does not depend on $N$; see \[Kne, Satz 7\] or [@maja Proposition 1].) Let us next establish Equation . To this end, we use , which we write in the following form: $$\tilde\zeta_A^{[N]}(s)=\frac{\zeta_A^{[N]}(s)-{\delta}^{s-N}|A_{\delta}|_N}{N-s},$$ for all $s\in{\mathbb{C}}$ with ${\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA$ and $s\ne N$. Making use of Equation , we deduce that $$\begin{aligned}\label{Aida} \frac{\zeta_{A\times[0,1]}^{[N+1]}(s)-{\delta}^{s-N-1}|(A\times[0,1])_{\delta}|_{N+1}}{(N+1)-s}&=\frac{\zeta_A^{[N]}(s-1)-{\delta}^{(s-1)-N}|A_{\delta}|_N}{N-(s-1)}\\ &\phantom{=}+\frac{\zeta_A^{[N+1]}(s)-{\delta}^{s-(N+1)}|(A\times\{0\})_{\delta}|_{N+1}}{(N+1)-s}, \end{aligned}$$ for all $s\in{\mathbb{C}}$ with ${\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA$ and $s\ne N+1$. Since, in light of , we have $|(A\times[0,1])_{\delta}|_{N+1}=|A_{\delta}|_N+|(A\times\{0\})_{\delta}|_{N+1}$, we conclude from after a short computation that $$\zeta_{A\times[0,1]}^{[N+1]}(s)=\zeta_A^{[N]}(s-1)+\zeta_A^{[N+1]}(s),$$ for all $s\in{\mathbb{C}}$ with ${\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA+1$, where we have also used the principle of analytic continuation. Note that, according to Theorem \[an\], both $\zeta_A^{[N]}(s-1)$ and $\zeta_{A\times[0,1]}^{[N+1]}(s)$ are holomorphic on $\{{\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA+1\}$ (recall that ${\overline{\dim}}_B(A\times[0,1])={\overline{\dim}}_BA+1$, see [@falc]), while, according to the same theorem, the function $\zeta_{A\times[0,1]}^{[N+1]}(s)-\zeta_A^{[N]}(s-1)=\zeta_A^{[N+1]}(s)$ is holomorphic on $\{{\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA\}$. Therefore, since $D(\zeta_A^{[N+1]})={\overline{\dim}}_BA<{\overline{\dim}}_BA+1=D(\zeta_A^{[N]}(\,\cdot\,-1))$, it follows from Definition \[simeq1\] that $\zeta_{A\times[0,1]}^{[N+1]}(s)\simeq\zeta_A^{[N]}(s-1)$. The remaining part of Lemma \[cartesian\] can be deduced from part $(c)$ of Lemma \[simeq2\] by noting that since $\zeta_A(s)$ can be meromorphically continued to the set $U$, then $\zeta_A(s-1)$ can be meromorphically continued to the set $U+1$. Hence, by Lemma \[simeq2\]$(c)$, we have $\zeta_{A\times[0,1]}^{[N+1]}(s)\sim\zeta_A^{[N]}(s-1)$ in the sense of Definition \[equ\], and therefore, $${{\mathop{\mathcal P}}}_c\big(\zeta_{A\times[0,1]}^{[N+1]}\big)={{\mathop{\mathcal P}}}_c\big(\zeta_A^{[N]}(\,\cdot\,-1)\big)={{\mathop{\mathcal P}}}_c\big(\zeta_A^{[N]}\big)+1,$$ or, equivalently, $\dim_{PC}(A\times[0,1])=\dim_{PC}A+1$. This completes the proof of the lemma. \[Axm\] Let $A$ be a bounded subset of ${\mathbb{R}}^N$ and let $d$ be a positive integer. Then the following properties hold: $($a$)$ The distance and tube zeta functions of $A\times[0,1]^d{\subset}{\mathbb{R}}^{N+d}$ are given, respectively, by $$\label{md} \zeta_{A\times[0,1]^d}^{[N+d]}(s)=\sum_{k=0}^d \binom dk \zeta_A^{[N+k]}(s-d+k)$$ and $$\label{m} \tilde\zeta_{A\times[0,1]^d}^{[N+d]}(s)=\sum_{k=0}^d \binom dk \tilde\zeta_A^{[N+k]}(s-d+k),$$ for all $s\in{\mathbb{C}}$ with ${\mathop{\mathrm{Re}}}s>{\overline{\dim}}_BA+d$. $($b$)$ If the distance zeta function $\zeta_A$ $($or, equivalently, the tube zeta function $\tilde\zeta_A$$)$ can be meromophically extended to a connected open set containing the critical line $\{{\mathop{\mathrm{Re}}}s={\overline{\dim}}_BA\}$, then $$\label{Axmsim} \zeta_{A\times[0,1]^d}^{[N+d]}(s)\sim\zeta_A^{[N]}(s-d),{\quad}\tilde\zeta_{A\times[0,1]^d}^{[N+d]}(s)\sim\tilde\zeta_A^{[N]}(s-d)$$ and $\mathcal{P}_c(\zeta_{A\times[0,1]^d})=\mathcal{P}_c(\zeta_A)+d$; that is, $$\label{dimCm} \dim_{PC}(A\times[0,1]^d)=\dim_{PC} A+d.$$ In particular, if ${\overline{\dim}}_BA<N$, then $$\begin{aligned} D(\zeta_{A\times[0,1]^d}^{[N+d]})&=D(\zeta_A^{[N]})+d=D(\tilde\zeta_A^{[N]})+d=D(\tilde\zeta_{A\times[0,1]^d}^{[N+d]})\\ &={\overline{\dim}}_B(A\times[0,1]^d)={\overline{\dim}}_BA+d. \end{aligned}$$ $(a)$ Let us first prove Equation . We do so by using mathematical induction on $d$. The case when $d=1$ has already been established in Lemma \[cartesian\]. Now, let us assume that the claim holds for some fixed positive integer $d\ge1$. From we see that $$\zeta_{A\times[0,1]^{d+1}}^{[N+d+1]}(s)=\zeta_{A\times[0,1]^{d}}^{[N+d]}(s-1)+\zeta_{A\times[0,1]^{d}}^{[(N+1)+d]}(s).$$ Therefore, $$\nonumber \begin{aligned} \zeta_{A\times[0,1]^{d+1}}^{[N+d+1]}(s)&=\sum_{k=0}^d \binom dk \zeta_A^{[N+k]}(s-1-d+k)+ \sum_{k=0}^d \binom dk \tilde\zeta_A^{[N+1+k]}(s-d+k)\\ &=\zeta_A^{[N]}(s-d-1)+\sum_{k=0}^{d-1}\binom d{k+1}\zeta_A^{[N+k+1]}(s-d+k)\\ &\phantom{=}+\sum_{k=0}^{d-1} \binom dk \zeta_A^{[N+1+k]}(s-d+k)+\zeta_A^{[N+1+d]}(s)\\ &=\sum_{k=0}^{d+1} \binom {d+1}k \zeta_A^{[N+k]}(s-(d+1)+k), \end{aligned}$$ where in the last equality we have used the fact that $\binom dk+\binom d{k+1}=\binom{d+1}{k+1}$. This completes the proof of Equation . Equation can be proved by mathematical induction in much the same way as in the case of the distance zeta function. This completes the proof of part $(a)$ of the theorem. $(b)$ To prove that $\zeta_{A\times[0,1]^d}^{[N+d]}(s)\sim\zeta_A^{[N]}(s-d)$, it suffices to note that, by Equation , the function $$h(s):=\zeta_{A\times[0,1]^d}^{[N+d]}(s)-\zeta_A^{[N]}(s-d)=\sum_{k=1}^d \binom dk \zeta_A^{[N+k]}(s-d+k)$$ has for abscissa of convergence $D(h)={\overline{\dim}}_BA+(d-1)\}<{\overline{\dim}}_BA+d=D(\zeta_A^{[N]}(\,\cdot\,-d))$, so that $\zeta_{A\times[0,1]^d}^{[N+d]}(s)\simeq \zeta_A^{[N]}(s-d)$. Using part $(c)$ of Lemma \[simeq2\], we deduce that $\zeta_{A\times[0,1]^d}^{[N+d]}(s)\sim \zeta_A^{[N]}(s-d)$ in the sense of Definition \[equ\], which proves the first relation in . The second relation in can be proved along the same lines. This completes the proof of claim $(b)$, as well as of the entire theorem. The relations appearing in can be written in a less precise form as follows: $$\label{Axmsim1} \zeta_{A\times[0,1]^d}(s)\sim\zeta_A(s-d){\quad}\mbox{and}{\quad}\tilde\zeta_{A\times[0,1]^d}(s)\sim\tilde\zeta_A(s-d).$$ We propose to call these two properties the [*shift properties*]{} of the distance and tube zeta functions, respectively. \[Cmae\] Let $C^{(m,a)}$ be the two-parameter generalized Cantor set introduced in Definition \[Cma\] below and let $d$ be a positive integer. Then, using and below, we obtain that $$\zeta_{C^{(m,a)}\times[0,1]^d}(s)\sim\frac1{1-ma^{s-d}}.$$ Furthermore, we conclude from that $$\label{CmaPC} \dim_{PC}(C^{(m,a)}\times[0,1]^d)=(\log_{1/a}m+d)+\frac{2\pi}{\log (1/a)}\,{\mathbbm{i}}{\mathbb{Z}}.$$ Moreover, by noticing that $\zeta_{C^{(m,a)}\times[0,1]^d}$ can be meromorphically extended to the whole complex plane, we conclude from Equation above and from the first part of Equation below that the set of all complex dimensions of $C^{(m,a)}\times[0,1]^d{\subset}{\mathbb{R}}^{1+d}$ is well defined in ${\mathbb{C}}$ and given by $${{\mathop{\mathcal P}}}(\zeta_{C^{(m,a)}\times[0,1]^d})=\{0,1,\dots,d\}\cup\bigcup_{k=0}^d\Big((\log_{1/a}m+k)+\frac{2\pi}{\log (1/a)}\,{\mathbbm{i}}{\mathbb{Z}}\Big).$$ The sets of the form $C^{(m,a)}\times[0,1]^d$ (with $m:=2$, $a:=1/3$, $d:=1$) appear, for example, in the study of the Smale horseshoe map; see, e.g., [@smale]. They also arise in the study of the singularities of Sobolev functions and of weak solutions of elliptic equations; see, e.g., [@lana], where they are called the ‘Cantor grills’. \[combs\] Similarly as in Example \[Cmae\], sets of the form ${\partial}{\Omega}\times[0,1]^{N-1}$, where ${\Omega}={\Omega}_a$ is a geometric realization of a fractal string (for example, the so-called $a$-string, ${\Omega}=\cup_{j=1}^{\infty}((j+1)^{-a},j^{-a})$), where $a>0$ and for which ${\partial}{\Omega}=\{j^{-a}:j\ge1\}\cup\{0\}$ satisfies ${\overline{\dim}}_B{\partial}{\Omega}=1/(a+1)$, are used in the study of fractal drums to extend certain results from one to higher dimensions $N\ge2$; see [@Lap1 Examples 5.1 and 5.1’]. The boundary of the open set ${\Omega}\times(0,1)^{N-1}$ is given by $$({\partial}{\Omega}\times[0,1]^{N-1})\cup\big([0,1]\times{\partial}((0,1)^{N-1})\big),$$ where ${\partial}\big(([0,1]^{N-1}\big)$ is taken in the space ${\mathbb{R}}^{N-1}$. The subset ${\partial}\big((0,1)^{N-1}\big)$ of ${\mathbb{R}}^{N-1}$ is an $(N-2)$-dimensional Lipschitz surface (which for $N=2$ degenerates to a pair of points), so that the box dimension of $[0,1]\times{\partial}((0,1)^{N-1})$ is equal to $N-1$. Therefore, by the property of ‘finite stability’ of the upper box dimension (see [@falc]), we have ${\overline{\dim}}_B({\Omega}\times(0,1)^{N-1})=\max\{{\overline{\dim}}_B({\partial}{\Omega}\times[0,1]^{N-1}),N-1\}={\overline{\dim}}_B({\partial}{\Omega}\times[0,1]^{N-1})={\overline{\dim}}_B{\partial}{\Omega}+N-1$. Since, according to [@lapidusfrank12 Theorem 6.21] (along with Example \[L\] and Remark \[entirely\]), $$\label{-rho} {{\mathop{\mathcal P}}}(\zeta_{{\partial}({\Omega}_a)})=\{\rho,-\rho,-2\rho,-3\rho,\dots\},$$ where $\rho:=1/(a+1)$, we deduce from Theorem \[Axm\] that $$\label{OaPC} \begin{gathered} {{\mathop{\mathcal P}}}(\zeta_{{\partial}({\Omega}_a\times(0,1)^{N-1})})={{\mathop{\mathcal P}}}(\zeta_{{\partial}({\Omega}_a)\times[0,1]^{N-1}})\\ =\{N-1+\rho,N-1-\rho,N-1-2\rho,N-1-3\rho,\dots\}, \end{gathered}$$ still with $\rho=1/(a+1)$. Furthermore, all of these complex dimensions are simple. More precisely, it could be that beside $\rho$, which is always a (simple) pole of $\zeta_{{\partial}{\Omega}}$, some of the numbers $-n\rho$ ($n\ge1$) appearing in are not poles of $\zeta_{{\partial}{\Omega}}$ (because the corresponding residue of $\zeta_{{\partial}{\Omega}}$ happens to vanish, for some arithmetic reason connected with the value of $a$). And, hence, similarly, in . Note that if, in Example \[combs\] just above, ${\Omega}={\Omega}_{CS}$ is the Cantor string (i.e., the complement of the classic ternary Cantor set in $[0,1]$), then according to [@lapidusfrank12 Equation (1.30)] and Equation , we have $$\dim_{PC}{\partial}({\Omega}\times(0,1)^{N-1})=\big((N-1)+\log_32\big)+\frac{2\pi}{\log3}{\mathbbm{i}}{\mathbb{Z}},$$ which is the special case of corresponding to $m:=2$, $a:=1/3$ and $d:=N-1$. Transcendentally $n$-quasiperiodic sets and their distance zeta functions {#quasi0} ========================================================================= The goal of this section is to describe a construction of some of the simplest classes of quasiperiodic sets, a notion which we introduce in Definition \[quasiperiodic\] below. The main result is obtained in Theorem \[quasi1\]. The construction will be carried out by using a class of generalized Cantor sets depending on two auxiliary parameters. We note that, as will be briefly discussed in Subsection \[hyperfractal\] below, this construction and its natural generalizations will play a key role in future developments of the present higher-dimensional theory of complex dimensions of fractals; see the corresponding discussion in Remark \[remr4\] and Subsection \[hyperfractal\] below. Generalized Cantor sets defined by two parameters {#cantor_ma} ------------------------------------------------- Let us introduce a class of generalized Cantor sets $C^{(m,a)}$, depending on two parameters. As a special case, we obtain the Cantor sets of the form $C^{(a)}:=C^{(2,a)}$ discussed in Example \[res-cantor\]. The classical ternary Cantor set $C^{(1/3)}$ corresponds to the case when $m:=2$ and $a:=1/3$. \[Cma\] The generalized Cantor sets $C^{(m,a)}$ are determined by an integer $m\ge2$ and a positive real number $a$ such that $ma<1$. In the first step of the analog of Cantor’s construction, we start with $m$ equidistant, closed intervals in $[0,1]$ of length $a$, with $m-1$ holes, each of length $(1-ma)/(m-1)$. In the second step, we continue by scaling by the factor $a$ each of the $m$ intervals of length $a$; and so on, ad infinitum. The $($two-parameter$)$ [*generalized Cantor set*]{} $C^{(m,a)}$ is defined as the intersection of the decreasing sequence of compact sets constructed in this way. It can be shown that the generalized Cantor sets $C^{(m,a)}$ have the following properties, which extend the ones established for the sets $C^{(a)}$. Apart from the proof of , which is easily obtained, the proof of the proposition is similar to that for the standard Cantor set (see [@lapidusfrank12 Equation (1.11)]), and therefore, we omit it. \[Cmap\] If $C^{(m,a)}{\subset}{\mathbb{R}}$ is the generalized Cantor set introduced in Definition \[Cma\], then $$\label{2.1.1} D:=\dim_B C^{(m,a)}=D(\zeta_A)=\log_{1/a}m.$$ Furthermore, the tube formula associated with $C^{(m,a)}$ is given by $$\label{Cmat} |C^{(m,a)}_t|=t^{1-D}G(\log t^{-1})$$ for all $t\in(0,\frac{1-ma}{2(m-1)})$, where $G=G(\tau)$ is the following nonconstant, positive and bounded periodic function, with minimal period equal to $T=\log (1/a)$, and defined by $$\label{Gtau} G(\tau)=c^{D-1}(ma)^{g\left(\frac{\tau-c}{T}\right)}+2\,c^Dm^{g\left(\frac{\tau-c}{T}\right)}.$$ Here, $c=\frac{1-ma}{2(m-1)}$, and $g:{\mathbb{R}}\to{\mathbb{R}}$ is the $1$-periodic function defined by $g(x)=1-x$ for $x\in(0,1]$. Moreover, the lower and upper Minkowski contents of $C^{(m,a)}$ are respectively given by $$\label{CmaM} \begin{aligned} {{\cal M}}_*^D(C^{(m,a)})&=\min G=\frac1D\left(\frac{2D}{1-D}\right)^{1-D},\\ {{\cal M}}^{*D}(C^{(m,a)})&=\max G=\left(\frac{1-ma}{2(m-1)}\right)^{D-1}\frac{m(1-a)}{m-1}. \end{aligned}$$ Therefore, $C^{(m,a)}$ is Minkowski nondegenerate but is not Minkowski measurable. Finally, if we assume that $\delta\ge\frac{1-ma}{2(m-1)}$, then, the distance zeta function of $A:=C^{(m,a)}$ is given by $$\label{zetaCma} \zeta_A(s):=\int_{-\delta}^{1+\delta}d(x,A)^{s-1}{\mathrm d}x=\left(\frac{1-ma}{2(m-1)}\right)^{s-1}\frac{1-ma}{s(1-ma^s)}+\frac{2\delta^s}s.$$ As a result, $\zeta_A(s)$ admits a meromorphic continuation to all of ${\mathbb{C}}$, given by the last expression in $(\ref{zetaCma})$. In particular, $$\label{zetasim} \zeta_A(s)\sim\frac1{1-ma^s},$$ and the set of poles of $\zeta_A$ $($in ${\mathbb{C}})$ and the residue of $\zeta_A$ at $s=D$ are respectively given by$$\label{2.1.6} \begin{aligned} {{\mathop{\mathcal P}}}(\zeta_A)&=(D+\mathbf p{{\mathbbm{i}}}{\mathbb{Z}})\cup\{0\},\\ {\operatorname{res}}(\zeta_A,D)&=\frac{1-ma}{DT}\left(\frac{1-ma}{2(m-1)}\right)^{D-1}, \end{aligned}$$ where $\mathbf p:=2\pi/T=2\pi/\log(1/a)$ is the oscillatory period of $C^{(m,a)}$. Finally, each pole in ${{\mathop{\mathcal P}}}(\zeta_A)$ is simple. According to the terminology introduced in [@lapidusfrank12], the value of $\mathbf p=2\pi/\log(1/a)$, appearing in Proposition \[Cmap\], is called the [*oscillatory period*]{}\[osc\_period\] of the generalized Cantor set $A=C^{(m,a)}$. As we see from Equation and from the equivalence in , the set of all complex dimensions of the generalized Cantor set $A=C^{(m,a)}$ and the set of principal complex dimensions of $A$ are given, respectively, by $${{\mathop{\mathcal P}}}(\zeta_A)=(D+\mathbf p{{\mathbbm{i}}}{\mathbb{Z}})\cup\{0\} {\quad}\hbox{\rm and}{\quad}{{\mathop{\mathcal P}}}_c(\zeta_A)=D+\mathbf p{{\mathbbm{i}}}{\mathbb{Z}}.$$ Construction of transcendentally $2$-quasiperiodic sets {#qp_sets} ------------------------------------------------------- In Example \[transcendent\] below, we provide some basic ideas for further definitions and constructions. The main result of this subsection is obtained in Theorem \[trans\]. \[transcendent\] Let us define two generalized Cantor sets $A=C^{(a)}:=C^{(2,a)}{\subset}[0,1]$, $a\in(0,1/2)$, and $B=C^{(3,b)}{\subset}[2,3]$, where $b\in(0,1/3)$. We choose $b$ so that $D:=\log_{1/a} 2=\log_{1/b}3$. We may take, for example, $a=1/3$ and $b=3^{-\log_23}$. Note that we then have $3b=3^{1-\log_23}<1$. Also, we have $$|A_t|=t^{1-D}G_1(\log t^{-1}),{\quad}|B_t|=t^{1-D}G_2(\log t^{-1}).$$ The functions $G_1$ and $G_2$ corresponding to $A$ and $B$ are $T$ and $S$-periodic, respectively, with $T=\log(1/a)=\log3$ and $S=\log(1/b)$. Furthermore, the quotient $T/S=\log3/\log(1/b)=\log_32$ is transcendental, which is a well-known result going back to F. von Lindemann\[lindemann\] and K. Weierstrass;\[weierstrass\] see [@baker p. 4]. For our later needs, it will be convenient to introduce the following definition, which partly follows [@enc]. \[quasip\] We say that a function $G=G(\tau):{\mathbb{R}}\to{\mathbb{R}}$ is [*transcendentally $n$-quasiperiodic*]{} if it is of the form $G(\tau)=H(\tau,\dots,\tau)$, where $H:{\mathbb{R}}^n\to{\mathbb{R}}$ is a function which is nonconstant and $T_k$-periodic in its $k$-th component, for each $k=1,\dots,n$, and the periods $T_1,\dots, T_n$ are [*algebraically*]{} independent (that is, linearly independent over the field of algebraic real numbers). The values of $T_i$ are called the [*quasiperiods of G*]{}. The least positive integer $n$ for which this definition is valid is called the [*order of quasiperiodicity*]{} of $G$. \[quasir\] It is possible to define analogously a class of [*algebraically $n$-quasiperiodic functions*]{}, but we do not study them here; see [@fzf]. If $G(\tau)=G_1(\tau)+G_2(\tau)$, where the functions $G_i$ are nonconstant and $T_i$-periodic (for $i=1,2$), such that $T_1/T_2$ is transcendental, then $G$ is transcendentally $2$-quasiperiodic (in the sense of Definition \[quasip\]). In this case and in the notation of Definition \[quasip\], we have $H(\tau_1,\tau_2):=G_1(\tau_1)+G_2(\tau_2)$. In the sequel, we shall need a classic result due to Gel’fond and Schneider (see [@gelfond]), proved independently by these two authors in 1934. We state it in a form that will be convenient for our purposes. \[gs\] Let $\rho$ be a positive algebraic number different from one, and let $x$ be an irrational algebraic number. Then $\rho^x$ is transcendental. \[quasiperiodic\] Given a bounded subset $A{\subset}{\mathbb{R}}^N$, we say that a function $G:{\mathbb{R}}\to{\mathbb{R}}$ [*is associated with the set A*]{} (or [*corresponds to $A$*]{}) if $A$ has the following tube formula: $$\label{quasiperiodictf} |A_t|=t^{N-D}(G(\log t^{-1})+o(1))\textrm{ as }t\to0^+,$$ where $0<\liminf_{\tau\to\infty}G(\tau)\le \limsup_{\tau\to\infty}G(\tau)<{\infty}$. Note that it then follows that $\dim_BA$ exists and is equal to $D$. In addition, we say that $A$ is a [*transcendentally $n$-quasiperiodic set*]{} if the corresponding function $G=G(\tau)$ is transcendentally $n$-quasiperiodic. Generalizing the idea of Example \[transcendent\] above, we obtain the following result. \[trans\] Let $A_1=C^{(m_1,a_1)}{\subset}[0,1]$ and $A_2=C^{(m_2,a_2)}{\subset}[2,3]$ be two generalized Cantor sets $($see Definition \[Cma\]$\,)$ such that their box dimensions coincide, with the common value $D\in(0,1)$. Let $\{p_1,p_2,\dots,p_k\}$ be the set of all distinct prime factors of $m_1$ and $m_2$, and write $$m_1=p_1^{\alpha_1}p_2^{\alpha_2}\dots p_k^{\alpha_k},\quad m_2=p_1^{\beta_1}p_2^{\beta_2}\dots p_k^{\beta_k},$$ where $\alpha_i,\beta_i\in{\mathbb{N}}\cup\{0\}$\[n\_0\] for $i=1,\ldots,k$. If the exponent vectors $$(\alpha_1,\alpha_2,\dots,\alpha_k){\quad}\mathrm{and}{\quad}(\beta_1,\beta_2,\dots,\beta_k),$$ corresponding to $m_1$ and $m_2$, are linearly independent over the rationals, then the function $G=G_1+G_2$, associated with $A=A_1\cup A_2$, is transcendentally $2$-quasiperiodic; that is, the quotient $T_1/T_2$ of the quasiperiods of $G$ $($i.e., of the periods of $G_1$ and $G_2$$)$ is transcendental. Moreover, we have that $$\zeta_{A}(s)\sim \frac1{1-m_1a_1^s}+\frac1{1-m_2a_2^s},{\quad}D(\zeta_{A})=D,{\quad}D_{\rm mer}(\zeta_{A})=-{\infty},$$ and hence, the set $\dim_{PC}A={{\mathop{\mathcal P}}}_c(\zeta_A)$ of principal complex dimensions of $A$ coincides with the following nonarithmetic set$:$ $$\dim_{PC} A=D+\Big(\frac{2\pi}{T_1}\,{\mathbb{Z}}\,\cup\,\frac{2\pi}{T_2}\,{\mathbb{Z}}\Big){{\mathbbm{i}}}.$$ Besides $(\dim_{PC}A)\cup\{0\}$, there are no other poles of the distance zeta function $\zeta_{A}$. In other words, ${{\mathop{\mathcal P}}}(\zeta_A)={{\mathop{\mathcal P}}}_c(\zeta_A)\cup\{0\}$. Furthermore, all of the complex dimensions are simple. Finally, exactly the same results hold for the tube zeta function $\tilde\zeta_A$ $($instead of $\zeta_A$$)$. First of all, using (\[Cmat\]), applied to both $A_1$ and $A_2$, we conclude that for all $t\in(0,1/2)$, $$|(A_1\cup A_2)_t|=t^{1-D}\left(G_1(\log t^{-1})+G_2(\log t^{-1})\right).$$ It thus suffices to show that the quotient $T_1/T_2$ of the quasiperiods $T_1$ and $T_2$ of the function $G(\tau):=G_1(\tau)+G_2(\tau)$ is transcendental. From $D=\log_{1/a_1}m_1=\log_{1/a_2}m_2$ and $T_i=\log m_i$, $i=1,2$, we deduce that $x:=T_1/T_2$ satisfies the equation $(m_2)^x=m_1$. The exponent $x$ cannot be an irrational algebraic number, since otherwise, by the Gel’fond-Schneider theorem (Theorem \[gs\]), $(m_2)^x$ would be transcendental. If $x$ were rational, say, $x=b/a$, with $a,b\in{\mathbb{N}}$ (note that $x>0$, since $m_1\ge2$), this would then imply that $(m_1)^a=(m_2)^b$; that is, $$p_1^{a\alpha_1}p_2^{a\alpha_2}\dots p_k^{a\alpha_k}=p_1^{b\beta_1}p_2^{b\beta_2}\dots p_k^{b\beta_k}.$$ Therefore, using the fundamental theorem of arithmetic, we would have $$a(\alpha_1,\alpha_2,\dots,\alpha_k)=b(\beta_1,\beta_2,\dots,\beta_k).$$ However, this is impossible due to the assumption of linear independence over the rationals of the above exponent vectors. Consequently, $x$ is transcendental. The claims about the zeta function $\zeta_{A_1\cup A_2}$ follow from Proposition \[Cmap\] applied to both $A_1$ and $A_2$. Indeed, since $A_1$ and $A_2$ are subsets of two disjoint compact intervals, then $\zeta_A(s)\sim\zeta_{A_1}(s)+\zeta_{A_2}(s)$, and on the other hand, $\zeta_{A_1}(s)+\zeta_{A_2}(s)\sim (1-m_1a_1^s)^{-1}+(1-m_2a_2^s)^{-1}$, in light of applied separately to $A_1$ and $A_2$. This completes the proof of the theorem. Theorem \[trans\] provides a construction of the set $A=A_1\cup A_2$, such that the set $\dim_{PC} A:={{\mathop{\mathcal P}}}_c(\zeta_A)$ of principal complex dimensions of $A$ is equal to the union of two (discrete) sets of complex dimensions, each of them composed of poles in infinite vertical arithmetic progressions, but with algebraically incommensurable [*oscillatory quasiperiods*]{}\[oscqp\] $\mathbf p_1=2\pi/T_1$ and $\mathbf p_2=2\pi/T_2$ of $A_1$ and $A_2$, respectively; that is, such that $\mathbf p_1/\mathbf p_2$ is transcendental. These oscillatory quasiperiods of $A$ are equal to the oscillatory periods of $A_1$ and $A_2$, respectively. Transcendentally $n$-quasiperiodic sets and Baker’s theorem {#qp_sets_baker} ----------------------------------------------------------- The main result of this subsection is stated in Theorem \[quasi1\] below, which extends Theorem \[trans\] to any integer $n\ge2$ and also provides further helpful information. In the sequel, we shall need the following important theorem from transcendental number theory, due to Baker [@baker Theorem 2.1]. It represents a nontrivial extension of Theorem \[gs\], due to Gel’fond and Schneider [@gelfond]. Recall that an algebraic number is a complex root of a polynomial with integer coefficients and that the field of algebraic numbers is isomorphic to the algebraic closure of ${\mathbb{Q}}$, the field of rational numbers. \[baker0\] Let $n\in{\mathbb{N}}$ with $n\geq 2$. If $m_1,\dots, m_n$ are positive algebraic numbers such that $\log m_1,\dots,\log m_n$ are linearly independent over the rationals, then $$1,\log m_1,\dots,\log m_n$$ are linearly independent over the field of all algebraic numbers. We now state the main result of this subsection, which can be considered as a fractal set-theoretic interpretation of Baker’s theorem. It extends Theorem \[trans\] even in the case when $n:=2$. \[quasi1\] Let $n\in{\mathbb{N}}$ with $n\geq 2$. Assume that $A_i=C^{(m_i,a_i)}$, $i=1,\dots,n$, are generalized Cantor sets $($in the sense of Definition \[Cma\]$)$ such that their box dimensions are all equal to a fixed number $D\in(0,1)$. Assume that there is a disjoint family of closed unit intervals $I_1,\dots,I_n$ on the real line, such that $A_i{\subset}I_i$ for each $j=1,\dots,n$. Let $T_i:=\log(1/a_i)$ be the associated periods, and $G_i$ be the corresponding $($nonconstant$)$ $T_i$-periodic functions, for $i=1,\dots, n$. Let $\{p_j:j=1,\dots,k\}$ be the union of all distinct prime factors which appear in the integers $m_i$, for $i=1,\dots, n$; that is, $m_i=p_1^{{\alpha}_{i1}}\dots p_k^{{\alpha}_{ik}}$, where ${\alpha}_{ij}\in{\mathbb{N}}\cup\{0\}$. If the exponent vectors $e_i$ of the numbers $m_i$, $$\label{2.1.111/2} e_i:=({\alpha}_{i1},\dots,{\alpha}_{ik}),\quad i=1,\dots,n,$$ are linearly independent over the rationals, then the numbers $$\label{Ts} \frac 1D,T_1,\dots, T_n$$ are linearly independent over the field of all algebraic numbers. It follows that the set $A:=A_1\cup\dots\cup A_n{\subset}{\mathbb{R}}$ is transcendentally $n$-quasiperiodic; see Definition \[quasiperiodic\]. Furthermore, in the notation of Definition \[quasiperiodic\], an associated transcendentally $n$-quasiperiodic function $G$ is given by $G:=G_1+\cdots+G_n$. Moreover, we have that $$\zeta_{A}(s)\sim \sum_{i=1}^n\frac1{1-m_ia_i^s},{\quad}D(\zeta_A)=D,{\quad}D_{\rm mer}(\zeta_A)=-{\infty},$$ and hence, the set $\dim_{PC}A={{\mathop{\mathcal P}}}_c(\zeta_A)$ of principal complex dimensions of $A$ consists of simple poles and coincides with the following nonarithmetic set$:$ $$\dim_{PC} A=D+\Big(\bigcup_{i=1}^n\frac{2\pi}{T_i}\,{\mathbb{Z}}\Big){{\mathbbm{i}}}.$$ Besides $(\dim_{PC}A)\cup\{0\}$, there are no other poles of the distance zeta function $\zeta_A$. That is, ${{\mathop{\mathcal P}}}(A)={{\mathop{\mathcal P}}}_c(A)\cup\{0\}$. Furthermore, all of these complex dimensions are simple. Finally, exactly the same results hold for the tube zeta function $\tilde\zeta_A$ $($instead of $\zeta_A$$)$. As in the proof of Theorem \[trans\], using (\[Cmat\]), applied to each $A_i$, for $i=1,\dots,n$, we see that for all $t>0$ small enough, $$|A_t|=t^{1-D}\sum_{i=1}^n G_i(\log t^{-1}),$$ and for each $i=1,\ldots,n$, $G_i=G_i(\tau)$ is $T_i$-periodic, where $T_i:=\log a_i^{-1}$. We next proceed in three steps: [*Step*]{} 1: It is easy to check that the numbers $\log p_j$ (for $j=1,\dots,n$) are rationally independent. Indeed, if we had $\sum_{j=1}^k{\lambda}_j\log p_j=0$ for some integers ${\lambda}_j$, then $\prod_{j=1}^k p_j^{{\lambda}_j}=1$. This implies that ${\lambda}_j=0$ for all $j$, since otherwise it would contradict the fundamental theorem of arithmetic.[^1] [*Step*]{} 2: Let us show that $\log m_1,\dots,\log m_n$ are linearly independent over the rationals. Indeed, assume that for $i=1,\ldots,n$, $\mu_i\in{\mathbb{Q}}$ are such that $\sum_{i=1}^n\mu_i\log m_i=0$. Then $$\sum_{i=1}^n\mu_i\sum_{j=1}^k{\alpha}_{ij}\log p_j=0.$$ Changing the order of summation, we have $$\sum_{j=1}^k\left(\sum_{i=1}^n \mu_i{\alpha}_{ij}\right)\log p_j=0.$$ Since, by Step 1, the numbers $\log p_j$ are rationally independent, we have that for all $j=1,\dots,k$, $$\sum_{i=1}^n \mu_i{\alpha}_{ij}=0;$$ that is, $\sum_{i=1}^n\mu_i e_i=0$, where the $e_i$’s are the exponent vectors given by . According to the hypotheses of the theorem, the exponent vectors $e_i$ are rationally independent, and we therefore conclude that $\mu_i=0$ for all $i=1,\dots,n$, as desired. [*Step*]{} 3: Using [@baker Theorem 2.1], that is, Theorem \[baker0\] above, we conclude that $1,\log m_1,\dots,\log m_n$ are linearly independent over the field of algebraic numbers. Since $T_i=\frac1D\log m_i$, for $i=1,\dots,n$, it then follows that the numbers listed in (\[Ts\]) are also linearly independent over the field of algebraic numbers. Therefore, the function $$G:=G_1+\dots+G_n,\quad G(\tau)=G_1(\tau)+\dots +G_n(\tau),$$ associated with $A$, is transcendentally $n$-quasiperiodic; that is, the set $A$ is transcendentally $n$-quasiperiodic. Note that here, $H(\tau_1,\dots,\tau_n):=G_1(\tau_1)+\dots+ G_n(\tau_n)$, in the notation of Definition \[quasip\]. The last claim, about the distance zeta function $\zeta_A$ and its complex dimensions, now follows from Proposition \[Cmap\] applied to each of the bounded sets $A_i$ ($i=1,\ldots,n$). This concludes the proof of the theorem. \[remr4\] In Theorem \[quasi1\], we have constructed a class of bounded subsets of the real line possessing an arbitrary prescribed finite number of algebraically incommensurable quasiperiods. As will be further discussed in Subsection \[hyperfractal\], this result is extended in [@memoir], where we construct a bounded subset $A_0$ of the real line which is [*transcendentally ${\infty}$-quasiperiodic set*]{}; that is, $A_0$ contains infinitely many algebraically incommensurable quasiperiods. In the following proposition, by a [*quasiperiodic set*]{} we mean a set which has one of the following [*types of quasiperiodicity*]{}: it is either $n$-transcendentally quasiperiodic (see Definition \[quasiperiodic\]), or $n$-algebraically quasiperiodic (see Remark \[quasir\]), for some $n\in \{2,3,\dots\}\cup\{{\infty}\}$ (the case when $n={\infty}$ is treated in \[LapRaŽu1,3\]). We adopt a similar convention for the quasiperiodic functions $G=G(\tau)$ appearing in Definition \[quasip\]. \[0L\] Assume that $A$ is a quasiperiodic set in ${\mathbb{R}}^N$ of a given type, with an associated quasiperiodic function $G=G(\tau)$. If $d$ is a positive integer and $L>0$, then the subset $A\times[0,L]^d$ of ${\mathbb{R}}^{N+d}$ is also quasiperiodic of the same type, with the associated quasiperiodic function equal to $L^d\cdot G$. In particular, if $n\ge2$ is an integer and $A$ is the $n$-quasiperiodic subset of ${\mathbb{R}}$ constructed in Theorem \[quasi1\], then the subset $A\times[0,L]^d$ of ${\mathbb{R}}^{1+d}$ is also $n$-quasiperiodic. Let us first prove the claim for $d=1$. By assumption, we have that $$|A_t|_N=t^{N-D}(G(\log t^{-1})+o(1)){\quad}\mbox{as $t\to0^+$,}$$ where $G=G(\tau)$ is a quasiperiodic function; see Equation . Much as in Equation , we can write $$\begin{aligned} |(A\times[0,L])_t|_{N+1}&=|A_t|_N\cdot L+|(A\times\{0\})_t|_{N+1}\\ &=t^{(N+1)-(D+1)}(L\cdot G(\log t^{-1})+o(1))+|(A\times\{0\})_t|_{N+1} \end{aligned}$$ as $t\to0^+$, where $A\times\{0\}{\subseteq}{\mathbb{R}}^{N+1}$ (so that $(A\times\{0\})_t$ is the $t$-neighborhood of $A\times\{0\}$ taken in ${\mathbb{R}}^{N+1}$). Since, obviously, $|(A\times\{0\})_t|_{N+1}\le|A_t|_N\cdot 2t$, we have that $$\begin{aligned} |A_t|_{N+1}&\le t^{N+1-D}(G(\log t^{-1})+o(1))=t^{(N+1)-(D+1)}\cdot t(G(\log t^{-1})+o(1))\\ &=t^{(N+1)-(D+1)}\cdot O(t){\quad}\mbox{ as $t\to0^+$.} \end{aligned}$$ Therefore, $$\begin{aligned} |(A\times[0,L])_t|_{N+1}&=t^{(N+1)-(D+1)}(L\cdot G(\log t^{-1})+o(1)+O(t))\\ &=t^{(N+1)-(D+1)}(L\cdot G(\log t^{-1})+o(1)){\quad}\mbox{as $t\to0^+$.} \end{aligned}$$ Hence, by Definition \[quasiperiodic\], the set $A\times[0,L]$ is quasiperiodic, with the associated quasiperiodic function $L\cdot G$. This completes the proof of the proposition for $d=1$. The general case is easily obtained by induction on $d$. Future applications and extensions: ${\infty}$-quasiperiodic sets, hyperfractals, and the notion of fractality {#hyperfractal} -------------------------------------------------------------------------------------------------------------- The results of Section \[quasi0\] and their various generalizations (and, especially, the construction of $n$-quasiperiodic sets carried out in Subsection \[qp\_sets\_baker\] above, once it has been extended to the case when $n={\infty}$, as described in Remark \[remr4\] above) will play a key role in the applications of the higher-dimensional theory of complex dimensions developed in the present paper and in our later work. This will be so, in particular, in relation to the construction of (transcendentally) ${\infty}$-quasiperiodic, [*maximally hyperfractal*]{} sets for which, by definition, the associated fractal zeta functions have a natural boundary along the critical line $\{{\mathop{\mathrm{Re}}}s={\overline{\dim}}_BA\}$ and, in fact, have a singularity at every point of that line. Such sets are as “fractal” as possible since, in some sense, they have a continuum of nonreal “complex dimensions” (interpreted here as singularities of the fractal zeta functions attached to $A$), in striking contrast with the more usual case where the fractal zeta functions can be meromorphically extended to an open connected neighborhood of the critical line $\{{\mathop{\mathrm{Re}}}s={\overline{\dim}}_BA\}$ and therefore have at most countably nonreal complex dimensions. Recall that following [@lapidusfrank12 Sections 12.1 and 13.4] (naturally extended to higher dimensions within the framework of our new theory), a bounded subset $A$ of ${\mathbb{R}}^N$ is said to be “fractal” if its associated fractal zeta function (here, $\zeta_A$ or $\tilde\zeta_A$) has a [*nonreal*]{} complex dimension or else, if it has a natural boundary along a suitable curve (a screen $\bm S$, in the sense of Subsection \[eqzf\] above); that is, the tube zeta function $\tilde\zeta_A$ (or, equivalently, the distance zeta function $\zeta_A$ if ${\overline{\dim}}_BA<N$) cannot be meromorphically extended beyond $\bm S$. We close these comments by noting that throughout Section \[quasi0\] (with the exception of Proposition \[0L\]), we have worked with bounded subsets of the real line, ${\mathbb{R}}$. However, by using the results of Subsection \[dtx\] (especially, Theorem \[Axm\]), one can easily obtain corresponding constructions of transcendentally ${\infty}$-quasiperiodic compact sets $A$ in ${\mathbb{R}}^N$ (for any $N\ge1$), with ${\overline{\dim}}_BA\in(N-1, N)$. (See also Proposition \[0L\] at the end of Subsection \[qp\_sets\_baker\].) Likewise, using Theorem \[Axm\], one can construct ${\infty}$-quasiperiodic maximally hyperfractal compact subsets $A$ of ${\mathbb{R}}^N$ (for any $N\ge1$) such that ${\overline{\dim}}_BA\in(N-1,N)$. (Actually, by considering the Cartesian product of the original subset of ${\mathbb{R}}$ by $[0,1]^d$, with $0\le d\le N-1$, one may assume that ${\overline{\dim}}_BA\in(d,N)$; the same comment can be made about all of the results obtained in Section \[quasi0\].) Finally, these results can also be applied in a key manner in order to establish the optimality of certain inequalities associated with the meromorphic continuations of the spectral zeta functions of (relative) fractal drums (see [@fzf Section 4.3] and [@brezish Section 6]). More specifically, as is pointed out in [@Lap3], the sharp error estimates obtained in [@Lap1] for the eigenvalue counting functions of Dirichlet (or, under appropriate assumptions, Neumann) Laplacians (and more general elliptic operators of order $2m$ with possibly variable coefficients) imply that the corresponding spectral zeta functions admit a meromorphic continuation to a suitable open right half-plane $\{{\mathop{\mathrm{Re}}}s>{\delta}_{{\partial}{\Omega}}\}$, where ${\delta}_{{\partial}{\Omega}}$ is the inner (upper) Minkowski dimension of the boundary ${\partial}{\Omega}$. Our construction of $n$-quasiperiodic sets, as given in Section \[quasi0\] of this paper, and extended to $n={\infty}$ (as suggested above), enables us to deduce that this inequality is sharp, in general. That is, we construct a bounded open set ${\Omega}$ in ${\mathbb{R}}^N$ with boundary $A:={\partial}{\Omega}$, such that the compact set $A{\subset}{\mathbb{R}}^N$ is ${\infty}$-quasiperiodic. It follows that ${\sigma}_{\rm mer}$, the abscissa of meromorphic continuation of the corresponding spectral zeta function, satisfies ${\sigma}_{\rm mer}\ge{\delta}_{{\partial}{\Omega}}$. For example, for the Dirichlet Laplacian on ${\Omega}$, we have ${\sigma}_{\rm mer}={\delta}_{{\partial}{\Omega}}$; i.e., $\{{\mathop{\mathrm{Re}}}s>{\delta}_{{\partial}{\Omega}}\}$ is the largest open right half-plane to which the associated spectral zeta function can be meromorphically continued. In fact, a much stronger statement is true in this case. Namely, the spectral zeta function $\zeta_{\nu}(s)$ has a nonisolated singularity at every point of the vertical line $\{{\mathop{\mathrm{Re}}}s={\delta}_{{\partial}{\Omega}}\}$. [**Acknowledgement.**]{} We express our gratitude to the referee for several useful remarks and suggestions. Appendix A: Equivalence relation and extended Dirichlet-type integrals {#appendix .unnumbered} ====================================================================== One problem with the notion of “equivalence” provided in Definition \[equ\] of Subsection \[eqzf\] is that, strictly speaking, it is not an equivalence relation or is not even well defined if (as can be very useful) one wishes to allow $g$ to be a meromorphic function (rather than a DTI) because, a priori, $f$ and $g$ no longer belong to the same class of functions. (Indeed, $f$ is then a Dirichlet-type integral, abbreviated DTI in the sequel, while $g$ is merely assumed to be meromorphic; in particular, the abscissa of convergence of $g$ need not be well defined.) The situation is very analogous, in spirit, to the evaluation of the “leading part” ($g=g(s)$, in the present case) of a function ($f=f(s)$, here) in the theory of asymptotic expansions. In that situation, the “leading part” $g$ belongs to a scale of typical functions (describing the possible asymptotic behaviors of the function $f$ in the given asymptotic limit). In our present situation, just as in the theory of asymptotic expansions, formally, the relation $\sim$ is both reflexive and (when it makes sense) transitive. Of course, it is also symmetric when it acts on the same class of functions (for example, DTIs). However, it is also possible to modify both the definition of $\sim$ and the class of functions on which it acts so that it becomes a true equivalence relation on a single space of functions, namely, the class of extended DTIs. The latter class of (tamed) extended DTIs contains the class of (tamed) DTIs (hence, all of the functions $f$ we wanted to work with in Definition \[equ\]) and it also contains (essentially) all of the functions $g$ occurring in practice (when applying Definition \[equ\]). By definition, given $r\in(0,1)$, a DTI [*of base*]{} $r$ is a function of the form $$g(s)=\zeta_{E,{\varphi},\mu}(r^{-s}),\tag{A.1}$$ where $f(s):=\zeta_{E,{\varphi},\mu}(s)$ is a (standard) DTI defined by $$\tag{A.2} \zeta_{E,{\varphi},\mu}(s):=\int_E{\varphi}(x)^s{\mathrm d}\mu(x).$$ (See also Definition \[abscissa\_f\].) It is then easy to check (using the analogous result for ordinary DTIs) that if $g$ is tamed (i.e., if $f$ is tamed), then the abscissa of convergence $D(g)$ of $g$ and the half-plane of convergence $\Pi(g):=\{{\mathop{\mathrm{Re}}}s>D(g)\}$ are not well defined. Indeed, note that $${\varphi}(x)^{r^s}={\varphi}(x)^{r^{{\mathop{\mathrm{Re}}}s}(\cos((\log r){\mathop{\mathrm{Im}}}s)+{\mathbbm{i}}\sin((\log r){\mathop{\mathrm{Im}}}s)},$$ so that the open set $V$ of complex numbers $s$ for which ${\varphi}(x)^{r^s}$ is Lebesgue integrable on $E$ (typically) consists of countably many connected components, and, hence, does not have the form of a half-plane. The indicated open set $V$ is analyzed in [@fzf Appendix A, Section A.4]. [**Definition A.1.**]{} An [*extended Dirichlet-type integral*]{} (an [*extended*]{} DTI or EDTI, in short) $h=h(s)$ is either of the form $$h(s):=\rho(s)\zeta_{E,{\varphi},\mu}(s)\tag{A.3}$$ or of the form $$h(s):=\rho(s)\zeta_{E,{\varphi},\mu}(r^{-s}),{\quad}\mbox{for some $r\in(0,1)$,}\tag{A.4}$$ where $\rho=\rho(s)$ is a nowhere vanishing entire function and $\zeta_{E,{\varphi},\mu}=\zeta_{E,{\varphi},\mu}(s)$ is a DTI. More generally, $\rho$\[rho\] can be a holomorphic function which does not have any zeros in the given domain $U{\subseteq}{\mathbb{C}}$ under consideration, where $U$ contains the closed half-plane $\{{\mathop{\mathrm{Re}}}s>D(\zeta_{E,{\varphi}\mu})\}$. If the extended DTI is of the form (A.3), it is said to be of [*type*]{} I, and if it is of the form (A.4), it is said to be of [*type*]{} II (or of type II$_r$ if one wants to keep track of the underlying base $r$). Note that EDTIs of type I include all ordinary DTIs as a special case (by taking $\rho\equiv1$). Let us denote by $f(s):=\zeta_{E,{\varphi},\mu}(s)$ the (standard) DTI and by $g(s):=\zeta_{E,{\varphi},\mu}(r^{-s})$ the DTI of base $r$ occurring in (A.4). Then, by definition (and in accordance with Definition A.1), if $h$ is of the form (A.3), its [*abscissa of convergence*]{} $D(h)$ is given by $D(h):=D(f)$, while if $h$ is of the form (A.4), then $D(h)=+{\infty}$, that is, $\Pi(h)=\emptyset$. If the DTI $f(s):=\zeta_{E,{\varphi},\mu}$ is tamed, then the extended DTI $h$ from Definition A.1 (either in (A.3) or in (A.4)) is said to be [*tamed*]{}. Finally, given any tamed extended DTI of type I, $h=h(s)$ (as in the first part of Definition A.1), we call $$\tag{A.5} \Pi(h):=\{{\mathop{\mathrm{Re}}}s>D(h)\}$$ the [*half-plane of convergence*]{} of $h$ (which is maximal, in an obvious sense), and (assuming that $D(h)\in{\mathbb{R}}$) we call $\{{\mathop{\mathrm{Re}}}s=D(h)\}$ the [*critical line*]{} of $h$. (The tameness condition enables us to show that this half-plane exists and is indeed, maximal.) Using a classic theorem about the holomorphicity of integrals depending on a parameter, one can show that $h$ is holomorphic on $\Pi(h)$. Hence, $D_{\rm hol}(h)\le D(h)$. Here, much as in Definition \[abscissa\_f\], $D(h)$ and $D_{\rm hol}(h)$ denote, respectively, the [*abscissa of $($absolute$)$ convergence*]{} and the [*abscissa of holomorphic continuation*]{} of $h$. Furthermore, if $h$ is given by (A.3) above, we set $D(h)=D(\zeta_{E,{\varphi},\mu})$ and $D_{\rm hol}(h)=D_{\rm hol}(\zeta_{E,{\varphi},\mu})$, where $D(\zeta_{E,{\varphi},\mu})$ and $D_{\rm hol}(\zeta_{E,{\varphi},\mu})$ are defined in Definition \[abscissa\_f\]. Moreover, if $h=h(s)$ admits a meromorphic continuation to an open connected set $U$ containing the closed half-plane $\{{\mathop{\mathrm{Re}}}s\ge D(h)\}$, we denote (much as was done in Definition \[dimc\] for the special case of DTIs) by ${{\mathop{\mathcal P}}}_c(h)$ the set of [*principal complex dimensions*]{} of $h$; that is, the set of poles of $h$ (in $U$) located on the critical line $\{{\mathop{\mathrm{Re}}}s=D(h)\}$ of $h$: $${{\mathop{\mathcal P}}}_c(h):=\{{\omega}\in U: \mbox{${\omega}$ is a pole of $h$ and ${\mathop{\mathrm{Re}}}{\omega}=D(h)$}\}.\tag{A.6}$$ Clearly, ${{\mathop{\mathcal P}}}_c(h)$ does not depend on the choice of the domain $U$ satisfying the above condition. We define similarly ${{\mathop{\mathcal P}}}(h)={{\mathop{\mathcal P}}}(h,U)$, the [*set of*]{} (visible) [*complex dimensions*]{} of $h$, relative to $U$: $${{\mathop{\mathcal P}}}(h):=\{{\omega}\in U:\mbox{${\omega}$ is a pole of $h$}\}.\tag{A.7}$$ Clearly, since $h$ is of type I (i.e., is given as in (A.3)), then ${{\mathop{\mathcal P}}}_c(h)={{\mathop{\mathcal P}}}_c(f)$ and ${{\mathop{\mathcal P}}}(h)={{\mathop{\mathcal P}}}(f)$, where $f(s):=\zeta_{E,{\varphi},\mu}(s)$. We can now modify as follows the definition of the “equivalence relation” provided in Definition \[equ\] of Subsection \[eqzf\]. [**Definition A.2.**]{} Let $h_1$ and $h_2$ be arbitrary tamed, extended DTIs of type I (as in Definition A.1) such that $D(h_1)=D(h_2)=:D$, with $D\in{\mathbb{R}}$. Assume that each of $h_1$ and $h_2$ admits a (necessarily unique) meromorphic continuation to an open connected neighborhood $U$ of the closed half-plane $\{{\mathop{\mathrm{Re}}}s\ge D\}$. Then the functions $h_1$ and $h_2$ are said to be [*equivalent*]{}, and we write $h_1\sim h_2$, if the sets of poles of $h_1$ and $h_2$ on their common vertical line $\{{\mathop{\mathrm{Re}}}s=D\}$ (and the corresponding poles have the same multiplicities): ${{\mathop{\mathcal P}}}_c(h_1)={{\mathop{\mathcal P}}}_c(h_2)$ (where the equality holds between multisets). We conclude this appendix by providing a class of tamed extended DTIs which can be used to determine the “leading behavior” of most of the fractal zeta functions used in the present theory. [**Theorem A.3.**]{} *Let $P\in{\mathbb{C}}[x]$ be a polynomial with complex coefficients. Then $f(s):=1/P(s)$ is a tamed DTI of type I.* More specifically, if $\deg P=:n\ge1$, then $$f(s):=\frac1{P(s)}=\zeta_{E,{\varphi},\mu}(s),\tag{A.8}$$ where $E:=[1,+{\infty})^n$, ${\varphi}(x):=(x_1\cdots x_n)^{-1}$ for all $x\in E$, and $$\tag{A.9} \mu({\mathrm d}x_1,\dots,{\mathrm d}x_n):=c\,x_1^{a_1}\frac{{\mathrm d}x_1}{x_1}\dots x_n^{a_n}\frac{{\mathrm d}x_n}{x_n},$$ so that its total variation measure $|\mu|$ $($in the sense of local measures$)$ is given by $$\nonumber |\mu|({\mathrm d}x_1,\dots,{\mathrm d}x_n):=c\,x_1^{{\mathop{\mathrm{Re}}}a_1}\frac{{\mathrm d}x_1}{x_1}\dots x_n^{{\mathop{\mathrm{Re}}}a_n}\frac{{\mathrm d}x_n}{x_n},$$ where $c:=\frac{1}{n!}P^{(n)}(0)$ and $a_1,\dots,a_n$ are the zeros of $P=P(s)$ $($counted according to their multiplicities, so that $P(s)=c\,\Pi_{m=1}^n(s-a_m)$$)$. Moreover, $D(f)=D(\zeta_{E,{\varphi},\mu})\le\max\{{\mathop{\mathrm{Re}}}a_1,\dots,{\mathop{\mathrm{Re}}}a_n\}$. If, in Theorem A.3, we assume that $\deg P=0$, i.e., if $P$ is constant, say $P\equiv1$, then clearly, $f(s)=1/P(s)=1=\zeta_{E,{\varphi},\mu}(s)$, where $E:=[1,+{\infty})$, ${\varphi}(t):=1$ for all $x\in E$, and $\mu:={\delta}_1$ (the Dirac measure concentrated at $1$). In particular, $f$ is also tamed in this case. Theorem A.3 is a consequence of the following two facts: $(i)$ If $f_a(s):=1/(s-a)$, where $a\in{\mathbb{C}}$ is arbitrary, then $f$ is a tamed DTI of type I, given by $$f(s):=\frac1{s-a}=\zeta_{E,{\varphi}_a,\mu_a}(s),\tag{A.10}$$ where $E:=[1,+{\infty})$, ${\varphi}_a(x):=x^{-1}$ for all $x\in E$, and $$\mu_a({\mathrm d}x):=x^a\frac{{\mathrm d}x}x;\tag{A.11}$$ so that $|\mu_a|({\mathrm d}x)=x^{{\mathop{\mathrm{Re}}}a}{\mathrm d}x/x$. Furthermore, $D(f_a)={\mathop{\mathrm{Re}}}a$. Note that $f_a:=\zeta_{E,{\varphi}_a,\mu_a}$ is obviously tamed because ${\varphi}_a(x)\le 1$ for all $x\ge1$. An entirely analogous comment can be made about $f=\zeta_{E,{\varphi},\mu}$ in the theorem. $(ii)$ The tensor product of two tamed DTIs is tamed. More specifically, if the DTIs $\zeta_{E,{\varphi},\mu}$ and $\zeta_{F,\psi,\eta}$ are tamed, then their [*tensor product*]{} is given by the following tamed DTI: $$\tag{A.12} h(s):=(\zeta_{E,{\varphi},\mu}\otimes\zeta_{F,\psi,\eta})(s)=\zeta_{E\times F,{\varphi}\otimes\psi,\mu\otimes\eta}(s),$$ where the tensor product ${\varphi}\otimes\psi$ is defined by $({\varphi}\otimes\psi)(x,y):={\varphi}(x)\,\psi(y)$ for $(x,y)\in E\times F$ and the tensor product $\mu\otimes\eta$ is the product measure of $\mu$ and $\eta$ (see, e.g., [@cohn]). It is easy to check that the DTI $h$ is tamed because (since $\zeta_{E,{\varphi},\mu}$ and $\zeta_{F,\psi,\eta}$ are tamed), we have $0\le{\varphi}(x)\le C({\varphi})$ $|\mu|$-a.e. on $E$ and $0\le\psi(x)\le C(\psi)$ $|\eta|$-a.e. on $F$, so that $0\le({\varphi}\otimes\psi)(x,y)\le C({\varphi})\,C(\psi)$ $|(\mu\otimes\eta)|$-a.e. on $E\times F$. Furthermore, $D(h)\le\max\{D(\zeta_{E,{\varphi},\mu}),D(\zeta_{F,\psi,\eta})\}$. Statement $(i)$ above follows from a direct computation, while statement $(ii)$ is proved by an application of the Fubini–Tonelli theorem (for iterated integrals with respect to positive measures) combined with the inequality (between local positive measures) $|\mu\otimes\eta|\le|\mu|\otimes|\eta|$, followed by an application of the classic Fubini theorem (for iterated integrals with respect to possibly signed or complex measures). [**Corollary A.4.**]{} [*The meromorphic function on all of ${\mathbb{C}}$ given by $$\tag{A.13} h_2(s):=\frac{\rho(s)}{P(r^{-s})},$$ where $r\in(0,1)$, $P\in{\mathbb{C}}[x]$ is an arbitrary polynomial with complex coefficients and $\rho$ is a nowhere vanishing entire function, is a tamed extended DTI of type II. More specifically, $h_2(s)=\rho(s)\zeta_{E,{\varphi},\mu}(r^{-s})$, where $E$, ${\varphi}$ and $\mu$ are given in Theorem A.3 above.*]{} As was alluded to earlier, in practice, when we apply the (modified) definition of the equivalence relation (see Definition A.2 above), $$h_1\sim h_2\tag{A.14}$$ the meromorphic function $h_1$ is a fractal zeta function (an ordinary DTI of type I), as well as the function $h_2$ (which gives the “leading behavior” of $h_1$, to mimick the terminology of the theory of asymptotic expansions). Hence, the importance of Theorem A.3 in the theory developed in the present paper as well as in its future developments. (See, however, Definition A.6 below and the comments surrounding it.) We refer the interested reader to [@fzf Appendix A] for more details about the topics discussed in the present appendix, along with detailed proofs of the main results. [*Remark*]{} A.5. The two definitions of the notion of equivalence $\sim$ provided in Definition \[equ\] and Definition A.2 are compatible in the sense that if, in Definition \[equ\], we assume that $f$ (denoted by $h_1$ in Definition A.2) is a DTI (as is the case in Definition \[equ\]), the meromorphic function $g$ is an extended DTI, then $f\sim g$ in the sense of Definition A.2. Note that the functions $f$ and $g$ of Definition \[equ\] are denoted by $h_1$ and $h_2$ in Definition A.2. (In particular, $D(g)$ and ${{\mathop{\mathcal P}}}_c(g)$ are well defined, $D(f)=D(g)$ and ${{\mathop{\mathcal P}}}_c(f)={{\mathop{\mathcal P}}}_c(g)$.) The converse statement clearly holds as well. Finally, it is possible, even likely, that in future applications of the current theory of fractal zeta functions developed in this paper and in our later work, we will need to deal with functions $g$ which are no longer extended DTIs (of type I), but are meromorphic functions of a suitable kind. In that case, we propose to use the following definition, which is a suitable modification of Definition \[equ\] and seems well suited to various applications. Strictly speaking, it no longer gives rise to an equivalence relation (since $f$ and $g$ belong to different classes of functions) but in this new sense, the statement $f\overset{\rm asym}{\sim} g$ captures appropriately the idea that “$f$ is asymptotic to $g$”. [**Definition A.6.**]{} Let $f$ be a tamed EDTI and let $g$ be a meromorphic function, both defined and meromorphic on an open and connected subset $U$ of ${\mathbb{C}}$ containing the closed right half-plane $\{{\mathop{\mathrm{Re}}}s\ge D(f)\}$. Then, the function $f$ is said to be [*asymptotically equivalent*]{} to $g$, and we write $f\overset{\rm asym}{\sim} g$, if $D(f)=D_{\rm hol}(g)$ (and this common value is a real number), and the poles of $f$ and $g$ located on the convergence critical line $\{{\mathop{\mathrm{Re}}}s>D(f)\}$ of $f$ (which, by assumption, is also the holomorphy critical line of $g$) coincide and have the same multiplicities. More succinctly, and with the obvious notation (compare with Equation in Definition \[equ\] above), we have $$\tag{A.15} f\overset{\rm asym}{\sim} g{\quad}\overset{\mbox{\tiny def.}}\Longleftrightarrow{\quad}D(f)=D_{\rm hol}(g)\,\,(\in{\mathbb{R}})\,\,\mbox{and}\,\,{{\mathop{\mathcal P}}}_c(f)={{\mathop{\mathcal P}}}_{c,\rm hol}(g). $$ More specifically, we let $$\nonumber {{\mathop{\mathcal P}}}_{c,\rm hol}(g):=\{{\omega}\in U:\mbox{${\omega}$ is a pole of $g$ and ${\mathop{\mathrm{Re}}}{\omega}=D_{\rm hol}(g)$}\}.$$Furthermore, much as in Definition \[equ\], $D(f)$ and $D_{\rm hol}(g)$ are viewed as multisets in Equation (A.15). [*Remark*]{} A.7. Observe that even if $g$ is assumed to be a tamed EDTI, Definition A.6 may differ from its counterpart used in the rest of this appendix (Definition A.2) or, in particular, in Definition \[equ\]. Indeed, there are examples of tamed DTIs $g$ for which $D_{\rm hol}(g)<D(g)$. 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--- abstract: 'We present a new multiphase-field theory for describing pattern formation in multi-domain and/or multi-component systems. The construction of the free energy functional and the dynamic equations is based on criteria that ensure mathematical and physical consistency. We first analyze previous multiphase-field theories, and identify their advantageous and disadvantageous features. On the basis of this analysis, we introduce a new way of constructing the free energy surface, and derive a generalized multiphase description for arbitrary number of phases (or domains). The presented approach retains the variational formalism; reduces (or extends) naturally to lower (or higher) number of fields on the level of both the free energy functional and the dynamic equations; enables the use of arbitrary pairwise equilibrium interfacial properties; penalizes multiple junctions increasingly with the number of phases; ensures non-negative entropy production, and the convergence of the dynamic solutions to the equilibrium solutions; and avoids the appearance of spurious phases on binary interfaces. The new approach is tested for multi-component phase separation and grain coarsening.' author: - 'Gyula I. Tóth' - Tamás Pusztai - László Gránásy title: '[A consistent multiphase-field theory for interface driven multi-domain dynamics]{}' --- Introduction ============ Despite recent advances in atomic scale continuum modeling of crystalline freezing [@Elder2002; @Elder2007; @Wu2010; @NanaXPFC2012; @Emmerich2012], and efforts relying on the orientation field models [@KobayashiWarren1998; @Granasy2002; @Plapp2012; @Granasy2014; @Haataja2014], the phase-field theoretical methods based on the multiphase-field (MPF) concept remain the method of choice, when addressing complex polycrystalline or multiphase/multi-component problems, such as multi-component phase separation or grain coarsening. A common feature of these models is that the individual physical phases / chemical components / solid grains are described by separate fields $\mathbf{u}(\mathbf{r},t) = [u_1(\mathbf{r},t),u_2(\mathbf{r},t),\dots,u_N(\mathbf{r},t)]$. A variety of this kind of models is available in the literature ranging from the early formulations by Chen and Yang [@ChenYang1994], Steinbach [@Steinbach1996], Steinbach and Pezzola [@SteinbachPezzola1999], Chen and coworkers [@FanChen1997a; @FanChen1997b], via later descendants by Nestler and coworkers [@Nest1; @Nest2; @Nest3; @Nest4; @Nest5], Moelans and coworkers [@Moelans1; @Moelans2; @Moelans3; @Moelans4], to more recent developments by Folch and Plapp [@FolchPlapp2003; @FolchPlapp2005], Kim [*et al.*]{} [@Kim2006], Takaki [*et al.*]{} [@Takaki2008], Steinbach [@Steinbach2009], Ofori-Opoku and Provatas [@NanaNick2010], Cogswell and Carter [@Cogswell2011], Bollada [*et al.*]{} [@Bollada2012], by Emmerich and coworkers [@Kundin2013], and Kim [*et al.*]{} [@Kim2014]. These models differ in important details that improve the individual models in various respects relative to the others. It is, therefore, desirable to compare them from a theoretical viewpoint, and identify the possible advantages / disadvantages they have relative to each other, to see whether a more general formulation that unifies the advantageous features can indeed be constructed on the basis of the work done so far in this field.\ In attempting to develop a consistent description of interface driven multi-domain dynamics, we need to first identify the criteria the models have to satisfy. A few of such criteria have already been formulated along the development of the MPF models: \(i) The *multiphase-field* descriptions view $u_i(\mathbf{r},t)$ as the local and temporal volume/mass/mole fraction of the component/grain, prescribing thus $\sum_{i=1}^N u_i(\mathbf{r},t)= 1$. (This work concentrates exclusively on these MPF models; the *multi order parameter* theories [@ChenYang1994; @FanChen1997a; @FanChen1997b; @Moelans1; @Moelans2; @Moelans3; @Moelans4; @NanaNick2010], which do not require this criterion, will be addressed elsewhere.) \(ii) A further natural requirement is that the physical results should be independent of the labeling of the variables. This condition is termed the ”principle of formal indistinguishability” of the fields. \(iii) The solution of the dynamic equations should tend towards the equilibrium solution obtained from the respective Euler-Lagrange equations (ELEs) based on the free energy functional, where the equilibrium solution minimizes then the free energy of the system. \(iv) As time evolves the free energy of the total system should decrease monotonically (second law of thermodynamics). \(v) It is an evident requirement that the formulations for different numbers of phases or grains should be consistent with each other; i.e., it should be possible to recover the respective models from each other, when adding or removing a new phase/grain/orientation. It has been suggested recently that the usual variational approach to the MPF model does not satisfy this condition [@FolchPlapp2005; @Bollada2012]. \(vi) Another fairly general requirement, formulated by several authors, is that (a) the two-phase planar interfaces should represent a (stable) equilibrium, and should be free of additional phases. This requirement can be extended to the dynamics (b) as follows: if a phase is not present, it should not appear deterministically (in the absence of fluctuations) at any time. This is often called the condition of “no spurious phase generation”. The applicability of this requirement makes depends on the problem addressed: In the case of grain coarsening, for instance, an uncontrolled appearance of new grains / orientations at the grain boundaries needs to be avoided. (In other problems, however, metastable structures [@tenWolde1995; @tenWolde1996; @tenWolde1997] or precipitates [@Provatas1996] may appear at the solid-liquid interface, requiring models that are able to describe such phenomena [@ShenOxtoby1996; @GranasyOxtoby2000; @Toth2007; @Toth2011].) This criterion for the absence of a third phase has been enforced different ways in different models. For example, Folch and Plapp [@FolchPlapp2005] have defined the free energy surface so that in the binary equilibrium the system stays in a two-phase subspace of the respective Gibbs simplex. Bollada [*et al.*]{} [@Bollada2012], in turn, used the mobility matrix to force the system to avoid the formation of the third phase irrespectively of the free energy surface (and thus the equilibrium states of the system). Finally, a practical requirement: \(vii) The model should allow the prescription of independent data for the interfacial properties and the kinetic coefficient of the individual phase pairs (including their possible anisotropy).\ While most of these criteria formulate natural/self-evident requirements, some of them were neglected when developing previous MPF models. In the present paper, we formulate an MPF approach that obeys all the criteria defined above. [Herein, we address [*interface driven*]{} phenomena, in which the free energy density incorporates exclusively the ”interfacial” contributions, comprising the gradient energy and multi-well terms, whereas driving forces associated with tilting functions[@FolchPlapp2005] are not considered. (An extension that includes the latter will be outlined elsewhere.)]{} The structure of our paper is as follows. In Section II, we present the mathematical formulation of criteria (i) to (vii). In Section III, we first investigate which of these criteria are satisfied and which are not by the existing MPF models. We also point out which features of the individual approaches can be adopted in developing a consistent theory. In Section IV we outline the generalized MPF formulation (henceforth abbreviated as XMPF) that satisfies criteria (i) to (vii). In Section V, we perform illustrative simulations using the XMPF approach to demonstrate the robustness of the theory. Section VI is devoted to a comparison with other models. Finally, in Section VII, we offer a few concluding remarks. Criteria of physical consistency ================================ In this section, we present and discuss mathematical formulations of criteria (i) to (vii) identified above in details. Free energy functional formalism -------------------------------- In the multiphase approach, the following local constraint \[criterion (i)\] applies for the variables: $$\label{eq:gencond} \sum_{i=1}^N u_i(\mathbf{r},t) = 1 \enskip .$$ As result of this constraint Eq. (\[eq:genfunc\]) is not an order parameter model, since normally different order parameters capturing various aspects of symmetry breaking are coupled to each other via physical laws, whereas here Eq. (\[eq:genfunc\]) prescribes a rather specific relationship: $u_i(\mathbf{r},t)$ represents the local fraction of the $i-$th phase, not identifiable as a quantity, whose magnitude is associated with the extent of symmetry breaking.\ The general interface contribution of a multiphase free energy functional is usually written as: $$\label{eq:genfunc} F[\mathbf{u}] = \int dV \left\{ \frac{\epsilon^2}{2} \sum_{i=1}^N A_{ij}(\nabla u_i \cdot \nabla u_j) + w \, g(\mathbf{u}) \right\} \enskip ,$$ where $\mathbf{u}(\mathbf{r},t)$ is the vector of the variables, $g(\mathbf{u})$ is the free energy density landscape, and $\mathbb{A}$ is a coefficient matrix of the general quadratic term for the gradients. For example, choosing $\mathbb{A}=\mathbb{I}$ (where $\mathbb{I}$) is the identity matrix yields a simple sum of the gradient square terms, $\mathbb{A}=\mathbb{I}-\mathbf{e} \otimes \mathbf{e}$ \[where $\mathbf{e}=(1,1,\dots,1)$\] results in a pure pairwise construction, while $A_{i,i}=\sum_{j \neq i}u_j^2$, $A_{ij \neq i}=-u_i u_j$ corresponds to the anti-symmetrized (Landau-type) gradient term. The (generally) $\mathbf{u}$-dependent coefficients $\epsilon^2$ and $w$ can be related to the pairwise interfacial properties (the interfacial free energy and the interface thickness).\ The equilibrium solution can be found by solving the multiphase Euler-Lagrange equations: $$\label{eq:genEL} \frac{\delta F}{\delta u_i} = \lambda(\mathbf{r}) \enskip , \quad i=1\dots N \enskip ,$$ where $\delta F/\delta u_i$ is the functional derivative of the free energy functional with respect to $u_i(\mathbf{r})$, and $\lambda(\mathbf{r})$ is a Lagrange multiplier emerging from the local constraint described by Eq. (\[eq:gencond\]). Eliminating the Lagrange multiplier results in $$\label{eq:genELv} \frac{\delta F}{\delta u_i} = \frac{\delta F}{\delta u_j} \quad \text{for} \quad i,j=1\dots N \enskip ,$$ thus offering the most general description of equilibrium.\ For non-conserved variables, the dynamic equations are written in the following general form: $$\label{eq:gendin} - \frac{\partial u_i}{\partial t} = \sum_{j=1}^N L_{ij} \frac{\delta F}{\delta u_j} \enskip ,$$ where the mobility matrix can be determined from the following conditions: 1. The time derivatives sum up to zero \[it follows from criterion (i)\]: $$\quad \sum_{i=1}^N \frac{\partial u_i}{\partial t} = 0 \enskip .$$ 2. The variables are not labeled, i.e., none of them is distinguished on the basis of its index \[criterion (ii)\]. 3. The solutions of the Euler-Lagrange equations must be stationary solutions of the dynamic equations \[a requirement that follows from criterion (iii)\]: $$\left. \frac{\partial \mathbf{u}}{\partial t}\right|_{\mathbf{u}^*(\mathbf{r})}=0 \enskip ,$$ where $\mathbf{u}^*(\mathbf{r})$ stands for a solution of Eq. (\[eq:genELv\]). Applying condition 1 for Eq. (\[eq:gendin\]) results in the general form $$\label{eq:gendin2} - \frac{\partial u_i}{\partial t} = \sum_{j=1}^N \kappa_{ij} \left( \frac{\delta F}{\delta u_i} - \frac{\delta F}{\delta u_j} \right) \enskip ,$$ where $\kappa_{ij}>0$ still can be arbitrary. Furthermore, using condition 2 yields $$\label{eq:gendin3} \sum_{i=1}^N \kappa_{ij} = 0$$ for $j=1 \dots N$. Note that Eq. (\[eq:gendin2\]) and (\[eq:gendin3\]) resulted in a mobility matrix, whose elements sum up to zero in each row and column. Finally, condition 3 results in a symmetry condition, i.e. $$\kappa_{ij}=\kappa_{ji}$$ (for the derivation, see Appendix A). Since $\kappa_{ii}=-\sum_{j\neq i}\kappa_{ij}$, $N(N-1)/2$ mobilities can be chosen arbitrarily.\ Next we have to test the mobility matrix against the time dependence of the total free energy. The main reason of applying a linear approximation for the dynamic equations is to establish a free energy minimizing behavior \[criterion (iv)\]. Using Eq. (\[eq:gendin\]), the condition for the time derivative of the total free energy reads as $$\begin{split} \frac{dF}{dt} &= \int dV \left\{ \sum_{i=1}^N \frac{\delta F}{\delta u_i} \frac{\partial u_i}{\partial t} \right\} =\\ &= - \int dV \left\{ (\delta F_\mathbf{u}) \mathbb{L} (\delta F_\mathbf{u})^T \right\} \leq 0 \enskip , \end{split}$$ where $\mathbb{L}$ is the mobility matrix, and we use the short notation $\delta F_\mathbf{u}=\left( \frac{\delta F}{\delta u_1},\frac{\delta F}{\delta u_2},\dots,\frac{\delta F}{\delta u_N} \right)$. Since the free energy must decrease in any volume, we can write $$\label{eq:Lcond} (\delta F_\mathbf{u}) \mathbb{L} (\delta F_\mathbf{u})^T \geq 0 \enskip ,$$ indicating that the mobility matrix must be *positive semidefinite*. A special, frequently made choice for the mobility matrix is the *Lagrangian* mobility $\kappa_{ij}\equiv 1/N$. Although this matrix is used quite widely, the dynamic equations are necessarily $N-dependent$ \[i.e., criterion (v) is not satisfied\], and it does not solve the problem of spurious phase appearance in non-equilibrium processes in general, as will be discussed later. Spurious phases --------------- Herein we address pattern coarsening phenomena, for which the requirement of ’no spurious phase appearance’ needs to be satisfied \[criterion (vi)\]. In general, this criterion means that any $p$–phase equilibrium solution (i.e., when exactly $p$ fields are present) must be stable against the appearance of a new phase. Accordingly, assuming that $q=N-p$ of the $N$ phases (namely, $i_1,i_2,\dots,i_q$) are missing in an equilibrium solution $\mathbf{u}^*(\mathbf{r})$, the condition can be re-formulated as: $$\delta F = F[\mathbf{u}^*(\mathbf{r})+\delta \mathbf{u}(\mathbf{r})]-F[\mathbf{u}^*(\mathbf{r})] \geq 0$$ for *any* small perturbation for which $\sum_{k=1}^N\delta u_k=0$ and at least one of $\delta u_{i_1}(\mathbf{r}),\delta u_{i_2}(\mathbf{r}),\dots,\delta u_{i_q}(\mathbf{r})$ is not equal to zero. In other words, leaving the $p=N-q$ dimensional subspace (together with keeping the local constraint, naturally) always has to result in higher energy. This condition is satisfied for the equilibrium solutions representing minima of the free energy functional: Since $$F[\mathbf{u}^*(\mathbf{r})+\delta \mathbf{u}(\mathbf{r})] = F[\mathbf{u}^*]+\frac{\delta F}{\delta \mathbf{u}}\cdot \delta \mathbf{u}^T + \frac{\delta \mathbf{u} \cdot \mathbb{D} \cdot \delta \mathbf{u}^T}{2} + \dots \enskip ,$$ the second term on the right hand side vanishes for a solution of the Euler-Lagrange equation: $(\delta F/\delta \mathbf{u})\cdot\delta\mathbf{u}=\lambda(\mathbf{r})\sum_{i=1}^N\delta u_i=0$. Therefore, if $\mathbb{D}$ is positive definite, the equilibrium solution is a minimum. Consequently, if the binary planar interfaces represent minima of the multiphase functional, they are stable against the appearance of additional phases, which is a crucial requirement from the viewpoint of criterion (vi).\ To fulfill criterion (vi), we also have to ensure the proper dynamic behavior of the system. The condition of ’no spurious phase appearance’ can be generalized for the non-equilibrium regime as follows: If a phase is not present in the system, it must not appear deterministically, i.e. $$\label{eq:spuricondd} \left.\frac{\partial u_i}{\partial t}\right|_{u_i(\mathbf{r},t)=0}=0 \enskip .$$ Unfortunately, in an *arbitrary*, $p$–phase non-equilibrium state $\mathbf{u}(\mathbf{r})$, the condition $\delta F=F[\mathbf{u}(\mathbf{r})+\delta\mathbf{u}(\mathbf{r})]-F[\mathbf{u}(\mathbf{r})]>0$ cannot be satisfied for all possible perturbations. Therefore, Eq. (\[eq:spuricondd\]) cannot be guaranteed on the level of the free energy functional *in general*. Note, however, that the mobility matrix defines a ’conditional functional derivative’, which allows the system to leave the $p$–dimensional subspace only in particular directions \[or, in other words, not any $\mathbf{u}(\mathbf{r})+\delta\mathbf{u}(\mathbf{r})$ state is available from $\mathbf{u}(\mathbf{r})$ in the dynamics\]. For special mobility matrices, like the Lagrangian matrix, one may find such a free energy functional that satisfies Eq. (\[eq:spuricondd\]) [@FolchPlapp2005]. Nevertheless, the free energy functional should not depend on the form of the mobility matrix in general. This problem is resolved in a recent work [@Bollada2012], in which the authors choose a mobility matrix having vanishing rows (and columns) for the fields not being present. [Although this concept trivially results in Eq. (\[eq:spuricondd\]), the application of such a matrix together with a particular free energy functional can be ’dangerous’ in the sense that it may generate stationary solution from a non-equilibrium state, a possibility that has to be checked for all solutions obtained.]{} Analysis of previous MPF descriptions ===================================== In this section, we analyze the most frequently used multiphase-field theories from the viewpoint of equilibrium solutions. We check whether the trivial extension of the planar interface emerging from the binary reduction of the free energy functional is a solution of the multiphase problem too. Summarizing, we require that the equilibrium solution for the pure binary planar interface in the multiphase-field problem ($N \le 3$) coincides with the equilibrium solution for the binary planar interface of the binary ($N$ = 2) problem \[criterion (iv)\]. The methodology for testing this feature consists of the following steps: 1. Take the free energy functional in the two-phase limit; 2. Solve the respective Euler-Lagrange equation of the two-phase problem for planar interface geometry, resulting in $u(x)$; 3. Make a natural multiphase extension of $u(x)$ via adding zero additional fields as needed for the multiphase case, i.e. $u_i(x):=u(x)$, $u_{j\neq i}(x):=1-u(x)$, and $u_{k \neq i,j}(x)=0$, where $1\leq i,j,k \leq N$; 4. Plug the extended solution into the Euler-Lagrange equations of the multiphase problem described by Eq. (\[eq:genEL\]), and check whether it satisfies them. The test can be simplified in case of free energy functionals constructed exclusively from pairwise contributions, i.e. $$F = \int dV \left\{ \sum_{i<j} \hat{f}(u_i,u_j) \right\} \enskip ,$$ where $\sum_{i<j}=\sum_{i=1}^{N-1} \sum_{j=i+1}^N$. $\hat{f}(u,v)$ is called *generator*. The functional derivatives read as: $$\label{eq:pairfunc} \frac{\delta F}{\delta u_i} = \sum_{j \neq i} \delta\hat{f}(u_i,u_j) \enskip ,$$ where $$\delta\hat{f}(u_i,u_j) = \frac{\partial \hat{f}}{\partial u_i} - \nabla \frac{\partial \hat{f}}{\nabla\partial u_i} \enskip .$$ *Assuming that $\delta\hat{f}(u_i,0)=0$*, and plugging in the extended planar interface solution into the Euler-Lagrange equations of the multiphase problem yield $$\begin{aligned} \frac{\delta F}{\delta u_i} &=& \delta\hat{f}[u(x),1-u(x)] = \lambda(x) \\ \frac{\delta F}{\delta u_j} &=& \delta\hat{f}[1-u(x),u(x)] = \lambda(x) \\ \frac{\delta F}{\delta u_k} &=& \delta\hat{f}[0,u(x)]+\delta\hat{f}[0,1-u(x)] = \lambda(x) \enskip ,\end{aligned}$$ where $u_i(x)=u(x)$, $u_{j\neq i}(x)=1-u(x)$, and $u_{k\neq i,j}(x)=0$, and $u(x)$ represents the planar binary solution. From the equations above follows that $$\label{eq:simplified} \delta\hat{f}(u,1-u) = \delta\hat{f}(1-u,u) = \delta\hat{f}(0,u)+\delta\hat{f}(0,1-u)$$ must apply. If Eq. (\[eq:simplified\]) does not apply, the binary planar interface is not an equilibrium solution of the multiphase problem. Steinbach et al. ---------------- In 1996, Steinbach [*et al.*]{} [@Steinbach1996] proposed the following free energy functional for multiphase systems, which serves as the basis for the worldwide used phase-field software MICRESS [@MICRESS]: $$\label{eq:sfunc} F = \int dV \left\{ f_{\rm intf}(\mathbf{u},\nabla\mathbf{u}) + f_{\rm df}(\mathbf{u}) \right\} \enskip ,$$ where $\sum_{i=1}^N u_i(\mathbf{r},t)=1$, whereas the term $f_{\rm df}(\mathbf{u})$ is responsible for the thermodynamic driving force (i.e. free energy difference between the bulk phases) and $f_{\rm intf}(\mathbf{u},\nabla\mathbf{u})$ denotes the interface energy consisting of a gradient \[$f_{\rm gr}(\mathbf{u},\nabla\mathbf{u})$\] and a multi-well \[$f_{\rm mw}(\mathbf{u})$\] contribution: $$\label{eq:sintf} f_{\rm intf}(\mathbf{u},\nabla\mathbf{u})=f_{\rm gr}(\mathbf{u},\nabla\mathbf{u})+f_{\rm mw}(\mathbf{u}) \enskip ,$$ where the terms are given in the following specific forms: $$\begin{aligned} \nonumber f_{\rm gr}(\mathbf{u},\nabla\mathbf{u}) &=& \sum_{i<j} \frac{\epsilon_{ij}^2}{2}(u_i\nabla u_j-u_j\nabla u_i)^2 \\ \label{eq:smulti} f_{\rm mw}(\mathbf{u}) &=& \sum_{i<j} \frac{w_{ij}}{2}(u_i u_j)^2 \enskip .\end{aligned}$$ The functional naturally reduces to the standard binary form $$F = \int dV \left\{ \frac{\epsilon^2_{ij}}{2}(\nabla u)^2 + \frac{w_{ij}}{2} [u(1-u)]^2 \right\} \enskip .$$ The 1D Euler-Lagrange equation reads as: $\delta F/\delta u=w_{ij}u(1-u)(1-2u)-\epsilon_{ij}^2 \partial_x^2 u=0$, thus resulting in the usual $$u(x) = \frac{1+\tanh[x/(2\,\delta_{ij})]}{2}$$ planar interface solution, where $\delta_{ij}^2=\epsilon_{ij}^2/w_{ij}$. As a first step, we investigate whether the extension of this solution minimizes the multiphase problem. Since $f_{intf}(\mathbf{u},\nabla\mathbf{u})$ is a pure pairwise construction, it is enough to take the generator, which reads as: $$\hat{f}(u_i,u_j) = \frac{\epsilon_{ij}^2}{2}(u_i\nabla u_j-u_j\nabla u_i)^2 + \frac{w_{ij}}{2}u_i^2 u_j^2 \enskip ,$$ yielding $$\label{eq:selgen} \begin{split} \delta\hat{f}(u_i,u_j) &= w_{ij} u_i u_j^2 + \epsilon^2_{ij} \left[ 2(u_i\partial_x u_j-u_j\partial_x u_i)\partial_x u_j+ \right. \\ & + \left. (u_i\partial_x^2 u_j - u_j\partial_x^2 u_i)u_j \right] \enskip . \end{split}$$ Note that $\delta\hat{f}(u,0)=0$. Substituting $u_i(x)=u(x)$, $u_{j\neq i}(x)=1-u(x)$ and $u_{k \neq i,j}(x)=0$ into Eq. (\[eq:selgen\]), yields $$\begin{aligned} \label{eq:sbfd1}&&\frac{\delta F}{\delta u_i} = \delta\hat{f}(u,1-u) = \frac{w_{ij}}{4} \text{sech}^4[x/(2\,\delta_{ij})] \\ \label{eq:sbfd2}&&\frac{\delta F}{\delta u_j} = \delta\hat{f}(1-u,u) = \frac{w_{ij}}{4} \text{sech}^4[x/(2\,\delta_{ij})] \\ \label{eq:sbfd3}&&\frac{\delta F}{\delta u_k} = \delta\hat{f}(0,u)+\delta\hat{f}(0,1-u) = 0 \enskip .\end{aligned}$$ These equations clearly show that *the planar binary interfaces are not equilibrium solutions of the multiphase problem*.\ The dynamic equations read as: $$-\frac{\partial u_i}{\partial t}=\sum_{i=1}^N \kappa_{ij} \left( \frac{\delta F}{\delta u_i}-\frac{\delta F}{\delta u_j} \right) \enskip ,$$ which are variational, therefore, the planar binary interfaces are not a stationary solutions of these. To avoid the problem, the authors used a “binary approximation” of $\frac{\delta F}{\delta u_i}-\frac{\delta F}{\delta u_j}$, in which all terms of $k \neq i,j$ indices are neglected [@Steinbach1996]: $$\frac{\delta F}{\delta u_i}-\frac{\delta F}{\delta u_j} \approx w_{ij}u_i u_j (u_j-u_i)+\epsilon_{ij}^2(u_i\nabla^2 u_j - u_j \nabla^2 u_i)$$ Although the planar binary interfaces represent stationary solutions of the resulting *non-variational* dynamics, Eq. (\[eq:Lcond\]) does not apply, therefore, *the dynamics is not energy minimizing in principle*, as it will be demonstrated later. Steinbach-Pezzolla ------------------ In the Steinbach-Pezzolla formalism (published first in 1999 [@SteinbachPezzola1999], and adopted in various works [@Takaki2008; @Steinbach2009; @Cogswell2011] including the OpenPhase software [@OpenPhase]), the interface contribution to the free energy is given by the following simplified form of Eq. (\[eq:sintf\]): $$\label{eq:spmodel} f_{\rm intf}^{\rm SP} = \sum_{i<j} \frac{4 \sigma_{ij}}{\eta} \left( |u_i| \, |u_j| - \frac{\eta^2}{\pi^2} \nabla u_i \cdot \nabla u_j \right) \enskip ,$$ where the gradient term $-\nabla u_i \cdot \nabla u_j$ is the linear approximation of $(u_i\nabla u_j-u_j\nabla u_i)^2$, $\sigma_{ij}$ the interfacial free energy of the equilibrium $(i,j)$ planar interface, while the local term $|u_i|\, |u_j|$ is responsible for the finite interface width given by $\eta$. Reducing the model to the $N=2$ case yields $$\label{eq:sp2red} f_{\rm intf}^{\rm SP} = \frac{4 \sigma_{ij}}{\eta} \left[ \frac{\eta^2}{\pi^2} (\nabla u)^2 + | u |\, | 1-u | \right] \enskip ,$$ where we used $u(x):=u_i(x)$, $u_j(x):=1-u(x)$, and $u_{k \neq i,j}(x)=0$. The 1D Euler-Lagrange equation reads as $${\rm sign}(u)\, |1-u| - {\rm sign}(1-u)\, |u| = \left( \frac{2 \eta^2}{\pi^2} \right) \partial_x^2 u(x) \enskip ,$$ from which $$u(x) = \frac{1+\sin[(\pi/\eta) \cdot x]}{2}$$ emerges for the planar binary interface. The generator function reads as $$\hat{f}(u_i,u_j) = \frac{4 \sigma_{ij}}{\eta} \left( |u_i|\,|u_j| - \frac{\eta^2}{\pi^2} \nabla u_i \cdot \nabla u_j \right) \enskip ,$$ therefore, $$\delta\hat{f}(u_i,u_j) = \frac{4\sigma_{ij}}{\eta} \left( \text{sign}(u_i)\,|u_j| + \frac{\eta^2}{\pi^2} \nabla^2 u_j \right) \enskip ,$$ yielding $$\begin{aligned} && \delta\hat{f}(u,1-u) = \frac{2}{\eta}\sigma_{ij} \\ && \delta\hat{f}(1-u,u) = \frac{2}{\eta}\sigma_{ij} \\ && \delta\hat{f}(0,u)+\delta\hat{f}(0,1-u) = \frac{2}{\eta}\sin\left(\frac{\pi}{\eta}x\right)(\sigma_{kj}-\sigma_{ki}) \quad\quad\end{aligned}$$ for the extension $u_i(x)=u(x)$, $u_{j=\neq i}(x)=1-u(x)$ and $u_{k \neq i,j}(x)=0$, showing that the planar binary interfaces do not minimize the free energy functional for non-zero interfacial free energies.\ The dynamic equations of Steinbach and Pezzolla [@SteinbachPezzola1999] also read as: $$-\frac{\partial u_i}{\partial t} = \sum_{j \neq i} \kappa_{ij} \left[ \frac{\delta F}{\delta u_i} - \frac{\delta F}{\delta u_j} \right] \enskip ,$$ with $\kappa_{ij}=\kappa_{ji}>0$ constants, therefore, the mobility matrix satisfies conditions 1–3 of Section II.C. Considering that the planar binary interfaces represent no equilibrium solution, they are not stationary. The problem is resolved again by replacing the gradient term $-\nabla u_i \cdot \nabla u_j$ by $(u_i\nabla u_j-u_j\nabla u_i)^2$, and using the ’binary approximation’ [@Steinbach2009]: $$\label{eq:twophaseapprox} \begin{split} \frac{\delta F}{\delta u_i}-\frac{\delta F}{\delta u_j} \approx \enskip & \text{sign}(u_i)|u_j|-\text{sign}(u_j)|u_i| \\ & - (u_j\Delta u_i - u_i\Delta u_j) \enskip . \end{split}$$ Although the resulting non-variational dynamics stabilizes the extension of the planar interface solution $u(x)=\{1+\sin[(\eta/\pi)x]\}$, unfortunately it does not minimize the free energy functional. A remarkable improvement of the Steinbach-Pezzola model has been put forward by Kim [*et al.*]{} [@Kim2006]. Introducing step functions that are $S_i = 1$ for $u_i > 0$ and $S_i = 0$ otherwise, the mobility matrix has been assumed to have the form shown below $$-\frac{\partial u_i}{\partial t} = \sum_{j \neq i} \kappa_{ij} S_i S_j \left[ \frac{\delta F}{\delta u_i} - \frac{\delta F}{\delta u_j} \right] \enskip .$$ This change leads to an important step ahead: it retains the variational formalism, while stabilizing the flat interface. Nevertheless, it is not yet a solution of the Euler-Lagrange equation of the multiphase problem, therefore, this mobility matrix is ’dangerous’ in the sense that it generates stationary solution from a non-equilibrium state. This approach has recently been applied for an asymmetric case taking the grain boundary energies from a database [@Kim2014].\ Finally, we mention that the derivations presented above can be trivially repeated for using $u_i\,u_j$ instead of $|u_i|\,|u_j|$ in the free energy functional described by Eq. (\[eq:spmodel\]), resulting in the same qualitative results, i.e. the planar binary interfaces do not minimize the multiphase functional. In addition, the absence of the absolute value function terminates the bulk $u_i=1$ equilibrium solution too. Nestler-Wheeler --------------- Another descendant of the original Steinbach [*et al.*]{} model [@Steinbach1996] is the general Nestler-Wheeler type formalism [@Nest1; @Nest2; @Nest3; @Nest4; @Nest5]: $$\label{eq:nwfunc} f_{\rm intf}^{\rm NW} = \sum_{i<j} \left[ \frac{\epsilon^2_{ij}}{2} (u_i \nabla u_j-u_j\nabla u_i)^2 + \frac{w_{ij}}{2} (|u_i|\cdot |u_j|)^p \right] \enskip ,$$ where $p=1$ or $2$. For $p=2$ [@Nest1; @Nest2; @Nest3; @Nest4; @Nest5], Eq. (\[eq:nwfunc\]) recovers Eqs. (\[eq:sfunc\])-(\[eq:smulti\]) (Steinbach *et al.*), whereas in case of $p=1$ [@Nest2; @Nest4; @Nest5], it reduces to Eq. (\[eq:sp2red\]) for $N=2$ with the solution $$u(x) = \frac{1+\sin(x/\delta_{ij})}{2} \enskip .$$ For $p=1$, the derivative of the generator function reads as $$\begin{split} \delta\hat{f}(u_i,u_j) =& (w_{ij}/2) \, {\rm sign}(u_i) |u_j| + \\ &+ \epsilon^2_{ij} \left[ 2(u_i\partial_x u_j-u_j\partial_x u_i)\partial_x u_j+ \right. \\ & + \left. (u_i\partial_x^2 u_j - u_j\partial_x^2 u_i)u_j \right] \enskip , \end{split}$$ which, in the case of $u_i(x):=u(x)$, $u_{j\neq i}(x):=1-u(x)$, and $u_{k \neq i,j}(x):=0$, yields $$\begin{aligned} &&\delta\hat{f}(u,1-u) = \frac{3w_{ij}}{4} \cos^2(x/\delta_{ij}) \\ &&\delta\hat{f}(1-u,u) = \frac{3w_{ij}}{4} \cos^2(x/\delta_{ij}) \\ && \delta\hat{f}(0,u) + \delta\hat{f}(0,1-u) = 0 \enskip ,\end{aligned}$$ showing that the binary planar interfaces do not minimize the free energy functional again. This, together with the fact that the mobility matrix was chosen to be Lagrangian, means that we have here the same problem as in the case of the Steinbach-Pezzolla formalism, which can be resolved by using Eq. (\[eq:twophaseapprox\]), i.e., by adopting non-variational dynamics.\ In a recent variant of the $p=1$ Nestler-Wheeler model by Ankit [*et al*]{} [@Ankit2013] the multi-obstacle free energy landscape contains a triplet term of the form: $$\label{eq:triplet} f_3 = \sum_{i<j<k} \gamma_{ijk} |u_i|\,|u_i|\,|u_k| \enskip ,$$ where the triple sum runs for all different $(i,j,k)$ triplets, and the authors use $\gamma_{ijk}$ to control the appearance of the third phase at the binary interfaces. We note, however, that Eq. (\[eq:triplet\]) has no effect on the existence of the planar interface solution, since the derivative $\partial f_3/\partial u_i=\text{sign}(u_i)\sum_{j<k}|u_i|\,|u_j|$ vanishes for binary planar interfaces. In other words, $f_3$ is not suitable for generating equilibrium planar interface solutions. Nevertheless, choosing $\gamma_{ijk}\to \infty$ results in two-phase interfaces free of additional fields, but for any finite $\gamma_{ijk}$, additional fields are always present at the interfaces mathematically.\ We note that, in models relying on the multi-obstacle potential, $f_{\rm intf} \to \infty$ is often prescribed out of the physical regime to prevent the evolution of the fields into the “unphysical” states $u_i<0$ and $u_i>1$. This is another way to stabilize the (otherwise non-equilibrium) two-phase interfaces. We recall furthermore that there is physical interpretation for $u_i<0$ and $u_i>1$. Comparison of the phase-field models to the Ginzburg-Landau model and/or to amplitude equations emerging from classical density functional theories [@TothProvatas] implies that $u_i<0$ can be simply associated with a negative amplitude of the first reciprocal lattice vector set in the crystal, which is a real perturbation of the liquid state for cubic crystal structures. Similarly, $u_i>1$ is nothing more than an amplitude larger than the equilibrium crystal amplitude.\ Folch-Plapp ----------- The term “Lagrange multiplier formalism” originates from Folch and Plapp [@FolchPlapp2005]. In their multiphase description, the free energy functional is based on several theoretical considerations including binary equilibrium solutions and the condition of no spurious phase generation. For three phases the interface contribution of the free energy functional reads as: $$\label{eq:FPfunc} f_{\rm intf}^{FP} = \frac{\epsilon^2}{2} \sum_{i=1}^3 (\nabla u_i)^2 + \frac{w}{2} f_{TW}(\mathbf{u}) \enskip ,$$ where the “triple-well” free energy density is $f_{TW}(\mathbf{u})=\sum_{i=1}^3 g(u_i)$, .where $g(u_i)=[u_i(1-u_i)]^2$. First, we analyze the binary planar interfaces. For $N=2$ the free energy functional reduces to $$f_{\rm intf}^{FP} = \epsilon^2 (\nabla u)^2 + w g(u) \enskip .$$ generating the usual 1D Euler-Lagrange equation $$\label{eq:FPRED} \frac{\delta F}{\delta u}=w g'(u) - \epsilon^2 \partial_x^2 u=0 \enskip .$$ The equilibrium planar interface solution is then $u(x)=\{1+\tanh[x/(2\delta)]\}/2$ with $\delta^2=\epsilon^2/w$. Since Eq. (\[eq:FPfunc\]) is not a pairwise construction, the generator function technique does not apply here. The general Euler-Lagrange equations read as: $$\label{eq:FPEL} \frac{\delta F}{\delta u_i} = \frac{1}{2} \left[ w \, g'(u_i)-\epsilon^2 \, \partial_x^2 u_i \right] =\lambda(\mathbf{r}) \enskip .$$ Comparing Eqs. (\[eq:FPRED\]) and (\[eq:FPEL\]) results in 2 important properties of the model: $$\frac{\delta F}{\delta u_i} \propto \left. \frac{\delta F}{\delta u}\right|_{u_i} \quad \text{and} \quad \left. \frac{\delta F}{\delta u_i}\right|_{u_i=0}=0 \enskip ,$$ indicating that *the equilibrium planar binary interfaces are solutions of the multiphase problem with $\lambda(x)=0$*. The proposed dynamics uses the [*Lagrangian mobility matrix*]{}, therefore, the Folch-Plapp model is the first model, which passes the binary interface criterion.\ Next, we discuss the appearance of spurious phases. The general condition reads as $u_k(\mathbf{r},t) = 0$, i.e. if phase $k$ is not present at $t=0$, it must not appear at any time. In the Folch-Plapp model, the time evolution of a phase, which is apparently not present reads as $$\left. \frac{\partial u_k}{\partial t} \right| _{u_k=0} \propto \left( \frac{\delta F}{\delta u_i} + \frac{\delta F}{\delta u_j} \right) \enskip ,$$ where $k \neq i,j$ and $i\neq j$. Using Eq. (\[eq:FPEL\]) it is obvious that $\delta F/\delta u_i+\delta F/\delta u_j \equiv 0$ for $u_i(\mathbf{r},t)+u_j(\mathbf{r},t)=1$, therefore, no spurious phases appear.\ The authors have also worked out an asymmetric version of the model, in which different binary interfacial free energies can be used, which also satisfies the basic criteria of physical consistency together with no spurious phase generation. Despite its advantageous features compared to former approaches, there are a few weaknesses of the model: (a) to avoid the appearance of spurious phases, a free energy functional is used whose form depends on the particular choice of the mobility matrix, which is clearly not physical, (b) no $N>3$ generalization of the model is available. Furthermore, (c) as the authors suggested, the necessary minimum exponent in the multi-well term might be proportional to the number of phases, making the model practically useless when a large number (e.g., thousands) of differently oriented dendrites or grains have to be simulated.\ *At this stage, it is clear that the Folch-Plapp model is a large step towards constructing a consistent multiphase description, since it satisfies almost all the criteria of physical consistency. Unfortunately, the price is high: it is not immediately clear how to generalize this model to $N > 3$, together with avoiding the appearance of spurious phases via introducing special terms in the free energy functional, which follow from the form of the dynamic equations.* Bollada-Jimack-Mullis --------------------- In the previous sections it has been demonstrated that the condition of no spurious phase generation works at two levels. First, if a solution of the Euler-Lagrange equation represents minimum of the free energy functional, it is stable against small perturbations (assuming variational dynamics with a positive semi-definite mobility matrix). In addition, to avoid the appearance of spurious phase outside of equilibrium may necessitate the adjustment of the free energy functional. In a recent work [@Bollada2012], however, Bollada [*et al.*]{} avoided the problem by introducing a field-dependent mobility matrix $$\label{eq:BJMmob} L_{ii} = \sum_{j \neq i}^N h(u_i,u_j) \quad \text{and} \quad L_{ij} = - h(u_i,u_j) \enskip \text{for} \enskip i \neq j \enskip ,$$ where $$h(u_i,u_j) = \left(\frac{u_i}{1-u_i}\right)\left(\frac{u_j}{1-u_j}\right) \enskip .$$ Apparently, Eq. (\[eq:BJMmob\]) satisfies the conditions of Section II.C; i.e., the dynamics ensures non-negative entropy production inside the simplex (i.e., when all $u_i \in [0,1]$, all $h(u_i,u_j) \geq 0$). In addition, the spurious phase generation is excluded, since for $u_k(\mathbf{r},t)=0$ the $k^{th}$ row (and column) of the mobility matrix vanishes, yielding $(\partial u_k/\partial t)|_{u_k(\mathbf{r},t)=0}=0$. Note that this is achieved without revising the free energy functional. Moreover, the description is $N$-independent, since the mobility matrix consistently reduces to the $N-1$ case. Despite the significant improvement, one has to be careful with the Bollada-Jimack-Mullis mobility matrices, since they can be dangerous with respect to the free energy functional in the sense that the mobility matrix may generate stationary solution of the dynamics from a non-equilibrium state (as it happens in case of a multi-obstacle potential). In addition, outside of the simplex (i.e., for $u_i<0$), the mobility matrix is indefinite, thus violating criterion (iv).\ [The results of the above analysis of the MPF theories are summarized in Table I. The following has been found:]{} [(A) The criteria for the sum of the fields and for labeling are satisfied by all the models investigated.]{} [(B) The lack of equilibrium planar two-phase interfaces (in the $N$-phase problem) can be resolved by employing: (a) non-variational dynamics (the models by Steinbach [*et al.*]{}[@Steinbach1996], Steinbach and Pezzola[@SteinbachPezzola1999], Nestler and Wheeler[@Nest1; @Nest2; @Nest3; @Nest4; @Nest5]), (b) a degenerate mobility matrix (the models by Kim [*et al.*]{}[@Kim2006], and by Bollada [*et al.*]{}[@Bollada2012]), or by (c) penalizing the triplet term (the model by Ankit [*et al.*]{}[@Ankit2013]). We identified the following problems associated with methods (a)-(c): the solution does not converge to the equilibrium solution; furthermore, in case (c) the third phase is unavoidably present (even if in a small amount).]{} [(C) When the equilibrium conditions are satisfied (as in the model by Folch and Plapp[@FolchPlapp2005]), we obtain an $N$-dependent approach without the possibility of prescribing freely the pairwise interfacial data.]{} [Considering these, we conclude that none of the MPF models investigated here satisfy all the consistency criteria specified. We stress furthermore that the introduction of additional thermodynamical driving force via an appropriate tilting function (as needed for describing polycrystalline solidification) would influence neither the validity of these criteria, nor the outcome of this analysis.]{} model $\backslash$ criterion i ii iii iv v vi(a) vi(b) vii ------------------------------ --- ---- ----- ---- --- ------- ------- ----- Steinbach [*et al.*]{} x x x x Steinbach & Pezzola x x x x Nestler & Wheeler x x x x Kim [*et al.*]{} x x x x x x Bollada [*et al.*]{} x x x x x x Ankit [*et al.*]{} x x x x x Folch & Plapp x x x x x x : Properties of different multiphase-field models from the viewpoint of criteria defined in this work. Consistent multiphase formalism =============================== General framework ----------------- Herein, we derive a multiphase description that satisfies criteria (i) to (vii). It is useful to start with the condition of formal reducibility \[criterion (iv)\]. First, setting $u_N(\mathbf{r},t)=0$ in the $N$-phase free energy functional $F^{(N)}$ should result in the $N-1$ phase functional, $F^{N-1}$. The same should apply for the dynamic equations $$-\frac{\partial u_i}{\partial t} = \sum_{i \neq j} \kappa_{ij}\left( \frac{\delta F}{\delta u_i}-\frac{\delta F}{\delta u_J}\right)$$ as well; i.e., $-\dot{\mathbf{u}}^{(N)}=\mathbb{L}^{(N)} \cdot \delta F_{\mathbf{u}}^{(N)}|_{u_{N}(\mathbf{r},t)=0}$ should reduce to $-\dot{\mathbf{u}}^{(N-1)}=\mathbb{L}^{(N-1)} \cdot \delta F_{\mathbf{u}}^{(N-1)}$, and $\dot{u}_N = 0$. Note that the latter satisfies the condition of no spurious phase generation \[criterion (ii)\]), since $\dot{u}_i|{u_i=0} = 0$. Here $\mathbb{L}^{(N)}$ is a general, symmetric, positive semidefinite $N$–phase mobility matrix, i.e. $L_{ij}=-\kappa_{ij}$ for $i \neq j$, while $L_{ii}=\sum_{j\neq i} \kappa_{ij}$, where $i,j=1\dots N$, and $\kappa_{ij}=\kappa_{ji}>0$. Since the (modified) Bollada-Jimack-Mullis matrix defined by Eq. (\[eq:BJMmob\]) satisfies the condition of formal reducibility, we choose this mobility matrix, namely, $$\kappa_{ij} := \kappa_{ij}^0 \left| \frac{u_i}{1-u_i}\right| \, \left| \frac{u_j}{1-u_j}\right| \enskip ,$$ where constant positive prefactors $\kappa_{ij}^0$ accounting for the mobility of the planar $i,j$ interface are also incorporated. Furthermore, we prescribe the condition of reduction also for the functional derivatives; i.e., the first $N-1$ components of $\delta F_{\mathbf{u}}^{N}$ should reduce to $\delta F_\mathbf{u}^{(N-1)}$ for $u_N(\mathbf{r},t)=0$. Since the $N^{th}$ row (and column) of the reduced Bollada-Jimack-Mullis matrix is $0$, it always results in $\dot{u}_N=0$, therefore, $(\delta F^{N}/\delta u_N)|_{u_N(\mathbf{r},t)=0}$ can be arbitrary. Since the $n$–phase Bollada-Jimack-Mullis matrix is positive semidefinite on an $n$–phase state (i.e. when none of the $n$ components vanishes) with multiplicity 1 for the eigenvalue 0 (it can be proven numerically), the $n$–phase stationary solutions coincide with the $n$ phase equilibrium states. Moreover, since the dynamic equations reduce naturally to the $(N-1)$–phase case, the stationary solutions of the reduced dynamics include the natural $N$ phase extensions of the $(N-1)$–phase equilibrium solutions (where none of the $N-1$ phases is missing). Therefore, the natural $N$–phase extensions of all $(N-1)$–phase equilibrium solutions emerging from $F^{(N-1)}$ should represent extrema of $F^{N}$. If this is true, the Bollada-Jimack-Mullis matrix is not dangerous with respect to the free energy functional, since all stationary states of the dynamics represent equilibrium. Since the condition must apply for arbitrary $N$, the general condition for the free energy functional reads as follows:\ *The $(p+q)$–phase trivial extensions of all $p$–phase equilibrium solutions (where all the $p$ phases are non-vanishing) emerging from the $p$–phase free energy functional $F^{(p)}$ must represent extrema of the $(p+q)$–phase free energy functional $F^{(p+q)}$ too, for any $p>0,q>0$.*\ For practical reasons, we introduce the following condition: for a field, which is not present, the functional derivative vanishes, i.e. $$\left.\frac{\delta F}{\delta u_i}\right|_{u_i(\mathbf{r},t)=0}=0 \enskip .$$ If the Lagrange multiplier also vanishes \[$\lambda(\mathbf{r})=0$\] for *all* $p-$phase equilibrium solutions of $F^{(p)}$ (where $p>1$ arbitrary), while the free energy functional (and the functional derivative) reduces naturally, all trivial extensions of all $p$-phase equilibrium solutions remain equilibrium solutions of the $N$-phase free energy functional. Consequently, here the Bollada-Jimack-Mullis matrix does not stabilize non-equilibrium solutions, while preventing the appearance of spurious phases. [Note, that this is obviously not true for the models of Steinbach *et al.*, Steinbach and Pezzola, Nestler and Wheeler, Ankit [*et al.*]{}, and for the model potential used in the work of Bollada, Jimack, and Mullis. Although there the free energy functionals and the functional derivatives reduce naturally, and $(\delta F/\delta u_i)|_{u_i=0}=0$ also applies, the planar two-phase interface solution generates different Lagrange multipliers for the vanishing and non-vanishing fields in the complete ($N$-phase) Euler-Lagrange problem, indicating that the natural extensions of the planar two-phase interfaces do not represent equilibrium. Yet, the application of the Bollada-Jimack-Mullis mobility matrix transforms them into stationary solutions, showing that in these cases the application of the Bollada-Jimack-Mullis matrix is ”dangerous”.]{} Free energy functional ---------------------- ### Symmetric system The main question is, how one should construct an interface term that satisfies the conditions given above. First, we consider the symmetric case, where all interface thicknesses and interfacial free energies are equal. Following Chen and co-workers [@ChenYang1994; @FanChen1997a; @FanChen1997b] and Moelans and co-workers [@Moelans1; @Moelans2; @Moelans3; @Moelans4], the interface term of the free energy functional is constructed as follows $$\label{eq:tothmodel} f_{\rm intf} = \frac{\epsilon^2}{2} \sum_{i=1}^N (\nabla u_i)^2 + w\, g(\mathbf{u}) \enskip ,$$ where we use the following new Ansatz for the multiphase barrier function: $$\label{eq:tothpoly} g(\mathbf{u}) := \frac{1}{12} + \sum_{i=1}^N \left( \frac{u_i^4}{4} - \frac{u_i^3}{3} \right) + \frac{1}{2} \sum_{i<j} u_i^2 u_j^2 \enskip .$$ The functional derivative reads as $$\label{eq:tothfder} \frac{\delta F}{\delta u_i} =w \, [u_i(\mathbf{u}^2-u_i)] - \epsilon^2 \nabla^2 u_i \enskip ,$$ which vanishes for $u_i=0$. The binary planar interface solution is $u(x)=\{1+\tanh[x/(2 \delta)]\}/2$ (where $\delta^2=\epsilon^2/w$), for which the multiphase Euler-Lagrange equations reduce to $$\begin{aligned} \frac{\delta F}{\delta u_i} &=& -\frac{\delta F}{\delta u_j} = w \, u(1-u)(1-2u) - \epsilon^2 \partial_x^2 u =0 \\ \frac{\delta F}{\delta u_k} &=& 0 \enskip .\end{aligned}$$ Here we used the trivial extension $u_i:=u(x)$, $u_{j\neq j}:=1-u(x)$ and $u_{k\neq i,j}=0$. Since the free energy functional and the functional derivatives reduce naturally, and $\delta F/\delta u_i$ vanishes for $u_i=0$, together with the fact that any trivial extension of the planar interface solution represents equilibrium, it is reasonable to assume that the Bollada-Jimack-Mullis matrix is not dangerous considering at least the planar interfaces, i.e., it does not stabilize a non-equilibrium planar interface, since all planar interfaces represent equilibrium. Naturally, the same investigation should be repeated for all $n$–phase equilibrium solutions of $F^{(n)}$ for any positive $n$, however, this kind of study is out of the scope of the present paper.\ It is important to mention, that Eq. (\[eq:tothpoly\]) shows a very practical feature $$\label{eq:tothtendency} g(\{1/N,1/N,\dots,1/N\}) = \frac{1}{12} \left( 1-\frac{1}{N^2} \right) \enskip,$$ i.e., the higher-order junctions are energetically increasingly less favorable. Note that this is not true for Eq. (\[eq:smulti\]). The tendency of increasing free energy is also ensured by the multi-well term defined in Eq. (\[eq:spmodel\]), however there, as we have shown previously, the binary planar interfaces do not minimize the free energy functional in the general $N$-phase case. It is worth noting that Eq. (\[eq:tothtendency\]) contradicts Folch and Plapp [@FolchPlapp2005], who expect that the polynomial degree of the $g(\mathbf{u})$ function that penalizes the high-order multiple junctions would increase with $N$. Eq. (\[eq:tothtendency\]) shows that the double-obstacle function is not the only one that realizes this tendency, furthermore, here \[see Eq. (\[eq:tothpoly\])\] the planar binary interface represents an equilibrium solution of the multiphase problem. ### Asymmetric extension Following Moelans [@Moelans2], the asymmetric extension of Eq. (\[eq:tothmodel\]) can be obtained by employing the Kazaryan-polynomials [@Kazaryan2000] $$\begin{aligned} \label{eq:totheps2} \epsilon^2(\mathbf{u}) &:=& \frac{\sum_{i,j} \epsilon^2_{ij} u_i^2 u_j^2}{\sum_{i,j} u_i^2 u_j^2}\\ w(\mathbf{u}) &:=& \frac{\sum_{i,j} w_{ij} u_i^2 u_j^2}{\sum_{i,j} u_i^2 u_j^2} \enskip .\end{aligned}$$ The free energy density then reads as $$\label{eq:tothasymm} f_{\rm intf} = \frac{\epsilon^2(\mathbf{u})}{2} \sum_{i=1}^N (\nabla u_i)^2 + w(\mathbf{u}) \, g(\mathbf{u}) \enskip ,$$ where $g(\mathbf{u})$ is defined by Eq. (\[eq:tothpoly\]). Since the Kazaryan polynomial is “quasi-constant” (the nominator and the denominator contain the same terms with different coefficients), it is reasonable to assume that this modification does not change the structure of extrema of $g(\mathbf{u})$. Although it will be demonstrated for asymmetric $N=4$ and $N=5$ systems, a strict mathematical derivation for arbitrary number of phases is out of the scope of the present paper. The binary reduction of Eq. (\[eq:tothasymm\]) reads as $$f_{\rm intf} = \epsilon_{ij}^2 (\nabla u)^2 + w_{ij} [u(1-u)]^2 \enskip ,$$ generating the equilibrium planar binary interface with $\delta^2 = \epsilon^2_{ij}/w_{ij}$. The general functional derivative is $$\label{eq:tothgender} \begin{split} \frac{\delta F}{\delta u_i} & = w \,\frac{\partial g}{\partial u_i} + \frac{\partial w}{\partial u_i} \, g(\mathbf{u}) - \epsilon^2 \, \nabla^2 u_i \\ & + \sum_{j=1}^N \left[ \frac{1}{2} \frac{\partial \epsilon^2}{\partial u_i} \nabla u_j - \frac{\partial \epsilon^2}{\partial u_j} \nabla u_i \right] \cdot \nabla u_j \enskip , \end{split}$$ where $$\begin{aligned} \frac{\partial \epsilon^2}{\partial u_j} &=& (2 u_j) \frac{\sum_{l \neq j} (\epsilon_{lj}^2-\epsilon^2)u_l^2}{\sum_{k,l} u_k^2 u_l^2}\\ \frac{\partial w}{\partial u_j} &=& (2 u_j) \frac{\sum_{l \neq j} (w_{lj}-w)u_l^2}{\sum_{k,l} u_k^2 u_l^2} \enskip .\end{aligned}$$ Note that $\left. \frac{\partial \chi}{\partial u_k} \right|_{u_k=0}=\left. \frac{\partial \chi}{\partial u_{i,j}} \right|_{u_i+u_j=1}=0$, where $\chi=\epsilon^2,w$, therefore, $(\delta F/\delta u_k)_{u_k=0}=0$, and the second line of Eq. (\[eq:tothgender\]) also vanishes for $u_i=u$, $u_j=1-u$, $u_{k \neq i,j}=0$, therefore, *the planar binary interfaces are equilibrium solutions of the multiphase problem for arbitrary pairwise $\epsilon_{ij}^2$ and $w_{ij}$* fitted to the interfacial free energy $\gamma_{ij}$ and interface thickness $\delta_{ij}$ as follows: $$\epsilon_{ij}^2 = 3 \, (\delta_{ij} \cdot \gamma_{ij}) \quad \text{and} \quad w_{ij} = 3 \, (\gamma_{ij}/\delta_{ij}) \enskip .$$ ### Introducing anisotropy In various practically important cases, including dendritic solidification and grain coarsening, the interfacial free energy between two phases displays anisotropy, which can be formulated mathematically as: $$\epsilon_{ij} \to \epsilon_{ij} \left[ 1 + a_{ij} \cdot \eta_{ij}(\mathbf{n}_{ij}) \right] \enskip ,$$ where $a_{ij}$ is the amplitude (strength) of the anisotropy, $$\mathbf{n}_{ij} = \frac{\nabla u_i - \nabla u_j}{| \nabla u_i - \nabla u_j |}$$ is a unit vector characterizing the $(i,j)$ binary interface, while $\eta_{ij}(\mathbf{n}_{ij})$ reflects the crystal symmetry. This extension modifies Eq. (\[eq:totheps2\]) and the functional derivative as follows $$\begin{split} \frac{\delta F}{\delta u_i} & = g(\mathbf{u}) \frac{\partial w}{\partial u_i} + w(\mathbf{u}) \frac{\partial g}{\partial u_i} + \frac{\partial \epsilon^2}{\partial u_i} \left[ \frac{1}{2} \sum_{j=1}^N (\nabla u_j)^2 \right] -\\ & - \nabla \cdot \left\{ \frac{\partial \epsilon^2}{\partial \nabla u_i} \left[ \frac{1}{2} \sum_{j=1}^N (\nabla u_j)^2 \right] + \epsilon^2 \cdot \nabla u_i \right\} \enskip . \end{split}$$ Here the extra term reads as $$\frac{\partial \epsilon^2}{\partial \nabla u_i} = \frac{\sum_{k,l} \frac{\partial \epsilon_{kl}^2}{\partial \nabla u_i} u_k^2 u_l^2}{\sum_{k,l} u_k^2 u_l^2} \enskip ,$$ where $\frac{\partial \epsilon_{kl}^2}{\partial \nabla u_i} \propto \delta_{ki}+\delta_{li}$, therefore, $\frac{\partial \epsilon^2}{\partial \nabla u_i} \propto u_i^2$, which means $(\delta F/\delta u_k)|_{u_k=0}=0$. In addition, for $u_i+u_j=1$ $\frac{\partial \epsilon^2}{\partial \nabla u_{i,j}} = \frac{\partial \epsilon_{ij}^2}{\partial \nabla u_{i,j}}$, therefore, $\frac{\delta F}{\delta u_{i,j}} = \frac{1}{2} \frac{\delta F}{\delta u}$, where $\delta F/\delta u$ is the functional derivative in the reduced model, therefore, the equilibrium binary interfaces emerging from $\delta F/\delta u=0$ are stationary solutions of the multiphase problem.\ Testing the XMPF model ====================== In this section, we review whether the proposed model satisfies indeed criteria (i) to (vii). Several of these criteria are satisfied owing to the specific formulation of our model \[these are (i), (iii) – (vi), and (vii)\] as summarized in sub-section A. Fulfillment of practical criterion (ii), however, needs further investigation, which is undertaken in sub-section B. [Finally, illustrative simulations are presented for grain coarsening in sub-section C.]{} Consistency criteria satisfied ------------------------------ It can be shown that the proposed model satisfies the following criteria:\ (i) $\sum_{i=1}^N u_i(\mathbf{r},t)= 1$ (see Section IV.A);\ (ii) Since the mobility matrix is symmetric, the physical results are invariant to formally exchanging pairs of field indices, $i \leftrightarrow j$, i.e. the variables are not labeled (see Section IV.A);\ (iii) Under appropriate boundary conditions, any trivial multiphase extension of the equilibrium binary solution represents equilibrium solution of the multiphase problem, which is then a stationary solution of the dynamic equations towards which the time dependent solution evolves (see Section IV.D);\ (iv) Since the mobility matrix is positive semidefinite, non-negative entropy production is ensured for both conserved and non-conserved dynamics (see Sections IV.A and VI.B);\ (v) Reduction/extension of the $N$-field theory to $N-1$ or $N+1$ fields is trivial on the level of both the free energy functional (and the functional derivative) and the mobility matrix (see Sections IV.A and IV.D);\ (vi) No additional phases appear at the equilibrium binary interfaces (see Sections IV.B, IV.C, IV.D and VI.A), and the dynamic spurious phase generation is also excluded (see Sections IV.A and V.B);\ (vii) Freedom for choosing independent interfacial ($\epsilon_{ij}$ and $w_{ij}$) and kinetic properties ($\kappa_{ij}$) for the individual binary boundaries, including their anisotropy (see the formulation in Sections IV.C and IV.D).\ Since the dynamic generation of spurious phases is excluded by the modified Bollada-Jimack-Mullis mobility matrix, the trivial extensions of the equilibrium planar solutions represent equilibrium, and the stationary solutions of the dynamic equations coincide with the equilibrium solutions, hence the mobility matrix is regarded as ’not dangerous’, when considering planar interfaces. Nevertheless, it still remains unclear, whether the same applies for [*all*]{} equilibrium solutions, such as the trivial extensions of equilibrium trijunctions, etc.. Therefore, next we investigate the time evolution of multi-domain systems in this respect. (a)![\[fig:timeevol\] (Color online) Time evolution in an $N = 5$ symmetric Cahn-Hilliard problem with starting conditions $u_1 = u_2 = u_3 = u_4 = u_5 = 0.2$: Different colors stand for different phases (where $u_i>0.99$). Snapshots taken at $t = (1, 3, 10, 30, 100, 300) \times 25,000$ dimensionless time are displayed. (Time increases from left to right, and from top to bottom.)](fig1a.png "fig:"){width="0.44\linewidth"} (b)![\[fig:timeevol\] (Color online) Time evolution in an $N = 5$ symmetric Cahn-Hilliard problem with starting conditions $u_1 = u_2 = u_3 = u_4 = u_5 = 0.2$: Different colors stand for different phases (where $u_i>0.99$). Snapshots taken at $t = (1, 3, 10, 30, 100, 300) \times 25,000$ dimensionless time are displayed. (Time increases from left to right, and from top to bottom.)](fig1b.png "fig:"){width="0.44\linewidth"}\ (c)![\[fig:timeevol\] (Color online) Time evolution in an $N = 5$ symmetric Cahn-Hilliard problem with starting conditions $u_1 = u_2 = u_3 = u_4 = u_5 = 0.2$: Different colors stand for different phases (where $u_i>0.99$). Snapshots taken at $t = (1, 3, 10, 30, 100, 300) \times 25,000$ dimensionless time are displayed. (Time increases from left to right, and from top to bottom.)](fig1c.png "fig:"){width="0.44\linewidth"} (d)![\[fig:timeevol\] (Color online) Time evolution in an $N = 5$ symmetric Cahn-Hilliard problem with starting conditions $u_1 = u_2 = u_3 = u_4 = u_5 = 0.2$: Different colors stand for different phases (where $u_i>0.99$). Snapshots taken at $t = (1, 3, 10, 30, 100, 300) \times 25,000$ dimensionless time are displayed. (Time increases from left to right, and from top to bottom.)](fig1d.png "fig:"){width="0.44\linewidth"}\ (e)![\[fig:timeevol\] (Color online) Time evolution in an $N = 5$ symmetric Cahn-Hilliard problem with starting conditions $u_1 = u_2 = u_3 = u_4 = u_5 = 0.2$: Different colors stand for different phases (where $u_i>0.99$). Snapshots taken at $t = (1, 3, 10, 30, 100, 300) \times 25,000$ dimensionless time are displayed. (Time increases from left to right, and from top to bottom.)](fig1e.png "fig:"){width="0.44\linewidth"} (f)![\[fig:timeevol\] (Color online) Time evolution in an $N = 5$ symmetric Cahn-Hilliard problem with starting conditions $u_1 = u_2 = u_3 = u_4 = u_5 = 0.2$: Different colors stand for different phases (where $u_i>0.99$). Snapshots taken at $t = (1, 3, 10, 30, 100, 300) \times 25,000$ dimensionless time are displayed. (Time increases from left to right, and from top to bottom.)](fig1f.png "fig:"){width="0.44\linewidth"} Multiphase separation --------------------- For practical reasons, the time evolution of multi-domain systems is investigated in two dimensions using conserved dynamics ($N-component$ Cahn-Hilliard systems), for which the dimensionless equations of motion read as $$\frac{\partial u_i}{\partial t} = \nabla \cdot \left[ \sum_{j\neq i} \kappa_{ij} \nabla \left( \frac{\delta F}{\delta u_i} - \frac{\delta F}{\delta u_j} \right) \right] \enskip .$$ Our choice is motivated by the fact that simulations with conserved dynamics are much less forgiving to possible numerical errors than those with non-conserved dynamics.\ First, we consider a symmetric system, and demonstrate that our free energy functional prescribes the hierarchy $F_{bulk} < F_{interface} < F_{trijunction} < F_{quadruple} < \dots$ for the equilibrium solutions, therefore, a 2-dimensional, multi-domain system displays a bulk–interface(–trijunction) topology. Accordingly, the additional phases near any bulk / interface / trijunction vanish with time, since these states are energetically not preferred in the vicinity of the equilibrium solutions, to which the system converges. *This happens independently of the particular choice of the mobility matrix*; the only requirement is that the mobility matrix has to be symmetric and positive semidefinite. Since $\sum_{i=1}^N (\delta F/\delta u_i) \equiv 0$ applies for the symmetric model, the Lagrangian mobility matrix $\kappa_{ij}=1$ yields: $$\frac{\partial u_i}{\partial t} = \nabla^2 \frac{\delta F}{\delta u_i} \enskip ,$$ a fairly simple system of dynamic equations. Since $(\delta F/\delta u_i)_{u_i=0}=0$, this dynamics satisfies the condition of “no spurious phase generation”. Indeed, we demonstrate that if a phase is not present in the beginning of the calculation, it never appears, even if the other phases are not in equilibrium.\ Next, we perform simulations in an $N = 4$ asymmetric system with two types of mobility matrices (the Lagrangian and the Bollada-Jimack-Mullis type) to demonstrate spurious phase generation. We show that the spurious phase appears with the Lagrangian mobility, whereas in the case of the Bollada-Jimack-Mullis mobility matrix the zero-amplitude initially prescribed for one of the field is retained throughout the simulation, even if the other fields are not in equilibrium. Finally, a similar behavior is demonstrated for anisotropic systems.\ If not stated otherwise, the dynamic equations were solved numerically, using a pseudo-spectral semi-implicit method based on operator splitting (see Appendix C), on a rectangular grid of size $512 \times 512$, applying periodic boundary conditions at the perimeters. The dimensionless time and spatial steps were $\Delta t = 1$, and $\Delta x = 1$. The computations were performed in double precision on GTX Titan GPU cards. As starting condition, we prescribe a spatially homogeneous state containing equal quantities of all the components, supplemented by a small amplitude of flux noise. For this composition, equilibrium requires the coexistence of $N$ pure phases. Accordingly, the transition process requires the formation and coarsening of these phases. We present illustrations for symmetric ($N = 5$), asymmetric ($N = 4$), and anisotropic ($N = 4$) cases. (a)![\[fig:2nonzero\] (Color online) $N = 5$ symmetric Cahn-Hilliard problem with starting conditions $u_1 = u_2 = 0.5$ and $u_3 = u_4 = u_5 = 0$. (a), (b): Snapshots of the phase fields $u_1$ and $u_2$ at $t = 2.5\time10^5$ dimensionless time; (c) in the left half panel $|u_3|+|u_4|+|u_5|$ is shown, whereas the right half panel shows $\sum_{i=1}^{5}u_i-1$. Note the absence of spurious phases \[criterion (ii)\] and that the sum of the phase fields is 1 \[criterion (i)\]. (d) Multiphase-field map at the same instant.](fig2a.png "fig:"){width="0.44\linewidth"} (b)![\[fig:2nonzero\] (Color online) $N = 5$ symmetric Cahn-Hilliard problem with starting conditions $u_1 = u_2 = 0.5$ and $u_3 = u_4 = u_5 = 0$. (a), (b): Snapshots of the phase fields $u_1$ and $u_2$ at $t = 2.5\time10^5$ dimensionless time; (c) in the left half panel $|u_3|+|u_4|+|u_5|$ is shown, whereas the right half panel shows $\sum_{i=1}^{5}u_i-1$. Note the absence of spurious phases \[criterion (ii)\] and that the sum of the phase fields is 1 \[criterion (i)\]. (d) Multiphase-field map at the same instant.](fig2b.png "fig:"){width="0.44\linewidth"}\ (c)![\[fig:2nonzero\] (Color online) $N = 5$ symmetric Cahn-Hilliard problem with starting conditions $u_1 = u_2 = 0.5$ and $u_3 = u_4 = u_5 = 0$. (a), (b): Snapshots of the phase fields $u_1$ and $u_2$ at $t = 2.5\time10^5$ dimensionless time; (c) in the left half panel $|u_3|+|u_4|+|u_5|$ is shown, whereas the right half panel shows $\sum_{i=1}^{5}u_i-1$. Note the absence of spurious phases \[criterion (ii)\] and that the sum of the phase fields is 1 \[criterion (i)\]. (d) Multiphase-field map at the same instant.](fig2c.png "fig:"){width="0.44\linewidth"} (d)![\[fig:2nonzero\] (Color online) $N = 5$ symmetric Cahn-Hilliard problem with starting conditions $u_1 = u_2 = 0.5$ and $u_3 = u_4 = u_5 = 0$. (a), (b): Snapshots of the phase fields $u_1$ and $u_2$ at $t = 2.5\time10^5$ dimensionless time; (c) in the left half panel $|u_3|+|u_4|+|u_5|$ is shown, whereas the right half panel shows $\sum_{i=1}^{5}u_i-1$. Note the absence of spurious phases \[criterion (ii)\] and that the sum of the phase fields is 1 \[criterion (i)\]. (d) Multiphase-field map at the same instant.](fig2d.png "fig:"){width="0.44\linewidth"} ### Symmetric case (a)![\[fig:4nonzero\] (Color online) $N = 5$ symmetric Cahn-Hilliard problem with starting conditions $u_1 = u_2 = u_3 = u_4 = 0.25$ and $u_5 = 0$. (a) – (b): Snapshots of the phase fields $u_1, u_2, u_3 $ and $u_4$ at $t = 10^6$ dimensionless time; (e) in the left half panel $u_5$ is shown, whereas the right half panel shows $\sum_{i=1}^{5}u_i-1$. Note the absence of spurious phases \[criterion (ii)\] and that the sum of the phase fields is 1 \[criterion (i)\]. (f) Multiphase-field map at the same instant.](fig3a.png "fig:"){width="0.44\linewidth"} (b)![\[fig:4nonzero\] (Color online) $N = 5$ symmetric Cahn-Hilliard problem with starting conditions $u_1 = u_2 = u_3 = u_4 = 0.25$ and $u_5 = 0$. (a) – (b): Snapshots of the phase fields $u_1, u_2, u_3 $ and $u_4$ at $t = 10^6$ dimensionless time; (e) in the left half panel $u_5$ is shown, whereas the right half panel shows $\sum_{i=1}^{5}u_i-1$. Note the absence of spurious phases \[criterion (ii)\] and that the sum of the phase fields is 1 \[criterion (i)\]. (f) Multiphase-field map at the same instant.](fig3b.png "fig:"){width="0.44\linewidth"}\ (c)![\[fig:4nonzero\] (Color online) $N = 5$ symmetric Cahn-Hilliard problem with starting conditions $u_1 = u_2 = u_3 = u_4 = 0.25$ and $u_5 = 0$. (a) – (b): Snapshots of the phase fields $u_1, u_2, u_3 $ and $u_4$ at $t = 10^6$ dimensionless time; (e) in the left half panel $u_5$ is shown, whereas the right half panel shows $\sum_{i=1}^{5}u_i-1$. Note the absence of spurious phases \[criterion (ii)\] and that the sum of the phase fields is 1 \[criterion (i)\]. (f) Multiphase-field map at the same instant.](fig3c.png "fig:"){width="0.44\linewidth"} (d)![\[fig:4nonzero\] (Color online) $N = 5$ symmetric Cahn-Hilliard problem with starting conditions $u_1 = u_2 = u_3 = u_4 = 0.25$ and $u_5 = 0$. (a) – (b): Snapshots of the phase fields $u_1, u_2, u_3 $ and $u_4$ at $t = 10^6$ dimensionless time; (e) in the left half panel $u_5$ is shown, whereas the right half panel shows $\sum_{i=1}^{5}u_i-1$. Note the absence of spurious phases \[criterion (ii)\] and that the sum of the phase fields is 1 \[criterion (i)\]. (f) Multiphase-field map at the same instant.](fig3d.png "fig:"){width="0.44\linewidth"}\ (e)![\[fig:4nonzero\] (Color online) $N = 5$ symmetric Cahn-Hilliard problem with starting conditions $u_1 = u_2 = u_3 = u_4 = 0.25$ and $u_5 = 0$. (a) – (b): Snapshots of the phase fields $u_1, u_2, u_3 $ and $u_4$ at $t = 10^6$ dimensionless time; (e) in the left half panel $u_5$ is shown, whereas the right half panel shows $\sum_{i=1}^{5}u_i-1$. Note the absence of spurious phases \[criterion (ii)\] and that the sum of the phase fields is 1 \[criterion (i)\]. (f) Multiphase-field map at the same instant.](fig3e.png "fig:"){width="0.44\linewidth"} (f)![\[fig:4nonzero\] (Color online) $N = 5$ symmetric Cahn-Hilliard problem with starting conditions $u_1 = u_2 = u_3 = u_4 = 0.25$ and $u_5 = 0$. (a) – (b): Snapshots of the phase fields $u_1, u_2, u_3 $ and $u_4$ at $t = 10^6$ dimensionless time; (e) in the left half panel $u_5$ is shown, whereas the right half panel shows $\sum_{i=1}^{5}u_i-1$. Note the absence of spurious phases \[criterion (ii)\] and that the sum of the phase fields is 1 \[criterion (i)\]. (f) Multiphase-field map at the same instant.](fig3f.png "fig:"){width="0.44\linewidth"} Snapshots illustrating the time evolution of an $N = 5$ symmetric Cahn-Hilliard problem (where $\epsilon_{ij}^2 = 1$ and $w_{ij} = 1$ for all phase pairs, $i,j = 1$ to $N$) are displayed in Fig. 1. The simulation started from a spatially homogeneous initial condition $\mathbf{u}(\mathbf{r},0) = \lbrace 0.2, 0.2, 0.2,$ $ 0.2, 0.2 \rbrace$, which was perturbed by a small amplitude of initial noise to induce phase separation. Note the coarsening of the various types of grains with time. To test whether spurious phase generation takes place, we have performed two $N = 5$ simulations with initial conditions: (1) $\mathbf{u}(\mathbf{r},0) = \lbrace 0.5, 0.5, 0, 0, 0 \rbrace$ and for (2) $\mathbf{u}(\mathbf{r},0) = \lbrace 0.25, 0.25, 0.25, 0.25, 0 \rbrace$. The individual phase-field maps are shown for $t = 250,000$ dimensionless time in Figs. 2 and 3. Apparently, $\sum u_j = 1$ is satisfied with a high accuracy \[criterion (i), see Figs. 2(c) and 3(f)\]. Furthermore, the zero amplitude fields retained accurately their zero amplitude status throughout the simulations, i.e., no third phase generation took place at the phase boundaries, indicating that, in agreement with the expectations, criterion (ii) is also fully satisfied. While in the effectively two-component case (Fig. 2), we observe the usual binary phase separation pattern, the patterns appearing in Figs. 1 and 3 are significantly different: multi-grain networks appear that are dominated exclusively by trijunctions and binary boundaries; higher-order junctions have not been observed at all. (a)![\[fig:timeevol\] (Color online) $N = 4$ asymmetric Cahn-Hilliard problem with initial condition $\mathbf{u} = \lbrace 1/3, 1/3, 1/3, 0 \rbrace$. The results on the left were obtained with a Lagrangian mobility matrix, whereas those on the right were obtained using the Bollada-Jimack-Mullis type mobility matrix. The upper four rows \[panels (a) – (h)\] show the maps for $u_1, u_2, u_3$ and $u_4$, respectively, whereas in the lowermost row \[panels (i) and (j)\] phase distribution maps are displayed. Results corresponding to $t = 10^6$ dimensionless time are presented. Note that using the Lagrangian mobility matrix spurious phase generation in the vicinity of the phase boundaries could not be avoided for $u_4$ \[see panel (g)\], whereas the Bollada-Jimack-Mullis mobility matrix suppresses the formation of spurious phases entirely \[see panel (h)\].](fig4a.png "fig:"){width="0.44\linewidth"} (b)![\[fig:timeevol\] (Color online) $N = 4$ asymmetric Cahn-Hilliard problem with initial condition $\mathbf{u} = \lbrace 1/3, 1/3, 1/3, 0 \rbrace$. The results on the left were obtained with a Lagrangian mobility matrix, whereas those on the right were obtained using the Bollada-Jimack-Mullis type mobility matrix. The upper four rows \[panels (a) – (h)\] show the maps for $u_1, u_2, u_3$ and $u_4$, respectively, whereas in the lowermost row \[panels (i) and (j)\] phase distribution maps are displayed. Results corresponding to $t = 10^6$ dimensionless time are presented. Note that using the Lagrangian mobility matrix spurious phase generation in the vicinity of the phase boundaries could not be avoided for $u_4$ \[see panel (g)\], whereas the Bollada-Jimack-Mullis mobility matrix suppresses the formation of spurious phases entirely \[see panel (h)\].](fig4b.png "fig:"){width="0.44\linewidth"}\ (c)![\[fig:timeevol\] (Color online) $N = 4$ asymmetric Cahn-Hilliard problem with initial condition $\mathbf{u} = \lbrace 1/3, 1/3, 1/3, 0 \rbrace$. The results on the left were obtained with a Lagrangian mobility matrix, whereas those on the right were obtained using the Bollada-Jimack-Mullis type mobility matrix. The upper four rows \[panels (a) – (h)\] show the maps for $u_1, u_2, u_3$ and $u_4$, respectively, whereas in the lowermost row \[panels (i) and (j)\] phase distribution maps are displayed. Results corresponding to $t = 10^6$ dimensionless time are presented. Note that using the Lagrangian mobility matrix spurious phase generation in the vicinity of the phase boundaries could not be avoided for $u_4$ \[see panel (g)\], whereas the Bollada-Jimack-Mullis mobility matrix suppresses the formation of spurious phases entirely \[see panel (h)\].](fig4c.png "fig:"){width="0.44\linewidth"} (d)![\[fig:timeevol\] (Color online) $N = 4$ asymmetric Cahn-Hilliard problem with initial condition $\mathbf{u} = \lbrace 1/3, 1/3, 1/3, 0 \rbrace$. The results on the left were obtained with a Lagrangian mobility matrix, whereas those on the right were obtained using the Bollada-Jimack-Mullis type mobility matrix. The upper four rows \[panels (a) – (h)\] show the maps for $u_1, u_2, u_3$ and $u_4$, respectively, whereas in the lowermost row \[panels (i) and (j)\] phase distribution maps are displayed. Results corresponding to $t = 10^6$ dimensionless time are presented. Note that using the Lagrangian mobility matrix spurious phase generation in the vicinity of the phase boundaries could not be avoided for $u_4$ \[see panel (g)\], whereas the Bollada-Jimack-Mullis mobility matrix suppresses the formation of spurious phases entirely \[see panel (h)\].](fig4d.png "fig:"){width="0.44\linewidth"}\ (e)![\[fig:timeevol\] (Color online) $N = 4$ asymmetric Cahn-Hilliard problem with initial condition $\mathbf{u} = \lbrace 1/3, 1/3, 1/3, 0 \rbrace$. The results on the left were obtained with a Lagrangian mobility matrix, whereas those on the right were obtained using the Bollada-Jimack-Mullis type mobility matrix. The upper four rows \[panels (a) – (h)\] show the maps for $u_1, u_2, u_3$ and $u_4$, respectively, whereas in the lowermost row \[panels (i) and (j)\] phase distribution maps are displayed. Results corresponding to $t = 10^6$ dimensionless time are presented. Note that using the Lagrangian mobility matrix spurious phase generation in the vicinity of the phase boundaries could not be avoided for $u_4$ \[see panel (g)\], whereas the Bollada-Jimack-Mullis mobility matrix suppresses the formation of spurious phases entirely \[see panel (h)\].](fig4e.png "fig:"){width="0.44\linewidth"} (f)![\[fig:timeevol\] (Color online) $N = 4$ asymmetric Cahn-Hilliard problem with initial condition $\mathbf{u} = \lbrace 1/3, 1/3, 1/3, 0 \rbrace$. The results on the left were obtained with a Lagrangian mobility matrix, whereas those on the right were obtained using the Bollada-Jimack-Mullis type mobility matrix. The upper four rows \[panels (a) – (h)\] show the maps for $u_1, u_2, u_3$ and $u_4$, respectively, whereas in the lowermost row \[panels (i) and (j)\] phase distribution maps are displayed. Results corresponding to $t = 10^6$ dimensionless time are presented. Note that using the Lagrangian mobility matrix spurious phase generation in the vicinity of the phase boundaries could not be avoided for $u_4$ \[see panel (g)\], whereas the Bollada-Jimack-Mullis mobility matrix suppresses the formation of spurious phases entirely \[see panel (h)\].](fig4f.png "fig:"){width="0.44\linewidth"}\ (g)![\[fig:timeevol\] (Color online) $N = 4$ asymmetric Cahn-Hilliard problem with initial condition $\mathbf{u} = \lbrace 1/3, 1/3, 1/3, 0 \rbrace$. The results on the left were obtained with a Lagrangian mobility matrix, whereas those on the right were obtained using the Bollada-Jimack-Mullis type mobility matrix. The upper four rows \[panels (a) – (h)\] show the maps for $u_1, u_2, u_3$ and $u_4$, respectively, whereas in the lowermost row \[panels (i) and (j)\] phase distribution maps are displayed. Results corresponding to $t = 10^6$ dimensionless time are presented. Note that using the Lagrangian mobility matrix spurious phase generation in the vicinity of the phase boundaries could not be avoided for $u_4$ \[see panel (g)\], whereas the Bollada-Jimack-Mullis mobility matrix suppresses the formation of spurious phases entirely \[see panel (h)\].](fig4g.png "fig:"){width="0.44\linewidth"} (h)![\[fig:timeevol\] (Color online) $N = 4$ asymmetric Cahn-Hilliard problem with initial condition $\mathbf{u} = \lbrace 1/3, 1/3, 1/3, 0 \rbrace$. The results on the left were obtained with a Lagrangian mobility matrix, whereas those on the right were obtained using the Bollada-Jimack-Mullis type mobility matrix. The upper four rows \[panels (a) – (h)\] show the maps for $u_1, u_2, u_3$ and $u_4$, respectively, whereas in the lowermost row \[panels (i) and (j)\] phase distribution maps are displayed. Results corresponding to $t = 10^6$ dimensionless time are presented. Note that using the Lagrangian mobility matrix spurious phase generation in the vicinity of the phase boundaries could not be avoided for $u_4$ \[see panel (g)\], whereas the Bollada-Jimack-Mullis mobility matrix suppresses the formation of spurious phases entirely \[see panel (h)\].](fig4h.png "fig:"){width="0.44\linewidth"}\ (i)![\[fig:timeevol\] (Color online) $N = 4$ asymmetric Cahn-Hilliard problem with initial condition $\mathbf{u} = \lbrace 1/3, 1/3, 1/3, 0 \rbrace$. The results on the left were obtained with a Lagrangian mobility matrix, whereas those on the right were obtained using the Bollada-Jimack-Mullis type mobility matrix. The upper four rows \[panels (a) – (h)\] show the maps for $u_1, u_2, u_3$ and $u_4$, respectively, whereas in the lowermost row \[panels (i) and (j)\] phase distribution maps are displayed. Results corresponding to $t = 10^6$ dimensionless time are presented. Note that using the Lagrangian mobility matrix spurious phase generation in the vicinity of the phase boundaries could not be avoided for $u_4$ \[see panel (g)\], whereas the Bollada-Jimack-Mullis mobility matrix suppresses the formation of spurious phases entirely \[see panel (h)\].](fig4i.png "fig:"){width="0.44\linewidth"} (j)![\[fig:timeevol\] (Color online) $N = 4$ asymmetric Cahn-Hilliard problem with initial condition $\mathbf{u} = \lbrace 1/3, 1/3, 1/3, 0 \rbrace$. The results on the left were obtained with a Lagrangian mobility matrix, whereas those on the right were obtained using the Bollada-Jimack-Mullis type mobility matrix. The upper four rows \[panels (a) – (h)\] show the maps for $u_1, u_2, u_3$ and $u_4$, respectively, whereas in the lowermost row \[panels (i) and (j)\] phase distribution maps are displayed. Results corresponding to $t = 10^6$ dimensionless time are presented. Note that using the Lagrangian mobility matrix spurious phase generation in the vicinity of the phase boundaries could not be avoided for $u_4$ \[see panel (g)\], whereas the Bollada-Jimack-Mullis mobility matrix suppresses the formation of spurious phases entirely \[see panel (h)\].](fig4j.png "fig:"){width="0.44\linewidth"} ### Asymmetric case Here, the parameters $\epsilon_{ij}^2$ and $w_{ij} $ are different for the individual binary interfaces (for the matrices used in the present work see Ref. [@matrices]). The simulations were performed for an $N = 4$ Cahn-Hilliard model. $\mathbf{u}(\mathbf{r},0) =$ $ \lbrace 1/3, 1/3, 1/3, 0\rbrace$ has been chosen as the initial condition, with a small amplitude of initial noise to induce phase separation. First a Lagrangian mobility matrix has been used. The phase-field maps corresponding to $t = 10^6$ dimensionless time are displayed in the left column of Fig. 4. Apparently, the solution is less satisfactory than the results for the symmetric case: substantial deviation from $u_4(\mathbf{r},t) = 0$ is observed \[Fig. 4(g)\]. However, this spurious phase generation disappears entirely, if the Bollada-Jimack-Mullis mobility matrix is used \[see Fig. 4(h)\], as expected. Again, the multiphase domain structure is dominated by trijunctions and binary boundaries at all times. It appears though that the structure obtained with the Bollada-Jimack-Mullis type mobility matrix contains chains of alternating $u_1$ and $u_4$ “bubbles” \[see Fig. 4(j)\], a feature that can be associated with the asymmetry of the kinetic coefficients ($\kappa_{ij}$) applied in this simulation. (a)![\[fig:timeevol\] (Color online) $N = 4$ asymmetric Cahn-Hilliard problem with anisotropy: (a) Time evolution towards the equilibrium shapes of $u_1$ and $u_2$ embedded into $u_3$ at $t=3.125\times10^6$ dimensionless time. (b)-(e): Snapshots of the phase fields $u_1, u_2, u_3,$ and $u_4$, respectively, taken at $t = 2\times10^5$ dimensionless time. Note the lack of third phase generation at the phase boundaries, and that $u_4$ remains absent even at the phase-boundaries.](fig5a.png "fig:"){width="0.9\linewidth"} (b)![\[fig:timeevol\] (Color online) $N = 4$ asymmetric Cahn-Hilliard problem with anisotropy: (a) Time evolution towards the equilibrium shapes of $u_1$ and $u_2$ embedded into $u_3$ at $t=3.125\times10^6$ dimensionless time. (b)-(e): Snapshots of the phase fields $u_1, u_2, u_3,$ and $u_4$, respectively, taken at $t = 2\times10^5$ dimensionless time. Note the lack of third phase generation at the phase boundaries, and that $u_4$ remains absent even at the phase-boundaries.](fig5b.png "fig:"){width="0.9\linewidth"} (c)![\[fig:timeevol\] (Color online) $N = 4$ asymmetric Cahn-Hilliard problem with anisotropy: (a) Time evolution towards the equilibrium shapes of $u_1$ and $u_2$ embedded into $u_3$ at $t=3.125\times10^6$ dimensionless time. (b)-(e): Snapshots of the phase fields $u_1, u_2, u_3,$ and $u_4$, respectively, taken at $t = 2\times10^5$ dimensionless time. Note the lack of third phase generation at the phase boundaries, and that $u_4$ remains absent even at the phase-boundaries.](fig5c.png "fig:"){width="0.9\linewidth"} (d)![\[fig:timeevol\] (Color online) $N = 4$ asymmetric Cahn-Hilliard problem with anisotropy: (a) Time evolution towards the equilibrium shapes of $u_1$ and $u_2$ embedded into $u_3$ at $t=3.125\times10^6$ dimensionless time. (b)-(e): Snapshots of the phase fields $u_1, u_2, u_3,$ and $u_4$, respectively, taken at $t = 2\times10^5$ dimensionless time. Note the lack of third phase generation at the phase boundaries, and that $u_4$ remains absent even at the phase-boundaries.](fig5d.png "fig:"){width="0.9\linewidth"} (e)![\[fig:timeevol\] (Color online) $N = 4$ asymmetric Cahn-Hilliard problem with anisotropy: (a) Time evolution towards the equilibrium shapes of $u_1$ and $u_2$ embedded into $u_3$ at $t=3.125\times10^6$ dimensionless time. (b)-(e): Snapshots of the phase fields $u_1, u_2, u_3,$ and $u_4$, respectively, taken at $t = 2\times10^5$ dimensionless time. Note the lack of third phase generation at the phase boundaries, and that $u_4$ remains absent even at the phase-boundaries.](fig5e.png "fig:"){width="0.9\linewidth"} ### Anisotropic case Next, we investigated an asymmetric $N = 4$ Cahn-Hilliard theory [@matrices], however, with anisotropic interface free energies and a Bollada-Jimack-Mullis type mobility matrix. The dynamic equations were solved on a rectangular grid of size $1024 \times 512$. Time and spatial steps of $\Delta t = 0.25$ and $\Delta x = 0.5$ were used. The starting conditions were as follows: Two circles filled with $u_1 = 1$ (left) and $u_2 = 1$ (right) were placed besides each other, while a zero value was assigned to these fields outside the circles. In contrast, $u_3 = 1$ was prescribed in the background, and $u_3 = 0$ inside the circles, whereas $u_4 = 0$ was assigned to the whole simulation domain (i.e., the fourth field was missing everywhere). All interfaces were assumed isotropic, except for the 1–3 interface, for which an anisotropy of $a_{13} = 0.1$ was prescribed that is larger than the critical anisotropy $a_c = 1/(2^k - 1) = 1/15$ for fourfold ($k =4$) symmetry [@Kobayashi2001; @Eggleston2001]. With elapsing time, the circle on the left evolved into a square-like object of curved sides, and four pointed corners (see Fig. 5), displaying missing orientations (following from $a_{13} > a_c$), as expected on the basis of the prescribed anisotropy function. Apparently, as found for the central finite differencing scheme [@Eggleston2001], the spectral discretization regularized the high anisotropy problem: the predicted numerical shape is very close to the analytical solution corresponding to this anisotropy. Remarkably, no spurious phase appearance was observed at the phase-boundaries, and $u_4 = 0$ has been satisfied throughout the simulation. We have obtained similar results using finite difference discretization. ![[(Color online) Snapshot of grain/phase-map taken at dimensionless rime, $t = 10^4$, for a simulation relying on misorientation dependent grain boundary energy obeying the Read-Shockley relationship[@ReadShockley1950]. The simulation was performed on a $4096^2$ rectangular grid. About 4000 grains can be distinguished at this stage.]{}](fig6.png){width="1.0\linewidth"} ![[(Color online) Limiting grain size distributions (LGSD): (a) for the isotropic simulation on a $8192^2$ grid. For comparison, experimental results[@Barmak2013] for metallic films (solid line: lognormal distribution fitted to experimental data[@Barmak2013]), and predictions by the theories of Mullins[@Mullins1956] and Hillert[@Hillert1965] (dashed and dotted lines, respectively) are also shown. (b) Comparison of the LGSD from XMPF (histogram) with predictions from previous MPFs by Kim [*et al.*]{}[@Kim2006] (triangles) and Schaffnit [*et al.*]{}[@Schaffnit2007] (circles). The solid line indicates a lognormal fit to the experimental results[@Barmak2013]. (c) The same as panel (b), except that LGSD from an asymmetric XMPF model (the simulation shown in Fig. 6) is displayed (histogram), in which the grain boundary energy follows the Read-Shockley relationship[@ReadShockley1950]. Note that some improvement relative to the previous MPF models has been achieved, but the population of the small grains is larger than desirable.]{}](fig7a.png "fig:"){width="1.0\linewidth"} ![[(Color online) Limiting grain size distributions (LGSD): (a) for the isotropic simulation on a $8192^2$ grid. For comparison, experimental results[@Barmak2013] for metallic films (solid line: lognormal distribution fitted to experimental data[@Barmak2013]), and predictions by the theories of Mullins[@Mullins1956] and Hillert[@Hillert1965] (dashed and dotted lines, respectively) are also shown. (b) Comparison of the LGSD from XMPF (histogram) with predictions from previous MPFs by Kim [*et al.*]{}[@Kim2006] (triangles) and Schaffnit [*et al.*]{}[@Schaffnit2007] (circles). The solid line indicates a lognormal fit to the experimental results[@Barmak2013]. (c) The same as panel (b), except that LGSD from an asymmetric XMPF model (the simulation shown in Fig. 6) is displayed (histogram), in which the grain boundary energy follows the Read-Shockley relationship[@ReadShockley1950]. Note that some improvement relative to the previous MPF models has been achieved, but the population of the small grains is larger than desirable.]{}](fig7b.png "fig:"){width="1.0\linewidth"} ![[(Color online) Limiting grain size distributions (LGSD): (a) for the isotropic simulation on a $8192^2$ grid. For comparison, experimental results[@Barmak2013] for metallic films (solid line: lognormal distribution fitted to experimental data[@Barmak2013]), and predictions by the theories of Mullins[@Mullins1956] and Hillert[@Hillert1965] (dashed and dotted lines, respectively) are also shown. (b) Comparison of the LGSD from XMPF (histogram) with predictions from previous MPFs by Kim [*et al.*]{}[@Kim2006] (triangles) and Schaffnit [*et al.*]{}[@Schaffnit2007] (circles). The solid line indicates a lognormal fit to the experimental results[@Barmak2013]. (c) The same as panel (b), except that LGSD from an asymmetric XMPF model (the simulation shown in Fig. 6) is displayed (histogram), in which the grain boundary energy follows the Read-Shockley relationship[@ReadShockley1950]. Note that some improvement relative to the previous MPF models has been achieved, but the population of the small grains is larger than desirable.]{}](fig7c.png "fig:"){width="1.0\linewidth"} Grain coarsening ---------------- In this subsection, we apply the XMPF model for grain coarsening in a two-dimensional (2D) polycrystalline system that contains a large number of differently oriented crystal grains that have equal free energy, therefore, the time evolution of the system is driven by the grain boundary energy. For the sake of simplicity, we distinguish only 30 orientations represented by $N=30$ fields. The respective non-conservative equations of motions read as: $$-\frac{\partial u_i}{\partial t} = \sum_{j\neq i}\kappa_{ij} \left(\frac{\delta F}{\delta u_i}-\frac{\delta F}{\delta u_j} \right) \enskip ,$$ which have been solved numerically on a rectangular grid, using a finite difference discretization and Euler forward time-stepping. As starting condition, the simulation box was covered by a large number of small random field patches arranged on a square lattice ($512\times512$ and $256\times256$, respectively, for the larger and smaller size simulations), mimicking athermal nucleation on a fine grid. \[We note that, during time evolution, the initial condition is fast forgotten: for example, after a transient period, very similar results were obtained with a uniform $1/N$ starting, while adding a small pixelwise Gaussian noise (a spatially random initiation).\] Two cases were investigated: (a) with an isotropic grain boundary energy (on a $8192^2$ grid), and (b) the misorientation dependence of the grain boundary energy follows the Read-Shockley relationship [@ReadShockley1950] (on a $4096^2$ grid). The corresponding mobilities were $\kappa_{ij}=1$, and $\kappa_{ij}=|u_i/(1-u_i)|\,|u_j/(1-u_j)|$, respectively. A typical grain map displaying the result of the simulation for case (b) at a dimensionless time $t=10^4$, when $\sim4000$ grains exist, is shown in Fig. 6. Similar grain maps have been obtained for the other case, except that there most of the trijunctions are close to symmetric, displaying angles $\sim120^{\circ}$. As observed in other MPF models[@Kim2006; @Schaffnit2007], in the experiments[@Barmak2013], and predicted by theory[@Feltham1950; @Mullins1956; @Hillert1965], after a transient period a limiting grain size distribution (LGSD) is established, which in the case of experiments on metallic film can be accurately fitted[@Barmak2013] by the lognormal distribution proposed by Feltham[@Feltham1950]. The models of Mullins and Hillert predict significantly different LGSDs (Fig. 7). The LGSD predicted by the XMPF model approximates the experimental results somewhat better than the previous MPF models[@Kim2006; @Schaffnit2007] \[see Figs. 7(b) and 7(c)\], and practically coincides with the results from the [*mutli order parameter*]{} approaches[@ChenYang1994; @FanChen1997a; @Moelans2014], yet the agreement is not particularly good with the experiments at small grain sizes. Apparently, in the experiments the small grains disappear faster than in the XMPF simulations. In the investigated cases, the time dependence of the average grain size \[$\langle r \rangle = A (t-t_0)^q$, where $A$ is a constant and $t_0$ the freezing time\] is described by an exponent $q = 0.5 (1 \pm 0.05)$, indicating an essentially diffusion controlled grain growth. We mention in this respect that a simple dynamical density functional theory, the Phase-Field Crystal approach[@Elder2002; @Emmerich2012], which incorporates a broad range of physical phenomena (elasticity, dislocation dynamics, grain rotation, etc.), reproduces the experimental LGSD fairly well[@Backofen2014]. Unfortunately, in the PFC studies, as in the case of experiments, the effect of different physical phenomena on the LGSD cannot be easily separated. It is expected, however, that the comparison of different models may contribute to the identification of the governing phenomena. Along these lines, the present study determined the LGSD the physically consistent XMPF model predicts. Apparently, further efforts are needed to improve the agreement between MPF models and experiments. Work is, underway[@Korbuly2015] to evaluate LGSD from phase-field models relying on orientation field(s)[@Warren2003; @Plapp2012; @Granasy2014] in describing different crystallographic orientations. Comparison with other models ============================ Having presented the essential properties of the XMPF model, it is desirable to compare it with other models from the following viewpoints:\ A. For a few of the most important multiphase-field models, we investigate whether the trivial $N = 3$ extension of the equilibrium binary solution is a stationary solution of the $N = 3$ dynamic problem \[part of criterion (vi), which in addition requires that the trivial extension be a solution of the $N = 3$ Euler-Lagrange equation too\].\ B. We explore, furthermore, for the best behaving models identified in sub-section A. whether the free energy decreases indeed monotonically with time \[criterion (v)\].\ (a)![\[fig:planar\_sin\] (Color online) $u(z)=[1+\sin(z)]/2$ models. (a) Initial condition (top panel), (b)-(g) final states after $10^6$ time steps. From second to bottom row, respectively: the model of Nestler and Wheeler $p=1$ [@Nest2; @Nest4; @Nest5], the model of Steinbach and Pezzola [@SteinbachPezzola1999], and the model of Steinbach and Pezzola with non-variational dynamics [@Steinbach2009]. (Left column: final states, right column: difference of final state and initial condition.)](fig8a.pdf "fig:"){width="0.9\linewidth"}\ (b)![\[fig:planar\_sin\] (Color online) $u(z)=[1+\sin(z)]/2$ models. (a) Initial condition (top panel), (b)-(g) final states after $10^6$ time steps. From second to bottom row, respectively: the model of Nestler and Wheeler $p=1$ [@Nest2; @Nest4; @Nest5], the model of Steinbach and Pezzola [@SteinbachPezzola1999], and the model of Steinbach and Pezzola with non-variational dynamics [@Steinbach2009]. (Left column: final states, right column: difference of final state and initial condition.)](fig8b.pdf "fig:"){width="0.44\linewidth"} (c)![\[fig:planar\_sin\] (Color online) $u(z)=[1+\sin(z)]/2$ models. (a) Initial condition (top panel), (b)-(g) final states after $10^6$ time steps. From second to bottom row, respectively: the model of Nestler and Wheeler $p=1$ [@Nest2; @Nest4; @Nest5], the model of Steinbach and Pezzola [@SteinbachPezzola1999], and the model of Steinbach and Pezzola with non-variational dynamics [@Steinbach2009]. (Left column: final states, right column: difference of final state and initial condition.)](fig8c.pdf "fig:"){width="0.44\linewidth"}\ (d)![\[fig:planar\_sin\] (Color online) $u(z)=[1+\sin(z)]/2$ models. (a) Initial condition (top panel), (b)-(g) final states after $10^6$ time steps. From second to bottom row, respectively: the model of Nestler and Wheeler $p=1$ [@Nest2; @Nest4; @Nest5], the model of Steinbach and Pezzola [@SteinbachPezzola1999], and the model of Steinbach and Pezzola with non-variational dynamics [@Steinbach2009]. (Left column: final states, right column: difference of final state and initial condition.)](fig8d.pdf "fig:"){width="0.44\linewidth"} (e)![\[fig:planar\_sin\] (Color online) $u(z)=[1+\sin(z)]/2$ models. (a) Initial condition (top panel), (b)-(g) final states after $10^6$ time steps. From second to bottom row, respectively: the model of Nestler and Wheeler $p=1$ [@Nest2; @Nest4; @Nest5], the model of Steinbach and Pezzola [@SteinbachPezzola1999], and the model of Steinbach and Pezzola with non-variational dynamics [@Steinbach2009]. (Left column: final states, right column: difference of final state and initial condition.)](fig8e.pdf "fig:"){width="0.44\linewidth"}\ (f)![\[fig:planar\_sin\] (Color online) $u(z)=[1+\sin(z)]/2$ models. (a) Initial condition (top panel), (b)-(g) final states after $10^6$ time steps. From second to bottom row, respectively: the model of Nestler and Wheeler $p=1$ [@Nest2; @Nest4; @Nest5], the model of Steinbach and Pezzola [@SteinbachPezzola1999], and the model of Steinbach and Pezzola with non-variational dynamics [@Steinbach2009]. (Left column: final states, right column: difference of final state and initial condition.)](fig8f.pdf "fig:"){width="0.44\linewidth"} (g)![\[fig:planar\_sin\] (Color online) $u(z)=[1+\sin(z)]/2$ models. (a) Initial condition (top panel), (b)-(g) final states after $10^6$ time steps. From second to bottom row, respectively: the model of Nestler and Wheeler $p=1$ [@Nest2; @Nest4; @Nest5], the model of Steinbach and Pezzola [@SteinbachPezzola1999], and the model of Steinbach and Pezzola with non-variational dynamics [@Steinbach2009]. (Left column: final states, right column: difference of final state and initial condition.)](fig8g.pdf "fig:"){width="0.44\linewidth"}\ (a)![\[fig:planar\_tanh\] (Color online) $u(z)=[1+\tanh(z/2)]/2$ models. (a) Initial condition (top panel), (b)-(g) final states after $2.5 \times 10^5$ time steps. From second to bottom row, respectively: the model of Steinbach [*et al.*]{} [@Steinbach1996] (coincides with the model of Nestler and Wheeler for $p=2$ [@Nest1; @Nest2; @Nest3; @Nest4; @Nest5]), the model of Steinbach [*et al.*]{} with non-variational dynamics [@Steinbach1996], and the XMPF model proposed in this work. (Left column: final states, right column: difference of final state and initial condition.)](fig9a.pdf "fig:"){width="0.9\linewidth"} (b)![\[fig:planar\_tanh\] (Color online) $u(z)=[1+\tanh(z/2)]/2$ models. (a) Initial condition (top panel), (b)-(g) final states after $2.5 \times 10^5$ time steps. From second to bottom row, respectively: the model of Steinbach [*et al.*]{} [@Steinbach1996] (coincides with the model of Nestler and Wheeler for $p=2$ [@Nest1; @Nest2; @Nest3; @Nest4; @Nest5]), the model of Steinbach [*et al.*]{} with non-variational dynamics [@Steinbach1996], and the XMPF model proposed in this work. (Left column: final states, right column: difference of final state and initial condition.)](fig9b.pdf "fig:"){width="0.44\linewidth"} (c)![\[fig:planar\_tanh\] (Color online) $u(z)=[1+\tanh(z/2)]/2$ models. (a) Initial condition (top panel), (b)-(g) final states after $2.5 \times 10^5$ time steps. From second to bottom row, respectively: the model of Steinbach [*et al.*]{} [@Steinbach1996] (coincides with the model of Nestler and Wheeler for $p=2$ [@Nest1; @Nest2; @Nest3; @Nest4; @Nest5]), the model of Steinbach [*et al.*]{} with non-variational dynamics [@Steinbach1996], and the XMPF model proposed in this work. (Left column: final states, right column: difference of final state and initial condition.)](fig9c.pdf "fig:"){width="0.44\linewidth"}\ (d)![\[fig:planar\_tanh\] (Color online) $u(z)=[1+\tanh(z/2)]/2$ models. (a) Initial condition (top panel), (b)-(g) final states after $2.5 \times 10^5$ time steps. From second to bottom row, respectively: the model of Steinbach [*et al.*]{} [@Steinbach1996] (coincides with the model of Nestler and Wheeler for $p=2$ [@Nest1; @Nest2; @Nest3; @Nest4; @Nest5]), the model of Steinbach [*et al.*]{} with non-variational dynamics [@Steinbach1996], and the XMPF model proposed in this work. (Left column: final states, right column: difference of final state and initial condition.)](fig9d.pdf "fig:"){width="0.44\linewidth"} (e)![\[fig:planar\_tanh\] (Color online) $u(z)=[1+\tanh(z/2)]/2$ models. (a) Initial condition (top panel), (b)-(g) final states after $2.5 \times 10^5$ time steps. From second to bottom row, respectively: the model of Steinbach [*et al.*]{} [@Steinbach1996] (coincides with the model of Nestler and Wheeler for $p=2$ [@Nest1; @Nest2; @Nest3; @Nest4; @Nest5]), the model of Steinbach [*et al.*]{} with non-variational dynamics [@Steinbach1996], and the XMPF model proposed in this work. (Left column: final states, right column: difference of final state and initial condition.)](fig9e.pdf "fig:"){width="0.44\linewidth"}\ (f)![\[fig:planar\_tanh\] (Color online) $u(z)=[1+\tanh(z/2)]/2$ models. (a) Initial condition (top panel), (b)-(g) final states after $2.5 \times 10^5$ time steps. From second to bottom row, respectively: the model of Steinbach [*et al.*]{} [@Steinbach1996] (coincides with the model of Nestler and Wheeler for $p=2$ [@Nest1; @Nest2; @Nest3; @Nest4; @Nest5]), the model of Steinbach [*et al.*]{} with non-variational dynamics [@Steinbach1996], and the XMPF model proposed in this work. (Left column: final states, right column: difference of final state and initial condition.)](fig9f.pdf "fig:"){width="0.44\linewidth"} (g)![\[fig:planar\_tanh\] (Color online) $u(z)=[1+\tanh(z/2)]/2$ models. (a) Initial condition (top panel), (b)-(g) final states after $2.5 \times 10^5$ time steps. From second to bottom row, respectively: the model of Steinbach [*et al.*]{} [@Steinbach1996] (coincides with the model of Nestler and Wheeler for $p=2$ [@Nest1; @Nest2; @Nest3; @Nest4; @Nest5]), the model of Steinbach [*et al.*]{} with non-variational dynamics [@Steinbach1996], and the XMPF model proposed in this work. (Left column: final states, right column: difference of final state and initial condition.)](fig9g.pdf "fig:"){width="0.44\linewidth"}\ Planar interfaces:\ comparing the MPF models ------------------------ Here, we investigate for several MPF models, whether a trivial $N = 3$ extension of the equilibrium interface of the two-phase problem (obtained by adding $u_3 = 0$ to the two-phase equilibrium solution) behaves like a local free energy minimum of the multiphase-field model: starting from the extended solution, we explore whether the $N = 3$ equations of motion keep the solution equal to the starting condition, or drive it away. For this test, we adopt non-conservative dynamics $$-\frac{\partial u_i}{\partial t} = \sum_{j \neq i} \left( \frac{\delta F}{\delta u_i}-\frac{\delta F}{\delta u_j} \right) \enskip ,$$ The initial condition is a liquid-solid-liquid slab, with periodic boundary condition at the two ends, while employing the respective analytic solutions in the interface regions, accompanied with $u_3(z) = 0$ throughout the computation box. (See Figs. 8(a) and 9(a) for the initial conditions used for the models that have binary equilibrium solutions of the forms $u(z)=[1+\sin(z)]/2$ and $u(z)=[1+\tanh(z/2)]/2$, respectively).\ The one-dimensional dynamic equations were solved numerically, using finite difference method, while employing dimensionless time and spatial steps, $h$ and $\Delta t$, as specified below.\ ### Models with sinusoidal equilibrium profile The long-time solutions of the dynamic equations ($10^6$ time steps, beyond which no further changes were perceptible) are shown in Fig. 8 for the Nestler-Wheeler $p=1$ model [@Nest2; @Nest4; @Nest5]\[Figs. 8(b) and 8(c)\], for the Steinbach–Pezzola model [@SteinbachPezzola1999] \[Figs. 8(d) and 8(e)\], and for the Steinbach–Pezzola model with non-variational dynamics [@Steinbach2009] \[Figs. 8(f) and 8(g)\] ($h = 0.1$ and $\Delta t = 0.0025$ were used.) The long-time interfacial field profiles are shown on the left \[Figs. 8(b), 8(d), and 8(f)\], together with their difference relative to the initial conditions on the right \[Figs. 8(c), 8(e), and 8(g)\]. While in the first and second cases, third-phase generation can be seen at the interface, the application of non-variational dynamics in the Pezzola-Steinbach model suppressed this phenomenon ($u_3 \approx 0$ was retained). ### Models with hyperbolic tangential equilibrium profile The long-time solutions ($2.5 \times 10^5$ time steps, beyond which no perceptible changes were seen) of the EOMs are shown in Fig. 9 for the following models: Figs. 9(b) and 9(c) – the model of Steinbach [*et al.*]{} [@Steinbach1996]; Figs. 9(d) and 9(e) – the model of Steinbach [*et al.*]{} [@Steinbach1996] with non-variational dynamics; Figs. 9(f) and 9(g) – the model proposed in the present paper. ($h = 0.25$ and $\Delta t = 0.01$.) The long-time interfacial field profiles are shown on the left \[Figs. 9(b), 9(d), and 9(f)\], together with their difference relative to the initial conditions on the right \[Figs. 9(c), 9(e), and 9(g)\]. While the model of Steinbach [*et al.*]{} [@Steinbach1996] leads to third-phase generation, the other two approaches are free of this problem. Remarkably, the predictions from the latter two models fall very close to each other. Yet, in the model of Steinbach [*et al.*]{} [@Steinbach1996], the trivial three-phase extension of the binary equilibrium solution is not a solution of the three-phase Euler-Lagrange equation (see Section III.C). In other words: although the same solution is a stationary solution of the non-variational EOM, stabilized by the non-variational dynamics, it is not a free energy minimum of the three-phase problem. While in this test, the results of the model of Steinbach [*et al.*]{} (with non-variational dynamics) are practically indistinguishable from those of the XMPF model proposed in this work, under other conditions significant differences can be seen. (a)![\[fig:energy\] (Color online) Comparison of the time evolution of phase transition (a), (c) in the non-variational model of Steinbach [*et al.*]{} [@Steinbach1996] (left), and (b), (d) in the present model (right). Snapshots of one of the fields are displayed. The time dependence of the free energy is shown in panel (e): dashed line – model of Steinbach [*et al.*]{}; solid line – the XMPF model. The symbols correspond to snapshots shown in panels (a)-(d).](fig10a.png "fig:"){width="0.43\linewidth"} (b)![\[fig:energy\] (Color online) Comparison of the time evolution of phase transition (a), (c) in the non-variational model of Steinbach [*et al.*]{} [@Steinbach1996] (left), and (b), (d) in the present model (right). Snapshots of one of the fields are displayed. The time dependence of the free energy is shown in panel (e): dashed line – model of Steinbach [*et al.*]{}; solid line – the XMPF model. The symbols correspond to snapshots shown in panels (a)-(d).](fig10b.png "fig:"){width="0.43\linewidth"}\ (c)![\[fig:energy\] (Color online) Comparison of the time evolution of phase transition (a), (c) in the non-variational model of Steinbach [*et al.*]{} [@Steinbach1996] (left), and (b), (d) in the present model (right). Snapshots of one of the fields are displayed. The time dependence of the free energy is shown in panel (e): dashed line – model of Steinbach [*et al.*]{}; solid line – the XMPF model. The symbols correspond to snapshots shown in panels (a)-(d).](fig10c.png "fig:"){width="0.43\linewidth"} (d)![\[fig:energy\] (Color online) Comparison of the time evolution of phase transition (a), (c) in the non-variational model of Steinbach [*et al.*]{} [@Steinbach1996] (left), and (b), (d) in the present model (right). Snapshots of one of the fields are displayed. The time dependence of the free energy is shown in panel (e): dashed line – model of Steinbach [*et al.*]{}; solid line – the XMPF model. The symbols correspond to snapshots shown in panels (a)-(d).](fig10d.png "fig:"){width="0.43\linewidth"}\ (e)![\[fig:energy\] (Color online) Comparison of the time evolution of phase transition (a), (c) in the non-variational model of Steinbach [*et al.*]{} [@Steinbach1996] (left), and (b), (d) in the present model (right). Snapshots of one of the fields are displayed. The time dependence of the free energy is shown in panel (e): dashed line – model of Steinbach [*et al.*]{}; solid line – the XMPF model. The symbols correspond to snapshots shown in panels (a)-(d).](fig10e.pdf "fig:"){width="0.9\linewidth"} Time dependence of the free energy ---------------------------------- In this test, we investigate the time evolution of the system in the non-variational model of Steinbach [*et al.*]{} [@Steinbach1996], and in the XMPF model presented in this work. A symmetric $N = 4$ Cahn-Hilliard model and Lagrangian mobility matrix have been chosen for the demonstration. The results are summarized in Fig. 10, which shows the map of one of the fields (the others are qualitatively similar) at dimensionless times $t = 2000$ and $20000$, computed on a grid $512 \times 512$. While in the XMPF model proposed here, the free energy decreases monotonically with time as expected, in the model of Steinbach [*et al.*]{} (when relying on a non-variational formalism), the free energy increases initially, reaching then a maximum at about $t = 3000$, followed by a slow decrease beyond. This behavior is presumably a consequence of the applied ’binary’ approximation, in which various terms of the variational equations of motion are omitted: once the free energy functional (Lyapunov functional) is defined, the variational dynamics ensures a monotonic reduction of the free energy with time. Any deviation from this approach raises the possibility of a non-monotonic time evolution of the free energy. Summary ======= In this work, we formulated a physically consistent multiphase-field theory for describing interface driven multi-domain processes. First, we identified a set of criteria, a physically consistent multiphase-field approach has to satisfy. These are: (i) the sum of the fields is 1 everywhere; (ii) the physical results should be invariant to exchanging pairs of field indices, $i \leftrightarrow j$; (iii) a trivial multiphase extension of the equilibrium binary solution should represent an equilibrium solution of the multiphase problem, which in turn should be a stationary solution of the dynamic equations towards which the time dependent solutions evolve; (iv) variational dynamic equations shall be used to ensure non-negative entropy production; (v) reduction/extension of the $N$-field theory to $N-1$ or $N+1$ fields should be straightforward, and happen consistently within the formalism; (vi) there should be no spurious third phase appearance at the equilibrium binary boundaries, and once a field is not present, it should not appear at any time in the dynamic equations; finally, (vii) freedom to choose the interfacial and kinetic properties for individual phase pairs. Next, considering these requirements, we have reviewed a range of the existing multiphase field models, and identified their advantageous and less advantageous features. Combining the advantageous features of the earlier multiphase-field models, we have constructed a multiphase-field approach (termed the XMPF model) obeying all criteria defined above. In addition, we performed illustrative simulations for $N = 4$ and 5 multiphase-field models that rely on conserved dynamics, describing thus multiphase separation problems ($N$-component Cahn-Hilliard problems). Symmetric (identical interface properties), asymmetric (pairwise different interface properties), and anisotropic (orientation dependent interfacial properties) cases were addressed, and it has been shown that using a suitable mobility matrix (Bollada-Jimack-Mullis type), the XMPF model avoids dynamic spurious phase generation. [We have performed further illustrative simulations for grain coarsening in polycrystalline systems using an $N=30$ XMPF model relying on non-conserved dynamics. While the predicted limiting grain size distribution is closer to the experimental results than those from the previous MPF models, further works is needed to improve the agreement.]{} The present work opens up the way towards physically consistent computations for microstructure evolution in multiphase / multigrain / multicomponent structures, and shall serve as a basis for developing a physically consistent [*quantitative*]{} multiphase-field approach that might be combined with melt flow and elasticity, and extended to fast processes along the lines described in Refs. [@Galenko1; @Archer2009; @Galenko2; @TothGranasyTegze2014], leading towards developing improved tools for knowledge based materials design. Work is underway to incorporate a phase-dependent thermodynamic driving force (a multiphase analogy of the ’tilting function’ in Ref. [@FolchPlapp2005]) into the XMPF model, which will be presented in a separate paper. [We note in this respect that the inclusion of thermodynamic driving force via a tilting function has no effect on the present results concerning the two- and multiphase equilibria. The existence of equilibrium two-phase planar interfaces in the multiphase problem is a basic requirement, which needs to be satisfied by a physically consistent model.]{} Appendix A1: Invariance of results to exchanging pairs of field indices, $i \leftrightarrow j$ {#appendix-a1-invariance-of-results-to-exchanging-pairs-of-field-indices-i-leftrightarrow-j .unnumbered} ============================================================================================== The general dynamic equations of a multiphase model read as: $$-\frac{\partial u_i}{\partial t} = \sum_{j\neq i} \kappa_{ij} \left( \frac{\delta F}{\delta u_i}-\frac{\delta F}{\delta u_j} \right) \enskip ,$$ where there are $N(N-1)$ mobilities ($\kappa_{ij}$). *The principle of formal indistinguishability* of the variables means that the variables are not “labeled”, i.e. none of them is distinguished formally on the basis of its index. This is true if the dynamic equations are invariant for the re-labeling of the variables, i.e. re-labeling of the variables on the level of the free energy functional results in the same as re-labeling the variables in the dynamic equations. This criterion is satisfied by symmetric mobility matrices, namely, $$\kappa_{ij}=\kappa_{ji} \enskip .$$ *Proof.* The dynamic equation for $u_J$ reads as $$\label{eq:AppA1} -\frac{\partial u_J}{\partial t} = \sum_{k\neq J} \kappa_{Jk} \left( \frac{\delta F}{\delta u_J}-\frac{\delta F}{\delta u_k} \right) \enskip .$$ The variables can be re-labeled by using the variable transformation $v_k:=u_k$ for $k\neq I,J$, $v_I:=u_J$, and $v_J:=u_I$. Using this in Eq. (\[eq:AppA1\]) yields then $$-\frac{\partial v_I}{\partial t} = \sum_{k\neq I,J} \left( \frac{\delta F}{\delta v_I}-\frac{\delta F}{\delta v_k} \right) + \kappa_{JI} \left( \frac{\delta F}{\delta v_I} - \frac{\delta F}{\delta v_J} \right) \enskip ,$$ where the chain rule for the functional derivative has also been used (see Appendix A2). Furthermore, re-labeling variables in the free energy functional first ($F[\mathbf{u}]\to F[\mathbf{v}]$), then deriving the dynamic equations simply results in $$-\frac{\partial v_I}{\partial t} = \sum_{k\neq I,J} \left( \frac{\delta F}{\delta v_I}-\frac{\delta F}{\delta v_k} \right) + \kappa_{IJ} \left( \frac{\delta F}{\delta v_I} - \frac{\delta F}{\delta v_J} \right) \enskip .$$ Comparing the two equations yields then $\kappa_{IJ}=\kappa_{JI}$.\ In order to illustrate the “no labeling” condition in practice, we choose a typical example of labeling the variables. Some authors eliminate of one of the variables even at the level of the free energy functional, i.e. they introduce the independent variables $v_i:=u_i$ for $i=1\dots N-1$, thus resulting in $u_N=1-\sum_{i=1}^{N-1}v_i$. Then, the following dynamic equations are used: $$-\tau_i\frac{\partial v_i}{\partial t}=\frac{\delta F}{\delta v_i} \enskip .$$ These can be written in terms of the old variables as: $$\label{eq:elim1} -\frac{\partial u_i}{\partial t} = \frac{1}{\tau_i} \left( \frac{\delta F}{\delta u_i} - \frac{\delta F}{\delta u_N} \right)$$ for $i=1\dots N-1$, and $$\label{eq:elim2} -\frac{\partial u_N}{\partial t} = - \sum_{j=1}^{N-1}\frac{1}{\tau_i}\left( \frac{\delta F}{\delta u_i} - \frac{\delta F}{\delta u_N} \right) \enskip .$$ It is straightforward to see, that Eqs. (\[eq:elim1\]) and (\[eq:elim2\]) prescribe the following mobility matrix: $$L_{ii}=\tau_i^{-1} \quad \text{and} \quad L_{iN}=-\tau_i^{-1}$$ for $i=1\dots N-1$, while the last row reads as $$L_{Ni}=-\tau_i^{-1} \quad \text{and} \quad L_{NN}=\sum_{i=1}^{N-1}\tau_i^{-1} \enskip ,$$ where the form $-\partial u_i/\partial t=\sum_{j=1}^N L_{ij} (\delta F/\delta u_j)$ is used. It is trivial that the elements of $\mathbb{L}$ sum up to 0 in each row and column, but the matrix is not symmetric! It means that the concept of eliminating a variable on the level of the free energy functional labels the variables, i.e. the eliminated variable is formally distinguished. Indeed, exchanging variables $I$ and $N$, deriving the dynamic equations, then exchanging $I$ and $N$ back result in a mobility matrix similar to the one described by Eqs. (\[eq:elim1\]) and (\[eq:elim2\]), however, the $N^{th}$ and the $I^{th}$ rows are exchanged. On the one hand, it means that the formal variable exchange corresponds to the elimination of phase $I$ instead of phase $N$. On the other hand, since the resulting mobility matrix is not identical to the original one, the eliminated variable is always labeled, therefore, this concept does not satisfy the condition of no labeling.\ Appendix A2: Chain rule for functional differentiation {#appendix-a2-chain-rule-for-functional-differentiation .unnumbered} ====================================================== Mathematically speaking, *the solution of the Euler-Lagrange equations is invariant to the variable transformation $\mathbf{Q}=\mathbf{T}[\mathbf{q}]$, if the transformation $\mathbf{T}[.]$ is unambiguous, i.e., if the inverse transform $\mathbf{T}^{-1}[.]$ also exists*. *Proof.* The Euler-Lagrange equations for the new variables read as: $$\label{eq:genFD} \frac{\delta F}{\delta Q_i} = \frac{\partial I}{\partial Q_i} - \nabla \frac{\partial I}{\partial \nabla Q_i} = 0 \enskip ,$$ where $I$ denotes the full integrand of Eq. (\[eq:genfunc\]). The terms on the right-hand side can be expanded as follows: $$\begin{aligned} \label{eq:ELtr1} \frac{\partial I}{\partial Q_i} &=& \sum_j \frac{\partial I}{\partial q_j}\frac{\partial q_j}{\partial Q_i} + \frac{\partial I}{\partial \nabla q_j}\frac{\partial \nabla q_j}{\partial Q_i} \\ \label{eq:ELtr2} \nabla \frac{\partial I}{\partial \nabla Q_i} &=& \nabla \left[ \sum_j \frac{\partial I}{\partial q_j}\frac{\partial q_j}{\partial \nabla Q_i} + \frac{\partial I}{\partial \nabla q_j}\frac{\partial \nabla q_j}{\partial \nabla Q_i} \right] \enskip .\end{aligned}$$ Since $q_j=T^{-1}_j(\mathbf{Q})$, $\partial q_j/\partial \nabla Q_i =0$. In addition, formally $\nabla q_j = \sum_i \frac{\partial q_j}{\partial Q_j} \nabla Q_i$, therefore, $\frac{\partial \nabla q_j}{\partial \nabla Q_i}=\frac{\partial q_j}{\partial Q_i}$. Using these together with Eqs. (\[eq:ELtr1\]) and (\[eq:ELtr2\]) in Eq. (\[eq:genFD\]) yields $$\label{eq:ELexpand} \begin{split} \frac{\delta F}{\delta Q_i} &= \sum_j \left( \frac{\partial I}{\partial q_j} - \nabla \frac{\partial I}{\partial \nabla q_j} \right) \frac{\partial q_j}{\partial Q_i} +\\ &+ \sum_j \frac{\partial I}{\partial \nabla q_j} \left( \frac{\partial \nabla q_j} {\partial Q_i} - \nabla \frac{\partial q_j}{\partial Q_i} \right) \enskip \end{split}$$ Finally, $\nabla q_j = \sum_k \frac{\partial q_j}{\partial Q_k} \nabla Q_k \Rightarrow \frac{\partial \nabla q_j}{\partial Q_i} = \sum_k \frac{\partial^2 q_j}{\partial Q_k \partial Q_i} \nabla Q_k$ and $\nabla \frac{\partial q_j}{\partial Q_i} = \sum_k \frac{\partial^2 q_j}{\partial Q_i \partial Q_k} \nabla Q_k$, therefore, the second sum on the right hand side of Eq. (\[eq:ELexpand\]) vanishes. The final result then reads as: $$\label{eq:ELfinal} \frac{\delta F}{\delta Q_i} = \sum_j \frac{\delta F}{\delta q_j} \frac{\partial q_j}{\partial Q_i} \enskip ,$$ i.e. *the chain rule of differentiation also applies for the functional derivative*. Let now $\mathbf{q}^*(\mathbf{r})$ denote the solution of $\delta F/\delta \mathbf{q}=0$. Apparently, the right hand side of Eq. (\[eq:ELfinal\]) vanishes at $\mathbf{q}^*(\mathbf{r})$. Formally $\mathbf{q}^*(\mathbf{r})=\mathbf{T}^{-1}[\mathbf{Q}^*(\mathbf{r})]$, indicating that $\mathbf{Q}^*(\mathbf{r}) = \mathbf{T}[\mathbf{q}^*(\mathbf{r})]$, i.e. the solution in $\mathbf{Q}$ is just the transformation of the solution in $\mathbf{q}$. In other words, the solution of the Euler-Lagrange equations is invariant to the choice of the generalized variables.\ Appendix B: Numerical method {#appendix-b-numerical-method .unnumbered} ============================ The dynamic equations were solved numerically on a periodic, two-dimensional domain by using an operator-splitting based, quasi-spectral, semi-implicit time stepping scheme as follows. The dynamic equations can be re-written in the form $$\label{eq:solve} \frac{\partial \mathbf{u}}{\partial t} = \mathbf{f}(\mathbf{u},\nabla\mathbf{u}) \enskip ,$$ where $\mathbf{f}(\mathbf{u},\nabla\mathbf{u})$ is the general, non-linear right-hand side. During time stepping $\mathbf{f}(\mathbf{u},\nabla\mathbf{u})$ is calculated at time point $t$, while $\partial u_i/\partial t$ is approximated as $$\label{eq:disct} \frac{\partial u_i}{\partial t} \approx \frac{u_i^{t+\Delta t}-u_i^t}{\Delta t} \enskip .$$ Next, we add a suitably chosen linear term $\hat{s}[u_i]= \sum_{i=1}^\infty (-1)^i s_i \nabla^{2i} u_i$ (where $s_i \geq 0$) to both sides of Eq. (\[eq:solve\]). We consider this term at $t+\Delta t$ at the left-hand side, but at $t$ on the right-hand side of the equation. This concept, together with Eq. (\[eq:disct\]) results in the following, explicit time stepping scheme in the spectrum: $$\label{eq:timestep} u_i^{t+\Delta t}(\mathbf{k}) = u_i^t(\mathbf{k}) + \frac{\Delta t}{1+s_i(\mathbf{k})\Delta t} \mathcal{F}\{f_i[\mathbf{u}^t(\mathbf{r}),\nabla\mathbf{u}^t(\mathbf{r})]\} \enskip ,$$ where $s_i(\mathbf{k})=\sum_{j=1}^\infty s^{(i)}_j (\mathbf{k}^2)^j$, and $\mathcal{F}\{.\}$ stands for the Fourier transform. 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--- abstract: | In this paper we consider the most common ABox reasoning services for the description logic ${\mathcal{DL}\langle \mathsf{4LQS^{R,\!\times}}\rangle({\mathbf{D}})}$ (${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$, for short) and prove their decidability via a reduction to the satisfiability problem for the set-theoretic fragment [$\mathsf{4LQS^R}$]{}. The description logic ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ is very expressive, as it admits various concept and role constructs, and data types, that allow one to represent rule-based languages such as SWRL. Decidability results are achieved by defining a generalization of the conjunctive query answering problem, called HOCQA (Higher Order Conjunctive Query Answering), that can be instantiated to the most widespread ABox reasoning tasks. We also present a [KE-tableau]{}based procedure for calculating the answer set from ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ knowledge bases and higher order ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ conjunctive queries, thus providing means for reasoning on several well-known ABox reasoning tasks. Our calculus extends a previously introduced [KE-tableau]{}based decision procedure for the CQA problem. author: - Domenico Cantone - 'Marianna Nicolosi-Asmundo' - | \ Daniele Francesco Santamaria bibliography: - 'biblioext.bib' title: 'A set-theoretic approach to ABox reasoning services (Extended Version)' --- Introduction ============ Recently, results from Computable Set Theory have been applied to knowledge representation for the semantic web in order to define and reason about description logics and rule languages. Such a study is motivated by the fact that Computable Set Theory is a research field plenty of interesting decidability results and that there exists a natural translation function between some set theoretical fragments and description logics and rule languages. In particular, the decidable four-level stratified fragment of set theory ${\ensuremath{\mathsf{4LQS^R}}}$, involving variables of four sorts, pair terms, and a restricted form of quantification over variables of the first three sorts (cf. [@CanNic2013]), has been used in [@CanLonNicSanRR2015] to represent the description logic ${\mathcal{DL}\langle \mathsf{4LQS^R}\rangle({\mathbf{D}})}$ (more simply referred to as ${\mathcal{DL}_{{\mathbf{D}}}^{4}}$). The logic ${\mathcal{DL}_{{\mathbf{D}}}^{4}}$ admits concept constructs such as full negation, union and intersection of concepts, concept domain and range, existential quantification and min cardinality on the left-hand side of inclusion axioms. It also supports role constructs such as role chains on the left hand side of inclusion axioms, union, intersection, and complement of abstract roles, and properties on roles such as transitivity, symmetry, reflexivity, and irreflexivity. As briefly shown in [@CanLonNicSanRR2015], ${\mathcal{DL}_{{\mathbf{D}}}^{4}}$ is particularly suitable to express a rule language such as the Semantic Web Rule Language (SWRL), an extension of the Ontology Web Language (OWL). It admits data types, a simple form of concrete domains that are relevant in real world applications. In [@CanLonNicSanRR2015], the consistency problem for ${\mathcal{DL}_{{\mathbf{D}}}^{4}}$-knowledge bases has been proved decidable by means of a reduction to the satisfiability problem for ${\ensuremath{\mathsf{4LQS^R}}}$, whose decidability has been established in [@CanNic2013]. It has also been shown that, under not very restrictive constraints, the consistency problem for ${\mathcal{DL}_{{\mathbf{D}}}^{4}}$-knowledge bases is **NP**-complete. Such a low complexity result is motivated by the fact that existential quantification cannot appear on the right-hand side of inclusion axioms. Nonetheless, ${\mathcal{DL}_{{\mathbf{D}}}^{4}}$ turns out to be more expressive than other low complexity logics such as OWL RL and suitable for representing real world ontologies. For example the restricted version of ${\mathcal{DL}_{{\mathbf{D}}}^{4}}$ allows one to express several ontologies, such as [@cilc15] classifying ancient pottery. In [@ictcs16], the description logic ${\mathcal{DL}\langle \mathsf{4LQS^{R,\!\times}}\rangle({\mathbf{D}})}$ (${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$, for short), extending ${\mathcal{DL}_{{\mathbf{D}}}^{4}}$ with Boolean operations on concrete roles and with the product of concepts, has been introduced and the *Conjunctive Query Answering* (CQA) problem for ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ has been proved decidable via a reduction to the CQA problem for ${\ensuremath{\mathsf{4LQS^R}}}$, whose decidability follows from that of ${\ensuremath{\mathsf{4LQS^R}}}$ (see [@CanNic2013]). CQA is a powerful way to query ABoxes, particularly relevant in the context of description logics and for real world applications based on semantic web technologies, as it provides mechanisms for interacting with ontologies and data. The CQA problem for description logics has been introduced in [@calvanese98; @calvanese08] and studied for several well-known description logics (cf. [@Calvanese:1998:DQC:275487.275504; @Calvanese2013335; @calvanese2007answering; @Ortiz:Calvanese:et-al:06a; @GlHS07a; @GliHoLuSa-JAIR08; @HorrSattTob-CADE-2000; @j.websem63; @HorrTess-aaai-2000; @DBLP:conf/ijcai/HustadtMS05; @Rosa07c; @DBLP:conf/cade/Lutz08; @Ortiz:2011:QAH:2283516.2283571]). Finally, we mention also a terminating [KE-tableau]{}based procedure that, given a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-query $Q$ and a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-knowledge base $\mathcal{KB}$ represented in set-theoretic terms, determines the answer set of $Q$ with respect to $\mathcal{KB}$. [KE-tableau]{}systems [@dagostino1999] allow the construction of trees whose distinct branches define mutually exclusive situations, thus preventing the proliferation of redundant branches, typical of semantic tableaux. In this paper we extend the results presented in [@ictcs16] by considering also the main ABox reasoning tasks for ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$, such as instance checking and concept retrieval, and study their decidability via a reduction to the satisfiability problem for ${\ensuremath{\mathsf{4LQS^R}}}$. Specifically, we define Higher Order (HO) ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-conjunctive queries admitting variables of three sorts: individual and data type values variables, concept variables, and role variables. HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-conjunctive queries can be instantiated to any of the ABox reasoning tasks we are considering in the paper. Then, we define the Higher Order Conjunctive Query Answering (HOCQA) problem for ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ and prove its decidability by reducing it to the HOCQA problem for ${\ensuremath{\mathsf{4LQS^R}}}$. Decidability of the latter problem follows from that of the satisfiability problem for ${\ensuremath{\mathsf{4LQS^R}}}$. ${\ensuremath{\mathsf{4LQS^R}}}$ representation of ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ knowledge bases is defined according to [@ictcs16]. ${\ensuremath{\mathsf{4LQS^R}}}$ turns out to be naturally suited for the HOCQA problem since HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-conjunctive queries are easily translated into ${\ensuremath{\mathsf{4LQS^R}}}$-formulae. In particular, individual and data type value variables are mapped into ${\ensuremath{\mathsf{4LQS^R}}}$ variables of sort 0, concept variables into ${\ensuremath{\mathsf{4LQS^R}}}$ variables of sort 1, and role variables into ${\ensuremath{\mathsf{4LQS^R}}}$ variables of sort 3. Finally, we present an extension of the [KE-tableau]{}presented in [@ictcs16], which provides a decision procedure for the HOCQA task for ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$. Preliminaries ============= The set-theoretic fragment [$\mathsf{4LQS^R}$]{} {#4LQS} ------------------------------------------------ It is convenient to first introduce the syntax and semantics of a more general four-level quantified language, denoted ${\ensuremath{\mathsf{4LQS}}}$. Then we provide some restrictions on the quantified formulae of ${\ensuremath{\mathsf{4LQS}}}$ to characterize [$\mathsf{4LQS^R}$]{}. The interested reader can find more details in [@CanNic2013] together with the decision procedure for the satisfiability problem for [$\mathsf{4LQS^R}$]{}. ${\ensuremath{\mathsf{4LQS}}}$ involves four collections, $\mathcal{V}_i$, of variables of sort $i=0,1,2,3$, respectively. These will be denoted by $X^i,Y^i,Z^i,\ldots$ (in particular, variables of sort $0$ will also be denoted by $x, y, z, \ldots$). In addition to variables, ${\ensuremath{\mathsf{4LQS}}}$ involves also *pair terms* of the form $\langle x,y \rangle$, for $ x,y \in \mathcal{V}_0$. *${\ensuremath{\mathsf{4LQS}}}$-quantifier-free atomic formulae* are classified as: - level 0:   $x=y$,   $x \in X^1$,   $\langle x,y \rangle = X^2$,   $\langle x,y \rangle \in X^3$; - level 1:   $X^1=Y^1$,   $X^1 \in X^2$; - level 2:   $X^2=Y^2$,   $X^2 \in X^3$. ${\ensuremath{\mathsf{4LQS}}}$-*purely universal formulae* are classified as: - [ level 1: $(\forall z_1)\ldots(\forall z_n) \varphi _0$, where $z_1,\ldots,z_n$ $\in \mathcal{V}_0$ and $\varphi _0$ is any propositional combination of quantifier-free atomic formulae of level 0;]{} - [ level 2: $(\forall Z^1_1)\ldots(\forall Z^1_m) \varphi _1$, where $Z^1_1,\ldots,Z^1_m $ $\in \mathcal{V}_1$ and $\varphi _1$ is any propositional combination of quantifier-free atomic formulae of levels 0 and 1, and of purely universal formulae of level 1;]{} - [level 3: $(\forall Z^2_1)\ldots(\forall Z^2_p) \varphi _2$, where $Z^2_1,\ldots,Z^2_p $ $\in \mathcal{V}_2$ and $\varphi _2$ is any propositional combination of quantifier-free atomic formulae and of purely universal formulae of levels 1 and 2.]{} ${\ensuremath{\mathsf{4LQS}}}$-formulae are all the propositional combinations of quantifier-free atomic formulae of levels 0, 1, 2, and of purely universal formulae of levels 1, 2, 3. The variables $z_1,\ldots,z_n$ are said to occur *quantified* in $(\forall z_1) \ldots (\forall z_n) \varphi_0$. Likewise, $Z^1_1,\ldots, Z^1_m$ and $Z^2_1, \ldots, Z^2_p$ occur quantified in $(\forall Z^1_1) \ldots (\forall Z^1_m) \varphi_1$ and in $(\forall Z^2_1) \ldots (\forall Z^2_p) \varphi_2$, respectively. A variable occurs *free* in a ${\ensuremath{\mathsf{4LQS}}}$-formula $\varphi$ if it does not occur quantified in any subformula of $\varphi$. For $i = 0,1,2,3$, we denote with ${\mathtt{Var}_i}(\varphi)$ the collections of variables of level $i$ occurring free in $\varphi$ and we put ${\mathtt{Vars}}(\varphi) {\coloneqq}\bigcup_{i=0}^{3} {\mathtt{Var}_i}(\varphi)$. A substitution $\sigma {\coloneqq}\{ \vec{x}/\vec{y}, \vec{X}^1/\vec{Y}^1, \vec{X}^2/\vec{Y}^2, \vec{X}^3/\vec{Y}^3\}$ is the mapping $\varphi \mapsto \varphi\sigma$ such that, for any given ${\ensuremath{\mathsf{4LQS}}}$-formula $\varphi$, $\varphi\sigma$ is the ${\ensuremath{\mathsf{4LQS}}}$-formula obtained from $\varphi$ by replacing the free occurrences of the variables $x_i$ in $\vec{x}$ (for $i = 1,\ldots, n$) with the corresponding $y_i$ in $\vec{y}$, of $X^1_j$ in $\vec{X}^1$ (for $j = 1,\ldots,m$) with $Y^1_j$ in $\vec{Y}^1$, of $X^2_k$ in $\vec{X}^2$ (for $k = 1,\ldots,p$) with $Y^2_k$ in $\vec{Y}^2$, and of $X^3_h$ in $\vec{X}^3$ (for $h= 1,\ldots,q$) with $Y^3_h$ in $\vec{Y}^3$, respectively. A substitution $\sigma$ is *free* for $\varphi$ if the formulae $\varphi$ and $\varphi\sigma$ have exactly the same occurrences of quantified variables. The *empty substitution*, denoted with $\epsilon$, satisfies $\varphi \epsilon = \varphi$, for every ${\ensuremath{\mathsf{4LQS}}}$-formula $\varphi$. A ${\ensuremath{\mathsf{4LQS}}}$-*interpretation* is a pair $\mathbfcal{M}=(D,M)$, where $D$ is a non-empty collection of objects (called *domain* or *universe* of $\mathbfcal{M}$) and $M$ is an assignment over the variables in $\mathcal{V}_i$, for $i=0,1,2,3$, such that:\ $MX^{0} \in D, ~~~ MX^1 \in {\mathcal{P}}(D), ~~~ MX^2 \in {\mathcal{P}}({\mathcal{P}}(D)), ~~~ MX^3 \in {\mathcal{P}}({\mathcal{P}}({\mathcal{P}}(D))),$ \ where $ X^{i} \in \mathcal{V}_i$, for $i=0,1,2,3$, and ${\mathcal{P}}(s)$ denotes the powerset of $s$. Pair terms are interpreted *à la* Kuratowski, and therefore we put\ $M \langle x,y \rangle {\coloneqq}\{ \{ Mx \},\{ Mx,My \} \}$. \ Quantifier-free atomic formulae and purely universal formulae are evaluated in a standard way according to the usual meaning of the predicates ‘$\in$’ and ‘$=$’. The interpretation of quantifier-free atomic formulae and of purely universal formulae is given in [@CanNic2013]. Finally, compound formulae are interpreted according to the standard rules of propositional logic. If $\mathbfcal{M} \models \varphi$, then $\mathbfcal{M} $ is said to be a ${\ensuremath{\mathsf{4LQS}}}$-model for $\varphi$. A ${\ensuremath{\mathsf{4LQS}}}$-formula is said to be *satisfiable* if it has a ${\ensuremath{\mathsf{4LQS}}}$-model. A ${\ensuremath{\mathsf{4LQS}}}$-formula is *valid* if it is satisfied by all ${\ensuremath{\mathsf{4LQS}}}$-interpretations. We are now ready to present the fragment [$\mathsf{4LQS^R}$]{} of ${\ensuremath{\mathsf{4LQS}}}$ of our interest. This is the collection of the formulae $\psi$ of ${\ensuremath{\mathsf{4LQS}}}$ fulfilling the restrictions: 1. for every purely universal formula $(\forall Z^1_1)\ldots(\forall Z^1_m) \varphi_1$ of level 2 occurring in $\psi$ and every purely universal formula $(\forall z_1)\ldots(\forall z_n) \varphi_0$ of level 1 occurring negatively in $\varphi_1$, $\varphi_0$ is a propositional combination of quantifier-free atomic formulae of level $0$ and the condition\ $\neg \varphi_0 \rightarrow \overset{n}{ \underset {i=1} \bigwedge} \; \overset {m} { \underset {j=1 }\bigwedge} z_i \in Z^1_j$ \ is a valid ${\ensuremath{\mathsf{4LQS}}}$-formula (in this case we say that $(\forall z_1)\ldots(\forall z_n) \varphi_0$ is *linked to the variables* $Z^1_1,\ldots,Z^1_m$); 2. for every purely universal formula $(\forall Z^2_1)\ldots(\forall Z^2_p) \varphi_2$ of level 3 in $\psi$: - every purely universal formula of level 1 occurring negatively in $\varphi_2$ and not occurring in a purely universal formula of level 2 is only allowed to be of the form\ $(\forall z_1)\ldots(\forall z_n) \neg( \overset {n}{ \underset {i=1} \bigwedge} \; \overset {n} { \underset {j=1}\bigwedge} \langle z_i,z_j \rangle=Y^2_{ij}),$ \ with $Y^2_{ij} \in \mathcal{V}^2$, for $i,j=1,\ldots,n$; - purely universal formulae $(\forall Z^1_1)\ldots(\forall Z^1_m) \varphi_1$ of level 2 may occur only positively in $\varphi_2$.[^1] Restriction 1 has been introduced for technical reasons concerning the decidability of the satisfiability problem for the fragment, while restriction 2 allows one to define binary relations and several operations on them. The semantics of [$\mathsf{4LQS^R}$]{}plainly coincides with that of ${\ensuremath{\mathsf{4LQS}}}$. The logic ${\mathcal{DL}\langle \mathsf{4LQS^{R,\!\times}}\rangle({\mathbf{D}})}$ {#dlssx} --------------------------------------------------------------------------------- The description logic ${\mathcal{DL}\langle \mathsf{4LQS^{R,\!\times}}\rangle({\mathbf{D}})}$ (which, as already remarked, will be more simply referred to as ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$) is an extension of the logic ${\mathcal{DL}\langle \mathsf{4LQS^R}\rangle({\mathbf{D}})}$ presented in [@CanLonNicSanRR2015], where Boolean operations on concrete roles and the product of concepts are defined. In addition to other features, ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ admits also data types, a simple form of concrete domains that are relevant in real-world applications. In particular, it treats derived data types by admitting data type terms constructed from data ranges by means of a finite number of applications of the Boolean operators. Basic and derived data types can be used inside inclusion axioms involving concrete roles. Data types are introduced through the notion of data type map, defined according to [@Motik2008] as follows. Let ${\mathbf{D}}= (N_{D}, N_{C},N_{F},\cdot^{{\mathbf{D}}})$ be a *data type map*, where $N_{D}$ is a finite set of data types, $N_{C}$ is a function assigning a set of constants $N_{C}(d)$ to each data type $d \in N_{D}$, $N_{F}$ is a function assigning a set of facets $N_{F}(d)$ to each $d \in N_{D}$, and $\cdot^{{\mathbf{D}}}$ is a function assigning a data type interpretation $d^{{\mathbf{D}}}$ to each data type $d \in N_{D}$, a facet interpretation $f^{{\mathbf{D}}} \subseteq d^{{\mathbf{D}}}$ to each facet $f \in N_{F}(d)$, and a data value $e_{d}^{{\mathbf{D}}} \in d^{{\mathbf{D}}}$ to every constant $e_{d} \in N_{C}(d)$. We shall assume that the interpretations of the data types in $N_{D}$ are nonempty pairwise disjoint sets. Let ${\mathbf{R_A}}$, ${\mathbf{R_D}}$, $\mathbf{C}$, $\mathbf{I}$ be denumerable pairwise disjoint sets of abstract role names, concrete role names, concept names, and individual names, respectively. We assume that the set of abstract role names ${\mathbf{R_A}}$ contains a name $U$ denoting the universal role. \(a) ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-data type, (b) ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-concept, (c) ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-abstract role, and (d) ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-concrete role terms are constructed according to the following syntax rules: - $t_1, t_2 \longrightarrow dr ~|~\neg t_1 ~|~t_1 \sqcap t_2 ~|~t_1 \sqcup t_2 ~|~\{e_{d}\}\, ,$ - $C_1, C_ 2 \longrightarrow A ~|~\top ~|~\bot ~|~\neg C_1 ~|~C_1 \sqcup C_2 ~|~C_1 \sqcap C_2 ~|~\{a\} ~|~\exists R.\mathit{Self}| \exists R.\{a\}| \exists P.\{e_{d}\}\, ,$ - $R_1, R_2 \longrightarrow S ~|~U ~|~R_1^{-} ~|~ \neg R_1 ~|~R_1 \sqcup R_2 ~|~R_1 \sqcap R_2 ~|~R_{C_1 |} ~|~R_{|C_1} ~|~R_{C_1 ~|~C_2} ~|~id(C) ~|~ $ $C_1 \times C_2 \, ,$ - $P_1,P_2 \longrightarrow T ~|~\neg P_1 ~|~ P_1 \sqcup P_2 ~|~ P_1 \sqcap P_2 ~|~P_{C_1 |} ~|~P_{|t_1} ~|~P_{C_1 | t_1}\, ,$ where $dr$ is a data range for ${\mathbf{D}}$, $t_1,t_2$ are data type terms, $e_{d}$ is a constant in $N_{C}(d)$, $a$ is an individual name, $A$ is a concept name, $C_1, C_2$ are ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-concept terms, $S$ is an abstract role name, $R, R_1,R_2$ are ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-abstract role terms, $T$ is a concrete role name, and $P,P_1,P_2$ are ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-concrete role terms. We remark that data type terms are introduced in order to represent derived data types. A ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-knowledge base is a triple ${\mathcal K} = (\mathcal{R}, \mathcal{T}, \mathcal{A})$ such that $\mathcal{R}$ is a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-$RBox$, $\mathcal{T}$ is a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-$TBox$, and $\mathcal{A}$ a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-$ABox$. A ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-$RBox$ is a collection of statements of the following forms:\ $R_1 \equiv R_2$, $R_1 \sqsubseteq R_2$, $R_1\ldots R_n \sqsubseteq R_{n+1}$, ${\mathsf{Sym}}(R_1)$, ${\mathsf{Asym}}(R_1)$, ${\mathsf{Ref}}(R_1)$, ${\mathsf{Irref}}(R_1)$, $\mathsf{Dis}(R_1,R_2)$, ${\mathsf{Tra}}(R_1)$, ${\mathsf{Fun}}(R_1)$, $R_1 \equiv C_1 \times C_2$, $P_1 \equiv P_2$, $P_1 \sqsubseteq P_2$, $\mathsf{Dis}(P_1,P_2)$, ${\mathsf{Fun}}(P_1)$, $ $\ where $R_1,R_2$ are ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-abstract role terms, $C_1, C_2$ are ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-abstract concept terms, and $P_1,P_2$ are ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-concrete role terms. Any expression of the type $w \sqsubseteq R$, where $w$ is a finite string of ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-abstract role terms and $R$ is an ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-abstract role term, is called a *role inclusion axiom (RIA)*. A ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-$TBox$ is a set of statements of the types: - $C_1 \equiv C_2$, $C_1 \sqsubseteq C_2$, $C_1 \sqsubseteq \forall R_1.C_2$, $\exists R_1.C_1 \sqsubseteq C_2$, $\geq_n\!\! R_1. C_1 \sqsubseteq C_2$,\ $C_1 \sqsubseteq {\leq_n\!\! R_1. C_2}$, - $t_1 \equiv t_2$, $t_1 \sqsubseteq t_2$, $C_1 \sqsubseteq \forall P_1.t_1$, $\exists P_1.t_1 \sqsubseteq C_1$, $\geq_n\!\! P_1. t_1 \sqsubseteq C_1$, $C_1 \sqsubseteq {\leq_n\!\! P_1. t_1}$, where $C_1,C_2$ are ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-concept terms, $t_1,t_2$ data type terms, $R_1$ a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-abstract role term, $P_1$ a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-concrete role term. Any statement of the form $C \sqsubseteq D$, with $C$, $D$ ${\mathcal{DL}_{{\mathbf{D}}}^{4}}$-concept terms, is a *general concept inclusion axiom*. A ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-$ABox$ is a set of *individual assertions* of the forms: $a : C_1$, $(a,b) : R_1$, $a=b$, $a \neq b$, $e_{d} : t_1$, $(a, e_{d}) : P_1$, with $C_1$ a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-concept term, $d$ a data type, $t_1$ a data type term, $R_1$ a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-abstract role term, $P_1$ a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-concrete role term, $a,b$ individual names, and $e_{d}$ a constant in $N_{C}(d)$. The semantics of ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ is given by means of an interpretation ${\mathbf{I}}= (\Delta^{\mathbf{I}}, \Delta_{{\mathbf{D}}}, \cdot^{\mathbf{I}})$, where $\Delta^{\mathbf{I}}$ and $\Delta_{{\mathbf{D}}}$ are non-empty disjoint domains such that $d^{\mathbf{D}}\subseteq \Delta_{{\mathbf{D}}}$, for every $d \in N_{D}$, and $\cdot^{\mathbf{I}}$ is an interpretation function. The definition of the interpretation of concepts and roles, axioms and assertions is illustrated in Table \[semdlss\]. [|&gt;m[2.5cm]{}|c|&gt;m[6.7cm]{}|]{} Name & Syntax & Semantics\ concept & $A$ & $ A^{\mathbf{I}}\subseteq \Delta^{\mathbf{I}}$\ ab. (resp., cn.) rl. & $R$ (resp., $P$ )& $R^{\mathbf{I}}\subseteq \Delta^{\mathbf{I}}\times \Delta^{\mathbf{I}}$ (resp., $P^{\mathbf{I}}\subseteq \Delta^{\mathbf{I}}\times \Delta_{\mathbf{D}}$)\ individual& $a$& $a^{\mathbf{I}}\in \Delta^{\mathbf{I}}$\ nominal & $\{a\}$ & $\{a\}^{\mathbf{I}}= \{a^{\mathbf{I}}\}$\ dtype (resp., ng.) & $d$ (resp., $\neg d$)& $ d^{\mathbf{D}}\subseteq \Delta_{\mathbf{D}}$ (resp., $\Delta_{\mathbf{D}}\setminus d^{\mathbf{D}}$)\ negative data type term & $ \neg t_1 $ & $ (\neg t_1)^{{\mathbf{D}}} = \Delta_{{\mathbf{D}}} \setminus t_1^{{\mathbf{D}}}$\ data type terms intersection & $ t_1 \sqcap t_2 $ & $ (t_1 \sqcap t_2)^{{\mathbf{D}}} = t_1^{{\mathbf{D}}} \cap t_2^{{\mathbf{D}}} $\ data type terms union & $ t_1 \sqcup t_2 $ & $ (t_1 \sqcup t_2)^{{\mathbf{D}}} = t_1^{{\mathbf{D}}} \cup t_2^{{\mathbf{D}}} $\ constant in $N_{C}(d)$ & $ e_{d} $ & $ e_{d}^{\mathbf{D}}\in d^{\mathbf{D}}$\ data range & $\{ e_{d_1}, \ldots , e_{d_n} \}$& $\{ e_{d_1}, \ldots , e_{d_n} \}^{\mathbf{D}}= \{e_{d_1}^{\mathbf{D}}\} \cup \ldots \cup \{e_{d_n}^{\mathbf{D}}\} $\ data range & $\psi_d$ & $\psi_d^{\mathbf{D}}$\ data range & $\neg dr$ & $\Delta_{\mathbf{D}}\setminus dr^{\mathbf{D}}$\ top (resp., bot.) & $\top$ (resp., $\bot$ )& $\Delta^{\mathbf{I}}$ (resp., $\emptyset$)\ negation & $\neg C$ & $(\neg C)^{\mathbf{I}}= \Delta^{\mathbf{I}}\setminus C$\ conj. (resp., disj.) & $C \sqcap D$ (resp., $C \sqcup D$)& $ (C \sqcap D)^{\mathbf{I}}= C^{\mathbf{I}}\cap D^{\mathbf{I}}$ (resp., $ (C \sqcup D)^{\mathbf{I}}= C^{\mathbf{I}}\cup D^{\mathbf{I}}$)\ valued exist. quantification & $\exists R.{a}$ & $(\exists R.{a})^{\mathbf{I}}= \{ x \in \Delta^{\mathbf{I}}: \langle x,a^{\mathbf{I}}\rangle \in R^{\mathbf{I}}\}$\ data typed exist. quantif. & $\exists P.{e_{d}}$ & $(\exists P.e_{d})^{\mathbf{I}}= \{ x \in \Delta^{\mathbf{I}}: \langle x, e^{\mathbf{D}}_{d} \rangle \in P^{\mathbf{I}}\}$\ self concept & $\exists R.\mathit{Self}$ & $(\exists R.\mathit{Self})^{\mathbf{I}}= \{ x \in \Delta^{\mathbf{I}}: \langle x,x \rangle \in R^{\mathbf{I}}\}$\ nominals & $\{ a_1, \ldots , a_n \}$& $\{ a_1, \ldots , a_n \}^{\mathbf{I}}= \{a_1^{\mathbf{I}}\} \cup \ldots \cup \{a_n^{\mathbf{I}}\} $\ universal role & U & $(U)^{\mathbf{I}}= \Delta^{\mathbf{I}}\times \Delta^{\mathbf{I}}$\ inverse role & $R^-$ & $(R^-)^{\mathbf{I}}= \{\langle y,x \rangle \mid \langle x,y \rangle \in R^{\mathbf{I}}\}$\ concept cart. prod. & $ C_1 \times C_2$ & $ (C_1 \times C_2)^I = C_1^I \times C_2^I$\ abstract role complement & $ \neg R $ & $ (\neg R)^{\mathbf{I}}=(\Delta^{\mathbf{I}}\times \Delta^{\mathbf{I}}) \setminus R^{\mathbf{I}}$\ abstract role union & $R_1 \sqcup R_2$ & $ (R_1 \sqcup R_2)^{\mathbf{I}}= R_1^{\mathbf{I}}\cup R_2^{\mathbf{I}}$\ abstract role intersection & $R_1 \sqcap R_2$ & $ (R_1 \sqcap R_2)^{\mathbf{I}}= R_1^{\mathbf{I}}\cap R_2^{\mathbf{I}}$\ abstract role domain restr. & $R_{C \mid }$ & $ (R_{C \mid })^{\mathbf{I}}= \{ \langle x,y \rangle \in R^{\mathbf{I}}: x \in C^{\mathbf{I}}\} $\ concrete role complement & $ \neg P $ & $ (\neg P)^{\mathbf{I}}=(\Delta^{\mathbf{I}}\times \Delta^{\mathbf{D}}) \setminus P^{\mathbf{I}}$\ concrete role union & $P_1 \sqcup P_2$ & $ (P_1 \sqcup P_2)^{\mathbf{I}}= P_1^{\mathbf{I}}\cup P_2^{\mathbf{I}}$\ concrete role intersection & $P_1 \sqcap P_2$ & $ (P_1 \sqcap P_2)^{\mathbf{I}}= P_1^{\mathbf{I}}\cap P_2^{\mathbf{I}}$\ concrete role domain restr. & $P_{C \mid }$ & $ (P_{C \mid })^{\mathbf{I}}= \{ \langle x,y \rangle \in P^{\mathbf{I}}: x \in C^{\mathbf{I}}\} $\ concrete role range restr. & $P_{ \mid t}$ & $ (P_{\mid t})^{\mathbf{I}}= \{ \langle x,y \rangle \in P^{\mathbf{I}}: y \in t^{\mathbf{D}}\} $\ concrete role restriction & $P_{ C_1 \mid t}$ & $ (P_{C_1 \mid t})^{\mathbf{I}}= \{ \langle x,y \rangle \in P^{\mathbf{I}}: x \in C_1^{\mathbf{I}}\wedge y \in t^{\mathbf{D}}\} $\ concept subsum. & $C_1 \sqsubseteq C_2$ & ${\mathbf{I}}\models_{\mathbf{D}}C_1 \sqsubseteq C_2 \; \Longleftrightarrow \; C_1^{\mathbf{I}}\subseteq C_2^{\mathbf{I}}$\ ab. role subsum. & $ R_1 \sqsubseteq R_2$ & ${\mathbf{I}}\models_{\mathbf{D}}R_1 \sqsubseteq R_2 \; \Longleftrightarrow \; R_1^{\mathbf{I}}\subseteq R_2^{\mathbf{I}}$\ role incl. axiom & $R_1 \ldots R_n \sqsubseteq R$ & ${\mathbf{I}}\models_{\mathbf{D}}R_1 \ldots R_n \sqsubseteq R \; \Longleftrightarrow \; R_1^{\mathbf{I}}\circ \ldots \circ R_n^{\mathbf{I}}\subseteq R^{\mathbf{I}}$\ cn. role subsum. & $ P_1 \sqsubseteq P_2$ & ${\mathbf{I}}\models_{\mathbf{D}}P_1 \sqsubseteq P_2 \; \Longleftrightarrow \; P_1^{\mathbf{I}}\subseteq P_2^{\mathbf{I}}$\ symmetric role & ${\mathsf{Sym}}(R)$ & ${\mathbf{I}}\models_{\mathbf{D}}{\mathsf{Sym}}(R) \; \Longleftrightarrow \; (R^-)^{\mathbf{I}}\subseteq R^{\mathbf{I}}$\ asymmetric role & ${\mathsf{Asym}}(R)$ & ${\mathbf{I}}\models_{\mathbf{D}}{\mathsf{Asym}}(R) \; \Longleftrightarrow \; R^{\mathbf{I}}\cap (R^-)^{\mathbf{I}}= \emptyset $\ transitive role & ${\mathsf{Tra}}(R)$ & ${\mathbf{I}}\models_{\mathbf{D}}{\mathsf{Tra}}(R) \; \Longleftrightarrow \; R^{\mathbf{I}}\circ R^{\mathbf{I}}\subseteq R^{\mathbf{I}}$\ disj. ab. role & $\mathsf{Dis}(R_1,R_2)$ & ${\mathbf{I}}\models_{\mathbf{D}}\mathsf{Dis}(R_1,R_2) \; \Longleftrightarrow \; R_1^{\mathbf{I}}\cap R_2^{\mathbf{I}}= \emptyset$\ reflexive role & ${\mathsf{Ref}}(R)$& ${\mathbf{I}}\models_{\mathbf{D}}{\mathsf{Ref}}(R) \; \Longleftrightarrow \; \{ \langle x,x \rangle \mid x \in \Delta^{\mathbf{I}}\} \subseteq R^{\mathbf{I}}$\ irreflexive role & ${\mathsf{Irref}}(R)$& ${\mathbf{I}}\models_{\mathbf{D}}{\mathsf{Irref}}(R) \; \Longleftrightarrow \; R^{\mathbf{I}}\cap \{ \langle x,x \rangle \mid x \in \Delta^{\mathbf{I}}\} = \emptyset $\ func. ab. role & ${\mathsf{Fun}}(R)$ & ${\mathbf{I}}\models_{\mathbf{D}}{\mathsf{Fun}}(R) \; \Longleftrightarrow \; (R^{-})^{\mathbf{I}}\circ R^{\mathbf{I}}\subseteq \{ \langle x,x \rangle \mid x \in \Delta^{\mathbf{I}}\}$\ disj. cn. role & $\mathsf{Dis}(P_1,P_2)$ & ${\mathbf{I}}\models_{\mathbf{D}}\mathsf{Dis}(P_1,P_2) \; \Longleftrightarrow \; P_1^{\mathbf{I}}\cap P_2^{\mathbf{I}}= \emptyset$\ func. cn. role & ${\mathsf{Fun}}(P)$ & ${\mathbf{I}}\models_{\mathbf{D}}{\mathsf{Fun}}(p) \; \Longleftrightarrow \; \langle x,y \rangle \in P^{\mathbf{I}}\mbox{ and } \langle x,z \rangle \in P^{\mathbf{I}}\mbox{ imply } y = z$\ data type terms equivalence & $ t_1 \equiv t_2 $ & $ {\mathbf{I}}\models_{{\mathbf{D}}} t_1 \equiv t_2 \Longleftrightarrow t_1^{{\mathbf{D}}} = t_2^{{\mathbf{D}}}$\ data type terms diseq. & $ t_1 \not\equiv t_2 $ & $ {\mathbf{I}}\models_{{\mathbf{D}}} t_1 \not\equiv t_2 \Longleftrightarrow t_1^{{\mathbf{D}}} \neq t_2^{{\mathbf{D}}}$\ data type terms subsum. & $ t_1 \sqsubseteq t_2 $ & $ {\mathbf{I}}\models_{{\mathbf{D}}} (t_1 \sqsubseteq t_2) \Longleftrightarrow t_1^{{\mathbf{D}}} \subseteq t_2^{{\mathbf{D}}} $\ concept assertion & $a : C_1$ & ${\mathbf{I}}\models_{\mathbf{D}}a : C_1 \; \Longleftrightarrow \; (a^{\mathbf{I}}\in C_1^{\mathbf{I}}) $\ agreement & $a=b$ & ${\mathbf{I}}\models_{\mathbf{D}}a=b \; \Longleftrightarrow \; a^{\mathbf{I}}=b^{\mathbf{I}}$\ disagreement & $a \neq b$ & ${\mathbf{I}}\models_{\mathbf{D}}a \neq b \; \Longleftrightarrow \; \neg (a^{\mathbf{I}}= b^{\mathbf{I}})$\ ab. role asser. & $ (a,b) : R $ & ${\mathbf{I}}\models_{\mathbf{D}}(a,b) : R \; \Longleftrightarrow \; \langle a^{\mathbf{I}}, b^{\mathbf{I}}\rangle \in R^{\mathbf{I}}$\ cn. role asser. & $ (a,e_d) : P $ & ${\mathbf{I}}\models_{\mathbf{D}}(a,e_d) : P \; \Longleftrightarrow \; \langle a^{\mathbf{I}}, e_d^{\mathbf{D}}\rangle \in P^{\mathbf{I}}$\ \ Let $\mathcal{R}$, $\mathcal{T}$, and $\mathcal{A}$ be as above. An interpretation ${\mathbf{I}}= (\Delta ^ {\mathbf{I}}, \Delta_{{\mathbf{D}}}, \cdot ^ {\mathbf{I}})$ is a ${\mathbf{D}}$-model of $\mathcal{R}$ (resp., $\mathcal{T}$), and we write ${\mathbf{I}}\models_{{\mathbf{D}}} \mathcal{R}$ (resp., ${\mathbf{I}}\models_{{\mathbf{D}}} \mathcal{T}$), if ${\mathbf{I}}$ satisfies each axiom in $\mathcal{R}$ (resp., $\mathcal{T}$) according to the semantic rules in Table \[semdlss\]. Analogously, ${\mathbf{I}}= (\Delta^ {\mathbf{I}}, \Delta_{{\mathbf{D}}}, \cdot^{\mathbf{I}})$ is a ${\mathbf{D}}$-model of $\mathcal{A}$, and we write ${\mathbf{I}}\models_{{\mathbf{D}}} \mathcal{A}$, if ${\mathbf{I}}$ satisfies each assertion in $\mathcal{A}$, according to the semantic rules in Table \[semdlss\]. A ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-knowledge base $\mathcal{K}=(\mathcal{A}, \mathcal{T}, \mathcal{R})$ is consistent if there is an interpretation ${\mathbf{I}}= (\Delta^ {\mathbf{I}}, \Delta_{{\mathbf{D}}}, \cdot^{\mathbf{I}})$ that is a ${\mathbf{D}}$-model of $\mathcal{A}$, $\mathcal{T}$, and $\mathcal{R}$. Decidability of the consistency problem for ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-knowledge bases was proved in [@CanLonNicSanRR2015] via a reduction to the satisfiability problem for formulae of a four level quantified syllogistic called [$\mathsf{4LQS^R}$]{}. The latter problem was proved decidable in [@CanNic2013]. Some considerations on the expressive power of ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ are in order. As illustrated in [@RR2017ext Table  1] existential quantification is admitted only on the left hand side of inclusion axioms. Thus ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ is less powerful than logics such as ${\mathcal{SROIQ}({\mathbf{D}})}\space$ [@Horrocks2006] for what concerns the generation of new individuals. On the other hand, ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ is more liberal than ${\mathcal{SROIQ}({\mathbf{D}})}\space$ in the definition of role inclusion axioms since roles involved are not required to be subject to any ordering relationship, and the notion of simple role is not needed. For example, the role hierarchy presented in [@Horrocks2006 page  2] is not expressible in ${\mathcal{SROIQ}({\mathbf{D}})}\space$ but can be represented in ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$. In addition, ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ is a powerful rule language able to express rules with negated atoms such as $Person(?p) \wedge \neg hasCar(?p, ?c) \implies CarlessPerson(?p)$. Notice that rules with negated atoms are not supported by the SWRL language. ABox Reasoning services for ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ knowledge base ======================================================================================== The most important feature of a knowledge representation system is the capability of providing reasoning services. Depending on the type of the application domains, there are many different kinds of implicit knowledge that is desirable to infer from what is explicitly mentioned in the knowledge base. In particular, reasoning problems regarding ABoxes consist in querying a knowledge base in order to retrieve information concerning data stored in it. In this section we study the decidability for the most widespread ABox reasoning tasks for the logic ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ resorting to a general problem, called Higher Order Conjuctive Query Answering (HOCQA), that can be instantiated to each of them. Let ${\mathsf{V}_{\mathsf{i}}}= \{{ {\mathsf{v}_{1}} }, { {\mathsf{v}_{2}} }, \ldots\}$, ${\mathsf{V}_{\mathsf{c}}}= \{{ {\mathsf{c}_{1}} }, { {\mathsf{c}_{2}} }, \ldots\}$, ${\mathsf{V}_{\mathsf{ar}}}= \{{ {\mathsf{r}_{1}} }, { {\mathsf{r}_{2}} }, \ldots\}$, and ${\mathsf{V}_{\mathsf{cr}}}= \{{ {\mathsf{p}_{1}} }, { {\mathsf{p}_{2}} }, \ldots\}$ be pairwise disjoint denumerably infinite sets of variables which are disjoint from ${\mathbf{Ind}}$, $\bigcup\{N_C(d): d \in N_{{\mathbf{D}}}\}$, ${\mathbf{C}}$, ${\mathbf{R_A}}$, and ${\mathbf{R_D}}$. A HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-*atomic formula* is an expression of one of the following types: $R(w_1,w_2)$, $P(w_1, u_1)$, $C(w_1)$, $\mathsf{r}(w_1,w_2)$, $\mathsf{p}(w_1, u_1)$, $\mathsf{c}(w_1)$, $w_1=w_2$, $u_1 = u_2$, where $w_1,w_2 \in {\mathsf{V}_{\mathsf{i}}}\cup {\mathbf{Ind}}$, $u_1, u_2 \in {\mathsf{V}_{\mathsf{i}}}\cup \bigcup \{N_C(d): d \in N_{{\mathbf{D}}}\}$, $R$ is a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-abstract role term, $P$ is a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-concrete role term, $C$ is a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-concept term, $\mathsf{r} \in {\mathsf{V}_{\mathsf{ar}}}$, $\mathsf{p} \in {\mathsf{V}_{\mathsf{cr}}}$, and $\mathsf{c} \in {\mathsf{V}_{\mathsf{c}}}$. A HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-atomic formula containing no variables is said to be *ground*. A HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-*literal* is a HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-atomic formula or its negation. A HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-*conjunctive query* is a conjunction of HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-literals. We denote with $\lambda$ the *empty* HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-conjunctive query. Let ${ {\mathsf{v}_{1}} },\ldots,{ {\mathsf{v}_{n}} } \in {\mathsf{V}_{\mathsf{i}}}$, ${ {\mathsf{c}_{1}} }, \ldots, { {\mathsf{c}_{m}} } \in {\mathsf{V}_{\mathsf{c}}}$, ${ {\mathsf{r}_{1}} }, \ldots, { {\mathsf{r}_{k}} } \in {\mathsf{V}_{\mathsf{ar}}}$, ${ {\mathsf{p}_{1}} }, \ldots, { {\mathsf{p}_{h}} } \in {\mathsf{V}_{\mathsf{cr}}}$, $o_1, \ldots, o_n \in {\mathbf{Ind}}\cup \bigcup \{N_C(d): d \in N_{{\mathbf{D}}}\}$, $C_1, \ldots, C_m \in {\mathbf{C}}$, $R_1, \ldots, R_k \in {\mathbf{R_A}}$, and $P_1, \ldots, P_h \in {\mathbf{R_D}}$. A substitution\ $\sigma {\coloneqq}\{{ {\mathsf{v}_{1}} }/o_1, \ldots, { {\mathsf{v}_{n}} }/o_n, { {\mathsf{c}_{1}} }/{C_1}, \ldots, { {\mathsf{c}_{m}} }/{C_m}, { {\mathsf{r}_{1}} }/{R_1}, \ldots, { {\mathsf{r}_{k}} }/{R_k}, { {\mathsf{p}_{1}} } /{P_1}, \ldots, { {\mathsf{p}_{h}} }/{P_h} \} $ is a map such that, for every HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-literal $L$, $L\sigma$ is obtained from $L$ by replacing the occurrences of ${ {\mathsf{v}_{i}} }$ in $L$ with $o_i$, for $i=1, \ldots, n$; the occurrences of ${ {\mathsf{c}_{j}} }$ in $L$ with $C_j$, for $j=1, \ldots, m$; the occurrences of ${ {\mathsf{r}_{\ell}} }$ in $L$ with $R_\ell$, for $\ell=1, \ldots, k$; the occurrences of ${ {\mathsf{p}_{t}} }$ in $L$ with $P_t$, for $t=1, \ldots, h$. Substitutions can be extended to HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-conjunctive queries in the usual way. Let $Q {\coloneqq}(L_1 \wedge \ldots \wedge L_m)$ be a HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-conjunctive query, and ${\mathcal{KB}}$ a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-knowledge base. A substitution $\sigma$ involving *exactly* the variables occurring in $Q$ is a *solution for $Q$ w.r.t. ${\mathcal{KB}}$* if there exists a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-interpretation ${\mathbf{I}}$ such that ${\mathbf{I}}\models_{{\mathbf{D}}} {\mathcal{KB}}$ and ${\mathbf{I}}\models_{{\mathbf{D}}} Q \sigma$. The collection $\Sigma$ of the solutions for $Q$ w.r.t. ${\mathcal{KB}}$ is the *higher order (HO) answer set of $Q$ w.r.t. ${\mathcal{KB}}$*. Then the *higher order conjunctive query answering* (HOCQA) problem for $Q$ w.r.t. ${\mathcal{KB}}$ consists in finding the HO answer set $\Sigma$ of $Q$ w.r.t. ${\mathcal{KB}}$. We shall solve the HOCQA problem just stated by reducing it to the analogous problem formulated in the context of the fragment ${\ensuremath{\mathsf{4LQS^R}}}$ (and in turn to the decision procedure for ${\ensuremath{\mathsf{4LQS^R}}}$ presented in [@CanNic2013]). The HOCQA problem for ${\ensuremath{\mathsf{4LQS^R}}}$-formulae can be stated as follows. Let $\phi$ be a ${\ensuremath{\mathsf{4LQS^R}}}$-formula and let $\psi$ be a conjunction of ${\ensuremath{\mathsf{4LQS^R}}}$-quantifier-free atomic formulae of level $0$ of the types $x=y$, $x \in X^1$, $ \langle x,y \rangle \in X^3$, or their negations. The *HOCQA problem for $\psi$ w.r.t. $\phi$* consists in computing the HO *answer set of $\psi$ w.r.t. $\phi$*, namely the collection $\Sigma'$ of all the substitutions $\sigma'$ such that ${\mathbfcal{M}}\models \phi \wedge \psi\sigma'$, for some ${\ensuremath{\mathsf{4LQS^R}}}$-interpretation ${\mathbfcal{M}}$. In view of the decidability of the satisfiability problem for ${\ensuremath{\mathsf{4LQS^R}}}$-formulae, the HOCQA problem for ${\ensuremath{\mathsf{4LQS^R}}}$-formulae is decidable as well. Indeed, let $\phi$ and $\psi$ be two ${\ensuremath{\mathsf{4LQS^R}}}$-formulae fulfilling the above requirements. To calculate the HO answer set of $\psi$ w.r.t. $\phi$, for each candidate substitution\ $\sigma' {\coloneqq}\{ \vec{x} / \vec{z}, \vec{X^1} / \vec{Y^1}, \vec{X^2} / \vec{Y^2}, \vec{X^3} / \vec{Y^3} \}$ \ one has just to check for satisfiability of the ${\ensuremath{\mathsf{4LQS^R}}}$-formula $\phi \wedge \psi\sigma'$. Since the number of possible candidate substitutions is $\left| {\mathtt{Vars}}(\phi) \right|^{\left| {\mathtt{Vars}}(\psi) \right|}$ and the satisfiability problem for ${\ensuremath{\mathsf{4LQS^R}}}$-formulae is decidable, the HO answer set of $\psi$ w.r.t. $\phi$ can be computed effectively. Summarizing, \[CQA4LQSR\] The HOCQA problem for ${\ensuremath{\mathsf{4LQS^R}}}$-formulae is decidable. The following theorem states decidability of the HOCQA problem for ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$. \[CQADL\] Given a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-knowledge base ${\mathcal{KB}}$ and a HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$- conjunctive query $Q$, the HOCQA problem for $Q$ w.r.t. ${\mathcal{KB}}$ is decidable. We first outline the main ideas and then we provide a formal proof of the theorem. In order to define a ${\ensuremath{\mathsf{4LQS^R}}}formula$ $\phi_{\mathcal{KB}}$, we recall the definition a function $\theta$ that maps the ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-knowledge base ${\mathcal{KB}}$ in the ${\ensuremath{\mathsf{4LQS^R}}}$-formula in Conjunctive Normal Form (CNF) $\phi_{{\mathcal{KB}}}$, introduced in [@ExtendedVersionICTCS2016]. The definition of the mapping $\theta$ is inspired to the definition of the mapping $\tau$ introduced in the proof of Theorem 1 in [@CanLonNicSanRR2015]. Specifically, $\theta$ differs from $\tau$ because it allows quantification only on variables of level $0$, it treats Boolean operations on concrete roles and the product of concepts, it constructs ${\ensuremath{\mathsf{4LQS^R}}}$-formulae in CNF and it is extended to ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-HO conjunctive queries. To prepare for the definition of $\theta$, we map injectively individuals $a\in {\mathbf{Ind}}$ and constants $e_d \in N_{C}(d)$ into level $0$ variables $x_a$, $x_{e_d}$, the constant concepts $\top$ and $\bot$, data type terms $t$, and concept terms $C$ into level $1$ variables $X_{\top}^1$, $X_{\bot}^1$, $X_{t}^1$, $X_{C}^1$, respectively, and the universal relation on individuals $U$, abstract role terms $R$, and concrete role terms $P$ into level $3$ variables $X_{U}^3$, $X_{R}^3$, and $X_{P}^3$, respectively.[^2] Then the mapping $\theta$ is defined as follows: $\theta(C_1 \equiv \top) {\coloneqq}(\forall z)( ( \neg(z \in X_{C_1}^1) \vee z \in X_{\top}^1) \wedge ( \neg(z \in X_{\top}^1) \vee z \in X_{C_1}^1))$, $\theta(C_1 \equiv \neg C_2) {\coloneqq}(\forall z)(( \neg(z \in X_{C_1}^1) \vee \neg(z \in X_{C_2}^1)) \wedge (z \in X_{C_2}^1 \vee z \in X_{C_1}^1))$, $\theta(C_1 \equiv C_2 \sqcup C_3 ) {\coloneqq}(\forall z)( ( \neg(z \in X_{C_1}^1) \vee (z \in X_{C_2}^1 \vee z \in X_{C_3}^1)) \wedge ( (\neg (z \in X_{C_2}^1) \vee z \in X_{C_1}^1) \wedge (\neg (z \in X_{C_3}^1) \vee z \in X_{C_1}^1 ))$, $\theta(C_1 \equiv \{a\}) {\coloneqq}(\forall z)( \neg(z \in X_{C_1}^1) \vee z = x_a) \wedge( \neg(z = x_a) \vee z \in X_{C_1}^1 )$, $\theta(C_1 \sqsubseteq \forall R_1.C_2) {\coloneqq}(\forall z_1)(\forall z_2)( \neg(z_1 \in X_{C_1}^1) \vee ( \neg(\langle z_1,z_2 \rangle \in X_{R_1}^3) \vee z_2 \in X_{C_2}^1))$, $\theta(\exists R_1.C_1 \sqsubseteq C_2) {\coloneqq}(\forall z_1)(\forall z_2)(( \neg(\langle z_1,z_2 \rangle \in X_{R_1}^3) \vee \neg( z_2 \in X_{C_1}^1)) \vee z_1 \in X_{C_2}^1)$, $\theta(C_1 \equiv \exists R_1.\{a\}) {\coloneqq}(\forall z)( ( \neg(z \in X_{C_1}^1) \vee \langle z,x_{a}\rangle \in X_{R_1}^3) \wedge ( \neg(\langle z,x_{a}\rangle \in X_{R_1}^3) \vee z \in X_{C_1}^1 ) )$, $\theta(C_1 \sqsubseteq \leq_n\!\! R_1.C_2) {\coloneqq}(\forall z)(\forall z_1)\ldots (\forall z_{n+1})( \neg(z \in X_{C_1}^1) \vee ( \overset{n+1}{\underset{i=1}\bigwedge}( \neg(z_i \in X_{C_2}) \vee \neg(\langle z,z_i\rangle \in X_{R_1}^3) \vee \underset {i<j} {\bigvee} z_i = z_j))$, $\theta(\geq_n\!\! R_1.C_1 \sqsubseteq C_2) {\coloneqq}(\forall z)(\forall z_1)\ldots (\forall z_{n})( \overset {n}{ \underset{i=1}\bigwedge }(( \neg(z_i \in X_{C_1}^1) \vee \neg( \langle z,z_i\rangle \in X_{R_1}^3)) \vee \underset {i<j} \bigvee z_i = z_j) \vee z \in X_{C_2}^1)$, $\theta(C_1 \sqsubseteq \forall P_1.t_1) {\coloneqq}(\forall z_1)(\forall z_2)( \neg(z_1 \in X_{C_1}^1) \vee ( \neg (\langle z_1,z_2 \rangle \in X_{P_1}^3) \vee z_2 \in X_{t_1}^1))$, $\theta(\exists P_1.t_1 \sqsubseteq C_1) {\coloneqq}(\forall z_1)(\forall z_2)(( \neg(\langle z_1,z_2 \rangle \in X_{P_1}^3) \vee \neg(z_2 \in X_{t_1}^1)) \vee z_1 \in X_{C_1}^1)$, $\theta(C_1 \equiv \exists P_1.\{e_{d}\}) {\coloneqq}(\forall z)( ( \neg(z \in X_{C_1}^1) \vee \langle z,x_{e_{d}}\rangle \in X_{P_1}^3) \wedge ( \neg(\langle z,x_{e_{d}}\rangle \in X_{P_1}^3) \vee z \in X_{C_1}^1) )$, $\theta(C_1 \sqsubseteq \leq_n\!\! P_1.t_1) {\coloneqq}(\forall z)(\forall z_1)\ldots (\forall z_{n+1})( \neg (z \in X_{C_1}^1) \vee ( \overset {n+1} { \underset{i=1}\bigwedge }( \neg(z_i \in X_{t_1}) \vee \neg(\langle z,z_i\rangle \in X_{P_1}^3) \vee \underset{i<j} {\bigvee} z_i = z_j))$, $\theta(\geq_n\!\! P_1.t_1 \sqsubseteq C_1) {\coloneqq}(\forall z)(\forall z_1)\ldots (\forall z_{n})( \overset {n} { \underset {i=1}\bigwedge}(( \neg(z_i \in X_{t_1}^1) \vee \neg(\langle z,z_i\rangle \in X_{P_1}^3)) \vee \underset {i<j} {\bigvee} z_i = z_j) \vee z \in X_{C_1}^1)$, $\theta(R_1 \equiv U) {\coloneqq}(\forall z_1)(\forall z_2)( ( \neg(\langle z_1,z_2\rangle \in X_{R_1}^3) \vee \langle z_1,z_2\rangle \in X_{U}^3) \wedge ( \neg(\langle z_1,z_2\rangle \in X_{U}^3) \vee \langle z_1,z_2\rangle \in X_{R_1}^3) )$, $\theta(R_1 \equiv \neg R_2) {\coloneqq}(\forall z_1)(\forall z_2)( ( \neg(\langle z_1,z_2\rangle \in X_{R_1}^3) \vee \neg (\langle z_1,z_2\rangle \in X_{R_2}^3 )) \wedge ( \langle z_1,z_2\rangle \in X_{R_2}^3 \vee \neg (\langle z_1,z_2\rangle \in X_{R_1}^3 )) )$, $\theta( R \equiv C_1 \times C_2 ) {\coloneqq}(\forall z_1)(\forall z_2) ( \neg (\langle z_1, z_2 \rangle \in X^3_R) \vee z_1 \in X^1_{C_1}) \wedge ( \neg (\langle z_1, z_2 \rangle \in X^3_R) \vee z_2 \in X^1_{C_2} ) \wedge (( \neg(z_1 \in X^1_{C_1}) \vee \neg(z_2 \in X^1_{C_2}) ) \vee \langle z_1, z_2 \rangle \in X^3_R ) )$ $\theta(R_1 \equiv R_2 \sqcup R_3) {\coloneqq}(\forall z_1)(\forall z_2)( ( \neg(\langle z_1,z_2 \rangle \in X_{R_1}^3) \vee (\langle z_1,z_2 \rangle \in X_{R_2}^3 \vee \langle z_1,z_2 \rangle \in X_{R_3}^3)) \wedge ( ( \neg(\langle z_1,z_2 \rangle \in X_{R_2}^3) \vee \langle z_1,z_2 \rangle \in X_{R_1}^3) \wedge ((\neg(\langle z_1,z_2 \rangle \in X_{R_3}^3) \vee \langle z_1,z_2 \rangle \in X_{R_1}^3) ) ))$, $\theta(R_1 \equiv R_2^{-}) {\coloneqq}(\forall z_1)(\forall z_2)( ( \neg(\langle z_1,z_2\rangle \in X_{R_1}^3) \vee \langle z_2,z_1\rangle \in X_{R_2}^3 ) \wedge ( \neg(\langle z_2,z_1\rangle \in X_{R_2}^3) \vee \langle z_1,z_2\rangle \in X_{R_1}^3 ) ) $, $\theta(R_1 \equiv id(C_1)) {\coloneqq}(\forall z_1)(\forall z_2)( ( ( \neg(\langle z_1,z_2\rangle \in X_{R_1}^3) \vee z_1 \in X_{C_1}^1 ) \wedge ( \neg(\langle z_1,z_2\rangle \in X_{R_1}^3) \vee z_2 \in X_{C_1}^1 ) \wedge (\neg(\langle z_1,z_2\rangle \in X_{R_1}^3) \vee z_1 =z_2) )\wedge ( ( \neg(z_1 \in X_{C_1}^1) \vee \neg(z_2 \in X_{C_1}^1) \vee z_1 \neq z_2) \vee \langle z_1,z_2\rangle \in X_{R_1}^3) )$, $\theta(R_1 \equiv R_{2_{C_1 |}}) {\coloneqq}(\forall z_1)(\forall z_2)( ( (\neg(\langle z_1,z_2\rangle \in X_{R_1}^3) \vee \langle z_1,z_2\rangle \in X_{R_2}^3) \wedge ( \neg(\langle z_1,z_2\rangle \in X_{R_1}^3) \vee z_1 \in X_{C_1}^1)) \wedge (( \neg(\langle z_1,z_2\rangle \in X_{R_2}^3) \vee \neg(z_1 \in X_{C_1}^1)) \vee \langle z_1,z_2\rangle \in X_{R_1}^3 ) )$, $\theta(R_1 \ldots R_n \sqsubseteq R_{n+1}) {\coloneqq}(\forall z)(\forall z_1)\ldots (\forall z_{n}) (( \neg(\langle z, z_1\rangle \in X_{R_1}^3) \vee \ldots \vee \neg(\langle z_{n-1},z_n\rangle \in X_{R_{n}}^3)) \vee \langle z, z_n\rangle\in X_{R_{n+1}}^3)$, $\theta({\mathsf{Ref}}(R_1)) {\coloneqq}(\forall z)(\langle z, z\rangle \in X_{R_1}^3)$, $\theta({\mathsf{Irref}}(R_1)) {\coloneqq}(\forall z)(\neg (\langle z,z\rangle \in X_{R_1}^3))$, $\theta({\mathsf{Fun}}(R_1)) {\coloneqq}(\forall z_1)(\forall z_2)(\forall z_3)(( \neg(\langle z_1,z_2\rangle \in X_{R_1}^3) \vee \neg(\langle z_1,z_3\rangle \in X_{R_1}^3)) \vee z_2 =z_3)$, $\theta(P_1 \equiv P_2) {\coloneqq}(\forall z_1)(\forall z_2)( ( \neg (\langle z_1,z_2\rangle \in X_{P_1}^3) \vee \langle z_1,z_2\rangle \in X_{P_2}^3) \wedge ( \neg(\langle z_1,z_2\rangle \in X_{P_2}^3) \vee \langle z_1,z_2\rangle \in X_{P_1}^3) )$, $\theta(P_1 \equiv \neg P_2) {\coloneqq}(\forall z_1)(\forall z_2)( ( \neg(\langle z_1,z_2\rangle \in X_{P_1}^3) \vee \neg(\langle z_1,z_2\rangle \in X_{P_2}^3)) \wedge (\langle z_1,z_2\rangle \in X_{P_2}^3 \vee \langle z_1,z_2\rangle \in X_{P_1}^3 ) )$, $\theta(P_1 \sqsubseteq P_2) {\coloneqq}(\forall z_1)(\forall z_2)( \neg(\langle z_1,z_2\rangle \in X_{P_1}^3) \vee \langle z_1,z_2\rangle \in X_{P_2}^3)$, $\theta({\mathsf{Fun}}(P_1)) {\coloneqq}(\forall z_1)(\forall z_2)(\forall z_3)( ( \neg(\langle z_1,z_2\rangle \in X_{P_1}^3) \vee \neg(\langle z_1,z_3\rangle \in X_{P_1}^3) \vee z_2 =z_3)$, $\theta(P_1 \equiv P_{2_{C_1 |}}) {\coloneqq}(\forall z_1)(\forall z_2) (( \neg(\langle z_1,z_2\rangle \in X_{P_1}^3) \vee \langle z_1,z_2\rangle \in X_{P_2}^3 ) \wedge ( \neg(\langle z_1,z_2\rangle \in X_{P_1}^3) \vee z_1 \in X_{C_1}^1) \wedge ( (\neg \langle z_1,z_2\rangle \in X_{P_2}^3) \vee \neg(z_1 \in X_{C_1}^1) \vee \langle z_1,z_2\rangle \in X_{P_1}^3) )$, $\theta(P_1 \equiv P_{2_{|t_1}}) {\coloneqq}(\forall z_1)(\forall z_2) ( ( \neg(\langle z_1,z_2\rangle \in X_{P_1}^3) \vee \langle z_1,z_2\rangle \in X_{P_2}^3 ) \wedge ( \neg(\langle z_1,z_2\rangle \in X_{P_1}^3) \vee z_2 \in X_{t_1}^1) \wedge ( ( \neg(\langle z_1,z_2\rangle \in X_{P_2}^3) \vee \neg(z_2 \in X_{t_1}^1)) \vee \langle z_1,z_2\rangle \in X_{P_1}^3 ) )$, $\theta(P_1 \equiv P_{2_{C_1|t_1}}) {\coloneqq}(\forall z_1)(\forall z_2) ( ( \neg(\langle z_1,z_2\rangle \in X_{P_1}^3) \vee \langle z_1,z_2\rangle \in X_{P_2}^3 ) \wedge ( \neg(\langle z_1,z_2\rangle \in X_{P_1}^3) \vee z_1 \in X_{C_1}^1) \wedge ( \neg(\langle z_1,z_2\rangle \in X_{P_1}^3) \vee z_2 \in X_{t_1}^1) \wedge ( \neg(\langle z_1,z_2\rangle \in X_{P_2}^3) \vee \neg(z_1 \in X_{C_1}^1) \vee \neg(z_2 \in X_{t_1}^1) \vee \langle z_1,z_2\rangle \in X_{P_1}^3) )$, $\theta(t_1 \equiv t_2){\coloneqq}(\forall z)(( \neg(z \in X_{t_1}^1) \vee z \in X_{t_2}^1) \wedge ( \neg(z \in X_{t_2}^1) \vee z \in X_{t_1}^1 ))$, $\theta(t_1 \equiv \neg t_2){\coloneqq}(\forall z)(( \neg(z \in X_{t_1}^1) \vee \neg (z \in X_{t_2}^1)) \wedge ( z \in X_{t_2}^1 \vee z \in X_{t_1}^1) )$, $\theta(t_1 \equiv t_2 \sqcup t_3){\coloneqq}(\forall z)( ( \neg(z \in X_{t_1}^1) \vee (z \in X_{t_2}^1\vee z \in X_{t_3}^1)) \wedge ( ( \neg (z \in X_{t_2}^1) \vee z \in X_{t_1}^1) \wedge (\neg (z \in X_{t_3}^1) \vee z \in X_{t_1}^1) ) )$, $\theta(t_1 \equiv t_2 \sqcap t_3){\coloneqq}(\forall z)(( \neg(z \in X_{t_1}^1) \vee (z \in X_{t_2}^1 \wedge z \in X_{t_3}^1)) \wedge ((( \neg(z \in X_{t_2}^1) \vee \neg(z \in X_{t_3}^1)) \vee z \in X_{t_1}^1) )$, $\theta(t_1 \equiv \{e_{d}\}){\coloneqq}(\forall z)((\neg (z \in X_{t_1}^1) \vee z = x_{e_{d}}) \wedge ( \neg(z = x_{e_{d}}) \vee z \in X_{t_1}^1) )$, $\theta(a : C_1) {\coloneqq}x_a \in X_{C_1}^1$, $\theta((a,b) : R_1) {\coloneqq}\langle x_a, x_b\rangle \in X_{R_1}^3$, $\theta((a,b) : \neg R_1) {\coloneqq}\neg(\langle x_a, x_b\rangle \in X_{R_1}^3)$, $\theta(a=b) {\coloneqq}x_a = x_b$, $\theta(a\neq b) {\coloneqq}\neg (x_a = x_b)$, $\theta(e_d : t_1) {\coloneqq}x_{e_d} \in X_{t_1}^1$, $\theta((a,e_d) : P_1) {\coloneqq}\langle x_a, x_{e_d}\rangle \in X_{P_1}^3$, $\theta((a,e_d) : \neg P_1) {\coloneqq}\neg(\langle x_a, x_{e_d}\rangle \in X_{P_1}^3)$, $\theta(\alpha \wedge \beta) {\coloneqq}\theta(\alpha) \wedge \theta(\beta)$. Let $\mathcal{KB}$ be our ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-knowledge base, and let ${\mathsf{cpt}_\mathcal{KB}}$, ${\mathsf{arl}_\mathcal{KB}}$, ${\mathsf{crl}_\mathcal{KB}}$, and ${\mathsf{ind}_\mathcal{KB}}$ be, respectively, the sets of concept, of abstract role, of concrete role, and of individual names in $\mathcal{KB}$. Moreover, let $N_{D}^\mathcal{KB} \subseteq N_{D}$ be the set of data types in $\mathcal{KB}$, $N_{F}^\mathcal{KB}$ a restriction of $N_{F}$ assigning to every $d \in N_{{\mathbf{D}}}^\mathcal{KB}$ the set $N_{F}^\mathcal{KB}(d)$ of facets in $N_{F}(d)$ and in $\mathcal{KB}$. Analogously, let $N_{C}^{\mathcal{KB}}$ be a restriction of the function $N_{C}$ associating to every $d \in N_{{\mathbf{D}}}^\mathcal{KB}$ the set $N_{C}^\mathcal{KB}(d)$ of constants contained in $N_{C}(d)$ and in $\mathcal{KB}$. Finally, for every data type $d \in N_{D}^\mathcal{KB}$, let ${\mathsf{bf}_\mathcal{KB}^{{\mathbf{D}}}}(d)$ be the set of facet expressions for $d$ occurring in $\mathcal{KB}$ and not in $N_{F}(d) \cup \{\top^{d},\bot_{d}\}$. We assume without loss of generality that the facet expressions in ${\mathsf{bf}_\mathcal{KB}^{{\mathbf{D}}}}(d)$ are in Conjunctive Normal Form. We define the [$\mathsf{4LQS^R}$]{}-formula $\phi_{\mathcal{KB}}$ expressing the consistency of $\mathcal{KB}$ as follows: [$$\phi_{\mathcal{KB}} {\coloneqq}\underset {H \in \mathcal{KB}}\bigwedge \theta(H) \wedge \bigwedge_{i=1}^{12}\xi_i \, ,$$]{} where $\xi_1 {\coloneqq}(\forall z)( ( \neg(z \in X_{{\mathbf{I}}}^1) \vee \neg(z \in X_{{\mathbf{D}}}^1)) \wedge (z \in X_{{\mathbf{D}}}^1 \vee z \in X_{{\mathbf{I}}}^1))\wedge (\forall z)(z \in $ $\hfill X_{{\mathbf{I}}}^1 \vee z \in X_{{\mathbf{D}}}^1)\wedge \neg (\forall z)\neg (z \in X_{{\mathbf{I}}}^1) \wedge \neg (\forall z)\neg (z \in X_{{\mathbf{D}}}^1)$,\ $\xi_2 {\coloneqq}((\forall z)( ( \neg(z \in X_{{\mathbf{I}}}^1) \vee z \in X_{\top}^1) \wedge (\neg (z \in X_{\top}^1) \vee z \in X_{{\mathbf{I}}}^1) ) \wedge (\forall z)\neg (z \in$ $\hfill X_{\bot})$,\ $\xi_3 {\coloneqq}\underset{A \in {\mathsf{cpt}_\mathcal{KB}}}\bigwedge (\forall z)( \neg(z \in X_{A}^1) \vee z \in X_{{\mathbf{I}}}^1)$,\ $\xi_4 {\coloneqq}( \underset{d \in N_{D}^\mathcal{KB}}\bigwedge((\forall z)( \neg(z \in X_{d}^1) \vee z \in X_{{\mathbf{D}}}^1) \wedge \neg (\forall z)\neg(z \in X_{d}^1)) \wedge (\forall z)$ $\hfill (\underset{(d_i,d_j \in N_{D}^\mathcal{KB}, i < j)}\bigwedge ( ( \neg(z \in X_{d_i}^1) \vee \neg (z \in X_{d_j}^1)) \wedge ( z \in X_{d_j}^1 \vee z \in X_{d_i}^1 ) )))$,\ $\xi_5 {\coloneqq}\underset{d \in N_{D}^\mathcal{KB}}\bigwedge((\forall z)( ( \neg(z \in X_{d}^1) \vee z \in X_{\top_d}^1) \wedge ( \neg(z \in X_{\top_d}^1) \vee z \in X_{d}^1 ) \wedge $ $\hfill (\forall z)\neg(z \in X_{\bot_d}^1))$,\ $\xi_6 {\coloneqq}\underset {\substack{\\ f_d \in N_{F}^\mathcal{KB}(d),\\ d \in N_{D}^\mathcal{KB}}}\bigwedge (\forall z)( \neg(z \in X_{f_d}^1) \vee z \in X_{d}^1)$,\ $\xi_7 {\coloneqq}(\forall z_1)(\forall z_2)( ( \neg(z_1 \in X_{{\mathbf{I}}}^1) \vee \neg(z_2 \in X_{{\mathbf{I}}}^1) \vee \langle z_1,z_2 \rangle \in X_{U}^3) \wedge ( (\neg(\langle z_1,z_2 \rangle \in$ $\hfill X_{U}^3) \vee z_1 \in X_{{\mathbf{I}}}^1) \wedge ( \neg(\langle z_1,z_2 \rangle \in X_{U}^3) \vee z_2 \in X_{{\mathbf{I}}}^1 ) ) )$,\ $\xi_8 {\coloneqq}\underset {R \in {\mathsf{arl}_\mathcal{KB}}} \bigwedge (\forall z_1)(\forall z_2) ( (\neg(\langle z_1,z_2\rangle \in X_R^3) \vee z_1 \in X_{{\mathbf{I}}}^1 ) \wedge ( \neg(\langle z_1,z_2\rangle \in X_R^3) \vee z_2 \in X_{{\mathbf{I}}}^1)))$,\ $\xi_9 {\coloneqq}\underset{T \in {\mathsf{crl}_\mathcal{KB}}} \bigwedge (\forall z_1)(\forall z_2) (\neg(\langle z_1,z_2\rangle \in X_T^3) \vee z_1 \in X_{{\mathbf{I}}}^1) \wedge ( \neg(\langle z_1,z_2\rangle \in X_T^3) \vee z_2 \in X_{{\mathbf{D}}}^1)))$,\ $\xi_{10} {\coloneqq}\underset{a \in {\mathsf{ind}_\mathcal{KB}}} \bigwedge(x_a \in X_{{\mathbf{I}}}^1) \wedge \underset { \substack{ \\ d \in N_{D}^\mathcal{KB}, \\{e_d \in N_{C}^\mathcal{KB}(d)}}} \bigwedge x_{e_d} \in X_{d}^1$,\ $\xi_{11} {\coloneqq}\underset {\{e_{d_1},\ldots, e_{d_n}\} \textrm{ in } \mathcal{KB}} \bigwedge (\forall z) ( ( \neg(z \in X_{\{e_{d_1},\ldots, e_{d_n}\}}^1) \vee \overset{n} { \underset {i=1} \bigvee }(z = x_{e_{d_i}})) \wedge ( \overset{n} { \underset {i=1} \bigwedge }(z \neq$ $\hfill x_{e_{d_i}} \vee z \in X_{\{e_{d_1},\ldots, e_{d_n}\}}^1 ) ) ) \wedge \quad \underset {\{a_{1},\ldots, a_{n}\} \textrm{ in } \mathcal{KB}} \bigwedge (\forall z)( (\neg(z \in X_{\{a_{1},\ldots, a_{n}\}}^1) \vee $ $\hfill \overset {n}{\underset {i=1} \bigvee}(z = x_{a_{i}})) \wedge ( \overset {n}{\underset {i=1} \bigwedge}(z \neq x_{a_{i}} \vee z \in X_{\{a_{1},\ldots, a_{n}\}}^1)) )$,\ $\xi_{12} {\coloneqq}\underset { \substack{d \in N_{{\mathbf{D}}}^\mathcal{KB},\\[1.5pt] {\psi_d \in {\mathsf{bf}_\mathcal{KB}^{{\mathbf{D}}}}(d)}}} \bigwedge (\forall z) ( \neg(z \in X_{\psi_d}^1) \vee z \in \zeta(X_{\psi_d}^1)) \wedge ( \neg(z \in \zeta(X_{\psi_d}^1)) \vee z \in X_{\psi_d}^1 )$ with $\zeta$ the transformation function from [$\mathsf{4LQS^R}$]{}-variables of level 1 to [$\mathsf{4LQS^R}$]{}-formulae recursively defined, for $d \in N_{\mathbf{D}}^\mathcal{KB}$, by $${ \zeta(X_{\psi_d}^1) {\coloneqq}\begin{cases} X_{\psi_d}^1 & \text{if } \psi_d \in N_{F}^\mathcal{KB}(d) \cup \{\top^{d},\bot_{d}\}\\ \neg \zeta(X_{\chi_d}^1) & \text{if } \psi_d = \neg \chi_d\\ \zeta(X_{\chi_d}^1) \wedge \zeta(X_{\varphi_d}^1) & \text{if } \psi_d = \chi_d \wedge \varphi_d\\ \zeta(X_{\chi_d}^1) \vee \zeta(X_{\varphi_d}^1) & \text{if } \psi_d = \chi_d \vee \varphi_d\,. \end{cases} }$$ In the above formulae, the variable $X_{{\mathbf{I}}}^1$ denotes the set of individuals ${\mathbf{Ind}}$, $X_{d}^1$ a data type $d \in N_{D}^\mathcal{KB}$, $X_{{\mathbf{D}}}^1$ a superset of the union of data types in $ N_{D}^\mathcal{KB}$, $X_{\top_d}^1$ and $X_{\bot_d}^1$ the constants $\top_d$ and $\bot_d$, and $X_{f_d}^1$, $X_{\psi_d}^1$ a facet $f_d$ and a facet expression $\psi_d$, for $d \in N_{D}^\mathcal{KB}$, respectively. In addition, $X_{A}^1$, $X_{R}^3$, $X_{T}^3$ denote a concept name $A$, an abstract role name $R$, and a concrete role name $T$ occurring in $\mathcal{KB}$, respectively. Finally, $X_{\{e_{d_1},\ldots,e_{d_n}\}}^1$ denotes a data range $\{e_{d_1},\ldots,e_{d_n}\}$ occurring in $\mathcal{KB}$, and $X_{\{a_{1},\ldots,a_{n}\}}^1$ a finite set $\{a_1,\ldots,a_n\}$ of nominals in $\mathcal{KB}$. The constraints $\xi_1-\xi_{12}$, slightly different from the constraints $\psi_1-\psi_{12}$ defined in the proof of Theorem 1 in [@CanLonNicSanRR2015], are introduced to guarantee that each model of $\phi_{{\mathcal{KB}}}$ can be easily transformed in a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-interpretation. The HOCQA problem for ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ can be solved via an effective reduction to the HOCQA problem for ${\ensuremath{\mathsf{4LQS^R}}}$-formulae, and then exploiting Lemma \[CQA4LQSR\]. The reduction is accomplished through the function $\theta$ extended in order to map also ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-conjunctive queries into ${\ensuremath{\mathsf{4LQS^R}}}$-formulae in conjunctive normal form (CNF), which can be used to map effectively HOCQA problems from the ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-context into the ${\ensuremath{\mathsf{4LQS^R}}}$-context. More specifically, given a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-knowledge base ${\mathcal{KB}}$ and a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-HO conjunctive query $Q$, using the function $\theta$ we can effectively construct the following ${\ensuremath{\mathsf{4LQS^R}}}$-formulae in CNF:\ $\phi_{{\mathcal{KB}}} {\coloneqq}\bigwedge_{H \in {\mathcal{KB}}} \theta(H) \wedge \bigwedge_{i=1}^{12} \xi_i, \qquad \psi_Q {\coloneqq}\theta(Q)\,.$ \ Then, if we denote by $\Sigma$ the higher order answer set of $Q$ w.r.t. ${\mathcal{KB}}$ and by $\Sigma'$ the higher order answer set of $\psi_Q$ w.r.t. $\phi_{{\mathcal{KB}}}$, we have that $\Sigma$ consists of all substitutions $\sigma$ (involving exactly the variables occurring in $Q$) such that $\theta(\sigma) \in \Sigma'$. Since, by Lemma \[CQA4LQSR\], $\Sigma'$ can be computed effectively, then $\Sigma$ can be computed effectively too. The mapping $\theta$ is extended for ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-HO conjuctive queries as follows.\ $\theta (R_1 (w_1,w_2)) {\coloneqq}\langle x_{w_1}, x_{w_2} \rangle \in X^3_{R_1}$,\ $\theta (P_1 (w_1,u_1)) {\coloneqq}\langle x_{w_1}, x_{u_1} \rangle \in X^3_{P_1}$,\ $\theta ( C_1(w_1) {\coloneqq}x_{w_1} \in X^1_{C_1}$,\ $\theta (w_1 =w_2) {\coloneqq}x_{w_1} = x_{w_2}$,\ $\theta (u_1 =u_2) {\coloneqq}x_{u_1} = x_{u_2}$.\ $\theta ({ {\mathsf{c}_{1}} } (w_1)) {\coloneqq}w_1 \in X^1_{{ {\mathsf{c}_{1}} }}$.\ $\theta ({ {\mathsf{r}_{1}} } (w_1,w_2)) {\coloneqq}\langle w_1, w_2 \rangle \in X^3_{{ {\mathsf{r}_{1}} }}$.\ $\theta ({ {\mathsf{p}_{1}} } (w_1,u_1)) {\coloneqq}\langle w_1, u_1 \rangle \in X^3_{{ {\mathsf{p}_{1}} }}$.\ To complete, we extend the mapping $\theta$ on substitutions $$\sigma {\coloneqq}\{ { {\mathsf{v}_{1}} }/o_1, \ldots { {\mathsf{v}_{n}} } / o_n, { {\mathsf{c}_{1}} } / C_1, \ldots, { {\mathsf{c}_{m}} } / C_m, { {\mathsf{r}_{1}} } / R_1, \ldots, { {\mathsf{r}_{k}} }, / R_k, { {\mathsf{p}_{1}} } / P_1, \ldots { {\mathsf{p}_{h}} } / P_h \}$$ with ${ {\mathsf{v}_{1}} },\ldots,{ {\mathsf{v}_{n}} } \in {\mathsf{V}_{\mathsf{i}}}$, ${ {\mathsf{c}_{1}} }, \ldots, { {\mathsf{c}_{m}} } \in {\mathsf{V}_{\mathsf{c}}}$, ${ {\mathsf{r}_{1}} }, \ldots, { {\mathsf{r}_{k}} } \in {\mathsf{V}_{\mathsf{ar}}}$, ${ {\mathsf{p}_{1}} }, \ldots, { {\mathsf{p}_{h}} } \in {\mathsf{V}_{\mathsf{cr}}}$, $o_1, \ldots, o_n \in {\mathbf{Ind}}\cup \bigcup \{N_C(d): d \in N_{{\mathbf{D}}}\}$, $C_1, \ldots, C_m \in {\mathbf{C}}$, $R_1, \ldots, R_k \in {\mathbf{R_A}}$, and $P_1, \ldots, P_h \in {\mathbf{R_D}}$. We put $$\label{sigma1} \begin{split} \theta(\sigma)= & \theta ( \{ { {\mathsf{v}_{1}} }/o_1, \ldots { {\mathsf{v}_{n}} } / o_n, { {\mathsf{c}_{1}} } / C_1, \ldots, { {\mathsf{c}_{m}} } / C_m, { {\mathsf{r}_{1}} } / R_1, \ldots, { {\mathsf{r}_{k}} }, / R_k, \\ & { {\mathsf{p}_{1}} } / P_1, \ldots { {\mathsf{p}_{h}} } / P_h \}) \\ = & \{ x_{{ {\mathsf{v}_{1}} }}/x_{o_1}, \ldots, x_{ {\mathsf{v}_{n}} } / x_{o_n}, X^1_{ {\mathsf{c}_{1}} } / X^1_{C_1}, \ldots, X^1_{ {\mathsf{c}_{m}} } / X^1_{C_m}, X^3_{ {\mathsf{r}_{1}} } / X^3_{R_1}, \ldots, X^3_{ {\mathsf{r}_{k}} }, / X^3_{R_k},\\ & X^3_{{ {\mathsf{p}_{1}} }} / X^3_{P_1}, \ldots, X^3_{ {\mathsf{p}_{h}} } / X^3_{P_h} \} \\ = & \sigma' \\ \end{split}$$ where $ x_{{ {\mathsf{v}_{1}} }}, \ldots x_{ {\mathsf{v}_{n}} }$, $x_{o_1}, \ldots, x_{o_n}$ are variables of level $0$, $X^1_{ {\mathsf{c}_{1}} }, \ldots, X^1_{ {\mathsf{c}_{m}} }$, $X^1_{C_1}, \ldots, X^1_{C_m}$ are variables of level $1$, $ X^3_{ {\mathsf{r}_{1}} }, \ldots, X^3_{ {\mathsf{r}_{k}} }$, $X^3_{ {\mathsf{p}_{1}} }, \ldots, X^3_{ {\mathsf{p}_{h}} }$, $X^3_{R_1}, \ldots, X^3_{R_k}$, and $X^3_{P_1}, \ldots, X^3_{P_h}$ are variables of level 3 in ${\ensuremath{\mathsf{4LQS^R}}}$. To prove the theorem, we show that $\Sigma$ is the higher order answer set for $Q$ w.r.t. ${\mathcal{KB}}$ iff $\Sigma$ is equal to $\overset{}{\underset{{\mathbfcal{M}}\models \phi_{{\mathcal{KB}}}}{\bigcup}} \Sigma_{{\mathbfcal{M}}}'$, where $\Sigma_{{\mathbfcal{M}}}'$ is the collection of substitutions $\sigma$ such that ${\mathbfcal{M}}\models \psi_{Q}\sigma$. Let us assume that $\Sigma$ is higher order the answer set for $Q$ w.r.t. ${\mathcal{KB}}$. We have to show that $\Sigma$ is equal to $ \Sigma' = \underset{{\mathbfcal{M}}\models \phi_{{\mathcal{KB}}}}{\bigcup} \Sigma'_{{\mathbfcal{M}}}$, where $\Sigma'_{{\mathbfcal{M}}}$ is the collection of all the substitutions $\sigma'$ such that ${\mathbfcal{M}}\models \psi_{Q}\sigma'$. By contradiction, let us assume that there exists a $\sigma \in \Sigma$ such that $\sigma \notin \Sigma'$, namely ${\mathbfcal{M}}\not\models \psi_{Q}\sigma$, for every ${\ensuremath{\mathsf{4LQS^R}}}$-interpretation ${\mathbfcal{M}}$ with ${\mathbfcal{M}}\models \phi_{{\mathcal{KB}}}$. Since $\sigma \in \Sigma$ there is a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-interpretation ${\mathbf{I}}$ such that ${\mathbf{I}}\models_{{\mathbf{D}}} {\mathcal{KB}}$ and ${\mathbf{I}}\models_{{\mathbf{D}}} Q\sigma$. Then, by the construction above, we can define a ${\ensuremath{\mathsf{4LQS^R}}}$-interpretation ${\mathbfcal{M}}_{{\mathbf{I}}}$ such that ${\mathbfcal{M}}_{{\mathbf{I}}} \models \phi_{{\mathcal{KB}}}$ and ${\mathbfcal{M}}_{{\mathbf{I}}}\models \psi_Q\theta{\sigma}$. Absurd. Conversely, let $\sigma' \in \Sigma'$ and assume by contradiction that $\sigma' \notin \Sigma$. Then, for all ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-interpretations such that ${\mathbf{I}}\models_D {\mathcal{KB}}$, it holds that ${\mathbf{I}}\not\models_D Q\sigma'$. Since $\sigma'\in \Sigma'$, there is a ${\ensuremath{\mathsf{4LQS^R}}}$-interpretation ${\mathbfcal{M}}$ such that ${\mathbfcal{M}}\models \phi_{{\mathcal{KB}}}$ and ${\mathbfcal{M}}\models \psi\sigma'$. Then, by the construction above, we can define a ${\mathcal{DL}_{{\mathbf{D}}}^{4}}$-interpretation ${\mathbf{I}}_{{\mathbfcal{M}}}$ such that ${\mathbf{I}}_{{\mathbfcal{M}}} \models_D {\mathcal{KB}}$ and ${\mathbf{I}}_{{\mathbfcal{M}}} \models_D Q\sigma'$. Absurd. In what follows we list the most widespread reasoning services for ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ -ABox and then show how to define them as particular cases of the HOCQA task. 1. [*Instance checking*: the problem of deciding whether or not an individual $a$ is an instance of a concept $C$.]{} 2. [*Instance retrieval*: the problem of retrieving all the individuals that are instances of a given concept.]{} 3. [*Role filler retrieval*: the problem of retrieving all the fillers $x$ such that the pair $(a,x)$ is an instance of a role $R$.]{} 4. [*Concept retrieval*: the problem of retrieving all concepts which an individual is an instance of.]{} 5. [*Role instance retrieval*: the problem of retrieving all roles which a pair of individuals $(a,b)$ is an instance of.]{} The instance checking problem is a specialization of the HOCQA problem admitting HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-conjunctive queries of the form $Q_{IC}=C(w_1)$, with $w_1 \in {\mathbf{Ind}}$. The instance retrieval problem is a particular case of the HOCQA problem in which HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-conjunctive queries have the form $Q_{IR}=C(w_1)$, where $w_1$ is a variable in ${\mathsf{V}_{\mathsf{i}}}$. The HOCQA problem can be instantiated to the role filler retrieval problem by admitting HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-conjunctive queries $Q_{RF}=R(w_1,w_2)$, with $w_1 \in {\mathbf{Ind}}$ and $w_2$ a variable in ${\mathsf{V}_{\mathsf{i}}}$. The concept retrieval problem is a specialization of the HOCQA problem allowing HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-conjunctive queries of the form $Q_{QR}=c(w_1)$, with $w_1 \in {\mathbf{Ind}}$ and $c$ a variable in ${\mathsf{V}_{\mathsf{c}}}$. Finally, the role instance retrieval problem is a particularization of the HOCQA problem, where HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-conjunctive queries have the form $Q_{RI}=r(w_1,w_2)$, with $w_1,w_2 \in {\mathbf{Ind}}$ and $r$ a variable in ${\mathsf{V}_{\mathsf{cr}}}$. Notice that the CQA problem for ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ defined in [@ictcs16] is an instance of the HOCQA problem admitting HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-conjunctive queries of the form $Q_{CQA} {\coloneqq}(L_1 \wedge \ldots \wedge L_m)$, with $L_i$ an atomic formula of any of the types $R(w_1,w_2)$, $C(w_1)$, and $w_1=w_2$ (or their negation), where $w_1,w_2 \in({\mathbf{Ind}}\cup{\mathsf{V}_{\mathsf{i}}})$. Notice also that problems 1, 2, and 3 are instances of the CQA problem for ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$, whereas problems 4 and 5 fall outside the definition of CQA. As shown above, they can be treated as specializations of HOCQA. An algorithm for the HOCQA problem for ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ ==================================================================================== In this section we introduce an effective set-theoretic procedure to compute the answer set of a HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-conjunctive query $Q$ w.r.t. a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ knowledge base ${\mathcal{KB}}$. Such procedure, called *HOCQA-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$*, takes as input $\phi_{\mathcal{KB}}$ (i.e., the ${\ensuremath{\mathsf{4LQS^R}}}$-translation of ${\mathcal{KB}}$) and $\psi_Q$ (i.e., the ${\ensuremath{\mathsf{4LQS^R}}}$-formula representing the HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-conjunctive query $Q$), and returns a [KE-tableau]{}${\mathcal{T}}_{\mathcal{KB}}$, representing the saturation of ${\mathcal{KB}}$, and the answer set $\Sigma'$ of $\psi_Q$ w.r.t. $\phi_{\mathcal{KB}}$, namely the collection of all substitutions $\sigma'$ such that ${\mathbfcal{M}}\models \phi_{\mathcal{KB}}\wedge \psi_Q\sigma'$, for some ${\ensuremath{\mathsf{4LQS^R}}}$-interpretation ${\mathbfcal{M}}$. Specifically, *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ constructs for each open branch of ${\mathcal{T}}_{\mathcal{KB}}$ a decision tree whose leaves are labelled with elements of $\Sigma'$. In the following we introduce definitions, notions, and notations useful for the presentation of Procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$. Assume without loss of generality that universal quantifiers in $\phi_{\mathcal{KB}}$ occur as inward as possible and that universally quantified variables are pairwise distinct. Let $S_1, \ldots, S_m$ be the conjuncts of $\phi_{\mathcal{KB}}$ having the form of ${\ensuremath{\mathsf{4LQS^R}}}$-purely universal formulae. For each $S_i {\coloneqq}(\forall z_1^i) \ldots (\forall z_{n_i}^i) \chi_i$, with $i=1,\ldots,m$, we put\ $Exp(S_i) {\coloneqq}\underset{ \{x_{a_1}, \ldots, x_{a_{n_i}}\} \subseteq {\mathtt{Var}_0}(\phi_{{\mathcal{KB}}})}{\bigwedge} S_i \{z_1^i / x_{a_1}, \ldots, z^i_{n_i} / x_{a_{n_i}} \}$. \ Let us also define the *expansion* $\Phi_{\mathcal{KB}}$ of $\phi_{\mathcal{KB}}$ by putting $$\label{phikb} \Phi_{\mathcal{KB}}{\coloneqq}\{ F_j : i=1,\ldots,k \} \cup \overset{m}{ \underset{i=1}{\bigcup}} Exp(S_i)\,,$$ where $F_1, \ldots, F_k$ are the conjuncts of $\phi_{\mathcal{KB}}$ having the form of ${\ensuremath{\mathsf{4LQS^R}}}$-quantifier free atomic formulae. To prepare for Procedure *HOCQA-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$* to be described next, a brief introduction on the [KE-tableau]{}system is in order (see [@dagostino1999] for a detailed overview of [KE-tableau]{}). [KE-tableau]{}is a refutation system inspired to Smullyan’s semantic tableaux [@smullyan1995first]. The main characteristic distinguishing [KE-tableau]{}from the latter is the introduction of an analytic cut rule (PB-rule) that permits to reduce inefficiencies of semantic tableaux. In fact, firstly, the classic tableau system can not represent the use of auxiliary lemmas in proofs; secondly, it can not express the bivalence of classical logic. Thirdly, it is extremely inefficient, as witnessed by the fact that it can not polynomially simulate the truth-tables. None of these anomalies occurs if the cut rule is permitted. For these reasons, Procedure *HOCQA-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$* constructs a complete [KE-tableau]{}${\mathcal{T}}_{{\mathcal{KB}}}$ for the expansion $\Phi_{\mathcal{KB}}$ of $\phi_{\mathcal{KB}}$ (cf. (\[phikb\])), representing the saturation of the ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-knowledge base ${\mathcal{KB}}$. Let $\Phi {\coloneqq}\{ C_1,\ldots, C_p\}$ be a collection of disjunctions of ${\ensuremath{\mathsf{4LQS^R}}}$-quantifier free atomic formulae of level $0$ of the types: $x =y$, $x \in X^1$, $\langle x,y\rangle \in X^3$. $\mathcal{T}$ is a *[KE-tableau]{}*for $\Phi$ if there exists a finite sequence $\mathcal{T}_1, \ldots, \mathcal{T}_t$ such that (i) $\mathcal{T}_1$ is a one-branch tree consisting of the sequence $C_1,\ldots, C_p$, (ii) $\mathcal{T}_t = \mathcal{T}$, and (iii) for each $i<t$, $\mathcal{T}_{i+1}$ is obtained from $\mathcal{T}_i$ either by an application of one of the rules in Fig. \[exprule\] or by applying a substitution $\sigma$ to a branch $\vartheta$ of $\mathcal{T}_i$ (in particular, the substitution $\sigma$ is applied to each formula $X$ of $\vartheta$; the resulting branch will be denoted with $\vartheta\sigma$). The set of formulae ${\mathcal{S}^{\overline{\beta}}_i}{\coloneqq}\{ \overline{\beta}_1,\ldots,\overline{\beta}_n\} \setminus \{\overline{\beta}_i\}$ occurring as premise in the E-rule contains the complements of all the components of the formula $\beta$ with the exception of the component $\beta_i$. $\infer[\textbf{E-Rule}] {\beta_i}{\beta_1 \vee \ldots \vee \beta_n & \quad {\mathcal{S}^{\overline{\beta}}_i}}$\ [ where ${\mathcal{S}^{\overline{\beta}}_i}{\coloneqq}\{ \overline{\beta}_1,...,\overline{\beta}_n\} \setminus \{\overline{\beta}_i\}$,]{}\ [ for $i=1,...,n$]{}             $\infer[\textbf{PB-Rule}] {A~~|~~\overline{A}}{}$\ [ with $A$ a literal]{} Let $\mathcal{T}$ be a [KE-tableau]{}. A branch $\vartheta$ of $\mathcal{T}$ is *closed* if it contains either both $A$ and $\neg A$, for some formula $A$, or a literal of type $\neg(x = x)$. Otherwise, the branch is *open*. A [KE-tableau]{}is *closed* if all its branches are closed. A formula $\beta_1 \vee \ldots \vee \beta_n$ is *fulfilled* in a branch $\vartheta$, if $\beta_i$ is in $\vartheta$, for some $i=1,\ldots,n$. A branch $\vartheta$ is *fulfilled* if every formula $\beta_1 \vee \ldots \vee \beta_n$ occurring in $\vartheta$ is fulfilled. A branch $\vartheta$ is *complete* if either it is closed or it is open, fulfilled, and it does not contain any literal of type $x=y$, where $x$ and $y$ are distinct variables. A [KE-tableau]{}is *complete* (resp., *fulfilled*) if all its branches are complete (resp., fulfilled or closed). A ${\ensuremath{\mathsf{4LQS^R}}}$-interpretation ${\mathbfcal{M}}$ *satisfies* a branch $\vartheta$ of a [KE-tableau]{}(or, equivalently, $\vartheta$ *is satisfied* by ${\mathbfcal{M}}$), and we write ${\mathbfcal{M}}\models \vartheta$, if ${\mathbfcal{M}}\models X$ for every formula $X$ occurring in $\vartheta$. A ${\ensuremath{\mathsf{4LQS^R}}}$-interpretation ${\mathbfcal{M}}$ satisfies a [KE-tableau]{}$\mathcal{T}$ (or, equivalently, $\mathcal{T}$ *is satisfied* by ${\mathbfcal{M}}$), and we write ${\mathbfcal{M}}\models \mathcal{T}$, if ${\mathbfcal{M}}$ satisfies a branch $\vartheta$ of $\mathcal{T}$. A branch $\vartheta$ of a [KE-tableau]{}$\mathcal{T}$ is *satisfiable* if there exists a ${\ensuremath{\mathsf{4LQS^R}}}$-interpretation ${\mathbfcal{M}}$ that satisfies $\vartheta$. A [KE-tableau]{}is satisfiable if at least one of its branches is satisfiable. Let $\vartheta$ be a branch of a [KE-tableau]{}. We denote with $<_{\vartheta}$ an arbitrary but fixed total order on the variables in $\mathsf{Var}_{0}(\vartheta)$. Procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ takes care of literals of type $x=y$ occurring in the branches of ${\mathcal{T}}_{\mathcal{KB}}$ by constructing, for each open and fulfilled branch $\vartheta$ of ${\mathcal{T}}_{\mathcal{KB}}$ a substitution $\sigma_{\vartheta}$ such that $\vartheta\sigma_{\vartheta}$ does not contain literals of type $x=y$ with distinct $x,y$. Then, for every open and complete branch $\vartheta':=\vartheta\sigma_{\vartheta}$ of ${\mathcal{T}}_{{\mathcal{KB}}}$, Procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ constructs a decision tree ${\mathcal{D}}_{\vartheta'}$ such that every maximal branch of ${\mathcal{D}}_{\vartheta'}$ induces a substitution $\sigma'$ such that $\sigma_{\vartheta}\sigma'$ belongs to the answer set of $\psi_{Q}$ with respect to $\phi_{{\mathcal{KB}}}$. ${\mathcal{D}}_{\vartheta'}$ is defined as follows. Let $d$ be the number of literals in $\psi_Q$. Then ${\mathcal{D}}_{\vartheta'}$ is a finite labelled tree of depth $d+1$ whose labelling satisfies the following conditions, for $i=0,\ldots,d$: - every node of ${\mathcal{D}}_{\vartheta'}$ at level $i$ is labelled with $(\sigma'_i, \psi_Q\sigma_{\vartheta}\sigma'_i)$; in particular, the root is labelled with $(\sigma'_0, \psi_Q\sigma_{\vartheta}\sigma'_0)$, where $\sigma'_0$ is the empty substitution; - [if a node at level $i$ is labelled with $(\sigma'_i, \psi_Q\sigma_{\vartheta}\sigma'_i)$, then its $s$ successors, with $s >0$, are labelled with $\big(\sigma'_i\varrho^{q_{i+1}}_1, \psi_Q\sigma_{\vartheta}(\sigma'_i\varrho^{q_{i+1}}_1)\big),\ldots,\big(\sigma'_i\varrho^{q_{i+1}}_s, \psi_Q\sigma_\vartheta(\sigma'_i\varrho^{q_{i+1}}_s)\big)$, where $q_{i+1}$ is the $(i+1)$-st conjunct of $\psi_Q\sigma_{\vartheta}\sigma'_i$ and $\mathcal{S}_{q_{i+1}}=\{\varrho^{q_{i+1}}_1, \ldots, \varrho^{q_{i+1}}_s \}$ is the collection of the substitutions $\varrho = \{v_1/o_1, \ldots, v_n/ o_n, c_1/C_1, \ldots, c_m/ C_m,\\ r_1/R_1, \ldots, r_k/ R_k, p_1/P_1, \ldots, p_h/ P_h \}$, with $\{v_1, \ldots, v_n\} = {\mathtt{Var}_0}(q_{i+1})$,]{} $\{c_1,\ldots, c_m\} = {\mathtt{Var}_1}(q_{i+1})$, and $\{p_1,\ldots, p_h,r_1,\ldots, r_k\} = {\mathtt{Var}_3}(q_{i+1})$, such that $t=q_{i+1}\varrho$, for some literal $t$ on $\vartheta'$. If $s = 0$, the node labelled with $(\sigma'_i, \psi_Q\sigma_{\vartheta}\sigma'_i)$ is a leaf node and, if $i = d$, $\sigma_\vartheta\sigma'_i$ is added to $\Sigma'$. We are ready to define Procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$. ; $\Sigma'$ := $\emptyset$; - let $\Phi_{\mathcal{KB}}$ be the expansion of $\phi_{\mathcal{KB}}$ (cf. (\[phikb\])); ${\mathcal{T}}_{{\mathcal{KB}}}$ := $\Phi_{\mathcal{KB}}$; - apply the E-Rule to $\beta_1 \vee \ldots \vee \beta_n$ and ${\mathcal{S}^{\overline{\beta}}_j}$ on $\vartheta$; ; ; $\sigma_{\vartheta} := \epsilon$ (where $\epsilon$ is the empty substitution); $\mathsf{Eq}_{\vartheta} := \{ \mbox{literals of type $x = y$, occurring in $\vartheta$}\}$; - select a literal $x = y$ in $\mathsf{Eq}_{\vartheta}$, with distinct $x$, $y$; $z := \min_{<_{\vartheta}}(x,y)$; $\sigma_{\vartheta} := \sigma_{\vartheta} \cdot \{x/z, y/z\}$; $\mathsf{Eq}_{\vartheta} := \mathsf{Eq}_{\vartheta}\sigma_{\vartheta}$; ; $\vartheta := \vartheta\sigma_{\vartheta}$; - initialize $\mathcal{S}$ to the empty stack; - push $(\epsilon, \psi_Q\sigma_\vartheta)$ in $\mathcal{S}$; - pop $(\sigma', \psi_Q\sigma_\vartheta\sigma')$ from $\mathcal{S}$; - let $q$ be the leftmost conjunct of $\psi_Q\sigma_\vartheta\sigma'$; $\psi_Q\sigma_\vartheta\sigma':= \psi_Q\sigma_\vartheta\sigma'$ deprived of $q$; $Lit^{M}_Q := \{ t \in \vartheta : t=q\rho$, for some substitution $\rho \}$; - let $t \in Lit^{M}_Q$, $t=q\rho$; $Lit^{M}_Q := Lit^{M}_Q \setminus \{t\}$; - push $(\sigma'\rho, \psi_Q\sigma_\vartheta\sigma'\rho)$ in $\mathcal{S}$; ; $\Sigma'$ := $\Sigma' \cup \{\sigma_\vartheta\sigma'\}$; ; ; ; ; $({\mathcal{T}}_{\mathcal{KB}}, \Sigma')$; ; For each open branch $\vartheta$ of ${\mathcal{T}}_{\mathcal{KB}}$, Procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ computes the corresponding ${\mathcal{D}}_{\vartheta}$ by constructing a stack of its nodes. Initially the stack contains the root node $(\epsilon,\psi_Q\sigma_\vartheta)$ of ${\mathcal{D}}_{\vartheta}$, as defined in condition (i). Then, iteratively, the following steps are executed. An element $(\sigma', \psi_Q\sigma_\vartheta\sigma')$ is popped out of the stack. If the last literal of the query $\psi_Q$ has not been reached, the successors of the current node are computed according to condition (ii) and inserted in the stack. Otherwise the current node must have the form $(\sigma',\lambda)$ and the substitution $\sigma_\vartheta\sigma'$ is inserted in $\Sigma'$. Correctness of Procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ follows from Theorems \[teo:correctness\] and \[teo:completeness\], which show that $\phi_{\mathcal{KB}}$ is satisfiable if and only if ${\mathcal{T}}_{{\mathcal{KB}}}$ is a non-closed [KE-tableau]{}, and from Theorem \[teo:proc2corr\], which shows that the set $\Sigma'$ coincides with the HO answer set of $\psi_Q$ w.r.t. $\phi_{\mathcal{KB}}$. Theorems \[teo:correctness\], \[teo:completeness\], and \[teo:proc2corr\] are stated below. In particular, Theorem \[teo:correctness\], requires the following technical lemmas. \[lemma:invariant\] Let $\vartheta$ be a branch of ${\mathcal{T}}_{{\mathcal{KB}}}$ selected at step $15$ of Procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$($\psi_{Q}$,$\phi_{{\mathcal{KB}}}$), let $\sigma_{\vartheta}$ be the associated substitution constructed during the execution of the while-loop 18–23, and let ${\mathbfcal{M}}= (D,M)$ be a [$\mathsf{4LQS^R}$]{}-interpretation satisfying $\vartheta$. Then $$\label{eq:invariant} Mx = Mx\sigma_{\vartheta}, \mbox{ for every } x \in \mathsf{Var}_0(\vartheta),$$ is an invariant of the while-loop 18–23. We prove the thesis by induction on the number $i$ of iterations of the while loop 18–23 of the procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$($\psi_{Q}$,$\phi_{{\mathcal{KB}}}$). For simplicity we indicate with $\sigma_{\vartheta}^{(i)}$ and with $Eq_{\sigma_{\vartheta}}^{(i)}$ the substitution $\sigma_{\vartheta}$ and the set $Eq_{\sigma_{\vartheta}}$calculated at iteration $i \geq 0$, respectively. If $i = 0$, $\sigma_{\vartheta}^{(0)}$ is the empty substitution $\epsilon$ and thus (\[eq:invariant\]) trivially holds. Assume by inductive hypothesis that (\[eq:invariant\]) holds at iteration $i \geq 0$. We want to prove that (\[eq:invariant\]) holds at iteration $i+1$. At iteration $i +1$, $\sigma_{\vartheta}^{(i+1)} = \sigma_{\vartheta}^{(i)} \cdot \{x/z,y/z\}$, where $z = \min_{<_{\vartheta}}\{x,y\}$ and $x = y$ is a literal in $Eq_{\sigma_{\vartheta}}^{(i)}$, with distinct $x,y$. We assume, without loss of generality, that $z$ is the variable $x$ (an analogous proof can be carried out assuming that $z$ is the variable $y$). By inductive hypothesis $Mw = Mw\sigma_{\vartheta}^{(i)}$, for every $w \in \mathsf{Var}_0(\vartheta)$. If $w\sigma_{\vartheta}^{(i)} \in \mathsf{Var}_0(\vartheta)\setminus \{y\}$, plainly $w\sigma_{\vartheta}^{(i)}$ and $w\sigma_{\vartheta}^{(i+1)}$ coincide and thus $Mw\sigma_{\vartheta}^{(i)} = Mw\sigma_{\vartheta}^{(i+1)}$. Since $Mw = Mw\sigma_{\vartheta}^{(i)}$, it follows that $Mw = Mw\sigma_{\vartheta}^{(i+1)}$. If $w\sigma_{\vartheta}^{(i)}$ coincides with $y$ we reason as follows. At iteration $i+1$ variables $x,y$ are considered because the literal $x=y$ is selected from $Eq_{\sigma_{\vartheta}}^{(i)}$. If $x=y$ is a literal belonging to $\vartheta$, then $Mx = My$. Since $w\sigma_{\vartheta}^{(i)}$ coincides with $y$, $w\sigma_{\vartheta}^{(i+1)}$ coincides with $x$, $My = Mx$, and by inductive hypothesis $Mw = Mw\sigma_{\vartheta}^{(i)}$, it holds that $Mw = Mw\sigma_{\vartheta}^{(i+1)}$. If $x = y$ is not a literal occurring in $\vartheta$, then $\vartheta$ must contain a literal $x' = y'$ such that, at iteration $i$, $x$ coincides with $x'\sigma_{\vartheta}^{(i)}$ and $y$ coincides with $y'\sigma_{\vartheta}^{(i)}$. Since $Mx' = My'$ and, by inductive hypothesis, $Mx' = Mx'\sigma_{\vartheta}^{(i)}$, and $My' = My'\sigma_{\vartheta}^{(i)}$, it holds that $Mx = My$, and thus, reasoning as above, $Mw = Mw\sigma_{\vartheta}^{(i+1)}$. Since (\[eq:invariant\]) holds at each iteration of the while loop, it is an invariant of the loop as we wished to prove. \[lemma:correctness\] Let ${\mathcal{T}}_0,\ldots,{\mathcal{T}}_h$ be a sequence of [KE-tableau]{}x such that ${\mathcal{T}}_0 = \Phi_{{\mathcal{KB}}}$, and ${\mathcal{T}}_{i+1}$ is obtained from ${\mathcal{T}}_i$ by applying either the rule of step 8, or the rule of step 10, or the substitution of step 24 of Procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$($\psi_{Q}$,$\phi_{{\mathcal{KB}}}$), for $i = 1,\ldots,h-1$. If ${\mathcal{T}}_i$ is satisfied by a [$\mathsf{4LQS^R}$]{}-interpretation ${\mathbfcal{M}}$, then ${\mathcal{T}}_{i+1}$ is satisfied by ${\mathbfcal{M}}$ as well, for $i = 1,\ldots,h-1$. Let ${\mathbfcal{M}}=(D,M)$ be a [$\mathsf{4LQS^R}$]{}-interpretation satisfying ${\mathcal{T}}_i$. Then ${\mathbfcal{M}}$ satisfies a branch $\bar{\vartheta}$ of ${\mathcal{T}}_i$. In case the branch $\bar{\vartheta}$ is different from the branch selected at step 6, if the E-rule (step 8) or the PB-rule (10) is applied, or at step 3, if a substitution for handling equalities (step 14) is applied, $\bar{\vartheta}$ belongs to ${\mathcal{T}}_{i+1}$ and therefore ${\mathcal{T}}_{i+1}$ is satisfied by ${\mathbfcal{M}}$. In case $\bar{\vartheta}$ is the branch selected and modified to obtain ${\mathcal{T}}_{i+1}$, we have to consider the following distinct cases. - $\bar{\vartheta}$ has been selected at step 6 and thus it is an open branch not yet fulfilled. Then, if step 8 is executed, the E-rule is applied to a not fulfilled formula $\beta_1 \vee \ldots \vee \beta_n$ and to the set of formulae ${\mathcal{S}^{\overline{\beta}}_j}$ on the branch $\bar{\vartheta}$ generating the new branch $\bar{\vartheta'} := \bar{\vartheta} ; \beta_i$. Plainly, if ${\mathbfcal{M}}\models \bar{\vartheta}$, ${\mathbfcal{M}}\models \beta_1 \vee \ldots \vee \beta_n$, ${\mathbfcal{M}}\models {\mathcal{S}^{\overline{\beta}}_j}$ and, as a consequence, ${\mathbfcal{M}}\models \beta_i$. Thus ${\mathbfcal{M}}\models \bar{\vartheta'}$ and finally, ${\mathbfcal{M}}$ satisfies ${\mathcal{T}}_{i+1}$. If step 10 is performed, the PB-rule is applied on $\bar{\vartheta}$ originating the branches (belonging to ${\mathcal{T}}_{i+1}$) $\bar{\vartheta'} := \bar{\vartheta} ; \overline{\beta}_h$ and $\bar{\vartheta''} := \bar{\vartheta} ; \beta_h$. Since either ${\mathbfcal{M}}\models \beta_h$ or ${\mathbfcal{M}}\models \overline{\beta}_h$, it holds that either ${\mathbfcal{M}}\models \bar{\vartheta'}$ or ${\mathbfcal{M}}\models \bar{\vartheta''}$. Thus ${\mathbfcal{M}}$ satisfies ${\mathcal{T}}_{i+1}$, as we wished to prove. - $\bar{\vartheta}$ has been selected at step 14 and thus it is an open and fulfilled branch not yet complete. Once step $24$ is executed the new branch $\bar{\vartheta} \sigma_{\bar{\vartheta}}$ is generated. Since ${\mathbfcal{M}}\models \bar{\vartheta}$ and, by Lemma \[lemma:invariant\], $Mx = Mx\sigma_{\bar{\vartheta}}$, for every $x \in \mathsf{Var}_0(\bar{\vartheta})$, it holds that ${\mathbfcal{M}}\models \bar{\vartheta} \sigma_{\bar{\vartheta}}$ and that ${\mathbfcal{M}}$ satisfies ${\mathcal{T}}_{i+1}$. Thus the thesis follows. Then we have: \[teo:correctness\] If $\phi_{{\mathcal{KB}}}$ is satisfiable, then ${\mathcal{T}}_{{\mathcal{KB}}}$ is not closed. Let us assume by contradiction that ${\mathcal{T}}_{{\mathcal{KB}}}$ is closed. Since $\Phi_{{\mathcal{KB}}}$ is satisfiable, there exists a ${\ensuremath{\mathsf{4LQS^R}}}$-interpretation ${\mathbfcal{M}}$ satisfying every formula of $\Phi_{{\mathcal{KB}}}$. Thanks to Lemma \[lemma:correctness\], any [KE-tableau]{}for $\Phi_{{\mathcal{KB}}}$ obtained by applying either step 8, or step 10, or step 24 of the procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$, is satisfied by ${\mathbfcal{M}}$. Thus ${\mathcal{T}}_{{\mathcal{KB}}}$ is satisfied by ${\mathbfcal{M}}$ as well. In particular, there exists a branch $\vartheta_c$ of ${\mathcal{T}}_{{\mathcal{KB}}}$ satisfied by ${\mathbfcal{M}}$. Since ${\mathcal{T}}_{{\mathcal{KB}}}$ is closed, by the absurd hypothesis, the branch $\vartheta_c$ is closed as well and thus, by definition, it contains either both $A$ and $\neg A$, for some formula $A$, or a literal of type $\neg (x = x)$. $\vartheta$ is satisfied by ${\mathbfcal{M}}$ and thus, either ${\mathbfcal{M}}\models A$ and ${\mathbfcal{M}}\models \neg A$ or ${\mathbfcal{M}}\models \neg (x = x)$. Absurd. Thus, we have to admit that the [KE-tableau]{}${\mathcal{T}}_{{\mathcal{KB}}}$ is not closed. \[teo:completeness\] If ${\mathcal{T}}_{{\mathcal{KB}}}$ is not closed, then $\phi_{{\mathcal{KB}}}$ is satisfiable. *Proof*. Since ${\mathcal{T}}_{{\mathcal{KB}}}$ is not closed, there exists a branch $\vartheta'$ of ${\mathcal{T}}_{{\mathcal{KB}}}$ which is open and complete. The branch $\vartheta'$ is obtained during the execution of the procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ from an open fulfilled branch $\vartheta$ by applying to $\vartheta$ the substitution $\sigma_{\vartheta}$ constructed during the execution of step 14 of the procedure. Thus, $\vartheta' = \vartheta\sigma_{\vartheta}$. Since each formula of $\Phi_{{\mathcal{KB}}}$ occurs in $\vartheta$, showing that $\vartheta$ is satisfiable is enough to prove that $\Phi_{{\mathcal{KB}}}$ is satisfiable. Let us construct a [$\mathsf{4LQS^R}$]{}-interpretation ${\mathbfcal{M}}_{\vartheta}=(D_{\vartheta},M_{\vartheta})$ satisfying every formula $X$ occurring in $\vartheta$ and thus $\Phi_{{\mathcal{KB}}}$. ${\mathbfcal{M}}_{\vartheta}=(D_{\vartheta},M_{\vartheta})$ is defined as follows. - $D_{\vartheta} {\coloneqq}\{x\sigma_{\vartheta} : x \in \mathsf{Var}_0(\vartheta) \}$; - $M_{\vartheta} x {\coloneqq}x\sigma_{\vartheta}$, $x \in \mathsf{Var}_0(\vartheta)$; - $M_{\vartheta} X^1 {\coloneqq}\{x\sigma_{\vartheta} : x \in X^1 \mbox{ occurs in } \vartheta\}$, $X^1 \in \mathsf{Var}_1(\vartheta)$; - $M_{\vartheta} X^3 {\coloneqq}\{\langle x\sigma_{\vartheta}, y\sigma_{\vartheta} \rangle : \langle x, y\rangle \in X^3 \mbox{ occurs in } \vartheta \}$, $X^3 \in \mathsf{Var}_3(\vartheta)$. In what follows we show that ${\mathbfcal{M}}_{\vartheta}$ satisfies each formula in $\vartheta$. Our proof is carried out by induction on the structure of formulae and cases distinction. Let us consider, at first, a literal $x = y$ occurring in $\vartheta$. By the construction of $\sigma_{\vartheta}$ described in the procedure, $x\sigma_{\vartheta}$ and $y\sigma_{\vartheta}$ have to coincide. Thus $M_{\vartheta}x = x\sigma_{\vartheta} = y\sigma_{\vartheta} = M_{\vartheta} y$ and then ${\mathbfcal{M}}_{\vartheta} \models x=y$. Next we consider a literal $\neg (z = w)$ occurring in $\vartheta$. If $z\sigma_{\vartheta}$ and $w\sigma_{\vartheta}$ coincide, namely they are the same variable, then the branch $\vartheta' = \vartheta\sigma_{\vartheta}$ must be a closed branch against our hypothesis. Thus $z\sigma_{\vartheta}$ and $w\sigma_{\vartheta}$ are distinct variables and therefore $M_{\vartheta} z= z\sigma_{\vartheta} \neq w\sigma_{\vartheta} = M_{\vartheta}w$, then ${\mathbfcal{M}}_{\vartheta} \not\models z = w$ and finally ${\mathbfcal{M}}_{\vartheta} \models \neg(z = w)$, as we wished to prove. Let $x \in X^1$ be a literal occurring in $\vartheta$. By the definition of $M_{\vartheta}$, $x\sigma_{\vartheta} \in M_{\vartheta}X^1$, namely $M_{\vartheta} x \in M_{\vartheta}X^1$ and thus ${\mathbfcal{M}}_{\vartheta} \models x \in X^1$ as desired. If $\neg(y \in X^1)$ occurs in $\vartheta$, then $y\sigma_{\vartheta}\notin M_{\vartheta}X^1$. Assume, by contradiction that $y\sigma_{\vartheta}\in M_{\vartheta}X^1$. Then there is a literal $z \in X^1$ in $\vartheta$ such that $z\sigma_{\vartheta}$ and $y\sigma_{\vartheta}$ coincide. In this case the branch $\vartheta'$, obtained from $\vartheta$ applying the substitution $\sigma_{\vartheta}$ would be closed, contradicting the hypothesis. Thus $y\sigma_{\vartheta}\notin M_{\vartheta}X^1$ implies that $M_{\vartheta}y \notin M_{\vartheta}X^1$, that ${\mathbfcal{M}}_{\vartheta} \not\models y \in X^1$, and finally that ${\mathbfcal{M}}_{\vartheta} \models \neg(y \in X^1)$. If $\langle x,y\rangle \in X^3$ is a literal on $\vartheta$, then by definition of $M_{\vartheta}$, $\langle x\sigma_{\vartheta}, y\sigma_{\vartheta}\rangle \in M_{\vartheta}X^3$, that is $\langle M_{\vartheta}x, M_{\vartheta}y\rangle \in M_{\vartheta}X^3$, and thus ${\mathbfcal{M}}_{\vartheta} \models \langle x,y\rangle \in X^3$. Let $\neg(\langle z,w\rangle \in X^3)$ be a literal occurring on $\vartheta$. Assume that $\langle z\sigma_{\vartheta},w\sigma_{\vartheta}\rangle \in M_{\vartheta}X^3$. Then a literal $\langle z',w'\rangle \in X^3$ occurs in $\vartheta$ such that $z\sigma_{\vartheta}$ coincides with $z'\sigma_{\vartheta}$ and that $w\sigma_{\vartheta}$ coincides with $w'\sigma_{\vartheta}$. But then the branch $\vartheta' = \vartheta\sigma_{\vartheta}$ would be closed contradicting the hypothesis. Thus we have to admit that $\langle z\sigma_{\vartheta},w\sigma_{\vartheta}\rangle \notin M_{\vartheta}X^3$, that is $\langle M_{\vartheta}z,M_{\vartheta}w\rangle \notin M_{\vartheta}X^3$. Thus ${\mathbfcal{M}}_{\vartheta} \not\models \langle x,y\rangle \in X^3$ and finally ${\mathbfcal{M}}_{\vartheta} \models \neg(\langle x,y\rangle \in X^3)$. Let $\beta = \beta_1 \vee \ldots \vee \beta_k$ be a disjunction of literals in $\vartheta$. Since $\vartheta$ is fulfilled, $\beta$ is fulfilled too and, therefore, $\vartheta$ contains a disjunct $\beta_i$, for some $i \in \{1,\ldots,k\}$ of $\beta$. By inductive hypothesis ${\mathbfcal{M}}_{\vartheta} \models \beta_i$ and thus ${\mathbfcal{M}}_{\vartheta} \models \beta$. We have shown that ${\mathbfcal{M}}_{\vartheta}$ satisfies each formula in $\vartheta$ and, in particular the formulae in $\Phi_{{\mathcal{KB}}}$. It turns out that $\Phi_{{\mathcal{KB}}}$ is satisfiable as we wished to prove. It is easy to check that the ${\ensuremath{\mathsf{4LQS^R}}}$-interpretation ${\mathbfcal{M}}_{\vartheta}$ defined in Theorem \[teo:completeness\] satisfies $\phi_{{\mathcal{KB}}}$, a collection of ${\ensuremath{\mathsf{4LQS^R}}}$-purely universal formulae and of ${\ensuremath{\mathsf{4LQS^R}}}$-quantifier free atomic formulae corresponding to a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-knowledge base ${\mathcal{KB}}$ and, therefore, that the following corollary holds. If ${\mathcal{T}}_{{\mathcal{KB}}}$ is not closed, then $\phi_{{\mathcal{KB}}}$ is satisfiable. In what follows, we state also a technical lemma which is needed in the proof of Theorem \[teo:proc2corr\]. \[lemma:proc2\] Let $\psi_Q {\coloneqq}q_1 \wedge \ldots \wedge q_d$ be a HO ${\ensuremath{\mathsf{4LQS^R}}}$-conjunctive query, let $({\mathcal{T}}_{\mathcal{KB}}, \Sigma')$ be the output of *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$($\psi_Q$,$\phi_{\mathcal{KB}}$), and let $\vartheta$ be an open and complete branch of ${\mathcal{T}}_{\mathcal{KB}}$. Then, for any substitution $\sigma$, we have $$\sigma \in \Sigma' \iff \{ q_1 \sigma, \ldots, q_d\sigma \} \subseteq \vartheta\,.$$ If $\sigma' \in \Sigma'$, then $\sigma'=\sigma_\vartheta\sigma'_1$ and the decision tree ${\mathcal{D}}_{\vartheta'}$ contains a branch $\eta$ of length $d+1$ having as leaf $(\sigma'_1, \lambda)$. Specifically, the branch $\eta$ is constituted by the nodes $(\epsilon, q_1\sigma_\vartheta \wedge \ldots \wedge q_d\sigma_\vartheta )$, $( \rho^{(1)}, q_2\sigma_\vartheta\rho^{(1)} \wedge \ldots \wedge q_d\sigma_\vartheta\rho^{(1)} )$, $\ldots$, $( \rho^{(1)} \ldots \rho^{(d)}, \lambda)$, and hence $\sigma'= \sigma_\vartheta \rho^{(1)} \ldots \rho^{(d)}$. Consider the node $ (\rho^{(1)} \ldots \rho^{(i+1)}, q_{i+2}\sigma_\vartheta \rho^{(1)} \ldots \rho^{(i+1)} \wedge \ldots \wedge q_d\sigma_\vartheta \rho^{(1)} \ldots \rho^{(i+1)})$ constructed from the father node $(\rho^{(1)} \ldots \rho^{(i+1)}, q_{i+1}\sigma_\vartheta \rho^{(1)} \ldots \rho^{(i)} \wedge \ldots \wedge q_d\sigma_\vartheta \rho^{(1)} \ldots \rho^{(i)})$ putting $q_{i+1}\sigma_\vartheta \rho^{(1)} \ldots \rho^{(i+1)} = t$, for some $t \in \vartheta'$. Since $q_{i+1}\sigma_\vartheta \rho^{(1)} \ldots \rho^{(i+1)}$ is a ground literal, $q_{i+1}\sigma_\vartheta \rho^{(1)} \ldots \rho^{(i+1)}$ coincides with $q_{i+1}\sigma'$, then $q_{i+1} \sigma'=t$, and hence $q_{i+1}\sigma' \in \vartheta'$. Given the generality of $i=0, \ldots, d-1$, $\{ q_1\sigma', \ldots, q_d \sigma' \} \subseteq \vartheta'$ as we wished to prove. We now prove the second part of the lemma. We show that the decision tree ${\mathcal{D}}_{\vartheta'}$ constructed by Procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$($\psi_{Q}$,$\phi_{{\mathcal{KB}}}$) has a branch $\eta$ of length $d+1$ having as leaf the node $(\sigma'_1,\lambda)$, with $\sigma_\vartheta\sigma'_1=\sigma' \in \Sigma'$. Since by hypothesis $\vartheta'= \vartheta\sigma_\vartheta$, the root of the decision tree ${\mathcal{D}}_{\vartheta'}$ is the node $(\epsilon, q_1\sigma_\vartheta \wedge \ldots \wedge q_d\sigma_\vartheta )$. At step $i$, the procedure selects a literal $q^{(i)}$, namely $q_{i}\sigma_{\vartheta}\rho^{(1)}\ldots\rho^{(i-1)}$, and finds a substitution $\rho^{(i)}$ such that $q_{i}\sigma_{\vartheta}\rho^{(1)}\ldots\rho^{(i)}$ coincides with $q_{i}\sigma'$. Then, the procedure constructs the node $(\rho^{(1)} \ldots \rho^{(i)}, q_{i+1}\sigma_\vartheta \rho^{(1)} \ldots \rho^{(i)} \wedge \ldots \wedge q_d\sigma_\vartheta \rho^{(1)} \ldots \rho^{(i)})$ At step $d-1$, the procedure constructs the leaf node $(\rho^{(1)}\ldots \rho^{(d)},\lambda)$, that is $(\sigma'_1,\lambda)$, as we wished to prove. \[teo:proc2corr\] Let $\Sigma'$ be the set of substitutions returned by Procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$($\psi_Q$, $\phi_{\mathcal{KB}}$). Then $\Sigma'$ is the HO answer set of $\psi_Q$ w.r.t. $\phi_{\mathcal{KB}}$. To prove the theorem we show that the following two assertions hold. 1. If $\sigma' \in \Sigma'$, then $\sigma'$ is an element of the HO answer set of $\psi_Q$ w.r.t. $\phi_{\mathcal{KB}}$. 2. If $\sigma'$ is a substitution of the HO answer set of $\psi_Q$ w.r.t. $\phi_{\mathcal{KB}}$, then $\sigma' \in \Sigma'$. We prove assertion (1) as follows. Let $\sigma' \in \Sigma'$ and $\vartheta'=\vartheta\sigma_\vartheta$ an open and complete branch of ${\mathcal{T}}_{\mathcal{KB}}$ such that ${\mathcal{D}}_{\vartheta'}$ contains a branch $\eta$ of $d+1$ nodes whose leaf is labelled $\langle \sigma_1', \lambda \rangle$, where $\sigma_1'$ is a substitution such that $\sigma' = \sigma_{\vartheta}\sigma_1'$. By Lemma \[lemma:proc2\], $\{ q_1\sigma', \ldots, q_d\sigma' \} \subseteq \vartheta'$. Let ${\mathbfcal{M}}_\vartheta$ be a ${\ensuremath{\mathsf{4LQS^R}}}$-interpretation constructed as shown in Theorem \[teo:completeness\]. We have that ${\mathbfcal{M}}_\vartheta \models q_{i}\sigma'$, for $i = 1,\ldots,d$ because $\{q_1\sigma', \ldots, q_d\sigma' \} \subseteq \vartheta'$ holds. Thus ${\mathbfcal{M}}_\vartheta \models \psi_Q\sigma'$, and since ${\mathbfcal{M}}_\vartheta \models \phi_{\mathcal{KB}}$, ${\mathbfcal{M}}_\vartheta \models \phi_{\mathcal{KB}}\wedge \psi_Q\sigma'$ holds. Hence $\sigma'$ is a substitution of the answer set of $\psi_Q$ w.r.t. $\phi_{\mathcal{KB}}$. To show that assertion (2) holds, let us consider a substitution $\sigma'$ belonging to the answer set of $\psi_Q$ w.r.t. $\phi_{\mathcal{KB}}$. Then there exists a ${\ensuremath{\mathsf{4LQS^R}}}$-interpretation ${\mathbfcal{M}}\models \phi_{\mathcal{KB}}\wedge \psi_Q\sigma'$. Assume by contradiction that $\sigma' \notin \Sigma'$. Then, by Lemma \[lemma:proc2\] $\{q_1\sigma, \ldots, q_d\sigma' \} \not\subseteq\vartheta'$, for every open and complete branch $\vartheta'$ of ${\mathcal{T}}_{\mathcal{KB}}$. In particular, given any open complete branch $\vartheta'$ of ${\mathcal{T}}_{\mathcal{KB}}$, there is an $i \in \{1,\ldots,d\}$ such that $q_i\sigma' \notin \vartheta' = \vartheta\sigma_{\vartheta}$ and thus ${\mathbfcal{M}}_\vartheta \not\models q_i\sigma'$. By the generality of $\vartheta'=\vartheta\sigma_\vartheta$, it holds that every ${\mathbfcal{M}}_\vartheta$ satisfying ${\mathcal{T}}_{\mathcal{KB}}$, and thus $\phi_{\mathcal{KB}}$, does not satisfy $\psi_Q\sigma'$. Since we can prove that ${\mathbfcal{M}}\models \phi_{\mathcal{KB}}\wedge \psi_Q\sigma'$, for some ${\ensuremath{\mathsf{4LQS^R}}}$-interpretation ${\mathbfcal{M}}$, by restricting our interest to the interpretations ${\mathbfcal{M}}_\vartheta$ of $\phi_{\mathcal{KB}}$ defined in the proof of Theorem \[teo:completeness\], it turns out that $\sigma'$ is not a substitution belonging to the answer set of $\psi_Q$ w.r.t. $\phi_{\mathcal{KB}}$, and this leads to a contradiction. Thus we have to admit that assertion (2) holds. Finally, since assertions (1) and (2) hold, $\Sigma'$ and the answer set of $\psi_Q$ w.r.t. $\phi_{\mathcal{KB}}$ coincide and the thesis holds. Termination of Procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ is based on the fact that the while-loops 5–13 and 14–44 terminate. Concerning termination of the while-loop 5–13, our proof is based on the following two facts. The E-Rule and PB-Rule are applied only to non-fulfilled formulae on open branches and tend to reduce the number of non-fulfilled formulae occurring on the considered branch. In particular, when the E-Rule is applied on a branch $\vartheta$, the number of non-fulfilled formulae on $\vartheta$ decreases. In case of application of the PB-Rule on a formula $\beta = \beta_1 \vee \ldots \vee \beta_n$ on a branch, the rule generates two branches. In one of them the number of non-fulfilled formulae decreases (because $\beta$ becomes fulfilled). In the other one the number of non-fulfilled formulae stays constant but the subset $B^{\overline{\beta}}$ of $\{\overline{\beta}_1,\ldots,\overline{\beta}_n\}$ occurring on the branch gains a new element. Once $|B^{\overline{\beta}}|$ gets equal to $n-1$, namely after at most $n-1$ applications of the PB-rule, the E-rule is applied and the formula $\beta = \beta_1 \vee \ldots \vee \beta_n$ becomes fulfilled, thus decrementing the number of non-fulfilled formulae on the branch. Since the number of non-fulfilled formulae on each open branch gets equal to zero after a finite number of steps and the E-rule and PB-rule can be applied only to non-fulfilled formulae on open branches, the while-loop 5–13 terminates. Concerning the while-loop 14–44, its termination can be proved by observing that the number of branches of the [KE-tableau]{}resulting from the execution of the previous while-loop 5–13 is finite and then showing that the internal while-loops 18–23 and 28–42 always terminate. Indeed, initially the set $\mathsf{Eq}_{\vartheta}$ contains a finite number of literals of type $x=y$, and $\sigma_\vartheta$ is the empty substitution. It is then enough to show that the number of literals of type $x=y$ in $\mathsf{Eq}_{\vartheta}$, with distinct $x$ and $y$, strictly decreases during the execution of the internal while-loop 18–23. But this follows immediately, since at each of its iterations one put $\sigma_{\vartheta} {\coloneqq}\sigma_{\vartheta} \cdot \{x/z, y/z\}$, with $z {\coloneqq}\min_{<_{\vartheta}}(x,y)$, according to a fixed total order $<_{\vartheta}$ over the variables of $\mathsf{Var}_{0}(\vartheta)$ and then the application of $\sigma_\vartheta$ to $\mathsf{Eq}_{\vartheta}$ replaces a literal of type $x=y$ in $\mathsf{Eq}_{\vartheta}$, with distinct $x$ and $y$, with a literal of type $x=x$. The while loop 28–42 terminates when the stack $\mathcal{S}$ of the nodes of the decision tree gets empty. Since the query $\psi_Q$ contains a finite number of conjuncts and the number of literals on each open and complete branch of ${\mathcal{T}}_{{\mathcal{KB}}}$ is finite, the number of possible matches (namely the size of the set $Lit^M_Q$) computed at step $(C )$ is finite as well. Thus, in particular, the internal while loop 34–38 terminates at each execution. Once the procedure has processed the last conjunct of the query, the set $Lit^M_Q$ of possible matches is empty and thus no element gets pushed in the stack $\mathcal{S}$ anymore. Since the first instruction of the while-loop at step $(i)$ removes an element from $\mathcal{S}$, the stack gets empty after a finite number of “pops”. Hence Procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ terminates, as we wished to prove. Next, we provide some complexity results. Let $r$ be the maximum number of universal quantifiers in each $S_i$ ($i=1,\ldots,m$), and put $k {\coloneqq}|{\mathtt{Var}_0}(\phi_{{\mathcal{KB}}})|$. Then, each $S_i$ generates at most $k ^r$ expansions. Since the knowledge base contains $m$ such formulae, the number of disjunctions in the initial branch of the [KE-tableau]{}is bounded by $m \cdot k^r$. Next, let $\ell$ be the maximum number of literals in each $S_i$. Then, the height of the [KE-tableau]{}(which corresponds to the maximum size of the models of $\Phi_{{\mathcal{KB}}}$ constructed as illustrated above) is $\mathcal{O}( \ell m k^r)$ and the number of leaves of the tableau, namely the number of such models of $\Phi_{{\mathcal{KB}}}$, is $\mathcal{O}(2^{\ell m k^r})$. Notice that the construction of $\mathsf{Eq}_{\vartheta}$ and of $\sigma_{\vartheta}$ in the lines 16–23 of Procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ takes $\mathcal{O}( \ell m k^r)$ time, for each branch $\vartheta$. Let $\eta({\mathcal{T}}_{{\mathcal{KB}}})$ and $\lambda({\mathcal{T}}_{{\mathcal{KB}}})$ be, respectively, the height of ${\mathcal{T}}_{{\mathcal{KB}}}$ and the number of leaves of ${\mathcal{T}}_{{\mathcal{KB}}}$ computed by Procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$. Plainly, $\eta({\mathcal{T}}_{{\mathcal{KB}}}) = \mathcal{O}(\ell m k^r)$ and $\lambda({\mathcal{T}}_{{\mathcal{KB}}})= \mathcal{O}(2^{\ell m k^r})$, as computed above. It is easy to verify that $s=\mathcal{O}(\ell k^r)$ is the maximum branching of ${\mathcal{D}}_\vartheta$. Since the height of ${\mathcal{D}}_\vartheta$ is $h$, where $h$ is the number of literals in $\psi_Q$, and the successors of a node are computed in $\mathcal{O}(\ell k^r)$ time, the number of leaves in ${\mathcal{D}}_\vartheta$ is $\mathcal{O}(s^{h})=\mathcal{O}((\ell k^r)^{h})$ and they are computed in $\mathcal{O}( s^{h} \cdot \ell k^r \cdot h) = \mathcal{O}(h \cdot (\ell k^r)^{(h+1)})$ time. Finally, since we have $\lambda({\mathcal{T}}_{{\mathcal{KB}}})$ of such decision trees, the answer set of $\psi_{Q}$ w.r.t. $\phi_{\mathcal{KB}}$ is computed in time $\mathcal{O}(h \cdot (\ell k^r)^{(h+1)} \cdot\lambda({\mathcal{T}}_{{\mathcal{KB}}})) =\mathcal{O}( h \cdot (\ell k^r)^{(h+1)} \cdot 2^{\ell m k^r})$. Since the size of $\phi_{\mathcal{KB}}$ and of $\psi_{Q}$ are related to those of ${\mathcal{KB}}$ and of $Q$, respectively (see the proof of Theorem \[CQADL\] in [@RR2017ext] for details on the reduction), the construction of the HO answer set of $Q$ with respect to ${\mathcal{KB}}$ can be done in double-exponential time. In case ${\mathcal{KB}}$ contains neither role chain axioms nor qualified cardinality restrictions, the complexity of our *HOCQA* problem is in EXPTIME, since the maximum number of universal quantifiers in $\phi_{{\mathcal{KB}}}$, namely $r$, is a constant (in particular $r = 3$). The latter complexity result is a clue of the fact that the *HOCQA* problem is intrinsically more difficult than the consistency problem (proved to be NP-complete in [@CanLonNicSanRR2015]). This is motivated by the fact that the consistency problem simply requires to guess a model of the knowledge base while the *HOCQA* problem forces the construction of all the models of the knowledge base and to compute a decision tree for each of them. We remark that such result compares favourably to the complexity of the usual CQA problem for a wide collection of description logics such as the Horn fragment of $\mathcal{SHOIQ}$ and of $\mathcal{SROIQ}$, respectively, EXPTIME- and 2EXPTIME-complete in combined complexity (see [@Ortiz:2011:QAH:2283516.2283571] for details). Conclusions and future work =========================== In this paper we have considered an extension of the CQA problem for the description logic ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ to more general queries on roles and concepts. The resulting problem, called HOCQA, can be instantiated to the most widespread ABox reasoning services such as instance retrieval, role filler retrieval, and instance checking. We have proved the decidability of the HOCQA problem by reducing it to the satisfiability problem for the set-theoretic fragment ${\ensuremath{\mathsf{4LQS^R}}}$. We have introduced an algorithm to compute the HO answer set of a ${\ensuremath{\mathsf{4LQS^R}}}$-formula $\psi_{Q}$ representing a HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-conjunctive query $Q$ w.r.t. a ${\ensuremath{\mathsf{4LQS^R}}}$-formula $\phi_{{\mathcal{KB}}}$ representing a ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ knowledge base. The procedure, called *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$, is based on the [KE-tableau]{}system and on decision trees. It takes as input $\psi_{Q}$ and $\phi_{{\mathcal{KB}}}$, and yields a [KE-tableau]{}${\mathcal{T}}_{{\mathcal{KB}}}$ representing the saturation of $\phi_{{\mathcal{KB}}}$ and the requested HO answer set $\Sigma'$. Procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ is proved correct and complete, and some complexity results are provided. Such procedure extends the one introduced in [@ictcs16] as it allows one to handle HO ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$-conjunctive queries. We are currently working at the implementation of Procedure *HOCQA*-${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$. We plan to increase the efficiency of the expansion rules and to extend reasoning with data types. Lastly, we intend to provide a parallel model of the procedure that we are implementing. We also plan to increase the expressive power of the set theoretic fragments we are working with. In particular, we intend to define a decidable $n$-level stratified syllogistic allowing to represent an extension of ${\mathcal{DL}_{{\mathbf{D}}}^{4,\!\times}}$ admitting data type groups. We also intend to extend the set-theoretic fragment presented in [@CanNic2013] with the construct of generalized union and with a restricted form of binary relational composition operator. The latter operator, in particular, turns out to be useful for the set-theoretic representation of various logics. The [KE-tableau]{}based procedure will be adapted to the new set-theoretic fragments by also making use of the techniques introduced in [@CanNicOrl11] and in [@CanNicOrl10] in the area of relational dual tableaux. On the other hand we think that [KE-tableau]{}x could be used in the ambit of relational dual tableaux to improve the performances of relational dual tableau-based decision procedures. [^1]: Definitions of positive occurrence and of negative occurrence of a formula inside another formula can be found in [@CanNic2013]. [^2]: The use of level $3$ variables to model abstract and concrete role terms is motivated by the fact that their elements, that is ordered pairs $\langle x, y \rangle$, are encoded in Kuratowski’s style as $\{\{x\}, \{x,y\}\}$, namely as collections of sets of objects.
--- abstract: 'We report intensity variations of different spectral components in the BL Lac object S5 0716+714 detected during coordinated [[*BeppoSAX* ]{}]{}and optical observations in 1996 and 1998. The transition between synchrotron and inverse Compton emission has been clearly detected as sharp X-ray spectral breaks at around 2-3 keV on both occasions. Correlated optical and soft X-ray variability was found during the second [[*BeppoSAX* ]{}]{}pointing when intensive optical monitoring could be arranged. The hard (Compton) component changed by a factor of 2 between the two observations, but remained stable within each exposure. During events of rapid variability S5 0716+714 showed spectral steepening with intensity, a behaviour rarely observed in BL Lacs. We interpret these findings as the probable consequence of a shift of the synchrotron peak emission from the IR/optical band to higher energies, causing the synchrotron tail to push into the soft X-ray band more and more as the source brightens.' author: - 'P. Giommi, E. Massaro, L. Chiappetti, E.C. Ferrara, G., Ghisellini, Minhwan Jang, M. Maesano, H.R. Miller, F. Montagni, R. Nesci, P. Padovani, E. Perlman, C.M. Raiteri, S. Sclavi, G. Tagliaferri, G. Tosti, M. Villata' date: 'Received ; accepted ' title: Synchrotron and Inverse Compton Variability in the BL Lacertae object S5 0716+714 --- epsf Introduction ============ It is generally agreed that the main mechanism powering BL Lacs is synchrotron emission followed by inverse Compton scattering in a relativistic beaming scenario (e.g. Kollgaard 1994, Urry & Padovani 1995, Ghisellini, Maraschi & Dondi 1996). The synchrotron component peaks (in a $\nu~vs~\nu~f_\nu$ representation) at energies ranging from infrared frequencies, for Low energy peaked (LBL) BL Lacs (mostly discovered in radio surveys), to hard X-rays for extreme High energy peaked (HBL) BL Lacs (typically discovered in X-ray surveys, Padovani & Giommi 1995, Sambruna, Maraschi & Urry 1996). S5 0716+714 is a strongly variable (e.g. Wagner & Witzel 1995, Otterbein [et al. ]{}1998) BL Lac object characterized by a spectral energy distribution (SED) peaking at intermediate frequencies ($\nu_{peak} \sim 10^{14}-10^{15}$ Hz) compared to LBL and HBL BL Lacs. To date no redshift has been measured for this object. In this paper we report the detection of intensity and spectral variations involving both the synchrotron and the inverse Compton components of S5 0716+714. The data are from two observations carried out with the Narrow Fields Instruments (NFI) of the [[*BeppoSAX* ]{}]{}satellite (Boella et al. 1997a), and from simultaneous optical observations made with a number of small telescopes in Italy and in Korea. Table 1. - Log of the [[*BeppoSAX* ]{}]{}observations of S5 0716+714 and image analysis results -------------- -------------- ------------------------ ------------------------ -------------- ----------------------- -- Date LECS Count rate Count rate MECS Count rate (2-10 keV) exposure (s) (0.1-2.keV)$ct~s^{-1}$ (2-10 keV) $ct~s^{-1}$ exposure (s) $ct~s^{-1}$ 14-NOV-1996 13122 $0.020\pm 0.001$ $0.006\pm 0.0008$ 122509 $0.0263^a\pm 0.0006$   7-NOV-1998  9475 $0.022\pm 0.002$ $0.011\pm 0.0010$   31317 $0.0344^b\pm 0.0012$ -------------- -------------- ------------------------ ------------------------ -------------- ----------------------- -- $^a$ Three MECS units;  $^b$ Two MECS units\ BeppoSAX observations and data analysis ======================================= S5 0716+714 was observed by [[*BeppoSAX* ]{}]{}twice on November 14, 1996 and on November 7, 1998. Preliminary results on the first [[*BeppoSAX* ]{}]{}observation have been reported in Chiappetti [et al. ]{}(1997). Screened event lists for the LECS (Parmar [et al. ]{}1997) and MECS (Boella et al 1997b) instruments, and the average PDS (Frontera [et al. ]{}1997) background-subtracted spectra were taken from the [[*BeppoSAX* ]{}]{}SDC on-line archive (Giommi & Fiore 1998). The data analysis was performed using the software available in the XANADU package (XIMAGE, XRONOS, XSPEC). The analysis of the [[*BeppoSAX* ]{}]{}X-ray images shows that S5 0716+714 was found in a rather faint state compared to previous observations (e.g. Urry [et al. ]{} 1996, Wagner [et al. ]{}1996). The count rates in the LECS and MECS instruments have been estimated using XIMAGE (Giommi et al. 1991), upgraded at the [[*BeppoSAX* ]{}]{}SDC to support [[*BeppoSAX* ]{}]{}imaging data. The observational parameters and the measured count rates for the two instruments are given in Table 1. Significant intensity variations were detected between and during each observation. In particular, the comparison of the 1996 and 1998 data shows that while the count rate in the soft band (0.1-2 keV) was found at the same level, the intensity in the harder 2-10 keV band changed by nearly a factor two both in the LECS and in the MECS [^1] instruments. This difference in variability implies that the X-ray spectral shape of S5 0716+714 changed significantly between the two observations. On time scales of hours X-ray variability was detected only in the low energy band (0.1-2 keV) as discussed in section 4. Although some signal is present in the PDS data in both observations the source was too faint to be detected above the confusion limit of $\approx 2-3 \times 10^{-12}$ cgs. XSPEC spectral fits (over the full 0.1-10. keV LECS and MECS bands) with a single power law model, modified by Galactic absorption as estimated from 21 cm measurements ($N_H = 3.8\times 10^{20}$ cm$^{-2}$, Dickey and Lockman 1990) gave best fit energy spectral indices $\alpha_x(1996)=1.1\pm0.08$ and $\alpha_x(1998)=0.9\pm 0.1$ and with high [$\chi_{\nu} ^{2}$]{}values ([$\chi_{\nu} ^{2}$]{}= 1.35, 55 d.o.f. and [$\chi_{\nu} ^{2}$]{}= 1.63, 53 d.o.f, respectively) due to a poor fit at soft energies. These slopes are much flatter than the one found in a previous ROSAT PSPC observation ($\alpha_x=1.8$, 0.1-2.0 keV) during which S5 0716+714 was also highly variable (Urry [et al. ]{}1996, Wagner [et al. ]{}1996). We next fitted the data with a broken power law model and $N_H$ fixed to the Galactic value. The spectral fit and the residuals for the 1996 observation are shown in Fig. 1. The resulting spectral parameters with the statistical ($1\sigma$) uncertainties are reported in Table 2. In both observations the spectral slope is much steeper ($\Delta \alpha = 0.6-0.7$ ) at lower energies (possibly consistent with the ROSAT slope), and the spectral break is around 2-3 keV. An F test shows that the broken power law model significantly improves (prob $> 99.99$%) the fit compared to the single power law model. The [$\chi_{\nu} ^{2}$]{}$~$ in the second observation ($\sim 1.5$) is still relatively high, likely because of the presence of non negligible rapid spectral variability (see below). Table 2. - Results of LECS+MECS spectral analysis - Broken power law model -------------- ----------------- ----------------- -------------------- ------------------------- ----------------------------- ----------------------------- Date $\alpha_{soft}$ $\alpha_{hard}$ Break energy (keV) [$\chi ^{2}$]{}(d.o.f.) Flux 0.1-2 kev Flux 2-10 kev (energy index) (energy index) [erg cm$^{-2}$ s$^{-1}~$]{} [erg cm$^{-2}$ s$^{-1}~$]{} 14-NOV-1996 $1.7 \pm 0.3$ $0.96 \pm 0.15$ $2.3\pm0.4$ 56(48) $2.0\times 10^{-12} $ $1.4\times 10^{-12} $   7-NOV-1998 $1.3 \pm 0.4$ $0.73 \pm 0.18$ $2.8\pm0.8$ 76(49) $1.8\times 10^{-12} $ $2.6\times 10^{-12} $ -------------- ----------------- ----------------- -------------------- ------------------------- ----------------------------- ----------------------------- Optical observations ==================== Photometric measurements of S5 0716+714 were performed with telescopes operated by the Roma, Perugia and Torino groups equipped with CCD cameras, two of them mounting a back illuminated SITe SIA502A chip. The bandpasses used were the standard Johnson B, V, the Cousins R$_C$, I$_C$ and the F86, F98 of the Arizona system (Johnson & Mitchell 1975). BVR$_C$I$_C$ magnitudes of three standard stars in the field of S5 0716+714 were taken from Ghisellini et al. (1997) and Villata [et al. ]{}(1998a), while the magnitude in the Arizona filters of the same stars were calibrated with the primary standard BS2527 (Johnson & Mitchell 1975). Other observations in the R$_C$ band were performed with the Kyung Hee University telescope in Korea at the beginning of the [[*BeppoSAX* ]{}]{}pointing. During both [[*BeppoSAX* ]{}]{}X-ray observations the source was not in bright states and at about the same magnitude R$_C\sim$13.8. The optical monitoring (Villata et al. 1998b, Massaro [et al. ]{}1999a, Raiteri [et al. ]{}1999) shows that in November 1996 S5 0716+714 was in a declining phase following a small burst occurred two weeks before, whereas in November 1998 it was in a mild brightening phase twenty days after the lowest recorded level (R$_C$=14.40) since January 1996. Short-term variability ====================== Figure 2 shows the soft (1.3-3.0 keV) and hard (5.0-10. keV) MECS lightcurve during the long 1996 [[*BeppoSAX* ]{}]{}observation. Clear variations of up to a factor of two are evident in the low energy curve, while above 5 keV the source flux remains constant within approximately 20%. During this observation we were able to perform simultaneous optical monitoring (in the F86 Arizona band) for about five hours starting at the beginning of the pointing, and covering only the initial part of the X-ray lightcurve. We observed a smooth decline of the luminosity: the F86 magnitude varied from 13.14 at 0.0h UT to 13.23 at 4.6h UT, a similar decrease was also observed in the V band. The decreasing trend is also clearly evident in the first segment of the soft X-ray light curve (Figure 2) and continued in this band for about six hours. A much better simultaneous coverage was achieved during the 1998 November 7 observation when the source was followed for about 12 hours in three optical bands and in the X-rays. The optical light curves are reported in Figure 3 (upper panel) and show a very similar behaviour with a variation of about 0.1 mag over a time interval of a few hours: the flux decreased from about 19h UT to a minimum at 24h UT and then increased again reaching the previous level in about two hours. The same general behaviour is clearly apparent in the 1.3-3.0 keV MECS data (Figure 3, lower panel), while at higher energies (5-10 keV) the count rate, similarly to the 1996 observation, is consistent with a constant value. The light curves of Figure 3 indicate the possibility that optical and X-ray variations are not strictly simultaneous: the soft X-ray luminosity could be declining since the beginning of the observation and could reach its lowest level about 2-3 hours before the optical minimum. The following rise, however, seems to start at the same time (about 25h) in all bands. This conclusion, however, cannot be firmly established because of poor statistics. Since the soft and hard X-rays follow different evolutions, the X-ray spectral shape must change with source intensity. This is clearly apparent from Figure 4 where the MECS hardness ratio (1.5-3.0/3.0-10 keV) in both [[*BeppoSAX* ]{}]{} observations is anti-correlated with intensity: the spectrum steepens when the source brightens. The LECS lightcurve is consistent with these findings, although, in this case the photon statistics is poor due to the lower exposure (Table 1) and less sensitivity of this instrument compared to the MECS. The correlation between the low energy X-ray and optical light curves observed by us is in contrast with the results of the 1996 April campaign on S5 0716+714 (Otterbein et al. 1998), but agrees with the 1991 observation (Wagner et al. 1996). Notice that the amplitude of the X-ray variation (a factor of $\approx 2$) is higher than at optical-IR frequencies, where it is $\sim$10 %. Discussion ========== During the two [[*BeppoSAX* ]{}]{}and optical observations presented here S5 0716+714 was detected in a faint state similar to that seen during other X-ray observations when the source was not flaring (Wagner [et al. ]{}1996, Otterbein [et al. ]{}1998). Nevertheless significant flux and spectral changes were detected, confirming the tendency of this object to be frequently variable. The X-ray spectrum of S5 0716+714 can be well represented by a broken power law with steep (energy) slope ($\alpha_x\gsim 1.5 $) until 2-3 keV where a much harder component ($\alpha_x \sim 0.8-0.9 $) emerges. Figure 5 shows the broad band $\nu-\nu~f_\nu$ SED of S5 0716+714, derived from our simultaneous optical and X-ray data together with nearly simultaneous radio data from the UMRAO database (Aller [et al. ]{}1999) and from non-simultaneous photometric data taken from NED. Figure 5 shows that the break detected by [[*BeppoSAX* ]{}]{}in the soft X-ray spectrum marks the merging of the steep tail of the synchrotron emission into the harder inverse Compton component. A similar transition between synchrotron and Compton emission has been recently detected in [[*BeppoSAX* ]{}]{}observations of another intermediate BL Lac object: ON 231 (Tagliaferri et al. 1999). The different spectral slopes detected in different luminosity states are then easily explained: the spectrum is steeper when the source is bright and the tail of the synchrotron component dominates the X-ray flux. When the source is faint most of the X-ray flux is due to the flat Compton emission. The fast flux variations detected only in the soft X-rays strongly suggest that rapid variability (correlated with the optical) comes from the high energy tail of the synchrotron component which peaks at a few times $10^{14} $Hz. Long term variability in the hard Compton component is instead apparent from the comparison of the 1996 and 1998 spectral energy distributions (Figure 5). The different variability timescales in the synchrotron (soft X-ray) and Compton (harder X-rays) emission explain in a natural way the surprisingly poor correlation between the simultaneous RXTE (2-10 keV) and ROSAT HRI (0.1-2.4 keV) light curves reported by Otterbein et al. (1998). Evidence for spectral steepening with intensity was found in the form of an anticorrelation between hardness ratio and source flux (Figure 4). Such a behaviour is opposite to that observed in many BL Lacs (Giommi [et al. ]{}1990, Takahashi [et al. ]{}1996, 1999) but can be simply explained as due to variability in the synchrotron tail that steepens the overall X-ray spectrum when the flux of this component increases. This effect was predicted by Padovani and Giommi (1996) to be detectable in intermediate BL Lac objects like S5 0716+714. If the optical and X-ray variability are associated, the presence of a lag between the minimum of the light curves in the different bands of the 1998 campaign can be ascribed to the different radiative cooling time of the electrons emitting in the optical and in the X-ray bands. If, instead, the delay is due to geometry (e.g. electrons diffusing out from an injection region while cooling, or a shock travelling in the jet), the cooling time must be shorter than the observed lag. In any case, a firm limit can be derived, by requiring that the cooling time is equal or shorter than the observed time lag. We therefore require that the synchrotron and self Compton cooling time for optical emitting electrons is equal to or shorter than $t_{lag}\delta/(1+z)$: $$t_{cool}\, = \, { 6\pi mc^2 \over \sigma_Tc\gamma_o B^2(1+U_S/U_B)} \, \le { t_{lag}\delta \over (1+z)},$$ where $\sigma_T$ is the Thomson scattering cross section, $\gamma_o m_ec^2$ is the energy of the particles emitting at the observed frequency, $U_B=B^2/(8\pi)$ is the magnetic energy density, and $U_S$ is the synchrotron radiation energy density (as measured in the comoving frame), and $\delta$ is the beaming or Doppler factor. The energy $\gamma_0$ is related to the observed frequency by $\nu_o \simeq 3.7\times 10^6 \gamma_o^2 B \delta/(1+z)$ Hz (assuming synchrotron radiation). We then obtain the two limits: $$B\, \ge \, 5.6 \, \nu_{15}^{-1/3} t_h^{-2/3}(1+U_S/U_B)^{-2/3} \left({1+z \over \delta}\right)^{1/3} \quad {\rm G}$$ $$\gamma_o\, \le \, 6.9\times 10^3 \nu_{15}^{2/3} \left[{ t_h (1+U_S/U_B)(1+z) \over \delta }\right]^{1/3}$$ where $t_h$ is the lag time measured in hours and $\nu_o=10^{15}\nu_{15}$ Hz. Assuming $\delta=10$, $t_h=4$, $\nu_{15}=0.5$ and $U_S/U_B=1$, we find $B\ge 0.9$ G and $\gamma_o\le 4.4\times 10^3$, in very good agreement with the estimates of Ghisellini [et al. ]{}(1997). With these parameters, we expect that the peak of the self Compton emission is at $h\nu_c\sim \gamma_o^2h\nu_o \lsim 50$ MeV. It has been shown in a number of occasions that the peak of the synchrotron power in BL Lacs tend to move to higher energies during flares (e.g. Giommi [et al. ]{}1990, Pian [et al. ]{}1998, Takahashi [et al. ]{}1999, Malizia [et al. ]{}1999, Giommi [et al. ]{}1999, Massaro [et al. ]{}1999b). In the case of S5 0716+714, a shift of $\nu_{peak}$ from the optical band to a somewhat higher energy, would cause a much larger synchrotron contribution to the X-ray flux. In case of large events, involving a shift of $\nu_{peak}$ of a factor 10 or more (as observed in MKN501, Pian et al. 1998, and 1ES2344+514, Giommi [et al. ]{}1999), the steep synchrotron emission would entirely dominate the X-ray spectrum and further flux increases would cause spectral flattening just as observed in many HBL BL Lacs where $\nu_{peak}$ is at the UV/X-ray frequencies. The results presented in this paper demonstrate that intermediate BL Lacs are potentially very important to study the relation and the interaction between the two main components in the Spectral Energy Distribution of BL Lacs. Important observational quantities are the determination of the position of synchrotron peak at different intensity levels, and the detailed evolution of the spectral shape in the X-ray band which includes in different mixtures both the Synchrotron and Compton components. Multi-wavelength campaigns on this type of BL Lacs, covering the near infrared and X-rays can lead to a deep understanding of the physical processes powering Blazars and can unravel the origin of the seed photons upscattered to high energies. This research has made use of the ASI-BeppoSAX SDC on-line database and archive system and of the NASA/IPAC National Extragalactic Database (NED). We thank F. Fiore for providing the code to de-redden the XSPEC unfolded spectra used to construct the Spectral Energy Distribution of S5 0716+714. This research has made use of data from the University of Michigan Radio Astronomy Observatory. Aller M.F, Aller H.D., Huges P.A., & Latimer G.E., 1999 ApJ 512, 601 Boella G. 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--- abstract: | We calculate exactly the quantum mechanical, temporal Wigner quasiprobability density for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz., a displaced-squeezed thermal state. The Wigner function allows us to calculate the fluctuations in photon number and the quadrature variance. We contrast the difference between the nonclassicality criteria, which is independent of the displacement parameter $\alpha$, based on the Glauber-Sudarshan quasiprobability distribution $P(\beta)$ and the classical/nonclassical behavior of the Mandel $Q_{M}(\tau)$ parameter, which depends strongly on $\alpha$. We find a phase transition as a function of $\alpha$ such that at the critical point $\alpha_{c}$, $Q_{M}(\tau)$, as a function of $\tau$, goes from strictly classical, for $|\alpha|< |\alpha_{c}|$, to a mixed classical/nonclassical behavior, for $|\alpha|> |\alpha_{c}|$.\ **Keywords**: degenerate parametric amplifier; displaced-squeezed thermal states; Wigner function; Mandel parameter; nonclassicality criteria author: - Moorad Alexanian title: 'Dynamically generated quadrature and photon-number variances for Gaussian states' --- Introduction ============ The generation of nonclassical radiation fields, e.g., quadrature-squeezed light, photon antibunching, sub-Poissonian statistics, etc., establishes the discrete nature of light and serves to study fundamental questions regarding the interaction of quantized radiation fields with matter [@GSA13]. In a recent work [@MA16], a detailed study was made of the temporal development of the second-order coherence function $g^{(2)}(\tau)$ for Gaussian states—displaced-squeezed thermal states—the dynamics of which is governed by a Hamiltonian for degenerate parametric amplification. The time development of the Gaussian state is generated by an initial thermal state and the system subsequently evolves in time where the usual assumption of statistically stationary fields is not made. Nonclassicality were observed [@MA16] for various values of the parameters governing the temporal development of the coherence function $g^{(2)}(\tau)$—such as the coherent parameter $\alpha$, squeeze parameter $\xi$, and the mean photon number $\bar{n}$ of the initial thermal state. Our characterization of nonclassicality was based solely on the coherence function violating inequalities satisfied by the classical correlation functions. More recently [@MAa16], we dwelt into the notion of nonclassicality based on the characteristic function $\chi(\eta)$ and its two-dimensional Fourier transform to determine the existence or nonexistence of the quasiprobability distribution $P(\beta)$ of the Glauber-Sudarshan coherent or P representation of the density of state. It was shown [@MAa16] that the nonclassicality criteria for the radiation field, based on the one-time function $P(\beta)$, cannot characterize the classical, quantum mechanical or mixed nature of the dynamical system as attested by the temporal behavior of the two-time $g^{(2)}(\tau)$ function. It is interesting that an analogous result was also obtained recently where it is argued that negative full counting statistics captures nonclassicality in the dynamics of the system in contrast to more conventional quasiprobability distributions that captures nonclassicality in the instantaneous state of the system [@HC16]. In this paper we show that the nonclassicality criteria based on the one-time $P(\beta)$ function cannot even characterize the classical/nonclassical behavior of the one-time Mandel $Q_{M}(\tau)$ parameter as a function of $\tau$. Generic compact expressions for the Wigner function for the one-mode electromagnetic field for general mixed Gaussian quantum states are well known [@DMM94]. The Wigner function is given in terms of five real parameters, viz. three of them are the variances and the covariance of photon quadrature components, while two others are the means of the quadratures. Of course, the importance of the present work is the exact time development of the dynamical system governed by a Hamiltonian for degenerate parametric amplification giving the explicit time dependence of all those five real parameters without assuming statistically stationary fields. We consider in Sec. II the general Hamiltonian of the degenerate parametric amplifier and present the result for the quantum degree of second-order coherence $g^{(2)}(\tau)$ of Ref. [@MA16]. In Sec. III, we give an explicit expression of the exact, time-dependent Wigner quasiprobability density. In Sec. IV, use is made of the Wigner function to calculate the field-quadrature variance. Sec. V gives the results for the photon-number variance. In Sec. VI, we present the differing criteria for nonclassicality. In Sec. VII, we compare numerically the behavior of the Mandel parameter and determine the existence of a phase transition. Sec. VIII summarizes our results. In Appendix A we give an explicit expression for the Wigner function in terms of the quadrature components $x_{\lambda}$ and $x_{\lambda + \pi/2}$ and express it in a form akin to the generic expression for the general mixed Gaussian quantum states. Finally, in Appendix B we present the relevant mathematical expressions that characterize the phase transition. Degenerate parametric amplification =================================== The Hamiltonian for degenerate parametric amplification, in the interaction picture, is $$\hat{H} = c \hat{a}^{\dag 2} + c^* \hat{a}^2 + b\hat{a} + b^* \hat{a}^\dag.$$ The radiation field is initially in a thermal state $\hat{\rho}_{0}$ and a after a preparation time $t$, the radiation field develops in time into a Gaussian state and so [@MA16] $$\hat{\rho}_{G}=\exp{(-i\hat{H}t/\hbar)}\hat{\rho}_{0} \exp{(i\hat{H}t/\hbar)}$$ $$= \hat{D}(\alpha) \hat{S}(\xi)\hat{\rho}_{0} \hat{S}(-\xi) \hat{D}(-\alpha),$$ with the displacement $\hat{D}(\alpha)= \exp{(\alpha \hat{a}^{\dag} -\alpha^* \hat{a})}$ and the squeezing $\hat{S}(\xi)= \exp\big{(}-\frac{\xi}{2} \hat{a}^{\dag 2} + \frac{\xi^*}{2} \hat{a}^{2} \big{ )}$ operators, where $\hat{a}$ ($\hat{a}^{\dag})$ is the photon annihilation (creation) operator, $\xi = r \exp{(i\theta)}$, and $\alpha= |\alpha|\exp{(i\varphi)}$. The thermal state is given by $$\hat{\rho}_{0} = \exp{(-\beta \hbar \omega\hat{n})}/ \textup{Tr}[\exp{(-\beta \hbar \omega \hat{n})}],$$ with $\hat{n}= \hat{a}^\dag \hat{a}$ and $\bar{n}= \textup{Tr}[\hat{\rho}_{0} \hat{n}]$ . The parameters $c$ and $b$ in the degenerate parametric Hamiltonian (1) are determined [@MA16] by the parameters $\alpha$ and $\xi$ of the Gaussian density of state (2) via $$tc = -i\frac{\hbar}{2} r\exp(i\theta)$$ and $$tb= -i\frac{\hbar}{2}\Big{(} \alpha \exp{(-i\theta)} + \alpha^* \coth (r/2)\Big{)} r,$$ where $t$ is the time that it takes the radiation field governed by the Hamiltonian (1) to generate the Gaussian density of state $\hat{\rho}_{G}$ from the initial thermal density of state $\hat{\rho}_{0}$. The quantum mechanical seconde-order, degree of coherence is given by [@MA16] $$g^{(2)}(\tau) = \frac{\langle \hat{a}^{\dag}(0) \hat{a}^{\dag}(\tau) \hat{a}(\tau) \hat{a}(0)\rangle }{\langle \hat{a}^{\dag}(0) \hat{a}(0)\rangle \langle \hat{a}^{\dag}(\tau)\hat{a}(\tau) \rangle},$$ where all the expectation values are traces with the Gaussian density operator, viz., a displaced-squeezed thermal state. Accordingly, the radiation field is initially in the thermal state $\hat{\rho}_{0}$. After time $t$, the radiation field evolves to the Gaussian state $\hat{\rho}_{G}$ and a photon is annihilated at time $t$, the system then develops in time and after a time $\tau$ another photon is annihilated [@MA16]. Therefore, two photon are annihilated in a time separation $\tau$ when the radiation field is in the Gaussian density state $\hat{\rho}_{G}$. It is important to remark that we do not suppose statistically stationary fields. Therefore, owing to the $\tau$ dependence of the number of photons in the cavity in the denominator of Equation (6), $g^{(2)}(\tau)$ asymptotically, as $\tau\rightarrow \infty$, approaches a finite limit without supposing any sort of dissipative processes [@MA16]. The coherence function $g^{(2)}(\tau)$ is a function of $\Omega \tau=(r/t)\tau$, $\alpha$, $\xi$, and the average number of photons $\bar{n}$ in the initial thermal state (3), where the preparation time $t$ is the time that it takes the system to dynamically generate the Gaussian density $\hat{\rho}_{G}$ given by (2) from the initial thermal state $\hat{\rho}_{0}$ given by (3). Note that the limit $r\rightarrow 0$ is a combined limit whereby $\Omega =r/t$ also approaches zero resulting in a correlation function which has a power law decay as $\tau/t \rightarrow \infty$ rather than an exponential law decay as $\tau/t \rightarrow \infty$ as is the case in the presence of squeezing when $r>0$ [@MA16]. Wigner quasiprobability density =============================== The dynamics of the system is governed by the degenerate parametric amplification Hamiltonian (1) that generates the Gaussian state $\hat{\rho}_{G}$ from the initial thermal state $\hat{\rho}_{0}$ and subsequently determines the temporal behavior of the system [@MA16]. One has that $$\hat{\rho}(t+\tau) =\exp\big{(}-i\hat{H}(t+\tau)\big{)} \hat{\rho}_{0}\exp \big{(}i\hat{H}(t+\tau)\big{)}$$ $$= \exp(-i\hat{H}\tau) \hat{\rho}_{G}\exp(i\hat{H}\tau).$$ Accordingly, for any operator function $\mathcal{\hat{O}}(\hat{a},\hat{a}^\dag)$, $$\textup{Tr}[\hat{\rho}(t+\tau) \mathcal{\hat{O}}(\hat{a},\hat{a}^\dag)] = \textup{Tr}[\hat{\rho}_{G} \mathcal{\hat{O}}\big{(}\hat{a}(\tau),\hat{a}^\dag (\tau)\big{)}]$$ $$\equiv \langle \mathcal{\hat{O}}\big{(}\hat{a}(\tau),\hat{a}^\dag (\tau)\big{)} \rangle .$$ One obtains for the characteristic function [@MAa16] $$\chi(\eta) = \textup{Tr}[\hat{\rho}(t+\tau)\exp{(\eta \hat{a}^\dag}-\eta^*\hat{a})]\exp{(|\eta|^2/2)}$$ $$=\textup{Tr}[\hat{\rho}(t+\tau)\exp{(\eta \hat{a}^\dag}) \exp{(-\eta^*\hat{a})}]$$ $$=\exp{(|\eta|^2/2)} \exp{\big{(}\eta A^*(\tau)- \eta^* A(\tau)\big{)}}$$ $$\times \exp{\big{(}-(\bar{n}+1/2)|\xi(\tau)|^2\big{)}},$$ where $$A(\tau) =\alpha \Bigg{(}\cosh(\Omega\tau)+\frac{1}{2}\coth(r/2) \sinh (\Omega \tau)$$ $$-\frac{1}{2} (\cosh(\Omega \tau)-1)+\exp[i(\theta -2 \varphi)]\Big{[} -\frac{1}{2}\sinh(\Omega\tau)$$ $$-\frac{1}{2}\coth(r/2)\big{(}\cosh(\Omega\tau)-1\big{)}\Big{]} \Bigg{)}$$ and $$\xi(\tau)= \eta\cosh(\Omega \tau +r) +\eta^* \exp(i\theta) \sinh(\Omega \tau +r),$$ with the displacement parameter $\alpha=|\alpha|e^{i\varphi}$, the squeezing parameter $\xi=re^{i\theta}$, and $t$ representing the time it takes the radiation field to dynamically evolve from the thermal state $\hat{\rho}_{0}$ to the Gaussian state $\hat{\rho}_{G}$. Define $$|\xi(\tau)|^2 = \eta^2 T^*(\tau) +\eta^{*2} T(\tau) +\eta \eta^* S(\tau),$$ with $$T(\tau)= \frac{1}{2} \exp{(i \theta)} \sinh [2(\Omega \tau + r)]$$ and $$S(\tau)= \cosh[2(\Omega \tau +r)].$$ The Wigner function [@GSA13] is defined by $$W(\beta) =\frac{1}{\pi^2} \int \textup{d}\eta^2\chi(\eta) e^{-|\eta|^2/2} \exp(-\beta^*\eta +\beta \eta^*).$$ Note the presence of the factor $e^{-|\eta|^2/2}$ in the Wigner function (15), which is absent in the definition of the quasiprobability distribution $P(\beta)$ given in Equation (16) of Ref. [@MAa16]. The integral (15) can be carried out for the characteristic function (9) and so $$W(\beta)=\frac{2}{\pi} \frac{1}{\sqrt{4a^2b^2-c^2}} e^{-(a^2f^2+b^2d^2+cfd)/(4a^2b^2-c^2)},$$ where $$a^2= (\bar{n}+1/2)\big{(}T(\tau) +T^*(\tau) +S(\tau)\big{)},$$ $$b^2= -(\bar{n}+1/2)\big{(}T(\tau) +T^*(\tau) -S(\tau)\big{)},$$ $$c=-2 i(\bar{n}+1/2)\big{(}T^*(\tau) -T(\tau)\big{)},$$ $$d= i(A(\tau) -A^*(\tau)-\beta+ \beta^*),$$ $$f= A(\tau)+A^*(\tau) -\beta-\beta^*.$$ Note the absence of the term $-\frac{1}{2}$ in both $a^2$ and $b^2$ in (17) which terms appear in the expression for the quasiprobability distribution $P(\beta)$ as given by Equation (18) in Ref. [@MAa16]. The absence of such terms has quite an important consequence for the nonclassicality criteria based on the Wigner function. The existence of a real-valued function $W(\beta)$ requires $$(4a^2b^2-c^2) = 4(\bar{n}+ 1/2)^2 \geq 0,$$ with the aid of (17), which is obviously satisfied. The existence of $W(\beta)$ requires, owing to the normalization condiition $\int W(\beta) \textup{d}\beta^2=1$, that $W(\beta) \rightarrow 0$ as $|\beta|\rightarrow \infty$. The bilinear form $(a^2f^2+b^2d^2+cfd)$ in the exponential in (16) can be diagonalized in the variables $\Re{(A(\tau)-\beta)}$ and $\Im{(A(\tau)-\beta)}$ resulting in the eigenvalues $(\bar{n}+1/2)\exp\big{(}-2(\Omega\tau +r)\big{)}$ and $(\bar{n}+1/2)\exp\big{(}2(\Omega\tau +r)\big{)}$ that must be nonnegative, which is obviously so for $0\leq \tau< \infty$. Accordingly, the Wigner function $W(\beta)>0$. The positive definiteness of $W(\beta)$ does not preclude, however, quantum behavior in the field-quadrature and photon number variances. The displaced thermal state follows directly from (16) in the limit $r \rightarrow 0$, where $\Omega =r/t \rightarrow 0$, and so $W(\beta)=(1/(\pi (\bar{n}+1/2))\exp{(-|\beta-\alpha|^2/(\bar{n}+1/2))}$. The result for the coherent state follows with $\bar{n}=0$ and for the thermal state with $\alpha=0$. One can express the Wigner function $W(\beta)$ in terms of quadratures of the field and so $$W(x_{\lambda}, x_{\lambda+\pi/2})=\frac{1}{2\pi^2}\int_{-\infty}^{\infty} \textup{d}\sigma\int_{-\infty}^{\infty} \textup{d}\kappa e^{i(\sigma x_{\lambda}+ \kappa x_{\lambda +\pi/2})}$$ $$\times \textup{Tr}[\hat{\rho}(t+\tau)e^{-i(\sigma\hat{x}_{\lambda} +\kappa \hat{x}_{\lambda+\pi/2})}]$$ $$= \frac{1}{\pi} \frac{1}{\sqrt{4a^2b^2-c^2}}e^{-(a^2f^2+b^2d^2+cfd)/(4a^2b^2-c^2)},$$ where the factor $1/\pi$ in (19), as compared to the factor $2/\pi$ in (16), is a direct consequence of the change of integration variables $\eta= (\kappa -i\sigma)e^{i\lambda} /\sqrt{2}$ from (15) to (19) whereby $(\textup{d} \Re \eta)(\textup{d}\Im \eta)=\frac{1}{2}(\textup{d}\sigma)(\textup{d}\kappa)$. The quadrature variables are $$x_{\lambda}=\frac{\beta e^{-i\lambda}+\beta^* e^{i\lambda}}{\sqrt{2}} \hspace{0.3in}x_{\lambda+\pi/2}=\frac{\beta e^{-i\lambda}-\beta^* e^{i\lambda}}{\sqrt{2}i},$$ where $$\beta + \beta^*= \sqrt{2}( x_{\lambda} \cos\lambda - x_{\lambda +\pi/2}\sin\lambda )$$ and $$\beta - \beta^*= \sqrt{2}i( x_{\lambda} \sin\lambda + x_{\lambda +\pi/2}\cos\lambda )$$ which are substituted in the expressions for $d$ and $f$ in Equation (17) when evaluating (19). Accordingly, the probability distribution for the two quadrature components $x_{\lambda}$ and $x_{\lambda+\pi/2}$ is given by $W( x_{\lambda}, x_{\lambda+\pi/2})$, which is a Gaussian function in both variables, that is, the exponential factor in (19) is a quadratic form in $x_{\lambda}$ and $x_{\lambda+\pi/2}$ . \[See Equation (A9) in Appendix A.\] Field-quadrature variance ========================= With the aid of successive derivatives of the characteristic function $\chi(\eta)$, one obtains for the quadrature $\langle\hat{x}_{\lambda}\rangle$ and the quadrature variance $\Delta x^2_{\lambda}$ $$\langle \hat{x}_{\lambda}\rangle = \textup{Tr}[\hat{\rho}(t+\tau) \hat{x}_{\lambda}] = \frac{1}{\sqrt{2}}[A(\tau) e^{-i\lambda} + A^*(\tau) e^{i\lambda}]$$ and $$\Delta x^2_{\lambda} =\textup{Tr}[\hat{\rho}(t+\tau)( \hat{x}_{\lambda} -\langle \hat{x}_{\lambda}\rangle)^2 ]$$ $$=(\bar{n} + 1/2) \Big{(}\exp{[2(\Omega \tau +r)]}\sin^2(\lambda-\theta/2)$$ $$+ \exp[-2(\Omega \tau +r)] \cos^2(\lambda-\theta/2) \Big{ )},$$ where the quadrature operator $\hat{x}_{\lambda} = (\hat{a}e^{-i \lambda} + \hat{a}^\dag e^{i \lambda})/\sqrt{2}$. The phase-sensitive quadrature operators represent a set of observables that can be measured for radiation modes, atomic motion in a trap, and other related systems [@WVO99]. The average (23) and variance (24) can also be evaluated with the aid of the probability distribution $W( x_{\lambda}, x_{\lambda+\pi/2})$. The expectation value of $\hat{x}_{\lambda}$ is determined by the coherent amplitude $\alpha$ as well as the squeezing parameter $\xi$ while the variance $\Delta x^2_{\lambda}$, and hence the squeezing, depends on the squeezing parameter $\xi$ only. The product of the variances of the two quadratures components $\hat{x}_{\lambda}$ and $\hat{x}_{\lambda + \pi/2}$ is bounded from below by the Heisenberg uncertainty principle since $$(\Delta x^2_{\lambda} )(\Delta x^2_{\lambda+ \pi/2})= (\bar{n}+1/2)^2 \big{(}\cosh^2[2(\Omega \tau +r)]$$ $$- \cos^2(\theta- 2 \lambda) \sinh^2[2(\Omega \tau +r)]\big{)}\geq (\bar{n}+1/2)^2 \geq \frac{1}{4},$$ where the first inequality becomes an equality for $\theta=2\lambda$. The signal-to-noise ratio [@RL00] is defined as $$\textup{SNR} = \frac{\langle \hat{x}_{\lambda}\rangle^2}{\Delta x^2_{\lambda}}.$$ Thus the maximum signal-to-noise ratio is $$\textup{SNR}_{\textup{max}} =|\alpha|^2 \frac{\big{[} \coth(r/2)(1- e^{-\Omega \tau}) +(1 +e^{-\Omega \tau})\big{]}^2}{(2\bar{n} +1) e^{-2(\Omega \tau +r)}},$$ for $\varphi =\lambda= \theta/2$. The result for the squeezed coherent state, $4 e^{2r}|\alpha|^2$, follows for $\tau=0$ and $\bar{n}=0$. Photon-number variance ======================= Similarly, with the aid of successive derivatives of the characteristic function $\chi(\eta)$, the time development of the photon number is given by $$\textup{Tr}[\hat{\rho}(t+\tau)\hat{a}^\dag \hat{a}] = \langle \hat{a}^\dag(\tau) \hat{a}(\tau)\rangle = \langle\hat{n}(\tau)\rangle$$ $$=(\bar{n}+1/2)\cosh[2(\Omega\tau+r)] + |A(\tau)|^2 -\frac{1}{2},$$ while the variance is $$\Delta n^2(\tau) = \textup{Tr}[\hat{\rho}(t+\tau) (\hat{n} - \langle \hat{n}\rangle)^2] = (\bar{n}+1/2)^2 \cosh[4(\Omega \tau +r)]$$ $$+(\bar{n}+1/2)\Big{(} 2\cosh[2(\Omega \tau +r)]|A(\tau)|^2 -\sinh[2(\Omega \tau+r)]$$ $$\times [e^{i\theta} A^{*2}(\tau)+e^{-i\theta} A^2(\tau)]\Big{)} -\frac{1}{4}.$$ Note, contrary to the quadrature variance (24), the photon-number variance (29) depends, in addition to the squeezing parameter $\xi$, also on the coherent amplitude $\alpha$ via $A(\tau)$ given by Equation (10). Nonclassicality criteria ======================== A sufficient conditions for nonclassicality is for the quadrature of the field to be narrower than that for a coherent state, that is, $$\Delta x^2_{\lambda} < \frac{1}{2}.$$ Another sufficient condition is determined by the Mandel $Q_{M}(\tau)$ parameter related to the photon-number variance [@GSA13] $$Q_{M}(\tau)= \frac{\Delta n^2(\tau) -\langle \hat{n}(\tau)\rangle}{\langle \hat{n}(\tau)\rangle},$$ where $-1 \leq Q_{M}(\tau) <0$ implies that the field must be nonclassical with sub-Poissonian statistics. In the Glauber-Sudarshan coherent state or P-representation, nonclassicality is signaled by the real function $P(\beta)$ assuming negative values or becoming more singular than a Dirac delta function. Note, however, that if both the Mandel $Q_{M}(\tau)$ parameter and the squeezing parameter $(\Delta x_{\lambda}^2-1/2)$ are positives, then no conclusion can be drawn on the nonclassical nature of the radiation field [@GSA13]. The necessary and sufficient condition [@MAa16] for the existence of a real-valued $P(\beta)$ is $$1\leq (2\bar{n}+1)e^{-2(\Omega \tau +r)}.$$ Accordingly, the necessary and sufficient condition for nonclassicality is then $$(2\bar{n}+1)e^{-2(\Omega \tau +r)}<1,$$ which is the same, with the aid of (24), as that given by condition (30) when $\theta= 2 \lambda$. If the nonclassicality condition (33) holds for $\tau=0$, then it holds for $\tau>0$. Therefore, the radiation field, if initially nonclassical remains so as time goes on. If the field is initially classical, that is, $(2\bar{n} +1) e^{-2r}\geq 1$, then for $\Omega \tau > \frac{1}{2} \ln [(2\bar{n} +1) e^{-2r}]$ the radiation field behaves nonclassically. Note, however, that $Q_{M}(\tau)$ need not mirror the classical/nonclassical behavior dictated by criteria (32)-(33) based on $P(\beta)$. \[See Figures. (1) and (2) below.\] Numerical comparisons ===================== Owing to the equivalence for $\theta=2\lambda$ of the nonclassical condition $\triangle x^2_\lambda <1/2$ given by (30) and the nonclassicality criteria (33) based on the Glauber-Sudarshan $P(\beta)$ function, we need study only numerically the relation of classicality or nonclassicality between the conditions based on the $P(\beta)$ and the $Q_{M}(\tau)$ functions. It is interesting that Equation (33) is independent of the coherent parameter $\alpha$ while the Mandel parameter $Q_{M}(\tau)$ is rather sensitive to the value of $\alpha$. This is so since the eigenvalues associated with the quadratic form, appearing in the exponential of the Wigner function $W(x_{\lambda}, x_{\lambda +\pi/2})$ in (19), do not depend on $A(\tau)$ and so are independent of the value of the displacement parameter $\alpha$. However, both the photon-number variance $\triangle n^2(\tau)$ given by (29) and the average photon number (28) do depend on the values of $\alpha$ and so does $Q_{M}(\tau)$. Figure 1 illustrates the case $(2\bar{n}+1)e^{-2r}\geq1$ where $Q_{M}(0)>0$ for all values of $\bar{n}$ and $r$ that satisfy inequality (B2), which indicates that the system behaves classically at $\Omega \tau=0$. The behavior of $Q_{M}(\tau)$ as a function of $|\alpha|$ shows how the system goes from a strictly classical behavior for $|\alpha| < |\alpha_{c}|=0.3494$ (blue graph) as a function of $\tau$ to a mixed classical/nonclassical behavior for $|\alpha| > |\alpha_{c}|$ (green graph) as a function of $\tau$. This behavior of $Q_{M}(\tau)$ characterizes a phase transition as a function of $|\alpha|$ when the system goes from a strictly classical behavior to one where the system exhibits, as a function of $\tau$, both classical and nonclassical behaviors. This behavior of the Mandel parameter $Q_{M}(\tau)$ as a function of $|\alpha|$ is reminiscent of the Van der Waals equation of state with the aid of the Maxwell construction whereby there exists a critical temperature, where $1/|\alpha_{c}|$ play the role of the critical temperature $T_{c}$, above which the system is in a single phase and below which there are two coexisting phases. ![Plot of $Q_{M}(\tau)$ for $\bar{n}=0.2$ and $r=0.1$ with $(2\bar{n}+1)e^{-2r}=1.1462>1$. (a) $|\alpha|=0.3$ (blue graph). (b) $|\alpha|=|\alpha_{c}|=0.3494$, where $\textup{d}Q_{M}(\tau)/\textup{d}\tau =0$ and $Q_{M}(\tau)=0$ at $\Omega \tau= 0.3857$ (red graph). (c) $|\alpha|=0.4$ (green graph).](char5fig1.eps) \[fig:theFig\] We consider in Figure 2 the case $(2\bar{n}+1)e^{-2r}<1$, where $Q_{M}(0)$, for given $\bar{n}$ and $r$, can assume either positive or negative values as determined by inequalities (B2) and (B3), respectively, and $Q_{M}(0)=0$ if the inequalities hold as equalities. Accordingly, depending on the value of $|\alpha|$, for given $\bar{n}$ and $r$, the system behaves classically or nonclassically at $\Omega \tau =0$. There is a critical value $|\alpha_{c}|$ such that for $|\alpha|<|\alpha_{c}|$, the system behaves strictly classically (green and blue graphs); whereas for $|\alpha|>|\alpha_{c}|$ the system is in a mixed classical/nonclassical state (red and black graphs). In contrast with the results of Figure 1, for $(2\bar{n}+1)e^{-2r} \geq 1$, in the case of Figure 2, for $(2\bar{n}+1)e^{-2r} <1$, we have an analogous transition at the critical point $|\alpha_{c}|$ but with the sign of $Q_{M}(0)$ changing from positive to negative values at the critical point. Note that $Q_{M}(0)$ remains positive for the case of Figure 1 as the transition point is traversed. ![Plot of $Q_{M}(\tau)$ for $\bar{n}=0.1$ and $r=0.2$ with $(2\bar{n}+1)e^{-2r}=0.8044<1$. (a) $|\alpha|=0.3$ (green graph). (b) $|\alpha|=|\alpha_{c}|=0.4961$, where $\textup{d}Q_{M}(\tau)/\textup{d}\tau =0$ and $Q_{M}(\tau)=0$ at $\Omega \tau= 0.2097$ (blue graph). (c) $|\alpha|=0.6507$, where $Q_{M}(0)=0$ (red graph). (d) $|\alpha|=1$ (black graph).](char5fig2.eps) \[fig:theFig\] . Figure 3 shows sequence of plots for $\bar{n}=1$ and $r=1$ as the amplitude $|\alpha|$ increases past the critical value $|\alpha_{c}|=9.7140$ whereby the Mandel parameter $Q_{M}(\tau)$, as a function of $\tau$, goes from exhibiting a strictly classical behavior for $|\alpha|<|\alpha_{c}|$ to a mixed classical/nonclassical behavior for $|\alpha|>|\alpha_{c}|$. It is important to remark that even though $Q_{M}(\tau)$ exhibits classical behavior, nonetheless, the radiation field is nonclassical since $(2\bar{n}+1)e^{-2r} <1$. There is no inconsistency since the negativity of $Q_{M}(\tau)$ is a sufficient condition for the field to be nonclassical, whereas, if $Q_{M}(\tau)>0$, no conclusion can be drawn about the nonclassicality of the radiation field [@GSA13]. ![Plot of $Q_{M}(\tau)$ for $\bar{n}=1$ and $r=1$ with $(2\bar{n}+1)e^{-2r}=0.4060<1$. (a) $|\alpha|=12$ (red graph). (b) $|\alpha|=|\alpha_{c}|=9.7140$, where $Q_{M}(0)=0$ (green graph). (c) $|\alpha|=8$ (blue graph). ](char5fig3.eps) \[fig:theFig\] . Summary and discussions ======================= We calculate the Wigner quasiprobability density (19) and the corresponding field-quadrature (24) and photon-number variances (29) for Gaussian states, *viz*., displaced-squeezed thermal states, where the dynamics is governed solely by the general, degenerate parametric amplification Hamiltonian (1). Our result (19) for the Wigner function is exact and is based on dynamically generating the Gaussian state first from an initial thermal state and subsequently determining the time evolution of the system without assuming statically stationary fields. We numerically analyze the conditions for nonclassicality as given by the Mandel parameter $-1\leq G_{M}(\tau)<0$, squeezing parameter $(\Delta x_{\lambda}^2-1/2)<0$, and the nonclassicality criteria (33) based on the Glauber-Sudarshan quasiprobability distribution $P(\beta)$. We show that the latter condition by itself, albeit determining the nonclassicality of the radiation field, does not determine the classical/nonclassical behavior exhibited by the Mandel parameter $Q_{M}(\tau)$. The nonclassicality criteria (33) for the radiation field based on $P(\beta)$ is independent of the value of the displacement parameter $\alpha$. However, the Mandel parameter $Q_{M}(\tau)$ has a sensitive dependence on the value of $\alpha$. We find a phase transition from classical to a mixed classical/nonclassical behavior for $Q_{M}(\tau)$, as a function of $\tau$, at $|\alpha_{c}|$ so that for $|\alpha|< |\alpha_{c}|$, the behavior of $Q_{M}(\tau)$ is strictly classical, whereas for $|\alpha|> |\alpha_{c}|$, $Q_{M}(\tau)$ exhibits both classical and nonclassical behaviors even though the radiation field is strictly nonclassical. Wigner function =============== In this appendix we express the quadratic form appearing in the exponential function in (19) explicitly in terms of the quadrature components $x_{\lambda}$ and $x_{\lambda +\pi/2}$. Now $$E(x_{\lambda}, x_{\lambda +\pi/2}) \equiv a^2f^2+b^2d^2+cfd$$ $$=\epsilon_{(x_{\lambda},x_{\lambda})} (x_{\lambda}-\langle\hat{x}_{\lambda}\rangle)^2+\epsilon_{(x_{\lambda+\pi/2},x_{\lambda+\pi/2})}(x_{\lambda+\pi/2}-\langle\hat{x}_{\lambda+\pi/2}\rangle)^2$$ $$+\epsilon_{(x_{\lambda},x_{\lambda+\pi/2})} (x_{\lambda}-\langle\hat{x}_{\lambda}\rangle )(x_{\lambda+\pi/2}-\langle\hat{x}_{\lambda+\pi/2} \rangle),$$ where $$\langle \hat{x}_{\lambda}\rangle= \frac{1}{\sqrt{2}}[A(\tau)e^{-i\lambda} +A^*(\tau)e^{i\lambda}],$$ $$\langle \hat{x}_{\lambda+\pi/2}\rangle= \frac{1}{i\sqrt{2}}[A(\tau)e^{-i\lambda} -A^*(\tau)e^{i\lambda}],$$ $$\epsilon_{(x_{\lambda},x_{\lambda})}=2(\bar{n}+1/2)\big{(}\cos(2\lambda-\theta)\sinh[2(\Omega \tau+r)]$$ $$+\cosh[(2(\Omega\tau+r)]\big{)},$$ $$\epsilon_{(x_{\lambda+\pi/2},x_{\lambda+\pi/2})}= -2(\bar{n}+1/2)\Big{(}\cos(\theta-2\lambda)\sinh[2(\Omega\tau+r)]$$ $$-\cosh[2(\Omega \tau+r)]\Big{)},$$ and $$\epsilon_{(x_{\lambda},x_{\lambda+\pi/2})}= 4(\bar{n}+1/2)\sin(\theta-2\lambda)\sinh[2(\Omega\tau+r)].$$ One has that the Wigner quasiprobability density is given by $$W(x_{\lambda}, x_{\lambda +\pi/2})= \frac{1}{\pi(\bar{n}+1/2)} e^{-E(x_{\lambda}, x_{\lambda +\pi/2})/(2\bar{n} +1)^2},$$ where the five real, time-dependent parameters (A2)-(A6) can be directly compared to the five real parameters appearing in the generic Gaussian Wigner function [@DMM94]. The probability distribution $P(x_{\lambda})$ for the quadrature component $\hat{x}_{\lambda}$ is $$P(x_{\lambda}) = \int_{-\infty}^{\infty} \textup{d}x_{\lambda+\pi/2} W(x_{\lambda}, x_{\lambda +\pi/2})$$ $$=\frac{1}{\sqrt{2\pi\triangle x^2_{\lambda}}} \exp{\Big{(}-\frac{(x_{\lambda}- \langle \hat{x}_{\lambda}\rangle)^2}{2\triangle x^2_{\lambda}}}\Big{)},$$ where $\langle \hat{x}_{\lambda}\rangle$ and $\triangle x^2_{\lambda}$ are given by (A2) and (24), respectively. Result (A1) for $E(x_{\lambda}, x_{\lambda +\pi/2})$ simplifies considerably for the case of maximum signal-to-noise ratio when $\theta=2\lambda$ and the cross term $\epsilon_{(x_{\lambda},x_{\lambda+\pi/2})}$ in (A6) vanishes, in which case $$W(x_{\lambda}, x_{\lambda +\pi/2})=\frac{1}{\pi(2\bar{n}+1)}\exp{\Big{(}-\frac{(x_{\lambda}- \langle \hat{x}_{\lambda}\rangle)^2}{(2\bar{n} +1)e^{-2(\Omega \tau +r)}}} \Big{)}$$ $$\times \exp{\Big{(}-\frac{(x_{\lambda+\pi/2}- \langle \hat{x}_{\lambda+\pi/2}\rangle)^2}{(2\bar{n} +1)e^{2(\Omega \tau +r)}}} \Big{)}.$$ phase transition ================ The Mandel parameter $Q_{M}(0)$ follows from (31) $$Q_{M}(0)=\Big{(}(\bar{n} +1/2)^2 \cosh(4r)+\big{(}(2\bar{n}+1)e^{-2r}-1\big{)}|\alpha|^2$$ $$-(\bar{n}+1/2)\cosh(2r) +1/4\Big{)}$$ $$\times \frac{1}{(\bar{n}+1/2)\cosh(2r)+|\alpha|^2 -1/2}$$ with $\theta=2\lambda$. The value of $Q_{M}(0)$ can assume either positive or negative values. If $Q_{M}(0)>0$, then $$[1-(2\bar{n}+1)e^{-2r}]|\alpha|^2$$ $$< (\bar{n}+1/2)^2 \cosh(4r)-(\bar{n}+1/2)\cosh(2r) +1/4.$$ Note that if $1-(2\bar{n}+1)e^{-2r}\leq 0$, then $Q_{M}(0)>0$ for all values of $|\alpha|$, $\bar{n}$ and $r$ provided $\bar{n}\geq(e^{2r} - 1)/2$. On the other hand, if $1-(2\bar{n}+1)e^{-2r}>0$, then $|\alpha|$ must satisfy inequality (B2). However, if $Q_{M}(0)<0$, then $$[1-(2\bar{n}+1)e^{-2r}]|\alpha|^2$$ $$> (\bar{n}+1/2)^2 \cosh(4r)-(\bar{n}+1/2)\cosh(2r) +1/4,$$ which requires $1>(2\bar{n}+1)e^{-2r}$. There are three possible behaviors of $Q_{M}(\tau)$ as a function of $\Omega \tau$ when $Q_{M}(0)>0$, (i) $Q_{M}(\tau)=0$ for two different values of $\Omega \tau$; (ii) $Q_{M}(\tau)$ vanishes for a single value of $\Omega \tau$, at which both $Q_{M}(\tau)$ and $\partial Q_{M}(\tau)/\partial \tau$ vanish; (iii) $Q_{M}(\tau) >0$ for $\Omega \tau\geq 0$. There is only one possible behavior for the system when $Q_{M}(0)<0$, $Q_{M}(\tau)$ as a function of $\tau$ crosses the axis once when it assumes a value of zero. Therefore, the Mandel parameter is negative and so nonclassical and remains so but as time goes the Mandel parameter becomes positive and so classical. Note that the radiation field is nonclassical since inequality (33) is satisfied but this does not preclude that $Q_{M}(\tau)$ can behave classically as is shown in both Figures 2 and 3. References {#refs .unnumbered} ---------- Agarwal, G.S. *Quantum Optics*, Cambridge University Press: Cambridge, **2013**. Alexanian, M. *J. Mod. Opt.* **2016**, 63, 961-967. Alexanian, M. *Phys. Rev. A* **2016**, 94, 043837. Hofer, P.P.; Clerk, A.A. *Phys. Rev. Lett.* **2016**, 116, 013603. Dodonov, V.V.; Man’ko, O.V.; Man’ko, V.I. *Phys. Rev. A* **1994**, 49, 2993-3001 and references therein. Welsch, D.-G; Vogel, W.; Opatrny, T. Homodyne detection and quantum state reconstruction, in Progress in Optics, edited by Emil Wolf (North Holland, Amsterdam, 1999), Vol. XXXIX, Cap. II, 63-210. Loudon, R. *The Quantum Theory of Light*, Oxford University Press: Oxford **2000**. Rice, P.R.; Carmichael, H.J. *Quantum IEEE J. Electron.* **1988**, 24, 1351-1366; Foster, G.T.; Mielke, S.L.; Orozco, L.A. *Phys. Rev. A* **2000**, 61, 053821. Richter, Th.; Vogel, W. *Phys. Rev. Lett.* **2002**, 89, 283601.
--- abstract: 'Two research groups have measured turbulent velocity statistics in superfluid helium using different techniques. The results were in conflict: one experiment revealed Gaussian distributions (as observed in ordinary turbulence), the other experiment determined power-laws. To solve the apparent puzzle, we numerically model quantum turbulence as a tangle of vortex filaments, and conclude that there is no contradiction between the two experiments. The transition from Gaussian to power-law arises from the different length scales which are probed using the two techniques. We find that the average distance between the quantum vortices marks the separation between quantum and quasi-classical length scales.' author: - 'A.W. Baggaley and C.F. Barenghi' title: Quantum turbulent velocity statistics and quasiclassical limit --- Turbulence in superfluid helium ($^4$He) [@Manchester; @Prague] consists of a tangle of discrete, thin vortex filaments, each carrying one quantum of circulation $\kappa=h/m$ (where $h$ is Planck’s constant and $m$ the mass of one atom). This ‘quantum turbulence’ [@Vinen; @Halperin] is also studied in superfluid $^3$He [@Lancaster; @Helsinki] and in atomic Bose-Einstein condensates [@Bagnato]. The absence of viscosity and the discrete nature of the vorticity distinguish quantum turbulence from ordinary (classical) turbulence. Nevertheless, experiments have also shown remarkable similarities: the same pressure drop along and pipes [@VanSciver-pressure], drag on a moving body [@VanSciver-sphere], vorticity decay [@Oregon], and distribution of kinetic energy over the length scales (Kolmogorov energy spectrum) [@Tabeling; @Salort2010]. The relation between quantum turbulence and classical turbulence has thus become the focus of attention. This report is concerned with the statistical properties of the components of the turbulent superfluid velocity field ${\mathbf {v}}$. In ordinary turbulence, experiments [@Noullez] and numerical simulations [@Vincent; @Gotoh] confirm that velocity statistics are Gaussian. The usual interpretation of this property is the following [@Davidson]. At any time $t$, the velocity field ${\mathbf {v}}$ at the point ${\mathbf {r}}$ is determined by the vorticity field ${{\mbox{\boldmath $\omega$}}}=\nabla \times {\mathbf {v}}$ via the Biot-Savart law [@Saffman] $${\mathbf {v}}({\mathbf {r}},t)=\frac{1}{4 \pi} \int \frac{{{\mbox{\boldmath $\omega$}}}({\mathbf {r}}',t) \times ({\mathbf {r}}-{\mathbf {r}}')}{\vert {\mathbf {r}}-{\mathbf {r}}'\vert^3}\, d{\mathbf {r}}'. \label{eq:BSomega}$$ Thus, if the point ${\mathbf {r}}$ is surrounded by many, randomly oriented vortical structures, Gaussianity results from the application of the Central Limit Theorem. In the last few years, the development of particle tracking visualization techniques suitable for liquid helium [@VanSciver; @Bewley] has opened the way to the measurement of quantum turbulence statistics. Using solid hydrogen tracers, Paoletti et al. [@Paoletti] at the University of Maryland deduced that, in superfluid helium, the velocity components of the turbulent velocity follow power-law distributions. Large velocity fluctuations are thus relatively more frequent than in ordinary turbulence, where these distributions are Gaussian [@note]. The authors suggested that power-laws distributions arise from vortex reconnections, clearly high-velocity events. The experiment of Paoletti et al. motivated theoretical work on the problem. White et al. [@White] solved numerically the Gross-Pitaevskii equation for a Bose-Einstein condensate and determined velocity distributions in a variety of quantum vortex systems at zero temperature: two-dimensional and three-dimensional trapped atomic condensates, three-dimensional homogeneous condensates, Onsager’s two-dimensional vortex gas. In all systems, they found power-law distributions, in agreement with Paoletti et al. [@Paoletti], even in the absence of vortex reconnections. Using results from previous work by Min and Leonard [@Leonard] on the numerical analysis of vortex methods (later confirmed by Weiss et al. [@Weiss]), White et al. [@White] concluded that the non-Gaussianity arises from the singular nature of the quantum vorticity, which causes an anomalously slow convergence to Gaussian behaviour. Further theoretical work at non-zero temperatures seemed to strengthen this conclusion. Baggaley and Barenghi [@Baggaley-vortexdensity] used the vortex filament model to numerically study intense vortex tangles. They confirmed the Kolmogorov spectrum and the power-laws distributions of the velocity components. The vortex filament model was also used by Adachi and Tsubota [@Adachi], who found power-law distributions in counterflow turbulence (quantum turbulence driven by a heat flow). In a recent experiment, Salort et al. [@Salort2011] at CNRS Grenoble measured the turbulent velocity in a superfluid wind tunnel using a small Pitot tube. They confirmed the Kolmogorov spectrum, but reported Gaussian velocity distributions, in apparent contradiction with Paoletti et al. [@Paoletti] and with the numerical simulations [@White; @Baggaley-vortexdensity; @Adachi]. To solve this puzzle, firstly we consider the size of the probe relative to the distance between vortices in the two experiments. The radius $a$ of the tracer particles used by Paoletti et al. was $a\approx 10^{-4}~\rm cm$. The turbulence was created by a heat flux $\dot Q$ which drove a counterflow current [@Adachi2010] $v_{ns}=\vert v_n-v_s \vert={\dot Q}/(\rho_s S T)$ between superfluid and normal fluid velocity components $v_n$ and $v_s$, where $\rho_s$ is the superfluid density and $S$ the specific entropy. For example, at $T=1.9~\rm K$ the largest heat flux was ${\dot Q}=0.17~\rm W/cm^2$, corresponding to $v_{ns}=1.5~\rm cm/s$, with $\rho_s=0.0831~\rm g/cm^3$ and $S=0.709~\rm J/(gK)$. Using the relation $L = \gamma^2 v_{ns}^2$ with $\gamma \approx 140~\rm s/cm^2$ as reported in Ref. [@Adachi2010], one finds $\ell \approx 4.7 \times 10^{-3}~\rm cm$, where $L$ is the vortex line density (vortex length per unit volume) and $\ell\approx L^{-1/2}$. Paoletti et al. [@Paoletti] took their velocity measurements during the decay of the turbulence, after switching off the heater, so the actual inter-vortex distance must have been larger than the above value of $\ell$. We conclude that, in the experiment of Paoletti et al. [@Paoletti], $a/\ell \ll 1$; this means that, although the tracer particles must have altered in some measure the motion of the vortex lines, the velocity field was probed at very small length scales, between $\ell$ and $a$. In the experiment of Salort et al. [@Salort2011], the nozzle of the Pitot tube had diameter $0.06~\rm cm$. The authors estimated [@Roche] that the inter-vortex separation was between $\ell \approx 5 \times 10^{-4}$ and $20 \times 10^{-4} \rm cm$. We conclude that in this experiment $a/\ell \gg 1$ (even without considering that the effective resolution in the streamwise direction, set by time dynamics of the probe, was perhaps much larger than $a$ [@Roche]). In summary, the experiment of Paoletti et al. [@Paoletti] probed the velocity field between vortices, at length scales less than $\ell$, whereas the experiment of Salort et al. [@Salort2011] detected motion at scales larger than $\ell$, containing many vortex lines. Secondly, we perform the following numerical calculation. In both experiments the length scales involved were much larger than the vortex core radius, $a_0=10^{-8}~\rm cm$, so it is appropriate to use the vortex filament model [@Schwarz]. We assume that the vorticity is concentrated in space curves ${\mathbf {s}}(\xi,t)$ around which the circulation is $\kappa=10^{-3}\rm cm^2/s$. Eq. (\[eq:BSomega\]) reduces to $$\frac{d{\bf s}}{dt}=-\frac{\kappa}{4 \pi} \oint_{\cal L} \frac{({\mathbf {s}}-{\mathbf {r}}) } {\vert {\mathbf {s}}- {\mathbf {r}}\vert^3} \times {\bf d}{\mathbf {r}}, \label{eq:BS}$$ where the line integral extends over all vortex lines. The numerical techniques to discretize the vortex lines, to compute the time evolution, to de-singularize the Biot-Savart integral and to algorithmically perform reconnections when two vortex lines become close to each other are described in the literature [@Schwarz; @Adachi] or in our previous papers [@Baggaley-cascade; @Baggaley-vortexdensity]. The computational domain is a periodic box of size $D=0.075~\rm cm$. The distance between discretization points along the vortex filaments is held at approximately $\delta/2$ where $\delta=0.001~\rm cm$. To speed-up the evaluation of Biot-Savart integrals we use a tree method [@Baggaley-long] with critical opening angle $\theta=0.4$. For the sake of simplicity, we assume that the temperature is $T=0$. This means that no external forcing is needed to sustain the turbulence, because the total kinetic energy is conserved during the evolution (at least for the time scale of interest here; the finite discretization and the reconnection algorithm introduce small energy losses which model phonon emission as described in Ref. [@Baggaley-cascade]). The initial condition at time $t=0$ consists of 100 straight, randomly oriented, vortex filaments discretized over $N=10^4$ Lagrangian points ${\mathbf {s}}_j$ ($j=1,\cdots N$). During the evolution the vortex lines interact, become curved, reconnect, and a vortex tangle is quickly formed. We find that the vortex line density $L$ initially grows, then saturates to the value $L\approx 2.04 \times 10^4~\rm cm^{-2}$. We check that the average curvature saturates too. Fig. \[fig1\] shows the vortex tangle when we stop the calculation at $t=0.1~\rm s$. We examine the energy spectrum $E(k)$, defined by $$E=\frac{1}{V} \int \frac{1}{2} \vert {\mathbf {v}}\vert^2 dV=\int_0^{\infty} E(k)\,dk, \label{eq:spectrum}$$ ![ Snapshot of the turbulent vortex tangle at $t=0.1\,$s after the vortex line density has saturated, the box size $D=0.075~\rm cm$. []{data-label="fig1"}](fig1.eps){width="45.00000%"} where $E$ is the total energy per unit mass. Fig. \[fig2\] shows that most of the energy is contained in the large scales, and that the spectrum is qualitatively consistent with the Kolmogorov scaling $E_k \sim k^{-5/3}$ in the region $k \ll 2 \pi/\ell \approx 900~\rm cm^{-2}$, in agreement with experiments [@Tabeling; @Salort2010] and with previous numerical calculations [@Nore; @Tsubota-vd; @Baggaley-vortexdensity; @Tsubota-GP]. The shallow spectrum at large values of $k$ is also consistent with existing similar calculations [@Tsubota-vd]. Secondly, we examine the velocity components and compute their probability density functions (normalized histograms, or PDF for short). Rather than sampling the values at points of the domain, we compute averages over regions of size $\Delta$, varying $\Delta$. This is done by calculating the velocity field, induced by the vortices, on a $128^3$ Cartesian mesh, and then averaging cells over the appropriate length scale. The result is shown in Fig. \[fig3\], where we compare the data against their Gaussian fits $${\rm gPDF}(v_i)=\frac{1}{\sqrt{2 \pi \sigma^2}} {\rm exp}(-(v_i-\mu)^2/(2 \sigma^2)), \label{eq:Gaussian}$$ ![ (Color online) Energy spectrum $E_k$ (arbitrary units) vs wavenumber $k$ (${\rm cm}^{-1}$) corresponding to Fig. \[fig1\]. The dashed line is the Kolmogorov scaling $E_k \sim k^{-5/3}$; the solid grey line shows wavenumber corresponding to the intervortex spacing $k_{\ell}\approx 900~\rm cm^{-1}$. []{data-label="fig2"}](fig2.eps){width="45.00000%"} where $\mu \approx 0$ and $\sigma$ is the standard deviation. It is apparent that, if $\Delta <\ell$, the PDFs are not Gaussian; by fitting ${\rm PDF}(v_i) \sim v_i^{-b}$ we obtain $b=3.2$, $3.2$ and $3.1$ for $i=x,y,z$ respectively, in agreement with the power-law statistics found experimentally by Paoletti et al. [@Paoletti] and numerically by White et al [@White], Baggaley and Barenghi [@Baggaley-vortexdensity], and Adachi and Tsubota [@Adachi]. If $\Delta >\ell$, however, the distributions are Gaussian, and we recover the same (classical) statistics measured by Salort et al. [@Salort2011]. In conclusion, the different velocity statistics observed by the Maryland and Grenoble experiments are consistent with each other. Taken together, these experiment and this work support the interpretation that, at scales larger than $\ell$, quantum turbulence exhibits quasi-classical behaviour (in terms of both energy spectrum and velocity statistics), whereas at scales smaller than $\ell$ (but still orders of magnitude larger than the quantum coherence length $a_0$), the discrete nature of quantized vorticity affects the distribution of energy over the length scales and the frequency of high-velocity events. ![ (Color online) Probability density functions (PDF) of turbulent velocity components $v_i$ ($i=x,y,z)$ vs $v_i/\sigma_i$ calculated by averaging over regions of size $\Delta=2 \ell$ (top), $\Delta=\ell$ (middle) and $\Delta=\ell/6$ (bottom). (Green) asterisks, (blue) circles and (red) triangles refer respectively to $i=x$, $i=y$ and $i=z$ components. The solid lines ares the Gaussian fits, defined by Eq. \[eq:Gaussian\], where $\bar{\sigma}=0.041$ (top), $0.048$ (middle) and $0.082~\rm cm/s$ (bottom) respectively. []{data-label="fig3"}](fig3a.eps "fig:"){width="40.00000%"} ![ (Color online) Probability density functions (PDF) of turbulent velocity components $v_i$ ($i=x,y,z)$ vs $v_i/\sigma_i$ calculated by averaging over regions of size $\Delta=2 \ell$ (top), $\Delta=\ell$ (middle) and $\Delta=\ell/6$ (bottom). (Green) asterisks, (blue) circles and (red) triangles refer respectively to $i=x$, $i=y$ and $i=z$ components. The solid lines ares the Gaussian fits, defined by Eq. \[eq:Gaussian\], where $\bar{\sigma}=0.041$ (top), $0.048$ (middle) and $0.082~\rm cm/s$ (bottom) respectively. []{data-label="fig3"}](fig3b.eps "fig:"){width="40.00000%"} ![ (Color online) Probability density functions (PDF) of turbulent velocity components $v_i$ ($i=x,y,z)$ vs $v_i/\sigma_i$ calculated by averaging over regions of size $\Delta=2 \ell$ (top), $\Delta=\ell$ (middle) and $\Delta=\ell/6$ (bottom). (Green) asterisks, (blue) circles and (red) triangles refer respectively to $i=x$, $i=y$ and $i=z$ components. The solid lines ares the Gaussian fits, defined by Eq. \[eq:Gaussian\], where $\bar{\sigma}=0.041$ (top), $0.048$ (middle) and $0.082~\rm cm/s$ (bottom) respectively. []{data-label="fig3"}](fig3c.eps "fig:"){width="40.00000%"} We thank J. Salort and P.E. Roche for providing us with experimental data and comments in advance of publication. This work was funded by the HPC-EUROPA2 project 228398, with the support of the European Community (Research Infrastructure Action of the FP7), and by the Leverhulme Trust (Grants F/00125/AH and RPG-097). [99]{} P.M. Walmsley, and A.I. 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--- --- [**Statistical Properties and Pre-hit Dynamics of Price Limit Hits in the Chinese Stock Markets** ]{}\ Yu-Lei Wan^1,2^, Wen-Jie Xie^2,3,4^, Gao-Feng Gu^2,3^, Zhi-Qiang Jiang^2,3^, Wei Chen^5^, Xiong Xiong^6,7,\*^, Wei Zhang^6,7^, Wei-Xing Zhou^1,2,3,\*^ **[1]{} Department of Mathematics, School of Science, East China University of Science and Technology, Shanghai 200237, China\ **[2]{} Research Center for Econophysics, East China University of Science and Technology, Shanghai 200237, China\ **[3]{} Department of Finance, School of Business, East China University of Science and Technology, Shanghai 200237, China\ **[4]{} Postdoctoral Research Station, School of Social and Public Administration, East China University of Science and Technology, Shanghai 200237, China\ **[5]{} Shenzhen Stock Exchange, Shenzhen 518010, China\ **[6]{} Collage of Management and Economics, Tianjin University, Tianjin 300072, China\ **[7]{} China Center for Social Computing and Analytics, Tianjin University, Tianjin 300072, China\ ************** \* xxpeter@tju.edu.cn (XX); wxzhou@ecust.edu.cn (WXZ) Abstract {#abstract .unnumbered} ======== Price limit trading rules are adopted in some stock markets (especially emerging markets) trying to cool off traders’ short-term trading mania on individual stocks and increase market efficiency. Under such a microstructure, stocks may hit their up-limits and down-limits from time to time. However, the behaviors of price limit hits are not well studied partially due to the fact that main stock markets such as the US markets and most European markets do not set price limits. Here, we perform detailed analyses of the high-frequency data of all A-share common stocks traded on the Shanghai Stock Exchange and the Shenzhen Stock Exchange from 2000 to 2011 to investigate the statistical properties of price limit hits and the dynamical evolution of several important financial variables before stock price hits its limits. We compare the properties of up-limit hits and down-limit hits. We also divide the whole period into three bullish periods and three bearish periods to unveil possible differences during bullish and bearish market states. To uncover the impacts of stock capitalization on price limit hits, we partition all stocks into six portfolios according to their capitalizations on different trading days. We find that the price limit trading rule has a cooling-off effect (object to the magnet effect), indicating that the rule takes effect in the Chinese stock markets. We find that price continuation is much more likely to occur than price reversal on the next trading day after a limit-hitting day, especially for down-limit hits, which has potential practical values for market practitioners. Introduction {#introduction .unnumbered} ============ In many stock markets, price limit rules are set expecting to reduce remarked swings by cooling off traders’ irrational mania. A stable stock market has lower risks and thus attracts more people to participate. This is certainly increase the resource reallocation function of stock markets and benefits the economies. Price limit rules constrain intraday prices to move within a preset price interval embraced by a price up limit and a price down limit. Usually, the limit prices are determined by fixed fluctuation percentages in reference to the closing price of the previous day. In most stock markets, the fluctuation percentages for up-limit and down-limit are symmetric. However, there are also examples for asymmetric price limits especially in certain market states. After the price reaches its limit, a circuit breaker may be triggered causing trading halt in some markets, while in other markets the traders can continue to trade shares [@Subrahmanyam-1994-JF]. The effectiveness of the price limit rules is controversial. It is expected to have a cooling-off effect to reduce the volatility of stocks [@Ma-Rao-Sears-1989-JFSR]. On the contrary, it may also cause a magnet effect, which refers to the phenomenon that the price limit acts as a magnet to attract more trades leading to higher trading intensity and price volatility and increases the probability of price rise or fall when the price is closer to the limit price [@Subrahmanyam-1994-JF]. The magnet effect occurs when the traders fear of the lack of liquidity and possible position lock caused by imminent price limit hits, and the traders are thus eager to protect themselves through submitting aggressive sub-optimal orders, which usually induces large price variations and heavy trading volumes. Since the cooling effect or the magnet effect takes place at the intraday level, studies of the presence of either effect, the evolution of pre-hit dynamics and the performance of price limit rules are of great interest to academics, investors and regulators to gain a better understanding of the mechanisms of how the market structure and the investors’ trading behavior affects price discovery. The study of price limits started on futures markets [@Telser-1981-JFutM; @Brennan-1986-JFE]. Empirical analysis has been carried out for different markets at different time periods. There is no consensus on the presence of a magnet effect or a cooling-off effect. Arak and Cook investigated if price behavior is infected by price limits on the treasury bond futures market and found no evidence of a magnet effect but rather a reversal effect [@Arak-Cook-1997-JFSR]. Berkman and Steenbeek compared the price formation processes under different price limits between Osaka Securities Exchange and Nikkei 225 index on the Singapore International Monetary Exchange and found no significant arbitrage opportunities between the two markets [@Berkman-Steenbeek-1998-JFinM]. In recent years, empirical studies about the magnet effect concentrated on stock markets. Cho et al. studied the 5-min return time series of 345 stocks traded on the Taiwan Stock Exchange from 1998/01/03 to 1999/03/20 and reported a statistically and economically significant magnet effect for stock prices to accelerate towards the up-limit and weak evidence of acceleration towards the down-limit [@Cho-Russell-Tiao-Tsay-2003-JEF]. Hsieh et al. analyzed the transaction data of 439 stocks traded on the Taiwan Stock Exchange in 2000 using logit models and found evidence of the magnet effect on both up-limt and down-limit [@Hsieh-Kim-Yang-2009-JEF]. Du et al. also observed the magnet effect when the stock prices are approaching the price limits in the Korean market [@Du-Liu-Rhee-2005-WP]. The Chinese stock markets also set price limits, which varied over time. The current $\pm10\%$ price limits were fixed since 1996/12/16 for A-share common stocks. There are several studies conducted on the presence of the magnet or cooling-off effect. However, empirical results lead to controversial conclusions [@Li-2005-cnJCUFE; @Meng-Jiang-2008-cnTE; @Fang-Chen-2007-cnSE; @Wong-Liu-Zeng-2009-CER; @Zeng-She-2014-cnASM; @Zhang-Zhu-2014-cnJCQUT]. According to the data released by the China Securities Regulatory Commission, by June 2014, there are 2540 listed companies and the total market capitalization is about 24.412 trillion Chinese Yuan. Due to its huge capitalization and representativeness as an emerging market, research on Chinese stocks is of great importance and remarkable interest. In this work, we will perform detailed analyses on the statistical properties of variables associated with price limit hits and the pre-hit dynamics of important financial variables before limit hits in the Chinese stock markets. These issues are less studied in previous works. To obtain conclusive results about the presence of a magnet or cooling-off effect in the Chinese stock markets, one needs to adopt different methods proposed in the literature and consider possible evolution of the effect (if present) per se. We leave this topic in a future work. Materials and Methods {#materials-and-methods .unnumbered} ===================== The Chinese stock markets {#the-chinese-stock-markets .unnumbered} ------------------------- There are two stock exchanges in mainland China. The first market for government approved securities was founded in Shanghai on 1990/11/26 and started operation on 1990/12/19 under the name of the Shanghai Stock Exchange (SHSE). Shortly after, the Shenzhen Stock Exchange (SZSE) was established on 1990/12/01, and started its operations on 1991/07/03. There are two separate markets for A-shares and B-shares on both exchanges. A-shares are common stocks issued by mainland Chinese companies, subscribed and traded in Chinese currency Renminbi (RMB), listed on mainland Chinese stock exchanges, bought and sold by Chinese nationals and approved foreign investors. B-shares are issued by mainland Chinese companies, traded in foreign currencies and listed on mainland Chinese stock exchanges. B-shares carry a face value denominated in RMB. The B-share market was launched in 1992 and was restricted to foreign investors before 2001/02/19. It has been open to Chinese investors since. The microstructure of the two markets has been changed on several aspects, such as the daily price up/down limit rules imposed since 1996/12/16. The price limits are $\pm10\%$ for common stocks and $\pm5\%$ for specially treated (ST and ST\*) stocks. We note that, before 1996/12/16, there were also periods with different intervals of price limits or without price limits. On each trading day, the trading time period is divided into three parts: opening call auction, cooling periods, and continuous double auction. The market opens at 9:15 a.m. and enters the opening call auction until 9:25 a.m, during which the trading system accepts order submissions and cancelations, and all matched transactions are executed at 9:25 a.m. This is followed by a cooling period from 9:25 to 9:30 a.m. During the cooling period, the exchanges are open to order routing from members, but does not accept the cancelation of orders. All matched orders are executed in real time. However, the information is not released to trading terminals during the cooling period and is publicly available at the end of the cooling period. The continuous double auction operates from 9:30 to 11:30 and from 13:00 to 15:00 (for SZSE, 14:57-15:00 is a closing call auction period to form the close price) and transaction occurs automatically by matching due to price and time priority. The time interval between 11:30 a.m. and 13:00 p.m. is a trade halt period. Outside these opening hours, unexecuted orders will be removed by the system. Data sets {#data-sets .unnumbered} --------- Our data sets were provided by RESSET (http://resset.cn/), which is a leading financial data provider supporting academic research. The data sets contain all common A-share stocks traded on the Shanghai Stock Exchange and Shenzhen Stock Exchange. The price limits for these stocks are $\pm10\%$. Specially treated stocks with price limits of $\pm5\%$ are not included in our analysis. The sample covers the period from 2000/01/04 to 2011/12/30, totally 12 years. Because the stocks have different initial public offering dates, the lengthes of stocks are not fixed. The quote frequency is about 5 seconds before 27 June 2011 and 3 seconds afterwards. Due to different liquidities of the stocks, the quote frequencies of different stocks can be lower. For each stock, we have a unique stock code that is a sequence of six digital numbers, the trading time, the trading price, the trading volume, and the prices and standing volumes at the three best levels before 5 December 2003 or 5 levels afterwards on both the buy and sell sides of the limit order book. Determining daily price up/down limits {#determining-daily-price-updown-limits .unnumbered} -------------------------------------- The records of stocks do not contain any indicator of price hits. Hence, we need to identify when the price of a stock hits the up-limit or the low limit. Because the tick size of all stocks is one cent (0.01 Chinese Yuan), no matter they have high prices or low prices, we are able to identify price limit hits. Indeed, for each stock, we can determine the price up limit and price down limit for each trading day. Let $P_i(T)$ denotes the closing price of stock $i$ on day $T$. The up-limit $P^+_i(T+1)$ and the down-limit $P^-_i(T+1)$ of stock $i$ on day $T+1$ are determined as follows, $$P^{\pm}_i(T+1) = {\mathcal{R}\left[100P_i(T)(1\pm10\%)\right]}/100, \label{Eq:P+-}$$ where $\mathcal{R}\left[x\right]$ is a round operator of $x$ such that the daily price limits are rounded to the tick size according to the [*Trading Rules of Shanghai Stock Exchange*]{} (2003, 2006) and the [*Shenzhen Stock Exchange Trading Rules*]{} (2003, 2006). Determining bullish and bearish periods of the Chinese stock markets {#determining-bullish-and-bearish-periods-of-the-chinese-stock-markets .unnumbered} -------------------------------------------------------------------- A representative measure of the status of the Chinese stock markets is the Shanghai Stock Exchange Composite (SSEC) Index. Figure \[Fig:SSEC:Bull:Bear\] illustrates the evolution of the SSEC Index from 1999 to 2011. The SSEC Index rose from 1406 on 2000/01/04 to its historical intraday high of 2242.4 on 2001/06/13 and since plummeted 32.2% to 1520.7 on 2001/10/22. The Chinese stock markets entered a bearish antibubble state since June of 2011 [@Zhou-Sornette-2004a-PA], which did not synchronize the U.S. antibubble after the burst of the so-called New Economy Bubble in 2000 [@Johansen-Sornette-2000a-EPJB; @Sornette-Zhou-2002-QF]. The SSEC Index hit its all-time intraday low at 998.23 on 2005/06/04 during the time period investigated in this work. After that, A huge bubble formed and the index reached its historical intraday high at 6124.04 on 2007/10/16, which was followed by a severe crash [@Jiang-Zhou-Sornette-Woodard-Bastiaensen-Cauwels-2010-JEBO] and the index plummeted to 1666.93 on 2008/10/28 as a all-time low since the crash. Although the whole individual sentiment was bearish in the last six years, there was a mediate-size bubble started after 2008/10/28 and the index reached 3478.01 on 2009/08/04. In summary, the time period under investigation in this work can be divided into alternating periods of bullish and bearish states. The stock market was bullish during the three time periods: 2000/1/4 - 2001/6/13, 2005/06/04 - 2007/10/16, and 2008/10/28 - 2009/08/04. Other periods are recognized as being bearish. ![\[Fig:SSEC:Bull:Bear\] [**Evolution of the Shanghai Stock Exchange Composite Index from January 2000 to December 2011.**]{} The data shown are the closing prices. The historical highs and lows usually occurred intraday. The index has been divided into alternating bullish and bearish periods. The red parts stand for bullish periods and the green parts correspond to bearish periods.](Fig1.eps){width="8cm" height="6cm"} In this work, we will compare the statistics of price limit hits in bullish and bearish market states. It is true that individual stocks may evolve differently. However, it is hard to recognize bullish and bearish states for individual stocks and it is not irrational to make such a comparison based on the SSEC Index. We leave this more complicated stock-by-stock classification of bulls and bears in future research. Fitting procedure {#fitting-procedure .unnumbered} ----------------- The empirical distribution of the number of limit hits for individuals is truncated on the left. We find that most of the distributions investigated in this work can be fitted by the left truncated normal distribution: $$p(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{\frac{(x-\mu)^2}{2\sigma^2}}, ~~~x>0. \label{Eq:Normal:Left}$$ Both ordinary least squares and maximum likelihood estimation are applied in curve fitting. When we use maximum likelihood estimation to estimate parameters such as mean and variance, we cannot use the sample mean and variance to substitute the whole population. A feasible method is the following [@Randall-J.Olsen-1980-Em]: $$\label{Eq:Ex:Qr} \left\{ \begin{array}{l} E(X|X > 0) = \mu + \sigma Q(\mu /\sigma )\\ Q(r) = {{f(r)}}/{{F(r)}} \end{array} \right.$$ where $f(\cdot)$ and $F(\cdot)$ are respectively the density function and the cumulative density function of the standard normal distribution. Denoting $\mu' = E(X|X > 0)$ the mean of the truncated normal distribution, one has $$\label{Eq:mu:var} \left\{ \begin{array}{l} \mu' = \mu + \sigma Q(\mu /\sigma )\\ {\mathrm{Var}}(X|X>0) = {\sigma^{2}}\{1-Q(\mu /\sigma )[\mu /\sigma + Q(\mu /\sigma )]\} \end{array} \right.$$ Denoting ${\sigma'^{2}} = {\mathrm{Var}}(X|X > 0)$ the variance of the truncated normal distribution, one obtains $$\label{Eq:Mu:Sigma} \mu'/\sigma'= \frac{\mu/\sigma + Q(\mu/\sigma)}{\sqrt{\{1-Q(\mu /\sigma )[\mu /\sigma + Q(\mu /\sigma )]\}}}.$$ Defining $r=\mu/\sigma$, $r'=\mu'/\sigma'$, one gets $$\label{Eq:R:result} r'= \left[r+ \frac{{f(r)}}{{F(r)}} \right]\div\sqrt{1-\frac{{f(r)}}{{F(r)}}\left[r + \frac{{f(r)}}{{F(r)}}\right]}.$$ In practical applications, we find that the function including $r$ on the right side of Eq. (\[Eq:R:result\]) is monotonically increasing. Hence the equation has a unique solution. We can determine $r$ firstly, and then $\sigma$ and $\mu$. Finally, we use $\sigma$ and $\mu$ to estimate the population distribution. Results {#results .unnumbered} ======= Basic statistics of the numbers of limit-hitting days {#basic-statistics-of-the-numbers-of-limit-hitting-days .unnumbered} ----------------------------------------------------- We provide statistical properties of the numbers of trading days with different types of limit hits. The variables are the following. $N$ is the total number of trading days with limit hits. $\langle{N}\rangle$ is the average number of limit-hitting days for individual stocks. $N_{\rm{con}}$ is the number of limit-hitting days with continued next-day opening prices. It contains up (down) limit hitting days with the opening prices on next trading days being higher (lower) than the closing prices. $N_{\rm{rev}}$ is the number of limit-hitting days with price reversal on the next day. It contains up (down) limit hitting days with the opening prices on next trading days being lower (larger) than the closing prices. $N_{\rm{open}}$, $N_{\rm{am}}$, $N_{\rm{pm}}$ and $N_{\rm{close}}$ are respectively the numbers of days with limit hits occurred in the opening call auction (9:15, 9:30\], in the continuous double auction session (9:30,11:30\] in the morning, in the continuous double auction session \[13:00, 15:00\] in the afternoon and at the closure of the trading days. $N_{\rm{close,con}}$ and $N_{\rm{close,rev}}$ are the numbers of trading days that closed at limit prices and the price continued rising up or falling down on the successive trading days. It is possible that for some stocks there are both up- and down-limit hits within the same day. In this case, the first limit hit is used in the calculation of different numbers. We further delete IPO days and ex-dividend days. Since trading halt may trigger after a price limit-hitting day and thus there is no followup open price, we do not count these days in $N_{\rm{con}}$, $N_{\rm{rev}}$, $N_{\rm{close,con}}$, and $N_{\rm{close,rev}}$. We also partition evenly all stocks with limit hits on a given day into six portfolios based on their capitalizations (Portfolio 1 with the smallest capitalizations and Portfolio 6 with the largest capitalizations) and count the defined numbers for each portfolio. The numbers are determined for the whole period, and the bullish and bearish periods as well. The capitalization of a stock is calculated as the product of the amount of shares times the price. The basic statistics of limit hits in the whole sample period and in bullish and bearish periods are presented in Table \[Tb:Statistics:ManyStocks\]. It is trivial to observe that $$N_{\pm} = \sum_{j=1}^6 N_{\pm,j}~~{\mathrm{and}}~~N_{\pm,i}\approx N_{\pm,j},$$ where the subscripts $+$ and $-$ represent respectively price up limit hit and price down limit hit, and $j$ stands for the portfolio serial number. [cccccccccccccccccccccccccccccc]{}\ && && && && && && &&\ &&Up & Down && Up & Down && Up & Down && Up & Down && Up & Down && Up & Down && Up & Down\ $N$ && 62346& 44216&& 10393& 7369&& 10393& 7369&& 10390& 7368&& 10389& 7368&& 10387& 7368&& 10394&7374\ $\langle{N}\rangle$ && 20.58 & 14.60 && 3.43 & 2.43 && 3.43 & 2.43 && 3.43 & 2.43 && 3.43 & 2.43 && 3.43 & 2.43 &&3.43 & 2.43\ $N_{\rm{con}}$ && 39696& 33601&& 6595& 6242&& 6819& 5706&& 6662& 5569&& 6538& 5514&& 6443& 5324&& 6639& 5246\ $N_{\rm{rev}}$ &&22627& 10603&& 3798& 1127&& 3574& 1662&& 3724& 1798&& 3848& 1852&& 3937& 2040&& 3746& 2124\ $N_{\rm{open}}$ &&6718& 2126&& 1444& 441&& 1445& 332&& 1115& 334&& 898& 347&& 886& 316&& 930& 356\ $N_{\rm{am}}$ &&33204& 15694&& 5740& 3082&& 5831& 2460&& 5673& 2551&& 5522& 2674&& 5434& 2496&& 5004& 2431\ $N_{\rm{pm}}$ &&29157& 28525&& 4653& 4287&& 4562& 4909&& 4720& 4818&& 4871& 4695&& 4959& 4873&& 5392& 4943\ $N_{\rm{close}}$ &&40752& 22213&& 6207& 3203&& 6948& 3745&& 7137& 3882&& 7004& 3857&& 6871& 3791&& 6585& 3735\ $N_{\rm{close,con}}$ && 32340& 19067&& 5120& 2986&& 5610& 3252&& 5612& 3316&& 5480& 3278&& 5321& 3147&& 5197&3088\ $N_{\rm{close,rev}}$ &&8394& 3138&& 1087& 217&& 1338& 493&& 1521& 566&& 1522& 577&& 1544& 641&& 1382& 644\ \ && && && && && && &&\ &&Up & Down && Up & Down && Up & Down && Up & Down && Up & Down && Up & Down && Up & Down\ $N$ &&32593& 17898&& 5432& 2983&& 5432& 2983&& 5432& 2983&& 5432& 2983&& 5431& 2983&& 5434& 2983\ $\langle{N}\rangle$ && 26.52 & 14.56 && 4.42 & 2.43 && 4.42 & 2.43 && 4.42 & 2.43 && 4.42 & 2.43 && 4.42 & 2.43 && 4.42 & 2.43\ $N_{\rm{con}}$ &&22344& 13888&& 3501& 2563&& 3713& 2397&& 3690& 2315&& 3787& 2278&& 3775& 2190&& 3878& 2145\ $N_{\rm{rev}}$ && 10249& 4010&& 1931& 420&& 1719& 586&& 1742& 668&& 1645& 705&& 1656& 793&& 1556& 838\ $N_{\rm{open}}$ && 2908& 986&& 660& 161&& 508& 134&& 421& 175&& 432& 183&& 440& 162&& 447& 171\ $N_{\rm{am}}$ &&15785& 6730&& 2925& 1117&& 2657& 1005&& 2650& 1168&& 2626& 1191&& 2590& 1173&& 2337& 1076\ $N_{\rm{pm}}$ && 16809& 11168&& 2507& 1866&& 2775& 1978&& 2782& 1815&& 2806& 1792&& 2842& 1810&& 3097& 1907\ $N_{\rm{close}}$ &&20814& 8778&& 3035& 1278&& 3511& 1480&& 3644& 1559&& 3652& 1540&& 3577& 1493&& 3395& 1428\ $N_{\rm{close,con}}$ &&17394& 8011&& 2541& 1195&& 2923& 1371&& 2977& 1407&& 3062& 1406&& 2986& 1352&& 2905& 1280\ $N_{\rm{close,rev}}$ &&3420& 767&& 494& 83&& 588& 109&& 667& 152&& 590& 134&& 591& 141&& 490& 148\ \ && && && && && && &&\ &&Up & Down && Up & Down && Up & Down && Up & Down && Up & Down && Up & Down && Up & Down\ $N$ && 29753& 26318&& 4961& 4386&& 4960& 4386&& 4956& 4385&& 4957& 4385&& 4958& 4385&& 4961& 4391\ $\langle{N}\rangle$ && 16.53 & 14.62 && 2.76 & 2.44 && 2.76 & 2.44 && 2.75 & 2.44 && 2.75 & 2.44 && 2.75 & 2.44 && 2.76 & 2.44\ $N_{\rm{con}}$ &&17352& 19713&& 3062& 3678&& 3034& 3299&& 2810& 3247&& 2785& 3231&& 2824& 3162&& 2837& 3096\ $N_{\rm{rev}}$ && 12378& 6593&& 1899& 708&& 1926& 1086&& 2141& 1137&& 2167& 1152&& 2130& 1219&& 2115& 1291\ $N_{\rm{open}}$ &&3810& 1140&& 935& 282&& 931& 189&& 600& 155&& 437& 165&& 409& 163&& 498& 186\ $N_{\rm{am}}$ && 17419& 8964&& 3004& 1960&& 3231& 1434&& 2984& 1404&& 2843& 1445&& 2747& 1365&& 2610& 1356\ $N_{\rm{pm}}$ && 12348& 17357&& 1957& 2426&& 1730& 2952&& 1977& 2982&& 2118& 2941&& 2214& 3021&& 2352& 3035\ $N_{\rm{close}}$ && 19938& 13435&& 3244& 1924&& 3424& 2261&& 3432& 2324&& 3344& 2322&& 3333& 2311&& 3161& 2293\ $N_{\rm{close,con}}$ &&14946& 11056&& 2621& 1788&& 2644& 1871&& 2502& 1900&& 2423& 1857&& 2435& 1824&& 2321& 1816\ $N_{\rm{close,rev}}$ &&4974& 2371&& 623& 136&& 780& 390&& 925& 424&& 917& 463&& 895& 484&& 834& 474\ \[Tb:Statistics:ManyStocks\] Panel A of Table \[Tb:Statistics:ManyStocks\] shows the results for the whole period from 2000 to 2011. There are more limit-up hits than limit-down hits, which is partially caused by the rapid growth of China’s economy and the absence of short mechanism. The limit-down days are more likely to have next-day price continuation than limit-up days since $N^+_{\rm{con}}/N^+=39696/62346=63.7\%$ and $N^-_{\rm{con}}/N^-=33601/44216=75.9\%$. In addition, a limit hitting day is more likely to have price continuation than price reversal because $N^{\pm}_{\rm{con}}\gg N^{\pm}_{\rm{rev}}$. For all other numbers concerning all stocks and different portfolios, we also observe more limit-up days than limit-down days. For limit-down days, the probability of next-day price continuation decreases with increasing average capitalization of the portfolio, while the probability of next-day price reversal increases with the capitalization. For limit-up days, the probabilities of next-day price continuation and price reversal do no have any clear trend. For $N^{\pm}_{\rm{open}}$ and $N^{\pm}_{\rm{am}}$, we observe decreasing trends with capitalization. For $N^{\pm}_{\rm{pm}}$, we observe increasing trends with capitalization. For $N^{\pm}_{\rm{close}}$, they increase first and then decrease. In all the seven cases (all stocks and the six portfolios), limit-up events are more likely to occur in the morning than in the afternoon, while limit-down events are more likely to happen in the afternoon than in the morning. Limit-up events have higher probability (65.3%) to close at limit price than limit-down events (51.0%) according to $N_{\rm{close}}$. When a trading day closes at the up-limit or the down-limit, the next-day opening price will rise with a very high probability (79.4% for limit-up events and 85.8% for limit-down events). In addition, $N^{\pm}_{\rm{close,con}}$ and $N^{\pm}_{\rm{close,rev}}$ increase first and then decrease. For the bullish periods in Panel B of Table \[Tb:Statistics:ManyStocks\], in all seven cases (all stocks and the six portfolios), there are more limit-up days for $N$, $N_{\rm{open}}$, $N_{\rm{am}}$ and $N_{\rm{close}}$. For the bullish periods in Panel C of Table \[Tb:Statistics:ManyStocks\], there are more limit-up days than limit-down days for $N$, $N_{\rm{open}}$, $N_{\rm{am}}$, and $N_{\rm{close}}$, with an exception that there are more limit-down days than limit-up days in the afternoon ($N^-_{\rm{pm}}>N^+_{\rm{pm}}$). However, the occurrence difference between limit-up and limit-down days is smaller in the bearish periods than in the bullish periods. For bullish periods in Panel B, $N^+_{\rm{open}}$ and $N^+_{\rm{am}}$ decrease with increasing capitalization, $N^+_{\rm{pm}}$ increases with capitalization, and no evident trends have been found in $N^-_{\rm{open}}$, $N^-_{\rm{am}}$, $N^-_{\rm{pm}}$ and $N^{\pm}_{\rm{close}}$. For bearish periods in Panel C, $N^{\pm}_{\rm{open}}$ and $N^{\pm}_{\rm{am}}$ decrease with capitalization, $N^+_{\rm{pm}}$ and $N^-_{\rm{pm}}$ increase with capitalization, and $N^+_{\rm{close}}$ and $N^{-}_{\rm{close}}$ do not exhibit any clear trend with respect to capitalization. For both bullish and bearish periods, price continuation is more likely to occur than price reversal for both limit-up and limit-down days, because $N^{\pm}_{\rm{con}}>N^{\pm}_{\rm{rev}}$ and $N^{\pm}_{\rm{close,con}}>N^{\pm}_{\rm{close,rev}}$. In addition, the probability of price continuation is higher for limit-down days than for limit-up days, that is $N^{+}_{\rm{con}}/N^+<N^-_{\rm{con}}/N^-$ and $N^{+}_{\rm{close,con}}/N^+_{\rm{close}}<N^-_{\rm{close,con}}/N^-_{\rm{close}}$. For bullish periods, the probability of price continuation increases with capitalization for limit-up days ($N^{+}_{\rm{con}}/N^+$) and decreases with capitalization for limit-down days ($N^-_{\rm{con}}/N^-$). These trends are absent for $N^{\pm}_{\rm{close}}$. For bearish period, we observe that $N^-_{\rm{con}}/N^-$, $N^+_{\rm{close,con}}/N^+_{\rm{close}}$ and $N^-_{\rm{close,con}}/N^-_{\rm{close}}$ all have a decreasing trend with respect to the capitalization. These findings have potential applications for practitioners in the Chinese stock markets, keeping in mind that one cannot short and there is a $T+1$ trading rule. When the price of a stock hits the up-limit, no matter in the intraday or at the close, she can buy right before market closure at 15:00 and sell with the opening price on the next trading day. Regardless of the transaction cost, the return will have high probability to be positive. The probability of earning money is even high if the price is locked to the up-limit and when the capitalization of the stock is low. Certainly, in this case, the liquidity is usually very low and it is hard to buy shares. On the contrary, if one holds a stock whose price experiences intraday down-limit hits, it is better to sell it to reduce losses. Advanced statistics of intraday limit hits {#advanced-statistics-of-intraday-limit-hits .unnumbered} ------------------------------------------ For each stock $i$ traded in $T_i$ days in our sample, we identify $K_i$ trading days on which the prices have hit either the up-limit or the down-limit at least once. We denote the set of limit-up days (trading days that have up-limit hits) of stock $i$ as ${\bf{U}}_i=\{u_{i,k}: k=1,2,\cdots,K^u_i\}$, where $K^u_i=\#{\bf{U}}_i$ is the number of limit-up trading days. Similarly, the set of limit-down days of stock $i$ is denoted as ${\bf{D}}_i=\{d_{i,k}: k=1,2,\cdots,K^d_i\}$, where $K^d_i=\#{\bf{D}}_i$ is the number of limit-down trading days. The intersection of ${\bf{U}}_i$ and ${\bf{D}}_i$ is not necessary to be empty, because, although very rare, it is possible that stock $i$ hits its up-limit and down-limit on the same day. In other words, $K_i\leq K_i^u+K_i^d$. The percents of limit-hitting days are calculated as follows $$\tilde{n}_i=\frac{{K_i}}{{T_i}},~~\tilde{n}_i^u=\frac{{K_i^u}}{T_i},~~{\mathrm{and}}~~\tilde{n}_i^d=\frac{{K_i^d}}{T_i}, \label{Eq:Prob:Hit:per:Day}$$ where $\tilde{n}_i$, $\tilde{n}_i^u$ and $\tilde{n}_i^d$ can be regarded as the empirical probabilities that stock $i$ will hit either the upper or the down-limit, the up-limit, and the down-limit on a trading day, respectively. Figure \[Fig:MagnetEffect:Prob:Hit:per:Day\] shows the empirical distributions of limit-hitting probability per day for individual stocks. As shown in Fig. \[Fig:MagnetEffect:Prob:Hit:per:Day\](a), for most of the stocks, the daily limit-hitting probability is less than 6%, with the most probability value around 2.5%. However, there are also a few stocks having very large limit-hitting probability up to 17%. Such seemingly outliers include 002606, 002632, 002635, 002643, 002644, and 002646. These stocks have relatively large numbers of price down limit hits, as shown in Fig. \[Fig:MagnetEffect:Prob:Hit:per:Day\](c). The large values of $\tilde{n}_i^d$ are mainly caused by the small values of $T_i$ because these stocks come into the market for a short time period. For most stocks, the daily up-limit hit is less than 4% according to Fig. \[Fig:MagnetEffect:Prob:Hit:per:Day\](b) and the daily dow-limit hit is less than 3% according to Fig. \[Fig:MagnetEffect:Prob:Hit:per:Day\](c). This finding is consistent with the observation that $N^+>N^-$ in Table \[Tb:Statistics:ManyStocks\]. It seems that the curves obtained by the OLS method fit the empirical data better than those by the MLE method. ![[**Empirical probability density functions $P(\tilde{n}_{i})$, $P(\tilde{n}^{u}_{i})$ and $P(\tilde{n}^{d}_{i})$ of daily price limit hitting probabilities $\tilde{n}_{i}$, $\tilde{n}^{u}_{i}$ and $\tilde{n}^{d}_{i}$ for individual stocks.**]{} The dots are empirical data, the black dash-dotted curves are the MLE fits to the truncated normal distribution in Eq. (\[Eq:Normal:Left\]), and the black dash-dotted curves are the OLS fits to the truncated normal distribution. (a) All limit hits; (b) Price up limit hits; (c) Price down limit hits.[]{data-label="Fig:MagnetEffect:Prob:Hit:per:Day"}](Fig2.eps){width="96.00000%" height="25.00000%"} Consider a limit-hitting day $u_{i,k}$ (or $d_{i,k}$) of stock $i$. The price may hit the price limit for $M^{u}_{i,k}$ (or $M^{d}_{i,k}$ ) times at intraday moments $t^{u}_{i,k,m}$ (or $t^{d}_{i,k,m}$) with $m=1, \cdots, M^{u}_{i,k}$ (or $M^{d}_{i,k}$). We define the average number of limit hits of stock $i$ as follows, $$M^{u}_i = \frac{{1}}{K^u_i}\sum_{k=1}^{K^u_i} M^u_{i,k} ~~{\mathrm{and}}~~ M^{d}_i = \frac{{1}}{K^d_i}\sum_{k=1}^{K^d_i} M^{d}_{i,k}. \label{Eq:M:ud:i}$$ where the denominators are not the total number of trading days, $T_i$, but the number of price up limit hitting days and of price down limit hitting days. According to Fig. \[Fig:MagnetEffect:PDF:Num:Hits\](a) and Fig. \[Fig:MagnetEffect:PDF:Num:Hits\](b), the distributions of $M^{u}_{i,k}$ and $M^{d}_{i,k}$ decrease sharply. About 30% of limit-hitting days have only one up-limit hit or down-limit hit. However, there are also limit-hitting days with very large numbers of limit hits in one day. For instance, stock 600863 hit the up-limit for 151 times on 2009/07/14, while stock 600102 hit the down-limit for 149 times on 2009/04/27, which has been delisted from the Shanghai Stock Exchange due to it poor performance. Surprisingly, the two distributions $P(M^{u}_{i})$ and $P(M^{d}_{i})$ are not monotonically decreasing functions. Instead, they can be well fitted by the truncated normal distribution. Speaking differently, if the price limit (10% or -10%) of a stock has been reached on certain trading day, the number of limit hits on that day is usually more than once. ![[**Number of limit hits on individual limit-hitting days.**]{} (a,b) Probability density functions $P(M^{u}_{i,k})$ and $P(M^{d}_{i,k})$ of $M^{u}_{i,k}$ and $M^{d}_{i,k}$. (c,d) Probability density functions $P(M^{u}_{i})$ and $P(M^{d}_{i})$ of $M^{u}_i$ and $M^{d}_i$ for individual stocks. []{data-label="Fig:MagnetEffect:PDF:Num:Hits"}](Fig3.eps){width="66.00000%" height="52.00000%"} When stock $i$ hits its price limit on day $u_{i,k}$ (or $d_{i,k}$), the $m$-th limit-up (or limit-down) started at $t^{u}_{i,k,m}$ (or $t^{d}_{i,k,m}$) may be opened at $t^{u}_{i,k,m}+\Delta{t}^{u}_{i,k,m}$ (or $t^{d}_{i,k,m}+\Delta{t}^{d}_{i,k,m}$). Hence, we have a sequence of limit-hitting durations $\Delta{t}^{u}_{i,k,m}$ (or $\Delta{t}^{d}_{i,k,m}$). The total duration on that day can be calculated as follows: $$\Delta{t}^{u}_{i,k} = \sum_{m=1}^{M^{u}_{i,k}} \Delta{t}^{u}_{i,k,m} ~~{\mathrm{and}}~~ \Delta{t}^{d}_{i,k} = \sum_{m=1}^{M^{d}_{i,k}} \Delta{t}^{d}_{i,k,m}. \label{Eq:Delta:ud:i:k}$$ Considering only the limit-hitting days of stock $i$, we can also define the average limit hit duration as follows: $$\Delta{t}^{u}_i = \frac{1}{K^u_i} \sum_{k=1}^{K^u_i} \sum_{m=1}^{M^{u}_{i,k}} \Delta{t}^{u}_{i,k,m} ~~{\mathrm{and}}~~ \Delta{t}^{d}_i = \frac{1}{K^d_i}\sum_{k=1}^{K^d_i} \sum_{m=1}^{M^{d}_{i,k}} \Delta{t}^{d}_{i,k,m}. \label{Eq:Delta:ud:i}$$ We stress that these two quantities are defined for individual stocks and the denominators are not $T_i$. Plots (a) and (d) of Fig. \[Fig:MagnetEffect:PDF:Duration\] show the empirical distributions of the durations $\Delta{t}^{u}_{i,k,m}$ and $\Delta{t}^{d}_{i,k,m}$ of individual limit hits for all stocks. The two distributions exhibit a similar “L” shape. In each distribution, there are three obvious peaks at $\Delta{t}^{u,d}_{i,k,m}=7200$s, 10800s and 12600s, which correspond respectively to 2 hours, 3 hours and 3.5 hours. These peaks are mainly caused by limit hits at 13:00 p.m., 10:30 a.m. and 10:00 a.m. with the prices remaining at the limit prices till market closure. These patterns contain significant information contents. The peak around 13:00 reflects the fact that the majority of the traders or some informed traders may obtain cumulated important information about the holding stock during the market closure at noon (11:30-13:00) and form a collective behavior to buy or sell their stock to push the price to its limit. This interpretation applies certainly to the peak at 14400s, corresponding to the case that the stock opens at its limit price due to overnight information and does not fluctuate during the whole trading day. The peak at 10800s are caused by the one-hour opening trading halts in which the stock usually bears abnormal information pit and traders’ trading habits and round number preference. The peak at 12600s is less significant, which is very likely caused by traders’ trading habits and round number preference. ![[**Limit hit duration.**]{} (a,d) Probability density functions $P(\Delta{t}^{u}_{i,k,m})$ and $P(\Delta{t}^{d}_{i,k,m})$ of the individual limit hit durations $\Delta{t}^{u}_{i,k,m}$ and $\Delta{t}^{d}_{i,k,m}$. (b,e) Probability density functions $P(\Delta{t}^{u}_{i,k})$ and $P(\Delta{t}^{d}_{i,k})$ of the daily limit hit durations $\Delta{t}^{u}_{i,k}$ and $\Delta{t}^{d}_{i,k}$. (c,f) Probability density functions $P(\Delta{t}^{u}_{i})$ and $P(\Delta{t}^{d}_{i})$ of the average daily limit hit durations $\Delta{t}^{u}_i$ and $\Delta{t}^{d}_i$. The unit of the variables is second.[]{data-label="Fig:MagnetEffect:PDF:Duration"}](Fig4.eps){width="96.00000%" height="52.00000%"} Plots (b) and (e) of Fig. \[Fig:MagnetEffect:PDF:Duration\] show the empirical distributions of the daily limit hit durations $\Delta{t}^{u}_{i,k}$ and $\Delta{t}^{d}_{i,k}$. The overall shapes of these two distributions are very similar with those in Fig. \[Fig:MagnetEffect:PDF:Duration\] (a) and (d). The main differences are that the distributions of daily limit hit durations have lower heights at the two edges and larger values in the bold parts. Plots (c) and (f) of Fig. \[Fig:MagnetEffect:PDF:Duration\] present the empirical distributions of the average daily durations $\Delta{t}^{u}_i$ and $\Delta{t}^{d}_i$ for individual stocks. The distributions are bimodal with an extra peak close to $\Delta{t}^{u,d}_i=0$s. We further define the total intraday limit hit duration on trading day $u_k$, which is the time elapse from the moment of the first limit hit to the last moment that prices stay at limit in a limit-hitting day: $$\Delta{T}^{u}_{i,k} = t^{u}_{i,k,M^u_{i,k}}+\Delta{t}^{u}_{i,k,M^u_{i,k}}-t^{u}_{i,k,1}~~{\mathrm{and}}~~ \Delta{T}^{d}_{i,k} = t^{d}_{i,k,M^d_{i,k}}+\Delta{t}^{d}_{i,k,M^d_{i,k}}-t^{d}_{i,k,1}. \label{Eq:totalDelta:ud:i:k}$$ By definition, we have $\Delta{T}^{u,d}_{i,k}\geq \Delta{t}^{u,d}_{i,k}$. We also define the average of the total intraday limit-hitting duration for individual stocks as follows, $$\Delta{T}^{u}_i = \frac{1}{K^u_i}\sum_{k=1}^{K^u_i}\Delta{T}^{u}_{i,k}~~{\mathrm{and}}~~ \Delta{T}^{d}_i = \frac{1}{K_i^d}\sum_{k=1}^{K^d_i} \Delta{T}^{d}_{i,k}. \label{Eq:totalDelta:ud:i}$$ Similarly, we have $\Delta{T}^{u,d}_{i}\geq \Delta{t}^{u,d}_{i}$. The upper panel of Fig. \[Fig:MagnetEffect:PDF:totalDuration\] shows the empirical distributions of the total intraday limit hit durations $\Delta{T}^{u}_{i,k}$ and $\Delta{T}^{d}_{i,k}$ for all stocks. The distribution in Fig. \[Fig:MagnetEffect:PDF:totalDuration\](a) has two local maxima around $\Delta{T}^{u}_{i,k}=7200$s and 10800s as in Fig. \[Fig:MagnetEffect:PDF:Duration\](b) for $\Delta{t}^{u}_{i,k}$. The distribution in Fig. \[Fig:MagnetEffect:PDF:totalDuration\](b) has a similar overall shape. However, it is less smoother. The lower panel of Fig. \[Fig:MagnetEffect:PDF:totalDuration\] shows the empirical distributions of $\Delta{T}^{u}_i$ and $\Delta{T}^{d}_i$. It is hard to find a suitable function form to fit the data. ![[**Total intraday limit hit duration.**]{} (a,b) Probability density functions $P(\Delta{T}^{u}_{i,k})$ and $P(\Delta{T}^{d}_{i,k})$ of the total intraday limit hit durations $\Delta{T}^{u}_{i,k}$ and $\Delta{T}^{d}_{i,k}$. (c,d) Probability density functions $P(\Delta{T}^{u}_{i})$ and $P(\Delta{T}^{d}_{i})$ of the averages of total intraday limit hit durations $\Delta{T}^{u}_i$ and $\Delta{T}^{d}_i$. The unit of the variables is second. []{data-label="Fig:MagnetEffect:PDF:totalDuration"}](Fig5.eps){width="66.00000%" height="52.00000%"} As a complementary to the distributions in Fig. \[Fig:MagnetEffect:PDF:Num:Hits\] to Fig. \[Fig:MagnetEffect:PDF:totalDuration\], we investigate the maxima, the medians and the means of the three kinds of daily variables, the daily numbers of limit hits $M^{u,d}_{i,k}$, the daily limit hit durations $\Delta{t}^{u,d}_{i,k}$ and the total intraday limit hit durations $\Delta{T}^{u,d}_{i,k}$. We compare the results for the whole time period, the bullish periods and the bearish periods. We also divide all the stocks into six portfolios as defined in Table \[Tb:Statistics:ManyStocks\] to investigate the impacts of stock capitalization. The results are presented in Table \[Tb:Intraday:statistics:IndividualStocks\]. [cccccccccccccccccccccccccccccccccccccccccccccc]{}\ && && && && && && &&\ &&Up & Down && Up & Down && Up & Down && Up & Down && Up & Down && Up & Down && Up & Down\ ${\max}\{M^{u,d}_{i,k}\}$ && 151& 149&& 88& 87&& 114& 102&& 113& 96&& 125& 119&& 133& 99&& 151& 149\ ${\rm{mean}}\{M^{u,d}_{i,k}\}$ &&7.47 & 8.27 && 5.74 & 6.67 && 7.13 & 7.86 && 7.70 & 8.62 && 7.95 & 8.55 && 8.06 & 8.83 && 8.23 & 9.09\ ${\rm{med}}\{M^{u,d}_{i,k}\}$ && 4& 5&& 3& 4&& 4& 5&& 4& 6&& 4& 6&& 4& 6&& 4& 6\ ${\rm{mean}}\{\Delta{t}^{u,d}_{i,k}\}$ && 4391 & 2189 && 4285 & 2014 && 4723 & 2036 && 4696 & 2291 && 4542 & 2347 && 4300 & 2247 && 3801 & 2202\ ${\rm{med}}\{\Delta{t}^{u,d}_{i,k}\}$ &&2096& 815&& 1720& 552&& 2386& 745&& 2397& 906&& 2282& 911&& 2113& 866&& 1815& 917\ ${\rm{mean}}\{\Delta{T}^{u,d}_{i}\}$ &&5740 & 3888 && 5553 & 3793 && 6241 & 3661 && 6059 & 4035 && 5822 & 4054 && 5590 & 3924 && 5175 & 3857\ ${\rm{med}}\{\Delta{T}^{u,d}_{i}\}$ &&4118& 1937&& 3496& 1785&& 5020& 1796&& 4740& 2035&& 4324& 2077&& 4004& 1953&& 3408& 1982\ \ && && && && && && &&\ &&Up & Down && Up & Down && Up & Down && Up & Down && Up & Down && Up & Down && Up & Down\ ${\max}\{M^{u,d}_{i,k}\}$ && 151& 149&& 88& 87&& 97& 102&& 114& 96&& 113& 72&& 125& 97&& 151& 149\ ${\rm{mean}}\{M^{u,d}_{i,k}\}$ &&7.34 & 7.76 && 6.42 & 7.16 && 7.19 & 7.67 && 7.71 & 8.18 && 7.64 & 7.83 && 7.46 & 7.90 && 7.61 & 7.83\ ${\rm{med}}\{M^{u,d}_{i,k}\}$ && 4& 5&& 3& 4&& 4& 5&& 4& 5&& 4& 5&& 4& 5&& 4& 5\ ${\rm{mean}}\{\Delta{t}^{u,d}_{i,k}\}$ && 4064 & 2403 && 3956 & 1757 && 4276 & 1991 && 4327 & 2643 && 4288 & 2858 && 4011 & 2719 && 3525 & 2452\ ${\rm{med}}\{\Delta{t}^{u,d}_{i,k}\}$ &&1778 & 790&& 1290 & 408&& 1901& 603&& 2070 & 956&& 2093 & 1064&& 1850& 1016&& 1580& 965\ ${\rm{mean}}\{\Delta{T}^{u,d}_{i}\}$ &&5251 & 4011 && 5123 & 3566 && 5410 & 3521 && 5524 & 4307 && 5493 & 4388 && 5225 & 4286 && 4736 & 3997\ ${\rm{med}}\{\Delta{T}^{u,d}_{i}\}$ &&3415 & 2018&& 2689& 1539&& 3675& 1605&& 3982 & 2187&& 3877 & 2264&& 3495& 2213&& 2853 & 2089\ \ && && && && && && &&\ &&Up & Down && Up & Down && Up & Down && Up & Down && Up & Down && Up & Down && Up & Down\ ${\max}\{M^{u,d}_{i,k}\}$ && 133& 125&& 65& 59&& 88& 71&& 108& 73&& 133& 119&& 131& 99&& 124& 125\ ${\rm{mean}}\{M^{u,d}_{i,k}\}$ &&7.61 & 8.62 && 4.84 & 6.30 && 7.23 & 8.02 && 7.96 & 8.99 && 8.28 & 9.05 && 8.55 & 9.58 && 8.81 & 9.78\ ${\rm{med}}\{M^{u,d}_{i,k}\}$ && 3& 6&& 2& 4&& 4& 6&& 4& 6&& 4& 6&& 4& 7&& 4& 7\ ${\rm{mean}}\{\Delta{t}^{u,d}_{i,k}\}$ && 4750 & 2044 && 4912 & 2193 && 5227 & 2042 && 5065 & 2013 && 4764 & 1988 && 4468 & 1979 && 4064 & 2049\ ${\rm{med}}\{\Delta{t}^{u,d}_{i,k}\}$ &&2520& 829&& 2848& 707&& 3093& 878&& 2640& 855&& 2541& 817&& 2305& 819&& 2043& 885\ ${\rm{mean}}\{\Delta{T}^{u,d}_{i}\}$ &&6275 & 3804 && 6460 & 3950 && 7179 & 3717 && 6470 & 3830 && 6104 & 3825 && 5838 & 3753 && 5597 & 3747\ ${\rm{med}}\{\Delta{T}^{u,d}_{i}\}$ &&5030& 1871&& 5509& 1919&& 7172& 1917&& 5480& 1869 && 4905& 1814&& 4305& 1845&& 4002 & 1885\ \[Tb:Intraday:statistics:IndividualStocks\] For the maximal number of limit hits, we find that ${\max}\{M^{u}_{i,k}\}>{\max}\{M^{d}_{i,k}\}$ for all cases, except for Portfolio 2 in bullish periods. However, for the means, we have ${\rm{mean}}\{M^{u}_{i,k}\}<{\rm{mean}}\{M^{d}_{i,k}\}$ in all the cases. On average, down-limit hits are less stable than up-limit hits, indicating that the fear sentiment of traders fluctuates more than the greed sentiments in the Chinese stock markets. Similarly, we observe that ${\rm{med}}\{M^{u}_{i,k}\}<{\rm{med}}\{M^{d}_{i,k}\}$ in all the cases. However, the difference ${\rm{med}}\{M^{d}_{i,k}\}-{\rm{med}}\{M^{u}_{i,k}\}$ is larger in the bearish periods than in the bullish periods. It indicates that the fighting between long positions and short positions is more severe when the price hits the down-limit than the up-limit, because trades can make profits when the market rises up while they can only deduce losses when the market falls down. No clear correlation is found between ${\rm{mean}}\{M^{u,d}_{i,k}\}$ and capitalization in the bullish periods. In contrast, both ${\rm{mean}}\{M^{u}_{i,k}\}$ and ${\rm{mean}}\{M^{d}_{i,k}\}$ are positively correlated with capitalization in the bearish periods. For the other variables, we find that the quantities for up-limits are greater than their counterparts for down-limits. Specifically, we observe that ${\rm{mean}}\{\Delta{t}^{u}_{i,k}\}>{\rm{mean}}\{\Delta{t}^{d}_{i,k}\}$, ${\rm{med}}\{\Delta{t}^{u}_{i,k}\}>{\rm{med}}\{\Delta{t}^{d}_{i,k}\}$, ${\rm{mean}}\{\Delta{T}^{u}_{i,k}\}>{\rm{mean}}\{\Delta{T}^{d}_{i,k}\}$, and ${\rm{med}}\{\Delta{T}^{u}_{i,k}\}>{\rm{med}}\{\Delta{T}^{d}_{i,k}\}$ for different periods and different portfolios. Together with the results for ${\rm{mean}}\{M^{u,d}_{i,k}\}$ , we find that the average duration of individual up-limit hits is about twice longer than that of individual down-limit hits. This is probably caused by the no-short trading rule because traders can make money only when the market rises. There are also traders known as “dare-to-die corps for up-limit hits”, who are actively engaged in pushing prices to the up-limits. It is also found that these quantities have a bell-like shape or monotonically decrease with respect to the capitalization. Intraday patterns of the occurrence of price limit hits {#intraday-patterns-of-the-occurrence-of-price-limit-hits .unnumbered} ------------------------------------------------------- We divide the continuous double auction period in each trading day into $N$ intervals of equal length of $\Delta{t}$ minutes. If an interval starts at $t_0$ and ends at $t_1=t_0+\Delta{t}$, we can count the number of occurrences of up-limit hits in this interval as $$C^u_i = \sum_{k=1}^{K^u_i}{\rm{I}}_{t^u_{i,k}\in(t_0,t_1]}, \label{Eq:C:u:i}$$ and similarly the number of occurrences of down-limit hits $$C^d_i = \sum_{k=1}^{K^d_i}{\rm{I}}_{t^d_{i,k}\in(t_0,t_1]}, \label{Eq:C:d:i}$$ where ${\rm{I}}_x$ is an indicator function of event such that the value of ${\rm{I}}_x$ is 1 if the event $x$ is true and 0 otherwise. We can further calculate the following quantities $$C^{u,d} = \sum_{i=1}^I{C^{u,d}_i}, \label{Eq:C:u:d}$$ where $I$ is the number of stocks in the investigated sample. The intraday patterns of the occurrence of limit hits in intervals of size $\Delta{t}=5$ min are illustrated in Fig. \[Fig:Interval:C:ud\]. For each intraday interval, we have $C^{u,d} = C^{u,d}_{\rm{bull}} + C^{u,d}_{\rm{bear}}$. The occurrence of up-limit hits is extremely high at the opening of the market around 9:30 a.m., while the occurrence of down-limits is relatively high at the opening. There are also other local peaks around 09:30, 09:45, 10:00, 10:30, 11:30, 13:30, 14:30 and 14:45. Up-limit hits have different intraday patterns from down-limit hits. However, the intraday patterns do not differ much when comparing the bullish periods with the bearish periods. In all the six cases, limit hits are more frequent in the last hour, especially for down-limit hits. For up-limits, the occurrence number is quite stable over the period from 10:30 a.m. to 14:20 p.m., except for the afternoon opening of the market. ![[**Intraday patterns of the occurrence of limit hits.**]{} The three columns of plots correspond respectively to the whole period (a,d), the bullish periods (b,e) and the bearish periods (c,f). The time interval $\Delta{t}$ is 5 min.[]{data-label="Fig:Interval:C:ud"}](Fig6.eps){width="96.00000%" height="52.00000%"} Dynamics of financial variables before limit hits {#dynamics-of-financial-variables-before-limit-hits .unnumbered} ------------------------------------------------- Limit hits are rare events. The dynamics of financial variables before limits can enrich our understanding of trading activities of investors around such extreme events. We investigate here the evolution of several important financial variables before stock prices hit the up-limits and down-limits. Limit hits occurred at the opening of the market are excluded from analysis, because one cannot trace the pre-event dynamics of any financial variables. We first study the average velocity of price change for all limit-hitting events in isometric intervals of price changes during bullish periods and bearish periods. For each limit-hitting event $i$, we consider the cases when the price rises above 5% or falls below -5%, compared with the close price of the previous trading day. We divide each of the two intervals $(5\%,10\%]$ and $[-10\%,-5\%)$ into 10 subintervals $((5 + 0.5m)\% ,(5.5 + 0.5m)\%]$ and $[-(5.5 + 0.5m)\%,-(5 + 0.5m)\%)$, where $m = 0,1,2,\cdots,9$. Let $\Delta{t}_{im}$ denotes the time duration for the stock price rising from $(5 + 0.5m)\%$ to $(5.5 + 0.5m)\%]$ or dropping from $-(5 + 0.5m)\%$ to $-(5.5 + 0.5m)\%]$. We consider four classes of limit hits: up-limit hits in bullish periods, up-limit hits in bearish periods, down-limit hits in bullish periods, and down-limit hits in bearish periods. For each class of limit hits, we define the dimensionless velocity of price change as follow, $$\label{Eq:period:yield:velocity} V_{m} = \frac{1}{{\frac{1}{\mathcal{N}}\sum\limits_{i = 1}^{\mathcal{N}} {\frac{{\Delta {t_{im}}}}{{\sum\limits_{m = 0}^9 {\Delta {t_{im}}} }}} }}, ~~ ~~ m = 0,1,2,\cdots,9$$ where $\mathcal{N}$ is the number of limit hits of the class under investigation. If $\Delta{t}_{im}$ is independent of $m$ for all $i$, the velocity is constant such that $V_m=9$. If the average $\langle\Delta{t}_{im}\rangle$ over $i$ increases (decreases) with $m$, $V_{m}$ decreases (increases) when the price approaches the price limit, showing evidence of a cooling off (magnet) effect. As shown in Fig. \[Fig:Prehit:Price:Acceleration\], when stock price approaches the limit price, the price movement velocity decreases, except that the velocity does not change much when the price rise is less than 9% for up-limit hits in the bullish periods. It suggests that there exists cooling-off effects when stock price gets close to the limit price. In this sense, the price limit rule is effective in the Chinese stock markets. The cooling-off effect is more remarkable in bearish periods than in bullish periods before up-limit hits and down-limit hits. However, the effectiveness of the cooling-off effect is mixed between up-limit hits and down-limit hits. The asymmetry of the cooling-off effect between bullish and bearish periods is probably caused by the no-short trading rule and the special lift forces in the bullish periods, such as the “dare-to-die corps for up-limit hits”. ![\[Fig:Prehit:Price:Acceleration\] [**Cooling-off effect of the price limit rule.**]{} (a) Comparison of price movement velocity $V_m^u$ before up-limit hits in the bullish and bearish periods. (b) Comparison of price movement velocity $V_m^d$ before down-limit hits in the bullish and bearish periods.](Fig7.eps){width="96.00000%" height="36.00000%"} We now study the evolution of another four financial measures associated with the last 100 trades till hitting the price limits. We first investigate the evolution of sizes of buyer-initiated trades and seller-initiated trades on the last 100 trades including the one pushing the price to its limit. The last trade causes the price to hit its limit, labelled as the 100th trade. If the $k$-th trade was initiated by a buyer before an up-limit hit $i$ or by a seller before a down-limit hit $i$, we denote $s_i^+(k)$ as its size and, in this case, $s_i^-(k)=0$. Alternatively, if the $k$-th trade was initiated by a seller before an up-limit hit $i$ or by a buyer before a down-limit hit $i$, we denote $s_i^-(k)$ as its size and $s_i^+(k)=0$. Hence, $s_i^+(k)$ and $s_i^-(k)$ are respectively the sizes of same-direction (momentum) and opposite-direction (contrarian) trades that move the price towards and away from the limit price. We compare up-limit hits and down-limit hits. We also consider separately bullish periods and bearish periods. The average logarithmic trade sizes of the last 100 transactions are calculated as follows, $$s^{\pm}(k) = \frac{1}{\mathcal{N}}\sum\limits_{i = 1}^{\mathcal{N}} \ln\left[s_i^{\pm}(k)\right], \label{Eq:Prehit:TradeSizes}$$ where $\mathcal{N}$ is the number of limit hits in one of the four cases (up-limit in bullish periods, down-limit in bullish periods, up-limit in bearish periods, and down-limit in bearish periods). Since the last trade is by definition the same-direction trade that push the price to the limit, we have $s^-_i(100)\equiv 0$ in all cases. For simplicity, we use the notation $s^-(100)=0$ instead of $s^-(100)=-\infty$ in the following. Figure \[Fig:Prehit:100trades:4quantities\](a) illustrates the evolution of the same-direction trade sizes $s^+(k)$ and the opposite-direction trade sizes $s^-(k)$ for different cases. The same-direction trade sizes $s^+(k)$ have similar evolutionary trajectories in all the four cases, so do the opposite-direction trade sizes $s^-(k)$. We find that $s^+(k)$ increases from $k=1$ to reach the local maximum at $k=96$ and then decreases till $k=99$ followed by a very large $s^+(100)$ pushing the price to the limit, while $s^-(k)$ decreases from $k=1$ to reach the local minimum at $k=96$ and then increases till $k=99$, ended with $s^-(100)=0$. For all $k$, we have $s^+(k)>s^-(k)$, indicating that the pushing force of the same-direction traders is stronger than that of the opposite-direction traders. The trade size difference $s^+(k)-s^-(k)$ continuously increases and reaches its local maxima at $k=96$ and then decreases till $k=99$. These findings also provide clues about the cooling down of the last three trades right before limit hits. The trade sizes $s^+(k)$ are larger when the price is close to the up-limit than to the down-limit, which is true for both bullish and bearish periods. The trade sizes $s^+(k)$ for the case of down-limit hits in bearish periods are relatively the lowest. The relative position of the four $s^+(k)$ curves provides evidence of a well-known trait of Chinese traders that they tend to buy rising stocks and hate to sell holding shares when the stock price drops. ![\[Fig:Prehit:100trades:4quantities\] [**Evolution of four important financial quantities along the last 100 trades right before limit hits.**]{} (a) Evolution of the logarithmic same-direction trade sizes $s^+(k)$ (the upper bundle) and the logarithmic opposite-direction trade sizes $s^-(k)$ (the lower bundle) for up-limit hits and down-limit hits in the bullish periods and bearish periods. (b) Evolution of the average trade-by-trade return $R(k)$. (c) Evolution of the average trade-by-trade volatility $v(K)$. (d) Evolution of the average bid-ask spread right before individual trades. For each financial quantity, we consider four cases, that is, up-limit hits in bullish periods, down-limit hits in bullish periods, up-limit hits in bearish periods, and down-limit hits in bearish periods.](Fig8.eps){width="80.00000%" height="60.00000%"} As shown in Fig. \[Fig:Prehit:100trades:4quantities\](a), the average size $s^+(100)$ of the last trade that pushed the price to hit the limit is extraordinary larger than the size of any preceding trades. We show the empirical distributions of $s^+(100)$ for the four cases. All the distributions are unimodal and have similar shapes. However, they have different peak heights. The peak is the highest for the case of up-limit hits in the bullish periods, the second largest for the case of up-limit hits in the bearish periods. We denote $p_i(k)$ the transaction price of the $k$-th trade before the $i$-th limit hit, $\left\{p^a_{i,j}(k)|j=1,\cdots,J\right\}$ and $\left\{V^a_{i,j}(k)|j=1,\cdots,J\right\}$ the prices and standing volumes at the first $J$ price levels of the sell-side limit order book right before the $k$-th trade, and $\left\{p^b_{i,j}(k)|j=1,\cdots,J\right\}$ and $\left\{V^b_{i,j}(k)|j=1,\cdots,J\right\}$ the prices and standing volumes on the first $J$ price levels of the buy-side limit order book right before the $k$-th trade. The second financial variable investigated is the average trade-by-trade return which is defined as follows, $$R(k) = \frac{1}{\mathcal{N}}\sum\limits_{i = 1}^{\mathcal{N}} \ln p_{i}(k) - \ln p_{i}(k-1), ~~ k= 1,2,3,\cdots,100. \label{Eq:Prehit:100trades:Return}$$ Figure \[Fig:Prehit:100trades:4quantities\](b) shows the evolution of average trade-by-trade return $R(k)$ before price limit hits. It is found that $R(k)$ increases superlinearly before $k$ close to 100 and then decreases sharply before up-limit hits in both bullish and bearish periods. Almost symmetrically, $R(k)$ decreases superlinearly before $k$ close to 100 and then increases sharply before down-limit hits in both bullish and bearish periods. These patterns are consistent with the behaviors of the average trade sizes, because large trade sizes usually cause large price movements [@Lillo-Farmer-Mantegna-2003-Nature; @Lim-Coggins-2005-QF; @Zhou-2012-QF; @Zhou-2012-NJP]. The third variable is the average trade-by-trade volatility which is defined as follows, $$v(k) = \frac{1}{\mathcal{N}}\sum\limits_{i = 1}^{\mathcal{N}} |\ln p_{i}(k) - \ln p_{i}(k-1)|, ~~ k= 1,2,3,\cdots,100. \label{Eq:Prehit:100trades:Volatility}$$ Figure \[Fig:Prehit:100trades:4quantities\](c) presents the evolution of average trade-by-trade volatility $v(k)$ before price limit hits. The volatility $v(k)$ before up-limit hits for both bullish and bearish periods is relatively stable when $k$ is less than about 50, then increases rapidly to reach a maximum one or two trades before limit hits, and finally drops to some extent. About five trades before up-limit hits, the volatility is higher during bullish periods than in bearish periods. The volatility about 70 trades before down-limit hits is higher than that before up-limit hits and exhibits a mild decreasing trend. The volatility increases afterwards and decreases again around $k=95$. The fourth variable investigated is the average bid-ask spread right before the $k$-th trade which is defined as follows, $$S(k) = \frac{1}{\mathcal{N}}\sum\limits_{i = 1}^{\mathcal{N}} \frac{p^a_{i,1}(k) - p^b_{i,1}(k)}{\frac{1}{2}\left[p^a_{i,1}(k) + p^b_{i,1}(k)\right]}, ~~ k= 1,2,3,\cdots,100. \label{Eq:Prehit:100trades:Spread}$$ As illustrated in Fig. \[Fig:Prehit:100trades:4quantities\](d), the four curves of the average bid-ask spread $S(k)$ decrease before $k\approx60$. The two spread curves for the bullish periods have very similar shapes. After the initial decrease, they increase during the next twenty trades or so and decrease again before the limit hits. For the curve associated with up-limit hits in bearish periods, the spread increases continuously after $k\approx75$. The comparison of the four curves is also quite intriguing. The spread before up-limit hits is narrower than that before down-limit hits, indicating higher liquidity before up-limit hits. On average, the spread is the narrowest before up-limit hits in bearish periods and the widest before down-limit hits in bearish periods. Discussion {#discussion .unnumbered} ========== Stock markets are complex systems in which humans interact by buying and selling shares. The evolutionary trajectories of stock markets are fully determined by the behaviors of human being. It is widely accepted that people in emerging markets are less skilled and more irrational and thus these markets are much riskier. Due to different factors such as imitation, global news, as well as collusive manipulation, traders may herd to push the price rise up or drop down rapidly in very short time intervals through positive feedback loops. Such kind of collective behaviors might be caused partially by the somewhat reciprocity among a small amount of traders through price manipulation, which is reminiscent of the cooperation phenomena among human beings [@Nowak-2005-Science; @Perc-Szolnoki-2010-Bs; @Perc-GomezGardenes-Szolnoki-Floria-Moreno-2013-JRSI; @Szolnoki-Perc-2013-PRX]. To cool off traders’ intraday mania to avoid the price deviating much from its fundamental value, a number of stock exchanges pose price limit rules. Empirical evidence is controversial about the presence of a magnet effect or a cooling off effect even for the Chinese stock markets [@Zhang-Zhu-2014-cnJCQUT]. Our preliminary result on the price movement velocity favors the presence of an intraday cooling-off effect for both up-limits and down-limits in both bullish and bearish periods. The evolution of other financial variables such as trade size, trade-by-trade return, trade-by-trade volatility and bid-ask spread seems to support the conclusion that the traders are cooled off right before stock prices hit the price limits, as shown by the anti-trend behavior just before hitting the price limits. It is well documented that the limit order books are thin near price limits [@Gu-Chen-Zhou-2008c-PA] and thus the liquidity is worse. When the price is pushed towards its limit, traders submit larger and larger orders and the price changes enlarge. However, the population of traders who place opposite orders to realize their gains or act as bottom fishers also increases. When the price is close to its limit, the force of opposite traders reverses the trends. In this sense, the price limit rule works in the Chinese stock markets. We have investigated the statistical properties of characteristic variables of up-limit and down-limit hits in bullish and bearish periods. 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--- abstract: 'We use machine learning to identify in color images of high-redshift galaxies an astrophysical phenomenon predicted by cosmological simulations. This phenomenon, called the blue nugget (BN) phase, is the compact star-forming phase in the central regions of many growing galaxies that follows an earlier phase of gas compaction and is followed by a central quenching phase. We train a Convolutional Neural Network (CNN) with mock “observed” images of simulated galaxies at three phases of evolution: pre-BN, BN and post-BN, and demonstrate that the CNN successfully retrieves the three phases in other simulated galaxies. We show that BNs are identified by the CNN within a time window of $\sim0.15$ Hubble times. When the trained CNN is applied to observed galaxies from the CANDELS survey at $z=1-3$, it successfully identifies galaxies at the three phases. We find that the observed BNs are preferentially found in galaxies at a characteristic stellar mass range, $10^{9.2-10.3} M_\odot$ at all redshifts. This is consistent with the characteristic galaxy mass for BNs as detected in the simulations, and is meaningful because it is revealed in the observations when the direct information concerning the total galaxy luminosity has been eliminated from the training set. This technique can be applied to the classification of other astrophysical phenomena for improved comparison of theory and observations in the era of large imaging surveys and cosmological simulations.' author: - 'M. Huertas-Company' - 'J. R. Primack' - 'A. Dekel' - 'D. C. Koo' - 'S. Lapiner' - 'D. Ceverino' - 'R. C. Simons' - 'G. F. Snyder' - 'M. Bernardi' - 'Z. Chen' - 'H. Domínguez-Sánchez' - 'C. T. Lee' - 'B. Margalef-Bentabol' - 'D. Tuccillo' title: 'Deep Learning Identifies High-z Galaxies in a Central Blue Nugget Phase in a Characteristic Mass Range' --- Introduction {#sec:intro} ============ Over the past years, we have acquired a detailed view of the statistical properties of galaxies at different cosmic epochs, thanks in particular to large scale imaging and spectroscopic surveys (e.g. SDSS; [@2000AJ....120.1579Y], CANDELS; [@Koekemoer2011]). However, establishing causal connexions between galaxy populations remains an important challenge (e.g. [@LillyCarollo2016]). This is obviously because of the timescales involved, which do not allow observations to follow the evolution of individual galaxies and also because of the degenerate link between commonly used observables and astrophysical processes. This is particularly true for the processes leading to morphological transformations of galaxies, which remain largely unconstrained despite the large quantities of available data. A fundamental question, how bulges form and grow in galaxies at different cosmic times, is still largely debated. One of the reasons is that morphological observables extracted from images are rather simplistic and have essentially remained unchanged for many years. The characterization of galaxies is essentially limited to the prominence of the bulge and disk components based on the measurement of the central density (e.g. [@2017ApJ...840...47B]), a parametric decomposition (e.g. [@1968adga.book.....S; @Peng2002]) or a ratio between enclosed light at different radii (e.g. [@1996MNRAS.279L..47A]). The interpretation of these observables to constrain an assembly history is a very degenerate problem, i.e. there are many different processes that can lead to the same observables.\ Our community is about to generate unprecedentedly large imaging datasets (e.g EUCLID, LSST, WFIRST). Hydro cosmological numerical simulations are also growing rapidly. It is thus timely to investigate alternative ways to extract a maximum amount of information from polychromatic images that might help break degeneracies with physics and improve the comparison between observations and simulated datasets. This is precisely the goal of this work. Ideally, one would like to have morphological measurements that directly correlate with some astrophysical process as predicted by theory and detected in simulations. That way, it would be possible to isolate objects from large surveys with high probability of experiencing a physical process and enable a more complete follow up. This is easily understandable for galaxy-galaxy mergers since it is a relatively well defined process associated with expected morphological features, at least at a first approximation. As a result, many efforts have been made to characterize merging galaxies from images (e.g. [@2000ApJ...529..886C; @2008ApJ...672..177L]) and to calibrate their observability timescale to constrain the merger history ([@2008MNRAS.391.1137L; @2017MNRAS.468..207S]). In that respect, it is important to calibrate with simulations that closely match the properties of the observed samples. For example, as shown in [@2015ApJ...805..181C], the morphological signatures of mergers at $z>1$ differ from those of mergers at $z\sim0$, and parametric classifications that robustly identify low-z mergers fail at $z>1$. Generalizing to other processes is less obvious since one needs to find the appropriate tracers from the multi wavelength pixel distribution. In recent years, there has been significant progress in the image processing community with the emergence of the so-called unsupervised feature learning techniques or deep learning (DL). These algorithms allow the user to automatically extract observables (or features) from the pixel space without any a-priori dimension reduction. As in many other disciplines, deep learning is rapidly being adopted in astronomy as well. It has been successfully used for several standard classification (e.g.[@Huertas2015; @2015MNRAS.450.1441D; @2017arXiv171105744D]) and regression problems (e.g [@2017arXiv171103108T]). We aim at investigating here an alternative way of using these advanced machine learning techniques to extract more physically relevant features from images and help establish a more solid link between theory and observations.\ In this exploratory proof-of-concept work, we explore whether deep learning can be used to detect a phenomenon dubbed as blue nugget (BN), recently found in numerical simulations of high redshift galaxies. Indeed these cosmological simulations [@2015MNRAS.450.2327Z; @2016MNRAS.458..242T; @2016MNRAS.457.2790T] reveal that a large fraction of the simulated galaxies undergo events of gaseous compaction, triggered, e.g., by mergers or counter-rotating inflowing streams, which leads to a central *blue nugget* (BN) at a characteristic stellar mass of $10^{9.2-10.3} M\odot$. The BN phase in turn triggers a central gas depletion and central quenching of star formation, sometimes surrounded by an extended, freshly formed, gaseous, star-forming ring/disc. Most of the structural, kinematic and compositional galaxy properties undergo significant transitions as the galaxy evolves through the BN phase ([@2015MNRAS.453..408C], Dekel et al., in prep.). One way to investigate whether these gaseous compactions are frequent in the observed galaxies would be then to directly detect features in the data (stellar distribution in our case) unequivocally associated with the BN phase. This is what we attempt in this paper. One main advantage of high resolution numerical simulations over, for example, semi-analytical models or low-resolution large volume simulations, is that we can use them to generate realistic *observed* simulated images for which the evolution history is known by construction (e.g. [@2015MNRAS.451.4290S]). One can therefore isolate a sample of simulated galaxies in the BN phase, as well as in the pre-BN or post-BN phases. In this work, we use state-of-the art zoom-in cosmological simulations with high spatial resolution [@2014MNRAS.442.1545C] to generate mock images as observed by HST of galaxies in a BN phase. We then use this dataset to train deep neural nets and explore whether the network is able to automatically find morphological proxies associated with the different phases in the observed mock data. We then apply the trained network to observed CANDELS data.\ The paper proceeds as follows. Sections \[sec:sims\] and \[sec:data\] describe the simulations and data used in this work. The main methodology is discussed in section \[sec:train\]. We show the main results on simulations and observations in sections \[sec:res\] and \[sec:real\] respectively. Simulations {#sec:sims} =========== Main properties of the simulations ---------------------------------- We use a set of zoom-in hydro cosmological simulations of 35 intermediate mass galaxies among which 31 are used in this work. The typical stellar mass of the simulated galaxies at $z\sim2$ is $10^{10}$ solar masses as shown in table \[tbl:table1\]. This is part of the VELA simulation suite which has been described and analyzed in several previous works [@2014MNRAS.442.1545C; @2015MNRAS.453..408C; @2015MNRAS.450.2327Z; @Tacchella2015; @2016MNRAS.458.4477T; @2016MNRAS.458..242T]. We refer the reader to the aforementioned works for a detailed description of the simulations. We summarize here only the most relevant properties. The initial conditions for the simulations are based on dark matter haloes that were drawn from dissipationless N-body simulations. The simulations were run with the AdaptiveRefinement Tree (ART) code [@1997ApJS..111...73K; @2003ApJ...590L...1K; @2009ApJ...695..292C] and the maximum resolution is $17-35$ pc at all times, which is achieved at densities of $\sim 10^{4}-10^{3}$cm$^{-3}$. The code includes several physical processes relevant for galaxy formation: gas cooling by atomic hydrogen and helium, metal and molecular hydrogen cooling, photoionization heating by the UV background with partial self-shielding, star formation, stellar mass loss, metal enrichment of the ISM and stellar feedback. In particular, the high spatial resolution allows tracing the cosmological streams that feed galaxies at high redshift, including mergers and smooth flows, and they resolve the Violent Disk Instabilities (VDI) that governs high-z disc evolution and bulge formation [@2009ApJ...703..785D]. This is important for this work focused on the growth of bulges and the reason why this small set of simulations is preferred to larger but lower resolution runs like Illustris. We recall that the gravitational softening for baryons in the Illustris series is of the order of $\sim 1kpc$ which means that any physical process that acts in smaller scales is unresolved. This is the case of the BN phase explored in this work. However, as with all state-of-the art numerical simulations, the VELA simulations suffer from several limitations specially related to sub-grid physics. Like other simulations, the treatment of star formation and feedback processes still depends on uncertain recipes. The code assumes indeed a SFR efficiency per free fall time without following in detail the formation of molecules and the effect of metallicity on the SFR [@2012ApJ...753...16K]. Additionally, no AGN feedback is yet included in the run used in this work. As a result, the full quenching observed in the data is not reached in many galaxies by the end of the simulations at $z\sim1$. Since we are more interested here in the BN phase that occurs when the galaxy is still star-forming, we do not expect that AGN will have a big impact on our results. However a color mismatch between simulated and observed galaxies might be expected. Besides that, as shown in [@2014MNRAS.442.1545C] and [@2016MNRAS.458..242T], the SFRs, gas fractions, and stellar-to-halo mass ratios are all close to the constraints imposed by observations, providing a better match to observations than earlier simulations.The uncertainties and any possible remaining mismatches by a factor of order 2 are comparable to the observational uncertainties. We stress that we are fully aware that the simulations present many limitations and that they are still very far from capturing all the complex physics of galaxy formation. This is mainly why the present work needs to be considered as a proof-of-concept work in that respect. However, we are at a stage at which we can produce fairly realistic galaxies that capture some of the physical processes governing the assembly history, and we have good reasons to think that this will be improved in the future. This enables a comparison with observations in a more general way that we explore in this work. Mock *Candelized* images {#sec:mock_images} ------------------------ The full output of the simulation is saved at many time steps and analyzed at steps of $\Delta a=0.01$ of the expansion factor, which roughly correspond to $\sim 100$ Myrs at $z\sim2$. For every snapshot between $z\sim4$ and $z\sim1$ , we generate mock 2D images as they would be observed by the HST. They are generated using the radiative transfer code <span style="font-variant:small-caps;">sunrise</span>[^1] [@2006MNRAS.372....2J; @2010NewA...15..509J; @2010MNRAS.403...17J] by propagating the light of stars through the dust. We refer to [@2015MNRAS.451.4290S] for details on the procedure followed. Very briefly, a spectral energy distribution (SED) is assigned to every star particle in the simulation based on its mass, age, and metallicity. The dust density is assumed to be directly proportional to the metal density predicted by the simulations. We set a dust-to-metals mass ratio of 0.4 (e.g. [@1998ApJ...501..643D; @2002MNRAS.335..753J]), and the dust grain size distribution from updated by [@2007ApJ...657..810D]. <span style="font-variant:small-caps;">Sunrise</span> then performs dust radiative transfer using a Monte Carlo ray-tracing technique. As each multiwavelength ray emitted by every star particle and HII region (according to its SED) propagates through the ISM and encounters dust mass, its energy is probabilistically absorbed or scattered until it exits the grid or enters one of the viewing apertures (*cameras*). The output of this process is then the SED at each pixel in all cameras. For this run we set 19 cameras from which five are fixed with respect to the angular momentum vector of each galaxy, seven are fixed in the simulations coordinates and the remaining seven are fully random at each time step. The camera numbers are summarized in table \[tbl:table2\]. Finally, from these data cubes, we create raw mock images by integrating the SED in each pixel over the spectral response functions of the CANDELS WFC3 filters ($F160W$, $F125W$ and $F105W$) in the observer frame. Images are then convolved with the corresponding HST PSF at a given wavelength. We finally add a random real noise stamp taken from the CANDELS data. This ensures that the galaxies are simulated at the same depth than the real CANDELS data and also that the correlated noise from the HST pipeline is well reproduced. We call this process *CANDELization*. For each 3D snapshot ($\Delta a=0.01$), we therefore generate 19 different 2D images corresponding to the 19 different camera orientations. The resulting dataset corresponds to approximately $\sim10000$ images in each of three filters. Even if the CANDELS filters probe the optical rest-frame up to $z\sim3$, we included galaxies up to $z\sim4$ since the most intense compaction events tend to happen at higher redshift in the VELA simulations. Given that the gas fractions (stellar to halo mass relations) are slightly underestimated (overestimated) in the simulations as previously stated, including higher redshift is justified and increases the size of our training set. We have checked however that the main results of the paper remain unaltered if only galaxies up to $z\sim3$ are used. We emphasize that the same procedure has been used to generate mock JWST galaxies in the different filters and therefore a similar analysis as the one presented in this work can be applied to this dataset in order to prepare JWST observations. camera number orientation --------------- --------------------------------------------------- cam00/02 Angular momentum face-on (opposite directions) cam01/03 Angular momentum edge-on (opposite directions) cam04 Angular momentum 45 degrees cam05/06/07 Fixed to x,y and z axis simulation box cam08-11 Random (same simulation coord. for all snapshots) cam12-18 Fully random : Explanation of the 19 camera orientations used to generate mock 2D images from the simulations.[]{data-label="tbl:table2"} Data {#sec:data} ==== We also use HST observational data to test our model in section \[sec:real\]. We use CANDELS images in the three infrared filters (F105W, F125W, F160W) from the 2 GOODS fields (North and South, [@Grogin2011; @Koekemoer2011]). The selection is based on the morphological catalog presented in [@Huertas2015], which is essentially a selection of the brightest galaxies (F150W&lt;24.5) from the official CANDELS catalogs ([@Guo2013], Barro et al. 2017). This is required to have enough S/N to measure morphological information from images. For this work, we select only galaxies in the redshift range $1-3$ to match the simulated redshift range. As shown in [@Huertas2016], the sample is mass complete down to $10^{9}$ solar masses at $z\sim1$ and $10^{10}$ at $z\sim3$. We restrict our analysis to galaxies more massive than $10^9$ solar masses to have enough statistics and match the typical stellar masses from the simulations. The sample might therefore suffer from incompleteness at high redshift. This is not critical for the illustrative purpose of this work. In addition to the reduced images, we also use official CANDELS redshifts [@2013ApJ...775...93D] which are a combination of photometric redshifts computed with several codes and spectroscopic when available. Stellar masses and star-formation rates from SED fitting are also used. Stellar masses are computed through SED fitting using the best redshift adopting a [@2003PASP..115..763C] IMF. SFRs are computed by combining IR and UV rest-frame luminosities ([@1998ApJ...498..541K; @2005ApJ...625...23B]) with [@2003PASP..115..763C] IMF (see [@2011ApJS..193...13B] for more details). The following relation was used: $SFR_{UV+IR}=1.09\times10^{-10}(L_{IR}+3.3L_{2800})$. Total IR luminosities are obtained using [@2001ApJ...556..562C] templates fitting MIPS $24\mu m$ fluxes and applying a *Herschel based* recalibration. For galaxies undetected in $24\mu m$, SFRs are estimated using rest-frame UV luminosities [@2011ApJ...742...96W]. We also compute for the selected dataset the central mass density ($\Sigma_1$) following the methodology of [@2017ApJ...840...47B]. Methods: Training the network {#sec:train} ============================= Training set: using the simulation metadata to label images {#sec:labels} ----------------------------------------------------------- The final goal is to train a deep neural network to identify, from the mock images, the BN phase (and consequently the pre and post-BN phases as well). As put forward by previous analysis of the same simulated dataset [@2015MNRAS.450.2327Z], almost all the simulated galaxies seem to evolve in three characteristic phases. They go from diffuse to compact star-forming objects through wet gas compaction to then quench in the central regions and build a central bulge that will in most of the cases rebuild a surrounding stellar disk. We notice that the intensity of the compaction depends on stellar mass, and while most of the simulations go through a BN phase, only the most massive become compact star-forming galaxies. As part of the training set, we then first define these 3 phases for all the galaxies in the simulation. The identification of the 3 phases is performed in an individual basis for each galaxy using the gas density evolution in the central galactic regions as explained in [@2015MNRAS.450.2327Z] and Dekel et al. (in prep). Basically we identify the *peak of the BN phase* as the time at which the gas density in the central kpc is maximum. We define the end of the BN phase when the central stellar density stops increasing, which is a signature that star-formation has been quenched in the center of the galaxy. The onset of the BN phase is considered to start when the central gas density starts to increase toward the BN peak. Naturally, this is more complicated than selecting the peak. In our current approach the selection is done by eye using also the 2D projection of the gas density to confirm the choice. Figure \[fig:comp\_def\] shows the cold gas and stellar mass evolution in the central kpc for some galaxies for illustrative purposes. We also show the dark matter content in the central kpc. The key take-away message from these plots is that compaction is not always well defined and that it comes in many different flavors. There are for instance some *clean* cases as VELA12 in which there is a single peak of the gas mass. However, there are other cases such as VELA25 for which the peak is not so pronounced and identifying the boundaries of the BN phase is not obvious and somewhat arbitrary. Notice also that many galaxies experience several BN phases as for example discussed in [@2016MNRAS.457.2790T]. In this work, we define a maximum of 3 BN phases for each galaxy as shown in table \[tbl:table1\]. Table \[tbl:table1\] summarizes indeed the redshifts of the BN phase of all galaxies analyzed. This is to say that the network that will be trained needs to somehow capture this heterogeneity in the process. It is important to keep this in mind when analyzing the results.\ As can be seen in table \[tbl:table1\], in the simulations, the BN phase tends to happen at a characteristic galaxy stellar mass $\sim 10^{9.2-10.3}M_\odot$ (e.g. [@2015MNRAS.450.2327Z]). Given the existing correlation between mass and luminosity, this implies that there is a brightness gradient between pre-BN, BN and post-BN, with pre-BN galaxies being generally fainter than post-BN. The difference in luminosity also implies a difference in S/N when the HST noise is added. A deep learning approach, as the one used in this work, has the unique power to automatically extract the optimal tracers from the data to minimize the classification error. It also implies a risk since the network can potentially use any available information. In our case, given the properties of the training set, there is therefore a potential risk that the network uses the S/N difference existing between the different phases to classify. Since we do not want the network to learn based on S/N but rather learn characteristic features of the BN phase, we artificially shuffle the magnitudes of the galaxies given to the network. To do so, before adding the noise (see section \[sec:mock\_images\]), we associate a random magnitude to all snapshots in the $F160W$ filter ($19-25$ in order to match the CANDELS distribution). This way, galaxies in the different phases have similar luminosities and S/N distributions. By doing so, the characteristic mass information is also washed out preventing the network from using that information to learn. We will discuss the effect of this choice in section \[sec:real\]. We remark that all other properties are kept unchanged. It includes obviously the spatial distribution of pixels which measure the degree of compactness and also the relative luminosities in each filter which is correlated with the SFR. We thus use this 3-class classification (pre-BN, BN or post-BN) to associate a unique label to every simulated image. Pre-BN includes all galaxies before experiencing any compaction event i.e. with a redshift larger than the maximum of ($z^1_{onset}, z^3_{onset}, z^3_{onset}$). Galaxies in the BN phase are the ones with redshift between $z^y_{onset}$ and $z^y_{post}$, with $y={1,2,3}$. Finally, all remaining images are labelled as post-BN. So galaxies with several compaction events are classified as *post-BN* between two events. As a result of this labelling process, every mock image has an associated label corresponding to its evolutionary phase. The final dataset consists therefore of $\sim 10000$ labelled images with the simulation assembly history that will be used to train and test a convolutional neural network model. Figure \[fig:ex\_stamps\] shows some random example stamps of galaxies in the three phases in the HST/WFC3 F160W filter. Pre-BN galaxies generally look smaller and post-BN tend to have a diffuse disk structure. However, no obvious visual difference is apparent. This underlines the challenge of this work, which is to train a CNN capable of distinguishing between the different phases. ------------ ---------------- -------------- ---------------- -------------- ---------------- -------------- ------------------- ------------------- simulation $z^1_{onset} $ $z^1_{post}$ $z^2_{onset} $ $z^2_{post}$ $z^3_{onset} $ $z^3_{post}$ $Log M_*/M_\odot$ $Log M_*/M_\odot$ $z=z_{comp}$ $z=2$ VELA01 1.86 1.38 – – – – 10.05 9.39 VELA02 1.70 1.00 – – – – 9.72 9.32 VELA03 3.00 1.94 1.27 0.96 – – 9.47 9.70 VELA04 2.23 1.63 1.50 1.17 – – 9.18 9.07 VELA05 1.38 1.08 – – – – 9.47 9.09 VELA06 5.25 3.17 2.57 1.86 – – 9.60 10.42 VELA07 3.55 2.57 4.88 3.35 – – 10.39 10.83 VELA08 2.23 1.50 0.96 0.69 – – 9.79 9.79 VELA09 4.00 3.00 1.63 1.33 – – 9.73 10.09 VELA10 3.17 2.13 1.44 1.13 – – 9.59 9.83 VELA11 4.00 2.85 2.12 1.70 – – 9.67 10.05 VELA12 4.56 3.17 – – – – 9.98 10.33 VELA13 2.85 2.03 – – – – 9.76 10.06 VELA14 2.33 1.56 – – – – 10.26 10.19 VELA15 2.70 2.13 1.70 1.38 – – 9.70 9.77 VELA17 7.33 3.55 3.76 2.57 – – 9.63 – VELA19 9.00 4.56 2.70 2.13 – – 9.75 – VELA20 4.00 2.85 5.67 3.76 – – 10.33 10.62 VELA21 3.55 2.57 4.88 3.35 7.33 4.56 10.51 10.65 VELA22 4.88 3.55 – – – – 10.02 10.67 VELA25 2.33 1.86 3.76 2.57 1.86 1.50 9.89 9.91 VELA26 3.17 2.13 5.25 3.55 – – 9.82 10.25 VELA27 2.23 1.70 3.35 2.57 – – 9.90 10.01 VELA28 1.63 1.22 – – – – 9.71 9.51 VELA30 5.67 3.17 – – – – 9.87 10.25 VELA32 7.33 4.00 – – – – 9.71 10.52 VELA33 4.88 3.00 3.35 2.45 2.33 1.78 9.61 10.73 VELA34 3.00 1.78 4.26 2.70 – – 10.06 10.32 ------------ ---------------- -------------- ---------------- -------------- ---------------- -------------- ------------------- ------------------- $\begin{array}{c c} \includegraphics[width=0.45\textwidth]{plots/comp_V12.jpg} & \includegraphics[width=0.45\textwidth]{plots/comp_V30.jpg} \\ \includegraphics[width=0.45\textwidth]{plots/comp_V20.jpg} & \includegraphics[width=0.45\textwidth]{plots/comp_V25.jpg}\\ \end{array}$ ![image](plots/stamps_paper.png){width="95.00000%"}\ Architecture ------------ We use a very simple sequential CNN architecture with only 3 convolutional layers followed by 2 fully connected layers implemented in Keras[^2] with a Theano backend (figure \[fig:arch\]). The main reason to use a relatively shallow network is the limited size of the training set. The architecture is inspired by previous configurations which were successful in detecting strong lenses in space-based images [@2018arXiv180203609M] and also for classical morphological classification [@2017arXiv171105744D]. We then add 2 fully connected layers to perform the classification. The last layer has a *softmax* activation function to ensure that the 3 outputs behave like probabilities and add to one. We use a *categorical crossentropy* as loss function and the model is optimized with the *adam* optimizer. The network is fed with images (fits format) of fixed size ($64\times64$ pixels) with 3 channels corresponding with the 3 main NIR CANDELS filters (F160W, F125W and F105W). We also tried to include bluer filters ($F850LP$), but the results do not change significantly. For simplicity in this illustrative work, we thus used the 3 redder filters since the pixel scale is the same and hence no interpolation is required. In principle the number of filters could be increased without any significant modification of the methodology. The input size is a trade-off between properly probing the galaxy outskirts ($\sim30$ kpc in the redshift range $1-3$) and having a small enough number of input parameters to prevent overfitting. In addition to this, we also use standard techniques to avoid overfitting at first order. Firstly after each convolutional layer we apply a 50% dropout. Secondly, we include a Gaussian noise layer at the entrance of the network to avoid that the model learns from the noise pattern given that our training set is small. Finally, we use real-time data augmentation. Images are randomly rotated (within 45 degrees), flipped and slightly off centered by 5 pixels maximum at every iteration so that the network does never see exactly the same image. ![image](plots/architecture.png){width="95.00000%"}\ Training and validation strategy -------------------------------- One obvious limitation we face in this work is that our training dataset is made of only $\sim28$ galaxies. Even though we increase the number of available images by using different camera orientations as well as data augmentation, there is a potential risk that the network learns how to identify the different phases for this particular set of galaxies without generalizing. To avoid this situation, we have designed a custom training strategy which slightly differs from the classical approaches typically used in machine learning. Among the 28 galaxies, we use 24 galaxies for training (i.e. $\sim9000$ images), 2 for real-time validation during the training and 2 additional completely independent galaxies for testing at the end of the training process. It is important to keep in mind that, when we say 2 galaxies it does not mean 2 images. Each galaxy corresponds to the full assembly history of the galaxy between $z=4$ and $z=1$ with 19 images at each time step. Therefore the test and validation sets contain $\sim1000$ galaxies each. We then train for a maximum of 250 epochs. The novelty is that every 50 epochs we move 2 galaxies from the training set to the validation sample and add the validation galaxies to the training. This helps the network not to overfit on the first sample of 24 galaxies while training for enough number of epochs to enable convergence. The 2 test galaxies are obviously never used in the process. Finally, in order to have more than 2 galaxies to test the classification accuracy, we repeat 5 times the training procedure just described, using two different galaxies for the test sample at every run. The final test dataset thus contains 10 galaxies, classified with 5 slightly different models. Figure \[fig:learn\_histo\] illustrates the learning history parametrized by the evolution of the accuracy as a function of the number of epochs of one of the five runs for illustration purposes. We plot the accuracy computed on the training and validation datasets. As expected the training curve monolithically increases and reaches roughly an accuracy of $80\%$. Notice however some small discontinuities every 50 epochs corresponding with the modification of the training set. The fact that the discontinuity is small suggests that small modifications of the training sample do not significantly alter the network performance. In other words, there is no over-fitting. The validation curve shows a particular behavior. This again is consequence of the adopted training strategy. Every 50 epochs there is a clearly noticeable jump. The break is larger than for the training because the validation is only made of 2 galaxies and the sample is fully changed every 50 epochs. So the break somehow reflects the accuracy variation between galaxies which can go from 100% for some galaxies to $\sim 60\%$. As previously stated, compaction is not a very well defined process and some galaxies have complex assembly histories with multiple BN phases. The red curve also presents large jumps between epochs. This is also most probably a consequence of the size and redundancy of the sample. Given that there are 19 images per snapshot a change in the classification of a few snapshots implies big changes in the accuracy value. ![Learning history resulting from the strategy described in the text. The blue solid line shows the accuracy on the training set and the red solid line is the accuracy for the validation set. Every 50 epochs the validation and training datasets are modified which explains the discontinuities. The accuracy on the validation is generally unstable because it is only made of 2 galaxies. See text for details. []{data-label="fig:learn_histo"}](plots/learning_curve_lumind.jpg "fig:"){width="45.00000%"}\ Results {#sec:res} ======= In this section we analyze the classification accuracy. We use for that purpose the test dataset (10 galaxies) which was not used in the training process (see section \[sec:train\]) throughout all the section. Detection of BNs ---------------- We first analyze the average accuracy of the trained model to detect pre-BNs, post-BNs and BNs. The global accuracy, defined as the fraction of images correctly classified, computed on the test dataset is $\sim70\%$, which means that $30\%$ of the objects are misclassified. This is certainly not very high. Recall however that there is a significant amount of redundancy in the test set. It is helpful to look into more detail to better understand what is going on before drawing conclusions. We first compute a standard confusion matrix showing the relation between input and output classes (figure \[fig:conf\_matrix\]) for different probability thresholds. At the lower probability threshold (0.5) most of the confusion comes from true pre-BN (or post-BN) that are predicted as BN. This is probably because, as previously stated, the compaction event is not very well defined. The duration and intensity strongly depend on the galaxy. As expected, the degree of contamination decreases when a higher probability threshold is used to select galaxies. At the highest threshold (0.8) 25% of true BNs are predicted to be post-BN. In fact one should keep in mind that the test set contains snapshots in steps of $\Delta_a=0.01$. A galaxy might be mis-classified as post-BN just before the compaction event ends for example or where there are multiple compactions closely followed in time, reducing the accuracy of the classification. However the classification might still be meaningful. To investigate this further, in figure \[fig:pred\_time\] we plot the output probabilities for a subset of individual galaxies from the test sample as a function of time. In this figure, the lines show the average probability over all camera orientations at a given snapshot. The shaded regions show the scatter due to different camera orientations. For illustration purposes, we show three cases with increasing complexity. The first example (VELA30) has one single intense BN phase. VELA11 is less massive and has 2 events of smaller intensity. Finally VELA08 is a low-mass galaxy with a very weak compaction.These three examples bracket the diversity of assembly histories the network needs to capture. As can be seen, there is a good correlation between the evolution of the probability values and the evolutionary phase. We observe that typically the probability of pre-BN tends to decrease before the compaction event, while the compaction probability increases. Towards the end of the BN phase, the probability of post-BN increases. This is true even for galaxies with complex assembly histories. This result indicates two main things. Firstly, it shows that the machine has learned somehow that there is a sequential order between the 3 phases. This is not obvious since all images were randomly included in the training process with random luminosities and, as seen in table \[tbl:table1\], the BN phase can happen at very different redshifts and can have very different durations. Secondly, it shows that despite the relatively low global accuracy, the confusion seems to come essentially from the snapshots taken at the transition phases. This is important because it means that when the machine misclassifies it is not fully random. The misclassification therefore is a reflection of the difficulty to define the different phases. It is also worth noticing that the scatter due to different camera orientations is generally not large ($\sim0.1-0.2$ in terms of probability). It suggests a mild impact of the projection in the classification accuracy. $\begin{array}{c} \includegraphics[width=0.45\textwidth]{plots/confusion_matrix_0-5.jpg} \\ \includegraphics[width=0.45\textwidth]{plots/confusion_matrix_0-6.jpg} \\ \includegraphics[width=0.45\textwidth]{plots/confusion_matrix_0-8.jpg} \\ \end{array}$ $\begin{array}{c c} \includegraphics[width=0.41\textwidth]{plots/compaction_summaryVELA30.jpg} & \includegraphics[width=0.41\textwidth]{plots/comp_V30.jpg} \\ \includegraphics[width=0.41\textwidth]{plots/compaction_summaryVELA11.jpg} & \includegraphics[width=0.41\textwidth]{plots/comp_V11.jpg} \\ \includegraphics[width=0.41\textwidth]{plots/compaction_summaryVELA08.jpg} & \includegraphics[width=0.41\textwidth]{plots/comp_V08.jpg} \\ \end{array}$ Impact of camera orientation ---------------------------- We investigate this further in figure \[fig:conf\_matrix\_cam\], which shows the confusion matrix divided by camera orientation. Despite some statistical fluctuations, no significant differences are appreciated as already suggested by the results shown in figure \[fig:pred\_time\]. This is also quantified in figure \[fig:acc\_camera\], which shows the global accuracy as a function of the camera number (see table \[tbl:table2\] for an explanation of the different numbers). The figure confirms that there is no systematic trend with the orientation. The global accuracy increases equally for all cameras when the probability threshold is increased. $\begin{array}{c} \includegraphics[width=0.45\textwidth]{plots/confusion_matrix_cam00.jpg} \\ \includegraphics[width=0.45\textwidth]{plots/confusion_matrix_cam01.jpg} \\ \includegraphics[width=0.45\textwidth]{plots/confusion_matrix_cam17.jpg} \\ \end{array}$ ![Measured accuracy on the test dataset as a function of the camera orientation. The numbers indicate the orientation (see Table 2). The different colors indicate different probability thresholds as labeled. The accuracy does not depend on the camera orientation. []{data-label="fig:acc_camera"}](plots/acc_ortientation.jpg "fig:"){width="45.00000%"}\ Calibration of observability timescales {#sec:obs_time} --------------------------------------- In fact, in view of applying the model to real data, probably the most interesting property to investigate is whether we can calibrate the observability timescales of the features learned by the classifier. In other words, what is the typical time window in which the network detects BNs. This is important because it allows us to better interpret the classification in terms of an evolutionary sequence and also to compute a *BN rate* from the observations as usually done for mergers typically. To do so, we take the test sample and classify all galaxies in the 3 classes according to the output probabilities. We simply add each image to the class of maximum probability and require that the probability value is larger than 0.5. We then compute, for each galaxy, the time difference with the main BN phase (computed as a fraction of the Hubble time at the BN peak, i.e $1/H(t)$, $H(t)$ being the Hubble constant. Figure \[fig:time\_comp\] shows the histograms for the 3 classes. We confirm that the 3 classes tend to probe a different regime although with some overlap as expected from the results of the previous sections. Pre-BN galaxies are on average selected $\sim0.40/H(t)$ before the event and post-BN galaxies are typically observed $\sim0.80/H(t)$ after the compaction. The galaxies classified are centered on the BN phase ($0.05\pm0.3$ Hubble times). Although there is some overlap between the different histograms, it is worth noticing that all galaxies which passed the BN phase by more than half a Hubble time are classified as post-BN galaxies. Also there are no galaxies classified as BN or pre-BN objects for which the event is more than $\sim0.5$ Hubble times away. This means that our classifier can indeed establish some temporal constraints regarding the BN phase based only on the stellar distributions. This is extremely important because it shows that there is a temporal sequence implied in the classification. So when applied to real data one can more easily interpret the results in terms of evolution as will be discussed in section \[sec:real\]. ![Observability of the BN phase with the calibrated classifier. The histograms show the distributions of time (relative to the Hubble time at the time of the peak of the BN phase). The blue, green and red histograms show the pre-BN, BN and post-BN phases. The dashed vertical lines show the average values for each class with the same color code. Despite some overlap, the classifier is able to establish temporal constraints on the different phases. The darker regions indicate overlapping histograms. []{data-label="fig:time_comp"}](plots/comp_time_test.jpg "fig:"){width="45.00000%"}\ Inside the network ------------------ An important caveat of the machine learning approach presented above is that it somehow behaves as a black box. It is thus difficult to precisely determine what are the features the machine is using to decide the output classification. This is a general problem for all deep learning applications. However, there exist more and more *network interrogation* techniques which identify the pixels in the input image that most contributed to the final classification. One recent method is called integrated gradients [@2017arXiv170301365S]. It is based on the measurement of the differences between gradients computed by the network in an input image as compared to a test image (usually a blank image with only zeros). We tested this method in our model and computed the integrated gradients for some of the galaxies. Figure \[fig:int\_grad\] shows one example for each class. The interpretation is not straightforward. However some useful information can be extracted from this exercise. It is interesting to see that the model automatically segments all the pixels belonging to the galaxy and takes the decision based on all the galaxy pixels. It also means it understood there is no information in the noise and confirms that the model is not over-fitting on the noise pattern. Also, as pointed out in previous works, after the BN phase a gaseous disk is often built in the simulations [@2015MNRAS.450.2327Z; @2016MNRAS.458..242T]. The bottom panels of the figure show clearly that the machine detects the diffuse disk component even if faint and probably uses this information to make the decision concerning the post-BN and sometimes the BN phase. For galaxies in the BN phase, the relevant pixels are more concentrated in the center since the galaxies are generally more compact as the obvious signature of this phase. It is also worth noticing that the gradient tends to have values of different sign in the center and in the outskirts as if the machine was using difference in flux between the center and the outskirts to classify. This is expected since the compaction event is by definition accompanied by a burst of central star-formation and the sSFR profiles evolve from decreasing to rising, indicating quenching outside-in in the pre-BN phase and inside-out in the post-BN phase [@2016MNRAS.458..242T]. The model is capturing all these correlations automatically. This is the strength of the presented methodology. Although the information that can be extracted from integrated gradients is quite limited at this stage, it is reasonable to think that interrogation techniques will become more advanced, and therefore there is potentially information that can be learned from a post-processing of the model outputs in the future. ![image](plots/image_2.png){width="90.00000%"}\ Identifying blue nuggets in the observations {#sec:real} ============================================ We now apply the model to the HST/CANDELS sample presented in section \[sec:data\]. We simply cut stamps around the selected galaxies in the three infrared filters ($F160W$, $F125W$, $F105W$) and classify them into three classes using the trained models. Since 10 models were produced (see section \[sec:train\]), we use each of them to classify all galaxies. Each real galaxy has therefore 10 different classifications using slightly different models. We then compute the average probability to increase the robustness of the classification. We stress that there is a general good agreement between the different models which confirms that the classification does not strongly depend on the specific subset of simulated galaxies used for training. The typical scatter in the probability values is of the order of $\sim0.1$.\ ![image](plots/CANDELS_stamps_paper_scale.png){width="95.00000%"}\ The first thing to notice is that the classification applied to real data returns objects with high probability values in the 3 classes. The fraction of galaxies with all probabilities lower than 0.5 is only 2% of the total sample. It means that the model found galaxies that resemble the galaxies in the simulation with high confidence.This reflects that the simulated galaxies are fairly similar to the observed ones and that the network found characteristic features learned in the simulations, in the CANDELS observations. Figure \[fig:ex\_stamps\_CANDELS\] shows some example stamps of observed galaxies in the three phases. It is not obvious to establish what would happen if galaxies from the training were very different from real datasets. This will be explored in future work. In order to have a first idea of how the network would behave when confronted to very different objects, we perform a simple exercise. We take the real observed galaxies from CANDELS and first randomly shuffle the central pixels of the galaxy and then shuffle all the pixels in the galaxy (inner+outskirts). This creates two fake datasets with different degrees of perturbation which are given to the network. Figure \[fig:p\_shuffle\] shows the probability distributions for the 3 classes when these fake datasets are given. The figure shows that the first effect of shuffling the center is that the number of galaxies with a compaction probability larger than $0.5$ almost drops to zero. This is somehow expected as most of the compaction features are naturally seen in the central parts. It confirms that the network is significantly using this information to classify. Since the probabilities need to add to 1, central shuffling provokes also an increase in the number of galaxies with large probability of post-BN. Given that post-BNs tend to be more extended, the fact of shuffling the central pixels pushes the network to boost the post-BN probability since it focuses on the outer pixels. However, the values remain low ($\sim0.6$) indicating a moderate level of confidence. When both outskirts and inner pixels are shuffled, both probability distributions, BN/post-BN, significantly narrow and peak at $\sim0.4-0.5$, meaning that the network is not able to clearly assign galaxies to classes. This exercise shows that the probability distributions somehow reflect the similarity between the simulations and the observations. We notice however, that even in the shuffled images, there is a fraction of galaxies with high post-BN probabilities. A visual inspection shows that these are bright galaxies for which the shuffling has pushed bright pixels towards very large distances. The network most likely interprets this as a very extended disk. The fact that the distributions on CANDELS galaxies resemble to the ones obtained on the test simulated sample (red solid/dashed lines in figure \[fig:p\_shuffle\]), suggests therefore that simulated and observed galaxies look similar to the network. This allows us to push the analysis a bit further by exploring the properties of galaxies in the three phases (BN, post-BN and pre-BN) in the observations. $\begin{array}{c c c} \includegraphics[width=0.33\textwidth]{plots/P_pre_sh.jpg} & \includegraphics[width=0.33\textwidth]{plots/P_comp_sh.jpg} & \includegraphics[width=0.33\textwidth]{plots/P_post_sh.jpg}\\ \end{array}$ A characteristic mass range for the BN phase {#sec:cmass} -------------------------------------------- In figure \[fig:mass\_evol\] we first look at the stellar mass distributions of CANDELS galaxies in the three different phases. Recall that the simulations used for training stop at $z\sim1$, so we only show galaxies above this redshift in the observations. The abundance of galaxies in different phases strongly depends on stellar mass. Pre-BN galaxies tend to increase at low stellar masses ($M_*/M_\odot<10^{9-9.5}$) and post-BN galaxies dominate at large stellar masses ($M_*/M_\odot>10^{10.5}$). BNs are most frequent at intermediate masses and peak at $\sim10^{9.2-10.3}$. Interestingly the position of the peak seems to be relatively independent of redshift with a small tendency to move towards lower masses at lower redshifts. We notice that at this characteristic stellar mass, the CANDELS dataset is affected by incompleteness as indicated by the vertical line in the plots. This should not affect the result in the sense that there are no reasons to think that post-BN galaxies are more difficult to detect. This characteristic mass for compaction is a prediction from the VELA simulations as first reported in [@2015MNRAS.450.2327Z] and [@2016MNRAS.458.4477T] and also reflected in table \[tbl:table1\] (see also [@2016MNRAS.457.2790T] and Dekel et al. in prep.). The fact that it appears clearly in the observations confirms that the network is automatically extracting the correlations existing in the simulations. It is worth recalling that the luminosity has been removed from the training set which ensures that the network is not classifying based on luminosity that is directly correlated with the stellar mass. The network is necessarily using other information such as spatial distribution, shape or color to identify the different phases. The characteristic mass naturally emerges in the observations. The network successfully identifies a population that resembles simulated galaxies experiencing compaction in the feature space learned and these galaxies tend to be near a characteristic stellar mass similar to the characteristic mass seen in the simulations. For comparison purposes, we also show in the appendix \[app:lum\] the resultant mass distributions in the observations when the luminosity is left in the training set. The results are similar, confirming that luminosity is not the main parameter used by the network. There is a tendency to find more pre-BN galaxies however. We speculate that this is because the algorithm uses some S/N related information if available. Since pre-BN are generally fainter, they also have lower S/N in the observed mock images so the network will tend to classify fainter objects as pre-BN. It highlights both the strengths and risks of the deep learning approach, in the sense that all information is taken into account in our unsupervised learning. An analogous behavior is also seen in figure \[fig:z\_evol\], where the redshift evolution of the fractions of galaxies in the three phases at fixed stellar mass is shown. Both plots are complementary. As expected the redshift evolution strongly depends on stellar mass. The galaxies that are more frequently potentially in the BN phase in the CANDELS redshift range are in the stellar mass range of $10^{9.2}<M_*/M_\odot<10^{10.3}$. The massive compact star-forming galaxies identified in the previous works might be the high-mass tail of the BN population. More massive galaxies tend indeed to be in the post-compaction phase at all redshifts. This means that if one wants to observe the progenitors of these most massive galaxies in the process of compaction, it is required to probe $\sim10^{9.5}$ solar mass galaxies at $z>3$. That will be straightforward with JWST. $\begin{array}{c c c} \includegraphics[width=0.32\textwidth]{plots/mass_dist1_comp_LUM.jpg} & \includegraphics[width=0.32\textwidth]{plots/mass_dist2_comp_LUM.jpg} & \includegraphics[width=0.32\textwidth]{plots/mass_dist3_comp_LUM.jpg}\\ \end{array}$ $\begin{array}{c c c} \includegraphics[width=0.32\textwidth]{plots/z_evol2_comp.jpg} & \includegraphics[width=0.32\textwidth]{plots/z_evol3_comp.jpg} & \includegraphics[width=0.32\textwidth]{plots/z_evol4_comp.jpg}\\ \end{array}$ The L-shape in sSFR vs. $M_*$ ----------------------------- The previous section has shown that the network successfully identifies a characteristic galaxy stellar mass range for the BN phase in the CANDELS data. This is remarkable given the known limitations of the simulations (see section \[sec:sims\]) and suggests that there are important similarities between simulated and observed galaxies. In future work, we plan to analyze in more detail how the different classes relate to other physical properties. As a preliminary step, we attempt a first look at the $sSFRs$ and central mass densities ($\Sigma_1$, [@2017ApJ...840...47B]) of galaxies in pre-BN, BN and post-BN phases. This is motivated because in the simulations, the compaction, BN, and quenching sequence puts the galaxy into a characteristic L-shape track in $sSFR-\Sigma_1$ with the BN phase at the turning point (e.g. [@2015MNRAS.450.2327Z]). This L-shape is similar to the observed distribution [@2013ApJ...765..104B; @2017ApJ...840...47B]. We show in figure \[fig:delta\_delta\] the $sSFR-\Sigma_1$ plane for pre-BN, BN and post-BN galaxies in CANDELS as defined by the CNN trained on the simulations. As previously reported, galaxies form a characteristic L-shape distribution in the plane. At first approximation, the median position (large dots in the figure) of pre-BN, BN and post-BN galaxies is different, and crudely follow the expected evolutionary sequence from the simulations. Pre-BN galaxies tend to be indeed in the main-sequence and have low central density values while post-BN galaxies have lower specific star-formation rates and larger central densities. BN galaxies lie in between. Given the observability timescales calibrated in section \[sec:obs\_time\], this suggests that there is an evolutionary sequence in the plane and that galaxies tend to move from left to right. We observe however that there is also significant overlap between the different phases in the three quadrants of the $sSFR-\Sigma_1$ diagram. For example, several galaxies are classified as post-BN while they have low $\Sigma_1$ values. Also, there is mixing of low sSFR and high sSFR compact galaxies that is not fully consistent with the distinction between the BN and post-BN phases in the simulations. For comparison, we show the same plot for the VELA simulations which shows a clearer separation, namely a stronger correlation between the three phases as defined based on the gas/SFR distribution and the distribution to three quadrants in the $sSFR-\Sigma_1$ diagram as derived from the stellar distribution. We emphasize that the main purpose of this work is to illustrate the methodology. We thus keep for future work a detailed investigation of the reasons of this increased confusion in CANDELS. One possible explanation resides in the definition of the BN phase used for training. We recall that several galaxies in the simulation present complex assembly histories, with many wet-compaction events of different intensities (see figure \[fig:comp\_def\]). A similar behavior is also reported in [@2016MNRAS.457.2790T], i.e. compaction and quenching events confine the galaxy to the main sequence, until a major BN event that is followed by long-term quenching as a result of a hot massive halo. Therefore, according to our labelling of the training set explained in section \[sec:labels\], galaxies can be still considered as post-BN (see for example VELA11 in figure \[fig:pred\_time\]) in between several events which could also contribute explaining the overlap we see in CANDELS. A way to explore the effects of minor compaction events would be to train a network with only major compactions and see how the classification changes. To do that a larger and more diverse training set is needed and also at higher redshift, in the JWST range, where major events tend to happen in the simulations. We keep this for future work. $\begin{array}{c c c} \includegraphics[width=0.33\textwidth]{plots/CANDELS_sSFR_sigma1_ssize0.jpg} & \includegraphics[width=0.33\textwidth]{plots/CANDELS_sSFR_sigma1_ssize1.jpg} & \includegraphics[width=0.33\textwidth]{plots/CANDELS_sSFR_sigma1_ssize2.jpg} \\ \includegraphics[width=0.33\textwidth]{plots/VELA_sSFR_sigma1_2.jpg} & \includegraphics[width=0.33\textwidth]{plots/VELA_sSFR_sigma1_3.jpg} & \includegraphics[width=0.33\textwidth]{plots/VELA_sSFR_sigma1_4.jpg} \\ \includegraphics[width=0.33\textwidth]{plots/VELA_DL_sSFR_sigma1_2.jpg} & \includegraphics[width=0.33\textwidth]{plots/VELA_DL_sSFR_sigma1_3.jpg} & \includegraphics[width=0.33\textwidth]{plots/VELA_DL_sSFR_sigma1_4.jpg} \\ \end{array}$ Summary and conclusions ======================= We have explored a new approach to classify galaxy images using deep learning calibrated on numerical simulations. The general methodology consists first of generating mock images of galaxies reproducing the observing conditions from hydro cosmological simulations which are then labelled based on the known evolution of gas, SFR and stars. The images are then fed to an unsupervised feature learning machine that automatically learns the features to detect a given evolution pattern. We have applied the method for detecting the characteristic blue nugget (BN) phase as seen in cosmological simulations, near a critical mass and preferentially at high redshifts, following a wet compaction process and followed by central quenching. We have used for that purpose a suite of high resolution zoom-in hydro numerical simulations of intermediate mass galaxies in the redshift range $1<z<3$. We have shown that a simple CNN is able to detect galaxies in the BN phase with $\sim80\%$ accuracy within a time window of $\pm0.2$ Hubble times and hence establish temporal constraints in the data. The described methodology presents several key advantages over more traditional approaches. First of all, it does not require any image pre-processing. Only the pixel distributions are fed into the network which automatically extracts the relevant information. This does not prevent however to combine the automatically extracted features with other standard measurements such as colors or sizes. Moreover there is no need of an a-priori assumption of the *optimal* observables for a given physical process. The procedure will automatically extract the best tracers if present in the data.\ We have then applied the trained model to observed galaxy multicolor images from the CANDELS survey observed with HST in the same redshift range and classify them into three main classes: pre-BN, BN and post-BN. The key results are: - The network finds galaxies with high probability of being in the three classes indicating similarity between simulated and observed galaxies. - The classification recovers a characteristic stellar mass for the BN phase of $\sim10^{9.2-10.3}$ solar masses mostly independent of redshift. More massive compact galaxies are found to be preferentially in the post-BN class, so they are compatible with having gone through the BN phase more than $0.5$ Hubble times before the time of observation. - Pre-BN, BN and post-BN galaxies occupy different regions in the $sSFR-\Sigma_1$ plane, suggesting an evolutionary sequence in the plane as predicted by the simulations. There is however some degree of confusion, i.e. post-BN galaxies with low central densities that will be investigated in future work. In particular, one important point that will be addressed in forthcoming works is the impact of the specific set of simulations used for training. Despite the similarities between simulations and observations suggested in section \[sec:cmass\], the VELA simulations used in this work might be still too limited to adequately represent the entire CANDELS data set, not only because of the lack of AGN but also because the sample is small and covers a limited mass range. Additionally, the assumptions regarding the sub-grid astrophysics are not well constrained by theory or observations as discussed in section \[sec:sims\]. To further investigate the impact of these limitations, we plan to enlarge our training sets by using new available simulated datasets with the same VELA initial conditions but different sub-grid astrophysics as well as other independent simulated datasets including AGN. The presented methodology could then be adapted to other robust physical processes captured in simulations and could constitute a useful tool to better compare future imaging surveys with forthcoming simulations. Acknowledgments {#acknowledgments .unnumbered} =============== The authors are grateful to Google for the unrestricted gift given to the University of California Santa Cruz to carry out the project: “deep learning for Galaxies” that greatly contributed to make this work possible. We also appreciate helpful discussions with Sander Dielemann, Daniel Freedman, Eric Hayashi and Jon Shlens at Google. We also thank Frédéric Bournaud for refereeing this work and providing interesting suggestions. This work was partly supported by the grants France-Israel PICS, US-Israel BSF 2014-273, and NSF AST-1405962. JRP acknowledges support from HST-AR-14578.001-A. AD also acknowledges support from GIF I-1341-303.7/2016, DIP STE1869/2-1 GE625/17-1, and I-CORE PBC/ISF 1829/12. MHC acknowledges support from the ANR ASTROBRAIN. DC has been funded by the ERC Advanced Grant, STARLIGHT: Formation of the First Stars (project number 339177). The VELA simulations were performed at the National Energy Research Scientific Computing Center (NERSC) at Lawrence Berkeley National Laboratory, and at NASA Advanced Supercomputing (NAS) at NASA Ames Research Center. 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This is to ensure that all galaxies have similar S/N and that the network does not learn based on that. As a matter of fact, since pre-BN galaxies in the simulations are found at higher redshift and have lower stellar masses than post-BN galaxies, they will be more noisy in the *CANDELized* images. The network might therefore use this information. To check the effect of this in the final classification, we show in figure \[fig:mass\_evol\_lum\] the same stellar mass distributions of galaxies in the three different phases in CANDELS as in figure \[fig:mass\_evol\] but obtained with a training set without randomizing the magnitudes. As can be seen, the distribution is similar, i.e. a BN peak at a characteristic stellar mass. However, the code tends to find more pre-BN galaxies at low mass. This is because it is learning some information from the S/N distribution. This exercice shows the strength of the deep-learning approach since it demonstrates that the network uses all available information. It highlights however the risks too. One needs to control the information that should not be used by the net. $\begin{array}{c c c} \includegraphics[width=0.32\textwidth]{plots/mass_dist1_comp_wLUM.jpg} & \includegraphics[width=0.32\textwidth]{plots/mass_dist2_comp_wLUM.jpg} & \includegraphics[width=0.32\textwidth]{plots/mass_dist3_comp_wLUM.jpg}\ \end{array}$ [^1]: sunrise is freely available at thttps://bitbucket.org/lutorm/sunrise. [^2]: https://keras.io/
--- abstract: 'In this article we prove a generalization of Weyl’s criterion for the essential spectrum of a self-adjoint operator on a Hilbert space. We then apply this criterion to the Laplacian on functions over open manifolds and get new results for its essential spectrum.' address: - 'Department of Mathematics and Statistics, University of Cyprus, Nicosia, 1678, Cyprus, and Deparment of Mathematics, Instituto Tecnológico Autónomo de México, Mexico D.F. 01000, Mexico' - 'Department of Mathematics, University of California, Irvine, Irvine, CA 92697, USA' author: - Nelia Charalambous - Zhiqin Lu date: 'September 1, 2012' title: On the spectrum of the Laplacian --- [^1] Introduction ============ Let $M$ be a complete noncompact Riemannian manifold of dimension $n$ and denote by $\Delta$ the Laplacian acting on $\mathcal C_0^\infty(M)$. It is well known that the self-adjoint extension of $\Delta$ on $L^2(M)$ exists and is a unique nonpositive definite and densely defined linear operator. We will also use $\Delta$ to denote this extension for the remaining paper. The spectrum of $-\Delta$, $\sigma(-\Delta)$, consists of all points $\lambda\in \mathbb{C}$ for which $\Delta+\lambda I$ fails to be invertible. Since $-\Delta$ is nonnegative definite, its $L^2$-spectrum is contained in $[0,\infty)$. The essential spectrum of $-\Delta$, $\sigma_\textup{ess}(-\Delta)$, consists of the cluster points in the spectrum and of isolated eigenvalues of infinite multiplicity. The following result is due to Donnelly [@Don2]: if there exists an infinite dimensional subspace $G$ in the domain of $\Delta$ such that $$\label{sdf} \|\Delta u+\lambda u\|_{L^2}\leq\sigma\|u\|_{L^2}$$ for all $u\in G$, then $$\sigma_\mathrm{ess}(-\Delta)\cap(\lambda-\sigma,\lambda+\sigma)\neq\emptyset.$$ The functions $u$ are referred to as the approximate eigenfunctions corresponding to the eigenvalue $\lambda$. The above criterion is simple to apply and has directed the study of the essential spectrum of the Laplacian for the last three decades. A related result of the above is as follows: let $u$ be a nonzero smooth function with compact support. If   is satisfied, then $$\sigma(-\Delta)\cap(\lambda-\sigma,\lambda+\sigma)\neq\emptyset.$$ We remark that for the above criteria to be valid, we do not have to assume the completeness of the manifold $M$. They can be applied [to]{} closed manifolds or open manifolds with boundary (with either Dirichlet or Neumann boundary conditions) as well as complete noncompact manifolds. If $M$ is compact, the criterion gives the eigenvalue estimates.\ In most problems the ideal space to work on is the $L^2$ function space when compared to the $L^q$ spaces. However, this is not the case when considering the spectrum of the Laplacian. On a Riemannian manifold, most of the approximate eigenfunctions we can write out explicitly must be related to the distance function. It is well known however, that the Laplacian of the distance function is locally bounded in $L^1$, but not in $L^2$. We can see this in the following simple example. Take $M=S^1\times (-\infty,\infty)$, letting $(\theta,x)$ be the coordinates. Then the [radial]{} function with respect to the point $(0,0)$ is given by $$r=\sqrt{x^2+(\min (\theta, 2\pi-\theta))^2}.$$ A straightforward computation gives $$\Delta r=-\frac{2\pi}{\sqrt{x^2+\pi^2}}\delta_{\{\theta=\pi\}}+\text{ a smooth function},$$ where $\delta_{\{\theta=\pi\}}$ is the Delta function along the submanifold $\{\theta=\pi\}$. Therefore $\Delta r$ is not locally $L^2$. The failure of the $L^2$ integrability of the Laplacian of the distance function was one of the main difficulties in applying the classical criterion above. In fact, it was not possible to prove that the $L^2$ essential spectrum of the Laplacian on a manifold with nonnegative Ricci curvature is $[0,\infty)$ by directly computing the $L^2$ spectrum. Additional assumptions on the curvature and geometry of the manifold were necessary (see for example [@Z; @jli; @C-L; @donnelly-1; @Esc86; @EF92]). Donnelly [@donnelly-1] proved that the essential spectrum of the Laplacian is $[0,\infty)$ for manifolds of nonnegative Ricci curvature and maximal volume growth. J-P. Wang [@Wang97], by using the seminal theorem of K. T. Sturm [@sturm], removed the maximal volume growth condition. Wang’s result confirms the conjecture that the spectrum of manifolds with nonnegative Ricci curvature is $[0,\infty)$. In [@Lu-Zhou_2011], Lu-Zhou gave a technical generalization of Wang’s result which includes the case of manifold of finite volume. In this article, we introduce a new method for computing the spectrum of a self-adjoint operator on a Hilbert space (see Theorem \[Thm.Weyl.bis-2\]) which has the following application in the case of the Laplacian \[thm00\] Let $M$ be a Riemannian manifold and let $\Delta$ be the Laplacian. Assume that for $\lambda\in \R^+$, there exists a nonzero function $u$ in the domain of $\Delta$ such that $$\label{mnb} \|u\|_{L^\infty}\cdot\|\Delta u+\lambda u\|_{L^1}\leq\sigma\|u\|_{L^2}^2$$ for some positive number $\sigma>0$. Then $$\sigma (-\Delta)\cap (\lambda-{\varepsilon},\lambda+{\varepsilon})\neq\emptyset,$$ where $${\varepsilon}=\min (1,(\lambda+1)\sigma^{1/3}).$$ Moreover, $$\sigma_{\textup{ess}} (-\Delta)\cap (\lambda-{\varepsilon},\lambda+{\varepsilon})\neq\emptyset,$$ if for any compact subset $K$ of $M$, there exists a nonzero function $u$ in the domain of $\Delta$ satisfying   whose support is outside $K$. We expect the above result to have many applications in spectrum theory (for example, on manifolds with corners, where the test functions are usually not smooth). In this paper, we concentrate on applying the criterion to the computation of the essential spectrum of complete noncompact manifolds. Theorem \[thm00\] proves to be a powerful tool in expanding the set of manifolds for which the $L^2$ essential spectrum is the nonnegative real line. In the case of shrinking Ricci solitons, we are able to prove the following result. \[ricci-solition\] The $L^2$ essential spectrum of a complete shrinking Ricci soliton is $[0,\infty)$. Note that no curvature assumption is needed here. For a large class of manifolds (for example, the shrinking Ricci solitons), we are able to control the volume growth about a fixed point, but it is difficult to prove the uniform volume growth. Without the uniform volume growth property, the theorem of Sturm does not apply and the results of Wang [@Wang97] or Lu-Zhou [@Lu-Zhou_2011] cannot be used. Therefore, the following result may be practically useful: \[thm11\] Let $M$ be a complete noncompact Riemannian manifold. Suppose that, with respect to a fixed point $p$, the radial Ricci curvature is asymptotically nonnegative (see Lemma \[DeltarEstim\]), and if the volume of the manifold is finite we additionally assume that its volume does not decay exponentially at $p$. Then the $L^2$ spectrum of the Laplace operator on functions is $[0,\infty)$. We shall also use Theorem \[thm00\] to modify a result of Elworthy-Wang [@ElwW04] on manifolds that posses an exhaustion function (Theorem \[72\]). We replace the $L^2$ norm assumption by an $L^1$ norm assumption.\ In the last section, we show that it is possible to work with continuous test functions in Theorem \[thm00\]. By using them instead we avoid the repetitive choosing of cut-off functions.\ The essential spectrum of the Laplacian on noncompact Riemannian manifolds is interesting and important as it reveals a lot of information about the geometry of the manifold. Although there are lot of interesting open problems in this direction, the authors believe that answering the following conjecture is the most important one. Let $M$ be a complete noncompact Riemannian manifold with Ricci curvature bounded below. Then the $L^2$ essential spectrum of the Laplacian on functions is a connected subset of the real line. In other words, the essential spectrum set is of the form $[a,\infty)$, where $a$ is a nonnegative real number. As is well-known, the essential spectrum of a Schr" odinger operator could be very complicated (cf. [@rs4]\*[Chapter XIII]{}) and it certainly need not be a connected set. For the case of the Laplacian on a complete manifold however, in all known examples the $L^2$ essential spectrum is a connected set. In this paper, we answer the above conjecture in some special cases. We believe that the analysis of the wave kernel is needed to answer the conjecture in full.\ [**Acknowledgement.**]{} The authors thank Rafe Mezzeo and Jiaping Wang for their interest in and discussions of the essential spectrum problem. They particularly thank David Krejčiř[í]{}k for the discussion on the alternative versions of Weyl’s Criterion which led to the proof of Theorem \[Thm.Weyl.bis-2\]. The Weyl Criterion for Quadratic Forms ====================================== Let $H$ be a self-adjoint operator on a Hilbert space ${\mathcal{H}}$. The norm and inner product in ${\mathcal{H}}$ are respectively denoted by $\|\cdot\|$ and $(\cdot,\cdot)$. Let $\sigma(H), \sigma_\mathrm{ess}(H)$ be the spectrum and the essential spectrum of $H$, respectively. Let ${\mathfrak{D}}(H)$ be the domain of $H$. The Classical Weyl criterion states that \[Thm.Weyl\] A nonnegative real number $\lambda$ belongs to $\sigma(H)$ if, and only if, there exists a sequence $\{\psi_n\}_{n \in {\mathbb{N}}} \subset {\mathfrak{D}}(H)$ such that 1. $ \forall n\in{\mathbb{N}}, \quad \|\psi_n\|=1 $, 2. $ (H-\lambda)\psi_n \to 0, \text{ as } n\to\infty \text{ in }\mathcal H.$ Moreover, $\lambda$ belongs to $\sigma_\mathrm{ess}(H)$ of $H$ if, and only if, in addition to the above properties 3. $ \psi_n \to 0 \text{ weakly as } n\to\infty \text{ in }\mathcal H. $ The above theorem is still true if the convergence in (2) is replaced by weak convergence, the statement of which can be found (without proof) in [@DDi] and later in [@KK]. This version of the Weyl criterion was applied for the first time to the Laplacian on curved Euclidean domains in [@KK]. The authors are grateful to David Krejčiř[í]{}k for informing them of the results. We have the following functional analytic result, which generalizes the weak Weyl criterion. To the authors’ knowledge, this result seems to be new. \[Thm.Weyl.bis-2\] Let $f$ be a bounded positive continuous function over $[0,\infty)$. A nonnegative real number $\lambda$ belongs to the spectrum $\sigma(H)$ if, and only if, there exists a sequence $\{\psi_n\}_{n \in {\mathbb{N}}} \subset {\mathfrak{D}}(H)$ such that 1. $ \forall n\in{\mathbb{N}}, \quad \|\psi_n\|=1 $, 2. $ (f(H) (H-\lambda)\psi_n, (H-\lambda)\psi_n)\to 0, \text{ as } n\to\infty \quad {and} $ 3. $ (\psi_n, (H-\lambda)\psi_n) \to 0, \text{ as } n\to\infty. $ Moreover, $\lambda$ belongs to $\sigma_\mathrm{ess}(H)$ of $H$ if, and only if, in addition to the above properties 4. $ \psi_n \to 0, \text{ weakly as } n\to\infty $ $\mathcal H$. Since $H$ is a densely defined self-adjoint operator, the spectral measure $E$ exists and we can write $$\label{decomp} H=\int_0^\infty \lambda\, dE.$$ Assume that $\lambda\in\sigma(H)$. Then by Weyl’s criterion, there exists a sequence $\{\psi_n\}$ such that $$\|(H-\lambda)\psi_n\|\to 0, \quad \|\psi_n\|=1$$ as $n\to\infty$. We write $$\psi_n=\int_0^\infty d E(t)\psi_n$$ as its spectral decomposition. Then $$(f(H) (H-\lambda)\psi_n, (H-\lambda)\psi_n)=\int_0^\infty f(t)(t-\lambda)^2 d\|E(t)\psi_n\|^2.$$ Since $f$ is a bounded positive function, we have $$(f(H) (H-\lambda)\psi_n, (H-\lambda)\psi_n)\leq C\int_0^\infty (t-\lambda)^2 d\|E(t)\psi_n\|^2=C\|(H-\lambda)\psi_n\|^2.$$ Similarly, $$(\psi_n, (H-\lambda)\psi_n)\leq C\,\|\psi_n\|\cdot\|(H-\lambda)\psi_n\|.$$ Thus the necessary part of the theorem is proved. Now assume that $\lambda>0$ and $\lambda\notin\sigma(H)$. Then there is a $\lambda>{\varepsilon}>0$ such that $E(\lambda+{\varepsilon})-E(\lambda-{\varepsilon})=0$. We write $$\label{fgh} \psi_n=\psi_n^1+\psi_n^2,$$ where $$\psi_n^1=\int_0^{\lambda-{\varepsilon}} dE(t)\psi_n,$$ and $\psi_n^2=\psi_n-\psi_n^1$. Then $$\begin{aligned} & (f(H) (H-\lambda)\psi_n, (H-\lambda)\psi_n) \\ &=(f(H) (H-\lambda)\psi_n^1, (H-\lambda)\psi_n^1) +(f(H) (H-\lambda)\psi^2_n, (H-\lambda)\psi_n^2)\\ & \geq c_1\|\psi_n^1\|^2+(f(H) (H-\lambda)\psi^2_n, (H-\lambda)\psi_n^2)\geq c_1\|\psi_n^1\|^2,\end{aligned}$$ where the positive number $c_1$ is the infimum of the function $f(t)(t-\lambda)^2$ on $[0,\lambda-{\varepsilon}]$. Therefore $$\|\psi_n^1\|\to 0$$ by [*(2)*]{}. On the other hand, we similarly get $$(\psi_n, (H-\lambda)\psi_n)\geq {\varepsilon}\|\psi_n^2\|^2-\lambda\|\psi_n^1\|^2.$$ If the criteria [*(2), (3)*]{} are satisfied, then, by the two [estimates]{} above, we conclude that both $\psi_n^1, \psi_n^2$ go to zero. This contradicts $\|\psi_n\|=1$, and the theorem is proved. Note that for $\lambda=0$, $\psi_n^1$ is automatically zero, and the second half of the proof would give the contradiction. We apply Theorem \[Thm.Weyl.bis-2\] to the Laplacian on functions. In this setting two particular cases of the function $f$ will be of interest. \[cor-Thm.Weyl.bis-2\] A nonnegative real number $\lambda$ belongs to the spectrum $\sigma(H)$ if, and only if, there exists a sequence $\{\psi_n\}_{n \in {\mathbb{N}}} \subset {\mathfrak{D}}(H)$ such that 1. $ \forall n\in{\mathbb{N}}, \quad \|\psi_n\|=1 $, 2. $ ((H+1)^{-1}\psi_n, (H-\lambda)\psi_n)\to 0, \text{ as } n\to\infty \quad {and} $ 3. $ (\psi_n, (H-\lambda)\psi_n) \to 0, \text{ as } n\to\infty. $ Moreover, $\lambda$ belongs to $\sigma_\mathrm{ess}(H)$ of $H$ if, and only if, in addition to the above properties 4. $ \psi_n \to 0, \text{ weakly as } n\to\infty $ $\mathcal H$. We take $f(x)=(x+1)^{-1}$. The corollary follows from the identity $$(H+1)^{-1}(H-\lambda)=1-(\lambda+1)(H+1)^{-1}.$$ In a similar spirit, taking $f(x)=(x+\alpha)^{-(N+1)}$ for a natural number $N$ and a positive number $\alpha>1$, we also obtain the following generalization \[cor-Thm.Weyl.bis-3\] A nonnegative real number $\lambda$ belongs to the spectrum $\sigma(H)$ if, and only if, there exists a sequence $\{\psi_n\}_{n \in {\mathbb{N}}} \subset {\mathfrak{D}}(H)$ such that 1. $ \forall n\in{\mathbb{N}}, \quad \|\psi_n\|=1 $, 2. $ ((H+\alpha)^{-i}\psi_n, (H-\lambda)\psi_n)\to 0 \text{ as } n\to\infty \ $ for two consecutive natural numbers i=N, N+1, and 3. $ (\psi_n, (H-\lambda)\psi_n) \to 0, \text{ as } n\to\infty. $ Moreover, $\lambda$ belongs to $\sigma_\mathrm{ess}(H)$ of $H$ if, and only if, in addition to the above properties 4. $ \psi_n \to 0, \text{ weakly as } n\to\infty $ $\mathcal H$. Using the Cauchy inequality, the above two corollaries reduce to Donnelly’s criterion  when we consider the case $H=-\Delta$. A spectrum estimate result ========================== In this section we will prove a special version of Theorem \[Thm.Weyl.bis-2\] for the Laplacian on functions. We begin [with]{} the fact that its resolvent is always bounded on $L^\infty$. \[lem42\] We have $$(-\Delta+1)^{-1}$$ is bounded from $L^\infty(M)$ to itself and the operator norm is no more than $1$. The lemma follows from the proof of Lemma 3.1 in [@CharJDE]. The resolvent is bounded on $L^\infty$ because the heat kernel is bounded on $L^\infty$. This is a property that Davies proves for any nonnegative self-adjoint operator that satisfies Kato’s inequality like the Laplacian  [@Davies]\*[Theorems 1.3.2,1.3.3]{}. It is due to the well-known fact that the Laplacian on functions is a self-adjoint operator that satisfies Kato’s inequality. [Together with Corollary \[cor-Thm.Weyl.bis-2\] this lemma allows us to obtain an even simpler criterion for the essential spectrum of the Laplacian on functions:]{} By the above lemma, we have $$\begin{aligned} & |(u, (-\Delta-\lambda)u)|\leq\|u\|_{L^\infty}\cdot \|(-\Delta-\lambda)u\|_{L^1}\\ & |((-\Delta+1)^{-1}(-\Delta-\lambda)u, (-\Delta-\lambda)u)|\leq \lambda\|u\|_{L^\infty}\cdot \|(-\Delta-\lambda)u\|_{L^1}\end{aligned}$$ We write $$u=u_1+u_2$$ according to the spectrum decomposition of the operator $-\Delta$ (cf.  ). Then we have $$\begin{aligned} & \|u_1\|_{L^2}^2\leq \frac{\lambda(\lambda+1)}{{\varepsilon}^2}\sigma\|u\|_{L^2}^2;\\ &{\varepsilon}\|u_2\|_{L^2}^2-\lambda\|u_1\|_{L^2}^2\leq\sigma\|u\|^2_{L^2}.\end{aligned}$$ Thus we have $${\varepsilon}\|u_2\|_{L^2}^2+{\varepsilon}\|u_1\|_{L^2}^2\leq \left(\frac{\lambda(\lambda+1)(\lambda+{\varepsilon})}{{\varepsilon}^2}+1\right)\sigma\|u\|_{L^2}^2.$$ The conclusion follows since $\|u_2\|_{L^2}^2+\|u_1\|_{L^2}^2=\|u\|_{L^2}^2$. The essential spectrum result of the theorem follows from the classical Weyl criterion (Theorem \[Thm.Weyl\], (3)). An Approximation Theorem ======================== Let $M$ be a complete noncompact Riemannian manifold. Let $p\in M$ be a fixed point. Define $$r(x)=d(x,p)$$ to be the radial function on $M$. It is well known that 1. $r(x)$ is continuous; 2. $|\nabla r(x)|=1$ almost everywhere and $r(x)$ is a Lipschitz function; 3. $\Delta r$ exists on $M\backslash \{p\}$ in the sense of distribution. In general, it is not possible to find smooth approximations of a Lipschitz function under the $\mathcal C^1$ norm. The following Proposition, which is a more precise version of  [@Lu-Zhou_2011]\*[Proposition 1]{}, implies that this can be done up to a function with small $L^1$ norm. Such kind of result is essential in Riemannian geometry and should be well-known, but given that we were not able to find a reference, we include a proof. \[pp-1\] For any positive continuous decreasing function $\eta: \mathbb R^+\to\mathbb R^+$ such that $$\lim_{r\to\infty} \eta(r)=0,$$ there exist $C^\infty$ functions $\tilde r(x)$ and $b(x)$ on $M$ such that 1. $\|b\|_{L^1(M\backslash B_{{p}}(R))}\leq\eta(R-1)$; 2. $\|\nabla \tilde r-\nabla r\|_{L^1(M\backslash B_{{p}}(R))}\leq\eta(R)$ and for any $x\in M$ with $r(x)>2$ 1. $|\tilde r(x)-r(x)|\leq \eta(r(x))$ and $|\nabla\tilde r (x)|\leq 2$; 2. $\Delta\tilde r(x)\leq \max_{y\in B_x(1)} \Delta r(y)+\eta(r(x))+|b(x)|$ [in the sense of distribution]{}. Without loss of generality, we assume that $\eta(r)<1$. Let $\{U_i\}$ be a locally finite cover of $M$ and let $\{\psi_i\}$ be the partition of unity subordinate to the cover. Let ${\bf x_i}=(x_i^1,\cdots,x_i^n)$ be the local coordinates of $U_i$. Define $r_i=r|_{U_i}$. Let $\xi(\bf x)$ be a non-negative smooth function on $\R^n$ whose support is within the unit ball. Assume that $$\int_{\mathbb R^n}\xi=1.$$ Without loss of generality, we assume that each $U_i$ is an open subset of the unit ball of $\mathbb R^n$ with coordinates ${\bf x_i}$. Then for any ${\varepsilon}>0$, $$r_{i,{\varepsilon}}=\frac{1}{{\varepsilon}^n}\int_{\mathbb R^n}\xi\left(\frac{{\bf x_i}-{\bf y_i}}{{\varepsilon}}\right)r_i({\bf y_i}) d{\bf y_i}$$ is a smooth function on $U_i$ and hence on $M$. Let $\{\sigma_i\}$ be a sequence of positive numbers such that $$\sum_i\sigma_i(|\Delta\psi_i(x)|+4|\nabla\psi_i (x)|+\psi_i(x))\leq\eta(r(x)).$$ By  [@gt]\*[Lemma 7.1, 7.2]{}, for each $i$, we can choose ${\varepsilon}_i<1$ small enough so that $$\begin{aligned} \label{oio} \begin{split} &|r_{i,{\varepsilon}_i}(x)-r_i(x)|\leq\sigma_i;\\ &\|\nabla r_{i,{\varepsilon}_i}-\nabla r_i\|_{L^1{(U_i)}}\leq\sigma_i. \end{split} \end{aligned}$$ We also have $$\label{oio-2} \Delta r_{i,{\varepsilon}_i}(x)\leq \max_{y\in B_x(1)}\Delta r_i(y).$$ Define $$\tilde r=\sum_i\psi_i r_{i,{\varepsilon}_i}, \quad b=2 { \sum_i} \nabla\psi_i\cdot \nabla r_{i,{\varepsilon}_i}.$$ Since $\sum_i (\nabla\psi_i\cdot \nabla r_i)=(\sum_i\nabla \psi_i) \cdot \nabla r=0$ [almost everywhere on $M$]{}, we have $$b=2\sum_i \nabla\psi_i\cdot (\nabla r_{i,{\varepsilon}_i}-\nabla r_i)$$ almost everywhere. Thus [*(a)*]{} follows. Similarly, observing that $$\tilde r-r=\sum_i\psi_i(r_{i,{\varepsilon}_i}-r_i), {\quad{\rm and}\ \ |\nabla r_{i,{\varepsilon}_i}|<2,}$$ we obtain [*(b), (c)*]{}. To prove [*(d)*]{}, we compute $$\Delta\tilde r=\sum_i[(\Delta\psi_i)\, r_{i,{\varepsilon}_i}+2\nabla\psi_i\nabla r_{i,{\varepsilon}_i}+\psi_i\Delta r_{i,{\varepsilon}_i}],$$ and since $$\sum_i(\Delta\psi_i ) r_i=\sum_i (\Delta\psi_i) r=0,$$ we have $$\Delta\tilde r=\sum_i[\Delta\psi_i(r_{i,{\varepsilon}_i}- r_i)+b+\psi_i\Delta r_{i,{\varepsilon}_i}].$$ By , we obtain [*(d)*]{} and the Proposition is proved. Manifolds with $\Delta r$ Asymptotically Nonpositive ==================================================== [As we have mentioned in the previous section,]{} the Laplacian of the radial function [$r(x)=d(x,p)$]{} exists in the sense of distribution (except at $p$). That is, for any nonnegative smooth function $f$ with compact support in $M\backslash\{p\}$, the integral $$\int_Mf \Delta r$$ [is defined]{}. The following simple observation is due to Wang [@Wang97] and is crucial in our estimates. \[wang\] The function $\Delta r$ is locally integrable away from $p$. Let $W$ be any compact set of the form $B_p(R)-B_p(r)$ [for $R>r>0$]{}. Then by the Laplacian comparison theorem, there is a constant $C$, depending only on the dimension, $r$, $R$, and the lower bound of the Ricci curvature on $B_p(R)$, such that $$\Delta r\leq C$$ on $W$ in the sense of distribution. Thus we have $$|\Delta r|=|C-\Delta r-C|\leq 2C-\Delta r$$ and therefore $$\int_W|\Delta r|\leq 2C\,{{\rm vol}\,}(W) -\int_W\Delta r.$$ Using Stokes’ Theorem, we obtain $$\int_W |\Delta r|\,\leq 2C\,{{\rm vol}\,}(W)-\int_{{\partial}W}\frac{{\partial}r}{{\partial}n}\leq 2C{{\rm vol}\,}(W)+{{\rm vol}\,}({\partial}W),$$ and the lemma is proved. In this section, we study manifolds with the following property $$\label{DeltarAsy} \overline{\lim_{r\to \infty}}\; \Delta r\leq 0$$ in the sense of distribution, where $r(x)$ is the radial distance of $x$ to a fixed point $p$. We shall give a precise estimate of the $L^1$ norm of $\Delta r$ in terms of the volume growth of the manifold. But before we do that, we first provide an important example where the above technical condition holds. We note that for a fixed point $p\in M$ the cut locus ${\rm Cut}(p)$ is a set of measure zero in $M$. The manifold can be written as the disjoint union $M=\Omega\cup {\rm Cut}(p)$, where $\Omega$ is star-shaped with respect to $p$. That is, if $x\in \Omega$, then the geodesic line [segment]{} $\overline {px}\subset \Omega$. $\p r= \p /\p r$ is well defined on $\Omega$. We have the following result \[DeltarEstim\] Let $r(x)$ be the radial function with respect to $p$. Suppose that there exists a continuous function $\delta(r)$ on $\mathbb{R}^+$ such that 1. ${\displaystyle \lim_{r\to\infty} \delta(r)=0}$ 2. $\delta(r)>0$ and 3. ${\rm Ric}(\p r, \p r)\geq -(n-1) \delta(r)$ on $\Omega$. Then   is valid in the sense of distribution. On $\Omega$, we have the following Bochner formula $$0=\frac 12\Delta|\nabla r|^2=|\nabla^2r |^2+\nabla r\cdot \nabla(\Delta r)+{\rm Ric}({\partial}r,{\partial}r).$$ Since $\nabla ^2 r({\partial}r, {\partial}r)=0$, using the Cauchy inequality, we have $$\label{bochner} 0\geq\frac 1{n-1}(\Delta r)^2+\frac{{\partial}}{{\partial}r}(\Delta r) -(n-1)\delta(r).$$ Since $\Omega$ is star-shaped, for any fixed direction ${\partial}/{\partial}r$, we obtain  by comparing the above differential inequality with the Riccati equation. On the points where $r$ is not smooth, we may use the trick of Gromov [as in Proposition 1.1 of [@SchoenYau_bk] to conclude the result in the sense of distribution.]{} Volume comparison theorems -------------------------- Let $p$ be the fixed point of the manifold. Denote $$B(r)=B_p(r),\quad V(r)={{\rm vol}\,}(B_p(r))$$ the geodesic ball of radius $r$ at $p$ and its volume respectively. The following volume comparison theorem is well-known. \[LemSubexp\] Let $r(x)$ be the radial function defined above. Assume that   is valid in the sense of distribution. Then the manifold has subexponential volume growth at $p$. In other words, for all ${\varepsilon}>0$ there exists a positive constant $C({\varepsilon})$, depending only on ${\varepsilon}$ and the manifold, such that for all $R>0$ $$V(R)\leq C({\varepsilon}) \, e^{{\varepsilon}\, R}.$$ Let $m(r)$ be a nonnegative continuous function such that $$\lim_{r\to\infty} m(r)=0,$$ and $$\Delta r\leq m(r)$$ in the sense of distribution. It follows that $$\int_{B(R)\backslash B(1)}\Delta r\leq \int_{B(R)\backslash B(1)} m(r)$$ which, by Stokes’ Theorem, implies that $${\rm vol}\,(\partial B(R))-{\rm vol}\,(\partial B(1))\leq \int_{B(R)\backslash B(1)} m(r).$$ Let ${\varepsilon}>0$. Then we can find an $R_{\varepsilon}$ such that $m(r)<{\varepsilon}$ for $r>R_{\varepsilon}$. Setting $f(R)=V(R)$, we obtain $$f'(R)\leq {\rm vol}\,(\partial B(1))+\int_{B(R_{\varepsilon})\backslash B(1)} m(r) +{\varepsilon}(f(R)-f(R_{\varepsilon}))$$ for any $R>R_{\varepsilon}$. Thus $$(e^{-{\varepsilon}R}(f(R)-f(R_{\varepsilon})))'\leq C_{\varepsilon}e^{-{\varepsilon}R}$$ for $R>R_{\varepsilon}$, where $C_{\varepsilon}$ is a constant depending on ${\varepsilon}$ and the manifold $M$. Integrating from $R_{\varepsilon}$ to $R$, we obtain $$f(R)< f(R_{\varepsilon})+C_{\varepsilon}{\varepsilon}^{-1}e^{-{\varepsilon}R_{\varepsilon}}e^{{\varepsilon}R}$$ for $R>R_{\varepsilon}$. Thus for any $R$, we have $$V(R)=f(R)<C({\varepsilon}) e^{{\varepsilon}R}$$ for $$C({\varepsilon})=f(R_{\varepsilon})+C_{\varepsilon}{\varepsilon}^{-1}e^{-{\varepsilon}R_{\varepsilon}}.$$ In other words, whenever the Laplacian of the radial function $r(x)=d(x,p)$ is asymptotically nonnegative in the sense of distribution, the manifold has subexponential volume growth with respect to the point $p$. In the case of finite volume for the manifold $M$, we will also need an assumption on the decay rate of the volume of a ball of radius $r$. We say that the volume of $M$ [*decays exponentially at $p$*]{}, if there exists an ${\varepsilon}_o>0$ such that $${{\rm vol}\,}(M)-V(r)\leq e^{-{\varepsilon}_o r}$$ for $r$ large. For the purposes of computing the $L^2$ essential spectrum, we will need that the volume does not decay exponentially. $L^1$ estimates for $\Delta \tilde{r}$. --------------------------------------- [We set $\tilde{r}$ to be the smoothing of $r$ from Proposition \[pp-1\]]{}. The following lemma is a more precise version of  [@Lu-Zhou_2011]\*[Lemma 2]{}. \[DeltarVolEst\] Let $r(x)$ be the radial function to a fixed point $p$ on $M$, and suppose that  is valid in the sense of distribution. Then we have the following two cases 1. Whenever ${{\rm vol}\,}(M)$ is infinite, for any ${\varepsilon}>0$ and $r_1>0$ [large enough]{}, there exists a $K=K({\varepsilon}, r_1)$ such that for any $r_2>K$, we have $$\label{wsx-1} \int_{B(r_2)\setminus B(r_1)} |\Delta \tilde{r}| \leq {\varepsilon}\, V(r_2+1) ;$$ 2. Whenever ${{\rm vol}\,}(M)$ is finite, for any ${\varepsilon}>0$ there exists a $K({\varepsilon})>0$ such that for any $r_2>K$, we have $$\int_{M \setminus B(r_2)} |\Delta \tilde{r}| \leq {\varepsilon}\, ({{\rm vol}\,}(M)-V(r_2))+2{{\rm vol}\,}({\partial}B(r_2)).$$ By Proposition  \[pp-1\] and using the idea in the proof of Lemma \[wang\], we obtain $$|\Delta \tilde r(x)|\leq 2(\max_{y\in B_x(1)} \Delta r(y)+\eta(r(x))+|b(x)|)-\Delta\tilde r(x)$$ in the sense of distribution. Using our assumptions on $\Delta r$ and $\eta$, we see that for any ${\varepsilon}>0$ we can find an $r_1>0$ large enough such that whenever $r(x)>r_1$, then $$2(\max_{y\in B_x(1)} \Delta r(y)+\eta(r(x))\,)<{\varepsilon}/2$$ [also in the sense of distribution.]{} Therefore for $r>r_1+2$, $$\int_{B(r)\setminus B(r_1)} |\Delta \tilde{r}| \leq\frac{\varepsilon}2\, (V(r)-V(r_1))+2\int_{M\setminus B(r_1)}|b|-\int_{B(r)\setminus B(r_1)}\Delta\tilde r.$$ Using Stokes’ Theorem, we get $$\int_{B(r)\setminus B(r_1)}|\Delta \tilde{r}|\leq \frac{\varepsilon}2\, (V(r)-V(r_1))+2\int_{M{\setminus B(r_1)}} |b|-\int_{\p B(r)}\frac{\p \tilde{r}}{\p n} + \int_{\p B(r_1)}\frac{\p \tilde{r}}{\p n},$$ where $\p/\p n$ is the outward normal direction on the boundary. Obviously, [the above implies that]{} $$\begin{aligned} \label{qaz-3} \int_{B(r)\setminus B(r_1)}|\Delta \tilde{r}| \leq & \frac{\varepsilon}2\, (V(r)-V(r_1))+2\int_{M{\setminus B(r_1)}}|b| \\ &+\int_{\p B(r)}\left|\frac{\p \tilde{r}}{\p n} -1\right|+ \int_{\p B(r_1)}\frac{\p \tilde{r}}{\p n}.\notag\end{aligned}$$ We first consider the case when the volume of $M$ is infinite. By Proposition  \[pp-1\], choosing $r_1$ large enough we obtain $$\int_{M{\setminus B(r_1)}}|b|< \frac{{\varepsilon}}{4}$$ and $$\label{qaz-2} \|\nabla \tilde r-\nabla r\|_{L^1(M\backslash B(r_1))}\leq 1$$ Since the volume of $M$ is infinite, then there exists $K=K({\varepsilon}, r_1)>r_1+2$ such that whenever $r>K$ $$\label{qaz-1} \int_{B(r)\setminus B(r_1)}|\Delta \tilde{r}|\leq \frac{3{\varepsilon}} 4\,( V(r){-V(r_1)\,})+\int_{{\partial}B(r)}\left|\frac{{\partial}\tilde r}{{\partial}n}-1\right|.$$ We choose an $r'$ such that $|r'-r|<1$ and $$\int_{{\partial}B(r')}\left|\frac{{\partial}\tilde r}{{\partial}n}-1\right|\leq \int_{r-1}^{r+1}\int_{{\partial}B(t)}\left|\frac{{\partial}\tilde r}{{\partial}n}-1\right|dt.$$ By , we have $$\int_{{\partial}B(r')}\left|\frac{{\partial}\tilde r}{{\partial}n}-1\right|<2.$$ Therefore, $$\int_{B(r')\setminus B(r_1)}|\Delta \tilde{r}|\leq \frac{3{\varepsilon}}{4}\, (V(r')-V(r_1))+2.$$ Choosing a possibly larger $K({\varepsilon},r_1)$ we get [*(a)*]{}. The proof of [*(b)*]{} is similar. We choose $\eta(r)$ decreasing to zero so fast so that $$\int_{M\backslash B(r_1)}|b|\leq \frac{\varepsilon}8({{\rm vol}\,}(M)- V(r_1)).$$ Since the volume of $M$ is finite, [ sending $r\to\infty$ in ]{} we have $$\int_{M\setminus B_p(r_1)}|\Delta \tilde{r}|\leq {\varepsilon}\, ({{\rm vol}\,}(M)-V(r_1))+ \int_{\p B_p(r_1)}\frac{\p \tilde{r}}{\p n}.$$ Since $|{{\partial}\tilde r}/{{\partial}n}|\leq 2$ by [*(c)*]{} of Proposition \[pp-1\], the lemma follows. \[corlDeltar\] Suppose that $(i), (ii), (iii)$ hold on $M$ as in Lemma \[DeltarEstim\]. Then the same integral estimates for $\Delta \tilde{r}$ hold as in Lemma \[DeltarVolEst\]. The $L^2$ Spectrum. {#sec5} =================== In this section,[we let $\tilde r(x)$ be the smoothing function defined in Proposition \[pp-1\]]{} of the radial function $r(x)=d(x,p)$. For each $i\in\mathbb N$, let $x_i, y_i, R_i, \mu_i$ be large positive numbers such that $x_i>2R_i>2\mu_i+4$ and $y_i>x_i+2R_i$. We take the cut-off functions $\chi_i: \mathbb{R}^+\to \mathbb{R}^+$, smooth with support on $[x_i/R_i-1, y_i/R_i+1]$ and such that $\chi_i=1$ on $[x_i/R,y_i/R]$ and $|\chi'_i|, |\chi''_i|$ bounded. Let $\lambda>0$ be a positive number. We let $$\label{4-equ} \phi_i(x)=\chi_i({\tilde{r}}/{R_i})\, e^{\sqrt{-1}\sqrt{\lambda}\,\tilde{r}}.$$ Setting $\phi=\phi_i$, $R=R_i$, $x=x_i$ and $\chi=\chi_i$, we compute $$\begin{aligned} \begin{split}& \Delta\phi +\lambda\phi = (R^{-2}\chi''(\tilde r/R)+2i\sqrt\lambda R^{-1}\chi'(\tilde r/R))e^{\sqrt{-1}\sqrt\lambda\tilde r}|\nabla\tilde r|^2\\& -\lambda\phi(|\nabla\tilde r|^2-1)+(R^{-1}\chi'(\tilde r/R)+i\sqrt{\lambda}\chi)e^{\sqrt{-1}\sqrt\lambda\tilde r}\Delta\tilde r. \end{split}\end{aligned}$$ Then we have $$\label{2} |\phi|\leq 1,\quad |\Delta\phi +\lambda\phi|\leq \frac{C}{R}+C|\Delta\tilde r| +C|\nabla\tilde r-\nabla r|,$$ [where $C$ is a constant depending only on $\lambda$ and $M$.]{} Denote the inner product on $L^2(M)$ by $( \cdot\,,\,\cdot)$. We have the following key estimates \[lem41\] Suppose that  is valid for the radial function $r$ in the sense of distribution. In the case that the volume of $M$ is finite, we make the further assumption that its volume does not decay exponentially at $p$. Then there exist sequences of large numbers $x_i, y_i, R_i, \mu_i$ such that the supports of the $\phi_i$ are disjoint and $$\frac{\|(\Delta+\lambda)\phi_i\|_{L^1}}{(\phi_i,\phi_i)}\to 0$$ as $i\to\infty$. [The proof is similar to that of  [@Lu-Zhou_2011].]{} We define $x_i,y_i,R_i,\mu_i$ inductively. If $(x_{i-1}, y_{i-1}, R_{i-1}, \mu_{i-1})$ are defined, then we only need to let $\mu_i$ large enough so that the support of $\phi_i$ is disjoint with the previous $\phi_j$’s. For simplicity we suppress the $i$ in our notation. The upper bound estimates for $|\phi|$ and $|\Delta\phi+\lambda\phi|$ given in imply that $$\begin{aligned} \label{1} \begin{split} \int_M(\phi, \Delta\phi+\lambda\phi) \leq & \frac CR \,[V(y+R) -V(x-R)] \\ & + C \int_{B(y+R)\setminus B(x-R)} |\Delta \tilde{r}| +\eta(x-R). \end{split}\end{aligned}$$ When the volume of $M$ is infinite, we choose a function $\eta$ as in Proposition \[pp-1\] such that $\eta\leq 1$. By Lemma \[DeltarVolEst\], if we choose $R, x$ large enough but fixed, then for any $y>0$ large enough we have $$\int_M(\phi, \Delta\phi+\lambda\phi) \leq 2 {\varepsilon}\,V(y+R+1).$$ Since $\|\phi\|_2^2\geq V(y)-V(x)$, if we choose $y$ large enough, $\|\phi\|_2^2\geq \frac 12 V(y).$ The subexponential volume growth of $M$ at $p$ that was proved in Lemma \[LemSubexp\] implies that there exists a sequence of $y_k\to \infty$ such that $V(y_k+R+1)\leq 2\, V(y_k)$. If not, then for a fixed number $y$ and for all $k\in \mathbb{N}$ we have that $$V(y+k(R+1))> 2^k\, V(y).$$ However, by the subexponential volume growth of the manifold $$2^k\, V(y) < V(y+k(R+1))\leq C({\varepsilon}_1) \, e^{{\varepsilon}_1 y} \, e^{k \,{\varepsilon}_1 (R+1)}$$ for any ${\varepsilon}_1>0$ and $k$ large. This leads to a contradiction when we choose ${\varepsilon}_1$ such that ${\varepsilon}_1 R <\log 2$. Therefore, there exists a $y$ such that $$V(y+R +1)\leq 2\,V(y)\leq 4 \|\phi\|_2^2.$$ Combing the above inequalities, we have $$\int_M(\phi, \Delta\phi+\lambda\phi) \leq 8 {\varepsilon}\|\phi\|_2^2.$$ We now consider the finite volume case. Using equation and Lemma \[DeltarVolEst\] we obtain for $x-R>K({\varepsilon})$ $$\begin{aligned} \int_M(\phi, \Delta\phi+\lambda\phi) \leq & (R^{-1} + {\varepsilon}) \,[{{\rm vol}\,}(M) -V(x-R)]\\ & + 2 C \,{{\rm vol}\,}(\p B(x-R)) +\eta(x-R).\end{aligned}$$ We set $h(r)={{\rm vol}\,}(M) -V(r)$, a [decreasing]{} function. We choose $\eta(r)$ as in Proposition \[pp-1\] so that $\eta(r)\leq \frac{\varepsilon}8 h(r)$. Making ${\varepsilon}$ even smaller and choosing $R$ and $x-R$ large enough, we get$$\int_M(\phi, \Delta\phi+\lambda\phi) \leq {\varepsilon}\,h(x-R) - 2 C \,h'(x-R).$$ Given that $\|\phi\|_2^2\geq h(x)-h(y)$ and the volume of $M$ is finite, we can choose $y$ large enough so that $$\|\phi\|_2^2\geq \frac 12 h(x).$$ We would like to prove in this case that there exists a sequence of $x_k\to \infty $ such that $${\varepsilon}\,h(x_k-R) - 2C\,h'(x_k-R)\leq 2 {\varepsilon}h(x_k).$$ If the above inequality does not hold, then for all $x$ large enough $${\varepsilon}\,h(x-R) -2C \,h'(x-R) > 2 {\varepsilon}h(x).$$ Replacing ${\varepsilon}$ by ${\varepsilon}/2C$, we obtain $${\varepsilon}\,h(x-R) - \,h'(x-R) > 2 {\varepsilon}h(x).$$ This implies that $$- \bigl(e^{-{\varepsilon}x}h(x-R)\bigr)' > 2 {\varepsilon}h(x) \, e^{-{\varepsilon}x}.$$ Integrating from $x$ to $x+R$ and using the monotonicity of $h$ we have $$h(x-R) >2(1 -e^{-{\varepsilon}R}) h(x+R).$$ Choosing $R$ even larger, we can make $2 {(1-e^{-{\varepsilon}R})}> 5/4$, therefore $$h(x-R) > \frac 54 h(x+R)$$ for all $x$ large enough. By iterating this inequality, we get for all positive integers $k$ $$h(x-R) > \left(\frac 54\right)^k\, h(x+(2k-1)R).$$ Therefore $${{\rm vol}\,}(M)-V(x-R) > \left(\frac 54\right)^k [{{\rm vol}\,}(M)-V(x+(2k-1)R)\, ]$$ which gives $${{\rm vol}\,}(M)-V(x+(2k-1)R) \leq \left(\frac 45\right)^k \, {{\rm vol}\,}(M).$$ Sending $k\to \infty$ this contradicts the nonexponential decay assumption on the volume. Corollary \[corlDeltar\] gives \[corl41\] Suppose that $(i), (ii), (iii)$ hold on $M$ as in Lemma \[DeltarEstim\]. In the case that the volume of $M$ is finite, we make the further assumption that its volume does not decay exponentially at $p$. Then there exist sequences of large numbers $x_i, y_i, R_i, \mu_i$ and cut-off functions $\chi_i$ such that the supports of the $\phi_i$ are disjoint and $$\frac{\|(\Delta+\lambda)\phi_i\|_{L^1}}{(\phi_i,\phi_i)}\to 0$$ as $i\to\infty$. Now we prove Theorem \[thm11\]. In fact we will be able to prove a more general, albeit more technical result \[thm12\] Let $M$ be a complete noncompact Riemannian manifold. Suppose that, with respect to a fixed point $p$, the radial function $r(x)=d(x,p)$ satisfies $$\overline{\lim_{r\to \infty}}\; \Delta r\leq 0$$ in the sense of distribution, and if the volume of the manifold is finite, we additionally assume that its volume does not decay exponentially at $p$. Then the $L^2$ spectrum of the Laplace operator on functions is $[0,\infty)$. Let $\phi_i$ be the sequence of functions as defined in  . Then by the construction of the functions and Corollary \[corl41\], the assumptions of Theorem \[thm00\] are satisfied. This completes the proof of the theorem. \[rmkwarp\] We note that a similar result should hold on warped product manifolds $M=\mathbb{R}\times_J \tilde{M}$ with metric $g=d\rho^2 + J^2(\rho,\theta) \, \tilde{g},$ where $(\tilde{M},\tilde{g})$ is a compact $(n-1)$-dimensional submanifold of $M$ and $\rho$ is the distance function from this submanifold. Under the same asymptotically nonnegative assumption on ${\rm Ric}(\p \rho, \p \rho)$ as in Lemma \[DeltarEstim\], we also get that the $L^2$ spectrum of the Laplace operator on functions is $[0,\infty)$. Complete Shrinking Ricci Solitons ================================= A noncompact complete Riemannian manifold $M$ with metric $g$ is called a gradient shrinking Ricci soliton if there exists a smooth function $f$ such that the Ricci tensor of the metric $g$ is given by $$R_{ij}+\n_i \n_j f = \rho \, g_{ij}$$ for some positive constant $\rho>0$. By rescaling the metric we may rewrite the soliton equation as $$R_{ij}+\n_i \n_j f = \frac 12 \, g_{ij}.$$ The scalar curvature $R$ of a gradient shrinking Ricci soliton is nonnegative, and the volume growth of such manifolds (with respect to the Riemannian metric) is Euclidean. Hamilton [@Ham] proved that the scalar curvature of a gradient shrinking Ricci soliton satisfies the equations $$\n_i R = 2 \, R_{ij}\, \n_j f,$$ $$R+|\n f|^2-f=C_o$$ for some constant $C_o$. We may add a constant to $f$ so that $$R+|\n f|^2-f=0.$$ In  [@Lu-Zhou_2011], the authors proved that 1. the $L^1$ essential spectrum of the Laplacian contains $[0, \infty)$; 2. the $L^2$ essential spectrum of the Laplacian is $[0, \infty)$, if the scalar curvature has sub-quadratic growth. Using our new Weyl Criterion, we are able to remove the curvature condition. It can be shown that $f(x)\geq 0$ and the key idea is to use $\rho(x)=2\sqrt{f(x)}$ as an approximate distance function on the manifold, because of the special properties that it satisfies. We define $$D(r)=\{x\in M : \rho(x)<r\}$$ and set $V(r) ={{\rm vol}\,}(D(r))$. For some positive number $y$ sufficiently large we consider the cut-off function $\chi: \mathbb{R}^+\to \mathbb{R}$, smooth with support in $[0, y +2]$ and such that $\chi=1$ on $[1,y+1]$ and $|\chi'|, |\chi''|\leq C$. For any $\lambda > 0$ and large enough constants $b, l$ we let $$\phi(\rho)=\chi\left(\frac{\rho-b}{l}\right)\, e^{\sqrt{-1}\sqrt{\lambda}\,\rho}$$ [which has support on $[b+l, b+l(y+1)]$]{}. Lu and Zhou [@Lu-Zhou_2011]\*[page 3289]{} demonstrate that for sufficiently large $l$ and $b$ $$\int_M |\Delta \phi +\lambda \phi| \leq {\varepsilon}V(b+(y+2)l).$$ At the same time $$\|\phi\|_{L^2}^2\geq V(b+(y+1)l)-V(b+l)$$ (note that the same holds true for the $L^1$ norm of $\phi$). Arguing as is  [@Lu-Zhou_2011]\*[Theorem 6]{} we conclude that there exists a $y$ large enough such that $$\int_M |\Delta \phi +\lambda \phi| \leq 4 {\varepsilon}\|\phi\|_{L^2}^2.$$ As in the previous section, we may also choose appropriate sequences of $b_i, l_i$ such that the supports of the $\psi_i$ are disjoint and condition [*(2)*]{} of Theorem \[thm00\] holds. Condition [*(1)*]{} is verified by the estimate above and the fact that $\|\phi_i\|_{L^\infty}=1$. Exhaustion functions on complete manifolds ========================================== From what we have seen so far, it is apparent that two things are important when computing the essential spectrum of the Laplacian: 1. The control of the $L^1$ norm of $\Delta r$; 2. The control of the volume growth and decay of geodesic balls. The same idea can be used for manifolds whose essential spectrum is not the half real line. In the spirit of the results above, we are also able to modify a theorem of Elworthy and Wang [@ElwW04]. We now consider manifolds on which there exists a continuous exhaustion function $\gamma\in \mathcal C(M)$ such that \(a) $\gamma$ is unbounded above and is $\mathcal C^2$ smooth in the domain $\{\gamma>R\}$ for some $R>0$ and \(b) ${{\rm vol}\,}(\{m_o<\gamma<n\})<\infty $ for some $m_o$ and any $n>m_o$ where the volume is measured with respect to the Riemannian metric. For $t>0$ and $c\in \mathbb{R}$ we define $B_t=\{\gamma(x)<t\}$ and set $dv_c=e^{-c\gamma}dv$. For $t\geq s$, let $U_c(s,t)={{\rm vol}\,}_c(B_t\setminus B_s)$ where ${{\rm vol}\,}_c$ is the volume with respect to the measure $dv_c$. We begin by stating the result of Elworthy and Wang for the sake of comparison. Suppose that there exists a function $\gamma \in \mathcal C(M)$ that satisfies $(a)$ and $(b)$ and a constant $c\in\mathbb{R}$ such that $$\label{exh2} \lim_{s\to\infty} \overline{\lim_{t\to\infty}}\; U_c(s,t)^{-1}\int_{B_t\setminus B_s} [(\Delta\gamma-c)^2+(|\n\gamma|^2-1)^2 ] \, dv_c=0$$ and $$\label{exh1} \lim_{t\to\infty} \max \{U_c(m_o,t), U_c(t,\infty)^{-1} \} \,e^{-{\varepsilon}t}=0 \qquad \rm{for \ any\ } {\varepsilon}>0.$$ Then $\sigma(-\Delta)\supset [c^2/4,\infty).$ When the above hold for $c=0$, then $\sigma(-\Delta)= [0,\infty).$\ Note that condition implies that when $c = 0$ the volume of the manifold grows and decays subexponentially, as was the case for us in the previous sections. The assumption here is that the weighted volume grows and decays subexponentially. Our result is as follows: \[72\] Suppose that there exists a function $\gamma \in \mathcal C(M)$ that satisfies $(a)$ and $(b)$ and a constant $c\in\mathbb{R}$ such that $$\label{exh3} \lim_{s\to\infty} \overline{\lim_{t\to\infty}}\; \,U_{c}(s,t)^{-1} \,\int_{B_t\setminus B_s} (|\Delta\gamma-c|+|\,|\n\gamma|^2-1|) \, dv_{c}=0$$ and $$\label{exh4} \lim_{t\to\infty} \max \{U_c(m_o,t), U_c(t,\infty)^{-1} \} \,e^{-{\varepsilon}t}=0 \qquad \rm{for \ any\ } {\varepsilon}>0.$$ If and hold for $c=0$, then $\sigma(-\Delta)=[0,\infty).$ In the case they hold for $c\neq 0$, we make the additional assumptions that the heat kernel of the Laplacian satisfies the pointwise bound $$\label{exh5} p_t(x,y)\leq C t^{-m}\,e^{-\frac{(\gamma(x)-\gamma(y))^2}{4C_1 t} -\frac{d(x,y)^2}{4C_2 t } +\beta_1|\gamma(x)-\gamma(y)|+\beta_2d(x,y)+\beta_3 t}$$ for some positive constants $m, C_1, C_2, \beta_1, \beta_2, \beta_3$, and that the Ricci curvature of the manifold is bounded below ${\rm Ric}(M)\geq -(n-1)K$ for a nonnegative number $K$. Then $\sigma(-\Delta) \supset [c^2/4,\infty).$ In the case $c=0$, the main difference between our result and Theorem 1.1 of [@ElwW04] is that we only need to control the $L^1$ norms of $|\Delta\gamma-c|$ and $| |\n\gamma|^2-1|$ as in , instead of their $L^2$ norms (compare to ). Our assumption is weaker in various cases, for example when $\gamma$ is the radial function where we know that its Laplacian is not locally $L^2$ integrable when the manifold has a cut-locus, but it is locally $L^1$ integrable. In the case $c\neq 0$, the additional assumption is similar to requiring a uniform Gaussian bound for the heat kernel, but now with respect to the $\gamma$ function as well. Such a bound is certainly true in the case of hyperbolic space with $\gamma$ the radial function. The proof uses similar estimates to those of Elworthy and Wang for the measures of annuli along the exhaustion function $\gamma$. We provide an outline of the argument with the necessary modifications. Set $\lambda\geq c^2/4$ be a fixed number. For any $t>s$ we let $\chi: \mathbb{R}^+\to \mathbb{R}^+$, be a smooth cut-off function with support on $[s-1, t+1]$ and such that $\chi=1$ on $[s,t]$ and $|\chi'|, |\chi''|$ bounded. Let $\lambda_c=\sqrt{\lambda-c^2/4}$ and define for $s\geq 0$ $$f(s)= e^{(i{\lambda_c} - c/2)\,s}.$$ Consider the test function $$\phi_{s,t}(x)=\chi(\gamma(x))\, f(\gamma(x)).$$ We compute $$\Delta\phi_{s,t} +\lambda\phi_{s,t} = (\chi''f+2\chi'f'+\chi f'')|\nabla\gamma|^2 + (\chi' f + \chi f')\Delta\gamma +\lambda \,\chi f.$$ Using the fact that $f'' +c f' +\lambda f=0$ we obtain $$\Delta\phi_{s,t} +\lambda\phi_{s,t} = (\chi''f+2\chi'f')|\nabla \gamma|^2 + (\chi' f )\Delta\gamma + \chi f'(\Delta \gamma -c |\n \gamma|^2) + \lambda \,\chi f(1-|\n \gamma|^2).$$ Therefore there exists a constant $C$ such that $$\label{73} |\Delta\phi_{s,t} +\lambda\phi_{s,t}|\leq Ce^{-c/2\gamma}\, {\bigl[}(|\Delta\gamma-c|+|\,|\n\gamma|^2-1|) 1_{\text{spt}(B_{{t+1}}\setminus B_{{s-1}})} +1_{\text{spt}(\chi')} {\bigr]}.$$ For the rest of the estimates, we will repeatedly use $$\label{74} \lim_{s,t\to\infty} (U_c(s-1,s)+U_c(t,t+1))/U_c(s,t)=0,$$ which follows from  . Using , we have $$\begin{aligned} \label{75} |(\phi_{s,t},\Delta\phi_{s,t} +\lambda\phi_{s,t})| \leq & C \int_{B_{{t+1}}\backslash B_{{s-1}}}(|\Delta\gamma-c|+|\,|\n\gamma|^2-1|)\,dv_c \\ &{+ C (U_c(s-1,s)+U_c(t,t+1))} .\notag\end{aligned}$$ We observe that $$\begin{aligned} \frac{1}{U_c(s,t)}\int_{B_{t+1}\backslash B_{s-1}}&(|\Delta\gamma-c|+|\,|\n\gamma|^2-1|)\,dv_c \\ =&\bigl[1+\frac{U_c(s-1,s)+U_c(t,t+1)}{U_c(s,t)}\bigr] \\ &\cdot\,\frac{1}{U_c(s-1,t+1)}\int_{B_{t+1}\backslash B_{s-1}}(|\Delta\gamma-c|+|\,|\n\gamma|^2-1|)\,dv_c,\end{aligned}$$ which tends to zero as $s,t\to\infty$ by   and assumption . Since $\|\phi_{s,t}\|^2_{L^2}\geq U_c(s,t)$, inequality , the above estimate and   imply that $$\label{76} \lim_{s,t\to\infty} |(\phi_{s,t}, \Delta\phi_{s,t} +\lambda\phi_{s,t})|/\|\phi_{s,t}\|^2_{L^2}=0.$$ When $c=0$, we choose appropriate sequences of $s_n, t_n \to \infty$ such that condition [*(2)*]{} of Theorem \[thm00\] holds. Condition [*(1)*]{} of the Corollary follows from and the fact that the functions $\phi_{s_n,t_n}$ are bounded. Therefore, $\lambda_0=\sqrt{\lambda}$ belongs to the essential $L^2$ spectrum. Given that $\lambda$ is any nonnegative number, the result follows. In the case $c\neq 0$, we will apply Corollary \[cor-Thm.Weyl.bis-3\]. For a fixed natural number $i>m$ and $\alpha>0$ we have that the integral kernel of $(-\Delta+\alpha)^{-i}$, $g_{\alpha}^{i}(x,y)$, is given by $$g_{\alpha}^{i}(x,y)= C(n)\int_0^\infty p_t(x,y)\, t^{i-1} \, e^{-\alpha t} \, dt.$$ On the other hand, it is a property of the exponential function that for any $\beta_4, \beta_5 \in \mathbb{R}$ $$e^{-\frac{(\gamma(x)-\gamma(y))^2}{4C_1 t}} \leq e^{-\beta_4|\gamma(x)-\gamma(y)|} e^{C_1\,\beta_4^2 t}$$ and $$e^{-\frac{d(x,y)^2}{4C_2 t}} \leq e^{-\beta_5 d(x,y)} e^{C_2\,\beta_5^2 t}.$$ Combining the above, we have that for any $N>m$ and $\beta_4, \beta_5>0$ there exists an $\alpha>0$ large enough, and a constant $C$ such that $$g_{\alpha}^{i}(x,y)\leq C\,e^{-\beta_4 \, |\gamma(x)-\gamma(y)|-\beta_5 d(x,y)}$$ for $i=N, N+1$. As a result, for any $t>s>2$ $$\begin{aligned} \int _{B_{t+1}\setminus B_{s-1}} g_{\alpha}^{i}(x,y) e^{-c/2\gamma(y)} dy&\leq C \int _{B_{t+1}\setminus B_{s-1}} e^{-\beta_4 |\gamma(x)-\gamma(y)| -\beta_5 d(x,y)} \, e^{-c/2\gamma(y)} dy \\ &\leq C\,e^{-c/2\gamma(x)}\end{aligned}$$ after choosing $\beta_4=|c|/2$ and $\beta_5>\sqrt{K}$. This estimate together with also give $$\begin{aligned} |(\,(-\Delta+\alpha)^{-i}\phi_{s,t},\Delta\phi_{s,t} +\lambda\phi_{s,t})| \leq & C \int_{B_{{t+1}}\backslash B_{{s-1}}}(|\Delta\gamma-c|+|\,|\n\gamma|^2-1|)\,dv_c \\ & + C (U_c(s-1,s)+U_c(t,t+1)) .\end{aligned}$$ As a result, $$\begin{aligned} \label{77} \lim_{s,t\to\infty} |(\,(-\Delta+\alpha)^{-i}\phi_{s,t},\Delta\phi_{s,t} +\lambda\phi_{s,t})|/\|\phi_{s,t}\|^2_{L^2}=0.\end{aligned}$$ Choosing appropriate sequences of $s_n, t_n \to \infty$ and setting $\psi_n = \phi_{s_n,t_n}/\|\phi_{s_n,t_n}\|_{L^2}$, conditions [*(1)*]{} and [*(4)*]{} of Corollary \[cor-Thm.Weyl.bis-3\] hold for the functions $\psi_n$. That [*(2)*]{} and [*(3)*]{} also hold follows from and respectively. The use of continuous test functions. ===================================== In this section we will see that it is not necessary to use cut-off functions in our test functions. We will do that by first proving yet another version of the generalized Weyl’s criterion (Corollary \[cor-Thm.Weyl.bis-8\]). This version of Weyl’s Criterion sometimes provides a cleaner method for computing the essential spectrum. Let $D$ be a bounded domain of $M$ with smooth boundary. We use the notation $\mathcal C_0^\infty(D)$ to denote the set of smooth functions on the closure $\bar D$ which vanish on the boundary ${\partial}D$. Let $\rho:D\to \mathbb R$ be the distance function to the boundary ${\partial}D$. \[def51\] We define $\mathcal C_0^+( D)$ to be the set of functions $f$ on $ D$ with the properties 1. $f$ is continuous, vanishing on ${\partial}D$; 2. $f$ is Lipschitz, $\nabla f$ is essentially bounded, and $|\Delta f|$ exists in the sense of distribution; 3. As ${\varepsilon}\to 0,$ $\int_{\{\rho \leq {\varepsilon}\}}|f|\leq \frac 12{\varepsilon}^2(\int_{{\partial}D} |\nabla f|+o(1))$, and $\int_{\{\rho \leq {\varepsilon}\}}|\nabla f|\leq {\varepsilon}(\int_{{\partial}D} |\nabla f|+o(1))$. Let $\mathcal C_0^+(M)$ be the set of continuous functions whose support is a bounded domain of $M$ with smooth boundary and $$f\in \mathcal C_0^+({\rm supp}{f}).$$ We have the following \[cor-Thm.Weyl.bis-8\] A nonnegative real number $\lambda$ belongs to the spectrum $\sigma(-\Delta)$, if there exists a sequence $\{\psi_n\}_{n \in {\mathbb{N}}}$ of functions in $\mathcal C_0^+(M)$ such that 1. ${\displaystyle \frac{\|\psi_n\|_{L^\infty(D_n)}\cdot(\|(-\Delta-\lambda)\psi_n\|_{L^1( D_n)}+\|\nabla\psi_n\|_{L^1({\partial}D_n)})}{\|\psi_n\|_{L^2(D_n)}^2} \to 0, \text{ as } n\to\infty,} $ where $D_n= {\rm supp}\,{\psi_n}$. Moreover, $\lambda$ belongs to $\sigma_{\rm ess}(-\Delta)$ of $\Delta$, if 2. For any compact subset $K$ of $M$, there exists an $n$ such that the support of $\psi_n$ is outside $K$. The above corollary can be proved using the following approximation result Let $f\in \mathcal C_0^+(M)$. Then for any ${\varepsilon}>0$, there exists a smooth function $h$ of $M$ such that 1. ${\rm supp}\,(h)\subset {\rm supp}\, (f)$; 2. $\|f-h\|_{L^1}+\|f-h\|_{L^2}\leq {\varepsilon}$; 3. $\|(-\Delta -\lambda) h\|_{L^1}\leq C (\|(-\Delta-\lambda) f\|_{L^1( D)}+\|\nabla f\|_{L^1({\partial}D)})$, where $C$ is a constant independent of $f$, and $D={\rm supp}\,(f)$. Let $\chi(t)$ be a cut-off function such that it vanishes in a neighborhood of $0$ and is $1$ for $t\geq 1$. Let $\delta>0$ be a small number. Consider $$g_\delta(x)=\chi\left(\frac{\rho(x)}{\delta}\right) f(x).$$ It is not difficult to prove [*(a), (b)*]{} in the Proposition when we replace $h$ by $g_\delta$. To prove [*(c)*]{} we compute $$(-\Delta-\lambda) g_\delta=\chi(-\Delta-\lambda) f-2\delta^{-1}\chi'\nabla \rho\nabla f -(\delta^{-2}\chi''+\delta^{-1}\chi'\Delta\rho) f.$$ Since ${\partial}D$ is smooth, $\rho$ is a smooth function near ${\partial}D$. Therefore by (3) of Definition \[def51\] we have $$\|(-\Delta-\lambda) g_\delta\|_{L^1}\leq C (\|(-\Delta-\lambda) f\|_{L^1( D)}+\|\nabla f\|_{L^1({\partial}D)})$$ for $\delta$ sufficiently small. The proof that $g_\delta$ can be approximated by a smooth [function]{} is similar to that of Proposition \[pp-1\]. We sketch the proof here. Let $D=\cup U_i$ be a finite cover of $D$. [Without loss of generality, we assume that those $U_i$’s which intersect with ${\partial}D$ are outside the support of $g_\delta$.]{} Let ${\bf x_i}=(x_i^1,\cdots,x_i^n)$ be the local coordinates of $U_i$. Define $g_i=g_\delta|_{U_i}$. Let $\xi(\bf x)$ be a non-negative smooth function of $\R^n$ whose support is within the unit ball. Assume that $$\int_{\mathbb R^n}\xi=1.$$ Without loss of generality, we assume that each $U_i$ is an open subset of the unit ball of $\mathbb R^n$ with coordinates ${\bf x_i}$. Then for any ${\varepsilon}>0$, $$g_{i,{\varepsilon}}=\frac{1}{{\varepsilon}^n}\int_{\mathbb R^n}\xi\left(\frac{{\bf x_i}-{\bf y_i}}{{\varepsilon}}\right)g_i({\bf y_i}) d{\bf y_i}$$ is a smooth function on $U_i$ and hence on $M$. Let $\{\sigma_i\}$ be a sequence of positive numbers such that $$\sum_i\sigma_i(|\Delta\psi_i(x)|+4|\nabla\psi_i (x)|+\psi_i(x))$$ is sufficiently small. By  [@gt]\*[Lemma 7.1, 7.2]{}, for each $i$, we can choose ${\varepsilon}_i<1$ small enough so that $$\begin{aligned} \label{oio-4} \begin{split} &|g_{i,{\varepsilon}_i}(x)-g_i(x)|\leq\sigma_i;\\ &\|\nabla g_{i,{\varepsilon}_i}-\nabla g_i\|_{L^1{(U_i)}}\leq\sigma_i. \end{split} \end{aligned}$$ We also have $$\label{oio-6} \|\Delta g_{i,{\varepsilon}_i}\|_{L^1}\leq \|\Delta g_i\|_{L^1}.$$ Define $$h=\sum_i\psi_i g_{i,{\varepsilon}_i}, \quad b=2 { \sum_i} \nabla\psi_i\cdot \nabla g_{i,{\varepsilon}_i}.$$ Since $\sum_i (\nabla\psi_i\cdot \nabla g_i)=(\sum_i\nabla \psi_i) \cdot \nabla g_\delta=0$ [almost everywhere on $D$]{}, we have $$b=2\sum_i \nabla\psi_i\cdot (\nabla g_{i,{\varepsilon}_i}-\nabla g_i).$$ We compute $$\Delta h=\sum_i[(\Delta\psi_i)\, g_{i,{\varepsilon}_i}+2\nabla\psi_i\nabla g_{i,{\varepsilon}_i}+\psi_i\Delta g_{i,{\varepsilon}_i}],$$ and since $$\sum_i(\Delta\psi_i ) g_i=\sum_i (\Delta\psi_i) g_\delta=0,$$ we have $$\Delta h=\sum_i[\Delta\psi_i(g_{i,{\varepsilon}_i}- g_i)+ 2\sum_i\nabla\psi_i\cdot(\nabla g_{i,{\varepsilon}_i}-\nabla g_i)+\psi_i\Delta g_{i,{\varepsilon}_i}].$$ By ,  , we may choose ${\varepsilon}_i$ to be sufficiently small so that $$\|{(-\Delta-\lambda)} h\|_{L^1{(D)}}\leq 2 \|(-\Delta-\lambda) g_\delta\|_{L^1(D)}.$$ [^1]: The first author was partially supported by CONACYT of Mexico and is thankful to the Asociación Mexicana de Cultura A.C. The second author is partially supported by the DMS-12-06748.
--- abstract: 'We offer some new applications of an extension of Abel’s lemma, as well as its more general form established by Andrews and Freitas. A nice connection is established between this lemma and series involving the Riemann zeta function.' author: - Alexander E Patkowski title: 'A note on some applications of the extension of Abel’s lemma' --- Introduction ============ In a paper by Andrews and Freitas \[4\], the extension of Abel’s lemma was further generalized and several new $q$-series were established. Recall that Abel’s lemma is the simple result that $\lim_{z\rightarrow 1^{-}}(1-z)\sum_{n\ge0}a_nz^n =\lim_{n\rightarrow\infty}a_n.$ We use the shifted factorial notation $(a)_n=a(a+1)\cdots (a+n-1)$ in this paper \[2\]. Their result may be stated as follows. (\[4, Proposition 1.2\]) Let $f(z)=\sum_{n\ge0} \alpha_n z^n$ be analytic for $|z|<1,$ and assume that for some positive integer $M$ and a fixed complex number $\alpha$ we have that (i) $\sum_{n\ge0}(n+1)_{M}(\alpha_{M+n}-\alpha_{M+n-1})$ converges, and (ii) $\lim_{n\rightarrow \infty} (n+1)_{M}(\alpha_{M+n}-\alpha)=0.$ Then $$\frac{1}{M}\lim_{z\rightarrow1^{-}}\left(\frac{\partial^M}{\partial z^M}(1-z)f(z)\right)=\sum_{n\ge0}(n+1)_{M-1}(\alpha-\alpha_{n+M-1}).$$ The formula being generalized here is given in \[3, Proposition 2.1\], where it was used to find generating functions for special values for certain $L$-functions. A corollary of this extension of Abel’s lemma was also given in \[7\]. In the work \[1\] we find a simple formula attributed there to a Christian Goldbach, $$\sum_{n\ge0}(1-\zeta(k+2))=-1.$$ Now it does not appear any connection has been made between the extension of Abel’s lemma and this result, but as we shall demonstrate, it is a simple consequence of it. To this end, we shall prove some more general formulas in the next section which we believe are interesting applications of the Andrews-Freitas formula. For this, we will use a result from the work \[6\]. For some relevant series identities of a similar nature see also \[5\]. Some new theorems ================= This section establishes some interesting theorems, which we hope will add value to the Andrews-Freitas formula. For convenience in our proofs, we decided to write down a simple lemma. If $f(z)$ has no factor $(1-z)^{-1},$ then we may write $$\lim_{z\rightarrow1^{-}}\frac{\partial^M}{\partial z^M}(1-z)f(z)=-Mf^{M-1}(1).$$ Put $f_1(z)=(1-z),$ and $f_2(z)=f(z).$ Then by the Leibniz rule, $$\lim_{z\rightarrow1^{-}}\frac{\partial^M}{\partial z^M}(1-z)f(z)=\lim_{z\rightarrow1^{-}}\sum_{j\ge0}\binom {M}{j} f_1^{(j)}f_2^{(M-j)}$$ $$=\lim_{z\rightarrow1^{-}}\binom {M}{1}f_1^{(1)}f_2^{(M-1)}$$ $$=-M\lim_{z\rightarrow1^{-}}f_2^{(M-1)}(z),$$ because if $j=0$ then the term in the sum, $f_1^{(0)},$ is 0 when $z\rightarrow1^{-},$ and for $j>1,$ $f_1^{(j)}=0.$ As usual, we denote $\gamma$ to be Euler’s constant \[2\]. We also define the polygamma function \[2\] to be the $(M+1)$-th derivative of the logarithm of the Gamma function: $\psi^{(M)}(z)=\frac{\partial^{M+1}}{\partial z^{M+1}}(\log\Gamma(z)).$ For positive integers $M,$ we have that $$\sum_{n\ge0}(n+1)_{M-1}(1-\zeta(n+M+1))=(-1)^M\sum_{j\ge0}\binom {M-1}{j} j!\psi^{(M-j-1)}(1)+(M-1)!(-1)^M+\gamma(-1)^{M}(M-1)!.$$ First we write down the well-known Taylor expansion of the digamma function \[1, 2\], for $|z|<1,$ $$\psi^{(0)}(z+1)=-\gamma-\sum_{k\ge1}\zeta(k+1)(-z)^k.$$ It is a trivial exercise to re-write (2.1) as $$-z^{-1} \psi^{(0)}(1-z)+z^{-1}\gamma=\sum_{k\ge0}\zeta(k+2)z^k.$$ Inserting the functional equation for $\psi^{(0)}(z),$ given by \[1, 2\] $$\psi^{(0)}(z+1)=\psi^{(0)}(z)+\frac{1}{z},$$ into (2.2) and multiplying by $(1-z)$ gives $$-z^{-1}(1-z)(\psi^{(0)}(2-z)-(1-z)^{-1})+z^{-1}(1-z)\gamma=(1-z)\sum_{k\ge0}\zeta(k+2)z^k,$$ Now applying Proposition 1.1 with $\alpha_n=\zeta(n+2),$ and involving (2.4) gives the theorem after applying Lemma 2.1. For $M=1$ Theorem 2.2 specializes to Goldbach’s formula (1.1). For positive integers $M$ and $N,$ we have that $$\sum_{n\ge0}(n+1)_{M-1}(n+M+1)^{N}(1-\zeta(n+M+1))=\sum_{l\ge1}^{N}S(N+1,l+1)(-1)^{l+1}g_{M,l}$$ $$+(-1)^M\sum_{j\ge0}\binom {M-1}{j} j!\psi^{(M-j-1)}(1)+(M-1)!(-1)^M+\gamma(-1)^{M}(M-1)!,$$ where for $l\ge0,$ $$g_{M,l}:=-\sum_{j\ge0}\binom {M-1}{j}(-1)^{M-1-j}\psi^{(l+M-1-j)}(1)\frac{(l+1)!}{(l+1-j)!}.$$ From \[6, Corollary 2\], we find the delightful formula for integers $N\ge1$ and $\Re(a)>0,$ $$\sum_{k\ge2}k^{N}z^{k}\zeta(k,a)=\sum_{l\ge1}^{N}S(N+1,l+1)l!\zeta(l+1,a-z)z^{l+1}-z(\psi^{(0)}(a-z)-\psi^{(0)}(a)),$$ for $|z|<|a|.$ Here $S(n,l)$ are the Stirling numbers of the second kind \[2\]. $\zeta(s,a)$ is the Hurwitz zeta function \[2\]. We have also corrected the stated formula by instead having $N\ge1.$ We have also shifted the sum by replacing $l$ by $l+1$ for our convenience. Now $\lim_{n\rightarrow0}\zeta(n,a)=0$ if $a>1,$ $1$ if $a=1,$ $+\infty$ if $0<a<1.$ Hence the formula (2.5) is of the type of interest to our study only if $a=1.$ So, in that case, we put $a=1,$ and re-write (2.5) as $$\sum_{k\ge2}k^{N}z^{k}\zeta(k)=\sum_{l\ge1}^{N}S(N+1,l+1)l!\zeta(l+1,1-z)z^{l+1}-z(\psi^{(0)}(1-z)-\psi^{(0)}(1)).$$ Differentiating (2.3) $l$ times we get that $$\psi^{(l)}(2-z)=\psi^{(l)}(1-z)+(1-z)^{-l-1}(-1)^l l!.$$ Now using equation \[1, eq.(2.15)\], we have $$\sum_{k\ge0}k^Nz^k=\sum_{k\ge1}k^Nz^k=\sum_{l\ge0}^{N}S(N+1, l+1)(-1)^ll!(1-z)^{-l-1}z^{l+1}.$$ Now $S(n,1)=1$ for all non-negative integers $n,$ so we may write (2.8) for $N\ge1$ as $$\sum_{k\ge0}k^Nz^k=z(1-z)^{-1}+\sum_{l\ge1}^{N}S(N+1, l+1)(-1)^ll!(1-z)^{-l-1}z^{l+1}.$$ Using $\psi^{(l)}(z)=(-1)^{l+1}l!\zeta(l+1,z),$ and (2.7), we re-write (2.6) as $$\sum_{k\ge2}k^{N}z^{k}\zeta(k)=\sum_{l\ge1}^{N}S(N+1,l+1)((-1)^{l+1}\psi^{(l)}(2-z)+(1-z)^{-l-1}(-1)^ll!)z^{l+1}$$ $$-z(\psi^{(0)}(1-z)-\psi^{(0)}(1)).$$ Now comparing equation (2.9) with (2.10), and noting $\psi^{(0)}(1)=-\gamma,$ we see that we have that $$\sum_{k\ge2}k^{N}z^{k}(\zeta(k)-1)=\sum_{l\ge1}^{N}S(N+1,l+1)(-1)^{l+1}\psi^{(l)}(2-z)z^{l+1}+z-z(1-z)^{-1}$$ $$-z(\psi^{(0)}(1-z)+\gamma).$$ Now we choose $\alpha_n=(n+2)^{N}(\zeta(n+2)-1)$ and note that since $1$ is removed from the first term in $\zeta(s)$ that $\lim_{n\rightarrow\infty}(n+2)^{N}(\zeta(n+2)-1)=0,$ since exponential growth is faster than polynomial growth. The far right side of equation (2.11) may be construed as (2.2). Multiplying both sides by $z^{-2},$ and applying Proposition 1.1 we use the formula $$\lim_{z\rightarrow1^{-}}\frac{\partial^M}{\partial z^M}((1-z)\psi^{(l)}(2-z)z^{l+1})$$ $$=-M\lim_{z\rightarrow1^{-}}\frac{\partial^{M-1}}{\partial z^{M-1}}(\psi^{(l)}(2-z)z^{(l+1)})$$ $$=-M\sum_{j\ge0}\binom {M-1}{j}(-1)^{M-1-j}\psi^{(l+M-1-j)}(1)\frac{(l+1)!}{(l+1-j)!}.$$ We employed the trivial formula $\lim_{z\rightarrow1^{-}}\frac{\partial^M}{\partial z^M}(z^{l})=l!/(l-M)!$ in the last line. This proves the theorem after noting that the $M$-th derivative of $(1-z)z^{-1}-z^{-1}=-1$ is $0.$ Note that since $N\ge1,$ Theorem 2.3 is not a generalization of Theorem 2.2 and so Theorem 2.2 is not redundant. Conclusion ========== The conclusion we have come to here is that the summation formula that was established to prove interesting $q$-series identities may also be used to prove identities for series involving the Riemann zeta function. Some further interest should be directed toward finding expressions for sums of the form $$\sum_{n\ge0}a_n(L(n+\sigma+1)-1),$$ where the $a_n$ are appropriately chosen for the series to converge, and $L(s)$ is a Dirichlet series which is assumed to have its first term to be $1$ and converges when $\Re(s)>\sigma.$ We believe this is a curious incidence where attractive results in one area of mathematics may be grouped as a consequence of a formula which has produced attractive results in another area. [9]{} V. S. Adamchik, H. M. Srivastava, *Some series of the zeta and related functions,* Analysis (Munich) 18 (1998) 131–144. G. E. Andrews, R. Askey, and R. Roy. *Special Functions,* volume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, New York, 1999. G. E. Andrews, J. Jimenez-Urroz and K. Ono, *q-series identities and values of certain L- functions,* Duke Math. Journal 108 (2001), 395-419. G. E. Andrews, and P. Freitas, *Extension of Abel’s lemma with q-series implications,* Ramanujan J. 10 (2005), 137–152. J. Choi, Y. J. Cho, H. M. Srivastava, *Series involving the Zeta function and multiple Gamma functions,* Appl. Math. Comput., 159 (2004), no. 2, 509–537. M. Hashimoto, S. Kanemitsu, Y. Tanigawa, M. Yoshimoto, W.-P. Zhang, On some slowly convergent series involving the Hurwitz zeta-function, J. Computational Applied Math. 160 (2003), 113–123. A. Patkowski, *An Observation on the extension of Abel’s Lemma,* Integers 10 (2010), 793–800. 1390 Bumps River Rd.\ Centerville, MA 02632\ USA\ E-mail: alexpatk@hotmail.com
--- abstract: 'In categorical compositional semantics of natural language one studies functors from a category of grammatical derivations (such as a Lambek pregroup) to a semantic category (such as real vector spaces). We compositionally build game-theoretic semantics of sentences by taking the semantic category to be the category whose morphisms are open games. This requires some modifications to the grammar category to compensate for the failure of open games to form a compact closed category. We illustrate the theory using simple examples of Wittgenstein’s language-games.' author: - Jules Hedges - Martha Lewis bibliography: - 'lg.bib' title: 'Towards Functorial Language-Games' --- Introduction ============ A key component of language is that it is used by agents to communicate about the world they live in, and that it forms part of a network of actions used to interact with the world. This notion was used by Wittgenstein in his description of language games as a way of exploring how signs and symbols could come to refer to objects in the world [@wittgenstein_philosophical_investigations]. It has also been used in the artificial intelligence community as a way of enabling artificial systems to develop their own meanings and to examine how those meanings arise [@steels2003]. Relatedly, game theory has been applied to language research. [@jager2008] provides a review of how game theory has been to formalise the idea of communication as a signalling game, to examine the notion of relevance, and, using evolutionary game theory, to investigate how signals evolve to correspond to a convex region of representational space. Another approach to language is the field of computational linguistics, and in particular compositional distributional semantics such as that presented in [@coecke2010; @paperno2014]. This approach seeks to unify the rich semantic representations that can be learnt from text corpora with the compositional and generative power of formal semantics. However, it is agent-free. There is no sense of language being used by an agent, of its relation to the world, or of its part in a network of actions. To use ideas from compositional distributional semantics in a system interacting with the world, the theory must be developed further. In this paper we begin to unify the categorical compositional distributional semantics of [@coecke2010] with the categorical approach to game theory introduced in [@hedges_etal_compositional_game_theory; @hedges_towards_compositional_game_theory]. We use the category of open games [@hedges_etal_compositional_game_theory] to represent the semantics of words and phrases. However, the grammar used in [@coecke2010] is not available to used, since the category of open games is not compact closed. We therefore develop a novel grammar, similar to Lambek’s protogroup grammar [@lambek1997], and which is based on the free teleological category introduced in [@hedges_coherence_lenses_open_games]. Finally, we give an example based on one of Wittgenstein’s language games. Background ========== Categorical Compositional Semantics {#sec:DisCoCat} ----------------------------------- In the study of language at least two distinct camps can be identified. On one side, the way that words compose to form phrases and sentences is the object of research. On the other, the meanings of the words themselves is seen as paramount. These two approaches can be labelled syntax and semantics respectively. There is then also pragmatics, the aspect of how words, phrases and so on are actually used in discourse, but we do not deal with this at present. The categorical compositional approach was introduced in the context of distributional semantics by [@coecke2010], and further elucidated in [@bolt2017]. The approach was outlined in [@bolt2017] as follows: 1. 1. Choose a compositional structure, such as a pregroup, or other categorial grammar. 2. Interpret this structure as a category, the [*grammar category*]{}. 2. 1. Choose or craft appropriate meaning or concept spaces, such as vector spaces. \[item:meaningcategory\] 2. Organize these spaces into a category, the [*semantics category*]{}, with the same abstract structure as the grammar category. \[item:interpret\] 3. Interpret the compositional structure of the grammar category in the semantics category via a functor preserving the type reduction structure. \[item:reduction\] 4. Bingo! This functor maps type reductions in the grammar category onto algorithms for composing meanings in the semantics category. In [@coecke2010], the authors instantiated this approach in the context of compact closed categories using pregroup grammar as the grammar category and ${\mathbf{FVect}}$ or ${\mathbf{Rel}}$ as the semantic category. [@coecke2013] show how the grammar category may be changed to Lambek categorial grammar. In [@piedeleu2015; @balkir2015], the same approach was instantiated using pregroup grammar and the category of completely positive maps on ${\mathbf{FVect}}$, and [@bolt2017] introduce a category of convex algebras and convex relations as the semantics category. The key concept in all of the above examples is that of a *monoidal category*. This is a category with a parallel composition operation $\otimes$, which is associative, has a unit, and acts bifunctorially on objects and morphisms. Monoidal categories have a convenient diagrammatic calculus, which we will make use of in later sections. For details of this calculus, see for example [@coeckepaquette]. Many of the examples work in the framework of compact closed categories. A compact closed category is a monoidal category with left and right adjoints $(-)^l$, $(-)^r$ on objects and the morphisms: $$\label{eq:unitsandcounits} \epsilon^l_A:A^l \otimes A \rightarrow I_A, \qquad \epsilon^r_A: A \otimes A^r \rightarrow I_A, \qquad \eta_A^l: I_A \rightarrow A \otimes A^l, \qquad \eta_A^r : I_A \rightarrow A^r \otimes A$$ termed counits and units respectively, which satisfy the *snake equations* $$\begin{aligned} &(1_A \otimes \epsilon^l )\circ (\eta^l \otimes 1_A) = 1_A &\:& (\epsilon^r \otimes 1_A) \circ (1_A \otimes \eta^r ) = 1_A \\ &(\epsilon^l \otimes 1_{A^l})\circ (1_{A^l} \otimes \eta^l)= 1_{A^l} &\:& (1_{A^r} \otimes \epsilon^r ) \circ (\eta^r \otimes 1_{A^r} ) = 1_{A^r}\end{aligned}$$ In compact closed categories, the diagrammatic calculus is enhanced by the fact that we can bend wires. In our linguistic examples, we frequently use the notion of a pregroup grammar. A pregroup grammar is built from the free compact closed category over a set of basic types. Consider the set $\{n, s\}$ of types corresponding to nouns and declarative sentences. The tensor product is denoted by concatenation, allowing us to build composite types. For example, the type of a transitive verb is $n^r s n^l$. A string of types is judged *grammatical* if it reduces to the type $s$ under application of the unit and counit morphisms. So, the sentence $\textit{Gamblers take chances}$ is typed: $n \;n^rsn^l\;n$ and reduces as follows: $$n \;n^rsn^l\;n \xrightarrow{\epsilon^r_n \; 1_s\;\epsilon^l_n} sn^l\;n \xrightarrow{1_s\;\epsilon^l_n} s$$ More precisely, a pregroup grammar is a triple $G = (\mathcal L, T, s, \mathcal D)$ where: - $\mathcal L$ is the *lexicon*, a set of words - $T$ is a set of *basic types* - $s \in T$ is the distinguished *sentence type* - $\mathcal D \subseteq \mathcal L \times P_T$ is the *dictionary*, where $P_T$ is (the set of elements of) the free pregroup on $T$. We model the grammar in the category $\mathbf{FVect}$ of finite dimensional vector spaces. We must choose a finite-dimensional vector space ${[\![t]\!]}$ for every $t \in T$. This induces a monoidal functor ${[\![-]\!]} : \mathcal C_T \to \mathbf{FVect}$, where $\mathcal C_T$ is the free compact closed category with generating objects $T$, by specifying that ${[\![t^l]\!]} = {[\![t^r]\!]} = {[\![t]\!]}^*$ (where $-^*$ denotes the dual vector space), and ${[\![t_1 t_2]\!]} = {[\![t_1]\!]} \otimes {[\![t_2]\!]}$. (Note the careful distinction between $P_T$ and $\mathcal C_T$, to avoid the argument in [@preller_logical_distributional_models] that monoidal functors out of $P_T$ are necessarily trivial.) We must also specify a word-vector ${[\![w]\!]} \in {[\![t]\!]}$ for each word $w \in \mathcal L$ with $(w, t) \in \mathcal D$; in practice these vectors will be computed from corpora of text by statistical or machine learning methods. Now given a sentence $w_1 \ldots w_n$, we compute its semantic vector ${[\![w_1 \ldots w_n]\!]} \in {[\![s]\!]}$ by the following steps: 1. Choose a type $(w_i, t_i) \in \mathcal D$ for each word (this is automatic if each word in $\mathcal L$ is assigned a unique type by $\mathcal D$) 2. Choose a morphism $f : t_1 \cdots t_n \to s$ in $\mathcal C_T$, which represents a grammatical parsing of the sentence 3. Apply the linear map ${[\![f]\!]} : {[\![t_1]\!]} \otimes \cdots \otimes {[\![t_n]\!]} \to {[\![s]\!]}$ to the bag-of-words vector ${[\![w_1]\!]} \otimes \cdots \otimes {[\![w_n]\!]}$ In the current paper, we instantiate the approach with the category ${\mathbf{OG}}$ of open games as the semantics category rather than $\mathbf{FVect}$. This allows us to express commands and subsequent actions, and is therefore a richer setting than previously, in which there was no concept of agent, nor of action. The category of open games, however, is not compact closed. In particular it has counits, but no units. We therefore use the notion of a *protogroup* [@lambek1997] as the basis for our grammar category. Language Games {#sec:language-games} -------------- The approach to language outlined in [@coecke2010] is focussed on solving a particular problem in computational linguistics. However, language in general is used by agents, in the world, and together with other actions. This attitude to language was outlined in [@wittgenstein_philosophical_investigations]. Meanings of words arise from the situations in which they are used. The term [*language game*]{} is used to talk about small fragments of language that can be used to highlight how language obtains its meaning. This approach to language has been used as a way of bootstrapping language learning in artificial intelligence approaches. These are summarised in [@steels2003]. In brief, rather than attempt to specify representations of concepts for AI systems, these systems are given the ability to form their own representations, based on their interactions with the world and with other systems (including humans). The Talking Heads experiment [@steels1998origins] was designed to show how a population of interacting agents could develop a shared lexicon, together with perceptually grounded categorizations corresponding to items of the lexicon. The game is played as follows. Both the speaker and the hearer can see the same set of shapes. The speaker chooses a shape, and says some words corresponding to that shape. Then, the hearer must point out which shape it guesses. If the hearer gets the shape wrong, the speaker indicates which shape is the correct one. This game is played a number of times, with each agent playing both the role of the speaker and the hearer in different rounds. As the game is played, representations are built up and altered, and the sets of words and concepts of the agents become coordinated, without central control or telepathy. The language game we will outline in the current paper is based on Wittgenstein’s builder and assistant game. This will be given in more detail in section \[sec:example\]. Briefly, the game runs as follows: > The language is meant to serve for communication between a builder A and an assistant B. A is building with building-stones: there are blocks, pillars, slabs and beams. B has to pass the stones, in the order in which A needs them. For this purpose they use a language consisting of the words “block", “pillar", “slab", “beam". A calls them out; — B brings the stone which he has learnt to bring at such-and-such a call. Conceive this as a complete primitive language. [@wittgenstein_philosophical_investigations §2] In contrast to the Talking Heads game, the game here is not played iteratively. The assistant $B$ has already learnt what each word means. We write down how the language game can be formalized game-theoretically, and give an extension of it, in which imperative verbs are introduced. Open games ---------- A game, informally speaking, is a model of (any number of) interacting agents, in which each individual agent acts *rationally* or *strategically* in order to optimise some outcome that is personal to them. The notion of *Nash equilibrium* is used to describe an assignment of behaviours to each agent which are mutually rational, that is, each is rational under the assumption that the others are fixed. An open game is a piece of a game, which can be composed together to build ordinary games [@hedges_towards_compositional_game_theory; @hedges_etal_compositional_game_theory]. There is a symmetric monoidal category $\mathbf{OG}$ whose morphisms are (equivalence classes of) open games, in which the categorical composition and monoidal product are respectively sequential and simultaneous play of open games. The objects of the category $\mathbf{OG}$ are pairs of sets $\binom X S$, which represent sets of ordinary forward-flowing information, and ‘counterfactual’ information that appears to flow backwards. Let $X, S, Y, R$ be sets. An *open game* $\mathcal G : \binom X S \to \binom Y R$ is a 4-tuple $(\Sigma_\mathcal G, \mathbf P_\mathcal G, \mathbf C_\mathcal G, \mathbf E_\mathcal G)$ where: - $\Sigma_\mathcal G$ is a set, the set of *strategy profiles* (roughly, possible behaviours of $\mathcal G$ unconstrained by rationality) - $\mathbf P_\mathcal G : \Sigma_\mathcal G \times X \to Y$ is the *play function*, which runs a strategy profile on an observation to produce a choice - $\mathbf C_\mathcal G : \Sigma_\mathcal G \times X \times R \to S$ is the *coplay function*, which propagates payoffs further backwards in time - $\mathbf E_\mathcal G : X \times (Y \to R) \to \mathcal P (\Sigma_\mathcal G)$ is the *equilibrium function*, which gives a subset of $\mathbf E_\mathcal G (h, k) \subseteq \Sigma_\mathcal G$ of strategy profiles that are Nash equilibria for each *context* $(h, k)$. The context consists of a *history* $h : X$ (which says what happened in the past) and a *continuation* $k : Y \to R$ (which says what will happen in the future given the present). The string diagram language for $\mathbf{OG}$ has directed strings, where a pair $\binom X S$ is represented by an $X$-labelled string directed forwards together with an $S$-labelled string directed backwards. A general open game $\mathcal G : \binom X S \to \binom Y R$ is represented by a string diagram of the form \(X) at (0, .5) [$X$]{}; (S) at (0, -.5) [$S$]{}; (Y) at (4, .5) [$Y$]{}; (R) at (4, -.5) [$R$]{}; (G) \[rectangle, minimum height=2cm, minimum width=1cm, draw\] at (2, 0) [$\mathcal G$]{}; (X) to (G.west |- X); (G.east |- Y) to (Y); (R) to (G.east |- R); (G.west |- S) to (S); A *closed game* is a scalar $\mathcal G : I \to I$ in $\mathbf{OG}$, where $I = \binom 1 1$ is the monoidal unit. A closed game $\mathcal G$ is determined by a set $\Sigma_\mathcal G$ of strategy profiles and a subset $\mathbf E_\mathcal G \subseteq \Sigma_\mathcal G$ of Nash equilibria. As an example, consider a single decision made by an agent who observes an element of $X$ before choosing an element of $Y$, in order to maximise a real number. This situation is represented by the open game $\mathcal A : (X, 1) \to (Y, \mathbb R)$ defined by the following data: - The set of strategy profiles is $\Sigma_\mathcal A = X \to Y$ - The play function is $\mathbf P_\mathcal A (\sigma, x) = \sigma (x)$ - The coplay function has codomain $1$, hence is trivial - The equilibrium function is $\mathbf E_\mathcal A (h, k) = \{ \sigma : X \to Y \mid \sigma (h) \in \arg\max k \}$ We represent $\mathcal A$ by the string diagram \(X) at (0, 0) [$X$]{}; (Y) at (4, .5) [$Y$]{}; (R) at (4, -.5) [$R$]{}; (G) \[rectangle, minimum height=2cm, minimum width=1cm, draw\] at (2, 0) [$\mathcal A$]{}; (X) to (G.west |- X); (G.east |- Y) to (Y); (R) to (G.east |- R); We do not have space to define the entire categorical structure of $\mathbf{OG}$, instead referring the reader to [@hedges_etal_compositional_game_theory; @hedges_towards_compositional_game_theory], but we define categorical composition as an illustration. Let $\mathcal G : \binom X S \to \binom Y R$ and $\mathcal H : \binom Y R \to \binom Z Q$ be open games. The composite open game $\mathcal H \circ \mathcal G : \binom X S \to \binom Z Q$ is defined as follows: - The set of strategy profiles is $\Sigma_{\mathcal H \circ \mathcal G} = \Sigma_\mathcal G \times \Sigma_\mathcal H$ - The play function is $\mathbf P_{\mathcal H \circ \mathcal G} ((\sigma, \tau), x) = \mathbf P_\mathcal H (\tau, \mathbf P_\mathcal G (\sigma, x))$ - The coplay function is $\mathbf C_{\mathcal H \circ \mathcal G} ((\sigma, \tau), x, q) = \mathbf C_\mathcal G (\sigma, x, \mathbf C_\mathcal H (\tau, \mathbf P_\mathcal G (\sigma, x), q))$ - The equilibrium function is $$\mathbf E_{\mathcal H \circ \mathcal G} (h, k) = \{ (\sigma, \tau) \mid \sigma \in \mathbf E_\mathcal G (h, k_\tau) \text{ and } \tau \in \mathbf E_\mathcal H (\mathbf P_\mathcal G (\sigma, h), k) \}$$ where $k_\tau : Y \to R$ is given by $k_\tau (y) = \mathbf C_\mathcal H (\tau, y, k (\mathbf P_\mathcal H (\tau, y)))$. A *zero-player open game* is an open game $\mathcal G$ is one which has exactly one strategy profile, which is an equilibrium for every context. $\mathbf{OG}$ is not a compact closed category, but it does have a ‘partial duality’ which can be defined on objects by $\binom X S^* = \binom S X$. For every $X$ there is a canonical (zero-player) open game $\varepsilon_X : \binom X X \to I$, but there is no canonical dual $I \to \binom X X$. We denote $\varepsilon_X$ by the string diagram (X1) at (0, 2) [$X$]{}; (X2) at (0, 0) [$X$]{}; (X1) to \[out=0, in=90\] (1.25, 1) to \[out=-90, in=0\] (X2); Given a set $X$, we write $\overline X$ for $\binom X 1$ and $\underline X$ for $\binom 1 X$. A function $f : X \to Y$ has both covariant and contravariant liftings as zero-player open games, which we call $\overline f : \overline X \to \overline Y$ and $\underline f : \underline Y \to \underline X$, which are dual in this sense; but no other open game has a dual. The former has play function $\mathbf P_{\overline f} (*, x) = f (x)$, and the latter has coplay function $\mathbf C_{\underline f} (*, *, x) = f (x)$. We denote these by (X1) at (0, 0) [$X$]{}; (Y1) at (3, 0) [$Y$]{}; (f1) at (1.5, 0) [$f$]{}; (X1) to (f1); (f1) to (Y1); (X2) at (8, 0) [$X$]{}; (Y2) at (5, 0) [$Y$]{}; (f2) at (6.5, 0) [$f$]{}; (X2) to (f2); (f2) to (Y2); Open games by example --------------------- The easiest way to show the expressive power of open games is by examples of closed games $\mathcal G : I \to I$, which are represented by string diagrams with no open strings passing the left or right boundaries. Given agents $\mathcal A_i : I \to \binom{X_i}{\mathbb R}$ as defined above (where $\mathcal A_i$ is a decision of an agent making an observation from $1 = \{ * \}$ and a choice from $X_i$), the $n$-fold monoidal product $\displaystyle \bigotimes_{i = 1}^n \mathcal A_i : I \to \begin{pmatrix} \prod_{i = 1}^n X_i \\ \mathbb R^n \end{pmatrix}$ represents a game in which $n$ player make choices simultaneously. The set of strategy profiles of this game is $\Sigma_{\bigotimes_{i = 1}^n \mathcal A_i} = \prod_{i = 1}^n X_i$ and, given a payoff function $k : \prod_{i = 1}^n X_i \to \mathbb R^n$, the equilibrium set $\mathbf E_{\bigotimes_{i = 1}^n \mathcal A_i} (*, k)$ is the set of Nash equilibria of the $n$-player game with payoff function $k$, namely $$\mathbf E_{\bigotimes_{i = 1}^n \mathcal A_i} (*, k) = \left\{ \left. \sigma : \prod_{i = 1}^n X_i\ \right|\ k (\sigma)_i \geq k (\sigma [i \mapsto x_i])_i \text{ for all } 1 \leq i \leq n \text{ and } x_i : X_i \right\}$$ We can extend this to a closed game by fixing some particular $k$, by lifting $k$ as a function and combining it with $\varepsilon$. For example, suppose we restrict to $n = 2$ and pick some payoff function $k : X \times Y \to \mathbb R^2$. This is the class of ‘bimatrix games’ and contains many famous examples such as the prisoner’s dilemma and chicken, depending on the choice of $X$, $Y$ and $k$. The 2-player closed game with payoff function $k$ is represented by the string diagram (D1) at (0, 3) [$\mathcal A_1$]{}; (D2) at (0, 0) [$\mathcal A_2$]{}; (U) at (3, 3) [$k$]{}; (d1) at (0, -.5) ; (d2) at (0, .5) ; (d3) at (0, 2.5) ; (d4) at (0, 3.5) ; (d5) at (0, 1) ; (d6) at (0, 2) ; (D1.east |- d4) to node \[above\] [$X$]{} (U.west |- d4); (D2.east |- d2) to \[out=0, in=180\] node \[above=5pt, very near start\] [$Y$]{} (U.west |- d3); (U.east |- d4) to \[out=0, in=90\] node \[above, near start\] [$\mathbb R$]{} (4.5, 2.5) to \[out=-90, in=0\] (3, .5) to \[out=180, in=0\] node \[above, very near end\] [$\mathbb R$]{} (D1.east |- d3); (U.east |- d3) to \[out=0, in=90\] node \[below, very near start\] [$\mathbb R$]{} (4.5, .5) to \[out=-90, in=0\] (3, -.5) to node \[below, very near end\] [$\mathbb R$]{} (D2.east |- d1); Recall that a closed game is defined by a set of strategy profiles and a subset of Nash equilibria. The closed game $\mathcal G : I \to I$ defined by this string diagram has strategy profiles $\Sigma_\mathcal G = X \times Y$, and Nash equilibria $$\mathbf E_\mathcal G = \{ (x, y) : X \times Y \mid k (x, y)_1 \geq k (x', y)_1 \text{ for all } x' : X \text{, and } k (x, y)_2 \geq k (x, y')_2 \text{ for all } y' : Y \}$$ Next, given a decision $\mathcal A : \binom X 1 \to \binom{Y}{\mathbb R}$ that observes an $X$ and chooses a $Y$, we form the composite $\mathcal A^\Delta : \binom X 1 \to \binom{X \times Y}{\mathbb R}$ defined by the string diagram \(X) at (-3, .75) [$X$]{}; (Y) at (2, .5) [$Y$]{}; (R) at (2, -.5) [$\mathbb R$]{}; (X2) at (2, 1.5) [$X$]{}; (G) at (0, 0) [$\mathcal A$]{}; (m) at (-1.75, .75) ; (X) to (m); (m) to \[out=-45, in=180\] (G); (m) to \[out=45, in=180\] (X2); (G.east |- Y) to (Y); (R) to (G.east |- R); where the node denotes the lifting of the copying function $\Delta : X \to X \times X$. This important gadget has set of strategy profiles $\Sigma_{\mathcal A^\Delta} = X \to Y$, play function $\mathbf P_{\mathcal A^\Delta} (\sigma, x) = (x, \sigma (x))$ and equilibrium function $$\mathbf E_{\mathcal A^\Delta} (h, k) = \{ \sigma : X \to Y \mid \sigma (h) \in \arg\max k (h, -) \}$$ Using this gadget we can build sequential games. For example, a 2-player sequential game in which the first player chooses from $X$, the second player observes this perfectly before choosing a $Y$, and payoffs being given by $k : X \times Y \to \mathbb R^2$ can be built from the decision $\mathcal A_1 : I \to \binom{X}{\mathbb R}$ and $\mathcal A_2 : \binom{X}{1} \to \binom{Y}{\mathbb R}$ using the string diagram (D1) at (0, 0) [$\mathcal A_1$]{}; (m) at (1.75, .5) ; (D2) at (3.5, 0) [$\mathcal A_2$]{}; (q) at (5.5, 1) [$k$]{}; (d1) at (0, -.5) ; (d2) at (0, 1.5) ; (D1.east |- m) to node \[above, near start\] [$X$]{} (m); (m) to \[out=45, in=180\] node \[above, very near end\] [$X$]{} (q.west |- d2); (m) to \[out=-45, in=180\] node \[below, near end\] [$X$]{} (D2); (D2.east |- m) to node \[above\] [$Y$]{} (q.west |-m); (q.east |- d2) to \[out=0, in=90\] node \[above, very near start\] [$\mathbb R$]{} (7, 0) to \[out=-90, in=0\] (3.5, -1.5) to \[out=180, in=0\] node \[below, very near end\] [$\mathbb R$]{} (D1.east |- d1); (q.east |- m) to \[out=0, in=90\] node \[above, near start\] [$\mathbb R$]{} (6.5, 0) to \[out=-90, in=0\] node \[above, very near end\] [$\mathbb R$]{} (D2.east |- d1); Similarly, if the second player cannot perfectly observe the first move $x$ but only its image under some function $f : X \to Z$, we can model this using the string diagram obtained from the previous one by inserting an $f$-labelled node between the copying node and $\mathcal A_2 : \binom{Z}{1} \to \binom{Y}{\mathbb R}$. As a typical example we impose an equivalence relation $\sim$ on $X$, and let $f : X \to X / \sim$ be the projection onto the quotient. Such equivalence classes are known as ‘information sets’ in game theory. If $f : X \to 1$ then we recover a simultaneous game, since deleting is the counit for the black comonoid. Note that each of the previous examples readily generalises to games of $n$ players. Process grammar {#sec:process-grammar} =============== We analyse the methodology of categorical compositional distributional semantics, arguing that we need to modify the concept of a pregroup grammar in order to compensate for the *semantic* category lacking a compact closed structure. Specifically, we argue that pregroup grammars rely on the semantic category satisfying state-process duality, a feature not present in the category of open games. While the grammatical parsing of sentences using a pregroup grammar relies only on the *counits* $x^l x \leq 1$, $x x^r \leq 1$ of a pregroup’s compact closed structure [@lambek1997 corollary 1], state-process duality in the semantic category relies on the *units* $I \to X \otimes X^*$. In order to capture this Lambek defines *protogroups* similarly to pregroups, but having only counits. Consider the typing of a noun phrase consisting of an adjective and a noun, such as *large slabs*. Typically, we have a primitive type of noun phrases $n \in T$, and the dictionary $\mathcal D$ contains the entries $(\text{slabs}, n)$ and $(\text{large}, n n^l)$. We have a semantic functor ${[\![-]\!]} : \mathcal C_T \to \mathbf{FVect}$, whose definition includes a choice of vector space $N = {[\![n]\!]}$, the *noun space*. By functorality, the type $n n^l$ of *large* is sent to the tensor product $N \otimes N^*$. However, we might imagine that the the semantics of an adjective is fundamentally a (linear) *function*, where for example the semantics of *large* is a function that takes the semantics of a noun phrase $x$ to the semantics of the noun phrase *large* $x$. The unit $\eta_N : I \to N \otimes N^*$ in $\mathbf{FVect}$ (where $I = \mathbb R$) is what allows us to encode a linear map $f : N \to N$ as a *vector* $\overline f \in N \otimes N^*$, such that function application $f (v)$ can be recovered (after encoding vectors in $V$ as linear maps $I \to V$) via the *counit* $\varepsilon_N : N^* \otimes N \to I$ $$I \cong I \otimes I \xrightarrow{\overline f \otimes v} (N \otimes N^*) \otimes N \cong N \otimes (N^* \otimes N) \xrightarrow{N \otimes \varepsilon_N} N \otimes I \cong N$$ This entire methodology fails when we replace $\mathbf{FVect}$ with a different semantic category, such as $\mathbf{OG}$, which does not have state-process duality. ($\mathbf{OG}$ is additionally not monoidal closed, which also rules out using a Lambek calculus.) In this section we develop an alternative. Notice that the following sets are equal: (1) Elements of the free pregroup on $T$; (2) Objects of the strict free compact closed category with generating objects $T$; (3) Elements of the free protogroup on $T$; (4) Objects of the free strict teleological category with generating objects $T$. See [@lambek1997; @preller_lambek_free_compact_2_categories; @kelly80; @hedges_coherence_lenses_open_games] for various characterisations of these free structures. A ‘teleological category’ is a category whose abstract structure is similar to $\mathbf{OG}$, in particular having counits but no units; posetal teleological categories are protogroups. A *process grammar* is a triple $G = (\mathcal L, T, s, \mathcal D)$ where - $\mathcal L$ is the *lexicon*, i.e. the set of words - $T$ is the set of primitive types - $s \in P_T$ is the *sentence type*, where $P_T$ is (the set of elements of) the free pregroup on $T$ - The dictionary $\mathcal D$ is a subset $\mathcal D \subseteq \mathcal L \times P_T \times P_T$ The difference between this definition and the standard definition of a pregroup grammar (see section \[sec:DisCoCat\]) is that words are associated to *pairs* of types, rather than single types. Given a dictionary entry $(w, t) \in \mathcal D$ in a pregroup grammar, we view the semantics ${[\![w]\!]}$ as a *state* $I \to {[\![t]\!]}$. Given a dictionary entry $(w, t, t') \in \mathcal D$ in a process grammar, we view the semantics ${[\![w]\!]}$ as a *process* ${[\![t]\!]} \to {[\![t']\!]}$. It was not necessary in the pregroup grammar to reprent the semantics ${[\![w]\!]}$ as a process, since in a compact closed category a morphism $A \rightarrow B$ can always be represented as a morphism $I \rightarrow B \otimes A^*$, and hence we only need keep track of one object. In contrast, in a process grammar we need both objects $A$ and $B$. We consider every pregroup grammar $(\mathcal L, T, s, \mathcal D)$ as a process grammar $(\mathcal L, T, s, \mathcal D')$ using the dictionary $\mathcal D' = \{ (w, 1, t) \mid (w, t) \in \mathcal D \}$. Let $G = (\mathcal L, T, s, \mathcal D)$ be a process grammar. We write $\mathcal C_G$ for the free teleological category with - The set of generating objects is $T$ - The set of generating morphisms[^1] is $\{ w_{t, t'} : t \to t' \mid (w, t, t') \in \mathcal D \}$ The free teleological category is characterised in [@hedges_coherence_lenses_open_games] as a category whose morphisms are equivalence classes of suitable string diagrams. With a pregroup grammar $G = (\mathcal L, T, s, \mathcal D)$, parsing a sentence $w_1 \ldots w_n$ is a 2-stage process: 1. Choose dictionary entries $(w_i, t_i) \in \mathcal D$ 2. Choose a morphism $t_1 \cdots t_n \to s$ in $\mathcal C_T$, where $\mathcal C_T$ is the free compact closed category with generating objects $T$ With a process grammar $\mathcal G$, these two stages become conflated: a choice of morphism $1 \to s$ in $\mathcal C_G$ contains both a choice of type for each word, and a reduction to the sentence type, since the types of each word represent the processes that comprise the reduction. This method of parsing is also considered in [@toumi_categorical_compositional_distributional_questions], where its equivalence to the previously described method is proved. There is a problem with this: the sentence has disappeared! Out of all of the morphisms $1 \to s$ in $\mathcal C_G$, how do we recognise those which are parsings of a particular sentence $w_1 \ldots w_n$? We must identify the morphisms $1 \to s$ that corresponds to the sentence we actually want to parse. A solution to this is to present the free teleological category $\mathcal C_G$ using a sequent calculus, which can be done in a standard way by identifying suitable proofs [@blute_scott_category_theory_linear_logicians; @lambek_scott_introduction_higher_order_categorical_logic], where the categorical composition is cutting proofs together. Specifically, consider a noncommutative linear calculus whose judgements are $\varphi \vdash \psi$ for $\varphi, \psi \in P_T$. For each $t \in T$ we have three identity axioms: $$(l\text{-id})\frac{}{t^l \vdash t^l} \qquad\qquad (\text{id})\frac{}{t \vdash t} \qquad\qquad (r\text{-id})\frac{}{t^r \vdash t^r}$$ For each $(w, \varphi, \psi) \in \mathcal D$ we have an axiom $$(w_{\varphi, \psi})\frac{}{\varphi \vdash \psi}$$ Only two of the four axioms for negation are present, reflecting that our grammar corresponds to protogroups rather than pregroups: $$(l\text{-intro})\frac{\varphi \vdash t \psi}{t^l \varphi \vdash \psi} \qquad\qquad (r\text{-intro})\frac{\varphi \vdash \psi t}{\varphi t^r \vdash \psi}$$ Finally, we have the cut and $\otimes$-introduction rules: $$(\text{cut})\frac{\varphi \vdash \psi \qquad \psi \vdash \chi}{\varphi \vdash \chi} \qquad\qquad (\otimes\text{-intro})\frac{\varphi \vdash \psi \qquad \varphi' \vdash \psi'}{\varphi \varphi' \vdash \psi \psi'}$$ With this setup, we can define a parsing of the sentence $w_1 \ldots w_n$ to be a derivation of $\vdash s$ such that, if we read the non-identity axioms used in the derivation from left to right, they are labelled by the $w_i$ in the correct order. Derivations then witness morphisms in $\mathcal C_G$, and so we can define the sentence semantics by applying a semantic functor directly to the derivation. As an example, suppose $\mathcal L$ contains the words *bring large slabs*, $T$ contains $n$ and $s$, and $\mathcal D$ contains the entries $(\text{bring}, 1, s n^l)$, $(\text{large}, n^l, n^l)$ and $(\text{slabs}, 1, n)$. Here is an example of a derivation of the sentence *bring large slabs*: Note that the restriction to the free teleological category does not restrict the sentences which are grammatical: Lambek’s switching lemma [@Lambek01] states that a sequence of epsilon and identity maps suffice for the representation of the grammatical structure of any sentence in a pregroup grammar, and hence the grammatical structure of any sentence in a pregroup grammar may also be represented in the teleological grammar. Language-games as functors ========================== Given the pieces we have set up, with the power of category theory our model can be stated very simply: We study functors from a protogroup (or more precisely, a free teleological category) to the category of open games. More precisely, we require that this functor preserves all of the structure of the protogroup, so it should be a monoidal functor and also respect duals. Functors with the required properties were named *teleological functors* in [@hedges_coherence_lenses_open_games], although we again need to generalise to the non-symmetric case. Let $G = (\mathcal L, T, s, \mathcal D)$ be a process grammar. A *functorial language-game* ${[\![-]\!]}$ on $G$ is determined by the following data: - For each type $t \in T$, a pair of sets ${[\![t]\!]}$ - For each dictionary entry $(w, t_1, t_2) \in \mathcal D$, a choice of open game ${[\![(w, t_1, t_2)]\!]} : {[\![t_1]\!]} \to {[\![t_2]\!]}$ Here ${[\![t_i]\!]}$ is the object of $\mathbf{OG}$ (pair of sets) computed functorially in the evident way: ${[\![t t']\!]} = {[\![t_1]\!]} \otimes {[\![t_2]\!]}$ (componentwise cartesian product), and ${[\![t^l]\!]} = {[\![t^r]\!]} = {[\![t]\!]}^*$ (swapping the pair). A functorial language-game ${[\![-]\!]}$ on $G$ induces a functor ${[\![-]\!]} : \mathcal C_G \to {\mathbf{OG}}$ in an evident way, since the data specifies its action on the generators of $\mathcal C_G$. In particular we take ${[\![\varepsilon_t]\!]} = \varepsilon_{{[\![t]\!]}}$. A functorial language-game associates a game to every sentence generated by a process grammar, in which the structure of the information flow in the game exactly reflects the grammatical structure of the sentence. In practice this is a very strong restriction, and it is unclear whether it is actually desirable; in this paper we are demonstrating that it is in principle possible, at least in simple cases. In particular, consider a sentence of the form $w_1 \ldots w_i \ldots w_j \ldots w_n$, in particular with $w_i$ appearing before $w_j$ in the list of words. In the resulting open game, the part corresponding to $w_i$ happens temporally before the part corresponding to $w_j$. Depending on the types, players in $w_j$ might or might not be able to observe the result of $w_i$; but there is no possible way for players in $w_i$ to observe $w_j$. This is awkward, and is also very sensitive to word order in the language we are modelling (English in our case). For example, in the next section we will model sentences such as *bring slabs*, where *bring* contains an agent who utters the order with a strategic preference for slabs to be brought. However, the agent representing *bring* is unable to observe *slabs*! Instead, we will define *slabs* as a zero-player open game, which means that the agent in *bring* is able to perfectly anticipate it. This is a subtle distinction, related to the strange categorical structure of $\mathbf{OG}$, but one simple consequence is that the agent’s strategy cannot be a function that chooses an order for every possible noun phrase. Example {#sec:example} ======= We will build an example based on [@wittgenstein_philosophical_investigations]’s example of the master builder and apprentice. Consider a universe that contains *slabs* and *planks*, both in two sizes, *small* and *large*. The master can order the apprentice either to *bring* an object, or to *cut* it. We build a simple categorial grammar to describe the master’s commands. (Imperative) sentences in this grammar will be interpreted as open games containing a single player, representing the master, who makes a move representing the order and receives utility if the appropriate action is carried out. This open game can be *closed* by composing it with a second player, the apprentice. We can study the Nash equilibria of the resulting game. We define a process grammar $G = (\mathcal L, T, s, \mathcal D)$ as follows. Let $\mathcal L = \{ \text{slabs}, \text{planks}, \text{large}, \text{bring}, \text{cut} \}$. Let $T = \{ n, s \}$ where $n$ is the basic type of noun phrases and $s$ is the basic type of imperative sentences. The dictionary is given by $$\text{slabs} : 1 \leq n \qquad\qquad \text{planks} : 1 \leq n \qquad\qquad \text{large} : n^l \leq n^l$$ $$\text{bring} : 1 \leq s n^l \qquad\qquad \text{cut} : 1 \leq s n^l$$ where $w : x \leq y$ means $(w, x, y) \in \mathcal D$. We interpret the nouns and adjectives as zero-player open games lifted from a simple Montague-like semantics. Let $U$ be a semantic universe, with chosen subsets $S, P, L \subseteq U$ of objects that are slabs, planks, and large things. We begin building a semantic functor ${[\![-]\!]} : \mathcal C_G \to \mathbf{OG}$. We firstly interpret ${[\![n]\!]} = \overline{\mathcal P (U)}$. The nouns *slabs* and *planks* are interpreted as the zero-player open games containing the constant function returning the subsets of $S$ and $P$ respectively. So ${[\![\text{slabs}]\!]} : I \to \overline{\mathcal P (U)}$ is the zero-player open game with play function $\mathbf P_{{[\![\text{slabs}]\!]}} (*, *) = S$, and similarly for *planks*. Consider the function $\cap L : \mathcal P (U) \to \mathcal P (U)$ given by $X \mapsto X \cap L$, i.e. it takes a set of things to the subset of those things which are large. *Large* has the grammatical process-type $n^l \leq n^l$, so ${[\![\text{large}]\!]}$ must be an open game $\underline{\mathcal P (U)} \to \underline{\mathcal P (U)}$. We define ${[\![\text{large}]\!]} = \underline{\cap L}$. We model an imperative verb as a player (representing the master) choosing from a choice of actions, which we think of as utterances in some language. This language could be defined to be the same language defined by the categorial grammar we are considering, but in general it need not be. Let $O$ be a set of possible utterances (orders) in this language, which we assume nothing about; defining $O$ to be the set of well-typed sentences of $G$ is one possible example. We need to define a preference structure for the master. He receives two outcomes: the first, which will be a constant, represents the set of objects referred to by the subsequent noun phrase. The second represents the action done by the apprentice. If the verb in question is *bring* then the outcome $(S, a)$ is preferred, where $S \subseteq U$ and $a$ is an action, precisely when $a$ represents bringing some element of $S$. Similarly if the verb is *cut* then $(S, a)$ is preferred when $a$ represents cutting some element of $S$. Let $A = \{ B (x) \mid x \in U \} \cup \{ C (x) \mid x \in U \}$ be the set of possible actions, where $B (x)$ represents bringing $x$ and $C (x)$ represents cutting it. We define the sentence space to be ${[\![s]\!]} = \binom O A$. This means that a sentence with derivation $1 \to s$ is represented as an open game $I \to \binom O A$ which chooses an element of $O$ with outcomes in $A$. The verb *bring* has grammatical process type $1 \leq s n^l$, which means its semantics must be an open game with type ${[\![\text{bring}]\!]} : I \to \binom O A \otimes \underline{\mathcal P (U)} = \binom{O}{A \times \mathcal P (U)}$. We define it as the following open game: - The set of strategy profiles is $\Sigma_{{[\![\text{bring}]\!]}} = O$ - The play function $\mathbf P_{{[\![\text{bring}]\!]}} : \Sigma_{{[\![\text{bring}]\!]}} \times 1 \to O$ is given by $\mathbf P_{{[\![\text{bring}]\!]}} (o, *) = o$ - The coplay function $\mathbf C_{{[\![\text{bring}]\!]}} : \Sigma_{{[\![\text{bring}]\!]}} \times 1 \times A \times \mathcal P (U) \to 1$ is given by $\mathbf C_{{[\![\text{bring}]\!]}} (o, *, a, S) = *$ - The equilibrium function $\mathbf E_{{[\![\text{bring}]\!]}} : 1 \times (O \to A \times \mathcal P (U)) \to \mathcal P (\Sigma_{{[\![\text{bring}]\!]}})$ is given by $$\mathbf E_{{[\![\text{bring}]\!]}} (*, k) = \{ o \mid k (o)_1 = B (x) \text{ for some } x \in k (o)_2 \}$$ The open game ${[\![\text{cut}]\!]} : I \to \binom{O}{A \times \mathcal P (U)}$ is defined in the same way, except that $B (x)$ is changed to $C (x)$. Consider the sentence *bring large slabs*, with the derivation $1 \leq s$ given at the end of section \[sec:process-grammar\]. Applying the functor ${[\![-]\!]}$ produces the open game ${[\![\text{bring large slabs}]\!]} : I \to \binom O A$ represented by the diagram: (bring) \[isosceles triangle, isosceles triangle apex angle=90, shape border rotate=180, minimum width=2cm, draw\] at (-2, 0) [${[\![\text{bring}]\!]}$]{}; (large) \[trapezium, trapezium left angle=75, trapezium right angle=0, shape border rotate=270, trapezium stretches=true, minimum height=1cm, minimum width=2cm, draw\] at (1, 1) [[\[\]]{}]{}; (slabs) \[isosceles triangle, isosceles triangle apex angle=90, shape border rotate=180, minimum width=2cm, draw\] at (3, 3) [${[\![\text{slabs}]\!]}$]{}; (O) at (6, 0) [$O$]{}; (A) at (6, -1) [$A$]{}; (slabs) to \[out=0, in=90\] node \[above, near start\] [$\mathcal P (U)$]{} (5, 2) to \[out=-90, in=0\] (3, 1) to node \[above\] [$\mathcal P (U)$]{} (large); (large) to node \[above\] [$\mathcal P (U)$]{} (bring.east |- large); (bring) to (O); (A) to (bring.east |- A); Explicitly, this open game is given by the following data, which can be computed compositionally given the operations on $\mathbf{OG}$: - The set of strategy profiles is $\Sigma_{{[\![\text{bring large slabs}]\!]}} = O$ - The play function is $\mathbf P_{{[\![\text{bring large slabs}]\!]}} (o, *) = o$ - The equilibrium function, which takes a parameter $k : O \to A$, is $$\mathbf E_{{[\![\text{bring large slabs}]\!]}} (k) = \{ o : O \mid k (o) = B (x) \text{ for some } x \in S \cap L \}$$ In other words, given a continuation $k : O \to A$ that converts orders into actions (which abstracts over the behaviour of an as-yet-unspecified apprentice), the good orders from the master’s perspective are the ones that the continuation will convert into the action of bringing some slab. It should be noted that this open game contains no built-in relationship between the orders $O$ and actions $A$, but rather takes this as a parameter through $k$; that is to say, our model of the master has not fixed a semantics of orders. This open game defines the master’s half of the situation, abstracting out everything else into the continuation $k$. In order to produce a genuine game we must compose with another open game representing the decision made by the apprentice. In our setup this is an extra post-processing step, and has not been produced functorially from the sentence’s grammar. In order to make the apprentice into a game-theoretic agent, we must consider what his objectives are. In Wittgenstein’s example the master utters the words “Bring large slabs”, and then the apprentice brings slabs; but even assuming the agent understands the master’s language, why should he follow the order? In order to model the situation game-theoretically, we must define the apprentice’s objectives such that following the order becomes rational. For now we will take a simple option: The apprentice has a built-in translation from $O$ to $A$, and has a preference to carry out the received order according to this endogenous language, in line with the Wittgenstein quote in section \[sec:language-games\]. Fix a function $f : O \times A \to \mathbb R$, which is the apprentice’s judgement of the similarity between orders and outcomes. We define the apprentice as an agent who, on receiving the order $o : O$, will choose some $a : A$ in order to maximise the similarity $f (o, a)$. Let $\mathcal G : \binom O A \to I$ be the open game defined by the diagram \(O) at (0, 2) [$O$]{}; (A) at (0, 0) [$A$]{}; (copy) \[circle, scale=.5, fill=black, draw\] at (2, 2) ; (copy2) \[circle, scale=.5, fill=black, draw\] at (9, 0) ; (argmax) \[rectangle, minimum height=2cm, draw\] at (4, 1.5) [$\arg\max$]{}; (f) \[trapezium, trapezium left angle=75, trapezium right angle=75, shape border rotate=90, trapezium stretches=true, minimum height=.75cm, minimum width=1.5cm, draw\] at (6, 1) [$f$]{}; (dummy) at (0, .5) ; (O) to (copy); (copy) to \[out=-45, in=180\] node \[below, near end\] [$O$]{} (argmax); (argmax.east |- O) to node \[above, very near start\] [$A$]{} (9, 2) to \[out=0, in=90\] (10.5, 1) to \[out=-90, in=0\] (copy2); (copy2) to \[out=135, in=0\] node \[below, very near end\] [$A$]{} (f.east |- argmax); (f) to node \[below\] [$\mathbb R$]{} (argmax.east |- f); (copy2) to \[out=-135, in=0\] (A); (copy) to \[out=45, in=180\] (9, 3.5) to \[out=0, in=90\] (11.5, 1) to \[out=-90, in=0\] (9, -1) to \[out=180, in=0\] node \[below, very near end\] [$O$]{} (f.east |- dummy); Concretely, it is given by the following data: - The set of strategy profiles is $\Sigma_\mathcal G = O \to A$ - The coplay function $\mathbf C_\mathcal G : \Sigma_\mathcal G \times O \times 1 \to A$ is $\mathbf C_\mathcal G (\sigma, o, *) = \sigma (o)$ - The equilibrium function $\mathbf E_\mathcal G : O \to \mathcal P (\Sigma_\mathcal G)$ is $$\mathbf E_\mathcal G (o) = \{ \sigma \mid \sigma (o) \in \arg\max_{a : A} f (o, a) \}$$ The final step is to form the composition $\mathcal G \circ {[\![\text{bring large slabs}]\!]} : I \to I$, a closed game. In a Nash equilibrium of the composite, the master uses the apprentice’s built-in language in order to choose an order. Concretely, the data specifying this game is: - The set of strategy profiles is $\Sigma_{\mathcal G \circ {[\![\text{bring large slabs}]\!]}} = O \times (O \to A)$ - The Nash equilibria, which are now simply a subset of $\Sigma_{\mathcal G \circ {[\![\text{bring large slabs}]\!]}}$, are $$\{ (o, \sigma) \mid \sigma (o) = B (x) \text{ for some } x \in S \cap L \text{ and } \sigma (o) \in \arg\max_{a : A} f (o, a) \}$$ Since this is a sequential game we can compute equilibria by a method known as backward induction, which also reveals the way that the agents themselves might reason. Although $o \mapsto \arg\max_{a : A} f (o, a)$ is a multi-valued function, each possible choice defines an optimal strategy $\sigma : O \to A$ for the apprentice. These $\sigma$ are the ways that the master can deduce the apprentice might behave, given the apprentice’s understanding of the language. It may also be reasonable to assume that $f$ specifies the language unambiguously in the sense that each $\arg\max_{a : A} f (o, a)$ is unique, in which case $\sigma$ is uniquely defined. A choice of $\sigma$ completely describes the apprentice’s expected behaviour, so now we can reason as the master. He would like to choose some order $o \in O$ such that $\sigma (o)$ is one of the actions that he will be satisfied with. In this case, the set of good actions is $B [S \cap L]$ (the forward image of $S \cap L$ under $B$), so the set of good orders given $\sigma$ is $\sigma^{-1} [B [S \cap L]]$. Outlook ======= The theory as developed so far describes a simple signalling game, but allows for compositional aspects of language to be included. Future work will introduce consequences for the apprentice, thereby enabling the master and apprentice to coordinate on a choice of language. There are several possibilities here. A nice idea is that there are *economic* consequences for the apprentice, such as being fired or having his pay docked. A more linguistic angle is that the master verbally thanks or scolds the apprentice depending on the action, and the apprentice has preferences over these utterances. Both of these have the attractive feature that the master and apprentice do not automatically share a language (precisely, a relationship between orders $O$ and actions $A$), but in each Nash equilibrium they successfully coordinate on a choice of language. Both of these require extending the game with further stages in the future, where the master moves again after the apprentice, possibly in an infinitely repeated game [@ghani_kupke_lambert_forsberg_compositional_treatment_iterated_open_games]. Doing this functorially will require more extensive changes to the grammar, so we leave it for future work. Game theoretic approaches to linguistics show that convex concepts emerge from agents’ interactions: using a more structured meaning space would allow us to investigate this. [^1]: Formally, a teleological category distinguishes ‘dualisable morphisms’ and ‘non-dualisable morphisms’. For simplicity, we consider all generators of $\mathcal C_G$ to be non-dualisable.
--- abstract: | Considering a finite Borel measure $ \mu $ on $ \mathbb{R}^d $, a pair of conjugate exponents $ p, q $, and a compatible semi-inner product on $ L^p(\mu) $, we introduce $ (p,q) $-Bessel and $ (p,q) $-frame measures as a generalization of the concepts of Bessel and frame measures. In addition, we define notions of $ q $-Bessel and $ q$-frame in the semi-inner product space $ L^p(\mu) $. Every finite Borel measure $\nu$ is a $(p,q)$-Bessel measure for a finite measure $ \mu $. We construct a large number of examples of finite measures $ \mu $ which admit infinite $ (p,q) $-Bessel measures $ \nu $. We show that if $ \nu $ is a $ (p,q) $-Bessel/frame measure for $ \mu $, then $ \nu $ is $ \sigma $-finite and it is not unique. In fact, by using convolutions of probability measures, one can obtain other $ (p,q) $-Bessel/frame measures for $ \mu $. We present a general way of constructing a $ (p,q) $-Bessel/frame measure for a given measure. **Keywords:** Fourier frame, Plancherel theorem, spectral measure, frame measure, Bessel measure, semi-inner product. **2010 AMS Subject Classification**: Primary 28A99, 46E30, 42C15. author: - | Fariba Zeinal Zadeh Farhadi$^{1}$, Mohammad Sadegh Asgari$^{2}$,\ Mohammad Reza Mardanbeigi$^{3^*}$ and Mahdi Azhini$^{4}$ title: '**Generalized Bessel and Frame Measures**' --- Introduction ============ According to [@5], a Borel measure $ \nu $ on $ \mathbb{R}^d $ is called a dual measure for a given measure $ \mu $ on $ \mathbb{R}^d $ if for every $ f\in L^2(\mu) $, $$\label{11} \int_{\mathbb{R}^d} |\widehat{f d\mu}(t)|^2 d\nu(t) \simeq \int_{\mathbb{R}^d} |f(x)|^2 d\mu(x),$$ where for a function $ f\in L^1(\mu) $ the Fourier transform is given by $$\widehat{f d\mu}(t) = \int_{\mathbb{R}^d} f(x) e^{-2\pi it\cdot x} d\mu(x) \qquad (t\in \mathbb{R}^d).$$ Precisely, the equivalence in Equation (\[11\]) means that there are positive constants $A$ and $B$ independent of the function $f(x)$ such that $$A\int_{\mathbb{R}^d} |f(x)|^2 d\mu(x) \leq \int_{\mathbb{R}^d} |\widehat{f d\mu}(t)|^2 d\nu(t) \leq B\int_{\mathbb{R}^d} |f(x)|^2 d\mu(x).$$ Therefore when $ A=B=1 $, by Plancherel’s theorem for Lebesgue measure $ \lambda $ on $ \mathbb{R}^d $, $ \lambda $ is a dual measure to itself. Dual measures are in fact a generalization of the concept of Fourier frames and they are also called frame measures. According to [@5], if $ \mu $ is not an $ F $-spectral measure, then there cannot be any general statement about the existence of frame measures $ \nu $. Nevertheless, the authors showed that if one frame measure exists, then by using convolutions of measures, many frame measures can be obtained, especially a frame measure which is absolutely continuous with respect to Lebesgue measure. Moreover, they presented a general way of constructing Bessel/frame measures for a given measure. In this paper we generalize the notion of Bessel/frame measure from Hilbert spaces $ L^2(\mu) $, $ L^2(\nu) $ to Banach spaces $ L^p(\mu) $, $ L^q(\nu) $ ($ p, q $ are conjugate exponents) via a compatible semi-inner product defined on $ L^p(\mu) $. Compatible semi-inner products are natural substitutes for inner products on Hilbert spaces. We introduce $ (p,q) $-Bessel and $ (p,q) $-frame measures, and we define notions of $ q $-Bessel and $ q $-frame in the semi-inner product space $ L^p(\mu) $. Then we investigate the existence and some general properties of them. The rest of this paper is organized as follows: In section $ 2 $ basic definitions and preliminaries are given. In section $ 3 $ we investigate the existence of $ (p,q) $-Bessel/frame measures. We show that every finite Borel measure $\nu$ is a $(p,q)$-Bessel measure for a finite measure $ \mu $. In addition, we construct a large number of examples of measures which admit infinite discrete $ (p,q) $-Bessel measures, by F-spectral measures and applying the Riesz-Thorin interpolation theorem. In general, for every spectral measure (B-spectral measure, or F-spectral measure respectively) $ \mu $, there exists a discrete measure $\nu = \sum_{\lambda\in\Lambda_\mu}\delta_\lambda $ which is a Plancherel measure (Bessel measure or frame measure respectively) for $ \mu $. Then the Riesz-Thorin interpolation theorem yields that $ \nu $ is also a $ (p,q) $-Bessel measure for $ \mu $, where $ 1 \leq p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. Moreover, this shows that if $ \mu $ is a spectral measure (B-spectral measure, or F-spectral measure), then the set $ \{e_\lambda\}_{\lambda\in\Lambda_\mu}$ forms a $ q $-Bessel for $ L^p(\mu) $. It is known [@13; @19] that if a measure $ \mu $ is an F-spectral measure, then it must be of pure type, i.e., $ \mu $ is either discrete, absolutely continuous or singular continuous. Therefore, we consider such measures in constructing the examples. The interested reader can refer to [@3; @6; @7; @9; @13; @16; @18; @19; @20; @21; @23; @24] to see examples and properties of spectral measures (B-spectral measures, or F-spectral measures) and related concepts. Besides discrete $ (p,q) $-Bessel measures $\nu = \sum_{\lambda\in\Lambda_\mu}\delta_\lambda $ associated to spectral measures (B-spectral measures, or F-spectral measures) $ \mu $, we prove that there exists an infinite absolutely continuous $ (p,q) $-Bessel measure $ \nu $ for a special finite measure $ \mu $. We show that if $ \nu $ is a $ (p_1,q_1) $-Bessel/frame measure and $ (p_2,q_2) $-Bessel/frame measure for $ \mu $, where $1 \leq p_1, p_2 < \infty$ and $ q_1, q_2 $ are the conjugate exponents to $ p_1, p_2 $, respectively, then $ \nu $ is a $ (p,q) $-Bessel measure for $ \mu $ too, where $ p_1 < p < p_2 $ and $ q $ is the conjugate exponent to $ p $. Consequently, if $ \nu $ is a Bessel/frame measure for $ \mu $, then it is a $ (p,q) $-Bessel measure for $ \mu $ too. In Proposition \[3.20\] we prove that there exists a measure $ \mu $ which admits tight $ (p,q) $-frame measures and $ (p,q) $-Plancherel measures.\ Section $ 4 $ is devoted to investigating properties of $ (p,q) $-Bessel/frame measures based on the results by Dutkay, Han, and Weber from [@5]. Preliminaries ============== Let $ t\in \mathbb{R}^d $. Denote by $ e_t $ the exponential function $$e_t(x) = e^{2\pi it\cdot x}\qquad (x\in \mathbb{R}^d).$$ Let $ H $ be a Hilbert space. A sequence $ \{f_i \}_{i \in I} $ of elements in $ H $ is called a *Bessel sequence* for $ H $ if there exists a positive constant $ B $ such that for all $ f \in H $, $$\sum_{i \in I }|\left<f, f_i\right>|^2 \leq B\| f\|^2.$$ Here $ B $ is called the *Bessel bound* for the Bessel sequence $ \{ f_i \}_{i\in I} $. The sequence $ \{f_i \}_{i \in I} $ is called a *frame* for $ H $, if there exist constants $ A, B > 0 $ such that for all $ f \in H $, $$A\| f\|^2 \leq \sum_{i \in I }|\left<f, f_i\right>|^2 \leq B\| f\|^2.$$ In this case, $ A $ and $ B $ are called *frame bounds*. Frames are a natural generalization of orthonormal bases. It is easily seen from the lower bound that a frame is complete in H, so every $ f $ can be expressed using (infinite) linear combination of the elements $ f_i $ in the frame [@2]. Let $ \mu $ be a compactly supported probability measure on $ \mathbb{R}^d $ and $\Lambda$ be a countable set in $ \mathbb{R}^d $, the set $ E(\Lambda)=\{e_\lambda : \lambda \in \Lambda \} $ is called a *Fourier frame* for $ L^2(\mu) $ if for all $ f\in L^2(\mu) $, $$A\| f\|_{L^2(\mu)}^2 \leq \sum_{\lambda \in \Lambda }|\left<f, e_\lambda \right>_{L^2(\mu)}|^2 \leq B\| f\|_{L^2(\mu)}^2.$$ When $ E(\Lambda) $ is an orthonormal basis (Bessel sequence, or frame) for $ L^2(\mu) $, we say that $ \mu $ is a *spectral measure (B-spectral measure, or F-spectral measure* respectively) and $ \Lambda $ is called a *spectrum (B-spectrum, or F-spectrum* respectively) for $ \mu $. We give the following definition from [@5], assuming that the given measure $ \mu $ is a finite Borel measure on $ \mathbb{R}^d $. Let $ \mu $ be a finite Borel measure on $ \mathbb{R}^d $. A Borel measure $ \nu $ is called a *Bessel measure* for $ \mu $, if there exists a positive constant $ B $ such that for every $ f\in L^2(\mu) $, $$\|\widehat{fd\mu}\|_{L^2(\nu)}^2 \leq B\| f\|_{L^2(\mu)}^2.$$ Here $ B $ is called a *(Bessel) bound* for $ \nu $. The measure $ \nu $ is called a *frame measure* for $ \mu $ if there exist positive constants $ A, B $ such that for every $ f\in L^2(\mu) $, $$A\| f\|_{L^2(\mu)}^2 \leq \|\widehat{fd\mu}\|_{L^2(\nu)}^2 \leq B\| f\|_{L^2(\mu)}^2.$$ In this case, $ A $ and $ B $ are called *(frame) bounds* for $ \nu $. The measure $ \nu $ is called a tight frame measure if $ A = B $ and Plancherel measure if $ A = B =1 $ (see also [@8]). The set of all Bessel measures for $ \mu $ with fixed bound $ B $ is denoted by $ \mathcal{B}_B(\mu) $ and the set of all frame measures for $ \mu $ with fixed bounds $ A, B $ is denoted by $ \mathcal{F}_{A,B}(\mu) $. \[2.5\] A compactly supported probability measure $ \mu $ is an F-spectral measure if and only if there exists a countable set $\Lambda $ in $ \mathbb{R}^d $ such that $ \nu=\sum_{\lambda \in \Lambda}\delta_\lambda $ is a frame measure for $ \mu $. A finite set of contraction mappings $ \{\tau_i\}_{i=1}^n $ on a complete metric space is called an *iterated function system (IFS)*. Hutchinson [@15] proved that, for the metric space $ \mathbb{R}^d $, there exists a unique compact subset $ X $ of $ \mathbb{R}^d $, which satisfies $ X=\bigcup_{i=1}^n \tau_i(X) $. Moreover, if the IFS is associated with a set of probability weights $ \{\rho_i\}_{i= 1}^n $ (i.e., $ 0 <\rho_i < 1 $, $ \sum_{i=1}^n \rho_i =1 $), then there exists a unique Borel probability measure $ \mu $ supported on $ X $ such that $ \mu=\sum_{i=1}^n \rho_i (\mu o\tau^{-1}) $. The corresponding $ X $ and $ \mu $ are called the *attractor* and the *invariant measure* of the IFS, respectively. It is well known that the invariant measure is either absolutely continuous or singular continuous with respect to Lebesgue measure. In an affine IFS each $ \tau_i $ is affine and represented by a matrix. If $R$ is a $ d \times d $ expanding integer matrix (i.e., all eigenvalues $ \lambda $ satisfy $ |\lambda|>1 $), and $\mathcal{A} \subset \mathbb{Z}^d $, with $ \#\mathcal{A} =: N \geq 2 $, then the following set (associated with a set of probability weights) is an affine iterated function system. $$\tau_a (x) = R^{-1}(x + a) \quad (x \in \mathbb{R}^d, a \in \mathcal{A}).$$ Since $R$ is expanding, the maps $\tau_a$ are contractions (in an appropriate metric equivalent to the Euclidean one). In some cases, the invariant measure $ \mu_\mathcal{A} $ is a *fractal measure* (see [@3]). For example singular continuous invariant measures supported on Cantor type sets are fractal measures (see [@15; @14]). (*Semi-inner product spaces*)\ Let $ X $ be a vector space over the filed $ F $ of complex (real) numbers. If a function $ [\cdot , \cdot] : X \times X \rightarrow F $ satisfies the following properties: 1. $[x + y, z] = [x, z] + [y, z], \;\;\ \text{for}\ x, y, z \in X;$ 2. $[\lambda x, y] = \lambda[x, y], \;\ \text{for} \ \lambda\in F \ \text{and} \ x, y \in X;$ 3. $[x, x] > 0, \;\ \text{for} \ x \neq 0;$ 4. $|[x, y]|^2 \leq [x, x][y, y],$ then $ [\cdot , \cdot] $ is called a *semi-inner product* and the pair$ (X, [\cdot, \cdot]) $ is called a *semi-inner product space*. It is easy to observe that $\|x\| = [x, x]^\frac{1}{2}$ is a norm on $ X $. So every semi-inner product space is a normed linear space. On the other hand, one can generate a semi-inner product in a normed linear space, in infinitely many different ways. As a matter of fact, semi-inner products provide the possibility of carrying over Hilbert space type arguments to Banach spaces. Every Banach space has a semi-inner product that is compatible. For example consider the Banach function space $ L^p(X , \mu),\; p \geq 1$, a compatible semi-inner product on this space is defined by (see [@12]) $$[f,g]_{L^p(\mu)} := \dfrac{1}{\| g\|_{L^p(\mu)}^{p-2}}\int_X f(x) |g(x)|^{p-1}\overline{sgn(g(x))} d\mu(x),$$ for every $ f, g \in L^p(X , \mu) $ with $ \|g\|_{L^p(\mu)} \neq 0 $, and $ [f,g]_{L^p(\mu)} = 0 $ for $ \|g\|_{L^p(\mu)} = 0 $.\ To construct frames in a Hilbert space $ H $ the sequence space $ l^2 $ is required. Similarly, to construct frames in a Banach space $ X $ one needs a Banach space of scaler valued sequences $ X_d $ (in fact a BK-space $ X_d $, see [@1] and the references therein). According to Zhang and Zhang [@26] frames in Banach spaces can be defined via a compatible semi-inner product in the following way: \[2.7\] Let $ X $ be a Banach space with a compatible semi-inner product $ [\cdot , \cdot] $ and norm $ \|\cdot\|_X $. Let $ X_d $ be an associated BK-space with norm $ \|\cdot\|_{X_d} $. A sequence of elements $ \{ f_ i \}_{i\in I} \subseteq X $ is called an *$ X_d $-frame* for $ X $ if $ \{[ f, f_ i ]\}_{i\in I} \in X_d $ for all $ f \in X $, and there exist constants $ A, B > 0 $ such that for every $ f \in X $, $$A\| f\|_X \leq \|\{[f,f_i]\}_{i\in I} \|_{X_d} \leq B\| f\|_X.$$ See also [@25]. Based on Definition \[2.7\], we present the next definition. We consider the function space $ L^p( \mu)$ and the sequence space $ l^q(I) $ (where $ p>1 $ and $ q $ is the conjugate exponent to $ p $) as the Banach space and the BK- space, respectively. Suppose that $ 1<p, q<\infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. Let $ \mu $ be a finite Borel measure on $ \mathbb{R}^d $ and let $ [\cdot ,\cdot] $ be the compatible semi-inner product on $ L^p(\mu) $ as defined above. We say that a sequence $ \{f_i\}_{i\in I} $ is a *$ q $-Bessel* for $ L^p(\mu) $ if there exists a constant $ B > 0 $ such that for every $ f\in L^p(\mu) $, $$\sum_{i\in I}|[f,f_i]_{L^p(\mu)}|^q\leq B\| f\|_{L^p(\mu)}^q.$$ We call B a *($ q $-Bessel) bound*. We say the sequence $ \{f_i\}_{i\in I} $ is a *$ q $-frame* for $ L^p(\mu) $ if there exist constants $A, B > 0 $ such that for every $ f\in L^p(\mu) $, $$A\| f\|_{L^p(\mu)}^q\leq\sum_{i\in I}|[f,f_i]_{L^p(\mu)}|^q\leq B\| f\|_{L^p(\mu)}^q.$$ We call $ A,B $ *($ q $-frame) bounds*. We call the sequence $ \{f_i\}_{i\in I} $ a *tight $ q $-frame* if $ A = B $ and *Parseval $ q $-frame* if $ A = B = 1 $. We extend the notions of Bessel and frame measures as follows. Suppose that $ 1\leq p<\infty, 1<q\leq\infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. Let $ \mu $ be a finite Borel measure on $ \mathbb{R}^d $, and let $ [\cdot ,\cdot] $ be the compatible semi-inner product on $ L^p(\mu) $ as defined above. We say that a Borel measure $ \nu $ is a *$ (p,q) $-Bessel measure* for $ \mu $, if there exists a constant $ B > 0 $ such that for every $ f\in L^p(\mu) $, $$\int_{\mathbb{R}^d} |[f,e_t]_{L^p(\mu)}|^q d\nu(t)\leq B\| f\|_{L^p(\mu)}^q \quad (p \neq 1, q\neq \infty)$$ and $$\|\widehat{fd\mu}\|_\infty \leq B\| f\|_{L^1(\mu)} \quad (p=1, q=\infty).$$ We call $ B $ a (*$ (p,q) $-Bessel*) *bound* for $ \nu $. We say the Borel measure $ \nu $ is a *$ (p,q) $-frame measure* for $ \mu $, if there exist constants $ A,B > 0 $ such that for every $ f\in L^p(\mu) $, $$A\| f\|_{L^p(\mu)}^q\leq \int_{\mathbb{R}^d} |[f,e_t]_{L^p(\mu)}|^q d\nu(t)\leq B\| f\|_{L^p(\mu)}^q \quad (p \neq 1, q \neq \infty)$$ and $$A\| f\|_{L^1(\mu)}\leq \|\widehat{fd\mu}\|_\infty \leq B\| f\|_{L^1(\mu)} \quad (p=1, q=\infty).$$ We call $ A,B $ (*$ (p,q) $-frame*) *bounds* for $ \nu $. If $ A = B $, we call the measure $ \nu $ a *tight $ (p,q) $-frame measure* and if $ A = B =1 $, we call it a *$ (p,q) $-Plancherel measure*. We denote the set of all $(p,q)$-Bessel measures for $ \mu $ with fixed bound $ B $ by $ \mathcal{B}_B(\mu)_{p,q} $ and the set of all $(p,q)$-frame measures for $ \mu $ with fixed bounds $A, B $ by $ \mathcal{F}_{A,B}(\mu)_{p,q} $. \[2.11\] Since $[f,e_t]_{L^p(\mu)}= \int_{R^d} f(x) e^{-2\pi it\cdot x} d\mu(x) = \widehat{f d\mu}(t)$ for any $ f\in L^p(\mu) $ and $ t \in \mathbb{R}^d$, we can also write $ \widehat{f d\mu}(t) $ instead of $ [f,e_t]_{L^p(\mu)} $. If there exists a $ (p,q) $-Bessel/frame measure $ \nu $ for $ \mu $, then the function $ T_\nu : L^p(\mu) \rightarrow L^q(\nu) $ defined by $ T_\nu f=\widehat{fd\mu} $ is linear and bounded. For $ p=1 $, $ q= \infty $, every $\sigma$-finite measure $ \nu $ on $ \mathbb{R}^d $ is a $(1, \infty)$-Bessel measure for $ \mu $, since we always have $ \| \widehat{fd\mu} \|_\infty \leq \| f \|_{L^1(\mu)} $. More precisely, $ \nu \in \mathcal{B}_1(\mu)_{(1, \infty)} $. *(Riesz-Thorin interpolation theorem)* Let $1 \leq p_0, p_1, q_0, q_1 \leq \infty$, where $p_0 \neq p_1$ and $q_0 \neq q_1$, and let $T$ be a linear operator. Suppose that for some measure spaces $(Y, \nu)$, $(X, \mu)$, $ T: L^{p_0}(X, \mu)\rightarrow L^{q_0}(Y, \nu) $ is bounded with norm $ C_0 $, and $ T: L^{p_1}(X, \mu)\rightarrow L^{q_1}(Y, \nu) $ is bounded with norm $ C_1 $. Then for all $ \theta \in (0, 1) $ and $ p, q $ defined by $ \dfrac{1}{p} = \dfrac{(1 - \theta)}{p_0} + \dfrac{\theta}{p_1}; \ \dfrac{1}{q} =\dfrac{1- \theta}{q_0} + \dfrac{\theta}{q_1}$, there exists a constant $ C $ such that $ C\leq C_0^{(1-\theta)} C_1^{\theta}$ and $ T: L^{p}(X, \mu)\rightarrow L^{q}(Y, \nu) $ is bounded with norm $ C $. All measures we consider in this paper, are Borel measures on $ \mathbb{R}^d $. Existence and Examples ====================== In this section we investigate the existence of $ (p,q) $-Bessel and $ (p,q) $-frame measures and also the existence of $ q $-Bessel and $ q $-frame sequences. In addition, we construct examples of measures which admit $ (p,q) $-Bessel measures. \[3.1\] Suppose that $ 1<p, q<\infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. Let $ \mu $ be a finite Borel measure. Then every finite Borel measure $\nu$ is a $(p,q)$-Bessel measure for $ \mu $. Take $f \in L^p (\mu)$ and $ t \in \mathbb{R}^d$. Then by applying Holder’s inequality $$|[f, e_t]_{L^p(\mu)}| \leq \int_{\mathbb{R}^d} |f(x) e^{-2\pi it\cdot x}| d\mu(x)\leq (\mu(\mathbb{R}^d))^{\frac{1}{q}}\| f\|_{L^p(\mu)} .$$ Thus, $$\int_{\mathbb{R}^d} |[f, e_t]_{L^p(\mu)}|^q d\nu(t) \leq \mu(\mathbb{R}^d) \nu(\mathbb{R}^d)\| f\|^q_{L^p(\mu)}.$$ Therefore $\nu \in \mathcal{B}_{\mu(\mathbb{R}^d) \nu(\mathbb{R}^d)}(\mu)_{(p, q)} $. For $ p=1 $, $ q= \infty $, as we mentioned in Remark \[2.11\] $ \nu \in \mathcal{B}_1(\mu)_{(1, \infty)} $. \[3.2\] Suppose that $ 1<p, q<\infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. Let $ \Lambda \subset \mathbb{R}^d $, $ \#\Lambda < \infty $ and $ \mu $ be a finite Borel measure. Then the finite sequence $ \{e_\lambda\}_{\lambda\in \Lambda} $ is a $ q $-Bessel for $ L^p(\mu) $. Consider the finite discrete measure $\nu = \sum_{\lambda\in\Lambda}\delta_\lambda $. Since $$\sum_{\lambda\in\Lambda}|[f, e_\lambda]_{L^p(\mu)}|^q =\int_{\mathbb{R}^d} |[f, e_t]_{L^p(\mu)}|^q d\nu(t),$$ then the assertion follows from Proposition \[3.1\]. Proposition \[3.1\] shows that the Bessel bound may change for different measures $ \nu $. So if we consider Borel probability measures $ \nu $, then we have a fixed Bessel bound $ \mu(\mathbb{R}^d) $ for all $ \nu $. Moreover, this Bessel bound does not depend on $p, q$, i.e., for every probability measure $\nu$ we have $\nu \in \mathcal{B}_{\mu(\mathbb{R}^d)}(\mu)_{(p, q)} $, where $ 1 < p< \infty $ and $ q $ is the conjugate exponent to $ p $. In addition, we obtain from Proposition \[3.1\] that for all conjugate exponents $ p, q > 1 $ the set $ \mathcal{B}_{\mu(\mathbb{R}^d)}(\mu)_{p,q} $ is infinite, since there are infinitely many probability measures $ \nu $ (such as every measure $ \nu=\frac{1}{\lambda(S)}\chi_S d\lambda $ where $ S \subset \mathbb{R}^d$ with the finite Lebesgue measure $ \lambda(S) $, every finite discrete measure $ \nu = \frac{1}{n}\sum_{a=1}^n \delta_a $ where $ \delta_a $ denotes the Dirac measure at the point $ a $, every invariant measure obtained from an iterated function system, and others). \[3.3\] Suppose that $ 1<p, q<\infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. Let $ \nu $ be a finite Borel measure. Then $ \nu $ is a $(p, q)$-Bessel measure for every finite Borel measure $ \mu $. In addition, $\nu \in \mathcal{B}_{\nu(\mathbb{R}^d)}(\mu)_{(p, q)} $ for all probability measures $ \mu $. See the proof of Proposition \[3.1\]. Suppose that $ 1<p, q<\infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. A finite Borel measure $ \nu $ is a $ (p,q) $-Bessel measure for a finite Borel measure $ \mu $, if and only if $ \mu $ is a $ (p,q) $-Bessel measure for $ \nu $. In particular, every finite Borel measure $ \mu $ is a $ (p,q) $-Bessel measure to itself. The statements are direct consequences of Propositions \[3.1\] and \[3.3\]. \[3.6\] Suppose that $ 1<p, q<\infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. Let $ \mu $ be a finite Borel measure. Then the following assertions hold. \(i) If there exists a countable set $\Lambda $ in $ \mathbb{R}^d $ such that $\{e_\lambda\}_{\lambda \in \Lambda} $ is a $ q $-frame for $L^p (\mu) $, then $\nu = \sum_{\lambda\in\Lambda}\delta_\lambda $ is a $ (p,q) $-frame measure for $\mu$. \(ii) If $ \nu $ is purely atomic, i.e. $\nu = \sum_{\lambda\in\Lambda} d_\lambda \delta_\lambda $, and a $(p,q)$-frame measure for the probability measure $ \mu $, then $\lbrace\sqrt [q]{d_\lambda}\; e_\lambda \rbrace_{\lambda\in\Lambda} $ is a q-frame for $L^p (\mu) $. \(i) Let $\nu = \sum_{\lambda\in\Lambda}\delta_\lambda $. Then for all $f \in L^p (\mu)$, $$\begin{aligned} \sum_{\lambda\in\Lambda} |[f,e_\lambda]_{L^p(\mu)}|^q =\int_{\mathbb{R}^d} |[f,e_t]_{L^p(\mu)}|^q d\nu(t).\end{aligned}$$ \(ii) Since for all $f \in L^p (\mu)$, $$\int_{\mathbb{R}^d} |[f,e_t]_{L^p(\mu)}|^q d\nu(t) = \sum_{\lambda\in\Lambda}d_\lambda|[f,e_\lambda]_{L^p(\mu)}|^q = \sum_{\lambda\in\Lambda} |[f,\sqrt [q]{d_\lambda} e_\lambda]|^q.$$ \[ex 4\] Suppose that $1\leq p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. If $ f \in L^p ([0 , 1]^d) $, then from the Hausdorff-Young inequality we have $ \hat{f} \in l^q(\mathbb{Z}^d) $ and $ \| \hat{f} \|_q \leq \| f\|_p $. Therefore the measure $ \nu=\sum_{t \in \mathbb{Z}^d} \delta_t $ is a $ (p,q) $-Bessel measure for $ \mu= \chi_{\{[0 , 1]^d\}}dx $. Besides, $\{e_t\}_{t \in \mathbb{Z}^d}$ is a $ q $-Bessel for $ L^p (\mu) $, since $ \sum_{t\in \mathbb{Z}^d} |[f , e_t]_{ L^p (\mu)}|^q \leq \| f\|_{p}^q $, where $1< p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. Let $1< p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. Let $ 0<a\leq \phi (x)\leq b <\infty $ on $ [0, 1]^d $ and $ \phi_t (x):=\phi (x) $ for all $ t \in \mathbb{Z}^d $. Then $\{\phi_te_t\}_{t \in \mathbb{Z}^d}$ is a $ q $-Bessel for $ L^p ([0 , 1]^d) $. Take $ f\in L^p \left([0 , 1]^d\right) $. We have $ \dfrac{1}{\| \phi \|_p^{p-2}}\phi^{p-1} f\in L^p ([0 , 1]^d) $, since $$\int_{[0 , 1]^d} |f(x)|^p \left| \dfrac{ \phi^{p-1}(x)}{\| \phi \|_p^{p-2}} \right|^p d(x) \leq \dfrac{b^{(p-1)p}}{a^{(p-2)p}} \int_{[0 , 1]^d} |f(x)|^p d(x)<\infty.$$ Hence by Example \[ex 4\], $$\begin{aligned} \sum_{t\in \mathbb{Z}^d} \left|[f , \phi_t e_t]_{L^p ([0 , 1]^d)} \right|^q &= \sum_{t\in \mathbb{Z}^d} \left| \dfrac{1}{\| \phi \|_p^{p-2}} \int_{[0 , 1]^d} f(x) \left|\phi(x)e_t(x) \right|^{p-1}e_{-t}(x)dx \right|^q \\ &\leq \left| \int_{[0 , 1]^d} |f(x)|^p \left| \dfrac{ \phi^{p-1}(x)}{\| \phi \|_p^{p-2}} \right|^p dx\right|^{q/p}\leq \dfrac{b^p}{a^{p-q}} \| f\|_p^q.\end{aligned}$$ Suppose that $ 1<p, q<\infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. Let $ \mu $ be a probability measure. Let $ 0<a\leq \phi (x)\leq b <\infty $ on $ supp\mu $ and $ \phi_i (x):=\phi (x) $ for all $ i\in I $. If $\{f_i\}_{i\in I}$ is a $ q $-frame for $ L^p (\mu) $, then $ \{\phi_if_i\}_{i \in I}$ is also a $ q $-frame for $ L^p (\mu) $ and for every $ f\in L^p(\mu) $, $$\dfrac{a^p}{b^{p-q}} A\parallel f\parallel_{L^p(\mu)}^q\leq\|\{[f,\phi_if_i]_{L^p(\mu)}\}_{i\in I} \|^q\leq \dfrac{b^p}{a^{p-q}} B\| f\|_{L^p(\mu)}^q.$$ Example \[ex 4\] cannot be extended to the case $ p>2 $, since there exist continuous functions $ f $ such that $ \sum_{n\in \mathbb{Z}} |[f , e_n]_{L^p(\mu)}|^{2-\epsilon} =\infty $ for all $ \epsilon>0 $. An example of such a function is\ $ f(x) = \sum_{n=2}^\infty \dfrac{e^{i n\log n}}{n^{1/2}(\log n)^2}e^{inx} $ (see [@17]). Therefore $\{e_n\}_{n \in \mathbb{Z}}$ is not a $ q $-Bessel for $ L^p ([0 , 1]) $ and also $ \nu=\sum_{n \in \mathbb{Z}} \delta_n $ is not a $ (p,q) $-Bessel measure for $ \mu= \chi_{[0 , 1] }dx $ where $ p>2 $. \[2.21\] Suppose that $ 1<p, q<\infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. Let $ \mu $ be a compactly supported Borel probability measure. Consider two subsets of $ \mathbb{R}^d $, $ \Lambda= \{ \lambda_n \; : n \in \mathbb{N} \} $ and $ \Omega= \{ \omega_n \; : n \in \mathbb{N}\} $ with the property that there exists a positive constant $ C $ such that $ |\lambda_n - \omega_n| \leq C $ for $ n\in \mathbb{N}$. \(i) If $\{e_{\lambda_n}\}_{n \in \mathbb{N}}$ is a $ q $-Bessel for $ L^p (\mu) $, then $\{e_{\omega_n}\}_{n \in \mathbb{N}} $ is a $ q $-Bessel too. \(ii) If $\{e_{\lambda_n}\}_{n \in \mathbb{N}}$ is a $ q $-frame for $ L^p (\mu) $, then there exists a $ \delta >0 $ such that if $ C\leq \delta $ then $\{e_{\omega_n}\}_{n \in \mathbb{N}} $ is a $ q $-frame too (see [@3]). We need only consider the case, when all $ \omega_n = \left( (\omega_n)_1, \ldots, (\omega_n)_d \right) $ differ from $ \lambda_n = \left( (\lambda_n)_1, \ldots, (\lambda_n)_d \right) $ just on the first component, then the assertion follows by induction on the number of components.\ Let supp$ \mu \subseteq [-M , M]^d $ for some $ M>0 $. Let $ f\in L^p (\mu) $ and $ x\in \mathbb{R}^d $. The function $ \widehat{fd\mu} $ is analytic in each variable $ t_1,\ldots, t_d $. Moreover, $$\dfrac{\partial^k \widehat{fd\mu}}{\partial t_1 ^k} (t) =\int f(x) (-2\pi i x_1)^k e^{-2\pi i t\cdot x} d\mu (x)= \left[(-2\pi i x_1)^k f, e_t\right]_{L^p (\mu)}, \quad ( t \in \mathbb{R}^d ).$$ Writing the Taylor expansion at $(\lambda_n)_1$ in the first variable and using Holder’s inequality, for all $ n\in \mathbb{N} $, $$\begin{aligned} \lvert \widehat{fd\mu}(\omega_n) -\widehat{fd\mu}(\lambda_n) \rvert^q &= \left| \sum_{k=1} ^\infty \dfrac{\dfrac{\partial^k \widehat{fd\mu}}{\partial t_1 ^k}(\lambda_n)}{k!} ( (\omega_n)_1 - (\lambda_n)_1 )^k \right|^q \\ &\leq \sum_{k=1}^\infty \dfrac{\left|\dfrac{\partial^k \widehat{fd\mu}}{\partial t_1 ^k}(\lambda_n)\right|^q}{k!}\cdot \left(\sum_{k=1}^\infty \dfrac{|(\omega_n)_1 - (\lambda_n)_1 |^{pk}}{k!}\right)^{q/p}\\ &\leq \sum_{k=1}^\infty \dfrac{\left|\dfrac{\partial^k \widehat{fd\mu}}{\partial t_1 ^k}(\lambda_n)\right|^q}{k!}\cdot \left( \sum_{k=1}^\infty \dfrac{C^{pk}}{k!}\right)^{q-1}=\sum_{k=1}^\infty \dfrac{\left|\dfrac{\partial^k \widehat{fd\mu}}{\partial t_1 ^k}(\lambda_n)\right|^q}{k!}\cdot \left(e^{C^p}-1\right)^{q-1}. \end{aligned}$$ Considering the $ q $-Bessel $\{e_{\lambda_n}\}_{n \in \mathbb{N}}$ with a bound $ B $, we obtain $$\sum_{n\in \mathbb{N}}\left|\dfrac{\partial^k \widehat{fd\mu}}{\partial t_1 ^k}(\lambda_n)\right|^q= \sum_{n\in \mathbb{N}} \left| \left[ (-2\pi i x_1)^k f, e_{\lambda_n} \right]_{L^p (\mu)}\right|^q\leq B\|(-2\pi i x_1)^k f\|_{L^p(\mu)}^q\leq B (2\pi M)^{qk} \| f\|_{L^p(\mu)}^q.$$ Then $$\begin{aligned} \sum_{n\in \mathbb{N}}\lvert \widehat{fd\mu}(\omega_n) -\widehat{fd\mu}(\lambda_n) \rvert^q &\leq B \left(e^{C^p}-1\right)^{q-1} \| f\|_{L^p(\mu)}^q \sum_{k=1}^\infty \dfrac{(2\pi M)^{qk}}{k!} \\ &= B \left(e^{C^p}-1\right)^{q-1} \left(e^{{(2\pi M)}^q}-1\right) \| f\|_{L^p(\mu)}^q. \end{aligned}$$ Hence by Minkowski’s inequality, $$\begin{aligned} \left(\sum_{n\in \mathbb{N}}| \widehat{fd\mu}(\omega_n) |^q \right)^{1/q} &\leq \left(\sum_{n\in \mathbb{N}}| \widehat{fd\mu}(\lambda_n) |^q \right)^{1/q} + \left(\sum_{n\in \mathbb{N}}\lvert \widehat{fd\mu}(\omega_n) -\widehat{fd\mu}(\lambda_n) \rvert^q\right)^{1/q}\\ &\leq \left( B^{1/q} + \left(B\left( e^{C^p}-1\right)^{q-1} \left(e^{{(2\pi M)}^q}-1\right)\right)^{1/q} \right) \| f\|_{L^p(\mu)}, \end{aligned}$$ and this implies that $\{e_{\omega_n}\}_{n \in \mathbb{N}} $ is a $ q $-Bessel for $ L^p (\mu) $.\ To show that $\{e_{\omega_n}\}_{n \in \mathbb{N}} $ is also a $ q $-frame for $ L^p (\mu) $, let $ A $ be a lower bound for $\{e_{\lambda_n}\}_{n \in \mathbb{N}}$. Take $ \delta > 0 $ small enough such that for $ 0 < C \leq \delta $, $$A^{1/q} - \left(B\left( e^{C^p}-1\right)^{q-1} \left(e^{{(2\pi M)}^q}-1\right)\right)^{1/q} > 0.$$ Then, by Minkowski’s inequality, $$\begin{aligned} \left(\sum_{n\in \mathbb{N}}| \widehat{fd\mu}(\omega_n) |^q \right)^{1/q} &\geq \left(\sum_{n\in \mathbb{N}}| \widehat{fd\mu}(\lambda_n) |^q \right)^{1/q} - \left(\sum_{n\in \mathbb{N}}\lvert \widehat{fd\mu}(\omega_n) -\widehat{fd\mu}(\lambda_n) \rvert^q\right)^{1/q}\\ &\geq \left( A^{1/q} - \left( B\left( e^{C^p}-1\right)^{q-1} \left(e^{{(2\pi M)}^q}-1\right)\right)^{1/q} \right) \| f\|_{L^p(\mu)}. \end{aligned}$$ Thus the assertion follows. \[3.9\] Suppose that $1 \leq p_0, p_1 < \infty$ and $ q_0, q_1 $ are the conjugate exponents to $ p_0, p_1 $ respectively. If $ \nu $ is a $ (p_0,q_0) $-Bessel measure and a $ (p_1,q_1) $-Bessel measure for $ \mu $, then $ \nu $ is also a $ (p,q) $-Bessel measure for $ \mu $, where $ p_0 < p < p_1 $ and $ q $ is the conjugate exponent to $ p $. If $ \nu $ is a $ (p_0,q_0) $-Bessel measure for $ \mu $ with bound $ C $ and also a $ (p_1,q_1) $-Bessel measure with bound $ D $, we have $$\forall f \in L^{p_0}( \mu) \;\;\; \| \widehat{fd\mu}\|^{q_0}_{ L^{q_0}(\nu)} \leq C \| f \|_{ L^{p_0}(\mu)}^{q_0},$$ and $$\forall f \in L^{p_1}( \mu) \;\;\; \|\widehat{fd\mu}\|^{q_1}_{ L^{q_1}(\nu)} \leq D \| f \|_{ L^{p_1}(\mu)}^{q_1} .$$ Now if $ \dfrac{1}{p} = \dfrac{(1 - \theta)}{p_0} + \dfrac{\theta}{p_1}; \ \dfrac{1}{q} =\dfrac{1- \theta}{q_0} + \dfrac{\theta}{q_1}$, where $ 0 < \theta < 1 $ (i.e., $ p_0 < p < p_1 $ and $ \dfrac{1}{p} + \dfrac{1}{q} = 1 $), then the Riesz-Thorin interpolation theorem yields $$\forall f \in L^{p}( \mu) \qquad \| \widehat{fd\mu}\|_{ L^q (\nu)}^q \leq B^q \| f \|_{ L^p (\mu)}^q.$$ where $ B\leq C^{\frac{1}{q_0}(1-\theta)} D^{\frac{1}{q_1}\theta}$ (Considering the fact that if $ p_0 =1 $ and $ q_0=\infty $, then $ C^{\frac{1}{q_0}} $ changes to $ C $, and if $ p_1 =1 $ and $ q_1=\infty $, then $ D^{\frac{1}{q_1}} $ changes to $ D $). Hence $ \nu $ is a $ (p,q) $-Bessel measure for $ \mu $, where $ p_0 < p < p_1 $ and $ q $ is the conjugate exponent to $ p $. If $ \nu $ is a Bessel/frame measure for $ \mu $, then $ \nu $ is also a $ (p,q) $-Bessel measure for $ \mu $, where $ 1\leq p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. Let $ p_0=1, q_0=\infty, p_1=2, q_1=2 $ in the assumption of Proposition \[3.9\], then the conclusion follows. \[3.13\] If $ \nu \in \mathcal{F}_{A,B}(\mu) $, then for any constant $\alpha > 0$, $ \nu $ is a frame measure for $ \alpha\mu $. More precisely $ \nu \in \mathcal{F}_{\alpha A,\alpha B}(\alpha\mu) $. Since $ \nu \in \mathcal{F}_{A,B}(\mu) $ for all $ f\in L^2(\mu) $, $$A\| f\|_{L^2(\mu)}^2 \leq \|\widehat{fd\mu}\|_{L^2(\nu)}^2 \leq B\| f\|_{L^2(\mu)}^2,$$ and we have $$\|\widehat{fd\alpha\mu}\|_{L^2(\nu)}^2 = \int_{\mathbb{R}^d} \left|\int_{\mathbb{R}^d} f(x) e_{-t}(x) d\alpha \mu(x)\right|^2 d\nu(t)=\|\widehat{\alpha fd\mu}\|_{L^2(\nu)}^2.$$ Since $ \alpha f\in L^2(\mu) $, $$A\| \alpha f\|_{L^2(\mu)}^2 \leq \|\widehat{\alpha fd\mu}\|_{L^2(\nu)}^2 \leq B\| \alpha f\|_{L^2(\mu)}^2\quad \text{for all } f\in L^2(\mu).$$ Therefore, $$\alpha A\| f\|_{L^2(\alpha \mu)}^2 \leq \|\widehat{fd\alpha \mu}\|_{L^2(\nu)}^2 \leq \alpha B\| f\|_{L^2(\alpha \mu)}^2\quad \text{for all } f\in L^2(\alpha \mu).$$ Hence $ \nu \in \mathcal{F}_{\alpha A,\alpha B}(\alpha\mu) $. \[2.17\] There exist positive constants $ c, C $ such that for every set $ S \subset \mathbb{R}^d $ of finite measure, there is a discrete set $ \Lambda \subset \mathbb{R}^d $ such that $ E(\Lambda) $ is a frame for $ L^2(S) $ with frame bounds $ c|S| $ and $ C|S| $, where $|S|$ denotes the measure of $S$. \[2.19\] Let $ S $ be a subset (not necessarily bounded) of $ \mathbb{R}^d $ with finite Lebesgue measure $|S|$. Then the probability measure $ \mu=\frac{1}{|S|}\chi_S dx $ has an infinite discrete $(p,q)$-Bessel measure $ \nu $, where $1\leq p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. By Theorem \[2.17\], There are positive constants $ c, C $ such that for every set $ S \subset \mathbb{R}^d $ of finite Lebesgue measure $|S|$, there is a discrete set $ \Lambda \subset \mathbb{R}^d $ such that $ E(\Lambda) $ is a frame for $ L^2(S) $ with frame bounds $ c|S|$ and $ C|S|$. Then by considering the upper bound of the frame, we have $$\sum_{\lambda \in \Lambda }|\left<f,e_\lambda\right>|^2 \leq C|S|\| f\|_{L^2(S)}^2 \quad \text{for all } f\in L^2(S).$$ Let $ \mu=\frac{1}{|S|}\chi_S dx $, and then by Proposition \[3.13\], $$\sum_{\lambda \in \Lambda }|\left<f,e_\lambda\right>|^2 \leq C\| f\|_{L^2(\mu)}^2 \quad \text{for all } f\in L^2(\mu).$$ In addition, $ \|\{[f,e_\lambda]_{L^1(\mu)}\}_{\lambda\in \Lambda} \|_\infty \leq \| f \|_{L^1(\mu)} $, for every $ f $ in $ L^1(\mu) $. Now if $ \dfrac{1}{p} = 1 -\dfrac{\theta}{2} ; \ \dfrac{1}{q} =\dfrac{\theta}{2}$, for $ 0 <\theta < 1 $ (i.e., $1 < p < 2 $ and $ q $ is the conjugate exponent to $ p $), then the Riesz-Thorin interpolation theorem yields $$\sum_{\lambda \in \Lambda }|[f,e_\lambda]_{L^p(\mu)}|^q \leq \mathcal{C}^q\| f\|_{L^p(\mu)}^q \quad \text{for all } f\in L^p(\mu),$$ where $ \mathcal{C} \leq C^{\frac{1}{2}\theta} $. Therefore, $ \nu=\sum_{\lambda \in \Lambda} \delta_\lambda $ is a $ (p,q) $-Bessel measure for $ \mu=\frac{1}{|S|}\chi_S dx $, and we have $ \nu \in \mathcal{B}_{\mathcal{C}^q}(\mu)_{(p, q)} $, where $1 < p < 2 $ and $ q $ is the conjugate exponent to $ p $. Moreover, $ \nu \in \mathcal{B}_{C}(\mu)_{(2, 2)} $ and $ \nu \in \mathcal{B}_{1}(\mu)_{(1, \infty)} $. On the other hand for every $1 < p < 2 $ and $ q $ (the conjugate exponent to $ p $), $ \{e_\lambda\}_{\lambda \in \Lambda} $ is a $ q $-Bessel for $ L^p(\mu) $, with bound $ \mathcal{C}^q $. If $ S \subset \mathbb{R}^d $ is a compact set with positive Lebesgue measure, then by Theorem \[2.17\], we always have the measure $ \mu=\frac{1}{|S|}\chi_S dx $ is an F-spectral measure, but whether it is a spectral measure, is related to Fuglede’s conjecture [@11]. In the following example we consider a spectral measure of this type. \[2.20\] Let $ \mu=\chi_{\left\{[0 , 1] \cup[2, 3]\right\}}dx $. The set of exponential functions $\{e_\lambda \; : \lambda \in \Lambda :=\mathbb{Z} \cup \mathbb{Z} +\frac{1}{4}\}$ is an orthogonal basis for $L^2(\mu)$ (see [@9]). We consider the probability measure $ \mu'=\dfrac{1}{2}\chi_{\left\{[0 , 1] \cup[2, 3]\right\}}dx $. Then for every $ f $ in $ L^2(\mu') $ we have $\sum_{\lambda \in \Lambda }|\left<f,e_\lambda\right>_{L^2(\mu')}|^2 = \| f\|_{L^2(\mu')}^2 $. In addition, for every $ f \in L^1(\mu') $, we have $ \|\{[f,e_\lambda]_{L^1(\mu')}\}_{\lambda \in \Lambda } \|_\infty \leq \| f \|_{L^1(\mu')} $. Now by applying the Riesz-Thorin interpolation theorem $ \sum_{\lambda \in \Lambda }|[f,e_\lambda]_{L^2(\mu')}|^q \leq \| f\|_{L^p(\mu')}^q$, for all $ f\in L^p(\mu') $, where $1\leq p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. Hence, $ \nu=\sum_{\lambda \in \Lambda} \delta_\lambda $ is a $ (p,q) $-Bessel measure for $ \mu' $, especially $ \nu \in \mathcal{B}_{1}(\mu')_{(p, q)} $, where $1\leq p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. Besides, $ \{e_\lambda\}_{\lambda \in \Lambda} $ is a $ q $-Bessel for $ L^p(\mu') $ with bound $ 1 $, where $1 < p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. \[3.18\] Let $ \mu(x) = \phi(x) dx $ be a compactly supported absolutely continuous probability measure. Then $ \mu $ is an F-spectral measure if and only if the density function $ \phi(x) $ is bounded above and below almost everywhere on the support (see also [@8]). If the density function of a compactly supported absolutely continuous probability measure $ \mu $ is essentially bounded above and below on the support, then the following assertions hold. \(i) There exists an infinite $ (p,q) $-Bessel measure $ \nu=\sum_{\lambda \in \Lambda_\mu} \delta_\lambda $ for $ \mu $, where $1\leq p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. Moreover, when $ \mu $ is a spectral measure, we have $ \nu \in \mathcal{B}_1(\mu)_{p, q}$, where $1\leq p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. \(ii) There exists a $ q $-Bessel $ \{e_\lambda\}_{\lambda \in \Lambda_{\mu}} $ for $ L^p(\mu) $, where $1< p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. In addition, when $ \mu $ is a spectral measure, $ \{e_\lambda\}_{\lambda \in \Lambda_{\mu}} $ is a $ q $-Bessel for $ L^p(\mu) $ with bound $ 1 $, where $1 < p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. The conclusion follows from Proposition \[3.18\] and the Riesz-Thorin interpolation theorem (see the proof of Theorem \[2.19\] and also, see Example \[2.20\]). By Proposition \[3.3\], if $ 1<p, q<\infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $, then a fixed finite Borel measure $ \nu $ is a $(p, q)$- Bessel measure for every finite measure $ \mu $, especially $\nu \in \mathcal{B}_{\nu(\mathbb{R}^d)}(\mu)_{(p, q)} $ for all probability measures $ \mu $, and by Proposition \[3.2\], if $ \nu=\sum_{\lambda \in \Lambda} \delta_\lambda $ is a finite discrete $ (p,q) $-Bessel measure for $ \mu $, then there exists a finite $ q $-Bessel for $ L^p(\mu)$. In the following we give an example of a discrete spectral measure $ \mu $ such that it has a finite discrete $ (p,q) $-Bessel measure $ \nu $ with Bessel bound less than $ \nu(\mathbb{R}^d) $, precisely $ \nu \in \mathcal{B}_{1}(\mu)_{(p, q)} $, where $1\leq p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. \[3.21\] Consider the atomic measure $ \mu :=\frac{1}{2} (\delta_0 +\delta_\frac{1}{2}) $, the set $ \{e_l :\l \in L:=\{0,1\} \} $ is an orthonormal basis for $ L^2 (\mu) $. Hence $ \sum_{l \in L } |\left<f,e_l \right>_{L^2(\mu)}|^2 = \| f\|_{L^2(\mu)}^2 $ for all $ f\in L^2(\mu) $. Moreover, for every $ f $ in $ L^1(\mu) $ we have $ \|\{[f, e_l]_{L^1(\mu)}\}_{l\in L} \|_\infty \leq \| f \|_{L^1(\mu)} $. Now by applying the Riesz-Thorin interpolation theorem $ \sum_{l \in L } |[f,e_l]_{L^p(\mu)}|^q \leq \| f\|_{L^p(\mu)}^q $, for all $ f\in L^p(\mu) $, where $1\leq p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. Therefore, $\{e_l\}_{l\in L} $ is a finite $ q $-Bessel for $ L^p(\mu)$ with bound 1, and $ \nu=\sum_{l \in L} \delta_l $ is a finite discrete $ (p,q) $-Bessel measure for $ \mu $, especially $ \nu \in \mathcal{B}_{1}(\mu)_{(p, q)} $, where $1\leq p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. When $ p > 2 $ and $ q $ is the conjugate exponent to $ p $, based on Proposition \[3.3\] $ \nu \in \mathcal{B}_{2}(\mu)_{(p, q)} $ and $\{e_l\}_{l\in L} $ is a finite $ q $-Bessel for $ L^p(\mu)$ with bound 2. \[3.22\] Let $ \mu= \sum_{c\in C}p_c\delta_c $ be a discrete probability measure on $ \mathbb{R}^d $. $ \mu $ is an F-spectral measure with an F-spectrum $ \Lambda $ if and only if $ \#C<\infty $ and $ \#\Lambda<\infty $. Let $ 1<p, q<\infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. If $ \mu $ is any probability measure, then we have the following two equivalent statements: \(i) A finite discrete measure $ \nu=\sum_{\lambda \in \Lambda} \delta_\lambda $ is a $ (p,q) $-Bessel measure for $ \mu $, precisely $\nu \in \mathcal{B}_{\nu(\mathbb{R}^d)}(\mu)_{(p, q)} $. If $ \mu= \sum_{c\in C}p_c\delta_c $ is an F-spectral measure, then there may exist a better $ (p,q) $-Bessel bound for $ \nu $, where $1< p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. \(ii) A finite sequence $ \{e_\lambda\}_{\lambda \in \Lambda} $ is a $ q $-Bessel for $ L^p(\mu) $ with bound $ \nu(\mathbb{R}^d) $. If $ \mu= \sum_{c\in C}p_c\delta_c $ is an F-spectral measure, then $ \{e_\lambda\}_{\lambda \in \Lambda} $ may admit a better $ q $-Bessel bound, where $1< p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. The conclusion follows from Propositions \[3.3\], \[3.22\], \[3.2\], and the Riesz-Thorin interpolation theorem (see also Example \[3.21\]). \[t.0\] Let $R$ be a $d \times d$ expansive integer matrix, $0 \in \mathcal{A} \subset \mathbb{Z}^d$. Let $ \mu_\mathcal{A} $ be an invariant measure associated to the iterated function system $$\tau_a (x) = R^{-1}(x + a) \quad (x \in \mathbb{R}^d, a \in\mathcal{A})$$ and the probabilities $(\rho_a)_{a\in \mathcal{A}}$. Then $ \mu $ has an infinite B-spectrum of positive Beurling dimension (Beurling dimension is used as a method of investigating existence of Bessel spectra for singular measures). \[t.1\] Any fractal measure $ \mu $ obtained from an affine iterated function system has an infinite discrete $(p,q)$-Bessel measure $ \nu $, where $1\leq p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. Suppose that $R$ is a $d \times d$ expansive integer matrix, $0 \in \mathcal{A} \subset \mathbb{Z}^d$. If $ \mu_\mathcal{A} $ is an invariant measure associated to the iterated function system $$\tau_a (x) = R^{-1}(x + a) \quad (x \in \mathbb{R}^d, a \in\mathcal{A})$$ and the probabilities $(\rho_a)_{a\in \mathcal{A}}$, then according to Theorem \[t.0\] there exist an infinite subset $ \Lambda $ of $ \mathbb{R}^d $ and a constant $ B > 0 $ such that $$\sum_{\lambda \in \Lambda }|\left<f,e_\lambda \right>_{L^2(\mu_\mathcal{A})}|^2 \leq B\| f\|_{L^2(\mu_\mathcal{A})}^2 \quad \text{for all } f\in L^2(\mu_\mathcal{A}).$$ We also have $ \| \{[f,e_\lambda]_{L^1(\mu_\mathcal{A})}\}_{\lambda \in \Lambda} \|_\infty \leq \| f \|_{L^1(\mu_\mathcal{A})} $, for every $ f \in L^1(\mu_\mathcal{A}) $. Now if $ \dfrac{1}{p} = 1 -\dfrac{\theta}{2} ; \ \dfrac{1}{q} =\dfrac{\theta}{2}$, for $ 0 <\theta < 1 $ (i.e., $1< p < 2 $ and $ q $ is the conjugate exponent to $ p $), then the Riesz-Thorin interpolation theorem yields $$\sum_{\lambda \in \Lambda }|[f,e_\lambda]_{L^p(\mu_\mathcal{A})}|^q \leq B'^q\| f\|_{L^p(\mu_\mathcal{A})}^q \quad \text{for all } f\in L^p(\mu_\mathcal{A}),$$ where $ B'\leq B^{\frac{1}{2}\theta} $. Thus, $ \nu=\sum_{\lambda \in \Lambda} \delta_\lambda $ is a $(p,q)$-Bessel measure for $ \mu_\mathcal{A} $, and $ \nu \in \mathcal{B}_{B'^q}(\mu_\mathcal{A})_{(p, q)} $, where $1 < p < 2 $ and $ q $ is the conjugate exponent to $ p $. Moreover, we have $ \nu \in \mathcal{B}_{B}(\mu_\mathcal{A})_{(2, 2)} $ and $ \nu \in \mathcal{B}_{1}(\mu_\mathcal{A})_{(1, \infty)} $. On the other hand for every $1 < p < 2 $ and $ q $ (the conjugate exponent to $ p $), $ \{e_\lambda\}_{\lambda \in \Lambda} $ is a $ q $-Bessel for $ L^p(\mu_\mathcal{A}) $ with bound $ B'^q $. If a measure $ \mu $ is an F-spectral measure, then it must be of pure type, i.e, $ \mu $ is either discrete, singular continuous or absolutely continuous [@19; @13]. The case when the measure $ \mu $ is singular continuous, is not precisely known. The first known example of a singular continuous spectral measure supported on a non-integer dimension set (a fractal measure), was given by Jorgensen and Pedersen [@16]. They showed that the measure $ \mu_4 $ (the Cantor measures supported on Cantor set of $1/4$ contraction), is spectral. A spectrum of $\mu_4 $ is $ \Lambda =\left \{ \sum_{m=0}^k 4^m d_m \; : d_m \in \{0, 1\}, k\in \mathbb{N} \right\} $. They also showed that $ \mu_{2k} $ (the Cantor measures with even contraction ratio) is spectral, but $ \mu_{2k +1} $ (the Cantor measures with odd contraction ratio) is not. Since Cantor type measures are fractal measures, by applying Theorem \[t.1\] one can obtain that every Cantor type measure $ \mu $ admits a $ (p,q) $-Bessel measure $ \nu=\sum_{\lambda \in \Lambda_{ \mu } } \delta_\lambda $, where $1\leq p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. Moreover, for every spectral Cantor type measure $ \mu_{2k} $, we have $ \nu \in \mathcal{B}_1(\mu_{2k})_{p, q}$, where $1\leq p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. In [@21] the author presents a method for constructing many examples of continuous measures $ \mu $ (including fractal ones) which have components of different dimensions, but nevertheless they are F-spectral measures. In the following we give some results from [@21]. By applying the Riesz-Thorin interpolation theorem, one can obtain infinite discrete $ (p,q) $-Bessel measures $ \nu=\sum_{\lambda \in \Lambda_{\mu}} \delta_\lambda $ (where $1\leq p \leq 2 $ and $ q $ is the conjugate exponent to $ p $), for such F-spectral measures $ \mu $. Let $ \mu $ and $ \mu' $ be positive and finite measures on $ \mathbb{R}^n $ and $ \mathbb{R}^m $, respectively. A *mixed type measure* $ \rho $ is a measure which is constructed on $ \mathbb{R}^{n+m} = \mathbb{R}^n \times \mathbb{R}^m $ and defined by $$\rho = \mu \times \delta_0 + \delta_0 \times \mu',$$ where $ \delta_0 $ denotes the Dirac measure at the origin. Equivalently, the measure $ \rho $ may be defined by the requirement that $$\int _{\mathbb{R}^n \times \mathbb{R}^m} f(x, y) d\rho (x, y) = \int _{\mathbb{R}^n} f(x, 0) d\mu(x) + \int _{\mathbb{R}^m} f(0, y) d\mu'(y),$$ for every continuous, compactly supported function $ f $ on $ \mathbb{R}^n \times \mathbb{R}^m $. Let $ \mu $ and $ \mu' $ be continuous F-spectral measures. Then the mixed type measure $ \rho = \mu \times \delta_0 + \delta_0 \times \mu' $ is also an F-spectral measure. If $ \mu $ is the sum of the $ k $-dimensional area measure on $[0, 1]^k \times \{0\}^{d-k}$, and the\ $j$-dimensional area measure on $\{0\}^{d-j} \times [0, 1]^j$ where $1 \leq j, k \leq d - 1$, then $ \mu $ is an F-spectral measure. The following theorem provides many examples of single dimensional measures which are F-spectral measures: Let $ \phi: \mathbb{R}^k \rightarrow \mathbb{R}^{d-k} $ be a smooth function $(1 \leq k \leq d - 1)$. If $ \mu $ is the k-dimensional area measure on a compact subset of the graph $\{(x, \phi(x)) : x \in \mathbb{R}^k\}$ of $ \phi $, then $ \mu $ is an F-spectral measure. The next proposition shows that if $ 1<p, q<\infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $, then considering any countable subset (finite or infinite) $ \Lambda $ of $ \mathbb{R}^d $, one can obtain tight $ (p,q) $-frame measures and $ (p,q) $-Plancherel measures $\nu_\Lambda $ for $ \delta_0 $. In addition, there exist tight and Parseval $ q $-frames for $ L^p(\delta_0) $. \[3.20\] Suppose that $ 1<p, q<\infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. Then there exists a measure $ \mu $ which admits tight $ (p,q) $-frame measures and $ (p,q) $-Plancherel measures. Moreover, there exist tight and Parseval $ q $-frames for $ L^p(\mu) $. Let $ \mu=\delta_0 $. For a countable subset $ \Lambda $ of $ \mathbb{R}^d $, Let $\nu_\Lambda =\sum_{\lambda\in\Lambda} c_\lambda \delta_{\lambda} $ where $ c_\lambda > 0 $. If $ \sum_{\lambda\in\Lambda} c_\lambda = m \neq 1 $, then for all $ f\in L^p(\mu) $, $$\int_{\mathbb{R}^d} |[f, e_t]_{L^p(\mu)}|^q d\nu(t) = \sum_{\lambda\in\Lambda} c_\lambda |f(0)|^q = m\| f \|_{L^p(\mu)}^q.$$ If $ 0 < c_\lambda <1 $ and $ \sum_{\lambda\in\Lambda} c_\lambda = 1 $, then for all $ f\in L^p(\mu) $, $$\int_{\mathbb{R}^d} |[f, e_t]_{L^p(\mu)}|^q d\nu(t) = \sum_{\lambda\in\Lambda} c_\lambda |f(0)|^q = \| f \|_{L^p(\mu)}^q.$$ On the other hand, for all $ f\in L^p(\mu) $ we have $$\int_{\mathbb{R}^d} |[f, e_t]_{L^p(\mu)}|^q d\nu(t) = \sum_{\lambda\in\Lambda} c_\lambda |[f, e_{\lambda}]_{L^p(\mu)}|^q = \sum_{\lambda\in\Lambda} |[f, \sqrt [q]c_\lambda e_{\lambda}]_{L^p(\mu)}|^q.$$ Hence If $ \sum_{\lambda\in\Lambda} c_\lambda = m \neq 1 $, then $ \{ \sqrt [q]c_\lambda e_{\lambda}\}_{\lambda \in \Lambda} $ is a tight $ q $-frame for $ L^p(\mu) $, and If $ 0 < c_\lambda <1 $, $ \sum_{\lambda\in\Lambda} c_\lambda = 1 $, then $ \{ \sqrt [q]c_\lambda e_{\lambda}\}_{\lambda \in \Lambda} $ is a Parseval $ q $-frame for $ L^p(\mu) $. Let $ \mu $ be a finite Borel measure and $ B $ be a positive constant. Then there exists a $ (p,q) $-Bessel measure $ \nu $ for $ \mu $ for all $ 1< p, q < \infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $, such that $ \nu \in \mathcal{B}_B(\mu)_{p,q} $. In addition, for every $ 1< p, q < \infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $, there exists a $ q $-Bessel with bound $ B $ for $ L^p(\mu) $. Let $ \nu =\sum_{i\in I}c_i \delta_{\lambda_i} $ for some $ \lambda_i\in \mathbb{R}^d $ such that $\sum_{i\in I}c_i \leq \dfrac{B}{\mu(\mathbb{R}^d)} $. Let $ p>1 $ and $ f\in L^p(\mu) $. If $ q $ is the conjugate exponent to $ p $, then by applying Holder’s inequality we have $$\label{4} \int_{\mathbb{R}^d} |[f, e_t]_{L^p(\mu)}|^q d\nu(t) =\sum_{i\in I}c_i\int_{\mathbb{R}^d} |[f, e_t]_{L^p(\mu)}|^q d\delta_{\lambda_i}(t)\leq\sum_{i\in I}c_i \parallel f\parallel_{L^p(\mu)}^q \mu(\mathbb{R}^d) \leq B\parallel f\parallel_{L^p(\mu)}^q.$$ Hence $\nu\in\mathcal{B}_B(\mu)_{p,q} $. Since $$\sum_{i\in I} |[f, \sqrt [q]c_i e_{\lambda_i}]_{L^p(\mu)}|^q = \sum_{i\in I}c_i |[f, e_{\lambda_i}]_{L^p(\mu)}|^q = \int_{\mathbb{R}^d} |[f, e_t]_{L^p(\mu)}|^q d\nu(t),$$ the second statement follows from (\[4\]). All infinite $ (p,q) $-Bessel measures $ \nu $ we observed were discrete. Now the question is whether we can find a finite measure $ \mu $ which admits a continuous infinite $ (p,q) $-Bessel measure $ \nu $. The following proposition shows that the answer is affirmative. \[3.26\] If $ \nu=\lambda $ is the Lebesgue measure on $ \mathbb{R}^d $ and $ \mu=\lambda |_{[0 , 1]^d} $, then $ \lambda $ is a $ (p,q) $-Bessel measure for $ \mu $ where $1\leq p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. According to Plancherel’s theorem the following equation is satisfied: $$\int_{\mathbb{R}^d} |\hat{f}(t)|^2 d\lambda(t) =\int_{\mathbb{R}^d} |f(x)|^2 d\lambda(x) \quad \text{for all } f\in L^2(\lambda).$$ If $ f $ is supported on $ [0 , 1]^d $, then $$\int_{\mathbb{R}^d} |\widehat{f d\mu}|^2 d\lambda(t) =\int_{[0, 1]^d} |f(x)|^2 d\mu (x) \quad \text{for all } f\in L^2(\mu).$$ Moreover, we have $ \| \widehat{fd\mu} \|_\infty \leq \| f \|_{L^1(\mu)} $ for all $ f $ in $ L^1(\mu) $. Now by applying the Riesz-Thorin interpolation theorem $$\int_{\mathbb{R}^d} |\widehat{f d\mu}|^q d\lambda(t) \leq \| f\|_{L^p(\mu)}^q \quad \text{for all } f\in L^p(\mu),$$ where $1\leq p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. Hence $ \lambda \in \mathcal{B}_1(\mu)_{p,q} $. The measure $ \mu=\lambda |_{[0 , 1]^d} $ has infinite continuous and discrete $ (p,q) $-Bessel measures, where $1\leq p \leq 2 $ and $ q $ is the conjugate exponent to $ p $. More precisely, if $ \nu_1 = \sum_{t\in \mathbb{Z}^d}\delta_t $ and $ \nu_2=\lambda $, then $ \nu_1, \nu_2 \in \mathcal{B}_1(\mu)_{p,q} $. The conclusion follows from Example \[ex 4\] and Proposition \[3.26\]. Properties and Structural Results ================================= In this section our assertions are based on the results by Dutkay, Han, and Weber from [@5]. We generalize the results and give some of the proofs for completeness. \[prop 2\] Let $ \mu $ be a Borel probability measure. Let $ 1< p, q < \infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. If $\nu$ is a $(p,q)$-Bessel measure for $\mu$, then there exists a constant $ C $ such that $ \nu(K) \leq C diam(K)^d $ for any compact subset $ K $ of $ \mathbb{R}^d $. Accordingly, $ \nu $ is $\sigma$-finite. It is easy to check that $\widehat{d\mu} : \mathbb{R}^d \rightarrow \mathbb{C}$ is uniformly continuous and $\widehat{d\mu}(0) =\mu(\mathbb{R}^d) =1$. So for every $ \eta>0 $ there exists $\epsilon > 0$ such that for $x\in \mathsf{B}(0,\epsilon)$ we have $|\widehat{d\mu}(0)|-|\widehat{d\mu}(x)|\leq|\widehat{d\mu}(0)-\widehat{d\mu}(x)|\leq \eta$, and then $ |\widehat{d\mu}(x)|\geq 1-\eta $. If $ \delta:= (1-\eta)^q $, then $|\widehat{d\mu}(x)|^q \geq \delta$ for $x\in \mathsf{B}(0,\epsilon)$. Thus, for any $t \in \mathbb{R}^d$, $$\begin{aligned} B=B\|e_t\|_{L^p(\mu)}^q &\geq \int_{\mathbb{R}^d} |[e_t , e_x]|^q d\nu(x) =\int_{\mathbb{R}^d} |[1 , e_{x-t}]|^q d\nu(x)= \int_{\mathbb{R}^d} |\widehat{d\mu}(x-t)|^q d\nu(x)\\ &\geq \int_{\mathsf{B}(t,\epsilon)}|\widehat{d\mu} (x-t)|^q d\nu(x) \geq \nu(\mathsf{B}(t,\epsilon))\delta. \end{aligned}$$ Now Let $ K\subseteq\mathbb{R}^d $ be compact and $ r= diam (K) $. Then there exists a point $ x = (x_1, \ldots, x_d) $ in $ \mathbb{R}^d $ such that $ K \subset \prod_{i=1}^d [x_i-r, x_i+r] $. We may assume that $ \epsilon < 2r $ and $ 2r/\epsilon \in \mathbb{N} $. Let $ M= 2r/\epsilon $. We have $ \prod_{i=1}^d [x_i-r, x_i+r] = \bigcup_{\alpha=1}^{M^d} C_\alpha $ where $ C_\alpha $s are d-dimensional cubes of side length $ \epsilon $. For any $ \alpha \in \{1,\ldots, M^d \} $, let $ t_\alpha $ be the center point of $ C_\alpha $. Then $ C_\alpha \subset \mathsf{B}(t_\alpha,\epsilon) $. Now if $ C:= (2/\epsilon)^d B/\delta $, then $$\nu(K) \leq \nu \left(\bigcup_{\alpha=1}^{M^d}\mathsf{B}(t_\alpha, \epsilon)\right) \leq\sum^{M^d}_{\alpha=1}\nu(\mathsf{B}(t_\alpha, \epsilon) )\leq \left(\dfrac{2r}{\epsilon}\right)^d \dfrac{B}{\delta} = r^d \left(\dfrac{2}{\epsilon}\right)^d \dfrac{B}{\delta} = C r^d.$$ Hence the assertion follows. Let $ 1< p, q < \infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. Let $ B > A > 0 $. Then the set $\mathcal{F}_{A,B}(\mu)_{p,q}$ is empty for some finite compactly supported Borel measures $ \mu $. Let $\mu=\chi_{[0,1]} dx+\delta_2$. Suppose $\nu \in \mathcal{F}_{A,B}(\mu)_{p,q}$. Let $f :=\chi_{\{2\}}$. Then $\| f\|_{L^p (\mu)} = 1$ and $|[f,e_t]| = 1$ for all $ t\in\mathbb{R}$. In addition, the upper bound implies that $\nu(\mathbb{R}) \leq B < \infty$. Then from the inner regularity of Borel measures we obtain that for any $\epsilon > 0$ there exist a compact set $ K\subset \mathbb{R} $ and a positive constant $ R $ such that $ \nu(\mathbb{R})- \epsilon < K \leq \nu (\mathsf{B}(0, R)) $. Therefore $\nu(\mathbb{R} \setminus \mathsf{B}(0, R)) < \epsilon$. Choose some $T$ large, arbitrary and let $g(x) := e^{-2\pi iTx}\chi_{[0,1]}$. Then $$|[g,e_t]_{L^p (\mu)}|^q = \left|\int_{[0, 1]} e^{-2\pi i (T + t)x} dx \right|^q = \left|\frac{sin(\pi(T+t))}{\pi(T+t)}\right|^q \qquad (t\in\mathbb{R}).$$ The substitution $ z:= -2\pi x $ gives the last equality. Consequently, $|[g,e_t]_{L^p(\mu)}|^q \leq 1$ for all $ t\in\mathbb{R}$, and if we take $T\geq2R$, then for all $t\in (-R,R)$ $$|[g,e_t]_{L^p(\mu)}|^q \leq \frac{1}{\pi^q(T-R)^q}.$$ Hence from the lower bound we obtain $$\begin{aligned} A=A\| g\|_{L^p(\mu)}^q & \leq \int_\mathbb{R} |[g,e_t]_{L^p(\mu)}|^q d\nu(t) = \int_{\mathsf{B}(0, R)}|[g,e_t]_{L^p(\mu)}|^q d\nu(t) +\int_{\mathbb{R} \setminus \mathsf{B}(0, R)}|[g,e_t]_{L^p(\mu)}|^q d\nu(t)\\&\leq \frac{1}{\pi^q(T-R)^q}.\nu(\mathbb{R}) + \epsilon.\end{aligned}$$ Now if $T \rightarrow \infty$ and $\epsilon \rightarrow 0$, then $ A=0 $. This is a contradiction. The next proposition shows that if there exists a $(p,q)$-Bessel/frame measure, then many others can be constructed. \[prop 4.3\] Let $ \mu $ be a finite Borel measure and $ A, B $ be positive constants. Let $ 1< p, q < \infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. Then both sets $ \mathcal{B}_B(\mu)_{p,q} $ and $ \mathcal{F}_{A,B}(\mu)_{p,q} $ are convex and closed under convolution with Borel probability measures. Let $ \nu_1 , \nu_2 \in\mathcal{B}_B(\mu)_{p,q} $ and $0 <\lambda <1$. For all $f \in L^p (\mu)$, $$\int_{\mathbb{R}^d} |\widehat{fd\mu}|^q d(\lambda\nu_1 +(1-\lambda)\nu_2)=\lambda\int_{\mathbb{R}^d} |\widehat{fd\mu}|^q d\nu_1 + (1-\lambda)\int_{\mathbb{R}^d} |\widehat{fd\mu}|^q d\nu_2 \leq B\| f\|_{L^p(\mu)}^q.$$ Then $ \lambda\nu_1 +(1-\lambda)\nu_2 \in\mathcal{B}_B(\mu)_{p,q} $. Similarly, if $ \nu_1 , \nu_2 \in\mathcal{F}_{A,B}(\mu)_{p,q}$, then $ \lambda\nu_1 +(1-\lambda)\nu_2 \in \mathcal{F}_{A,B}(\mu)_{p,q}$. Let $ s\in \mathbb{R}^d $. Then for all $f \in L^p (\mu)$, $$\|e_s f\|^p_{L^p(\mu)}=\int_{\mathbb{R}^d} |e_s(x)f(x)|^p d\mu(x)=\int_{\mathbb{R}^d} |f(x)|^p d\mu(x)=\|f\|^p_{L^p(\mu)}.$$ In addition, Let $ \nu\in\mathcal{B}_{B}(\mu)_{p,q} $ and $ \rho $ be a Borel probability measure on $\mathbb{R}^d$. Then for any $ t \in \mathbb{R}^d $ and $f \in L^p (\mu)$, $$\begin{aligned} [e_{-s}f, e_t]_{L^p(\mu)}&=\int_{\mathbb{R}^d} e_{-s}(x)f(x)e^{-2\pi it\cdot x} d\mu(x) =\int_{\mathbb{R}^d}f(x)e^{-2\pi i(s+t)\cdot x} d\mu(x) \\ &=[f ,e_{s+t}]_{L^p(\mu)}.\end{aligned}$$ Therefore, $$\begin{aligned} \int_{\mathbb{R}^d} |[f,e_t]_{L^p(\mu)}|^q d\nu \ast \rho(t) & = \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} |[f,e_{t+s}]_{L^p(\mu)}|^q d\nu(t) \ d\rho(s) = \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} |[e_{-s}f,e_t]_{L^p(\mu)}|^q d\nu(t) \ d\rho(s)\\&\leq \int_{\mathbb{R}^d} B\| e_{-s}f\|_{L^p(\mu)}^q d\rho(s) = B \int_{\mathbb{R}^d}\| f\|_{L^p(\mu)}^q d\rho(s) = B\| f\|_{L^p(\mu)}^q.\end{aligned}$$ For $ \nu\in\mathcal{F}_{A,B}(\mu)_{p,q} $ one can obtain the lower bound analogously. Let $ 1< p, q < \infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. If there exists a $(p,q)$-Bessel/frame measure for $\mu$, then there exists one which is absolutely continuous with respect to the Lebesgue measure and whose Radon-Nikodym derivative is $ C^\infty $. Let $ \nu $ be a $(p,q)$-Bessel/frame measure for $\mu$. Convoluting $ \nu $ with the Lebesgue measure on $ [0,1] $ we have $$\begin{aligned} \nu*\chi_{[0 ,1]}dm (E)&=\int_{\mathbb{R}}\int_{\mathbb{R}} \chi_E (x+y)d\nu(x) \chi_{[0 ,1]}(y) dm (y) =\int_{\mathbb{R}}\int_{\mathbb{R}} \chi_E (t) \chi_{[0 ,1]}(t-x) d\nu(x)dm (t-x)\\ &=\int_{\mathbb{R}}\chi_E (t) \nu([t-1 , t])dm(t) =\int_E \nu([t-1 , t])dm,\end{aligned}$$ where $ E $ is any Borel subset of $ \mathbb{R} $. Thus, we obtained a $(p,q)$-Bessel/frame measure for $ \mu $ which is absolutely continuous with respect to the Lebesgue measure. Now consider the following two propositions from [@10].\ $ (i) $ If $ d\mu= fdm $ and $ d\nu= gdm $, then $ d(\mu\ast \nu) = (f \ast g) dm $.\ $ (ii) $ If $ f\in L^1 $ (or $ f $ is locally integrable on $ \mathbb{R}^d $), $ g\in C^k $, and $ \partial^\alpha $ is bounded for $ |\alpha|\leq k $, then $ f \ast g \in C^k $ and $ \partial^\alpha (f \ast g) = f \ast(\partial^\alpha g) $ for $ |\alpha|\leq k $. Let $ g \geq 0 $ be a compactly supported $ C^\infty $-function with $ \int g(t) dt = 1 $. Let $ d\mu_0 = gdm $ and\ $ d\nu_0 = \nu \ast \chi_{[0 ,1]}dm $. Then we have $ d(\mu_0\ast \nu_0) = ( \nu([\cdot-1 , \cdot]) \ast g) dm $ and $\nu([\cdot-1 , \cdot]) \ast g \in C^\infty $. A sequence of Borel probability measures {$ \lambda_n\} $ is called an *approximate identity* if $$sup\{\parallel t\parallel : t\in supp \lambda_n\} \rightarrow 0 \quad \text{as}\quad n\rightarrow \infty .$$ \[lem 6\] Let $ \{\lambda_n\} $ be an approximate identity. If $ f $ is a continuous function on $\mathbb{R}^d $, then for any $ x\in\mathbb{R}^d $,$\int f(x+t)\ d\lambda_n(t) \rightarrow f(x)$ as $n \rightarrow \infty$. By Proposition \[prop 4.3\], if $ \nu $ is a $(p,q)$-Bessel/frame measure for $ \mu $, then $ \nu \ast \lambda $ is a $(p,q)$-Bessel/frame measure for $ \mu $ with the same bound(s), where $ \lambda $ is any Borel probability measure. An obvious question is under what conditions the converse is true. The next theorem gives an answer. Let $ 1< p, q < \infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. Let $ \{\lambda_n\} $ be an approximate identity. Suppose $ \nu $ is a $ \sigma $-finite Borel measure, and suppose all measures $ \nu \ast \lambda_n $ are $(p,q)$-Bessel/frame measures for $ \mu $ with uniform bounds, independent of $ n $. Then $ \nu $ is a $(p,q)$-Bessel/frame measure. Take $ f\in L^p(\mu) $. Since $| [f, e_.]_{L^p(\mu)}|^q $ is continuous on $\mathbb{R}^d $, by Lemma \[lem 6\] and Fatou’s lemma we have $$\begin{aligned} \int_{\mathbb{R}^d} |[f,e_x]_{L^p(\mu)}|^q d\nu(x) & \leq \liminf_n \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} |[f,e_{x+t}]_{L^p(\mu)}|^q d\lambda_n(t)\ d\nu(x)\\ &= \liminf_n \int_{\mathbb{R}^d} |[f,e_y]_{L^p(\mu)}|^q d(\nu \ast \lambda_n)(y)\\ &\leq B\| f \|_{L^p(\mu)}^q. \end{aligned}$$ Hence $ \nu $ is a $(p,q)$-Bessel measure with the same bound $ B $ as $ \nu \ast \lambda_n $. Now showing that $$\int_{\mathbb{R}^d} |[f,e_x]_{L^p(\mu)}|^q d(\nu \ast \lambda_n) \rightarrow \int_{\mathbb{R}^d} |[f,e_x]_{L^p(\mu)}|^q d\nu,$$ gives the lower bound (see [@5]). We need the following two propositions from [@5] to present a general way of constructing $ (p,q) $-Bessel/frame measures for a given measure. Let $ \mu $ and $ \mu' $ be Borel probability measures. For $ f\in L^1(\mu) $, the measure $ (f d\mu)\ast \mu' $ is absolutely continuous w.r.t. $ \mu \ast \mu' $ and if the Radon-Nikodym derivative is denoted by $ P f $, then $$P f = \frac{(f d\mu)\ast\mu'}{d(\mu\ast\mu')}.$$ Let $ \mu$, $ \mu' $ be two Borel probability measures and $ 1 \leq p\leq\infty $. if $ f\in L^p(\mu) $, then the function $ P f $ is also in $ L^p( \mu \ast \mu') $ and $$\| P f\|_{ L^p( \mu \ast \mu')}\ \leq\ \| f\|_{ L^p( \mu )}.$$ Now we show that if a convolution of two measures admits a $ (p,q) $-Bessel/frame measure, then one can obtain a $ (p,q) $-Bessel/frame measure for one of the measures in the convolution by using the Fourier transform of the other measure in the convolution. Let $ \mu $, $ \mu' $ be two Borel probability measures. Let $ 1< p, q < \infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. If $ \nu $ is a $ (p,q) $-Bessel measure for $ \mu \ast \mu' $, then $ | \hat{\mu'} |^q d\nu $ is a $ (p,q) $-Bessel measure for $ \mu $ with the same bound. If in addition $ \nu $ is a $ (p,q) $-frame measure for $ ( \mu \ast \mu') $ with bounds $ A $ and $ B $, and for all $ f \in L^p(\mu) $, $ c\| f \|_{L^p(\mu)}^q \leq \| P f\|_{ L^p( \mu \ast \mu')}^q $, then $ | \hat{\mu}' |^q d\nu $ is a $ (p,q) $-frame measure for $ \mu $ with bounds $ c A $ and $ B $. If $ \mu,\nu \in M(\mathbb{R}^d) $, then $ \widehat{\mu\ast\nu} = \hat{\mu}.\hat{\nu} $ (see[@10]). Take $ f \in L^p(\mu) $. Then $$\int_{\mathbb{R}^d} |\widehat{(f d\mu)}|^q . | \hat{\mu'} |^q d\nu = \int_{\mathbb{R}^d} |\widehat{(f d\mu)\ast \mu' |}^q d\nu = \int_{\mathbb{R}^d} |\widehat{P f d(\mu \ast \mu')}|^q d\nu.$$ Thus, we have $$cA\| f \|_{L^p(\mu)}^q\leq A \| P f\|_{ L^p( \mu \ast \mu')}^q \leq \int_{\mathbb{R}^d} |\widehat{P f d(\mu \ast \mu')}|^q d\nu \leq B \| P f\|_{ L^p( \mu \ast \mu')}^q \leq B\| f \|_{L^p(\mu)}^q.$$ In the next theorem we have some stability results. In fact, this theorem is a generalization of Proposition \[2.21\]. \[theo 219\] Let $ \mu $ be a compactly supported Borel probability measure. Let $ 1< p, q < \infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. If $ \nu$ is a $ (p,q) $-Bessel measure for $ \mu $, then for any $ r > 0 $ there exists a constant $ D > 0 $ such that $$\int_{\mathbb{R}^d} \sup_{|y|\leq r} |[f,e_{x+y}]_{L^p(\mu)}|^q d\nu(x) \leq D \| f \|_{L^p(\mu)}^q, \qquad \text{for all}\; f \in L^p(\mu).$$ If $ \nu $ is a $ (p,q) $-frame measure for $ \mu $, then there exist constants $ \delta > 0 $ and $ C > 0 $ such that $$C\| f \|_{L^p(\mu)}^q \leq \int_{\mathbb{R}^d} \inf_{|y|\leq \delta} |[f,e_{x+y}]_{L^p(\mu)}|^q d\nu(x), \qquad \text{for all}\; f \in L^p(\mu).$$ The approach is completely similar to the proof of Theorem $ 2.10 $ from [@5]. We show that by using this stability of $ (p,q) $-frame measures, one can obtain atomic $ (p,q) $-frame measures from a general $ (p,q) $-frame measure. \[4.12\] Let $ Q = [0,1)^d $ and $ r > 0 $. If $ \nu$ is a Borel measure on $ \mathbb{R}^d $ and if $ (x_k)_{k\in\mathbb{Z}^d} $ is a set of points such that for all $ k\in\mathbb{Z}^d $ we have $ x_k\in r(k +Q) $ and $ \nu(r(k +Q)) <\infty $, then a *discretization of the measure $ \nu $* is defined by $$\nu' := \sum_{k\in\mathbb{Z}^d} \nu(r(k +Q))\delta_{x_k}.$$ \[theo 220\] Let $ 1< p, q < \infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. If a compactly supported Borel probability measure $ \mu $ has a $(p,q)$-Bessel/frame measure $ \nu$, then it also has an atomic one. More precisely, if $ \nu$ is a $ (p,q) $-Bessel measure for $ \mu $ and if $ \nu' $ is a discretization of the measure $ \nu $, then $ \nu' $ is a $ (p,q) $-Bessel measure for $ \mu $. If $ \nu $ is a $ (p,q) $-frame measure for $ \mu $ and $ r > 0 $ is small enough, then $ \nu' $ is a $ (p,q) $-frame measure for $ \mu $. Let $ Q = [0,1)^d $. Let $ (x_k)_{k\in\mathbb{Z}^d} $ be a set of points such that $ x_k\in r(k +Q) $ for all $ k\in\mathbb{Z}^d $. For every $ x\in r(k +Q) $ define $ \epsilon(x) := x_k - x $. Thus, $ |\epsilon(x)| \leq r\sqrt{d} =:r' $ and for any $ f \in L^p(\mu) $, $$\begin{aligned} \int_{\mathbb{R}^d} |[f,e_{x +\epsilon(x)}]_{L^p(\mu)}|^q d\nu(x) &= \sum_{k\in\mathbb{Z}^d} \int_ {r(k +Q)} |[f,e_{x_k}]_{L^p(\mu)}|^q d\nu(x)\\ & = \sum_{k\in\mathbb{Z}^d}\nu(r(k +Q))|[f,e_{x_k}]_{L^p(\mu)}|^q.\end{aligned}$$ Since we have $$\begin{aligned} \int_{\mathbb{R}^d} \inf_{|y|\leq r'} |[f,e_{x+y}]_{L^p(\mu)}|^q d\nu(x) & \leq \int_{\mathbb{R}^d} |[f,e_{x +\epsilon(x)}]_{L^p(\mu)}|^q d\nu(x)\\ & \leq \int_{\mathbb{R}^d} \sup_{|y|\leq r} |[f,e_{x+y}]_{L^p(\mu)}|^q d\nu(x), \end{aligned}$$ the upper and lower bounds follow from Theorem \[theo 219\]. By Lemma \[3.6\], if there exists a purely atomic $ (p,q) $-frame measure $ \nu $ for a probability measure $ \mu $, then there exists a $ q $-frame for $ L^p(\mu) $. Now we conclude that if there exists a $ (p,q) $-frame measure $ \nu $ (not necessarily purely atomic) for a compactly supported probability measure $ \mu $, then there exists a $ q $-frame for $ L^p(\mu) $. Let $ \mu $ be a compactly supported Borel probability measure. Let $ 1< p, q < \infty $ and $ \dfrac{1}{p} + \dfrac{1}{q} =1 $. If $ \nu $ is a $ (p,q) $-frame measure for $ \mu $ with bounds $ A, B $ and $ r > 0 $ is sufficiently small, then there exist positive constants $ C, D $ such that $\{ c_ke_{x_k} : k\in\mathbb{Z}^d\} $ is a $ q $-frame for $ L^p(\mu) $ with bounds $ C, D $, where $ x_k\in r(k +Q) $ and $ c_k =\sqrt [q]{\nu(r(k +Q))} $. Let $ \nu \in \mathcal{F}_{A,B}(\mu)_{p, q} $. Then by Theorems \[theo 220\] and \[theo 219\], $ \nu'=\sum _{k\in\mathbb{Z}^d} c_k^q\delta_{x_k}$ is a $ (p,q) $-frame measure for $ \mu $. More precisely, $ \nu' \in\mathcal{F}_{C,D}(\mu)_{p, q} $. Hence for all $ f \in L^p(\mu) $, $$C\| f \|_{L^p(\mu)}^q\leq \int_{\mathbb{R}^d} |[f,e_t]_{L^p(\mu)}|^q d\nu'(t)=\sum_{k\in\mathbb{Z}^d}c_k^q |[f, e_{x_k}]_{L^p(\mu)}|^q= \sum_{k\in\mathbb{Z}^d} |[f,c_k e_{x_k}]_{L^p(\mu)}|^q\leq D\| f \|_{L^p(\mu)}^q.$$ Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to thank Dr. Nasser Golestani for his valuable guidance and helpful comments. [10]{} P. G. Casazza, O. Christensen, D. T. Stoeva, Frame expansions in separable Banach spaces, J. Math. Anal. Appl. 307 (2005) 710-723. https://doi.org/10.1016/j.jmaa.2005.02.015. O. Christensen, An Introduction to Frames and Riesz Bases, Applied and Numerical Harmonic Analysis, Birkh$\ddot{a}$user Boston Inc., Boston, MA, 2003. D. Dutkay, D. Han, Q. Sun, E. 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Wang, Fourier bases and fourier frames on self-affine measures, Preprint (2016). arXiv:1602.04750. G. B. Folland, Real analysis, second ed., John Wiley, New York, 1999. B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal. 16 (1974) 101-121. https://doi.org/10.1016/0022-1236(74)90072-X. J. R. Giles, Classes of semi-inner product spaces, Trans. Amer. Math. Soc. 129 (1967) 436-446. https://doi.org/10.1090/S0002-9947-1967-0217574-1. X.-G. He, C.-K. Lai, K.-S. Lau, Exponential spectra in $L^2(\mu)$, Appl. Comput. Harmon. Anal. 34 (3) (2013) 327-338. https://doi.org/10.1016/j.acha.2012.05.003. T.-Y. Hu, K.-S. Lau, X.-Y. Wang, On the absolute continuity of a class of invariant measures, proc. Amer. Math. Soc. 130 (3) (2001) 759-767. https://doi.org/10.2307/2699851. J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (5) (1981) 713-747. https://doi.org/10.1512/iumj.1981.30.30055. P. Jorgensen, S. 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Anal. 31 (2011) 1-25.\ https://doi.org/10.1016/j.acha.2010.09.007. $^{1}$Department of Mathematics , Science and Research Branch, Islamic Azad University, Tehran, Iran. *E-mail address*:\ $^{2}$Department of Mathematics, Faculty of Science, Islamic Azad University, Central Tehran Branch,Tehran, Iran. *E-mail address*:\ $^{3}$Department of Mathematics , Science and Research Branch, Islamic Azad University, Tehran, Iran. *E-mail address*:\ $^{4}$Department of Mathematics , Science and Research Branch, Islamic Azad University, Tehran, Iran. *E-mail address*:
--- abstract: 'When one constructs specific models with the fundamental scale as low as the TeV scale, there arise many difficulties. In this paper we examine the hybrid inflation due to bulk scalar fields, which has been proposed to solve the problem of fine-tuning in producing density perturbations. We find that the Kaluza-Klein modes play significant roles, which enhance the speed of the phase transition and alter the reheating process. We also argue that a lower bound must be put to the fundamental scale, in order to construct the successful hybrid inflation due to the bulk scalar fields.' --- [SIT-HEP/TM-10]{} 1.5 truecm .75 truecm **Tomohiro Matsuda [^1]** .4 truecm *Laboratory of Physics, Saitama Institute of Technology,* *Fusaiji, Okabe-machi, Saitama 369-0293, Japan* 1\. truecm addtoreset[equation]{}[section]{} 1. truecm Introduction ============ In spite of the great success in the quantum field theory, there is still no consistent scenario in which the quantum gravity is included. The most promising scenario in this direction would be the string theory, where the consistency is ensured by the requirement of additional dimensions. Initially the sizes of extra dimensions had been assumed to be as small as $M_p^{-1}$, but later it has been observed that there is no reason to believe such a tiny compactification radius[@Extra_1]. The idea of the large extra dimension may solve or weaken the hierarchy problem. Denoting the volume of the $n$-dimensional compact space by $V_n$, the observed Planck mass is obtained by the relation $M_p^2=M^{n+2}_{*}V_n$, where $M_{*}$ denotes the fundamental scale of gravity. If one assumes more than two extra dimensions, $M_{*}$ may be close to the TeV scale without conflicting any observable bounds. Although this new idea inspired creativity in many physicists to lead them to a new paradigm of phenomenology, a drastic modification is required for the conventional cosmological scenarios. Models of inflation and baryogenesis[@low-baryon] are especially sensitive to this low fundamental scale. To avoid extreme fine tuning, we should reconstruct the conventional scenarios of the standard cosmology. This requires inclusion of novel ideas that are quite different from the conventional one that was used for the models with large fundamental scale $M_* \sim M_p$, where $M_p$ denotes the Planck scale. This makes it difficult to construct a specific model for the early evolution of the Universe. For example, if one puts the inflaton fields on the brane, their masses are required to be unnaturally small[@fine-tune]. On the other hand, in generic cases the mass of the inflaton is bounded from below to achieve successful reheating. Thus it seems quite difficult to construct a model for successful inflation driven by a field on the brane. A way to avoid these difficulties is put forward by Arkani-Hamed et al.[@Arkani-inflation], where inflation is assumed to occur [**before**]{} the stabilization of the internal dimensions. In this case, however, the late oscillation of the radion is a problem, which will be solved by the second weak inflation. One can find other ways to solve problems of inflation in models with large extra dimensions. Due to some dynamical mechanisms, the extra dimension may be stabilized before the Universe exited from inflation. If the stabilization of the internal dimensions occurred before the end of inflation, inflation cannot be induced by the fields on the brane, since their energy densities are highly suppressed. In this case one may use the bulk field rather than a field on the brane[@Mohapatra-1; @Mazumdar-Bulk-Inflation]. One may find another interesting possibility, “brane inflation”[@brane-inflation], which uses the interbrane distance as the inflaton. In this paper we focus our attention to the second possibility, where the bulk field drives hybrid inflation after the radion stabilization. We first make a brief review of the hybrid bulk field inflation, then we will examine the effect of the Kaluza-Klein modes.[^2] In ref.[@Mazumdar-Bulk-Inflation], it is claimed that the phase transition becomes so slow that it is impossible to produce the required density perturbations after inflation with normal parameter values. However, we find that the phase transition is fast because of the huge number of the destabilized Kaluza-Klein modes at the end of inflation. The number of the destabilized Kaluza-Klein modes becomes up to about $O(M_p^2/M_*^2)$. Although the problem of the slow phase transition seems to be solved by the Kaluza-Klein modes, another problem is induced by the excited Kaluza-Klein modes. The overproduction of the excited Kaluza-Klein modes is the problem, because they efficiently emit Kaluza-Klein gravitons when they decay into lower excited modes[@Mohapatra-1]. Hybrid inflation due to bulk scalar fields ========================================== In ref.[@Mohapatra-1], it is argued that a single field inflationary model cannot provide adequate density perturbations, either with the inflaton on the brane or with the bulk field inflation. Thus it is natural to invoke a hybrid inflationary model in higher dimensions with the potential for the bulk field, $$V(\phi_5,\sigma_5)_{5D}=\lambda^2 M_* \left(\sigma_{5o}^2-\frac{1}{M_*}\sigma_5^2\right)^2 + \frac{m_\phi^2}{2} \phi_5^2 + \frac{g^2}{M_*} \sigma_5^2 \phi_5^2,$$ where $\phi_5$ is the inflaton field with a chaotic initial condition. The coupling constants $\lambda$ and $g$ are assumed to be $O(1)$. To keep things as simple as possible, here we have assumed five dimensional theory with only one extra dimension. Extensions to higher dimensional theories are straightforward. As the higher dimensional fields $\phi_5$ and $\sigma_5$ has a mass of dimension $3/2$, the interaction terms are nonrenormalizable. Higher dimensional fields $\phi_5,\sigma_5$ are related to the effective four dimensional fields $\phi, \sigma$ by a scaling $$\phi=\sqrt{R}\phi_5, \, \sigma=\sqrt{R}\sigma_5,$$ which leads to couplings suppressed by the four dimensional Planck mass $M_p$, where $M_p^2=M_*^{3}R$. Here $R$ denotes the volume of the extra dimensions. After dimensional reduction, the effective four dimensional potential for the 0-modes reads; $$\begin{aligned} \label{potential_0} V(\phi,\sigma)_{4D} &\sim& \lambda^2\frac{M_*^2}{M_p^2} \left(\frac{M_p^2 \sigma_0^2}{M_*^2} -\sigma^2\right)^2\nonumber\\ &&+\frac{m_{\phi^2}}{2}\phi^2 +g^2\left(\frac{M_*}{M_p}\right)^2 \phi^2 \sigma^2.\end{aligned}$$ Including the Kaluza-Klein modes, the potential becomes[^3] $$\begin{aligned} \label{potential_1} V(\phi_n, \sigma_n)_{KK} &\sim& \lambda^2\frac{M_*^2}{M_p^2} \left(\frac{M_p^2 \sigma_0^2}{M_*^2} -(\sigma^2 + \sum_n \sigma_n^2)\right)^2\nonumber\\ &&+\frac{m_{\phi^2}}{2}(\phi^2 +\sum_n\phi_n^2) +g^2\left(\frac{M_*}{M_p}\right)^2 (\phi^2 + \sum_n \phi^2_n) (\sigma^2 + \sum_n \sigma_n^2)\nonumber\\ &&+ \sum_n \frac{n^2}{R^2} \phi_n^2 + \sum_n \frac{n^2}{R^2} \sigma_n.\end{aligned}$$ Let us first ignore the Kaluza-Klein modes in the effective potential and discuss the hybrid bulk field inflation induced by the potential eq.(\[potential\_0\]). In ref.[@Mohapatra-1], the parameter values $\sigma_o \sim M_* = 10^5$GeV, $m_\phi\sim 10$GeV were considered. With these values, the density perturbation $$\frac{\delta \rho}{\rho} \sim \left(\frac{g}{2\lambda^{3/2}}\right) \times 10^{-5}$$ were obtained. Then the reheating temperature is calculated within the perturbation theory. In ref.[@Mohapatra-1], the decay rates of the inflaton field are already discussed. For the decay of the inflaton field into two Higgs fields on the brane, the decay width becomes $$\Gamma_{HH} \sim \frac{M_*^4}{32\pi M_p^2 m_\phi}.$$ On the other hand, their decay into excited Kaluza-Klein modes are highly suppressed, $$\Gamma_{\phi\phi\rightarrow \phi_n \phi_n} \sim \lambda^2 \frac{M_*^2}{M_p^2}.$$ Thus it seems appropriate to calculate the reheating temperature only by the decay into Higgs fields on the brane. Then the reheating temperature becomes $\sim 100$ MeV, which satisfies the requirement for the successful big bang nucleosynthesis, and at the same time solves the problem of overproduction of Kaluza-Klein gravitons. In this model, however, a serious problem was reported in ref.[@Mazumdar-Bulk-Inflation]. The authors of ref.[@Mazumdar-Bulk-Inflation] claimed that the phase transition becomes extremely slow and it is impossible to reproduce the present Universe with the above parameter values. Let us first make a brief review of the argument. The slope of the potential (\[potential\_0\]) in the $\phi$ direction is given by $$\label{slope1} \frac{dV}{d\phi}=\left[2g^2 \left(\frac{M_*}{M_p}\right)^2 \sigma^2 + m_{\phi}^2\right]\phi,$$ and the inflaton $\phi$ rolls down the potential with the Hubble parameter $$H=\sqrt{\frac{8\pi}{3}}\frac{\lambda \sigma_0^2}{M_*}.$$ The inflaton $\phi$ slowly rolls down the potential until it reaches the critical value $$\phi_c = \frac{\lambda}{g}\left(\frac{\sigma_0M_p}{M_*}\right),$$ where the potential in the $\sigma$ direction is destabilized. The evolution of the inflaton $\phi$ is given by $$\phi = \phi_c exp\left[-\frac{1}{\sqrt{24\pi}\lambda} \left(\frac{m_\phi}{\sigma_0}\right)^2 M_* t \right]$$ where we take the time variable $t$ as $t=0$ at $\phi=\phi_c$. In the $\sigma$ direction, the slope of the potential (\[potential\_0\]) is $$\label{slope2} \frac{dV}{d\sigma} = \left(\frac{M_*}{M_p}\right)^2 \left[\lambda^2 \sigma^2 + 2g^2 (\phi^2-\phi^2_c) \right] \sigma.$$ At $t=0$, the initial value of $\sigma$ is expected to be about $\sigma_{ini} \sim H \sim M_*$. The phase transition lasts until the first term in eq.(\[slope1\]) comes to dominate the evolution. It happens when $$\label{condition2} 2g^2\left(\frac{M_*}{M_p}\right)^2 \sigma^2\sim m_\phi^2,$$ where $\sigma$ becomes about $\sigma_{end} \sim 10^{14}$GeV. Solving the equation for $\sigma$, one obtains large e-foldings $N_e \sim 10^5$. During this period, $\sigma$ evolves from $\sigma_{ini} \sim 10^5$GeV to $\sigma_{end}\sim 10^{14}$GeV. However, the Kaluza-Klein modes are important in this case. Seeing the effective potential eq.(\[potential\_1\]), one can easily find that not only the 0-mode $\sigma$ but also the Kaluza-Klein modes $\sigma_n$ are destabilized to contribute the phase transition. The number of the destabilized modes at the end of inflation is estimated by the potential (\[potential\_1\]), $$N_{KK}\sim (R\, \lambda \sigma_0) \sim \left(\frac{M_p}{M_*}\right)^2.$$ In the presence of a large number of destabilized channels near $\sigma \simeq \sigma_{n} \simeq 0$, the phase transition becomes inevitably fast. In the original calculation of ref.[@Mazumdar-Bulk-Inflation], the initial value of $\sigma$ is $\sigma_{ini}\sim H \sim M_*\sim 10^5$GeV, which is much smaller than the critical value $\sigma_{end}$ in eq.(\[condition2\]). On the other hand, in the case when $\sigma^2$ in eq.(\[slope1\]) is replaced by the sum of the huge number of the Kaluza-Klein modes, the condition (\[condition2\]) is already satisfied as soon as the phase transition starts at $\phi\sim \phi_c$. In this case, each Kaluza-Klein mode $\sigma_n$ may have the initial value of $\sigma_n \sim H$, if the mass is smaller than the Hubble parameter. Although the problem of the slow phase transition is avoided by the huge numbers of the Kaluza-Klein modes, there arises another serious problem. As is discussed in ref.[@Mohapatra-1], the reheating after the bulk inflation is not a problem when only the 0-mode oscillates and decays after inflation. In this model, however, the decay products of the excited Kaluza-Klein modes will dominate the Universe after inflation, because the inflation ends with the oscillation and the production of the excited Kaluza-Klein modes. Then the Kaluza-Klein gravitons are produced when the excited Kaluza-Klein modes decay into lower modes. The decay rate is enhanced by the large number of accessible modes in the final state, thus the excited modes are very short lived[@Mohapatra-1]. Let us examine the condition for the Kaluza-Klein gravitons to decay safe before nucleosynthesis. The decay width of the Kaluza-Klein gravitons into fields on the brane is estimated in ref.[@kk-graviton-decay], $$\Gamma \sim \frac{E^3}{M_p^2}.$$ Here $E$ denotes the energy of the graviton propagating [**in the bulk**]{}. In the most optimistic case, when $E\sim M_*$, the Kaluza-Klein gravitons may decay before nucleosynthesis if the fundamental scale is larger than $10^6$GeV. Conclusions and Discussions =========================== In this paper, we have discussed the possibility of successful hybrid inflation due to the bulk scalar field, in models with large extra dimensions. Bulk field inflation is already discussed in papers[@Mohapatra-1; @Mazumdar-Bulk-Inflation], where the problem of Kaluza-Klein gravitons[@Mohapatra-1] and the problem of the slow phase transition[@Mazumdar-Bulk-Inflation] are discussed. In ref.[@Mohapatra-1], it is argued that the production rate of the Kaluza-Klein graviton is so small that the energy of the inflaton is safely drained into the standard model fields on the brane. However, including the Kaluza-Klein interactions, the production of the Kaluza-Klein gravitons becomes efficient. The problem of the slow phase transition is discussed in ref.[@Mazumdar-Bulk-Inflation]. However, because of the huge number of excited Kaluza-Klein states that becomes unstable at the end of inflation, the phase transition becomes fast. The remaining problem is the overproduction of the Kaluza-Klein gravitons, which puts a lower bound to the fundamental scale. Even in the most optimistic case, the bound becomes $M_* > 10^6$GeV. The significant effect of the Kaluza-Klein excited states, which we have discussed for the bulk inflation, is generically important in cosmological models that utilizes the phase transition of the bulk field. Acknowledgment ============== We wish to thank K.Shima for encouragement, and our colleagues in Tokyo University for their kind hospitality. [1]{} I.Antoniadis, N.A-Hamed, S.Dimopoulos, and G.R.Dvali, Phys.Lett.B436:257-263,1998; I.Antoniadis, Phys.Lett.B246:377-384,1990; N.A-Hamed, S.Dimopoulos and G.R.Dvali, Phys.Lett.B429:263-272,1998. G.R. Dvali, G. Gabadadze, Phys.Lett.B460:47-57,1999; T.Matsuda, Phys.Rev.D65:103501,2002, Phys.Rev.D65:103502,2002, Phys.Rev.D66:023508,2002, Phys.Rev.D65:107302,2002, hep-ph/0205331; A. Masiero, M. Peloso, L. Sorbo, and R. Tabbash, Phys.Rev.D62:063515,2000; A.Mazumdar, Nucl.Phys.B597(2001)561, Phys.Rev.D64(2001)027304; A. Mazumdar and A. Perez-Lorenzana, Phys.Rev.D65:107301,2002; R. Allahverdi, K. Enqvist, A. Mazumdar, and A. Perez-Lorenzana, Nucl.Phys.B618:277-300,2001; A.Pilaftsis, Phys.Rev.D60:105023,1999; R.Allahverdi, K.Enqvist, A.Mazumdar and A.P-Lorenzana, Nucl.Phys. B618:377,2001; S. Davidson, M. Losada, and A. Riotto, Phys.Rev.Lett.84:4284-4287,2000. N. Kaloper and A. D. Linde, Phys.Rev.D59:101303,1999; D. H. Lyth, Phys.Lett.B466:85-94,1999. N. Arkani-Hamed, S. Dimopoulos, N. Kaloper, and J. March-Russell, Nucl.Phys.B567:189-228,2000 R. N. Mohapatra, A. Perez-Lorenzana, and C. A. de S. Pires,\ Phys.Rev.D62:105030,2000 A. M. Green and A. Mazumdar, Phys.Rev.D65:105022,2002 P. Kanti, K. A. Olive, Phys.Rev.D60:043502,1999; Phys.Lett.B464:192-198,1999. Yun-Song Piao, Wen-bin Lin, Xin-min Zhang, and Yuan-Zhong Zhang, Phys.Lett.B528:188-192,2002. G. R. Dvali and S.H.Henry Tye, Phys.Lett.B450:72-82,1999; G. R. Dvali, Phys.Lett.B459:489-496,1999; N. Arkani-Hamed, S. Dimopoulos, and G.R. Dvali, Phys.Rev.D59:086004,1999. [^1]: matsuda@sit.ac.jp [^2]: In ref.[@kanti], interesting aspects of bulk inflation after the stabilization of the radion field is studied by using the KK modes in the 4D effective theory. See also ref.[@piao]. [^3]: We have assumed that the nonrenormalizable terms in the five dimensional potential (\[potential\_0\]) are obtained by integrating out massive modes. These massive modes, which mediate the interactions $\sigma^4_5$ or $\phi^2_5\sigma^2_5$, are assumed to have no Kaluza-Klein number.
--- abstract: 'Magnons in antiferromagnetic insulators couple strongly to conduction electrons in adjacent metals. We show that this interfacial tie can lead to superconductivity in a tri-layer consisting of a metal sandwiched between two antiferromagnetic insulators. The critical temperature is closely related to the magnon gap, which can be in the THz range. We estimate the critical temperature in Mn$\text{F}_2$-Au-Mn$\text{F}_2$ to be on the order of $1$K. The Umklapp scattering at metal-antiferromagnet interfaces leads to a d-wave superconductive pairing, in contrast to the p-wave superconductivity mediated by magnons in ferromagnets.' author: - Eirik Løhaugen Fjærbu - Niklas Rohling - Arne Brataas title: 'Superconductivity at metal-antiferromagnetic insulator interfaces' --- [^1] Introduction ============ Antiferromagnetic insulators (AFIs) offer several advantages over ferromagnets such as higher operating frequencies and the absence of stray magnetic fields [@NatPhys.14.200; @NatPhys.14.213]. Spin waves and their quanta, magnons, in AFIs couple strongly to electrons in adjacent normal metals (NMs) [@PhysRevLett.113.057601; @PhysRevB.90.094408; @PhysRevB.95.144408]. Importantly, this enables electric control of the antiferromagnetic spin dynamics. Even so, AFIs have received less attention than ferromagnetic insulators (FIs) in spintronics. A standard model for the interfacial tie is an exchange coupling between the itinerant electrons and the localized spins [@cheng2014aspects; @PhysRevB.90.094408; @PhysRevB.95.144408]. In this formalism, the electrons experience a staggered field and scatter through two different scattering channels: a regular channel and an Umklapp channel [@PhysRevB.90.094408; @PhysRevB.95.144408]. In this paper, we show that the electron-magnon coupling at the NM-AFI interfaces can lead to superconductivity. The magnons in the AFIs mediate the superconductive pairing of the itinerant electrons in the NM. The strong coupling between magnons and electrons enhances the superconductive pairing. The dispersions of the conduction electrons and the magnons influence the pairing significantly. Choosing different combinations of materials and tuning the interface quality controls the superconductive gap. Extensive studies on the interplay between antiferromagnetic ordering and superconductivity have been conducted. Experiments have shown that the two phenomena can coexist in several different materials [@PhysRevLett.41.1133; @PhysRevB.61.R14964] and even within the same electron bands [@PhysRevLett.60.615; @PhysRevLett.75.1178; @PhysRevB.56.11749]. Because many high-$T_C$ superconductors are created from antiferromagnetic insulators by doping [@RevModPhys.78.17], their discovery led to a renewed interest in the relation between superconductivity and antiferromagnetism. Even more recently, superconductivity has been found to coexist with antiferromagnetism in iron pnictide superconductors [@PhysRevB.78.214515; @PhysRevB.79.014506; @NewJPhys2009Rotter; @PhysRevLett.101.087001]. Theory predicts that magnons can mediate superconductivity in bulk antiferromagnets, with either p-wave or d-wave pairing symmetry [@PhysRevB.96.214409; @JPSJ.63.1861]. There are also suggestions that magnons mediate superconductive pairing in iron pnictides [@KAR201818; @WuJPC2011]. At topological insulator (TI)/FI interfaces, ferromagnetic magnons are predicted to mediate p-wave pairing of spin-momentum locked electrons, where the involved electrons can have equal momenta [@KagarianPRL2016]. For Bi/Ni bilayers, Ref.  developed a similar model, but with a d-wave pairing, to explain their experimental findings of superconductivity. At TI/AFI interfaces, there are predictions that magnons mediate the pairing of spin-momentum locked electrons with either equal or antiparallel momenta [@hugdal2018magnon]. We consider pairing between spin-degenerate electrons in a metal. In Ref. , we showed that magnons in FIs can mediate the p-wave pairing of electrons with opposite momenta in FI/NM/FI tri-layers. In this paper, we replace the ferromagnetic insulators with antiferromagnetic insulators and consider AFI/NM/AFI tri-layers. Magnons in ferromagnets and antiferromagnets significantly differ, resulting in distinctive magnon-induced pairings. For the AFI/NM/AFI system, we find d-wave pairing. Our paper is organized as follows. In Sec. \[sec:model\], we introduce the model describing the metallic layer, the antiferromagnetic layers, and the interaction between the layers. Sec. \[sec:gap\_equation\] presents the resulting magnon-mediated electron-electron interaction, the gap equation, and its solution. We conclude the paper in Sec. \[sec:conclusions\]. Appendix \[app:mat\] provides estimates for material parameters, and Appendix \[app:non-zero-momentum\] considers an alternative superconducting pairing with a non-zero sum of the electron momenta and p-wave symmetry. We will see that this pairing is suppressed compared to the d-wave pairing. Model {#sec:model} ===== Our model consists of three monolayers: a NM sandwiched between two identical easy-axis AFIs, as shown in Fig. \[fig:system\]. We denote the left (right) AFI by $\Gamma = {L}$ (${R}$) and the central NM by $\Gamma = {C}$. We assume that all three layers have identical square lattices with lattice constant $d$, where node $i$ has the same in-plane position vector ${\mathbf{r}}_{i}$ in all layers ${R}$, ${C}$, and ${L}$. We define the unit vectors $\hat{y}$ and $\hat{z}$ along the lattice vectors, and $\hat{x}$ is transverse to the monolayers. We characterize the spin directions with the coordinates $\chi$, $\upsilon$, and $\zeta$, where $\hat{\zeta}$ is parallel to the easy axis of the AFI. There are $N_y$ lattice nodes in the $y$ direction and $N_z$ lattice nodes in the $z$ direction. The total number of sites in the metal layer is $N = N_y N_z$. We use periodic boundary conditions along the $y$- and $z$-directions. ![(Color online) Tri-layer system: normal metal sandwiched between two antiferromagnetic insulators. (a) Electrons in the NM scatter at the interfaces, creating or annihilating a magnon. This leads to an effective electron-electron interaction. The spin of the electron is flipped in each scattering event. (b) Three-monolayer lattice structure and coordinate axes $x$, $y$, and $z$. []{data-label="fig:system"}](Figure1){width="\columnwidth"} We describe both AFIs using Heisenberg Hamiltonians with nearest-neighbor exchange interaction $J$ and easy-axis anisotropy $K_{\zeta}$, $$H_{{\rm{AFI}}}^{\Gamma} = \frac{J}{\hbar^2} \sum_{ {\left< i , j \right>}}{\mathbf{S}}_{i}^{\Gamma} \cdot {\mathbf{S}}_{j}^{\Gamma} + \frac{K_{\zeta}}{\hbar^2} \sum_{i} \left( S_{i \zeta}^{\Gamma} \right)^2 \, .$$ Here, $\hbar$ is the reduced Planck constant, ${\mathbf{S}}_{i}^{{L}}$ (${\mathbf{S}}_{i}^{{R}}$) is the spin at node $i$ in the left (right) AFI, and ${\left< i , j \right>}$ is a pair of nearest-neighbor nodes. Each AFI is divided into two sublattices: $A$ and $B$. When the AFI is in its classical ground state, all the spins on sublattice $A$ ($B$) point along $\hat{\zeta}$ ($-\hat{\zeta}$). We assume that the matching nodes in the left and right AFIs are in opposite sublattices so that ${\mathbf{S}}_{i}^{{L}} = - {\mathbf{S}}_{i}^{{R}}$ in the classical ground state; see Fig. \[fig:system\] (b). For the electronic states, we consider two different models. The plane-wave states $c_{{\mathbf{q}},\sigma} = \sum_{j} \exp( i {\mathbf{r}}_{j} \cdot {\mathbf{q}} ) c_{j \sigma} /\sqrt{N} $ are eigenstates of both models, but the energy dispersions differ. In the first case, the energy dispersion follows from the tight-binding model (${\rm{TB}}$). In the second case, we assume that the electron dispersion is quadratic (${\rm{Q}}$). The Hamiltonian of the tight-binding model is $$H_{{\rm{TB}}} = - t \sum_{\sigma} \sum_{ {\left< i , j \right>} } c^{\dag}_{i \sigma} c_{j \sigma} \, ,$$ where $c_{j \sigma}$ ($c^{\dag}_{j \sigma}$) annihilates (creates) a conduction electron with spin $\sigma$ along $\hat{\zeta}$ at node $j$. The plane-wave states are eigenstates of this Hamiltonian with the dispersion $E_{{\mathbf{q}}}^{{\rm{TB}}} = 2 t \left[ 2 - \cos(q_y d) - \cos(q_z d) \right]$. For the quadratic model (${\rm{Q}}$), we assume that the dispersion is $E^{{\rm{Q}}}_{{\mathbf{q}}} = \hbar^2 {\mathbf{q}}^2 / ( 2m )$. Here, $m$ is the effective electron mass. We assume half-filling in both models. The electron dispersion relations are illustrated in Fig. \[fig:dispersion\] (a). ![(Color online) Dispersion relations along $q_z = 0$ for (a) the conduction electrons and (b) the magnons, assuming $J/K_{\zeta} = 10$ in the antiferromagnets. The quadratic electron dispersion $E^{{\rm{Q}}}_{{\mathbf{q}}}$ is the red dashed line, and the tight-binding dispersion $E^{{\rm{TB}}}_{{\mathbf{q}}}$ is the blue solid line. []{data-label="fig:dispersion"}](Dispersions.pdf){width="\columnwidth"} The spins in the AFIs couple to the conduction electrons via an interfacial exchange coupling $J_I$, $$H_{{\rm{Int}}} = - \frac{J_I}{\hbar} \sum_{\sigma \sigma'} \sum_{j} \sum_{\Gamma={L},{R}} c^{\dag}_{j \sigma} {\boldsymbol{\sigma}}_{\sigma \sigma'} c_{j \sigma'} \cdot {\mathbf{S}}_j^{\Gamma} \, .$$ Here, ${\boldsymbol{\sigma}} = \hat{\chi}\sigma_x + \hat{\upsilon}\sigma_y + \hat{\zeta}\sigma_z$, and $\sigma_x$, $\sigma_y$, and $\sigma_z$ are the Pauli matrices. We perform a Holstein-Primakoff transformation, treating the sublattices $A$ and $B$ separately, and we define $S^{\Gamma}_{i \pm} = S^{\Gamma}_{i \chi} \pm i S^{\Gamma}_{i \upsilon}$. Assuming that the AFIs are close to their classical ground states, we find, for sublattice $A$, $S^{\Gamma}_{i +} = S^{\Gamma \dag}_{i -} = \hbar \sqrt{2s} a^{\Gamma}_{i} $ and $S^{\Gamma}_{i \zeta} = \hbar ( s - a^{\Gamma \dag}_{i} a^{\Gamma}_{i} )$, and, for $B$, $S^{\Gamma}_{i +} = S^{\Gamma \dag}_{i -} = \hbar \sqrt{2s} b^{\Gamma \dag}_{i}$ and $S^{\Gamma}_{i \zeta} = \hbar ( b^{\Gamma \dag}_{i} b^{\Gamma}_{i} - s )$. Using Fourier- and Bogoliubov transformations, we obtain the magnon eigenstates $$a^{\Gamma}_{{\mathbf{k}}} = \sqrt{\frac{2}{N}} \left( \sum_{i \in A} u_{{\mathbf{k}}} e^{-i {\mathbf{k}} {\mathbf{r}}_{i}} a_{i}^{\Gamma} - \sum_{i \in B} v_{{\mathbf{k}}} e^{i {\mathbf{k}} {\mathbf{r}}_{i }} b^{\Gamma \dag}_{i} \right) \, .$$ The expression for $b^{\Gamma}_{{\mathbf{k}}}$ is found by exchanging $a$ and $b$. The Bogoliubov constants $u_{{\mathbf{k}}}$ and $v_{{\mathbf{k}}}$ satisfy $u_{{\mathbf{k}}}^2 - v_{{\mathbf{k}}}^2 = 1$. We assume the anisotropy $K_{\zeta}$ is substantially smaller than the exchange $J$ so that [@PhysRevB.95.144408] $u_{{\mathbf{k}}} \approx - v_{{\mathbf{k}}} \approx \sqrt{ \varepsilon_J / \varepsilon_{{\mathbf{k}}} }/ \sqrt[4]{2} \gg 1$. Because the dominant contribution to the superconducting gap is expected to come from the long-wavelength magnons [@PhysRevB.97.115401], we will use this so-called exchange approximation throughout. In the long-wavelength limit, the magnon dispersion is $\varepsilon_{{\mathbf{k}}} = 2 s \sqrt{ 2 J \left( 2 K_{\zeta} + J {\mathbf{k}}^2 d^2 \right) }$. In terms of the magnon gap $\varepsilon_0 = 4 s \sqrt{ J K_{\zeta} }$ and the exchange energy scale $\varepsilon_J = 2 \sqrt{2} J s$, the dispersion is $\varepsilon_{{\mathbf{k}}} = \sqrt{\varepsilon_0^2 + \varepsilon_J^2 {\mathbf{k}}^2 d^2}$. The momenta (${\mathbf{q}}$) of the conduction electrons reside in the Brillouin zone, ${\mathcal{BZ}}$, of the lattice of the NM. By contrast, the magnon momenta (${\mathbf{k}}$) are defined in the reduced Brillouin zone of the sublattices, ${\mathcal{BZR}}$; see Fig. \[fig:Brillouin\] (a). At half-filling, the ${\mathcal{BZR}}$ matches the interior of the Fermi surface of the tight-binding model. ![(Color online) Fermi surfaces for (a) the tight-binding model (blue) and (b) the quadratic model (red). The Brillouin zone of the conduction electrons (${\mathcal{BZ}}$) is shown in yellow. The reduced (magnon) Brillouin zone (${\mathcal{BZR}}$) corresponds to the interior of the Fermi surface of the tight-binding model (light blue). The Umklapp momentum ${\mathbf{q}}^{{U}}$ is related to ${\mathbf{q}}$ by a reflection across the diagonal of the ${\mathcal{BZ}}$ (dashed line) and a subsequent reflection in the Fermi surface. []{data-label="fig:Brillouin"}](Figure3){width="\columnwidth"} We disregard terms of second order in the magnon operators from $H_{{\rm{Int}}}$. Then, the total Hamiltonian $H = H_{{\rm{AFI}}}^{{L}} + H_{{\rm{AFI}}}^{{R}} + H_{{\rm{NM}}} + H_{{\rm{Int}}}$ is given by [@PhysRevB.95.144408] $$\begin{aligned} H &= \sum_{\Gamma}\sum_{{\mathbf{k}}} \varepsilon_{{\mathbf{k}}} \left( a_{{\mathbf{k}}}^{\Gamma \dag} a_{{\mathbf{k}}}^{\Gamma} + b_{{\mathbf{k}}}^{\Gamma \dag} b_{{\mathbf{k}}}^{\Gamma} \right) + \sum_{{\mathbf{q}} \sigma}E_{{\mathbf{q}}}c_{{\mathbf{q}},\sigma}^\dag c_{{\mathbf{q}},\sigma} \\ &+ \sum_{\Gamma} \sum_{{\mathbf{k}} {\mathbf{q}}} {\bar{V}}_{{\mathbf{k}}} \left( a_{{\mathbf{k}}}^{\Gamma} c^{\dag}_{{\mathbf{q}}^{{U}},\downarrow} c_{{\mathbf{q}}-{\mathbf{k}}, \uparrow} + b_{{\mathbf{k}}}^{\Gamma}c^{\dag}_{{\mathbf{q}}^{{U}},\uparrow}c_{{\mathbf{q}}-{\mathbf{k}}, \downarrow} + \text{h.c.} \right) \notag ,\end{aligned}$$ where $$\label{eq:Umklapp} {\mathbf{q}}^{U}= {\mathbf{q}} + {\mathbf{q}}_{{\rm{AF}}}\text{ with }{\mathbf{q}}_{{\rm{AF}}} = \left(\hat{y} + \hat{z} \right) \pi/d$$ is the Umklapp momentum of ${\mathbf{q}}$ and ${\bar{V}}_{{\mathbf{k}}} = - \sqrt{s/2N} J_I u_{{\mathbf{k}}}$. Importantly, the interfacial coupling ${\bar{V}}_{{\mathbf{k}}}$ is enhanced by the Bogoliubov constants relative to the magnon-electron coupling in ferromagnets [@PhysRevB.97.115401]. To leading order in the exchange approximation, the conduction electrons only interact with magnons through Umklapp scattering. In contrast to NM-AFI bilayers, the contribution from the normal channel is negligible because the static spin-dependent potentials from the two AFIs compensate each other almost completely. In the electronic tight-binding model at half-filling, the Umklapp process ${\mathbf{q}} \to {\mathbf{q}}^{{U}}$ can be split into two steps; see Fig. \[fig:Brillouin\] (a). First, there is a reflection across one of the diagonals of the full Brillouin zone (${\mathcal{BZ}}$). Second, there is a reflection across the Fermi surface. The second reflection occurs at the surface parallel to the diagonal of the first reflection. For initial states on the Fermi surface, an Umklapp process takes a state ${\mathbf{k}}$ to another state ${\mathbf{k}}^{U}$ that is also on the Fermi surface. Next, we consider the approximate model with quadratic electron dispersion $E^{{\rm{Q}}}_{{\mathbf{q}}}$. To retain the main physics of the tight-binding model, we introduce a modified Umklapp momentum ${\mathbf{q}}^{{\rm MU}}$ that contains two analogous consecutive reflections. The first reflection is across one of the diagonals of the ${\mathcal{BZ}}$. The second reflection is across the circular Fermi surface corresponding to the quadratic electron dispersion; see Fig. \[fig:Brillouin\] (b). The definition of ${\mathbf{q}}^{{\rm MU}}$ depends on the choice of the diagonal where the first reflection occurs. We remove this ambiguity by requesting that $\operatorname{sgn}(q_y^{{\rm MU}})\operatorname{sgn}(q_z^{{\rm MU}}) = \operatorname{sgn}(q_y)\operatorname{sgn}(q_z)$. However, for the symmetries of the superconducting gap that we consider in the following section, all choices for the first reflection lead to the same results. In Sec. \[sec:quadratic\], we will see that the simplifications associated with the rotational symmetry of the quadratic dispersion together with the modified Umklapp process allow for exploration of a large range of parameters as the angular dependence of the gap can be treated analytically. Gap Equation {#sec:gap_equation} ============ Integrating over all the magnons, we find the magnon-mediated electron-electron interaction $$H_{{\rm{eff}}} = \sum_{{\mathbf{q}}{\mathbf{p}}{\mathbf{k}}} \tilde{V}_{{\mathbf{k}}{\mathbf{q}}{\mathbf{p}}} c^\dag_{ {\mathbf{p}}^{U},\downarrow} c^\dag_{{\mathbf{q}}-{\mathbf{k}} \uparrow} c_{{\mathbf{p}}-{\mathbf{k}} \uparrow} c_{ {\mathbf{q}}^{{U}}, \downarrow} \, . \label{eq:Hint}$$ The interaction of Eq. (\[eq:Hint\]) influences all the electrons. We focus on the possible formation of Cooper pairs. We consider the scenario, whereby the essential terms in Eq. (\[eq:Hint\]) satisfy ${\mathbf{p}} = -{\mathbf{q}}^{U}+ {\mathbf{k}}$. Then, the two electrons forming a pair have opposite momenta as in the BCS theory. Another possibility will be discussed in Appendix \[app:non-zero-momentum\]. The effective interaction simplifies to $$H = \sum_{{\mathbf{q}} {\mathbf{p}}} V_{{{\mathbf{q}}}, {\mathbf{p}}} c_{{{\mathbf{q}}} \downarrow}^{\dag} c_{-{\mathbf{q}} \uparrow}^{\dag} c_{-{\mathbf{p}}\uparrow} c_{{{\mathbf{p}}} \downarrow} \, ,$$ where the effective coupling is $$V_{{\mathbf{q}}, {\mathbf{p}}} = \frac{4 J_I^2 J s^2}{N_y N_z} \frac{ \theta_{ {\mathbf{q}}^{{U}} + {\mathbf{p}} } }{\varepsilon_{{\mathbf{q}}^{{U}}{+}{\mathbf{p}}}^2 - (E_{{\mathbf{q}}} - E_{{\mathbf{p}}})^2} \, . \label{eq:gap.effectiveinteraction}$$ Here, we have used the step function $\theta$, where $\theta_{ {\mathbf{q}} } = 1$ when ${\mathbf{q}}$ is inside the ${\mathcal{BZR}}$ and $\theta_{ {\mathbf{q}} } = 0 $ otherwise. We define a spin-singlet gap function $$\Delta_{{\mathbf{q}} } = \sum_{ {\mathbf{p}} } V_{ {\mathbf{q}} {\mathbf{p}} } \langle c_{-{\mathbf{p}}\uparrow} c_{ {\mathbf{p}} \downarrow} - c_{-{\mathbf{p}}\downarrow} c_{{\mathbf{p}} \uparrow} \rangle \, . \label{eq:gapopp}$$ The corresponding gap equations is $$\Delta_{{\mathbf{q}} } = - \sum_{{\mathbf{q}}'} V_{{{\mathbf{q}}}, {\mathbf{q}}'} \frac{\Delta_{{\mathbf{q}}'}}{2\tilde{E}_{{\mathbf{q}}'}} \tanh\left(\frac{\tilde{E}_{{\mathbf{q}}'}}{2k_B T}\right) \, , \label{eq:gap}$$ where $\tilde{E}_{{\mathbf{q}}} = \sqrt{(E_{{\mathbf{q}}}-E_F)^2+\abs{\Delta_{{\mathbf{q}}}}^2}$, $k_B$ is the Boltzmann constant, and $T$ is the conduction-electron temperature. In order to determine the symmetry of the gap function, we consider the case where the dominant part of $V_{{{\mathbf{q}}}, {\mathbf{q}}'}$ in Eq.  comes from the long-wavelength magnons ${\mathbf{q}}^{{U}}{+}{\mathbf{q}}' \approx {\mathbf{0}}$, as in Ref. . Then, we expect that $\Delta_{ -{\mathbf{q}}^{{U}} } \approx - \Delta_{ {\mathbf{q}} }$, where the minus follows from comparing the sign in Eq.  with the BCS theory or Ref. . In the tight-binding model, these relations are satisfied if the gap function $\Delta$ is of d-wave symmetry, i.e., it satisfies $$\Delta_{ (q_y,q_z) } = -\Delta_{ (-q_z,q_y) } = \Delta_{ (-q_y,q_z) } = \Delta_{ (q_y,-q_z) }. \label{eq:gapsymmopp}$$ We assume that the superconducting gap has the same symmetry in the quadratic model. To solve the gap equation, we replace the sum over momenta with integrals over the energy $E = E_{{\mathbf{q}}}$ and the angle $\varphi$, where ${\mathbf{q}} = {q} \left[\sin(\varphi),\cos(\varphi) \right]$. We assume that the dominant contribution to the effective coupling $V_{{{\mathbf{q}}}, {\mathbf{q}}'}$ in Eq.  stems from the regions where ${\mathbf{q}}^{{U}}{+}{\mathbf{q}}'$ lies within the reduced Brillouin zone ${\mathcal{BZR}}$. We therefore set $\theta_{{\mathbf{q}}^{{U}} + {\mathbf{p}}} = 1$ for all ${\mathbf{q}}$ and ${\mathbf{p}}$ in Eq. . We then introduce dimensionless variables in terms of the magnon gap, $\varepsilon_0$, such that $\delta = \Delta/\varepsilon_0$, $\tau = k_B T/\varepsilon_0$, $x = (E-E_F)/\varepsilon_0$, $\tilde{x} = \tilde{E}/\varepsilon_0$, and $\epsilon = \varepsilon/\varepsilon_0$. The gap $\delta=(x,\varphi)$ has to satisfy the self-consistent equation $$\delta(x,\varphi) = -\tilde\alpha\!\! \int\limits_{-x_B}^{x_B}\!\!\!\!dx' \! \! \int\limits_0^{2\pi}\!\!d\varphi'\, \frac{\delta(x',\varphi')v(x,x',\varphi,\varphi')}{\tilde x'} \tanh\left[\frac{\tilde x'}{2\tau}\right] \label{eq:gap_eeq2D}$$ with the dimensionless coupling strength $\tilde\alpha=J_I^2 s \varepsilon_J/ ( 2 \pi \sqrt{2} E_F \varepsilon_0^2 )$, $\tilde x' = \sqrt{(x')^2+|\delta(x',\varphi')|^2}$, and $$v(x,x',\varphi,\varphi') = \frac{1}{1+\varepsilon_J^2 |{\mathbf{k}}^U\!\!+{\mathbf{k}}'|^2d^2/ \varepsilon_0^2 - (x{-}x')^2} \propto V_{{\mathbf{k}},{\mathbf{k}}'} \label{eq:pot_2D}$$ where we approximate ${\mathbf{k}}$ by ${\mathbf{k}} = k_F(\hat{y} \sin\varphi + \hat{z} \cos\varphi)$ and $k_F = \sqrt{2\pi}/d$. The dependence of ${\mathbf{k}}$ on $x$ is disregarded since $x\ll E_F/\varepsilon_0$. This means that the magnon energy depends solely on the angles: $\varepsilon=\varepsilon(\varphi,\varphi')$. We restrict the energy $x'$ to an interval $[-x_{{\text{B}}},x_{{\text{B}}}]$, where $x_{{\text{B}}} > 1$ is chosen such that $\abs{\delta(x',\varphi')} \ll \max_\varphi\abs{\delta(0,\varphi)}$ for all $x'$ outside the interval. The remainder of this section is organized as follows. In Sec. \[sec:quadratic\], we solve the gap equation for a simplified model. In this model, we assume a quadratic electron dispersion together with the modified Umklapp momentum, ${\mathbf{q}}^{{\rm MU}}$, introduced at the end of Sec. \[sec:model\]. We explore the dependence of the superconducting gap on the coupling strength and on the temperature in the exchange limit where the magnon gap is smaller than the exchange energy, $\varepsilon_0/\varepsilon_J\ll1$. The purpose of obtaining these results is to give a basic understanding of the physics. In Sec. \[sec:2D\], we and solve the gap equation numerically for the actual Umklapp relation from Eq. (\[eq:Umklapp\]) and the quadratic dispersion. Sec. \[sec:tb\] discusses differences in the tight-binding model compared to the calculations with the quadratic dispersion. Finally, in Sec. \[sec:analysis\], we will analyze the differences between the simplified model (Sec. \[sec:quadratic\]), the quadratic dispersion model (Sec. \[sec:2D\]), and the tight-binding model (Sec. \[sec:tb\]). Simplified model: quadratic electron dispersion with modified Umklapp relation {#sec:quadratic} ------------------------------------------------------------------------------ Using the modified Umklapp relation is a great simplification because we can use the rotational symmetry. The gap equation has a d-wave solution $\delta(x,\varphi)$ satisfying Eq. . At the critical temperature, $\tau=\tau_c$, where the gap approaches zero, this state takes the form $\delta(x,\varphi) = f(x) \cos(2 \varphi) $, where $f$ satisfies $$f(x) = \alpha \! \! \! \int\limits_{-x_{{\text{B}}}}^{x_{{\text{B}}}} \! \! dx' \frac{ V(x{-}x') f(x')}{ \sqrt{x^{\prime 2}{+}f(x')^2} } \tanh \! \! \left[\frac{ \sqrt{x^{\prime 2}{+}f(x')^2} }{2\tau}\right] \! . \label{eq:gap.quadratic}$$ Here, $\alpha = \tilde\alpha \sqrt{\pi/2} \cdot ( \varepsilon_0 / \varepsilon_J ) $ is the coupling constant, the effective potential is $V(y) \approx -C_{V} + 1/\sqrt{1 - y^2 }$, and the constant $C_{V}=\sqrt{2}\varepsilon_0/(\sqrt{\pi}\varepsilon_J)$, which we will set to zero in the numerical calculations. Note that for $\tau<\tau_c$, $\tilde x'=\sqrt{(x')^2+|\delta(x',\varphi')|^2}$ depends in general on $\varphi'$. Consequently, the integration over the angle $\varphi'$ cannot be separated from the integration over $x'$ as was done for the derivation of Eq. (\[eq:gap.quadratic\]). Thus, for temperatures below the critical temperature, using Eq. (\[eq:gap.quadratic\]) represents a simplifying assumption compared to solving Eq. (\[eq:gap\_eeq2D\]) for $\delta(x',\varphi')$. However, the solution $f(x')$ to Eq. (\[eq:gap.quadratic\]) is approximately equal to the maximum amplitude of the d-wave gap for a given energy, $\max_{\phi'} \{ \delta(x',\varphi') \}$. Also, since Eq. (\[eq:gap.quadratic\]) is valid near the critical temperature, we can use it to calculate the critical temperature itself. The p-wave gap function of Appendix \[app:non-zero-momentum\] satisfies Eq. (\[eq:gap.quadratic\]) at all temperatures, so the results are also valid for this pairing. ![ (Color online) Numerical results for the energy dependence of the gap function $f(x)$ according to Eq.  at zero temperature ($\tau=0$), found by iterations starting with a Gaussian. The small constant $C_{V}$ was approximated as vanishing, $C_{V} = 0$. We consider four different values of the dimensionless coupling constant $\alpha=0.07$ (blue solid line), $\alpha=0.1$ (green dashed line), $\alpha=0.13$ (orange dotted line), and $\alpha=0.17$ (red dash-dot line). []{data-label="fig:quadratic_gap"}](gap_vs_energy){width="\columnwidth"} We solve the 1D gap equation numerically by iteration. $f$ is symmetric about the Fermi surface: $f(-x)=f(x)$. Fig. \[fig:quadratic\_gap\] shows the solutions of Eq.  for different coupling constants $\alpha$ at zero temperature. We find a relatively constant behavior around $x=0$ and, for small $\alpha$, a pronounced peak at $\abs{x}\approx1$. We compare the $\alpha$ dependence of $f_{\rm max} = \max_{x}f(x)$ and $f(0)$ to the standard BCS result $f\sim\exp(-1/\alpha)$; see Fig. \[fig:quadratic\] (a). The BCS result was derived for a potential $V(x,x')$ which is constant $V(x,x')= V_c$ if $|x|,|x'|<1$ and $0$ otherwise, for $V_c = \pi/2=\int_{-1}^1 dy V(y)/2$. The $\alpha$ dependence of the critical temperature $\tau_c$ is comparable to the one of $f(x{=}0)$; see Fig. \[fig:quadratic\] (b). The ratio $f(x{=}0)/\tau_c$ is slightly higher in our model than in standard BCS theory, where the ratio is approximately $1.76$; see Fig. \[fig:quadratic\] (c). Note that the angle dependence is already integrated out in Eq. (\[eq:gap.quadratic\]), and a constant potential would result in the $1.76$ ratio. When we vary the temperature $\tau$, $f_{\rm max}$ and $f(x{=}0)$ both vanish at $\tau_c$, as expected. As we see from Fig. \[fig:quadratic\] (d), $f_{\rm max}$ and $f(x{=}0)$ show similar $\tau$ dependencies. ![(Color online) Numerical results for $\alpha$ and for the temperature dependence of the gap function defined by Eq.  for the potential with a constant $C_{V}$ again set to be zero. (a) Semi-logarithmic plot of $f(x{=}0)$ (black squares) and $f_{\rm max}=\max_x f(x)$ (red circles) for $\tau=0$ in the dependence of $1/\alpha$. The gray dotted line refers to the BCS-like consideration of a constant potential $V(x,x')=V_c$ within $|x|,|x'|<1$, which results in $f(x)=2\exp(-1/(\alpha\pi))$ for $|x|<1$. (b) Semi-logarithmic plot of the dimensionless critical temperature $\tau_c$ as a function of the coupling $1/\alpha$. (c) Ratio of $f(x{=}0)$ at $\tau=0$ to the critical temperature $\tau_c$ as a function of $\alpha$. (d) Temperature dependence of $f(x{=}0)$ (black squares) and $f_{\rm max}$ (red circles) for $\alpha = 0.15$. []{data-label="fig:quadratic"}](alpha_temp_dependence){width="\columnwidth"} In making the model dimensionless, the magnon gap $\varepsilon_0$ is a natural choice of energy scale. In the resulting gap equation, the coupling $\alpha$ is inversely proportional to $\varepsilon_0$. As we observed in Fig. \[fig:quadratic\], $\tau_c$ scales similarly to $\exp(-1/\alpha)$. Therefore, $T_c$ might increase by reducing the magnon gap $\varepsilon_0$. However, if we increase $\alpha$, the system eventually enters a regime where higher order effects will have to be considered. For FI/NM/FI tri-layers, the exchange energy scale $\varepsilon_J$ plays the same role as $\varepsilon_0$ for AFI/NM/AFI tri-layers [@PhysRevB.97.115401]. Because $\varepsilon_J$ is typically larger than $\varepsilon_0$, $T_c$ should in many cases be higher for AFI/NM/AFI tri-layers than for FI/NM/FI tri-layers, assuming that the coupling $J_I$ is the same. However, the strong-coupling regime may set in at lower values of $J_I$ for AFIs than FIs since the coupling constant $\alpha$ is typically larger for AFIs. We estimate $\varepsilon_0$ and $\alpha$ for a Mn$\text{F}_2$-Au-Mn$\text{F}_2$ tri-layer in Appendix \[app:mat\]. We find $\varepsilon_0/k_B = 13$ K and the range of values $[0.02\text{--}0.18]$ for $\alpha$. For the simplified model, the corresponding critical temperatures are up to the order of one Kelvin. We assume that $J_I$ is similar in magnitude for AFI/NM interfaces as for FI/NM interfaces. Similar assumptions have been made in earlier work [@PhysRevLett.113.057601; @PhysRevB.90.094408]. In our model, $J_I$ represents the strength of the interfacial electron-magnon coupling. Spin transport across AFI/NM interfaces has been measured in several experiments [@PhysRevLett.115.266601; @PhysRevLett.116.097204; @JAppPhys.118.233907]. The spin transport between an FI and a NM can be enhanced by inserting an AFI in between, indicating that the coupling at AFI/NM interfaces is as strong as compared to FI/NM interfaces [@PhysRevLett.113.097202]. Quadratic electron dispersion with the actual Umklapp relation {#sec:2D} -------------------------------------------------------------- Now we consider the quadratic dispersion relation together with the actual Umklapp relation and solve Eq. (\[eq:gap\_eeq2D\]) numerically. ![(Color online) Iterative solution of the gap equation Eq. (\[eq:gap\_eeq2D\]) for the gap $\delta$ as a function of dimensionless energy $x$ and angle $\varphi$ at zero temperature and $\alpha=0.15$. The initial guess is $\delta_0(x,\varphi)=f(x) \cos(2\varphi)$, where $f(x)$ is the solution of Eq. (\[eq:gap.quadratic\]) obtained previously. (a) Gap after ten iterations, $\delta_{10}(x,\varphi)$. If the gap $\delta(x,\varphi)$ is known for $\varphi\in[0,\pi/4)$ and $x>0$, its values at all other points in k space follow from symmetry. (b) Gap as a function of $x$ for $\varphi=0$. (c) Gap as a function of $x$ at the angle $\varphi_m=\arcsin(\pi/2{-}1)/2\approx0.3$, where the Fermi surface and the boundary of the ${\mathcal{BZR}}$ intersect. (d) Gap as a function of $\varphi$ for $x=0$. In (b-d), the iterations are $j=0$ (black dashed line), and then, $j=1,\ldots,10$, shown in light blue (light gray) to red (darker gray). []{data-label="fig:2D"}](gap_x_phi.pdf) In Fig. \[fig:2D\], we present iterative results for Eq. (\[eq:gap\_eeq2D\]) for $\alpha=0.15$ at zero temperature. The initial guess for the iterations is the d-wave gap function $\delta_0(x,\varphi) = f(x)\cos(2\varphi)$, where $f(x)$ is the solution of Eq. (\[eq:gap.quadratic\]) with $C_V = 0$. As we see in Fig. \[fig:2D\](b-d), the gap function converges after a few iterations. The resulting function is smaller compared to the initial guess. We find the highest values at the angle $\varphi_m = \arcsin(\pi/2{-}1)/2$, where the Fermi surface intersects with the boundary of the ${\mathcal{BZR}}$. At this point in $k$ space, the modified Umklapp relation $Q^{{\rm MU}}$ used previously is equal to the actual Umklapp relation $Q^{{U}}$. The gap function $\delta(x,\varphi)$ does not have the $\cos(2\varphi)$ dependence on $\varphi$; see Fig. \[fig:2D\](d). The reason is the difference between the actual Umklapp relation and the simplified one used previously. We anticipate a similar behavior at finite temperature, as the simplified Umklapp relation remains only accurate at $\varphi=\varphi_m$. For the critical temperature, we find numerically $\tau_c=0.012$ and a ratio $\delta(x{=}0,\varphi{=}\varphi_m)/\tau_c=2.1$. The ratio is slightly larger than the results in Fig. \[fig:quadratic\_gap\] (c). To summarize the numerical results for the non-simplified model with quadratic dispersion relation: a solution of the gap equation with opposite-momentum pairing of d-wave type exists. Specifications of the gap equations in the tight-binding model {#sec:tb} -------------------------------------------------------------- We noticed that at half filling, the ${\mathcal{BZR}}$ is identical to the Fermi surface of the tight-binding model for the electrons. This means that for the tight-binding model, the Umklapp process relates one point at the Fermi surface to another one at the Fermi surface. This indicates that the pairing mechanism is efficient at the Fermi energy similar to the simplified model considered in Sec. \[sec:quadratic\]. However, there are differences in the tight-binding model compared to the simplified model that can have a significant impact on the superconductivity. In contrast to the circular Fermi surface of the quadratic dispersion, the tight-binding half-filling Fermi surface touches the boundary of the ${\mathcal{BZ}}$, implying additional boundary conditions. A d-wave gap symmetric gap function satisfies the additional boundary conditions in the sense that it is continuous at the edges of the ${\mathcal{BZ}}$. Thus, we conclude that the d-wave gap can be the dominant contribution to superconductivity; compare with Appendix B. We have observed that the d-wave gap is robust in the two models considered. We believe that it will remain robust even when including the full electronic tight-binding dispersion. The increased density of states near the corners of the Fermi surface may enhance the amplitude of the superconducting gap and the critical temperature. However, $\varepsilon_0$ remains the natural choice of energy scale. As we saw in section \[sec:2D\], the scale of the superconducting gap can be up to the order $\varepsilon_0/10$ for the quadratic dispersion. The gap in the tight-binding model is expected to be of the same order or higher. If the gap in the tight-binding model is of the order of $\varepsilon_0$ or larger, higher-order effects may have to be included. These effects, together with the tight-binding dispersion, add considerable complexity to the problem, which is beyond the scope of this initial work. Analysis of the solution of the gap equation {#sec:analysis} -------------------------------------------- As we see from comparing Secs. \[sec:quadratic\] and \[sec:2D\], the pairing symmetry depends on the details of the electron dispersion and its interplay with the Umklapp process, which we will analyze in the following. Umklapp scattering dominates the electron-magnon scattering in the scenario that we consider here. This situation differs when the antiferromagnetic sublattices couple unequally to the metal layer; see Ref. . From Sec. \[sec:2D\], we see that the opposite-momenta d-wave gap has the highest amplitude where the Fermi surface intersects with the ${\mathcal{BZR}}$. We assume that the same is the case for all electron dispersion relations. The energy scale of the superconducting pairing is given by the magnon gap $\varepsilon_0$. This differs from the results obtained for FI/NM/FI systems, where the relevant energy scale is the exchange energy between the spins in the FI layers [@PhysRevB.97.115401]. In the AFI/NM/AFI system, the exchange energy $\varepsilon_J$ drops out of the gap equation completely for the simplest case, as $\alpha$ in Eq. (\[eq:gap.quadratic\]) does not depend on it at all. The reason for this is an interplay of the Bogoliubov coefficients and the angular dependence of the gap equation together with the fact that $\varepsilon_J/\varepsilon_0\ll1$. A further difference between the AFI/NM/AFI system with respect to the FI/NM/FI tri-layer is in the dependence of the size of the superconducting gap and the critical temperature on the dimensionless coupling constant $\alpha$. Note that $\alpha$ is quadratic in the interfacial coupling $J_I$ ($\alpha\sim J_I^2$). For the FI/NM/FI system, we found a dependence close to $f(x)\sim \alpha^2$ [@PhysRevB.97.115401]; we find here a behavior similar to the constant-potential result $f(x)\sim\exp(-1/\alpha)$. The origin of this difference lies in the fact that here the width of the gap $f(x)$ is given approximately by $2\varepsilon_0$, whereas for FI/NM/FI, it was dependent on $\alpha$. Conclusions {#sec:conclusions} =========== In conclusion, we predict that magnons mediate superconductivity in antiferromagnetic insulator-metal-antiferromagnetic insulator tri-layers. The exchange interaction at the antiferromagnet insulator-normal metal interfaces couples the electrons to the magnons. The influence of the interaction is, therefore, most potent when the metal is thin. We find superconducting d-wave pairing of electrons with opposite momenta. The d-wave pairing dominates over p-wave finite momentum pairing, considered in Appendix \[app:non-zero-momentum\]. We find that the critical temperature is closely related to the magnon gap in the antiferromagnets. We estimate the critical temperature for a combination of Mn$\text{F}_2$ and Au to be on the order of Kelvin. We thank Asle Sudbø, Eirik Erlandsen, and Akashdeep Kamra for useful discussions. This work was partially supported by the European Research Council via Advanced Grant No. 669442 “Insulatronics”, the Research Council of Norway through its Centers of Excellence funding scheme, project number 262633, ”QuSpin”, and by the Deutsche Forschungsgemeinschaft (DFG) under project number 417034116. Material Parameters {#app:mat} =================== As a candidate AFI, we consider a (111)-layer of Mn$\text{F}_2$. Mn$\text{F}_2$ is an AFI with a large uniaxial anisotropy. The $s = 5/2$ Mn-ions in the (111)-layer form a square lattice with a lattice constant of $3.82$  [@JAUCH1983907]. Based on measurements of the spin-wave dispersion of Mn$\text{F}_2$, we find $J/k_B = 4.1$ K and $K_{\zeta}/k_B = 0.39$ K [@JAppPhys.35.998]. For the normal metal, we consider a monolayer of gold with the same lattice structure as Mn$\text{F}_2$. We estimate the effective mass using $m=2\pi g_{{\text{Sh}}}\hbar^2/E^{{\text{B}}}_F$, where $g_{{\text{Sh}}} = 12$ $\text{nm}^{-2}$ [@TserkovnyakRMP2005] is the Sharvin conductance and $E^{{\text{B}}}_{F} = 5.5$ eV [@ashcroft1976] is the bulk Fermi energy. We use the quadratic model and the assumption of half-filling to estimate the Fermi energy of the monolayer: $E_{F} = 1.6$ eV. As explained in Sec. \[sec:quadratic\], we assume that the interfacial exchange coupling $J_I$ is similar in magnitude at AFI/NM interfaces compared to FI/NM interfaces. We therefore estimate $J_I$ using experimental values for the FI/SC interfaces, where the superconductor (SC) is either aluminum or vanadium. Estimates for the exchange coupling [^2] within the range $[10\text{--}30]$ meV have been given for several such interfaces [@Tkaczyk_thesis; @Roesler_et_al_1994; @Miao_et_al_NatComm2013]. Using $\alpha = J_I^2 / ( 16 E_F \sqrt{\pi JK} )$, we find a range of values $[0.02\text{--}0.18]$ for $\alpha$. Non-zero-momentum pairing {#app:non-zero-momentum} ========================= In this appendix, we consider an alternative type of superconducting pairing, electron pairs with nonzero total momentum, where the important terms in Eq. (\[eq:Hint\]) are those with ${\mathbf{p}} = - {\mathbf{q}} + {\mathbf{k}}$; then, the Hamiltonian reduces to $$H = \sum_{{\mathbf{q}} {\mathbf{p}}} V_{{{\mathbf{q}}}^{{U}}, {\mathbf{p}}} c_{{{\mathbf{q}}}^{{U}} \downarrow}^{\dag} c_{-{\mathbf{q}} \uparrow}^{\dag} c_{-{\mathbf{p}}\uparrow} c_{{{\mathbf{p}}}^{{U}} \downarrow} \, ,$$ which means that the sum of the momenta of the paired electrons is ${\mathbf{q}}_{{\rm{AF}}}$. Here, the gap has a p-wave symmetry. Electron pairs with a total momentum of ${\mathbf{q}}_{{\rm{AF}}}$ were proposed in Ref.  for bulk s-wave superconductors with antiferromagnetic order. For the non-zero momentum pairing, the gap can be defined in two possible ways: $$\begin{aligned} \Delta_{{\mathbf{q}} } &= \label{eq:gapnz} \\ &\sum_{ {\mathbf{p}} } \! V_{ {\mathbf{q}}^{{U}} {\mathbf{p}} } \! \langle c_{{-}{\mathbf{p}}\uparrow} c_{{\mathbf{p}}^{{U}} \downarrow} {\pm} c_{{-}{\mathbf{p}}\downarrow} c_{{\mathbf{p}}^{{U}} \uparrow} {\pm} c_{{-}{\mathbf{p}}^{{U}}\uparrow} c_{{\mathbf{p}} \downarrow} {+} c_{{-}{\mathbf{p}}^{{U}}\downarrow} c_{{\mathbf{p}} \uparrow} \rangle \notag .\end{aligned}$$ Here the spins are either in the singlet or the antiparallel-spin triplet state. The gap equation reads $$\Delta_{{\mathbf{q}} } = - \sum_{{\mathbf{q}}'} V_{{{\mathbf{q}}}^{{U}}, {\mathbf{q}}'} \frac{\Delta_{{\mathbf{q}}'}}{2\tilde{E}_{{\mathbf{q}}'}} \tanh\left(\frac{\tilde{E}_{{\mathbf{q}}'}}{2k_B T}\right) \, ,$$ $\Delta$ has to satisfy p-wave symmetry, which means $$\Delta_{ (q_y,q_z) } = \Delta^{*}_{ (-q_y,q_z) } = - \Delta^{*}_{ (q_y,-q_z) }. \label{eq:gapsymmnz}$$ This symmetry was used to determine the spin states in Eq. . We now consider the simplified conditions of Sec. \[sec:quadratic\], where ${\mathbf{q}}^{{U}}$ is replaced by ${\mathbf{q}}^{{\rm MU}}$, and the electron dispersion is quadratic. The p-wave solution has the form $$\delta(x,\varphi) = f(x)\exp(\pm i\varphi) \, .$$ Then, we find that $f(x)$ satisfies Eq. (\[eq:gap.quadratic\]) with the potential $V(y) \approx -\varepsilon_0/(\sqrt{2\pi}\varepsilon_J) + 1/\sqrt{1 - y^2 }$. This means that in this simplified model, the d-wave state considered in the main text and the p-wave solution discussed in this appendix lead approximately to the same critical temperature. The p-wave symmetry is energetically slightly preferred, as the constant contribution to the potential $V(y)$ is smaller. However, when considering quadratic electron dispersion together with the actual Umklapp process, the non-zero-momentum pairing is strongly suppressed. This is because the Umklapp process does not map states on the Fermi surface to other states on the Fermi surface. Technically speaking, in Eq. (\[eq:pot\_2D\]), we would need to replace $(x-x')$ in the expression for the potential $v(x,x',\varphi,\varphi')$ by $(x^U-x')$, where $x^U\varepsilon_0$ is the energy at the Umklapp vector ${\mathbf{k}}^U$. As this will be in most cases far away from the Fermi surface, i.e., $x^U-E_F/\varepsilon_0\gg1$, there will be only small contributions to the integral on the right-hand side of the gap equation . Regarding the tight-binding model, the p-wave gap function must be either discontinuous or zero at the corners of the Fermi surface. We believe that this suppresses the p-wave solution. To summarize this appendix, while the simplified model seems to allow for the non-zero-momentum p-wave superconducting pairing, dropping the simplifying assumptions leads to a suppression of this type of pairing both for the quadratic dispersion as well as for the tight-binding model. [38]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](\doibase 10.1038/s41567-018-0063-6) [****,  ()](\doibase 10.1038/s41567-018-0063-6) [****,  ()](\doibase 10.1103/PhysRevLett.113.057601) [****,  ()](\doibase 10.1103/PhysRevB.90.094408) [****,  ()](\doibase 10.1103/PhysRevB.95.144408) **, @noop [Ph.D. thesis]{},  () [****,  ()](\doibase 10.1103/PhysRevLett.41.1133) [****,  ()](\doibase 10.1103/PhysRevB.61.R14964) [****,  ()](\doibase 10.1103/PhysRevLett.60.615) [****,  ()](\doibase 10.1103/PhysRevLett.75.1178) [****,  ()](\doibase 10.1103/PhysRevB.56.11749) [****,  ()](\doibase 10.1103/RevModPhys.78.17) [****,  ()](\doibase 10.1103/PhysRevB.78.214515) [****,  ()](\doibase 10.1103/PhysRevB.79.014506) [****, ()](http://stacks.iop.org/1367-2630/11/i=2/a=025014) [****,  ()](\doibase 10.1103/PhysRevLett.101.087001) [****,  ()](\doibase 10.1103/PhysRevB.96.214409) [****,  ()](\doibase 10.1143/JPSJ.63.1861) @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevLett.117.076806) [**** (), 10.1126/sciadv.1602579](\doibase 10.1126/sciadv.1602579) [****,  ()](\doibase 10.1103/PhysRevB.97.195438) [****,  ()](\doibase 10.1103/PhysRevB.97.115401) [****,  ()](\doibase 10.1103/PhysRevLett.115.266601) [****, ()](\doibase 10.1103/PhysRevLett.116.097204) [****,  ()](\doibase 10.1063/1.4937913) [****,  ()](\doibase 10.1103/PhysRevLett.113.097202) [ ()](https://arxiv.org/abs/1903.01470) [****,  ()](\doibase https://doi.org/10.1016/0038-1098(83)90146-1) [****,  ()](\doibase 10.1063/1.1713575) [****,  ()](https://doi.org/10.1103/RevModPhys.77.1375) [**](http://cds.cern.ch/record/102652) (, ) [Ph.D. thesis](http://hdl.handle.net/1721.1/17231), () [****,  ()](http://dx.doi.org/10.1117/12.179175) [****, ()](http://dx.doi.org/10.1038/ncomms4682) [****,  ()](\doibase 10.1103/PhysRevLett.46.614) [^1]: Currently at the Norwegian Defence Research Establishment (FFI), NO-2027 Kjeller, Norway; eirik-lohaugen.fjarbu@ffi.no [^2]: The exchange coupling as given in Ref. [@Miao_et_al_NatComm2013] corresponds to $J_I/2$.
--- abstract: 'We assess the probability of resonances between sufficiently distant states $\ux=(x_1, \ldots, x_N)$ and $\uy=(y_1, \ldots, y_N)$ in the configuration space of an $N$-particle disordered quantum system. This includes the cases where the transition $\ux \rightsquigarrow \uy$ “shuffles” the particles in $\ux$, like the transition $(a,a,b) \rightsquigarrow (a, b, b)$ in a $3$-particle system. In presence of a random external potential $V(\cdot, \omega)$ (Anderson-type models) such pairs of configurations $(\ux,\uy)$ give rise to local (random) Hamiltonians which are strongly coupled, so that eigenvalue (or eigenfunction) correlator bounds are difficult to obtain (cf. [@AW09a], [@CS09b]). This difficulty, which occurs for $N\ge 3$, results in eigenfunction decay bounds weaker than expected. We show that more efficient bounds, obtained so far only for $2$-particle systems [@CS09b], extend to any $N>2$.' address: | Département de Mathématiques\ Université de Reims, Moulin de la Housse, B.P. 1039\ 51687 Reims Cedex 2, France\ E-mail: victor.tchoulaevski@univ-reims.fr author: - Victor Chulaevsky title: 'A remark on charge transfer processes in multi-particle systems' --- Introduction. The model {#sec:intro} ======================= We study quantum systems in a disordered environment, usually referred to as Anderson-type models, due to the seminal paper by P. Anderson [@An58]. For nearly *fifty years* following its publication, the localization phenomena have been studied in the single-particle approximation, i.e. under the assumption that the interaction between particles subject to the common disordered (usually understood as “random”) external potential is “sufficiently weak” to be neglected in the analysis of the decay properties of eigenstates of the multi-particle system in question. A detailed discussion of recent developments in the physics of disordered media is most certainly beyond the scope of this paper; we simply refer to recent papers by Basko–Aleiner–Altshuler [@BAA05] and by Gornyi–Mirlin–Polyakov [@GMP05] (the order of citation is merely alphabetical) where it was shown, in the framework of physical models and methods, that the localization phenomena, firmly established in the non-interacting multi-particle disordered quantum systems, persist in presence of non-trivial interactions. The rigorous mathematical analysis of localization in strongly disordered quantum systems started approximately “Twenty Years After” the aforementioned Anderson’s paper, when Goldsheid and Molchanov [@GM76] first proved the existence of point spectrum of the Sturm–Liouville operator $$H_V = -\frac{d^2}{dt^2} + V(t, \omega), \; t\in\DR,$$ with random potential of the form $V(t,\omega) = F(X_t(\omega))$ generated by a sample of sufficiently regular Markov process $X_\bullet(\omega)$ and a “non-flat” smooth function $F$ on some auxiliary phase space, and then the complete spectral localization was established for such operators by Goldsheid, Molchanov and Pastur [@GMoP77]. A few years later, Kunz and Souillard [@KS80] proved a similar result (spectral localization) for a lattice (“tight-binding”) Anderson model in one dimension. Techniques of [@GM76; @GMoP77; @KS80] are applicable only to one-dimensional systems, or, with some modifications, to “quasi-one-dimensional” models, like “tubes” $\cD \times \DR \subset \DR^{d+1}$, $\cD \subset \DR^{d}$, $d\ge 1$, extended only in one dimension. For this reason, further progress in higher-dimensional Anderson-type models has been made only several years later, approximately “Thirty Years After” the publication of Anderson’s original paper, in the works by Fröhlich and Spencer [@FS83], Holden and Martinelli [@HM84], Fröhlich, Martinelli, Scoppola and Spencer [@FMSS85]. Later, Spencer [@Spe88] gave a *very short* and simple proof of exponential decay of the Green functions (i.e. the kernel of the resolvent $G(x,y;E) = (H-E)^{-1}(x,y)$). Combined with a result by Simon and Wolff [@SW86], this implied spectral localization. The MSA procedure was reformulated by von Dreifus and Klein [@DK89], giving rise not only to a simplified finite-volume criterion of localization, but also to a versatile approach which has later been adapted to various models, including those in the Euclidean space $\DR^d$, $d\ge 1$, both for standard Schrödinger operators $H_V = -\Delta + V$ and for more general second-order differential operators. A detailed discussion and an extensive bibliography can be found, e.g., in the monograph [@St01]. A survey of a multitude of techniques and results concerning Anderson-type models in the Euclidean space $\DR^d$, $d\ge 1$, is also beyond the scope of the present paper. We do not discuss either an elegant alternative approach to the localization developed by Aizenman and Molchanov [@AM93] (see also [@A94; @ASFH01]), and recently adapted by Aizenman and Warzel [@AW09a; @AW09b] to multi-particle tight binding models. For the sake of clarity of the presentation, we consider the Hamiltonian $\uH_{V,\uU}(\omega)$ in the Hilbert space ${\boldsymbol{\cH}}_N := \ell^2(\DZ^{Nd})$, of the form $$\label{eq:H} \uH_{V,U} = \sum_{j=1}^N \left( \Delta^{(j)} + V(x_j, \omega)\right) + \uU,$$ where $V:\DZ^d\times \Omega \to \DR$ is a random field relative to a probability space $(\Omega, {\mathcal}{F}, {\mathbb}{P})$, $$\Delta^{(j)} \Psi(x_1, \ldots, x_N) = \sum_{y\in\DZ^d:\, |y-x_j|=1} \Psi(x_1, \ldots, x_j + y, \ldots, x_N),$$ and $\uU$ is the multiplication operator by a function $\uU(\ux)$ which we assume bounded (this assumption can be relaxed). The symmetry of the function $\uU$ is not required, and we do not assume $\uU$ to be a “short-range” or rapidly decaying interaction. In fact, we focus here on restrictions of $\uH_{V,\uU}$ to finite subsets of the lattice, so that $\uH_{V,\uU}$ may or may not be well-defined on the entire lattice $\DZ^d$: this does not affect our main result. The assumptions on the random field $V$ are described below, in Section \[sec:EV.sep\]. Eigenvalue concentration bounds ------------------------------- We focus here on probabilistic bounds of certain eigenvalue correlators, or eigenvalue concentration bounds, known in the single-particle localization theory as Wegner-type bounds, due to the paper by Wegner [@We81]. It would not be an exaggeration to say that this bound is the heart of the MSA. (In a slightly disguised form, it also appears in the framework of the FMM both in its the single-particle and multi-particle version, as the reader can observe in [@AM93; @AW09a; @AW09b].) In essence, one may call a Wegner-type bound a (sufficiently explicite and suitable for applications) probabilistic bound of the form $$\pr{ \operatorname{dist}(E, \sigma(\uH_{\Lambda}(\omega))) \le \epsilon} \le f(|\Lambda|, \epsilon), \eqno({\rm W1})$$ where $\uH_{\Lambda}(\omega)$ is the restriction of $\uH(\omega)$ on a bounded subset $\Lambda$ with some self-adjoint boundary conditions, and $\sigma(\uH_{\Lambda}(\omega))$ is its spectrum (a finite number of random points, in the case of lattice models). The role and importance of such bounds can be easily understood: the MSA procedure starts with the analysis of the resolvents $(\uH_{\Lambda} - E)^{-1}$, so it is vital to know how unlikely it is to have the spectrum of $\uH_\Lambda$ $\epsilon$-close to a given value $E\in\DR$ . Unfortunately, the Wegner-type bound alone, in its original form given in (W1), does not suffice in the context of multi-particle (even two-particle) models. It does not matter, actually, how sharp is the “one-volume” bound of the type (W1): it simply does not provide a sufficient input for multi-particle adaptations of the MSA. For this reason, we do not discuss here a series of very interesting papers including [@CHK03], [@CHKR04], [@CHK07]. The knowledge of the so-called limiting integrated density of states (IDS, in short) of a multi-particle system is also insufficient for the multi-particle MSA (MPMSA). Such information is indeed available due to Klopp and Zenk [@KZ95] who proved that the limiting IDS of an $N$-particle system with a decaying interaction between particles is the same as without interaction. In the case where the marginal probability distribution function $F_V$ of an IID random field $V$ is analytic in a strip around the real axis (e.g., Gaussian, Cauchy) a Wegner-type bound was proven in our earlier work [@CS07] where we also proved the analyticity of finite-volume eigenvalue distributions. Kirsch [@K08] proved an analog of the finite-volume bound (W1), with an optimal volume dependence, for multi-particle systems under the assumption of bounded marginal density. Zenk [@Z08] established a Wegner-type bound for multi-particle systems in $\DR^d$, $d\ge 1$, with a realistic long-range particle-particle interaction. However, it should be emphasized again that a one-volume Wegner-type bound (W1), ***even if it were an exact, explicite equality*** and not just un upper bound, seems so far **insufficient** for the MPMSA to work. On the other hand, bounds of the form (W1) **are** necessary for the MPMSA, but for this purpose it suffices to have the volume dependence very far from optimal. For instance, $f$ may have the form $$\label{eq:W2.log} f(|\Lambda|,\epsilon) = C |\Lambda|^{B} \ln^{-A} \epsilon,$$ with any given $B <+\infty$ and sufficiently large $A>0$. A stronger bound with $f(|\Lambda|,\epsilon) = C |\Lambda|^{B} \epsilon^{b}$, $b>0$, which can usually be obtained under assumption of Hölder-continuity of the marginal distribution of an IID random field $V(x,\omega)$, is more than sufficient. The reader can see that it is much weaker than “the” Wegner bound with $f(|\Lambda|,\epsilon) = C |\Lambda|\, \epsilon$. Given any finite cube $\uC_{L}(\uu) := \{\ux\in\DZ^{Nd}\,|\, |\ux - \uu| \le L\}$, we will consider a finite-volume approximation of the Hamiltonian $\uH$ $$\uH_{\uC_{L}(\uu)} = \uH \upharpoonright_{\ell^2(\uC_{L}(\uu))} \text{ with Dirichlet boundary conditions on } \partial \uC_{L}(\uu).$$ acting in the finite-dimensional Hilbert space $\ell^2(\uC_{L}(\uu))$. In [@CS08] the following “two-volume” version of the Wegner bound was established for pairs of two-particle operators $\uH_{\uC_{L}(\uu)}$, $\uH_{\uC_{L'}(\uu')}$ such that $L \ge L'$ and $\operatorname{dist}(\uC_{L}(\uu), \uC_{L'}(\uu') ) \ge 8L$: if $\nu$ is the continuity modulus of the marginal distribution of the IID random field $V$, then $$\pr{ \operatorname{dist}( \sigma(\uH_{\uC_{L}(\uu)} ), \sigma( \uH_{\uC_{L'}(\uu')}) \le \epsilon}\\ \le \, (2L+1)^{2d} (2L'+1)^d \,\nu(2\epsilon). \eqno({\rm W2})$$ The proof given in [@CS08] is based on a geometrical notion of “separable” pairs of cubes, combined with Stollmann’s lemma on “diagonally monotone” functions. In [@CS07] a similar bound was proven in the case of IID random field $V$ with analytic marginal distribution. Unfortunately, starting from $N=3$, additional difficulties appear in the analysis of pairs of spectra $ \sigma(\uH_{\uC_{L}(\uu)} ), \sigma( \uH_{\uC_{L}(\uu')})$. To put it simply, no a priori lower bound on the distance $\operatorname{dist}(\uC_{L}(\uu), \uC_{L}(\uu') )>CL$ between two cubes of sidelength $O(L)$ can guarantee the approach of [@CS08] to work, no matter how large is the constant $C$. This gives rise to a significantly more sophisticated MPMSA procedure in the general case where $N\ge 3$. A similar difficulty arises in [@AW09a]. The main goal {#ssec:motiv} ------------- It is well-known that the FMM, when applicable, leads directly to the proof of the dynamical localization, while it is more natural for applications of the MSA to establish first the spectral localization, via probabilistic bounds of the kernels of resolvents $G_\Lambda(E) = (H_\Lambda-E)^{-1}$ in finite subsets (usually cubes) $\Lambda\subset \DZ^d$, and then derive dynamical localization from decay bounds of the resolvents $G_\Lambda(E)$. In [@CS09a; @CS09b] a multi-particle adaptation of the MSA was used to prove *spectral* localization (i.e., exponential decay of eigenfunctions) in the strong disorder regime. Aizenman and Warzel [@AW09a; @AW09b] used the FMM to prove directly *dynamical* localization (hence, spectral localization) in various parameter regions including strong disorder, “extreme” energies and weak interactions. Despite many differences between these two approaches, similar technical difficulties have been encountered in both cycles of papers. Namely, it turned out to be difficult to prove the decay bounds of eigenfunctions $\Psi_j^{(N)}(x_1, \ldots, x_n)$ of $N$-particle Hamiltonians in terms of some *norm* $\|\cdot\|$ in $\DR^{Nd}$: $$|\Psi_j^{(N)}(x_1, \ldots, x_n;\omega)| \le C_j(\omega) e^{-m \|\ux\|}.$$ If the interaction $\uU$ is symmetric (and so is, then, $\uU + \uV$), then it is natural to expect (or to fear ...) “resonances” and “tunneling” between a point $\ux = (x_1, \ldots, x_N)$ and the points $\tau(\ux) = (x_{\tau(1)}, \ldots, x_{\tau(N)})$ obtained by permutations $\tau\in \fS_N$. So, it is much more natural in this context to use the “symmetrized” distance $$d_S(\ux, \uy) := \min_{\tau\in \fS_N} \|\ux - \uy\|.$$ Note also that if the quantum particles are bosons or fermions, then the points $\tau(\ux)$ should even be treated as identical, or, more precisely, the spectral problem should be solved in the subspace of symmetric or anti-symmetric functions of variables $x_j$. However, due to a highly correlated nature of the potential of a multi-particle system, even the above concession did not suffice, and it was easier to use “Hausdorff distance” (see the definition below, in Section \[sec:config.WS\]) between points $\ux,\uy\in\DZ^{Nd}$. This resulted in weaker decay estimates than expected. (Note that the Hausdorf distance was not used directly in [@CS09b].) Aizenman and Warzel [@AW09a] analyzed explicitly the aforementioned technical problem and pointed out that, physically speaking, it was difficult to rule out the possibility of “tunneling” between points $\ux$ and $\uy$ related by a “partial charge transfer” process, e.g., between points $(a, a, b)$ and $(a, b, b)$, $a\ne b$, corresponding to the states: $$\begin{array}{l} \text{ state $\ux$: 2 particles at the point $a$ and $1$ particle \,\,at $b$}\\ \text{ state $\uy$: 1 particle \,\,at the point $a$ and $2$ particles at $b$.} \end{array}$$ Observe that the norm-distance between such states can be arbitrarily large. *In the present paper we address this problem and show that resonances between distant states in the configuration space, related by partial charge transfer processes, are unlikely, providing probabilistic estimates for such unlikely situations.* Structure of the paper ---------------------- - The main theorem of this paper is formulated in Section \[ssec:main.res\]. - In Section \[sec:config.WS\] we introduce basic notions and give a geometrical sufficient condition for “weak separability” of multi-particle configurations (see Lemma \[lem:dist.are.WS\]). - In Section \[ssec:W2.WS\] we formulate and prove the main technical result, Lemma \[lem:main.lemma\]. - The assertion of Theorem \[thm:W2.general\] follows from Lemma \[lem:main.lemma\] combined with Lemma \[lem:dist.are.WS\]; the proof (quite short) is given in Section \[ssec:proof.main.thm\]. The main result {#ssec:main.res} ---------------- Introduce the following notations. Given a parallelepiped $Q\subset \DZ^d$, we denote by $\xi_{Q}(\omega)$ the sample mean of the random field $V$ over the $Q$, $$\xi_{Q}(\omega) = \langle V \rangle_Q = \frac{1}{| Q |} \sum_{x\in Q} V(x,\omega)$$ and the “fluctuations” of $V$ relative to the sample mean, $$\eta_x = V(x,\omega) - \xi_{Q}(\omega), \; x\in Q.$$ We denote by $\fF_{Q}$ the sigma-algebra generated by $\{\eta_x, \,x\in Q\}$, and by $F_\xi( \cdot\,| \fF_{Q})$ the conditional distribution function of $\xi_Q$ given $\fF_{Q}$: $$F_\xi(s\,| \fF_{Q}) := \pr{ \xi_Q \le s\,|\, \fF_{Q} }.$$ For a given $s\in\DR$, $F_\xi(s\,| \fF_{Q})$ is a random variable, determined by the values of $\{\eta_x, x\in Q\}$, but we will often use inequalities involving it, meaning that these relations hold true for ${\mathbb}{P}$-a.e. condition. We will assume that the random field $V$ satisfies the following condition[^1] (CCM = Continuity of the Conditional Mean): : *There exist constants $C', C'', A', A'', b', b''\in(0,+\infty)$ such that $\forall\, Q\subset \DZ^d$ with $\operatorname{{\rm diam}}(Q)\le R$ the conditional distribution function $F_\xi( \cdot \,| \fF_{Q})$ satisfies for all $s\in(0,1)$ $$\label{eq:CMxi} \;\; \pr{ \sup_{t\in \DR} \; |F_\xi(t+s\,| \fF_{Q}) - F_\xi(t\,| \fF_{Q})| \ge C' R^{A'} s^{b'}) } \le C'' R^{A''} s^{b''}.$$* Note that in the case of an IID random field $V$ the probability distribution of $\xi_Q$ is the same for all finite subsets $Q$ of given cardinality $|Q|$. In the particular case of a Gaussian IID field $V$, e.g., with zero mean and unit variance, $\xi_Q$ is a Gaussian random variable with variance ${|Q|}^{-1}$, independent of the “fluctuations” $\eta_x$, so that its probability density is bounded: $$p_{\xi_{Q}}(s) = |Q|^{1/2} \, (2\pi)^{-1/2}\, e^{-\frac{{|Q|} s^2}{2 } } \le |Q|^{1/2} \, (2\pi)^{-1/2},$$ although the $L_\infty$-norm of its probability density grows as $|Q|\to\infty$, and so does the continuity modulus of the distribution function $F_{\xi_{Q}}$. \[thm:W2.general\] Let $V: \DZ^d\times \Omega \to \DR$ be a random field satisfying . Then for any pair of $N$-particle operators $\uH_{\uC_{L'}(\uu')}$, $\uH_{\uC_{L''}(\uu'')}$, $0 \le L', L'' \le L$, satisfying $d_S(\uu', \uu'') > 2(N+1)L$, and any $s>0$ the following bound holds: $$\label{eq:main.bound} \pr{ \operatorname{dist}(\sigma(\uH_{\uC_{L'}(\uu')}), \sigma(\uH_{\uC_{L''}(\uu'')})) \le s } = h_L(s)$$ with \[eq:def.h.L\] h\_L(s) := |\_[L”]{}()| |\_[L”]{}()| C’L\^[A’]{} s\^[b’]{} + C”L\^[A”]{} s\^[b”]{}. In general, the conditional distribution function $F_\xi(\cdot\,|\fF_{Q})$ is not necessarily uniformly continuous, let alone Hölder-continuous. Moreover, the following simple example shows that for some conditions the distribution of the sample mean can be extremely singular. \[ex:unif\] Let $v_1(\omega), v_2(\omega)$ be two independent random variables uniformly distributed in $[0,1]$. Set $\xi = (v_1+v_2)/2$, $\eta = (v_1-v_2)/2$. Conditioning on $\eta\ge 0$ induces a uniform probability distribution on the segment $I(\eta)=\{ (t+2\eta,t), t \in(0, 1-2\eta)\}$ of length $|I(\eta)| = 1-2\eta$, with constant probability density $(1-2\eta)^{-1}$, if $\eta< 1/2$. Obviously, these distributions are not uniformly continuous. Moreover, for $\eta=1/2$, $\xi$ takes a single value: $\xi = 1/2$, so that its conditional distribution is no longer continuous. Observe, however, that “singular” conditions have probability zero, and conditions which give rise to large conditional density of $\xi$ have small probability. Using the main idea of the above example, Gaume [@G10], in the framework of his PhD project, proved the property for the IID potential with a uniform marginal distribution, and later extended the proof to IID potentials with piecewise constant marginal probability density. Our estimate is certainly *not sharp* and can probably be improved in various ways. Nevertheless, it suffices for the purposes of the MPMSA and allows to substantially simplify its structure, while leading to stronger results. Finally, note that we consider here only lattice models for the sake of brevity and clarity of presentation. Configurations and weak separability {#sec:config.WS} ===================================== Basic definitions ----------------- Consider the lattice $(\DZ^{d})^N \cong \DZ^{Nd}$, $N>1$ . We denote by $\DD$ the “principal diagonal” in $(\DZ^{d})^N $: $$\DD = \{ \ux\in\DZ^{Nd}: \,\ux = (x, \ldots, x), \, x\in\DZ^d \}.$$ Intervals of integer values will often appear in our formulae, and it is convenient to use a standard notation $[[a,b]] := [a,b]\cap \DZ$. We identify vectors $\ux\in\DZ^{Nd}$ with configurations of $N$ distinguishable particles in $\DZ^d$: $\ux \equiv (x_1, \ldots, x_N) \in \DZ^d \times \cdots \times \DZ^d$. Working with norms of vectors $\ux\in\DZ^{Nd}$, we always use the canonical embedding $\DZ^{Nd} \subset \DR^{Nd}$. \[def:subconfig\] Let $\ux\in\DZ^{Nd}$ be an $N$-particle configuration and consider a subset $\cJ \subset[[1,N]]$ with $1\le |\cJ|=n < N$. A **subconfiguration** of $\ux$ associated with $\cJ$ is the pair $(\ux', \cJ)$ where the vector $\ux' \in \DZ^{nd}$ has the components $x'_i = x_{j_i}$, $i\in[[1, n]]$. Such a subconfiguration will be denoted as $\ux_{\cJ}$. The complement of a subconfiguration $\ux_{\cJ}$ is the subconfiguration $\ux_{\cJ^c}$ associated with the complementary index subset $\cJ^c := [[1,N]]\setminus \cJ$. By a slight abuse of notations, we will identify a subconfiguration $\ux_\cJ = (\ux', \cJ)$ with the vector $\ux'$. With $\cJ$ clearly identified (this will always be the case in our arguments), it should not lead to any ambiguity, while making notations simpler. \[def:sep.cubes\] (a) Let $N\ge 2$ and consider the set of all $N$-particle configurations $\DZ^{Nd}$. For each $j\in[[1,N]]$ the coordinate projection $\Pi_j:\DZ^{Nd}\to \DZ^d$ onto the coordinate space of the $j$-th particle is the mapping $$\Pi_j: (x_1, \ldots, x_N) \mapsto x_j.$$ (b) The support $\Pi \ux$ of a configuration $\ux\in\DZ^{nd}$, $n\ge 1$, is the set $$\Pi \ux := \cup_{j=1}^n \Pi_j \ux = \{x_1, \ldots, x_N \}.$$ Similarly, the support of a subconfiguration $\ux_\cJ$ is defined by $ \Pi \ux_\cJ := \cup_{j\in \cJ}^n \Pi_j \ux. $ \(c) Given a subset $\cJ\subset [[1,N]]$ with $|\cJ|=n$, the projection $\Pi_\cJ: \,\DZ^{Nd} \to \DZ^{d}$ is defined as follows: $$\Pi_\cJ \ux = \begin{cases} \Pi \ux_\cJ, & \text{ if } \cJ \ne \varnothing \\ \varnothing, & \text{ otherwise}. \end{cases}$$ Finally, for each subset $\uC\subset \DZ^{Nd}$ its support $\Pi \uC$ is defined by $$\Pi \uC := \bigcup_{j=1}^N \Pi_j \uC \subset \DZ^d.$$ We will not associate with the empty subconfiguration $\ux_\varnothing$ any object other than its support $\Pi \ux_\varnothing = \varnothing \subset \DZ^d$, so the above definitions and notations suffice for our purposes. It is worth noticing that we use the max-norm $\|\cdot\|_\infty$ in $\DR^{nd}$, $n\ge 1$, $$\|\ux \|_\infty = \| (x_1, \ldots, x_n) \|_\infty := \max_{j\in[[1, n]]} \max_{i\in[[1, d]]} |x_j^{(i)}|.$$ It is often more suitable for multi-particle geometric bounds than the Euclidean norm. This norm canonically induces the notion of diameter, denoted below as “$\operatorname{{\rm diam}}$”. Particle configurations being associated with point subsets of $\DZ^d$, one can introduce the distance between two arbitrary configurations $\ux'\in \DZ^{N'd}$, $\ux''\in \DZ^{N''d}$, $N', N'' \ge 1$, as the distance between the respective subsets of $\DZ^d$, induced by the max-norm: $$\begin{array}{c} \rho\left(\ux', \ux'' \right) \equiv \rho\left( \{x'_1, \ldots, x'_{N'} \}, \{x''_1, \ldots, x''_{N''} \} \right) \\ \quad = \displaystyle \min_{i\in[[1, N']]} \; \min_{j\in[[1, N'']]} \|x'_i - x''_j\|. \end{array}$$ Further, given a bounded subset $\cX \subset \DR^{d}$, we call the canonical envelope of $\cX$ the minimal parallelepiped $\cQ$ with edges parallel to coordinate hyperplanes and such that $\cX \subseteq \cQ$. It will often be denoted as $\cQ(\cX)$. It is readily seen that $\operatorname{{\rm diam}}(\cX) = \operatorname{{\rm diam}}( \cQ(\cX))$ (recall that “diam” is induced by the max-norm). Keeping in mind the canonical embedding $\DZ^{d} \subset \DR^{d}$, this notion applies also to bounded lattice subsets. Now we introduce a modified version of the distance $\rho$ which allows to simplify certain geometrical constructions used below; it suffices for our purposes to define it for bounded subsets $\cX, \cY \subset \DR^{d}$ as follows: $$\operatorname{{\rho_{\mathcal{C}}}}(\cX, \cY)) = \rho( \cQ(\cX), \cQ(\cY)),$$ where $\cQ(\cX)$ and $\cQ(\cY)$ are the canonical envelopes of sets $\cX$ and, respectively, $\cY$. (Here $\cC$ in $\operatorname{{\rho_{\mathcal{C}}}}$ stands for “convex”). For further references, we state the following elementary \[lem:geom.sep.paral\] Consider bounded subsets $\cX, \cY \subset\DZ^{d}\subset\DR^{d}$. Then: 1. $\operatorname{{\rho_{\mathcal{C}}}}(\cX, \cY) \ge R\ge 1$ $\Leftrightarrow$ there exist two hyperplanes $\cL',\cL''\subset\DZ^{d}$, parallel to one of the coordinate hyperplanes, at max-norm distance $R$ from each other and such that the layer with the border $\cL' \cup \cL''$ separates $\cX$ from $\cY$; 2. if $\operatorname{{\rho_{\mathcal{C}}}}(\cX, \cY) =0$, then $\operatorname{{\rm diam}}( \cQ( \cX \cup \cY)) \le \operatorname{{\rm diam}}(\cX) + \operatorname{{\rm diam}}(\cY)$. We would like to stress that the distance $\operatorname{{\rho_{\mathcal{C}}}}$, unlike the distance $\rho$, does *not always* separate disjoint sets. In particular, the assertion (B) can (and will) be applied to some disjoint pairs of sets, too. The proof is straightforward, so we omit it. \[def:part.decouple\] A particle configuration $\ux\in\DZ^{Nd}$, $N>1$, is called $R$-decoupled, with $R>0$, if it contains a pair of non-empty complementary subconfigurations $\ux_{\cJ}, \ux_{\cJ^c}$ with $\operatorname{{\rho_{\mathcal{C}}}}\left( \ux_{\cJ}, \ux_{\cJ^c}\right) > 2R$. The decoupling width of a configuration $\ux$ is the quantity $$D(\ux) := \max_{ \cJ, \cJ^c \subsetneq [[1,N]]} \operatorname{{\rho_{\mathcal{C}}}}(\ux_\cJ, \ux_{\cJ^c}).$$ The decoupling width $D(\uC_L(\ux))$ of a cube $\uC_L(\ux)$ is defined as follows: $$D(\uC_L(\ux)) := \max_{ \cJ, \cJ^c \subsetneq [[1,N]]} \operatorname{{\rho_{\mathcal{C}}}}\left( \bigcup_{i\in \cJ} C_L(x_i), \bigcup_{j\in {\cJ^c}} C_L(x_j) \right).$$ In other words, $R$-decoupling of $\ux$ means that $R$-neighborhoods of two complementary (and non-empty) subconfigurations $ \ux_{\cJ}, \ux_{\cJ^c}$ are disjoint. \[def:cluster\] An $R$-cluster of a configuration $\ux$ is a subconfiguration $\ux_\cJ$ which is not $R$-decoupled and is not contained in any non-$R$-decoupled subconfiguration $\ux_{\cJ'}$ with $|\cJ'| > |\cJ|$. The set of all $R$-clusters of a configuration $\ux$ is denoted by ${\mathbf}{\Gamma}(\ux,R)$. \[lem:clusters\] Fix a positive number $R$ and integers $N>1$, $d\ge 1$. 1. Every configuration $\ux\in\DZ^{Nd}$ can be decomposed into a family ${\mathbf}{\Gamma}(\ux,R)$ of $R$-clusters $\Gamma_1, \ldots, \Gamma_M$, with some $1\le M=M(\ux)\le N$, so that $(\cJ_{\Gamma_1}, \ldots, \cJ_{\Gamma_M})$ is a partition of $[[1,N]]$ and each particle $x_j$ is contained in exactly one cluster $\Gamma = \Gamma_\ux(x_j)$. 2. If $\,\Gamma', \Gamma''\in {\mathbf}{\Gamma}(\ux,R)$ and $\Gamma' \ne \Gamma''$, then $\operatorname{{\rho_{\mathcal{C}}}}( \Gamma', \Gamma'') > 2R$, i.e. the canonical envelopes obey $\rho( \cQ(\Gamma'), \cQ(\Gamma'')) > 2R$. 3. The diameter of any cluster $\Gamma$ is bounded by $2(N-1)R$, and so is the diameter of its envelope $\cQ(\Gamma)$. It is convenient here to make use of the canonical embedding $\DZ^d \subset \DR^d$. For each $i\in[[1, N]]$ introduce the cube $C'_{R}(x_j)\subset \DR^d$. The union $\cup_i C'_{R}(x_i)$ is uniquely decomposed into a union of maximal $\operatorname{{\rho_{\mathcal{C}}}}$-disjoint components $\{\widetilde{ C }_j\}$: for some partition $\{\cJ_j, j\in[[1,M]]\}$ of $[[1,N]]$ with $1 \le M \le N$, $$\bigcup_{i=1}^N C'_{R}(x_i) = \coprod_{j=1}^M \;\left( \bigcup_{i\in \cJ_j} C'_{R}(x_i) \right) =: \coprod_{j=1}^M \widetilde{ C }_j,$$ and for all $j \ne j'$, $\cQ(\widetilde{ C }_j)$, $\cQ(\widetilde{ C }_{j'})$ obey $$\label{eq:dist.Q} \rho\left( \cQ(\widetilde{ C }_j), \cQ(\widetilde{ C }_{j'}) \right) >0.$$ Set $$\Gamma_j = \widetilde{ C }_j \cap \Pi \ux \equiv \widetilde{ C }_j \cap \{ x_1, \ldots, x_N\} \subset \DZ^d, \quad j=1, \ldots, M.$$ Observe that implies $$\operatorname{{\rho_{\mathcal{C}}}}( \Gamma_{j'}, \Gamma_{j''} ) \ge \operatorname{{\rho_{\mathcal{C}}}}( \widetilde{ C }_{j'}, \widetilde{ C }_{j''} ) + 2R > 2R.$$ Finally, $\operatorname{{\rm diam}}(\Gamma_j) \le 2(|\Gamma_j|-1) R$: it suffices to combine the assertion (B) of Lemma \[lem:geom.sep.paral\], the fact that $\widetilde{ C }_j$ is a union of cubes of diameter $2R$, and the inequality $|\Gamma_j| \le N$. Weakly separable cubes {#ssec:def.PCT} ---------------------- A cube $\uC_L(\ux)$ is weakly separable from $\uC_L(\uy)$ if there exists a parallelepiped $Q\subset \DZ^d$ in the 1-particle configuration space, of diameter $R \le 2NL$, and subsets $\cJ_1, \cJ_2\subset [[1,N]]$ such that $|\cJ_1| > |\cJ_2|$ (possibly, with $\cJ_2=\varnothing$) and $$\label{eq:cond.WS} \begin{array}{l} \Pi_{\cJ_1} \uC_L(\ux) \cup \Pi_{\cJ_2} \uC_L(\uy) \; \subseteq Q,\\ \Pi_{\cJ^c_2} \uC_L(\uy) \cap Q = \varnothing. \end{array}$$ A pair of cubes $(\uC_L(\ux), \uC_L(\uy))$ is weakly separable if at least one of the cubes is weakly separable from the other. The physical meaning of the weak separability is that in a certain region of the one-particle configuration space the presence of particles from configuration $\ux$ is more important than that of the particles from $\uy$. As a result, some local fluctuations of the random potential $V(\cdot;\omega)$ have a stronger influence on $\ux$ than on $\uy$. Some upper bound on the diameter $R$ of the “separating parallelepiped” $Q$ is required in applications of the main theorem to the localization analysis of multi-particle systems. Indeed, the continuity modulus $\nu_L$ figuring in the main bound depends upon the size $L\ge L', L''$, and it grows with $L$. However, in most cases there is a substantial freedom for the choice of the upper bound of $\operatorname{{\rm diam}}Q$; cf. . Again, the case of a Gaussian IID random potential $V$ is instructive here. \[lem:dist.are.WS\] Cubes $\uC_L(\ux), \uC_L(\uy)$ with $d_S(\ux, \uy)> 2(N+1)L$ are weakly separable. Let ${\mathbf}{\Gamma}(\ux, 2L)$ and ${\mathbf}{\Gamma}(\uy, 2L)$ be the collections of $2L$-clusters for the configurations $\ux$ and $\uy$. Consider clusters $\Gamma_1, \ldots, \Gamma_M\in {\mathbf}{\Gamma}(\ux, 2L)$, $M=|{\mathbf}{\Gamma}(\ux, 2L)|$, and the disjoint parallelepipeds $$\label{eq:2.5} \displaystyle Q_i = \cQ\left( \bigcup_{k:\, x_k\in \Gamma_i} C_{L}(x_k) \right), \; i\in[[1,M]].$$ By virtue of assertion (C) of Lemma \[lem:clusters\], $$\operatorname{{\rm diam}}(Q_i) \le \operatorname{{\rm diam}}(\Gamma_i) + 2L \le 2(N-1)L+ 2L = 2NL.$$ Introduce the “occupancy numbers” of parallelepipeds $Q_i$ for configurations $\ux$ and $\uy$: $$\begin{aligned} n_i(\ux) = \operatorname{card}\left( \Pi \ux \cap Q_i \right), \; i\in[[1,M]],\\ n_i(\uy) = \operatorname{card}\left( \Pi \uy \cap Q_i \right), \; i\in[[1,M]].\end{aligned}$$ There can be two possible situations: (I) For all $i\in[[1,M]]$ we have $n_i(\ux) = n_i(\uy)$. Then there exists a permutation $\tau\in\mathfrak{S}_N$ such that for all $j\in[[1,N]]$, $$\|x_{\tau(j)} - y_j\| \le 2(N-1)L + 2L = 2NL,$$ yielding $$d_S(\ux, \uy) \le \|\tau(\ux) - \uy \| = \max_{1\le j \le N} \|x_{\tau(j)} - y_j \| \le 2NL.$$ If $d_S(\ux, \uy)> 2(N+1)L$, then the occupancy numbers $n_i(\ux), n_i(\uy) $ cannot be all identical, so this situation is impossible under the hypotheses of the lemma. (II) For some $i\in[[1,M]]$, $n_i(\ux) \ne n_i(\uy)$. By the definition of $Q_i$, it contains $|\Gamma_i|\ge 1$ particles of the configuration $\ux$, so that $n_i(\ux)\ge 1$ for all $i\in[[1,M]]$. Observe that $$\label{eq:occup.num} \sum_{i=1} ( n_i(\ux) - n_i(\uy)) = N - \sum_{i=1} n_i(\uy) \ge 0.$$ Since not all quantities $n_i(\ux) - n_i(\uy)$ vanish, there exists some $j_\circ\in [[1,M]]$ such that $n_{j_\circ}(\ux) - n_{j_\circ}(\uy)>0$ , otherwise the LHS of would be negative. Now setting $Q = Q_{j_\circ}$, we see that the conditions are fulfilled. Eigenvalue concentration bound for distant cubes {#sec:EV.sep} ================================================= Bounds for weakly separable cubes {#ssec:W2.WS} --------------------------------- \[lem:main.lemma\] Let $V: \cZ\times \Omega \to \DR$ be a random field satisfying the condition . Let $\ux,\uy\in \cZ$ be two configurations such that the cubes $\uC_{L}(\ux)$, $\uC_{L}(\uy)$ are weakly separable. Consider operators $\uH_{\uC_{L'}(\uy)}(\omega)$, $\uH_{\uC_{L''}(\uy)}(\omega)$, with $L', L''\le L$. Then for any $s>0$ the following bound holds for the spectra $\Sigma_\ux=\sigma(\uH_{\uC_{L'}(\ux)})$, $\Sigma_\ux=\sigma(\uH_{\uC_{L''}(\uy)})$ of these operators: $$\bal \pr{ \operatorname{dist}(\Sigma_\ux,\Sigma_\uy)) \le s } \le |\uC_{L}(\uu')| \, |\uC_{L}(\uu'')|\, C'L^{A'} (2s)^{b'} + C''L^{A''} (2s)^{b''}. \eal$$ Let $Q$ be a ball satisfying the conditions for some $\cJ_1, \cJ_2 \subset [[1,N]]$ with $|\cJ_1|=n_1 > n_2 =|\cJ_2|$. Introduce the sample mean $\xi=\xi_{Q}$ of $V$ over $Q$ and the fluctuations $\{\eta_x, \, x\in Q \}$ defined as in Section \[ssec:main.res\]. The operators $\uH_{\uC_{L'}(\ux)}(\omega)$, $\uH_{\uC_{L''}(\uy)}(\omega)$ read as follows: $$\label{eq:Ham.decomp} \begin{array}{l} \uH_{\uC_{L'}(\ux)}(\omega) = n_1 \xi(\omega) \, \operatorname{\mathbf{1}}+ \uA(\omega), \\ \uH_{\uC_{L''}(\uy)}(\omega) = n_2 \xi(\omega) \,\operatorname{\mathbf{1}}+ \uB(\omega) \end{array}$$ where operators $\uA(\omega)$ and $\uB(\omega)$ are $\fF_{Q}$-measurable. Let $ \{ \lambda_1, \ldots, \lambda_{M'}\}, \; M' = \,\,|\uC_{L'}(\ux)|, $ and $ \{ \mu_1, \ldots, \mu_{M''}\}, M'' = |\uC_{L''}(\uy)|, $ be the sets of eigenvalues of $\uH_{\uC_{L'}(\ux)}$ and of $\uH_{\uC_{L''}(\uy)})$, counted with multiplicities. Owing to , these eigenvalues can be represented as follows: $$\begin{array}{l} \lambda_j(\omega) = n_1\xi(\omega) + \lambda_j^{(0)}(\omega), \quad \mu_j(\omega) = n_2\xi(\omega) + \mu_j^{(0)}(\omega), \end{array}$$ where the random variables $\lambda_j^{(0)}(\omega)$ and $\mu_j^{(0)}(\omega)$ are $\fF_{Q}$-measurable. Therefore, $$\lambda_i(\omega) - \mu_j(\omega) = (n_1-n_2)\xi(\omega) + (\lambda_j^{(0)}(\omega) - \mu_j^{(0)}(\omega)),$$ with $n_1-n_2 \ge 1$, owing to our assumption. Further, we can write $$\bal \pr{ \operatorname{dist}(\Sigma_\ux, \Sigma_\uy)) \le s } &= \pr{ \exists\, i,j:\, |\lambda_i - \mu_j| \le s } \\ & \le \sum_{i=1}^{M'} \sum_{j=1}^{M''} \esm{ \pr{ |\lambda_i - \mu_j| \le s \,| \fF_{Q}}}. \eal$$ Note that for all $i$ and $j$ we have $$\bal \pr{ |\lambda_i - \mu_j| \le s \,|\, \fF_{Q}} & = \pr{ |(n_1 - n_2)\xi + \lambda_i^{(0)} - \mu_j^{(0)}| \le s \,| \fF_{Q}} \\ &\le \displaystyle \nu_L( 2|n_1 - n_2|^{-1} s \,|\, \fF_{Q}) \le \nu_L( 2 s \,| \fF_{Q}). \eal$$ Consider the event $$ \_L = { \_[t]{} |F\_(t+s| \_[Q]{}) - F\_(t| \_[Q]{})|C’L\^[A’]{} s\^[b’]{} } $$ By hypothesis (cf. ), $\pr{\cE_L} \le C''L^{A''} s^{b''}\}$. Therefore, $$\bal \pr{ \operatorname{dist}(\Sigma_\ux, \Sigma_\uy)) \le s } &= \esm{ \pr{ \operatorname{dist}(\Sigma_\ux, \Sigma_\uy) \le s \,| \fF_{Q}}} \\ &\le \esm{ \operatorname{\mathbf{1}}_{\cE^c_L} \pr{ \operatorname{dist}(\Sigma_\ux, \Sigma_\uy) \le s \,| \fF_{Q}}} + \pr{\cE_L} \\ &\le |\uC_{L''}(\ux)| \cdot |\uC_{L''}(\uy)|\, C'L^{A'} s^{b'} + C''L^{A''} s^{b''} = h_L(s) \eal$$ with $h_L$ defined in . Proof of the main result {#ssec:proof.main.thm} ------------------------ $\,$ By the hypothesis of Theorem \[thm:W2.general\], we have $d_S(\ux, \uy) > 2(N+1)L$; therefore, by Lemma \[lem:dist.are.WS\], cubes $\uC_{L'}(\ux)$ and $\uC_{L''}(\uy)$ are weakly separable. Now the assertion of the theorem follows from Lemma \[lem:main.lemma\]. [^1]: In an earlier version of this manuscript (1005.3387v2, 02.07.2010), we assumed a stronger condition: a uniform continuity of the conditional probability distribution function $F_\xi(\cdot\,| \fF_{Q})$, i.e., a uniform bound for a.e. condition.
--- abstract: 'We investigate an application in the automatic tuning of computer codes, an area of research that has come to prominence alongside the recent rise of distributed scientific processing and heterogeneity in high-performance computing environments. Here, the response function is nonlinear and noisy and may not be smooth or stationary. Clearly needed are variable selection, decomposition of influence, and analysis of main and secondary effects for both real-valued and binary inputs and outputs. Our contribution is a novel set of tools for variable selection and sensitivity analysis based on the recently proposed *dynamic tree* model. We argue that this approach is uniquely well suited to the demands of our motivating example. In illustrations on benchmark data sets, we show that the new techniques are faster and offer richer feature sets than do similar approaches in the static tree and computer experiment literature. We apply the methods in code-tuning optimization, examination of a cold-cache effect, and detection of transformation errors.' address: - | R. B. Gramacy\ M. Taddy\ University of Chicago Booth School\ of Business\ 5807 S. Woodlawn Avenue\ Chicago, Illinois 60637\ USA\ \ \ - | S. M. Wild\ Mathematics and Computer Science Division\ at Argonne National Laboratory\ 9700 S. Cass Avenue, Bldg. 240-1154\ Argonne, Illinois 60439\ USA\ and\ Computation Institute\ University of Chicago\ USA\ \ author: - - - title: 'Variable selection and sensitivity analysis using dynamic trees, with an application to computer code performance tuning' --- , Introduction {#sec1} ============ The optimization of machine instructions derived from source codes has long been of interest to compiler designers, processor architects, and code developers. Compilers such as `gcc`, for example, provide a myriad of flags, each allowing the programmer to choose the “level” of optimization. As codes and their optimization become more complex, however, it can be harder to know a priori what modifications will benefit or hinder performance in execution. Recent advances in the area have demonstrated that higher performance of a given code can be achieved through annotation scripts (e.g., Orio \[@Orio09\]), which directly apply code transformations such as loop reordering to the original source to generate a modified, but semantically equivalent, version of the code. The output code can then be compiled in various ways (e.g., by setting compiler optimization flags or by choosing different compilers), resulting in an executable that runs on a particular machine more or less quickly depending on the nature of the transformation, compilation, machine architecture, and original source code. Given detailed knowledge of each aspect of the process, from original source to final executable, one can obtain significant speedups in execution. But missteps can result in significant slowdowns. Modern high-performance computing facilities are increasingly complex, making it difficult and/or time-consuming for a scientific programmer to intimately understand or control the environment in which the source code is executed, and thereby affect its performance. For example, commercial cloud-computing services such as the Amazon Elastic Compute Cloud (EC2) provide only a limited description of the available hardware and accompanying resources; and scientific and governmental computing facilities are diverse. Thus, the need arises to tune codes automatically. In this paper we report on aspects of a performance-tuning effort being undertaken at Argonne National Laboratory to meet needs in scientific computing. Our aim, given a target machine and source code, is to study how a suite of given transformations, together with compiler options (e.g., `gcc` flags), can be used to minimize code execution times under the constraint that it yields correct output. As evidenced by the success of the ATLAS project (<http://math-atlas.sourceforge.net/>), involving a similar but more limited search set, even minor performance gains for basic computational kernels can be significant when called repeatedly. A performance-tuning computer experiment {#sec1.1} ---------------------------------------- We focus on data arising from a set of exploratory benchmarking experiments described by @Balaprakash20112136. In the design of each experiment (the input source code), a subset of the possible transformation and compilation options (inputs) was thought to yield correct numerical outputs, and these were varied in full enumeration over the input space to obtain execution times. Some of the inputs are ordinal and some categorical. Such full enumerations therefore result in combinatorially huge design spaces—too big to explore in a time that is reasonable to wait for a compiled executable. We investigate the extent to which a statistical model can be used to measure the relative importance of each input for predicting execution times, to explore how each relevant input contributes to the execution time marginally and (to the extent possible) conditionally, and to check for any predictable patterns of constraint violations arising from unsuccessful compilation or runtime errors in the executed code. Our results in Section \[secapp\] show that we can dramatically reduce the space of options in the search for fast executables: one of the five inputs is completely removed, and each of the other four has its range decreased by roughly a factor of $2$. We perform this analysis on a dramatically reduced design that, together with a thrifty inferential technique, means that such information can be gleaned in an amount of time that most programmers would deem acceptable to produce a final executable. We then provide a final iterative optimization, primed with the results of that analysis, to obtain a fast executable. The remainder of the fully enumerated design is then used for validation purposes, wherein we show that our solution is preferable to alternatives out-of-sample. Several aspects of this type of data make it unique in the realm of computer experiments, therefore justifying a noncanonical approach. The first is size. Even when using reduced designs, these experiments are large by conventional standards. Second, although some of the transformation options (i.e., inputs) are ordinal (e.g., a loop unrolling factor), there is no reason to expect an a priori smooth or stationary relationship between that input and the response: for some architectures it may be reasonably smooth, and for others it may have regime changes due to, for example, being memory-bound versus being compute-bound. Third, high-order interactions between the inputs are expected, a priori, which may prohibit the use of additive models. Fourth, checking for valid outputs requires a classification surrogate. Fifth, since (valid) responses are execution times, the experiment being modeled is inherently stochastic, whereas many authors define a computer experiment as one where the response is deterministic. This last point is perhaps more nuanced than it may seem at first. In actuality, many of the sources that can contribute to the “randomness” of an executable are known. For example, processor loads can be controlled; interruptions from operating system maintenance threads follow schedules; and locations in memory, which affect data movement, can be controlled. But these may more usefully, and practically, be modeled as random. However, one contributor to the nature of the “noise” in the experiment is of particular interest to the Argonne researchers: the *cold-cache effect*. This effect, due to compulsory cache misses sometimes arising from initial accesses to a cache block, is also referred to as *cold-start misses* @DPJLbook and can cause the first execution instance to run slower than subsequent instances. It would be useful to know whether acknowledging and controlling for this effect are necessary when searching for an optimal executable. The degree of the cold-cache effect varies greatly from problem to problem, and determining its significance is vital for designing an experimental setup (e.g., whether the cache needs to be warmed before each execution of a code configuration). Although recent works \[e.g., @PBSWBN11\] have focused on defining input spaces for performance tuning problems, formulating appropriate objectives in the presence of the cache effects and other operating system noise remains an unresolved issue, which application of our techniques can help inform. In Section \[secapp\] we show that the cold-cache effect is present but negligible for the particular problem examined. One can optimize the executable without acknowledging its effect because it is very small and does not vary as a function of the input parameters.=-1 Roadmap {#sec1.2} ------- The remainder of the paper is organized as follows. Given the unique demands of our motivating problem set, we make the case in Section \[seccase\] that a new, thrifty approach to modeling computer experiments and decomposing the influence of inputs is needed. We maintain that, without using such an analysis to first significantly narrow the search space, searches for optimal transformations and compilation settings cannot be performed in a time that is acceptable to practitioners. We propose that these needs are addressed by *dynamic tree* (DT) models, which (along with previous approaches to model-based decomposition of influence) are reviewed below and in the . In Sections \[secVS\] and \[secSA\] we improve upon standard methodology for variable selection and input sensitivity analysis by leveraging the unique aspects of DTs. Compared with previous tree modeling approaches, our new methodology offers sequential decision making and fully Bayesian evidence not previously enjoyed in these contexts. Compared with the canonical Gaussian process (GP) model for computer experiments, which serves as a straw man for many of our comparisons, our methods facilitate decompositions of input variable influence on problems that are several orders of magnitude larger than previously possible, while simultaneously avoiding assumptions of smoothness and stationarity and allowing for higher-order interactions. Both sections conclude with an illustration of the methods, in both classification and regression applications, and a brief comparison study in support of these observations. Section \[secapp\] contains a detailed analysis of our motivating performance-tuning example using such methods. We conclude in Section \[secdiscuss\] with a brief discussion. Background {#seccase} ========== We begin with a review of previous approaches to the analysis of input influence as relevant to applications in computer experiments, motivating our dynamic trees approach. These models and accompanying inferential techniques are then discussed in some detail. Decomposition of influence -------------------------- In any regression analysis, one must quantify the influences on the response by individual candidate explanatory variables. This assessment should cover an array of information, attributing direction, strength, and evidence to covariate effects, both when acting independently and when interacting. For linear statistical models, various well-known tools are available for the task. In ordinary least-squares, for example, there are $t$ and $F$ tests for the effect of predictor(s), ANOVA to decompose variance contributions, and leverages to measure influence in the input space. Such tools are fundamental to applied linear regression analysis and are widely available in modern statistical software packages. In contrast, analogous techniques for more complicated nonparametric regression methods, such as neural networks, other basis expansions, or GPs and other stochastic models, are far less well established. Many related techniques exist, and we provide a detailed review in our . However, they are not part of the conventional arsenal applied to the broad engineering problems that motivate this work—optimization under uncertainty and emulation of noisy computer simulators—where modeling is further complicated by nonstationarities manifesting in varying degrees of smoothness. A lack of fast, easy-to-apply tools (and readily available software) means that one typically treats the response surface model as a black-box prediction machine and neglects analyses essential for tackling the application motivating this paper. To resolve the tension between flexibility and interpretability, we present a framework that provides both. We argue that dynamic trees (DTs), introduced in @taddgrapols2011 and summarized below, are a uniquely appropriate platform for predictive modeling and analysis of covariance in complex regression and classification settings. Although aspects of DT modeling are just as opaque as, say, neural networks, they inherit many advantages from the well-understood features of classic trees. They take a fast and flexible divide-and-conquer approach to regression and classification by fitting piecewise constant, linear models, and multinomial models. Besides employing a prior that regularizes the nature of the patchwork fits that result, they make few assumptions about the nature of the data-generating mechanism. This approach is in contrast to GP models, which may disappoint when stationary modeling is inappropriate and which are burdened by daunting computational hurdles for large data sets. Part of our argument holds for tree models in general, of which DTs are just one modern example: partitioning of the covariate space, the same quality that is key to model flexibility, acts as an interpretable foundation for attribution of variable influence. Distinct from most other tree methods, however, DTs are accompanied by an efficient particle sequential Monte Carlo (SMC) method for posterior inference and can provide full uncertainty quantification for each metric of covariance analysis, hence allowing for proper consideration of statistical evidence. DT inference is also inherently on-line and naturally suited to the analysis of sequential data. This aspect is exploited in our final optimization of the motivating computer experiment. Our methodological contributions comprise two complementary analyses: *variable selection* and input *sensitivity analysis*. The first focuses on selecting the subset of covariates to be included in the model, in that they lead to predictions of low variance and high accuracy. The second characterizes how elements of this subset influence the response. As discussed in more detail in the , it is most common to focus on only one of these two analyses: variable selection is common in additive models, where the structure for covariance is assumed rather than estimated, whereas in more complicated functional sensitivity analysis settings, the set of covariates is taken as given. This methodological split is unfortunate, because variable selection and sensitivity analysis work best together, with sensitivities providing a higher-fidelity analysis that follows in-or-out decisions made after preselecting variables. Hence, we have found that the use of DTs as a platform for both analyses is a powerful tool in applied statistics and is ideal for our motivating performance-tuning application. Dynamic tree models {#secdynaTree} ------------------- The dynamic tree (DT) framework was introduced in @taddgrapols2011 to provide Bayesian inference for regression trees that change in time with the arrival of new observations. It builds directly on work by @chipgeormccu1998 ([-@chipgeormccu1998; -@chipgeormccu2002]), wherein prior models over the space of various decision trees are first developed. Since the Taddy et al. paper contains a survey of Bayesian tree models and full explanation of the DT framework, we focus here on communicating an intuitive understanding of DTs and refer the reader elsewhere for details. For those interested in using these techniques, software is available in the `dynaTree` \[@dynaTree\] package for , which also includes all the methods of this paper. Consider covariates ${\mathbf{x}}^t = \{{\mathbf{x}}_s\}_{s=1}^t$ paired with response ${\mathbf{y}}^t = \{y_s\}_{s=1}^t$, as observed up to time $t$ (the data need not be ordered, but it is helpful to think sequentially). A corresponding tree ${\mathcal{T}}_t$ consists of a hierarchy of *nodes* $\eta \in {\mathcal{T}}_t$ associated with different disjoint subsets of ${\mathbf{x}}^t$. This structure is built through a series of recursive *split rules* on the support of ${\mathbf{x}}_t$, as illustrated in the top row of Figure \[treefig\]: the left plot shows top-down sorting of observations into nodes according to variable constraints, and the right plot shows the partitioning at the bottom of the tree implied by such split rules. These terminal nodes are called *leaves*, and, in a regression tree, they are associated with a prediction rule for any new covariate vector. That is, new ${\mathbf{x}}_{t+1}$ will fall within a single leaf node $\eta({\mathbf{x}}_{t+1}$), and this provides a distribution for $y_{t+1}$. For example, a *constant tree* has simple leaf response functions ${\mathbb{E}}[y_{t+1} | {\mathbf{x}}_{t+1}] = \mu_{\eta({\mathbf{x}}_{t+1})}$, a [*linear tree*]{} fits the plane $y = \alpha_{\eta({\mathbf{x}}_{t+1})} + {\mathbf{x}}^\top{\boldsymbol{\beta}}_{\eta({\mathbf{x}}_{t+1})}$ through the observations in each leaf, and a *classification tree* uses within-leaf response proportions as the basis for classification. ![Prior possibilities for tree change ${\mathcal{T}}_{t} \rightarrow {\mathcal{T}}_{t+1}$ upon arrival of a new data point at ${\mathbf{x}}_{t+1}$.[]{data-label="treefig"}](590f01.eps) Bayesian inference relies on prior and likelihood elements to obtain a tree posterior, ${\mathrm{p}}({\mathcal{T}}_t | [{\mathbf{x}}, y]^t) \propto {\mathrm{p}}(y^t|{\mathcal{T}}_t, {\mathbf{x}}^t) \pi({\mathcal{T}}_t)$. Given independence across partitions, tree likelihood is available as the product of likelihoods for each terminal node; constant and linear leaves use normal additive error around the mean, while classification trees assume a multinomial distribution for each leaf’s response. This is combined with a product of conjugate or reference priors for each leaf node’s parameters to obtain a conditional model for leaves given the tree. Chipman et al. define the probability of splitting node $\eta$, with node depth $D_{\eta}$, as $p_{\mathrm{split}}({\mathcal{T}}_t, \eta) = \alpha(1+D_\eta)^{-\beta}$. Hence, the full tree prior is $ \pi({\mathcal{T}}_t) \propto \prod_{\eta \in {\mathcal{I}}_{{\mathcal{T}}_t}} p_{\mathrm{split}}({\mathcal{T}}_t, \eta) \prod_{\eta \in L_{{\mathcal{T}}_t}} [1-p_{\mathrm{split}}({\mathcal{T}}_t, \eta)]$, where ${\mathcal{I}}_{{\mathcal{T}}_t}$ is the set of internal nodes and $L_{{\mathcal{T}}_t}$ are the leaves. They show how a taxonomy of choices of $\alpha$ and $\beta$ map to prior distributions over trees via their depth. The DT model of Taddy et al. adopts this basic framework but combines it with rules for how a given tree can change upon the observation of new data. In particular, $\pi({\mathcal{T}}_{t+1})$ for a new tree is replaced with ${\mathrm{p}}({\mathcal{T}}_{t+1} | {\mathcal{T}}_{t}, {\mathbf{x}}^{t+1})$, where this conditional prior is proportional to Chipman et al.’s $\pi({\mathcal{T}}_{t+1})$ but restricted to trees that result from three possible changes to the neighborhood of the leaf containing ${\mathbf{x}}_{t+1}$: [*stay*]{} and keep the existing partitions, *prune* and remove the partition above $\eta({\mathbf{x}}_{t+1})$, or *grow* a new partition by splitting on this leaf. This evolution from ${\mathcal{T}}_{t}$ to ${\mathcal{T}}_{t+1}$ via ${\mathbf{x}}_{t+1}$ is illustrated in Figure \[treefig\]. The original DT paper contains much discussion of tree dynamics, but the founding idea is that this process leverages the assumed independence structure of trees to introduce stability in estimation: a new observation at ${\mathbf{x}}_{t+1}$ will change our beliefs only about the local area of the tree around $\eta({\mathbf{x}}_{t+1})$. While the moves from ${\mathcal{T}}_{t}$ to ${\mathcal{T}}_{t+1}$ are designed to be local to new observations, inference for these models must account for global uncertainty about ${\mathcal{T}}_{t}$. This is achieved through use of a filtering algorithm that follows a general particle learning recipe set out by @CarvJohaLopePols2009. In such methods, the posterior for ${\mathcal{T}}_{t}$ is approximated with a finite sample of potential tree *particles* ${\mathcal{T}}_{t}^{(i)} \in \{{\mathcal{T}}_{t}^{(1)} \cdots {\mathcal{T}}_{t}^{(N)}\}$, each of which contain the set of tree-defining partition rules. This posterior is updated to account for $[{\mathbf{x}}_{t+1},y_t]$ by first *resampling* particles proportional to the predictive probability $p(y_t | {\mathcal{T}}_{t}^{(i)}, {\mathbf{x}}_{t+1})$ and then *propagating* these particles by sampling from the conditional posterior ${\mathrm{p}}({\mathcal{T}}_{t+1} | {\mathcal{T}}_{t}^{(i)}, [{\mathbf{x}},y]^{t+1})$ (i.e., drawing from the three moves illustrated in Figure \[treefig\], proportional to each resulting tree’s prior multiplied by its likelihood). Hence, tree propagation is local, but resampling accounts for global uncertainty about tree structure. Although DTs’ inferential mechanics are tailored to sequential applications, such as sequential design or optimization, they can also provide a powerful tool for batch analysis. Since the data ordering can be arbitrary, it can be helpful to run several independent repetitions of the SMC method each with a different random pass through the data. This approach allows one to study the Monte Carlo error of the method, which can be mitigated by averaging inferences across repetitions. Such averaging is especially important for Bayes factor estimation \[@taddgrapols2011\]. Variable selection {#secVS} ================== Tree models engender basic variable selectionthrough the estimation of split locations: any variable not split on has been deselected. However, this binary determination does not provide any spectrum of variable importance, and the unavailable null distribution for tree splits can lead to inclusion of spurious variables. Hence, we need measures of covariate influence that are based on analysis of response variance. Combining these with the full probability model provided by DTs, one can obtain a probabilistic measure of variable importance and evidence for inclusion. Measuring the importance of predictors -------------------------------------- Following the basic logic of tree-based variable selection, variables contribute to reduction in predictive variance through each split location. We label the leaf model-dependent uncertainty reduction for each node $\eta$ as $\Delta(\eta)$. Grouping these by variable, we obtain the importance index for each covariate $k \in \{1,\ldots, p\}$ as $$\label{eqJ} J_k(\mathcal{T}) = \sum_{\eta \in \mathcal{I}_{\mathcal{T}}} \Delta(\eta) \mathbh{1}_{[v(\eta) = k]},$$ where $v(\eta) \in \{1,\ldots, p\}$ is the splitting dimension of $\eta$ and $\mathcal{I}_{\mathcal{T}}$ is the set of all internal tree nodes (i.e., split locations). Through efficient storage of data and split rules, these indices are inexpensive to calculate for any given tree. Given a filtered set of trees, as described in Section \[secdynaTree\], the implied sample of $J_k$ indices provides a full posterior distribution of importance for each variable; this can form a basis for model-based selection. For $\Delta(\eta)$ we consider the decrease in predictive uncertainty associated with the split in $\eta$. In regression, the natural choice is the average reduction in predictive variance, $$\Delta(\eta) = \int_{A_\eta} \sigma_\eta^2( {\mathbf{x}}) \,d{\mathbf{x}} - \int_{A_{\eta_\ell}} \sigma_{\eta_\ell}^2( {\mathbf{x}}) \,d{\mathbf{x}} - \int_{A_{\eta_r}} \sigma_{\eta_r}^2( {\mathbf{x}}) \,d{\mathbf{x}}, \label{eqredvar}\vadjust{\goodbreak}$$ where $\eta_\ell$ and $\eta_r$ are $\eta$’s children, $\sigma_\eta^2({\mathbf{x}})$ is the predictive variance at ${\mathbf{x}}$ in the node $\eta$, and $A_\eta$ is the bounding covariate rectangle for that node. Rectangles on the boundary of the tree are constrained to the observed variable support, and, from recursive partitioning, $A_\eta = A_{\eta_\ell} \cup A_{\eta_r}$. For constant leaf-node models, each integral in (\[eqredvar\]) is simply the area of the appropriate rectangle multiplied by that node’s predictive variance. For classification, we replace the predictive variance at each node with the predictive entropy based on $\hat{p}_c$, the posterior predicted probability of each class $c$ in node $\eta$. This leads to the entropy reduction $\Delta(\eta) = |A_\eta| H_{\eta} - |A_{\eta_\ell}| H_\ell - |A_{\eta_r}| H_r$, where $H_\eta = - \sum_c \hat{p}_c \log \hat{p}_c$. Since the rectangle area calculations involve high-dimensional recursive partitioning and can be both computationally expensive and numerically unstable, a Monte Carlo alternative is to replace $|A_\eta|$ with $n$, the number of data points in $\eta$. We find that this provides a fast and accurate approximation. A regression tree with linear leaves presents a more complex setting, since the reduction in predictive variance is not constant over each partition. In Appendix \[seclinint\], we show that the calculations in (\[eqredvar\]) remain available in closed form. However, since in this case covariates also affect the response through the linear leaf model, (\[eqJ\]) provides only a partial measure of variable importance. In Section \[secSA\] we describe other sensitivity metrics whose interpretations do not depend on leaf model specification. Selecting variables ------------------- An $N$-particle posterior sample $\{J_k(\mathcal{T}_t^{(i)})_{i=1}^N\}$ can be used to assess the importance of each predictor $k=1,\ldots, p$, through both graphical visualization and ranking of summary statistics. As a basis for deselecting variables, we advocate estimated relevance probability, $\mathbb{P}(J_k(\mathcal{T}) > 0) \approx \frac{1}{N} \sum_{i=1}^N \mathbb{I}_{\{J_k(\mathcal{T}_t^{(i)}) > 0 \}}$.[^1] A backward selection procedure based on this criterion, illustrated in the examples below, is to repeatedly refit the trees after deselecting variables whose relevance probability is less than a certain threshold. ![Prior splitting frequencies (light) and probabilities of at least one split (dark) for a 10-dimensional input space plotted by sample size. Since all inputs feature equally in the random design, the results for just one input are shown.[]{data-label="fprior"}](590f02.eps) We use a default relevance threshold of 0.5, such that a variable’s relevance posterior must be less than 50% negative to entertain deselection. However, as we comment in Section \[secccache\], this can be problematic for some designs, for example, with many categorical predictors. A more conservative 0.95 threshold has analogy to the familiar 5% level for evidence in hypothesis testing, and can be appropriate in such settings. For guidance and intuition, one can refer to the prior distribution on the probability that the tree splits on a particular input. Note that this is not the same as a prior distribution on relevance, which does not exist under our improper leaf-model priors; rather, the probability of splitting on a given variable is its probability of having a nonzero relevance. Figure \[fprior\] plots the average number of splits (lighter) and the probability of at least one split (darker) using the four pairs of $(\alpha, \beta)$ values explored by @chipgeormccu2002, plus the `dynaTree` default values $(0.95, 2)$, as a function of the sample size obtained uniformly in $[0,1]^{10}$. These quantities stabilize after about $t=10$ samples and indicate that, *for this uniform design*, there is about a 12% prior probability of splitting. Ultimately, the backstop for a proposed deselection is the Bayes factor of the old (larger) model over the proposed (smaller) one, terminating the full procedure when proposals longer indicate a strong preference for the simpler model. Reliable marginal likelihoods are available through the sequential factorization $p(y^T |{\mathbf{x}}^T) \approx \frac{1}{N} \sum_{i=1}^N \sum_{t=1}^T \log p(y_t | {\mathbf{x}}_t, {\mathcal{T}}_{t-1}^{(i)})$ and lead to useful Bayes factor estimates \[see @taddgrapols2011\], as we shall demonstrate. Examples {#secVSexamples} -------- ### Simple synthetic data {#simple-synthetic-data .unnumbered} We consider data first used by @fried1991 to illustrate multivariate adaptive regression splines (MARS) and then used by @taddgrapols2011 to demonstrate the competitiveness of DTs relative to modern (batch) nonparametric models. The input space is ten-dimensional, however, the response, given by $10 \sin(\pi x_1 x_2) + 20(x_3 - 0.5)^2 + 10x_4 + 5 x_5$ with ${\mathrm{N}}(0,1)$ additive error, depends only on five of the predictors. Although the true function is additive in a certain transformation of the inputs, we do not presume to know that a priori in this illustration. A particle set of size $N= 10\mbox{,}000$ was used to fit the DT model to $T=$ 1000 input-output pairs sampled uniformly in $[0,1]^{10}$. Following @taddgrapols2011, we repeated the process ten times to understand the nature of the Monte Carlo error on our selection procedure. ![Variable selection in the Friedman data. The boxplots on the *left* show the posterior relevance. The *right* two plots show (log) Bayes factors, first for the full predictor set versus the set reduced to the five relevant variables, and then with a relevant variable removed.[]{data-label="ffried"}](590f03.eps) The results are summarized in Figure \[ffried\]. The boxplots on the left show the cumulative 100,000 samples of the tallied relevance statistics for each variable. The first five all have relevances above zero with at least 99% posterior probability. The latter five useless variables are easily identified, since their relevance statistics tightly straddle zero. They average about 35% relevance above zero, cleanly falling below 95% or 50% thresholds. (Figure \[fprior\] is matched to this input domain.) After removing these variables we reran the fitting procedure and calculated (log) Bayes factors, treating the smaller model as the null (i.e., in the denominator). All ten (log) paths (*center* panel) eventually indicate that the larger model is not supported by the data. In fact, a decreasing trend in the Bayes factor suggests that the smaller model is actually a better fit. Thus, while deselecting irrelevant variables is not technically necessary, doing so becomes increasingly important as the data length grows relative to a fixed-sized ($N$) particle cloud (i.e., in order to ward off particle depletion). The right panel in the figure shows the (log) Bayes factor calculation that would have resulted if we had further considered the first input for deselection (i.e., suggesting only inputs 2–5 were important). Clearly, the larger model (in the numerator) is strongly preferred. ### Spam data {#spam-data .unnumbered} We turn now to the Spambase data set from the UCI Machine Learning Repository \[@Asuncion+Newman2007\]. The aim is not only to illustrate our selection procedure in a classification context but also to scale up to larger $n$ and $p$ with significant interaction effects. The data contains binary classifications of 4601 emails based on 57 attributes (predictors). The left panel of Figure \[fspam\] shows the results of a Monte Carlo experiment based on misclassification rates obtained using random fivefold cross-validation training/testing sets. This was repeated twenty times for 100 training/testing sets total producing 100 rates. The comparators are modern, regularized logistic regression models, including fully Bayesian (“`b`”) and maximum a posteriori (“`map`”) estimators via Gibbs sampling \[@grapols2012\], an estimator from the `glmnet` package \[“`glmn`”; @friedhasttibsh2009\], and the EM-based method \[“`krish`”; @krishetal2005\]. Results for these comparators on an interaction-expanded set of approximately 1700 predictors are also provided. Expansion is crucial to realize good performance from the logistic models. ![(*Left panel*) Boxplots of misclassification rates divided into two sections, depending on absence or presence of interaction terms in the design matrix. (*Right panels*) Posterior samples of relevance statistics and their means.[]{data-label="fspam"}](590f04.eps) Our DT contributions are `dt` and `dt2`, each using $N=$1000 particles and 30 replicates, which took about half the execution time of the interaction-expanded logistic comparators. The `dt2` estimator is the result of a single iteration of the selection procedure outlined above using a 50% threshold (explained below), leveraging the $\{J_k\}$ obtained from the initial `dt` run. This usually resulted in 25 (of 57) deselections. The subsequent Bayes factor calculation(s) indicated a preference for the small model in every case considered. Notable results include the following. The DT-based estimators perform as well as the interaction-expanded linear model estimators, without explicitly using the expanded predictor set. Trees benefit from a natural ability to exploit interaction—even a few three-way interactions were found that, for the other comparators in the study, would have required an enormous expansion of the predictor space. Without modification, our new selection procedure simultaneously allows variables not helpful for main effects or interactions to be culled. Hence, the estimator obtained after deselection (`dt2`) is just as good as the former (`dt`, using the entire set of predictors) but with lower Monte Carlo error. In fact, based on the worst cases in the experiment, `dt2` is the best estimator in this study. We found that marginal reductions in Monte Carlo error can be obtained with further deselection stages. The right panels of Figure \[fspam\] show the posterior samples of the entropy difference tallies for predictors whose median relevance was greater than zero; also shown is the corresponding posterior means by which the samples have been ordered. A similar plot is given for random forests in HTF (Figure 10.6). Our ordering of importance is similar, but importance drops off quickly because our single-tree model is more parsimonious than are the additive trees of random forests. As an advantage of our approach, the middle figure shows posterior uncertainty around these means: there is a large amount of variability, and evidence of multicollinearity shows in any given parameter’s potential effect ranging from zero to very large. This observation and an effort to match the size of the predictor set selected by HTF both contributed to our choice of the 50% threshold. Sensitivity analysis {#secSA} ==================== The importance indices of Section \[secVS\] provide a computationally efficient measure of a covariate’s first-order effect—variance reduction directly attributed to splits on that variable. These indices are not, however, appropriate for all applications of sensitivity analysis. First, with nonconstant leaf prediction models, such as for linear trees, focusing only on variance reduction through splits ignores potential influence in the leaf model; for example, a covariate effect that is perfectly linear will lead to $J_k$ near zero if fit with linear trees. Second, the importance indices depend on the entire sample and cannot easily be focused on local input regions, say, for optimization. Third, the importance indices provide a measure that is clearly interpretable in the context of tree models but does not correspond to any of the generic covariance decompositions in standard input sensitivity analysis. In this section we describe a technique for Monte Carlo estimation of these decompositions, referred to as *sensitivity indices* in the literature, that is model-free and can be constrained to subsets of the input space. Sensitivity indices {#secST} ------------------- The classic paradigm of input sensitivity analysis involves analysis of response variability in terms of its conditional and marginal variance. This occurs in relation to a given *uncertainty distribution* on the inputs, labeled $U({\mathbf{x}})$. It can represent uncertainty about future values of ${\mathbf{x}}$ or the relative amount of research interest in various areas of the input space \[see @TaddLeeGrayGrif2009\]. In applications, $U$ is commonly set as a uniform distribution over a bounded input region. Although one can adapt the type of sampling described here to account for correlated inputs in $U$ \[e.g., @SaltTara2002\], we treat only the standard and computationally convenient independent specification, $U({\mathbf{x}}) = \prod_{k=1}^p u_k(x_k)$. The sensitivity index for a set of covariates measures the variance, with respect to $U$, in conditional expectation given those variables. For example, the two most commonly reported indices concern *first-order* and *total* sensitivity: $$S_j = \frac{{\mathrm{\mathbb{V}ar}}\{{\mathbb{E}}\{y | x_j \}\}}{{\mathrm{\mathbb{V}ar}}\{y\}} \quad\mbox{and}\quad T_j = \frac{{\mathbb{E}}\{{\mathrm{\mathbb{V}ar}}\{y | {\mathbf{x}}_{-j}\}\}}{{\mathrm{\mathbb{V}ar}}\{y\}},\qquad j =1,\ldots, p, \label{eqinds1}$$ respectively. The first-order index represents response sensitivity to variable main effects and is closest in spirit to the importance metrics of Section \[secVS\]. From the identity ${\mathbb{E}}\{ {\mathrm{\mathbb{V}ar}}\{ y | {\mathbf{x}}_{-j} \} \} = {\mathrm{\mathbb{V}ar}}\{y\} - {\mathrm{\mathbb{V}ar}}\{ {\mathbb{E}}\{ y | {\mathbf{x}}_{-j} \} \}$, $T_j$ measures residual variance in conditional expectation and thus represents all influence connected to a given variable. Hence, $T_j - S_j$ measures the variability in $y$ due to the interaction between input $j$ and the other inputs, and a large difference $T_j-S_j$ can trigger additional local analysis to determine its functional form. Note that all moments in (\[eqinds1\]) are with respect to $U$; additional modeling uncertainty about $y |{\mathbf{x}}$ is accounted for in posterior simulation of the indices. We propose a scheme based on integral approximations presented by @Salt2002. Extra steps are taken to account for an unknown response surface: “known” responses are replaced with predicted values. Subsequent integration is repeated across each tree in a particle representation of the posterior and then averaged over all particles. Although we focus on first-order and total sensitivity, full posterior indices for any covariate subset are available through analogous adaptation of the appropriate routines of @Salt2002. In the remainder of this section calculations are presented for a given individual tree; we suppress particle set indexing. Everything is conditional on a given posterior realization for $y({\mathbf{x}})$. We begin to integrate the common ${\mathbb{E}}^2\{y\}$ terms by recognizing that $$S_j = \frac{{\mathbb{E}}\{{\mathbb{E}}^2\{y|x_j\}\} - {\mathbb{E}}^2\{y\}}{{\mathrm{\mathbb{V}ar}}\{y\}}\quad \mbox{and}\quad T_j = 1 - \frac{{\mathbb{E}}\{{\mathbb{E}}^2\{y|x_{-j}\}\} - {\mathbb{E}}^2\{y\}}{{\mathrm{\mathbb{V}ar}}\{y\}}. \label{eqinds2}\hspace*{-15pt}$$ Assuming uncorrelated inputs, an approximation can be facilitated by taking two equal-sized random samples with respect to $U$. Although any sampling method respecting $U$ may be used, we follow @TaddLeeGrayGrif2009 and use a Latin hypercube design for the noncategorical inputs to obtain a cheap space-filling sample on the margins, thereby reducing the variance of the resulting indices. Specifically, we create designs $M$ and $M'$ each of size $m$, assembled as matrices comprising $p$-length row-vectors ${\mathbf{s}}_k$ and ${\mathbf{s}}_k'$, for $k=1,\ldots,m$, respectively. The unconditional quantities use $M$: $$\widehat{{\mathbb{E}}\{y\}} = \frac{1}{m} \sum_{k=1}^m {\mathbb{E}}\{y|{\mathbf{s}}_k\} \quad\mbox{and}\quad \widehat{{\mathrm{\mathbb{V}ar}}\{y\}} = \frac{1}{m} {\mathbb{E}}\{y|M\}^\top{\mathbb{E}}\{y|M\} - \widehat{{\mathbb{E}}^2\{y\}}, \label{eq}\hspace*{-35pt}$$ where ${\mathbb{E}}\{y|M\}$ is the column vector $ [{\mathbb{E}}\{y|{\mathbf{s}}_1\}, \ldots, {\mathbb{E}}\{y|{\mathbf{s}}_m\} ]^\top$ and $\widehat{{\mathbb{E}}^2\{y\}} = \widehat{{\mathbb{E}}\{y\}}\widehat{{\mathbb{E}}\{y\}}$. Approximating the remaining components in (\[eqinds2\]) involves mixing columns of $M'$ and $M$, which is where the independence assumption is crucial. Let $M'_j$ be $M'$ with the $j$th column replaced by the $j$th column of $M$, and likewise let $M_j$ be $M$ with the $j$th column of $M'$. The conditional second moments are then $$\begin{aligned} \qquad\widehat{{\mathbb{E}}\bigl\{{\mathbb{E}}^2\{y|x_j\} \bigr\}} &=& \frac{1}{m-1}{\mathbb{E}}\{y|M\}^\top{\mathbb{E}}\bigl\{y|M'_j \bigr\}, \nonumber\hspace*{-30pt} \\[-8pt] \\[-8pt] \nonumber \widehat{{\mathbb{E}}\bigl\{{\mathbb{E}}^2\{y|x_{-j}\}\bigr\}} &=& \frac{1}{m-1}{\mathbb{E}}\bigl\{y|M'\bigr\}^\top{\mathbb{E}}\{y|M_j\} \approx \frac{1}{m-1}{\mathbb{E}}\{y|M\}^\top{\mathbb{E}}\bigl \{y|M'_j\bigr\},\hspace*{-30pt}\end{aligned}$$ the latter approximation saving us the effort of predicting at the locations in $M_j$. In total, the set of input locations requiring evaluation under the predictive equations is the union of $M$, $M'$, and $\{M'_j\}_{j=1}^p$. For designs of size $m$ this is $m(p+2)$ locations for each of $N$ particles. Together $m$ and $N$ determine the accuracy of the approximation. Usually $N$ is fixed by other, more computationally expensive, particle updating considerations. Particle-wide application of the above provides a sample from the posterior distribution for ${\mathbf{S}}$ and ${\mathbf{T}}$. Visualization of main effects {#secme} ----------------------------- A byproduct of the above procedure is information that can be used to estimate main effects. For each particle and input direction $j$, we apply a simple one-dimensional smoothing of the scatterplot of $[s_{1j}, \ldots, s_{mj}, s'_{1j}, \ldots, s'_{mj}]$ versus $[{\mathbb{E}}\{y|M\}, {\mathbb{E}}\{y|M'\}]$. This provides a realization of ${\mathbb{E}}\{y|x_j\}$ over a grid of $x_j$ values and therefore a draw from the posterior of the main effect curve. Note that we use here the posterior means ${\mathbb{E}}[y |{\mathbf{s}}]$, as opposed to the posterior realizations for $y|{\mathbf{s}}$ used in calculating sensitivity indices. Average and quantile curves from each particle can then be used to visualize the posterior mean uncertainty for the effect of each input direction as a function of its value. One-dimensional curve estimation is robust to smoother choice in such a large sample size ($2m$); we use a simple moving average. Examples {#secSAexamples} -------- Consider again the Friedman data from Section \[secVSexamples\], using the first six inputs. Ordinarily we would recommend an initial selection procedure before undertaking further sensitivity analysis to eliminate all irrelevant variables, but we keep one irrelevant input for illustrative purposes. ![Main effects (*first row*) and ${\mathbf{S}}$ (*second row*) and ${\mathbf{T}}$ (*third row*) indices for the Friedman data using dynamic trees and GPs.[]{data-label="ffriedSA"}](590f05.eps) Figure \[ffriedSA\] summarizes the analysis under constant and linear DTs (DTC and DTL, resp.), and under a GP (fit using `tgp`) for comparison. In all three cases the number of particles (or MCMC samples for the GP) and samples from $U$ were the same: $N=10\mbox{,}000$ and $m=1000$, without replicates. The main effects for DTL and GP are essentially identical. As evidenced in the plots, DTC struggles to capture the marginal behavior of every input; $x_3$ is particularly off. These observations carry over to the ${\mathbf{S}}$ and ${\mathbf{T}}$ indices. DTL displays the same average values as does the GP, but with greater uncertainty. DTC again shows less agreement and greater uncertainty. Whereas DTC works well for variable selection, DTL seems better for decomposing the nature of variable influence. With DTL and GP providing such similar sensitivity indices, why should one bother with DTL? The answer rests in the computational expense of the two procedures. The DT fit and sensitivity calculation stages each take a few minutes. The GP version, even using a multithreaded version of `tgp`, takes about six hours on the same machine and requires that the two stages occur simultaneously. Hence, if new ${\mathbf{x}}-y$ pairs are added or a new $U$ is specified, the entire analysis must be rerun from scratch. With DTs, the fit can be updated in a matter of seconds, and only the sensitivity stage must be rerun, leading to even greater savings. In sum, the DT analysis can give similar results to GPs but is hundreds of times faster. GPs also are good (but even slower) at classification (GPC). Perhaps this is why we could not find GPC software providing input sensitivity indices for comparison. Figure \[fexpclass\] shows the results of a sensitivity analysis for a three-class/2D data set \[see @gramacypolson2011 for details and GPC references\]. Fitting a GPC model from 200 ${\mathbf{x}}-y$ pairs takes about an hour, for example, with the `plgp` package. By contrast, fitting a DT with multinomial leaves using $N=10\mbox{,}000$ particles takes a few seconds; and the sensitivity postprocessing steps, which must proceed separately for each class, take a couple of minutes. The MAP class labels and predictive entropy shown on the left panel indicate the nature of the surface. Notice that the entropy is high near the misclassified points (red dots). The smooth transitions are difficult to capture with axis-aligned splits. ![(*Left panel*) Posterior predictive mean and entropy; misclassified points are shown as red dots. (*Right panels*) Sensitivity main effects and ${\mathbf{S}}$ and ${\mathbf{T}}$ indices for each class. The black lines and boxplots correspond to input $x_1$, and the red ones to $x_2$.[]{data-label="fexpclass"}](590f06.eps) The plots in the right panels show the main effects and ${\mathbf{S}}$ and ${\mathbf{T}}$ indices for each class. All three sets of plots indicate a dominant $x_1$ influence, which conforms to intuition because that axis spans three labels whereas $x_2$ spans only two. Lower $S$ and $T$ values for $x_2$ provide further evidence that its contribution to the variance is partly coupled with that of $x_1$. A computer experiment: Optimizing linear algebra kernels {#secapp} ======================================================== We now examine the data generated by linear algebra kernels from @Balaprakash20112136. The execution times for these experiments were obtained on Fusion, a 320-node cluster at Argonne National Laboratory. Each compute node contained a 2.6 GHz Pentium Xeon 8-core processor with 36 GB of RAM. We focus here on the GESUMMV experiment. The results obtained for the other two kernels (MATMUL and TENSOR) we examined are similar and are therefore omitted because of space constraints. GESUMMV, from the updated BLAS library \[@UBLAS\], carries out a sum of dense matrix-vector multiplies. The tuning design variables considered consist of two loop-unrolling parameters taking integer values in $\{1,\ldots , 30\}$ and three binary parameters associated with performing scalar replacement, loop parallelization, and loop vectorization, respectively. Argonne allowed an exceptional amount of computing resources to be assigned to these and a suite of similar performance-tuning examples in order to study aspects of the tuning apparatus and to enable initial explorations into elements of the online optimization of executables such as the one we describe below. In particular, resources were allocated for transformation, compilation, and obtaining the timings of 35 repeated (on the same dedicated node) execution trials at each design point in a full enumeration of the GESUMMV design space. These tests incurred over 30 CPU-hours (roughly half of which were devoted to transformation or compilation and half to execution). Although well beyond an acceptable budget for a one-off optimized compilation procedure, results from exhaustive enumerations are vital for performance benchmarking of analyses such as ours. They allow us to compare our automated procedures, made on the basis of much smaller searches, with out-of-sample quantities. They also help build a library of “rules of thumb” and functional and design parameter characteristics that can be useful for priming future searches whose tuning variables and input source codes are similar to those of the fully enumerated experiments. The fully enumerated GESUMMV problem (as well as MATMUL and TENSOR) is relatively small from a performance-tuning perspective, and hence is a prime candidate for our validation and benchmarking purposes. In \[@PBSWBN11\], problems with up to $10^{53}$ design points are posed, clearly indicating that practical tuning will require sampling of only a very small portion of the total design space. The GESUMMV experiment is summarized as follows. Of the $2^{3}30^2=7200$ total design points, 199 resulted in a compilation error or an improper memory access and thus were deemed to violate a constraint on correctness. The resulting 245,035 (successful) runtimes were between 0.15 and 0.68 seconds, the mean and median both being 0.22 seconds. Our focus here is on a carefully chosen subset of this data, described below, comprising about 1% of the full set of runs. The intention is to simulate a realistic scenario wherein variable selection and sensitivity analysis techniques can reasonably be expected to add value to an automated tuning and compilation optimization. We begin by examining the extent of the cold-cache effect by using selection techniques. We then turn to a full analysis of the sensitivity to inputs, leading to a localization and subsequent optimization. Next we explore the extent to which one can learn about, and avoid, constraint violations. We conclude with an out-of-sample comparison between DTs and GPs. ![(*Left*) Histograms for 4 particular trials with respect to the order statistics on decreasing runtimes. (*Right*) Frequency of trial number that yielded the maximum of the 35 runtimes.[]{data-label="fcoldcache"}](590f07.eps) Cold-cache effect and variable selection {#secccache} ---------------------------------------- Figure \[fcoldcache\] illustrates the cold-cache effect over the fully enumerated data. The left plot shows four histograms counting the number (out of the 7001 input locations that did not result in a constraint violation) of times the first, second, seventeenth, and last of the trials resulted in the $\kappa$th largest runtime of the 35 trials performed. While the first trial stands out as the slowest, results for the three other trials indicate that this effect does not persist for later trials. That is, the second (17th or 35th) does not tend to have the second (17th or 35th) largest runtime. However, the right histogram in the figure clearly shows that lower trial numbers tend to yield the maximum execution time more frequently than do higher ones. These results make clear the existence of a marginal cold-cache effect; indeed, a paired $t$-test squarely rejects the null hypothesis that no marginal effect occurs. However, our interest lies in determining whether the effect is influential enough to warrant inclusion in a model for predicting runtimes. In particular, the absolute average distance from the maximum to median runtime (among 35 trials for each of the 7001 input configurations) is about 0.01, compared with the full difference between the maximum and minimum execution in the entire data set at 0.53. Given the effect’s low magnitude and our limited available degrees of freedom, it is not clear whether estimating the cold-cache effect is worthwhile in statistical prediction.=1 The remainder of this subsection and Section \[secsa2\] work with a maxmin space-filling subsampled design of size 500 from the 7001, and just the first 5 of the 35 replicates (together a 99% reduction in the size of the data). First, we consider the following experiment on a further subset of the data comprising the first trial and the last (fifth) trial for every input in the space-filling design (1000 runs in total). The five inputs were augmented with a sixth indicator, which is zero for those from the first trial and one for those from the fifth. If the cold-cache effect is statistically significant, then this experiment should reveal so. ![(*Top-Left*) Relevance for the five inputs, plus the cold-cache indicator (sixth input). (*Top-Right*) Sequential Bayes factors comparing the model with the sixth input to the one without. (*Bottom*) Prior splitting frequencies (light) and probabilities of at least one split (dark) for the real-valued inputs (*left*) and categorical ones (*right*).[]{data-label="frelevant"}](590f08.eps) Figure \[frelevant\] summarizes our results, based on a constant leaf model with 1000 particles and 30 replicates. The top-left panel shows the posterior relevance samples; the focus, for now, is on the relevance of the sixth input, which is small. The scale of the $y$-axis is, however, somewhat deceiving: the posterior probability that relevance is greater than zero is 0.58, with mean relevance of $2.6\times 10^{-6}$, indicating that the cold-cache may have a tiny but possibly significant effect. It is helpful to consult the prior inclusion probabilities for further guidance here. The bottom-right figure shows the Boolean predictor’s relevance to be approaching 20% as the sample size gets large, nearly twice that of our earlier regression example. The Bayes factor in the the right panel shows a gradually decreasing trend, signaling that the cold-cache predictor is not helpful. Before turning to SA, having decided to ignore the cold-cache effect based on the above analysis, we observe that input three also shows low—in fact, negative—relevance. A similar Bayes factor calculation (not shown) strongly indicated that it too could be dropped from the model. The remaining four inputs have much greater, and entirely positive, posterior relevance; Bayes factors (also not shown) reinforce that these predictors are important to obtain a good fit. ![Main effects (*first row*) and ${\mathbf{S}}$ (*second row*) and ${\mathbf{T}}$ (*third row*) indices for GESUMMV.[]{data-label="fgesummvSA"}](590f09.eps) Sensitivity analysis {#secsa2} -------------------- To further inform an optimization of the automatic code tuning process, we perform a SA. Figure \[fgesummvSA\] summarizes main effects and ${\mathbf{S}}$ and ${\mathbf{T}}$ indices for the four remaining variables. The full reduced design (all five trials) was used—25,000 input-output pairs total, ignoring the cold-cache effect. Results for both constant and linear leaf models are shown. In contrast to our earlier results for the Friedman data, the differences between linear and constant leaves are negligible. Perhaps this is not surprising since both treat binary predictors identically.[^2] The ${\mathbf{S}}$ and ${\mathbf{T}}$ indices on the remaining predictors tell a similar, but richer, story compared with the relevance statistics. Input two has the largest effect, and input four the smallest, but we also see that the effect of the inputs, marginally, is small (since the $S$s are low and the $T$s are high). This result would lead us to doubt that a rule of thumb for optimizing the codes based on the main effects alone would bear fruit, namely, that inputs close to $\langle x_1=5, x_2=12, x_3=0\rangle$ are most promising. Although this may be a sensible place to start, intricate interactions among the variables, as suggested by the ${\mathbf{T}}$ indices in particular, mean that a search for optimal tuning parameters may benefit from a methodical iterative approach, say, with an expected improvement (EI) criterion \[@jonesschonlauwelch1998\] or another optimization routine. Before launching headlong in that direction, however, we first illustrate how a more localized sensitivity analysis may be performed without revisiting the computations used in the fitting procedure. The result can either be cached to prime future code optimizations having similar inputs or to initialize an iterative EI-like search on a dramatically reduced search space.=-1 ![Close-up of (constrained) main effects. ${\mathbf{S}}$ and ${\mathbf{T}}$ are similar to Figure \[fgesummvSA\].[]{data-label="fgesummvSAl"}](590f10.eps) Figure \[fgesummvSAl\] shows the main effects from a new sensitivity analysis (using DTC) whose uncertainty distribution $U'({\mathbf{x}})$ is constrained so that the first input is $\leq 15$, the second is $\geq 5$, and the third is fixed to zero (representing a tenfold reduction in the number of possible design points). The relevance indices indicated importance of the fourth input, so we allowed it to vary unrestricted in $U'$, suspecting that localizing the first three inputs might yield a more pronounced effect for the fourth. Note that only $U'({\mathbf{x}})$ is restricted, not the actual input-output pairs, and that the model fitting does not need to be rerun. In contrast to the conclusion drawn from Figure \[fgesummvSA\], the localized analysis strongly indicates that $x_4 = 0$ is required for a locally optimal solution. The other two inputs have a smaller marginal effect, locally. A finer iterative search may be useful for choosing among the $\geq 2$ local minima in the first two inputs. Many optimization methods are viable at this stage. We prefer to stay within the SMC framework, allowing thrifty DT updates to pick up where the size 500 space-filling design left off. Each subsequent design point is chosen by using a tree-based EI criterion \[@taddgrapols2011\] evaluated on all remaining candidates that meet criteria suggested by our final, zoomed-in, analysis, namely, all unevaluated locations from the fully-enumerated set having $\langle x_3, x_4 \rangle = \langle 0, 0 \rangle$ and $\langle x_1, x_2 \rangle \in [2, 12] \times [11,24]$. Ignoring the irrelevant third input, this results in 98% reduction in the search space compared with the original, fully enumerated design. After 100 such updates, an evaluation of the predictive distribution on the full 7001 design led to selecting $x^* = \langle 4, 22, 0, 0 \rangle$, giving a mean execution time of $17.9$. ![Histograms (same on *left* and *right* but with different $y$-axes) of the median of the 35 runs of each of 7001 non-`NA` evaluations, shown with the predicted execution time of $x^*$ found via localized EI (red dot). The *right* panel includes a kernel density estimate of the predicted responses at the localized design, $\langle x_3, x_4 \rangle = \langle 0, 0 \rangle$ and $\langle x_1, x_2 \rangle \in [2, 12] \times [11,24]$.[]{data-label="fopt"}](590f11.eps) Figure \[fopt\] shows how this solution is better than 98% of the median of the 35 runs from the fully enumerated set. Both panels show the same histogram of those times, with a red dot indicating $\hat{y}(x^*)$. The right panel augments with the kernel density of the predicted responses at the reduced/zoomed-in design, indicating the value of our variable selection and sensitivity pre-analysis. Even choosing $x^*$ uniformly at random in this region provides an output that is better than 88% of the total options. The final EI-based optimization ices the cake. Since it takes just seconds to perform, it represents an operation that, when more complete optimization is desired, can be bolted on at compile time for slight variations of the input source: say, for differently-sized matrices. The “compiler” could call up our reduced GESUMMV design and perform a quick search on the new input matrices. Constraint violation patterns and out-of-sample accuracy -------------------------------------------------------- We return to the original, fully enumerated design to check two possibilities: (1) whether Argonne engineers unknowingly created an inefficient timing experiment (i.e., with predictable regions of code failure); and (2) whether a GP-based analysis would have led to more accurate predictions, and subsequently a better variable selection, sensitivity analysis, and optimization, if a vastly greater computational resource had been available. For (1), the original 7200 design points, with (two) classification labels indicating `NA` values or positive real numbers (times), were used to fit a DT model with multinomial leaves. Otherwise the setup was similar to our earlier examples. The data has the feature that if one of the 35 trials was `NA`, then they all were. The reason is that failures were due to the transformed code failing to compile (precluding the code from running at all) or resulting in a segmentation fault upon execution. Other failures (e.g., due to hardware failures, soft faults, or the computed quantities differing more than a certain tolerance from reference quantities) are possible in practice but were not seen in the present data. Sequential updating of the DT classification model revealed a posterior distribution of the importance indices that was decidedly null. The importance probabilities (i.e., of having a positive index) were $0.003, 0.026, 0.000,\break 0.000$, and $0.000$ for the five inputs, respectively. We interpret these results as meaning that the DT model detects no spatial pattern in the 2.7% of code failures compared with the successful runs. This conclusion was backed up by a simple Bayes factor calculation where the null model disallowed any partitioning. These results are reassuring because the input space was designed to limit the number of correctness violations; if relationships between the inputs and these violations were known a priori, the design space would be adjusted accordingly to prevent failures at compile or runtime. For (2), we performed a 100-fold Monte Carlo experiment. In each fold, a DT constant model and a DT linear model were trained on a random maxmin design of size 100 subsampled from the fully enumerated 7001 locations using the first five replicates. Besides the smaller design, the SMC setup is identical to the one described earlier in this section. Then, a Bayesian GP with a separable correlation function and nugget was also fit (using the `tgp` package) by using a number of MCMC iterations deemed to give good mixing. The resulting computational effort was about ten times greater than for the DT fit, owing to the $100 \times 100$ matrices that required repeated inversion. We originally hoped to do a size 500 design as in the preceding discussion, but the $2500 \times 2500$ inversions were computationally infeasible in such nested Monte Carlo repetition. Finally, a BART model was trained using commensurate MCMC settings. For validation of the models in each fold a random maxmin testing design of size 100 was subsampled from the remaining 6901 locations. RMSEs were obtained by first calculating the squared deviation from the posterior mean predictors to actual timings of the 35 execution replicates associated with each testing location. The square root of the average of the 350 distances was then recorded. ------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------ ![RMSE comparison on GESUMMV Monte Carlo experiment, by boxplot (*left*) and empirical quantiles (*right*).[]{data-label="frmse"}](590f12.eps) **RMSE** **DTC** **DTL** **GP** **BART** ---------- --------- --------- -------- ---------- 5% 0.0398 0.0373 0.0394 0.0462 50% 0.0525 0.0503 0.0527 0.0591 95% 0.0675 0.0661 0.0668 0.0750 ------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------ Figure \[frmse\] summarizes RMSEs by boxplot and numbers: median and 90% quantiles. The absolute performance of the DT and GP methods are strikingly similar. In pairwise comparison, however, DTL is better than the DTC and GP comparators 85% and 71% of the time, respectively, emerging as a clear winner. Therefore, thrifty sequential variable selection, sensitivity analysis, and EI-based optimization notwithstanding, a DT can be at least as good as the canonical Gaussian process response model for computer experiments. This performance may be due to a slight nonstationarity or heteroskedasticity, which cannot be accommodated by the stationary GP. BART was included as a comparator to further explore this aspect. As noted by \[@taddgrapols2011\], BART will tend to outperform DTs (and sometimes GPs) when there is nonstationarity or nonsmoothness in the mean, but not the variance (i.e., under homoskedastic noise). The opposite is true in the heteroskedastic noise case, and this is what we observe here. DTC and DTL have lower RMSEs than BART 92% and 98% of the time, respectively. These results suggest our execution-times data may benefit from methods that can accommodate input-dependent noise. Discussion {#secdiscuss} ========== The advent of fast and cheap computers defined a statistical era in the late 20th century, especially for Bayesian inference. For computer experiments and other spatial data, modestly-sized data sets and clever algorithms allowed the use of extremely flexible nonparametric models. GP models typify the state of the art from that era, with many successful applications. In classification problems, latent variables were key to exploiting computation for modeling flexibility. Today, further technological advance is defining a new era, that of massive data generation and collection where computer and physical observables are gathered at breakneck pace. These huge data sets are testing the limits of the popular models and implementations. GPs are buckling under the weight of enormous matrix inverses, and latent variable models suffer from mixing (MCMC) problems. Although exciting inroads have recently been made toward computationally tractable, approximate GP (regression) inference in large data settings \[e.g., @haalandqian2012, @sanghuang2012\], their application to canonical computer experiments problems such as design and optimization remains a topic of future study. In this paper we suggest that the new method of dynamic regression trees, an update of classic partition tree techniques, has merit as an efficient alternative in nonparametric modeling. In particular, we perform many of the same experiment-analysis functions as do GP and latent variable models, at a fraction of the computational cost. By borrowing relevance statistics from classical trees and sensitivity indices from GPs, the end product is an exploratory data analysis tool that can facilitate variable selection, dimension reduction, and visualization. An open-source implementation is provided in a recent update of the `dynaTree` package for . Our illustrations included data sets from the recent literature and a new computer experiment on automatic code generation that is likely to be a hot application area for statistics and other disciplines as heterogeneous computing environments become more commonplace. Ultimately, the goal is to optimize code for the architecture “just in time,” when it arrives at the computing node. In order to be realistically achievable, that goal will require rules of thumb, as facilitated by selection and sensitivity procedures like those outlined in this paper, and iterative optimization steps like the EI approach we illustrated. We note that the input space for these types of experiments can, in practice, be much larger than the specific ones we study; indeed, the median size of the problems presented in \[@PBSWBN11\] is more than $10^{15}$ input configurations. This makes enumeration prohibitively expensive even for academic purposes, irregardless of acceptable compilation times. In those cases, variable reductions and localizations on the order of those we provide here will be crucial to enable any study of the search space, let alone a subsequent optimization. \[app\] Input analysis ============== Variable selection is largely equated with setting coefficients to zero. Hence, the approaches are predicated on a specific, usually additive, form for the influence of covariates on response. In the analysis of computer experiments, for example, @cantoniflemmingronchetti2011 and @maitylin2011 use the nonnegative garrote \[NNG, @breiman1995\] and @huanghorowitzwei2010 apply grouped LASSO. An advantage of these approaches is that they can leverage off-the-shelf software for variable selection. However, because of the complexity of the modeled processes and a need for high precision, researchers using statistical emulation for engineering processes are seldom content with a single additive regression structure for the entire input space. Moreover, the consideration of interaction terms in additive models can require huge, overcomplete bases, typically leading to burdensome computation. As a result, it is more common to rely on GP priors or other nonparametric regression techniques \[e.g., @BayaBergKennKott2009, @SansLeeZhouHigd2008\]. However, such modeling significantly complicates the task of selecting relevant variables. Although several approaches have been explored in recent literature \[e.g., @linkletteretal2006, @bastosohagan2009, @yishichoi2011, and references therein\], their complexity seems to have precluded the release of software for use by practitioners. An interesting middle ground is considered by @reichstorliebondell2009, who propose an additive model comprising univariate functions of each predictor and bivariate functions for all interactive pairs. Each is given a GP prior, and there is a catch-all (higher interactions) remainder term. This extends previous work wherein $B$-splines were proposed for a similar task \[e.g., @gu2002\]. Stochastic-search variable selection \[SSVS, @georgemccu1993\] is used for selecting main effects and interactions. Although perhaps more straightforward than performing SSVS directly on the lengthscale parameters of a GP \[@linkletteretal2006\], this approach has the added computational complexity of inverting many $(\mathcal{O}(m^2)) n \times n$ covariance matrices. Instead of a dedicated variable selection procedure, engineering simulators typically employ some form of input sensitivity analysis. Classically, as in examples from @SaltChanScot2000, @SaltEtAl2008, running the computer code to obtain a response is presumed to be cheap. When it is expensive, one must emulate the code with an estimated probability model \[see @santwillnotz2003 for an overview\]. In turn, researchers have proposed a variety of schemes for extension of classic sensitivity analysis to account for response surface uncertainty. GPs, because of their role as the canonical choice for modeling computer experiments, are combined with sensitivity analysis in applications \[e.g., @ziehntomlin2009 [@marelletal2009]\]. However, the associated methodology is usually based on restrictive stationarity and homoskedasticity assumptions needed to derive either empirical Bayes \[@oakleyohagan2004\] or fully Bayesian \[@MorrKottTaddFurfGana2008 [@farahkottas2011]\] estimates of sensitivity indices. Notable exceptions are presented by @storlieetal2009 and @TaddLeeGrayGrif2009. In the former, approximate bootstrap confidence intervals are derived for sensitivity indices based on a nonparametrically modeled response surface. In the latter, variability integration embedded within MCMC simulation yields samples from sensitivity indices’ full posterior distribution; a similar idea forms the basis for our framework in Section \[secSA\]. Partition trees \[e.g., CART: @brei1984\] provide a basis for regression that has both a simplicity amenable to selecting variables and the flexibility required for modeling computer experiments. Furthermore, partition trees overcome some well-known drawbacks of the more commonly applied GP computer emulators: expensive $\mathcal{O}(n^3)$ matrix inversion, involving special consideration for categorical predictors and responses and allowing for the possibility of nonstationarity in the response or heteroskedastic errors. @taddgrapols2011 provide extensive background on general tree-based regression and argue for its wider adoption in engineering applications. In the context of this paper’s goals, trees present a unique, nonadditive foundation for determining variable relevance. In their simplest form, with constant mean response at the tree leaves, variable selection is automatic: if a variable is never used to define a tree partition, it has been effectively removed from the regression. Indeed, this idea motivated some of the earliest work on the use of trees, as presented by @MorgSonq1963, for automatic interaction detection. In a more nuanced approach, @brei1984 introduce indices of variable importance that measure squared error reduction due to tree-splits defined on each covariate. @hastietibshfried2009 \[HTF; ([-@hastietibshfried2009]), Chapter 10\] promote these indices for sensitivity analysis and describe how the approach can be extended for their boosted trees. However, these techniques are purely algorithmic and lack a full probability model, hence, their use is especially problematic in analysis of computer experiments, where uncertainty quantification is often a primary objective. Moreover, the HTF importance indices are only point estimates of the underlying sensitivity metrics, thus, they preclude basing the variable selection criteria on posterior evidence and make it difficult to properly deduce and interpret just how each variable is contributing. Researchers have attempted to overcome some of these shortcomings through the use of Bayesian inference, most recently in schemes that augment the tree model to allow for better control or flexibility. @ChipGeorMcCu2010 describe a Bayesian additive regression tree (BART) model, and their `BayesTree` software includes a direct analogue of the HTF importance indices; and the method of @TaddLeeGrayGrif2009 is implemented in `tgp` \[see @gramacytaddy2010, Section 3\]. Variance integral for linear leaves {#seclinint} =================================== Here, we derive the variance integrals from (\[eqredvar\]) for a model with linear leaves. Dropping the node subscript ($\eta$, $\eta_\ell$, or $\eta_r$), we have $$\begin{aligned} \label{eqs2lin} \int_A \sigma^2({\mathbf{x}}) \,d{\mathbf{x}} &=& \int _A \frac{s^2 - \mathcal{R}}{n-p-1} \biggl(1 + \frac{1}{n} + {\mathbf{x}}^\top \mathcal{G}^{-1} {\mathbf{x}} \biggr) \,d{\mathbf{x}} \nonumber \\[-8pt] \\[-8pt] \nonumber &= &\frac{s^2 - \mathcal{R}}{n-p-1} \biggl(|A| \biggl(1 + \frac{1}{n} \biggr) + \int _A {\mathbf{x}}^\top \mathcal{G}^{-1} {\mathbf{x}} \,d{\mathbf{x}} \biggr), \nonumber\end{aligned}$$ where $s^2$ is the sum of squares, $\mathcal{R}$ is the regression sum of squares, $n\equiv|\eta|$ is the number of $({\mathbf{x}},y)$ pairs, $\mathcal{G}$ is the Gram matrix, and $|A|$ is the area of the rectangle. The remaining integral is just a sum of polynomials: with the intervals outlining the rectangle given by $(a_1,b_1),\ldots,(a_p, b_p)$ and $(g_{ij})$ the components of $\mathcal{G}^{-1}$, $$\begin{aligned} \label{eqint} \int_A {\mathbf{x}}^\top \mathcal{G}^{-1} {\mathbf{x}} \,d{\mathbf{x}} &=& \int_{a_1}^{b_1} \cdots \int _{a_p}^{b_p} \sum_{i=1}^p \sum_{j=1}^p x_i x_j g_{ij} \,d x_i \nonumber\\[-2pt] &=& \sum_{i=1}^p \frac{g_{ii}}{3} \bigl(b_i^3 - a_i^3\bigr) \prod _{k\ne i} (b_k - a_k) \nonumber \\[-9pt] \\[-9pt] \nonumber &&{}+ 2 \sum _{i=1}^p \sum_{j > i} \frac{g_{ij}}{4}\bigl(b_i^2 - a_i^2 \bigr) \bigl(b_j^2 - a_j^2\bigr) \prod_{k \ne i,j} (b_k - a_k) \\[-2pt] &=& |A| \Biggl( \sum _{i=1}^p \frac{g_{ii}(b_i^3 - a_i^3)}{3 (b_i - a_i)} + \sum _{i=1}^p \sum_{j > i} \frac{g_{ij} (b_i+ a_i)(b_j+a_j)}{2} \Biggr). \nonumber\end{aligned}$$ Acknowledgments {#secack .unnumbered} =============== We are grateful to Prasanna Balaprakash for providing the data from \[@Balaprakash20112136\]. Many thanks to the Editor, Associate Editor, and two referees for their valuable comments, which led to many improvements. [49]{} (). , (). . , (). . . (). . . , , , , , , , (). . . , , , , , , , , , , , (). . . (). . . , , (). . , . , (). . . , , (). . . , (). . . , (). . . , (). . . (). (). . . , (). . . (). . . (). . . (). . . (). . . (). . (). . , . (). . . , (). . In . , . , (). , ed. , . , (). . . , (). . . , , (). . . , , (). . . , , , (). . . (). . . , , (). . . (). . . , , , (). . . (). . . (). , ed. , . , (). . . (). . . , , eds. (). . , . (). . . , , , , , , (). . , . (). . . , (). . , . , , (). . . , (). . . , , (). . . , (). . . (). . . [^1]: Note that $\Delta(\eta)$, and thus $J_k(\mathcal{T}_t^{(i)})$ for particle $i$, may be negative for some $\eta\in \mathcal{I}_{\mathcal{T}_t^{(i)}}$ because of the uncertainty inherent in a Monte Carlo posterior sample. [^2]: It is important to flag the two remaining binary inputs as categorical in the `dynaTree` software when using the linear leaf model. This allows splitting on the binary input as usual but removes such inputs from the within-leaf calculations so that the resulting Gram matrices are nonsingular.
--- abstract: 'The stellar disk size of a galaxy depends on the ratio of the disk stellar mass to the halo mass, $m_\star \equiv M_\star/M_{\rm dh}$, and the fraction of the dark halo angular momentum transferred to the stellar disk, $j_\star \equiv J_\star/J_{\rm dh}$. Since $m_{\star}$ and $j_{\star}$ are determined by many star-formation related processes, measuring $j_\star$ and $m_\star$ at various redshifts is essential to understand the formation history of disk galaxies. We use the 3D-HST GOODS-S, COSMOS, and AEGIS imaging data and photo-$z$ catalog to examine $j_\star$ and $m_\star$ for star-forming galaxies at $z \sim$ 2, 3, and 4, when disks are actively forming. We find that the $j_\star/m_\star$ ratio is $\simeq 0.77\pm 0.06$ for all three redshifts over the entire mass range examined, $8\times 10^{10} < M_{\rm dh}/h^{-1} M_\odot < 2\times 10^{12}$, with a possible ($<30\%$) decrease with mass. This high ratio is close to those of local disk galaxies, descendants of our galaxies in terms of $M_{\rm dh}$ growth, implying a nearly constant $j_\star/m_\star$ over past 12 Gyr. These results are remarkable because mechanisms controlling angular momentum transfer to disks such as inflows and feedbacks depend on both cosmic time and halo mass and indeed theoretical studies tend to predict $j_\star/m_\star$ changing with redshift and mass. It is found that recent theoretical galaxy formation simulations predict smaller $j_{\star}/m_{\star}$ than our values. We also find that a significant fraction of our galaxies appears to be unstable against bar formation.' author: - 'Taku Okamura, Kazuhiro Shimasaku, and Ryota Kawamata' bibliography: - 'bibtex.bib' title: Angular momentum evolution of stellar disks at high redshifts --- Introduction {#sec1} ============ Within the $\Lambda$CDM paradigm, galaxies form in the center of hierarchically growing dark matter halos [@fall80]. In the tidal torque theory, gases and dark matter halos acquire angular momentum with log-normal distributions of the spin parameter through tidal gravitational fields [@peebles69]. The dimensionless spin parameter is given by $$\begin{aligned} \lambda \equiv \frac{J |E|^{1/2}}{GM^{5/2}},\end{aligned}$$ where $J$, $E$, and $M$ are the total angular momentum, total energy, and total mass of the system. Since gases and halos share initial tidal torque fields, it is expected that gases and dark matter halos have the same amount of specific angular momentum. Gases gradually radiate away the thermal energy and then cool and collapse toward the center of dark matter halos. Their angular momentum halts the collapse and leads to a rotationally supported disk galaxy [@fall80; @white91; @mo98]. In this formation scenario of disk galaxies, the disk size of a galaxy ($r_{\rm d}$) is given by $$\begin{aligned} r_{\rm d} = \frac{1.678}{\sqrt{2}}\left(\frac{j_{\rm d}}{m_{\rm d}}\right)\lambda r_{200} f_{\rm c}(c_{\rm vir})^{-1/2} f_{\rm R}(\lambda, c_{\rm vir}, m_{\rm d}, j_{\rm d}), \nonumber \\ \label{eq1_1} \end{aligned}$$ [@mo98]. Here $j_{\rm d}/m_{\rm d}$ $( j_{\rm d} \equiv J_{\rm d}/J_{\rm dh},\ m_{\rm d} \equiv M_{\rm d} / M_{\rm dh}; \ {\rm d}:$star+gas) is the angular momentum retention factor and displays how much angular momentum acquired via tidal torques is conserved during the disk formation, $r_{200}$ is the radius of the dark matter halo within which encloses 200 times critical density, and $f_{\rm c}$ and $f_{\rm R}$ show, respectively, the difference in the density profile from an exponential profile and the gravitational effect of the disk. By assuming that the angular momentum of a disk is fully conserved, $j_{\rm d}/m_{\rm d}=1$, this model successfully reproduces scaling relations of local disk galaxies: the stellar mass–size relation and the stellar mass–size scatter relation [@mo98; @dutton12; @fall12]. Because of the success of this picture, this model has been adopted in many semi-analytical models [e.g. @somerville08; @porter14; @croton16]. However, the assumption that $j_{\rm d}/m_{\rm d}$ equals to unity independent of mass and cosmic time is not trivial, because highly-complex baryonic processes such as cooling, dynamical friction, and various feedback processes can change the specific angular momentum of disk galaxies. These processes are closely dependent on the mass of host dark matter halos. For example, the mass of dark matter halos controls how much expelled gases, which exchange the angular momentum with hot halo gases, can return to the galaxies again. The accumulation of such processes may increase or decrease the disk specific angular momentum. This is why the information of angular momentum is essential for comprehensive understanding of galaxy formation and evolution. It is important to understand the evolution of the angular momentum of galaxies as a function of dark halo mass at various redshifts. In the present-day universe, since the pioneer work of @fall83, the angular momenta of galaxies with various morphological types and masses have been studied by observations and cosmological simulations [e.g. @steinmetz99; @governato07; @fall12; @fall13]. @fall12 and @fall13 have extended and updated the study of @fall83 with recent observational data. They have found that the specific angular momenta of spiral galaxies are not conserved, with $j_{\rm d}/ m_{\rm d} \simeq 0.6$ independent of halo mass. This implies that some baryonic processes mentioned above decrease the disk specific angular momentum. Recent semi-analytical and hydrodynamical galaxy formation models have also obtained low angular momentum retention factors [@sales12; @colin16; @stevens16]. The roles of baryonic processes that determine the disk specific angular momentum have been examined: they include various types of feedback processes and the formation of bulges by disk instabilities. On the other hand, beyond $z\sim1$, there are only a few studies that have observationally examined the specific angular momentum of galaxies because of the difficulty in obtaining kinematic measurements. @burkert16 have analyzed the angular momenta of 359 disk star-forming galaxies at $z\sim 0.8-2.6$ and found $j_{\rm d}/m_{\rm d} \simeq 1$. @contini16 have found in 28 low mass galaxies at $z\sim1$ almost the same stellar mass–angular momentum relation as the local one. However, some semi-analytical and hydrodynamical models predict that disk galaxies at $z\sim 1$ have smaller specific angular momenta than local galaxies [e.g. @sales12; @pedrosa15; @stevens16]. Some results of cosmological galaxy formation simulations support the picture in which disk galaxies gradually acquire specific angular momentum as they grow. A consensus has not been reached on the angular momentum evolution beyond $z\sim1$. More observational data are needed to test the model predictions. In this paper, to tackle the issue of the angular momentum evolution of disk galaxies and understand the formation and evolution of galaxy disks, we study the relation between the fraction of the dark-halo angular momentum transferred to the stellar disk ($j_{\star}:\ \star:$star) and the stellar to dark matter halo mass ratio ($m_{\star}$) at $z\sim$ 2, 3, and 4. We estimate dark halo masses by two independent methods: clustering analysis and abundance matching technique. In order to measure $j_{\star}$, it is popular to analyze galaxy kinematics with spectroscopy. However, it is very difficult to construct a large spectroscopic sample at high redshifts. Instead, we make use of the analytical model of @mo98 that connects disk size with angular momentum. By measuring the disk sizes of galaxies and assuming this analytic model, we estimate $j_{\star}$. @kravtsov13 has investigated stellar disk size to halo size ratios $(r_{\rm d}/r_{\rm dh})$, which also reflect angular momentum retention factors, for local galaxies with a similar approach. @kawamata15 and @shibuya15 have extended his study to high redshift galaxies and found that the disk size to halo size ratios are almost flat out to high redshift. Recently, @huang17 and @somerville17 have examined the disk size to halo size ratios as a function of stellar mass in more detail out to $z\sim3$ from CANDELS surveys using abundance matching. They have found that the disk sizes are proportional to the halo sizes from $z\sim0-3$ and the ratios slightly decrease toward $z\sim0$ and high stellar masses. Our studies are complementary to these studies. There are some new aspects in our work. We study the mass–angular momentum relation at high redshift. Moreover, while all previous studies have used abundance matching analysis, we use clustering analysis, which is independent of abundance matching analysis to estimate dark halo masses. We also compare our results with recent cosmological galaxy formation simulations. The structure of our paper is as follows. In Section 2, we construct galaxy samples for this study. After measuring sizes in Section 3, we derive the stellar mass–disk size relation at each redshift bin in Section 4. The evolution of disk sizes is also discussed. In Section 5, we estimate dark halo masses from clustering analysis and abundance matching results. In Section 6, we present $j_{\star}$ and $m_{\star}$ estimates and compare them with recent cosmological galaxy formation simulations. Disk instabilities are also discussed. Conclusions are shown in Section \[seq\_con\]. Throughout this paper, we adopt the cosmology ($\Omega_{\rm m}, \Omega_{\Lambda}, h$, $\sigma_8$) = (0.3, 0.7, 0.7, 0.8). Magnitudes are in the AB system [@oke83]. Galaxy sizes are given in the physical scale. Data and samples {#chap:data} ================ Data ---- We use data from the 3D-HST and CANDELS programs [@grogin11; @koekeoer11; @brammer12; @skelton14]. @skelton14 provide a photometric catalog of the 3D-HST and CANDELS imaging data for five sky fields (COSMOS, GOODS-North, GOODS-South, AEGIS, and UDS) with a total area of $\sim 900 \rm\ arcmin^2$. As these fields have wealthy available data of optical to near-infrared broadband photometry, one can obtain a precise spectral energy distribution (SED) for many high-redshift galaxies. The number of optical to near-infrared broadband filters ranges from 18 in UDS up to 44 in COSMOS. We make use of photometric redshift, stellar mass, and star formation rates (SFR), all of which are available through the 3D-HST Web site. Sources have been detected with $\tt{SExtractor}$ [@bertin96] from the combined F125W, F140W, and F160W images. Among the five fields we only use COSMOS, GOODS-South, and AEGIS fields because the clustering properties of galaxies in the remaining two fields appear to largely deviate from the cosmic average as detailed in Appendix. Photometric redshifts have been determined from the $\tt{EAZY}$ [@brammer08] package, a public photometric redshift code. From the output catalog of $\tt{EAZY}$, we adopt $\tt{z\_peak}$ as photometric redshifts. Stellar masses and SFRs have been obtained by using the $\tt{FAST}$ code [@kriek09]. See @skelton14 for details of the procedure. In this paper, we assume a @chabrier03 initial mass function (IMF). From here, we take photometric redshifts as redshifts. Sample selection ---------------- We limit our sample to $H_{160} < 26.0$, which is nearly equal to the 5$\sigma$ complete magnitude in the shallowest field COSMOS [@skelton14]. As size measurements need images with high signal to noise ratios (S/N), the 5$\sigma$ limit is marginally acceptable and slightly shallower compared to other size measurement studies [@vanderwel14; @shibuya15]. Stellar masses are limited to $M_{\star} > 10^{8.3} M_{\odot}$. In the $H_{160}$–$M_{\star}$ diagram (Figure \[fig2\_1\]), stellar masses are largely complete down to $M_{\star} \simeq 10^{9.0} M_{\odot}$ for $z\sim2$ and down to $M_{\star} \simeq 10^{10}M_{\odot}$ for $z\sim3$ and 4. Below those values, our samples are biased toward low $M/L$ galaxies. We exclude galaxies with $M_{\star} > 10^{10.4} M_{\odot}$ from our samples for $z\sim3$ and $4$ because the number of galaxies is insufficient for clustering analysis. We use the stellar mass–SFR diagram to remove quiescent galaxies. On the basis of the stellar masses and the SFRs obtained from the $\tt{FAST}$, we construct stellar mass–SFR diagrams for our samples, as shown in Figure \[fig2\_2\]. First, we fit the stellar mass–SFR distribution by a power law, which defines the main-sequence. At $z\sim$ 2 and 3, galaxies that lie above the $-2\sigma$ of the main-sequence are considered to be star-forming galaxies, where the standard deviation of the MS is $\sigma \simeq 0.33$ dex for both redshifts. For $z\sim4$, we remove galaxies that have small SFRs by eye. In this paper, we do not consider the effects of bulges because main sequence galaxies above $z\sim2$ have low B/T ratios [@brennan17]. We exclude regions that have a shallow or deep exposure time for each field because clustering analysis requires images with a uniform depth. We also construct masks to avoid the vicinity of bright stars and diffraction spicks. For each redshift, we divide the entire sample into four ($z\sim$ 2) or three ($z\sim$ 3 and 4) subsamples according to stellar mass. The number of galaxies in the final samples is summarized in Table \[table2\_1\]. [ccrr]{} 2.0 &$10.4 - 11.1$ & 264 & 198\ & $9.7-10.4$ & 1086 & 870\ & $9.0-9.7$ & 3267 & 2458\ & $8.3-9.0$ & 3173 & 1772\ 3.0 & $9.7-10.4$ & 805 & 560\ & $9.0-9.7$ & 1596 & 1060\ & $8.3-9.0$ & 838 & 412\ 4.0 &$9.7- 10.4$& 273 & 161\ & $9.0-9.7$ & 348 & 176\ & $8.3- 9.0$ & 133 & 70 Size measurements {#seq_size_measurements} ================= Size measurements with $\tt{GALFIT}$ {#seq_size_gal} ------------------------------------ Galaxy sizes are measured for the F160W imaging data provided by the 3D-HST. Position, flux, half-light radius ($r_{\rm d}$), Sérsic index ($n$), axis ratio ($q \equiv b/a$), and position angle are treated as free parameters to determine. In this paper, we use the half-light radius along the semi-major axis of the Sérsic profile to define the size of galaxies. We make $100$ pixels $\times$ 100 pixels cutout images around object galaxies before size measurement. We then run $\tt{GALFIT}$ [@peng02; @peng10] on those cutout images, where neighbors are masked as not to perturb the fitting of the target galaxies. The masks are created from $\tt{SExtractor}$ segmentation maps. As an initial guess of the free parameters, we use $\tt{SExtractor}$ output parameters given in the 3D-HST catalog. Results of $\tt{GALFIT}$ are not sensitive to initial values as long as they are not far from real values (Häussler et al., 2007). We vary individual parameters over the following ranges: $\Delta x,\ \Delta y < 3$ pixels, $0.3<r_{\rm d}<100$ pixel, $0.1<n<8$, $0.1<q<1$, where $\Delta x$ and $ \Delta y $ are the difference in the centroids between $\tt{SExtractor}$ and $\tt{GALFIT}$. We define galaxies whose best-fit parameters are within these ranges as “success”. We only use “success” galaxies in the following analysis in Sections \[seq\_size\_measurements\] and \[cap\_mass\_size\]. The number of “success” galaxies is summarized in Table \[table2\_1\]. While we obtain robust structural parameters of only a part of our clustering sample, the average $\tt{SExtractor}$ sizes of the “success” sample and the entire sample are nearly equal. Thus we use the $\tt{GALFIT}$ sizes of the “success” sample as the representatives of the entire sample. Deriving $r_{\rm d}$ at rest-frame 5000$\AA$ -------------------------------------------- We derive $r_{\rm d}$ at the rest-frame $5000\AA$ at all redshifts. While we measure sizes in observed $1.6\mu m$ (F160W band), there exists a color gradient that depends on stellar mass and redshift. We obtain rest $5000\AA$ $r_{\rm d}$ by using the formula given in @vanderwel14: $$\begin{aligned} r_{\rm d} = r_{{\rm d}, F160W} \left(\frac{1+z}{1+z_p}\right)^{\Delta \log r_{\rm d}/\Delta \log \lambda}.\end{aligned}$$ where $z_p$ is the “pivot redshift"(2.2 for F160W) and the wavelength dependence is given by: $$\begin{aligned} \frac{\Delta \log r_{\rm d}}{\Delta \log \lambda} = -0.35 + 0.12z - 0.25 \log \left( \frac{M_{\star}}{10^{10} M_{\odot}}\right).\end{aligned}$$ Although van del Wel et al. (2014) have only examined wavelength dependence over $0<z<2$, we extend this formula to $z \simeq 4$ because the redshift evolution of this relation looks linear as a function of redshift. In any case, the correction values at $z \sim$ 3 and 4 are relatively small. Stellar mass–size relation {#cap_mass_size} ========================== The stellar mass–size distributions of our star-forming galaxies are shown in Figure \[fig4\_1\]. In Section \[sec\_modeling\], we analyze these distributions by modeling them with a power law. Then, we discuss the results in Section \[seq\_sizeevolution\]. Analytical Model of the stellar mass–size relation {#sec_modeling} -------------------------------------------------- The stellar mass–size relation is usually modeled as a single power-law: $$\begin{aligned} \overline{r}_{\rm d} (M_{\star,10}) / {\rm kpc} = A \cdot M_{\star,10}^{\alpha},\end{aligned}$$ where $M_{\star,10} = M_{\star} / 1.0\times 10^{10} M_{\odot}$, and $\overline{r}_{\rm d}\ (M_{\star,10})$ is the median size at $M_{\star,10}$. For the size distribution at a given stellar mass, we adopt a log-normal distribution: $$\begin{aligned} p(r_{\rm d}|\sigma_{\ln r}, \overline{r}_{\rm d})dr_{\rm d} = \frac{1}{\sqrt{2\pi}\sigma_{\ln r} r_{\rm d}} \exp \left[ -\frac{(\ln r_{\rm d} - \ln \overline{r}_{\rm d})^2}{2\sigma^2_{\ln r}} \right] dr_{\rm d}, \nonumber \\\end{aligned}$$ where $p(r_{\rm d}|\sigma_{\ln r}, \overline{r}_{\rm d})dr_{\rm d}$ is the probability density that a galaxy has a size between $(r_{\rm d}, r_{\rm d}+dr_{\rm d})$ at the given stellar mass, and $\sigma_{\ln r}$ is the dispersion of the distribution. The reason for adopting a log-normal distribution comes from Equation (\[eq1\_1\]). The disk size is proportional to the dimensionless spin parameter $\lambda$, and the distribution of $\lambda$ is well approximated by a log-normal distribution according to $N$-body simulations [@barnes87; @bullock01]. We assume that each of the observed disk sizes has a gaussian error: $$\begin{aligned} g ( x |\delta r_{\rm d})dx = \frac{1}{\sqrt{2\pi} \delta r_{\rm d}} \exp \left(- \frac{ x^2}{2 \delta r_{\rm d}^2} \right)dx,\end{aligned}$$ where $g ( x | \delta r_{\rm d}) dx$ is the probability density that a galaxy has a intrinsic disk size between $x$ and $x+dx$. The probability of observing $(r_{\rm d}, \delta r_{\rm d})$ assuming the log-normal distribution $p(r_{\rm d}|\sigma_{\ln r}, \overline{r}_{\rm d})$ is given by the convolution of the two functions: $$\begin{aligned} (p * g)(r_{\rm d}) = \int p(x) g(r_{\rm d}-x) dx.\end{aligned}$$ We use the 1$\sigma$ error in $\tt{GALFIT}$ as $\delta r_{\rm d}$. For each redshift, the free parameters of this model are given by $\textbf{P} = (A, \alpha, \sigma_{\ln r,i})$, where, $i$ denotes $i$-th subsample; we assume that different stellar mass bins have different $\sigma_{\ln r}$ values. We have six free parameters at $z\sim2$, and five free parameters at $z\sim$ 3 and 4. We use the maximum likelihood estimation (MLE) to determine these parameters, where the estimated parameters make the observed $r_{\rm d}$ distribution the most probable. For subsample $i$ at a given redshift, the likelihood function is defined as $$\begin{aligned} \mathcal{L}_{i} = \prod_{j=1}^N (p * g)(r_{{\rm d},j}|\sigma_{\ln r,i}, \overline{r}_{\rm d}),\end{aligned}$$ where $j$ represents the $j$-th object. We determine the parameter set $\textbf{P}$ that maximizes the likelihood function $\mathcal{L} \equiv \prod \mathcal{L}_{i}$. The best-fit values are listed in Table \[table4\_1\]. We use the $\tt{scipy.optimize}$ package and the L-BFGS-B algorithm [@zhu97] to find the maximizing point. The uncertainties in the parameters are estimated by the Markov Chain Monte Carlo (MCMC) sampling. MCMC is a powerful algorithm to approximate multi-dimensional parameters using a Markov chain. We use the python package $\tt{emcee}$ [@foreman13] to run MCMC. In Figure \[fig4\_2\], we show for each parameter the best-fit values and the 68$\%$, 95$\%$, and 99$\%$ confidence intervals. This figure is made using the public python package $\tt{corner}$ [@foreman16]. [ccccccc]{} 2.0 & $2.51^{+0.03}_{-0.05}$ & $0.19^{+0.01}_{-0.01}$ & $0.46^{+0.01}_{-0.01}$ & $0.51^{+0.01}_{-0.01}$ & $0.50^{+0.02}_{-0.02}$ & $0.53^{+0.03}_{-0.05}$\ 3.0 &$1.94^{+0.06}_{-0.05}$ & $0.14^{+0.01}_{-0.03}$ & $0.42^{+0.03}_{-0.03}$ & $0.47^{+0.02}_{-0.02}$ & $0.47^{+0.02}_{-0.02}$ & …\ 4.0 & $1.57^{+0.11}_{-0.13}$ & $0.08^{+0.05}_{-0.05}$ & $0.45^{+0.18}_{-0.05}$ & $ 0.51^{+0.08}_{-0.07}$ & $0.47^{+0.09}_{-0.06}$ & … Size evolution {#seq_sizeevolution} -------------- The evolution of $A$, $\alpha$, and $\sigma_{\ln r}$ are shown in Figure \[fig4\_3\]. In this Section, we discuss the evolution of each parameter in detail. ### Median size evolution The size evolution at a fixed stellar mass is generally parameterized as $(1+z)^{-\beta_z}$, where $\beta_z$ is a constant expressing the strength of evolution (evolution slope). The top panel of Figure \[fig4\_3\] represents the median size evolution of disk star-forming galaxies at $M_{\star} = 1.0 \times 10^{10} M_{\odot}$. The solid blue line shows the best-fit function over $z\sim2-4$: $\overline{r}_{\rm d} (M_{\star,10}) / {\rm kpc} =6.88 (1+z)^{-0.91 \pm 0.01}$. @allen16 have measured the size evolution of a mass-complete sample $(\log (M_{\star}/M_{\odot}) > 10)$ of star-forming galaxies over redshifts $z = $ 1 $-$ 7, to find that the average size at a fixed mass of $\log (M_{\star}/M_{\odot}) = 10.1$ is expressed by $r_{\rm d} = 7.07(1+z)^{-0.89\pm 0.01}$. The slope we find is in agreement with @allen16’s value. @shibuya15 have also measured the stellar mass–median circularized size evolution of star-forming galaxies with $9.0 < \log (M_{\star}/M_{\odot}) < 11.0$ at $0<z<6$. The gray dotted line represents the average circularized half-light radius from their samples with the gray region showing the 16th and 84th percentiles. The evolution slope is consistent with our result. The difference in the amplitude is largely due to the different definition of galaxy sizes. We also note that @shibuya15 have used the Salpeter IMF [@salpeter55] to derive stellar masses. However, $\beta_z = 0.91 \pm 0.01$ is slightly steeper than the value by @vanderwel14. They have studied a mass complete sample of star-forming galaxies and have found $(1+z)^{-0.75}$ at a $\log (M_{\star}/M_{\odot}) = 10.7$ over the redshift range $0 < z < 3$. As their method of size measurements is the same as ours, we attribute this discrepancy to the difference in the redshift range. The evolution slope of star-forming galaxies appears to become steeper above $z\sim$ 2 or 3. @allen16’s sample also shows steeper slopes at higher redshifts [See Figure 3 of @allen16]. As size evolution is closely related to the evolution of hosting dark matter halos, $\beta_z$ contains information of dark matter halos. From Equation (\[eq1\_1\]), when $r_{\rm d}/r_{\rm 200}$ is constant irrespective of $z$ and $M_{\rm dh}$, $r_{\rm d}$ is given by $$\begin{aligned} r_{\rm d} &\propto& H(z)^{-1}V_{\rm c} \label{eq4_2_1} \\ &\propto& H(z)^{-2/3}M^{1/3}_{\rm dh}, \label{eq4_2_3}\end{aligned}$$ where $V_{c}$ is the circular velocity of dark matter halos. The Hubble parameter as a function of $z$, $H(z) = H_0 \sqrt{\Omega_{\rm m} (1+z)^3 + \Omega_{\Lambda}}$, is approximated as $H(z) \propto (1+z)^{1.5}$. According to Equations (\[eq4\_2\_1\]) and (\[eq4\_2\_3\]), $r_{\rm d} \propto (1+z)^{-1.5}$ means evolution at a constant circular velocity and $r_{\rm d} \propto (1+z)^{-1.0}$ means evolution at a constant virial mass [@ferguson04]. The $\beta_{z} = 0.91$ is close to the prediction for a constant virial mass. ### Slope evolution {#sub_slope} The middle panel of Figure \[fig4\_3\] shows the slope evolution in the stellar mass–size relation ($\alpha$). The slope evolution of the stellar mass–size relation for late-type galaxies was first investigated by @vanderwel14. They have found that the slope has nearly a constant value $\simeq 0.2$ over the redshift range $0<z<3$. Similarly @allen16 have found $\alpha = 0.15\pm 0.01$ for star-forming galaxies at $1<z<2.5$. Our results are consistent with those of @vanderwel14 and @allen16 at $z\sim$ 2 and 3, however, being slightly lower at $z\sim 4$. The slope evolution of the stellar mass–size relation is determined as a combination of the slope of the stellar mass–halo mass relation and the slope of the disk size–halo size relation. In this paper, We have measured all three slopes. We will discuss the relation between the three slopes in Section \[seq\_ang\]. ![Redshift evolution of the stellar mass–size relation of star-forming galaxies. Top: the size evolution at $M_{\star} = 1.0 \times 10^{10} M_{\odot}$. The blue diamond symbols indicate the results obtained in this paper, and the solid blue line shows the best-fit power law. The green solid line shows the average size of star forming galaxies from @allen16 at $10^{10.1}M_{\odot}$. The red solid line indicates the size evolution of late-type galaxies from @vanderwel14 at $10^{9.75} M_{\odot}$, and the red dashed line is its extrapolation. The gray dotted line and the shaded region indicate the median circularized size and the 16th and 84th percentiles distribution of star-forming galaxies with $9.5<\log M_{\star}/M_{\odot}<10.0$ [@shibuya15]. Middle: slope evolution. The blue and red symbols represent our galaxies and late-type galaxies from @vanderwel14, respectively. Bottom: the intrinsic scatter evolution from this work and previous studies. The blue symbols represent our galaxies. The orange symbols represent LBGs from @huang13 at $z\sim$ 4 and 5. The filled and open red symbols show the late-type galaxies of @vanderwel14. The green symbol shows the SDSS galaxies of @shen03 at the faint end.[]{data-label="fig4_3"}](size_evolution_h26_closex.pdf){width="0.7\linewidth"} ### Scatter evolution We present the evolution of the intrinsic scatter in the bottom panel of Figure \[fig4\_3\]: here, “intrinsic” means that measurement errors have been removed. The scatter for local galaxies is generally small. @shen03 have found $\sigma_{\ln r_{\rm d}} \sim 0.3$ for both late-type and early type galaxies from SDSS. This result has also been ascertained by the result of @courteau07, $\sigma_{\ln r_{\rm d}} \sim 0.3$, for local spiral galaxies. These studies have been extended by @vanderwel14 to the high-redshift universe and they have found that the intrinsic scatter dose not strongly evolve since $z\sim 2.75$ for both late-type and early-type galaxies. In their study, the scatter for late type galaxies is $0.16-0.19$ $\rm dex$, which is comparable to the result of @shen03 and @courteau07. We extend @vanderwel14’s study up to $z\sim4$, and find that the intrinsic scatter is constant with $0.4-0.6$ over $z\sim2-4$. The scatter of $\lambda$ has been specifically investigated by $N$-body simulations and found to be $\sigma_{\lambda} \sim 0.5$ [@bullock01]. Thus the disk formation model of Equation (\[eq1\_1\]) naively predicts that the intrinsic scatter of sizes is $\sim 0.5$. The results for local galaxies imply that the size scatter is smaller than that of the spin parameter $\lambda$. To explain the observed small scatters, some mechanisms are needed. One possible mechanism is bulge growth. The growth of bulges increases the specific angular momentum of disks and thus expands disk sizes. Low-spin galaxies selectively grow their bulges. Some kind of disk instability and feedback has also been proposed which remove galaxies with low-spin and high-spin halos. Our result, $\sigma_{\ln r_{\rm d}} \sim 0.4$ $-$ 0.6, is comparable with the scatter of the log-normal distribution of $\lambda$. This implies that for star-forming galaxies at $z\sim$ 2 $-$ 4 the size scatter at a given stellar mass is fully explained by the scatter of $\lambda$. Our result, however, does not agree with the large scatters, $\sigma_{\ln r_{\rm d}}\sim0.8-0.9$, found by @huang13 for the size–UV luminosity relations of $z\sim4-5$ LBGs. This may suggest that the UV luminosity–halo mass relation of LBGs has a considerably large scatter. Halo mass estimates {#seq_halo} =================== In this Section we estimate the masses of the dark matter halos hosting our galaxies by using two independent methods: clustering analysis and abundance matching technique. Clustering analysis utilizes the large scale clustering amplitude of observed galaxies to obtain their hosting dark matter halo masses. Clustering analysis is a popular way to estimate hosting dark matter halo masses, however the mass estimates in this paper have relatively large errors because the sizes of individual subsamples are not so large. To test the results of the clustering analysis, we use abundance matching technique, which connects the stellar mass of galaxies to that of dark matter halos. While abundance matching can easily estimate hosting dark matter halo masses, it does not consider that different galaxy types have different stellar mass dark halo mass ratios. We briefly explain the two methods and show the obtained dark matter halo masses. Clstering analysis ------------------ ### Angular correlation function We compute the two point angular correlation functions (ACFs), $\omega_{\rm true}(\theta)$, of star-forming galaxies. Here, we assume all of our galaxies as central galaxies. The observed ACFs, $\omega_{\rm obs}(\theta)$, are measured by counting the number of unique pairs of observed galaxies and comparing it with what is expected from random samples. We adopt the estimator proposed by @landy93: $$\begin{aligned} \omega_{\rm obs}(\theta) = \frac{ DD(\theta) - 2DR(\theta) + RR(\theta)}{RR(\theta)},\end{aligned}$$ where $DD(\theta)$, $DR(\theta)$, and $RR(\theta)$ are the normalized numbers of galaxy-galaxy, galaxy-random and random-random pairs, respectively, with separation $\theta$. We generate 1000 times as many random points as the number of galaxies accounting for the geometry of the observed area and the masks. The formal error in $\omega_{\rm true}$ is given by $$\begin{aligned} \sigma_{\omega} = \sqrt{[1+\omega_{\rm obs}]/DD(\theta)}.\end{aligned}$$ We assume a power low parameterization for the ACF, $$\begin{aligned} \omega_{\rm true} (\theta) = A_{\omega} \theta^{-\beta}. \label{eq1}\end{aligned}$$ We fix $\beta = 0.8$ following previous studies [e.g. @peebles75; @ouchi01; @ouchi04; @ouchi10; @foucaud03; @foucaud10; @harikane16]. It is known that $\omega_{\rm obs}$ is underestimated because we only use a finite survey area. This is compensated by introducing an integral constraint (IC) [@groth77]: $$\begin{aligned} \omega_{\rm true} = \omega_{\rm obs} + \rm{IC}.\end{aligned}$$ The IC value depends on the size and shape of the survey area, and is estimated using a random catalog: $$\begin{aligned} {\rm IC} = \frac{\sum_i RR(\theta_i) \omega_{\rm true}(\theta_i)}{\sum_i RR(\theta_i)} = \frac{\sum_i RR(\theta_i) A_{\omega} \theta^{-\beta}}{\sum_i RR(\theta_i)}.\end{aligned}$$ Because the three 3D-HST fields used in this paper have almost the same size, we obtain nearly the same IC value (${\rm IC}_{\rm GOODS-S} = 0.016A_{\rm \omega}$, ${\rm IC}_{\rm COSMOS} = 0.013A_{\rm \omega}$, and ${\rm IC}_{\rm AEGIS} = 0.010A_{\rm \omega}$). The amplitude $A_{\omega}$ is estimated through the ACFs of the three fields by minimizing $\chi^2$: $$\begin{aligned} \chi^2 = \displaystyle \sum_{i,\,j={\rm fields}} \frac{[A_{\omega} \theta_{i}^{-\beta} - (\omega_{{\rm obs},j}(\theta_{i}) + {\rm IC}_j)]^2}{\sigma_{\omega,j}^2(\theta_{i})},\end{aligned}$$ where ${\rm IC}_j$, $\omega_{{\rm obs}, j}$, and $\sigma_{\omega,j}^2(\theta)$ denote the IC, observed ACF, and errors in field $j$, respectively. We use data at $\theta > 10"$ for fitting because at $\theta < 10"$ the contribution of the one halo term cannot be ignored. In figure \[fig5\_1\] we plot the ACFs of our subsamples with the best-fit power laws. Then we estimate the spatial correlation function, $\xi(r)$, from the measured ACFs and the redshift distribution of galaxies. The spatial correlation function is usually assumed to be a single power low as $$\begin{aligned} \xi(r) = \left( \frac{r}{r_0} \right)^{-\gamma},\end{aligned}$$ where $r_0$ is the correlation length and $\gamma$ is the slope of the power low. These parameters are related to those of the two point angular correlation function via the Limber transform [@peebles80; @efstathiou91]. $$\begin{aligned} \beta &=& \gamma - 1, \\ A_{\omega} &=& \frac{r_0^{\gamma} B[1/2, (\gamma - 1)] \int^{\infty}_0 dz N(z)^2 F(z) D_{\theta}(z)^{1-\gamma} g(z)}{[\int^{\infty}_0 N(z)dz]^2 }, \nonumber \\ \\ g(z) &=& \frac{H_0}{c}(1+z)^2 \{1 + \Omega_{\rm m} z + \Omega_{\Lambda}[(1+z)^{-2} -1]\}^{1/2},\end{aligned}$$ where $D_{\theta}(z)$ is the angular diameter distance, $N(z)$ is the redshift distribution of galaxies, $B$ is the beta function, and $F(z)$ describes the redshift evolution of $\xi(r)$. $F(z)$ is often modeled as $F(z) = [(1+z)/(1+z_{\rm c})]^{-(3+\overline{\epsilon})}$ with $\epsilon = -1.2$ [@roche99], where $z_{\rm c}$ is the characteristic redshift of galaxies. We assume that the clustering evolution is fixed in comoving coordinates over the redshift range in question. ### Galaxy biases and halo masses To understand the relation between galaxies and hosting dark matter halos we use the halo model of @sheth01, which is obtained from the ellipsoidal collapse model. In the model of @sheth01 the bias factor of dark halos, $b_{\rm dh}$, is given by $$\begin{aligned} b_{\rm dh} = 1 + \frac{1}{\delta_{\rm c}} \left[ \nu'^{2} + b \nu'^{2(1-c)} - \frac{\nu'^{2c} / \sqrt{a}}{\nu'^{2c} + b(1-c)(1-c/2)} \right], \nonumber \\\end{aligned}$$ where $\nu' = \sqrt{a} \nu$, $a = 0.707$, $b = 0.5$, $c = 0.6$, and $\delta_{\rm c} = 1.69$ is the critical amplitude above which overdense regions collapse to form a virialized object. Here, $\nu$ is defined as $$\begin{aligned} \nu = \frac{\delta_{\rm c} }{\sigma(M,z)} = \frac{\delta_{\rm c} }{D(z) \sigma(M,0)},\end{aligned}$$ where $D(z)$ is the linear growth factor, $\sigma(z)$ is the mass rms. of the smoothed density field. We calculate $D(z)$ by the formula of @carroll92 and $\sigma(M,0)$ using an initial power spectrum of a power law index $n=1$ and the transfer function of @bardeen86. Then we define the linear galaxy bias, which is the relation between the clustering amplitude of galaxies and that of dark matter halos, at a large scale ($=8 h^{-1}_{100} \rm\ Mpc$) as $$\begin{aligned} b_{\rm g} = \sqrt{\frac{\xi_{\rm g}( r = 8 h^{-1}_{100} \rm\ Mpc)}{\xi_{\rm DM} (r=8 h^{-1}_{100} \rm\ Mpc)}} = \sqrt{\frac{[8 h^{-1}_{100} \rm\ Mpc / r_0]^{-\gamma}}{\xi_{\rm DM} (r=8 h^{-1}_{100} \rm\ Mpc)}}, \nonumber \\\end{aligned}$$ where $\xi_{\rm DM} (r=8 h^{-1}_{100} \rm\ Mpc)$ is the dark matter spatial correlation function. We calculate $\xi_{\rm DM} (r=8 h^{-1}_{100} \rm\ Mpc)$ using the non-linear model of @smith03. Assuming that the galaxy bias at large scales is almost the same as the halo bias ($b_{\rm g} \simeq b_{\rm dh}$), we obtain an estimate of dark halo masses. The correlation length and the estimated halo masses are summerized in Table \[table5\_1\]. [ccrcccc]{} 2.0 &10.58& 264 & $5.40^{+0.96}_{-0.96}$ & $12.30^{+1.18}_{-1.25}$ & $13.37^{+0.10}_{-0.12}$ & 12.23\ & 9.94 & 1086 & $0.69^{+0.25}_{-0.25}$ & $ 3.92^{+0.73}_{-0.87}$ & $11.69^{+0.32}_{-0.56}$ & 11.79\ & 9.30 & 3267 & $0.67^{+0.07}_{-0.07}$ & $ 3.86^{+0.21}_{-0.23}$ & $11.66^{+0.11}_{-0.13}$ & 11.51\ & 8.72 & 3173 & $0.51^{+0.08}_{-0.08}$ & $ 3.31^{+0.28}_{-0.30}$ & $11.32^{+0.18}_{-0.23}$ & 11.30\ 3.0 & 9.93 & 805 & $1.45^{+0.31}_{-0.31}$ & $ 5.18^{+0.58}_{-0.65}$ & $11.92^{+0.17}_{-0.23}$ & 11.81\ & 9.37 & 1596 & $0.86^{+0.15}_{-0.15}$ & $ 3.87^{+0.36}_{-0.39}$ & $11.40^{+0.17}_{-0.21}$ & 11.53\ & 8.78 & 838 & $0.51^{+0.31}_{-0.31}$ & $ 2.90^{+0.87}_{-1.18}$ & $10.79^{+0.56}_{-1.60}$ & 11.29\ 4.0 &10.01& 273 & $2.08^{+0.93}_{-0.93}$ & $ 5.57^{+1.27}_{-1.56}$ & $11.79^{+0.31}_{-0.56}$ & 11.78\ & 9.37 & 348 & $1.77^{+0.72}_{-0.72}$ & $ 5.09^{+1.06}_{-1.28}$ & $11.64^{+0.30}_{-0.51}$ & 11.45\ & 8.82 & 133 & $2.78^{+1.74}_{-1.74}$ & $ 6.54^{+2.03}_{-2.75}$ & $12.03^{+0.38}_{-0.91}$ & 11.22 Abundance Matching ------------------ In order to reinforce the results of the clustering analysis, we also use abundance matching analysis, which connects the number density of galaxies to that of dark halos to estimate the hosting dark halo mass for a given stellar mass. We adopt the abundance matching result of @behroozi13. Many researchers that study the angular momentum retention factor adopt the abundance matching analysis of @dutton10 and @behroozi13 to estimate halo masses [e.g. @fall12; @burkert16]. This makes easy to compare our results with previous results of angular momentum studies. The estimated halo masses are also summerized in Table \[table5\_1\]. Figure \[fig5\_2\] shows a comparison of the estimated dark matter halo masses. The estimated dark matter halo masses by the two independent methods are consistent within the error bars except for the highest stellar mass bins at $z\sim 2$. This makes the results of the clustering analysis more plausible. In the following Section, we display the results based on the both methods. Angular momentum {#seq_ang} ================ Estimation of the specific angular momentum {#sec_estimate} ------------------------------------------- In this Section, we briefly explain the way to estimate the disk specific angular momentum. As already mentioned in Section \[sec1\], the disk size of a galaxy reflects its specific angular momentum. According to the model of @mo98, the specific angular momentum of disk galaxies with an exponential profile ($n=1$) is given by: $$\begin{aligned} j_{\rm d} = \frac{\sqrt{2}}{1.678} r_{\rm d} m_{\rm d} \lambda^{-1} r_{200}^{-1} f_{\rm c}(c_{\rm vir})^{1/2} f_{\rm R}(\lambda, c_{\rm vir}, m_{\rm d},j_{\rm d})^{-1}. \nonumber \\ \label{eq6_1}\end{aligned}$$ If we assume $r_{\rm d}$ as the half-light radius of a Sérsic index $n$, we can expand this equation to: $$\begin{aligned} j_{\rm d} &=& f_n(n)^{-1} r_{\rm d} m_{\rm d} \lambda^{-1} r_{200}^{-1} f_{\rm c}(c_{\rm vir})^{1/2} f_{\rm R}(\lambda, c_{\rm vir}, m_{\rm d},j_{\rm d})^{-1}, \nonumber \\ \label{eq6_2} \\ f_n(n) &=& \frac{\sqrt{2} \Gamma(2n) \kappa^n}{\Gamma(3n)},\end{aligned}$$ where $\Gamma$ is a gamma function, and $\kappa$ is well approximated by $$\begin{aligned} \kappa &=& 2n - \frac{1}{3} + \frac{4}{405n} + \frac{46}{25525n^2} \nonumber \\ &\quad&+ \frac{131}{1148175n^3} + \mathcal{O}(n^{-4})\ (n>0.36),\\ \kappa &=& 0.01945 - 0.8902 n + 10.95 n^2 - 19.67n^3 \nonumber \\ &\quad&+ 13.43 n^4\ (n<0.36)\end{aligned}$$ [@ciotti99; @macarthur03]. The full functional forms of $f_{\rm c}$ and $f_{\rm R}$ are given in @mo98. The values of $\lambda$ and $c_{\rm vir}$ are well determined by $N$-body simulations [@vitvitska02; @davis09; @prada12; @puebla16]. We adopt $(\lambda, c_{\rm vir}) = (0.035,4.0)$ throughout the examined redshift range ($z \sim 2-4$). From the dark matter halo masses estimated in Section \[seq\_halo\], we can calculate $m_{\rm d}$ and $r_{200}$, where $r_{200}$ is calculated by $$\begin{aligned} r_{200} = \left(\frac{GM_{\rm dh}}{100 H(z)^2}\right)^{1/3}.\end{aligned}$$ Combined with $n$ and $r_{\rm d}$ measured in Sections \[seq\_size\_measurements\] and \[cap\_mass\_size\], we can estimate $j_{\rm d}$. Mass–angular momentum relation ------------------------------ ### Average $j_{\rm d}/m_{\rm d}$ ratio and its evolution Figure \[fig6\_3\] shows the angular momentum retention factor of star-forming galaxies as a function of hosting halo mass. We find $j_{\star}/m_{\star} = 0.77\pm0.06$ from clustering analysis and $j_{\star}/m_{\star} = 0.83\pm0.13$ from abundance matching at $z \sim$ 2, 3, and 4. No strong redshift evolution is confirmed. As we mention in Section \[sec1\], $j_{\star}/m_{\star} = 1$ means that the angular momentum is fully conserved and $j_{\star}/m_{\star} < 1$ means that galaxies lose their specific angular momentum during their formation and evolution. @fall12 have investigated kinematical structure for about 100 bright early and late-type galaxies at $z\sim0$. They have found that late-type galaxies typically have $j_{\rm d}/m_{\rm d} \simeq 0.6$ and early-type galaxies have $j_{\rm d}/m_{\rm d} \simeq 0.1$. A small $j_{\rm d}/m_{\rm d}$ value has also been reported by @dutton12. They have calculated angular momentum retention factor as a function of halo mass by constructing the mass models [@dutton11] tuned to observed scaling relations for SDSS galaxies. They have obtained a constant value $j_{\rm d}/m_{\rm d} = 0.61^{+0.13}_{-0.11}$ with halo masses $10^{11.3}M_{\odot} \lesssim M_{\rm dh} \lesssim 10^{12.7} M_{\odot}$. Our values at $z\sim$ 2, 3, and 4 are in rough agreement with these local values for late-type galaxies within errors. There exist a few studies that have investigated the mass–angular momentum relation at high redshifts. Recently, @burkert16 have investigated the relation for $\sim360$ star-forming galaxies at $z\sim0.8-2.6$, among which about 100 are at $z\sim2$, by H$\alpha$ kinematics based on KMOS and SINS/zC-SINF surveys. They have found $j_{\rm d}/m_{\rm d} = 1.0$ with a statistical uncertainty of $\pm0.1$ and a systematic uncertainty of $\pm0.5$. This $j_d/m_d$ value is consistent with our result at $z\sim2$. We then compare our results with those of @huang17 and @somerville17. These authors have derived disk size to halo size ratios $(r_{\rm d}/r_{\rm dh})$ as a function of stellar mass over $z \sim 0$ and 3 using the CANDELS data and mapping stellar masses to halo masses with abundance matching. At $z\sim2$, the $r_{\rm d}/r_{\rm dh}$ ratios obtained by @huang17 are consistent with ours, with values of $\sim0.03$ in the stellar mass range $10^9 M_{\odot} < M_{\star} < 10^{10.5} M_{\odot}$. We note that our method is very similar to theirs. Their definitions of disk sizes and halo sizes are the same as ours. They have used four abundance matching results including that of @behroozi13 which we also use. On the other hand, @somerville17 have obtained somewhat higher ratios of $r_{\rm d}/r_{\rm dh} \simeq 0.4$. They have adopted a different halo definition and also taken a different method to link stellar masses to halo masses; they have carried out “forward modeling" where halos are taken from an $N$-body simulation and are assigned to stellar masses taking account of a random scatter. These differences may be a cause of the inconsistency in $r_{\rm d}/r_{\rm dh}$ estimates. To connect our study to those for low redshifts, we use Extended Press-Schechter (EPS) formalism [@bond91; @bower91; @lacey93]. The EPS formalism is able to calculate the conditional probability mass function ($f(M_{2}|M_{1})$) of $z=z_{2}$ descendant halos for a given halo mass ($M_{1}$) at a high-redshift ($z_{1}$) universe by following their merger histories. We set $M_{1} = 5.0 \times 10^{11}h^{-1}M_{\odot}$ and $z_{1} = 3.0$ to follow the evolution of our halos. The lower 68 and upper percentiles of $f(M_{2}|M_{1})$ at $z_{2}=0$ are $2.0\times 10^{12}h^{-1}M_{\odot}$ and $5.6 \times 10^{12}h^{-1}M_{\odot}$, respectively. This implies that some fraction of our galaxies are the progenitors of objects in the @dutton12 sample in terms of mass growth. From the results we obtain, we can depict a unified view of the angular momentum evolution. Disk galaxies maintain high $j_{\rm d}/m_{\rm d}$ values during their evolution from cosmic noon to the present day, unless they lose angular momenta by some mechanisms like mergers and turn into early-type galaxies [@fall12]. ### Halo mass dependence of $j_{\rm d}/m_{\rm d}$ and the slope of the size–stellar mass relation When we introduce the disk size–halo mass relation in Equation (\[eq4\_2\_3\]), we assume that $r_{\rm d}/r_{\rm 200}$ is constant, which means that $j_{\star}/m_{\star}$ is constant irrespective of $z$ and $M_{\rm dh}$. However, it appears from Figure \[fig6\_3\] that $j_{\star}/m_{\star}$ weakly depends on both $M_{\rm dh}$ and $z$. Similar dependencies have also been shown in @huang17 and @somerville17: $r_{\rm d}/r_{\rm dh}$ weakly depends on both $M_{\rm dh}$ and $z$. We approximate the observed $j_{\star}/m_{\star}$–$M_{\rm dh}$ relation at each redshift by a power law, $j_{\star}/m_{\star} \propto M_{\rm dh}^{\gamma_{z}}$. We find $\gamma_{z} = -0.09\pm0.02$ for $z\sim2$, $\gamma_{z} = -0.13\pm0.01$ for $z\sim3$, and $\gamma_{z} = -0.29\pm0.02$ for $z\sim4$. A negative slope of $\gamma_{z} = -0.19 \pm 0.04$ has also been obtained by @burkert16 for $z\sim 0.8-2.6$ galaxies. With a non-zero slope $\gamma_{z}$, Equation (\[eq4\_2\_3\]) is replaced by: $$\begin{aligned} r_{\rm d} \propto H(z)^{-2/3} M_{\rm dh}^{\gamma_{z}+1/3}.\end{aligned}$$ We also approximate the stellar mass–halo mass relation by a single power-law, $M_{\star} \propto M_{\rm dh}^{\epsilon}$: $\epsilon \simeq1.6$, from the abundance matching results of @behroozi13. By combining these two relations, we obtain the disk size–stellar mass relation: $$\begin{aligned} r_{\rm d} &\propto& M_{\star}^{1/3\epsilon + \gamma_{z}/\epsilon},\\ r_{\rm d} &\propto& M_{\star}^{0.2+0.6 \gamma_{z}}. \label{eq6_3}\end{aligned}$$ The slope of the size–stellar mass relation of our galaxies is $\alpha$ = $0.19^{+0.01}_{-0.01}$ for $z\sim2$, $0.14^{+0.01}_{-0.03}$ for $z\sim3$, and $0.08^{+0.05}_{-0.05}$ for $z\sim4$ (see Section \[sub\_slope\]). The result that $\alpha$ is less than 0.2 for all three redshifts is explained by the negative $\gamma_{z}$ values obtained above. We also find that the decrease in $\alpha$ from $z\sim3$ to $z\sim4$ is due to the decrease in $\gamma_{z}$. Using a theoretical modified cooling model which includes disc instability, @dutton12 have predicted a slightly negative $\gamma_{z}$ for high redshift disk galaxies, in qualitative agreement with our results. Their negative slope reflects the fact that the mass loading factor decreases with increasing of halo mass. While this model is not consistent with their empirical model at $z\sim0$, this model may be applicable to high redshifts. The possible decrease in $\gamma_{z}$ from $z\sim3$ to $z\sim4$ found above may imply that feedback processes also change in this redshift range. As already seen in Figure \[fig4\_3\], @vanderwel14 have reported constant disk size–stellar mass slopes ($\sim 0.2$) since $z\sim2-0$. From the model of Equation (\[eq6\_3\]), this implies that the angular momentum–halo mass relations are also flat. This is quite in agreement with the empirical results of @dutton12 at the present-day universe. Thus Equation (\[eq6\_3\]) well represents the relation between angular momentum and disk size. Comparison with galaxy formation models {#sec_conparison} --------------------------------------- As the kinematics of galaxies provides us with important constraints on galaxy formation and evolution as well as do other global properties like stellar mass, star-formation rate, and metallicity, many modelers have attempted to reproduce the kinematic structures of galaxies. Early attempts concerning angular momentum with hydrodynamical simulations were in trouble with reproducing observations. They suffered from unexpected angular momentum loss. In those simulations, most of the angular momentum of galaxies was transferred to the background hosting halos. As a result, compact disk galaxies were produced [e.g. @navarro91; @navarro94]. This problem is known as the “angular momentum catastrophe”. This problem has been considerably improved by high-resolution hydrodynamical simulations with a proper treatment of feedback processes [@robertson06; @governato07; @scannapieco08]. In recent years, many galaxy formation simulations have succeeded in reproducing the mass–angular momentum relation for both early-type and late-type galaxies in the present-day universe [@genel15; @teklu15]. On the other hand, at high redshifts, there do not exist theoretical studies that compare with observational data. It is still unknown that these simulations are able to reproduce the observed mass–angular momentum relation beyond $z\sim 1$. Here, we first compare our observational angular momentum results with those of some galaxy formation simulations [@sales12; @pedrosa15; @stevens16]. In Figure \[fig6\_4\], we compare the mass–angular momentum distribution of star-forming galaxies obtained from clustering analysis and abundance matching analysis with predictions from hydrodynamical and semi-analytical galaxy formation models at $z\sim2$. To directly compare with two models which give only stellar plus gas properties, we also estimate the entire disk masses by correcting for gas masses using the gas fraction estimates given in @schinnerer16. They have investigated the gas masses for 45 massive star-forming galaxies observed with ALMA at redshifts of $z\sim3-4$. We extend their results to lower mass and lower redshift by the prediction of 2-SFM (2 star formation mode) model [@sargent14]. We correct $m_{\star}$ and $j_{\star}$ by the same factor assuming that the stellar and gas disks have the same $j$ value. The right panel of Figure \[fig6\_4\] shows the baryonic disk mass–angular momentum relation. @sales12 have presented baryonic mass–angular momentum relations with various types of feedback from large cosmological $N$-body/gasdynamical simulations at $z\sim 2$. They have found that regardless of the strength of the feedback process $m_{\rm d}$ vs. $j_{\rm d}$ follows the same relation (the yellow solid lines in Figure \[fig6\_4\]). When strong feedbacks push out most of the baryons from the galaxies, both $m_{\rm d}$ and $j_{\rm d}$ are reduced. @pedrosa15 have also analyzed the mass–angular momentum relation by decomposing disks and bulges with cosmological hydrodynamical simulations at $z\sim0-2$. They have found no significant evolution since $z\sim2 $ to $z\sim 0$. The relation for total baryonic components at $z\sim2$ is shown in Figure \[fig6\_4\]. @stevens16 have presented a semi-analytical model D[ARK]{} S[AGE]{}, which is designed for specific understanding of angular momentum evolution. They have investigated the evolution of the stellar mass–specific angular momentum relation over $0<z<4.8$. The solid cyan lines in Figure \[fig6\_4\] indicate the predicted mass–angular momentum relation at $z\sim2$. Here, we assume the abundance matching results by @behroozi13 to map stellar mass to dark halo mass and an analytical model by @fall12, which connects dark matter halo mass to their halo angular momentum: $$\begin{aligned} j_{\rm vir} = 4.23 \times 10^{4} \lambda \left(\frac{M_{\rm vir}}{10^{12} M_{\odot}} \right)^{2/3} {\rm km\,s^{-1}\,kpc}.\end{aligned}$$ Note that as the @fall12’s model uses cosmological parameters at the present day $(c_{\rm vir}=9.7,\Delta_{\rm vir}=319)$, we replace them with values $(c_{\rm vir}=4.0,\Delta_{\rm vir}=200)$. All of these simulations predict specific angular momenta systematically smaller than our values from both dark matter halo mass estimation methods. Our relations are almost parallel to the line of angular momentum conservation (dotted gray lines in Figure \[fig6\_4\]) regardless of mass scales, however, the simulations predict smaller specific angular momenta and the deviations are large for smaller $m_{\star}$ and $m_{\rm d}$. While the star+gas plots appear to have smaller deviations than those of the star only plots, note that we ignore a possible difference in the distribution of gases and stars within galaxies. In other words, we assume that gases and stars have the same specific angular momentum. However, @brook11_1 have shown that the angular momentum distributions of stars and $\rm H_{I}$ gases are different, with $\rm H_{I}$ gases having a tail of high angular momentum. Indeed, extended $\rm H_{I}$ gas disks are found in intermediate [@puech10] and high redshift [@daddi10] galaxies. Gases beyond star-forming regions serve as a high angular momentum reservoir [@brook11_1]. These gases should have a larger specific angular momentum than stars. In this case, the gaps on the right panels in Figure \[fig6\_4\] become larger. These deviations imply that these simulations produce too small disk sizes at high redshifts. Some mechanisms that increase disk specific angular momentum at high redshifts may be needed. For example, @brook12 have proposed that selective ejection of low angular momentum material from galaxies leads to a redistribution of angular momentum. This explains the difference in the distribution of angular momentum between dark matter halos and visible galaxies: dark matter halos have a large low angular momentum tail, while observed galaxies do not. This process reproduces large bulge-less high angular momentum galaxies. Whether or not these feedback related mechanisms are enough to solve the deviations seen in Figure \[fig6\_4\] is still unknown. More detailed observations and simulations are needed. Disk instability ---------------- The angular momentum of disks is also closely related to their global instabilities. Disks can be unstable against bar mode instability, because low angular momentum material forms a bar [@shen03]. @efstathiou82 have investigated this kind of instabilities for a exponential disk embedded in a variety of halos using $N$-body simulations and found a stellar disk is globally unstable against bar formation under the criterion: $$\begin{aligned} \epsilon_{\rm m} \equiv \frac{V_{\rm max}}{(GM_{\rm d} / r_{\rm d})^{1/2}} \lesssim 1.1, \label{eq6_9}\end{aligned}$$ where $V_{\rm max}$ is the maximum rotation velocity of the disk. The threshold for gaseous disks is $\epsilon_{\rm m} \simeq 0.9$. According to @mo98, for a NFW halo, this criterion is well approximated by $$\begin{aligned} \lambda' < m_{\rm d}, \label{eq6_10}\end{aligned}$$ where $\lambda' \equiv \lambda j_{\rm d} / m_{\rm d}$. We note that the criteria of Equations (\[eq6\_9\]) and (\[eq6\_10\]) are not strict. @guo11 have proposed an alternative criterion, $V_{\rm max} < \sqrt{GM_{\rm d}/3r_{\rm d}}$, which reflects that $V_{\rm max}$ of the real dark matter halo systems is smaller than that of ideal systems. In this paper, we use Equation (\[eq6\_10\]). We show in Figure \[fig6\_5\] the distribution in the $\lambda'$–$m_{\rm d}$ plane of our star-forming galaxies over $z \sim$ 2 $-$ 4. We find most of the data points to be near the line of instability over the entire redshift range regardless of the method to estimate dark halo masses. This implies some fractions of $z \sim 2-4$ galaxies may be dynamically changing the disk structure toward forming a bar and a bulge through bar formation. To compare with local spiral galaxies, we assume $\lambda = 0.04$ and $j_{\rm d}/m_{\rm d} \simeq 0.6$ [@fall12] in the present-day Universe. Then, the average value of $\lambda'$ is estimated as 0.024. The abundance matching result of @behroozi13 predicts $m_{\rm d}$ lower than 0.024 in a wide range of halo mass. This displays that local spiral galaxies appear to be more stable than high redshift galaxies. We have to keep in mind again that we should take into account a possible difference in angular momentum between gases and stars mentioned in Section \[sec\_conparison\]. In this case, the plots in Figure \[fig6\_5\] will move to more stable regions. Other than the global instability, there exist scenarios that form bars and bulges [@mo10]. For example, an interaction with a massive perturber leads to a bar-like structure [@noguchi87]. In addition to this, the migration of giant clumps, which are created by local Toomre Q instabilities [@toomre64], grows a bulge. Global instability may be one of the ways to explain galaxies with bars or bulges in the local Universe. Conclusion {#seq_con} ========== In this paper, we have used the 3D-HST GOODS-South, COSMOS, and AEGIS imaging data and galaxy catalog to analyze the relation between the ratio of the disk stellar mass to the halo mass, $m_\star \equiv M_\star/M_{\rm dh}$, and the fraction of the dark halo angular momentum transferred to the stellar disk, $j_\star \equiv J_\star/J_{\rm dh}$ for 11738 star-forming galaxies over the stellar mass range $8.3 < \log(M_{\star}/M_{\odot})< 11.1$ at $z\sim$ 2, 3, and 4. For each redshift, we have divided the catalog into several $M_{\star}$ bins and infer $M_{\rm dh}$ by two independent methods, clustering analysis and abundance matching, to obtain an average $m_{\star}$ value for each bin. We have confirmed that the two mass estimators give consistent results. For our objects we have also measured effective radii $r_{\rm d}$ at rest 5000Å with $\tt{GALFIT}$, and combined them with $m_{\star}$ and $M_{\rm dh}$ estimates to obtain $j_{\star}$ by applying @mo98 analytic model of disk formation. The followings are the main results of this paper.\ \ (i) We have found the median size evolution of disk star-forming galaxies $\overline{r}_{\rm d} (M_{\star,10}) / {\rm kpc} =6.88 (1+z)^{-0.91 \pm 0.01}$ at $M_{\star} = 1.0\times 10^{10}M_{\odot}$. This redshift evolution is in agreement with the results by @allen16 and @shibuya15. We have also analyzed the slope of the disk size–stellar mass relation. While the slope is consistent with the results by @vanderwel14 at $z\sim2$, we have found that the slope becomes shallower beyond $z\sim$ 2. The scatter of $r_{\rm d}$–$M_{\star}$ relation is $\sigma_{\ln r_{\rm d}} \sim 0.4$ $-$ 0.6 over the redshift range examined, which is comparable with the scatter of the log-normal distribution of $\lambda$.\ \ (ii) We have obtained the angular momentum retention factor $j_{\star}/m_{\star}$ averaged over mass and redshift to be $\simeq 0.77\pm0.06$ from clustering analysis and $\simeq 0.83\pm0.13$ from abundance matching. These values are in rough agreement with those of local late-type galaxies by @fall12 and those of star-forming galaxies at $z\sim0.8-2.6$ by @burkert16.\ \ (iii) Contrary to the star-forming galaxies at the present-day universe, $j_{\star}/m_{\star}$ appears to decrease with halo mass especially when abundance matching is used as the mass estimator. Combined with the slope of the $M_{\star}$–$M_{\rm dh}$ relation, this negative slope of the $j_{\star}/m_{\star}$–$M_{\rm dh}$ relation explains the shallow ($<0.2$) slopes of the $r_{\rm d}$–$M_{\star}$ relation obtained in this paper. We have also found a possible decrease in the $j_{\star}/m_{\star}$–$M_{\rm dh}$ slope from $z\sim2$ to $z\sim4$, which may imply that feedback processes also change over this redshift range.\ \ (iv) We have for the first time compared the observed mass–angular momentum relation with those of the recent galaxy formation simulations at $z\sim2$ by @sales12, @pedrosa15, and @stevens16. We have found that all of these simulations predict specific angular momenta systematically smaller than our values, which implies that these simulations produce too small disks at high redshifts while reproducing local measurements. We have also found that a significant fraction of our galaxies appear to be unstable against bar formation. Acknowledgments {#acknowledgments .unnumbered} =============== This work is based on observations taken by the 3D-HST Treasury Program (GO 12177 and 12328) with the NASA/ESA HST, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. This work is supported by KAKENHI (16K05286) Grant-in-Aid for Scientific Research (C) through Japan Society for the Promotion of Science (JSPS). R.K. acknowledges support from Grant-in-Aid for JSPS Research Fellow (16J01302). \[appen\] Before clustering analysis in Section \[seq\_halo\], we calculate the angular correlation functions for all five fields. We separate each sample to luminosity bins, and compare with previous results [@ouchi04; @lee06; @barone14]. Figure \[figap\_1\] shows the angular correlation functions for the GOODS-North and UDS fields. The clustering properties for these two fields are relatively smaller than the values by the previous results. The GOODS-North field has a negative correlation with luminosity. The UDS field has a smaller angular correlation function and there are no signals beyond 100 arcsec. Because of this strange behavior, we does not include these two fields for our analysis. The cause of this weak clustering properties is not clear. The small number of filters used for SED fitting may affect clustering properties.
--- abstract: 'When the backward shift operator on a weighted space $H^2_w=\{f=\sum_{j=0} ^{\infty} a_jz^j : \sum_{j=0}^{\infty} |a_j|^2w_j < \infty\}$ is an $n$-hypercontraction, we prove that the weights must satisfy the inequality $$\frac{w_{j+1}}{w_j} \leq {\frac{1+j}{n+j}}.$$ As an application of this result, it is shown that such an operator cannot be subnormal. We also give an example to illustrate the important role that the $n$-hypercontractivity assumption plays in determining the similarity of Cowen-Douglas operators in terms of the curvatures of their eigenvector bundles.' address: - 'Department of Mathematics, Hebei Normal University, Shijiazhuang, Hebei, 050016, China' - 'Department of Mathematics and Statistics, University at Albany- State University of New York, Albany, NY, 12222, USA' author: - Kui Ji - 'Hyun-Kyoung Kwon' - Jing Xu title: '$N$-hypercontractivity and similarity of Cowen-Douglas operators ' --- [^1] Introduction {#s1} ============ In order to generalize the much-celebrated model theorem of B. Sz.-Nagy and C. Foias, J. Agler in [@Agler], introduced the notion of an $n$-hypercontraction which extends that of a contraction. Let $\mathcal{H}$ be a separable Hilbet space and denote by $\mathcal{L}(\mathcal{H})$ the algebra of bounded, linear operators defined on $\mathcal{H}$. If $n$ is a positive integer, then an *$n$-hypercontraction* is an operator $T \in \mathcal{L}(\mathcal{H})$ with $$\sum_{j=0}^k (-1)^j{k \choose j}(T^*)^jT^j \geq 0,$$ for all $1 \leq k \leq n$. For a real number $r$ and an integer $0 \leq k \leq r$, set $${r\choose k} = \frac{r(r-1)\ldots(r-k+1)}{k!}.$$ One then considers the Hilbert space $\mathcal{M}_n$ of analytic functions on the unit disk $\mathbb{D}$ defined as $$\mathcal{M}_n:= \{ f= \sum_{k=0}^{\infty} \hat{f}(k)z^k: \sum_{k=0}^{\infty} |\hat{f}(k)|^2 \frac{1}{{n+k-1 \choose k}} < \infty \}.$$ As can be easily checked, different function spaces correspond to different $n$’s: the Hardy space for $n=1$ and the weighted Bergman spaces $A^2_{n-2}$ for $n \geq 2$. The space $\mathcal{M}_n$ is a reproducing kernel Hilbert space with kernel function given by $$K_n(z,w)=\frac{1}{(1-\overline{w}z)^{n}},$$ for $ z, w \in \mathbb{D}$. The vector-valued spaces $\mathcal{M}_{n, \mathcal{E}}$ with values in a separable Hilbert space $\mathcal{E}$ can also be naturally defined. The (forward) shift operator $S_{n,\mathcal{E}}$ on $\mathcal{M}_{n, \mathcal{E}}$ is defined as $$S_{n,\mathcal{E}}f(z):=zf(z),$$ and the backward shift operator $S^*_{n,\mathcal{E}}$ is its adjoint. We are now ready to state the following theorem by J. Agler: \[model\] For $T \in \mathcal{L}(\mathcal{H})$, there exist a Hilbert space $\mathcal{E}$ and an $S^*_{n, \mathcal{E}}$-invariant subspace $\mathcal{N} \subseteq \mathcal{M}_{n, \mathcal{E}}$ such that $T$ is unitarily equivalent to $S^*_{n, \mathcal{E}}|\mathcal{N}$ if and only if $T$ is an $n$-hypercontraction and $\lim\limits_{j\rightarrow \infty} \|T^j h\|=0$ for all $h \in \mathcal{H}$. The functionality of the $n$-hypercontractivity assumption is also apparent in the study of similarity. By using Theorem \[model\], the second author, with R. G. Douglas and S. Treil, proved a similarity theorem between an $n$-hypercontractive Cowen-Douglas operator $T \in \mathcal{L}(\mathcal{H})$ and the backward shift operator $S^*_n$ on $\mathcal{M}_n$ [@DKT]. Let us now recall the definition of a Cowen-Douglas operator. Let $\Omega$ be an open connected set of the complex plane $\mathbb{C}$ and let $m$ be a positive integer. The Cowen-Douglas class $ B_{m}(\Omega)$ consists of operators $ T\in \mathcal{L}(\mathcal{H})$with the following conditions: 1. ${\Omega}\text{ }{\subset}\text{ }{\sigma}(T)=\{w{\in} \mathbb{C} \text{}: T-w {\mbox { is not invertible}} \};$ 2. ${\operatorname{ran}}(T-w)$ is closed $\text{ for every } w\in\Omega$; 3. $\bigvee \limits_{w{\in}{\Omega}} \ker(T-w)=\mathcal H$; and 4. $\dim \ker (T-w)=m \text{ for every } w\in\Omega$. One of the main results of [@CD] states that each operator $T\in B_m(\Omega)$ induces a Hermitian holomorphic eigenvector bundle $$\mathcal{E}_T:=\{(w, x)\in \Omega\times {\mathcal H}: x \in \ker (T-w)\},$$ over $\Omega$. Since condition (4) implies that $\mathcal{E}_T$ is a bundle of rank $m$, we set $\{e_j(w)\}^{m}_{j=1}$ to be its holomorphic frame. Letting $$h(w):=(\langle e_j(w),e_i(w \rangle)_{m \times m},$$ for each $w\in \Omega,$ the curvature function $\mathcal{K}_T$ of $\mathcal{E}_T$ is defined as $$\mathcal{K}_T=-\overline{\partial}(h^{-1}\partial h).$$ For $T\in B_1(\Omega)$, the curvature function is much simpler to calculate as it is equivalent to $$\mathcal{K}_T(w)=-\partial \bar{\partial} \log ||\gamma(w)||^2,$$ where $\gamma(w)\in \ker(T-w)$ is a holomorphic cross section of $\mathcal{E}_T$ [@CD]. More recently, the first two authors, along with Y. Hou, showed that the results of [@DKT] can be rephrased. The following are equivalent: 1. An $n$-hypercontractive Cowen-Douglas operator $T \in B_m(\mathbb{D})$ is similar to $S^*_{n, \mathbb{C}^m}$ on $\mathcal{M}_{n,\mathbb{C}^m}$. 2. $\partial \overline{\partial} \psi (w) \geq \text{trace }\mathcal{K}_{\bigoplus\limits^{m}_{j=1}S^*_{n}}(w)-\text{trace }\mathcal{K}_T(w),$ for some positive, bounded, subharmonic fiunction $\psi$ defined on $\mathbb{D}$ and for every $w \in \mathbb{D}$. When $n=m=1$, $T$ is a contraction and $S^*_1$ is just the adjoint of the shift operator on the Hardy space. In [@KT1], the second author and S. Treil gave an example of a backward shift operator $T$ (that is not a contraction) defined on a weighted space that is not similar to $S_1^*$ but such that it still satisfies the inequality $$\partial \overline{\partial} \psi (w) \geq \mathcal{K}_{S_1^*}(w)- \mathcal{K}_T(w).$$ This means that one cannot ignore the contraction assumption when considering the similarity to the backwad shift operator on the Hardy space in terms of curvature. We try to do something analogous here and consider weighted spaces and $n$-hypercontractions. In particular, we give a necessary condition for the backward shift operator defined on a weighted space to be an $n$-hypercontraction. The first two cases are trivial to show. For $n \geq 3$, we make clever use of certain systems of linear equations with solutions that have negative entries. This work is done through the two lemmas in the next section. As corollaries of this result, we consider the subnormality problem of these weighted backward shift operators and also state a related result involving curvature. In the last section, we use $K$-theory to show that without the $n$-hypercontractivity assumption, the similarity criteria given in [@HJK] fails for the higher rank cases as well. $n$-hypercontractive backward shift operators ============================================= The following theorem is our main result of the paper. \[main\] Let $T$ be the backward shift operator on the space $$H^2_w=\{f=\sum_{j=0} ^{\infty} a_jz^j : \sum_{j=0}^{\infty} |a_j|^2w_j < \infty\},$$ where $w_j >0$, $\liminf_j |w_j|^{\frac{1}{j}}=1$, and $\sup_j \frac{w_{j+1}}{w_j} < \infty$. If $T$ is an $n$-hypercontraction, then we have for every nonnegative integer $j$, $$\frac{w_{j+1}}{w_j} \leq {\frac{1+j}{n+j}}.$$ Note that the condition $\liminf_j |w_j|^{\frac{1}{j}}=1$ makes $H^2_w$ a space of analytic functions on the unit disk $\mathbb{D}$, while the condition $\sup_j \frac{w_{j+1}}{w_j} < \infty$ guarantees the boundedness of the shift operator on the space. It is also easy to see that $T$ should be of the form $$T \left (\sum_{j=0} ^{\infty} a_jz^j \right )=\sum_{j=0} ^{\infty} \frac{w_{j+1}}{w_j} a_{j+1}z^j.$$ Based on the definition given by A. L. Shields in [@SH], $T$ is a weighted shift operator with weight sequence given by $\left \{ \sqrt {{\frac{w_{j+1}}{w_j}}} \right \}_{j=0} ^{\infty}$. To give a proof of the above theorem, we need a few lemmas. \[x1\] Let $n \geq 2$ be a positive integer. For each $1 \leq j \leq k-1$, set $$x_{j}=-{n+(j-2)\choose j}\frac{k-j}{k},$$ where $2 \leq k \leq n$. Then, $\begin{bmatrix}\begin{smallmatrix}\setlength{\arraycolsep}{0.2pt} \begin{array}{ccccccc} -{n \choose 1}&1&0&\cdots&0&0\\ {n \choose 2}&-{n \choose 1}&1&\cdots&0&0\\ -{n \choose 3}&{n \choose 2}&-{n \choose 1}&\cdots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ (-1)^{j}{n \choose j}&(-1)^{j-1}{n \choose j-1}&(-1)^{j-2}{n \choose j-2}&\cdots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ (-1)^{k-2}{n \choose k-2} & (-1)^{k-3}{n \choose k-3} & (-1)^{k-4}{n \choose k-4} & \cdots & -{n \choose 1} & 1\\ (-1)^{k-1}{n \choose k-1} & (-1)^{k-2}{n \choose k-2} & (-1)^{k-3}{n \choose k-3} & \cdots & {n \choose 2} & -{n \choose 1}\\ \end{array} \end{smallmatrix}\end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_j \\ \vdots \\x_{k-2} \\ x_{k-1} \end{bmatrix} = \begin{bmatrix}(-1)^{2}{n\choose 2}\\ (-1)^{3}{n\choose 3}\\ (-1)^{4}{n\choose 4}\\ \vdots \\ (-1)^{j+1}{n\choose j+1}\\ \vdots \\ (-1)^{k-1}{n\choose k-1}\\ (-1)^{k}{n\choose k}\end{bmatrix}$. The conclusion is equivalent to $$\linespread{0.5}\selectfont \left\{ \begin{array}{cc} {n \choose 2}=-{n \choose 1}x_{1}+x_{2}\\ -{n \choose 3}={n \choose 2}x_{1}-{n \choose 1}x_{2}+x_{3}\\ \vdots\\ (-1)^{j+1}{n \choose j+1}=(-1)^{j}{n \choose j}x_{1}+(-1)^{j-1}{n \choose j-1}x_{2}+\cdots+(-1)^{1}{n \choose 1}x_{j}+x_{j+1}\\ \vdots\\ (-1)^{k-1}{n \choose k-1}=(-1)^{k-2}{n \choose k-2}x_{1}+(-1)^{k-3}{n \choose k-3}x_{2}+\cdots+(-1)^{1}{n \choose 1}x_{k-2}+x_{k-1}\\ \\ (-1)^{k}{n \choose k}=(-1)^{k-1}{n \choose k-1}x_{1}+(-1)^{k-2}{n \choose k-2}x_{2}+\cdots+(-1)^{2}{n \choose 2}x_{k-2}+(-1)^{1}{n \choose 1}x_{k-1}. \end{array} \right.$$ We proceed by strong induction. Let $x_{1}=-{n+(1-2)\choose 1}\frac{k-1}{k}$ and substituting this into the first equation, we obtain $$x_{2}=-{n+(2-2)\choose 2}\frac{k-2}{k}.$$ For $2 \leq m \leq k-1$, we will set for each $1 \leq j \leq m-1$, $$x_{j}=-{n+(j-2)\choose j}\frac{k-j}{k},$$ and prove that $$x_{m}=-{n+(m-2)\choose m}\frac{k-m}{k}.$$ This means that $$x_{j}=-{n+(j-2)\choose j}\frac{k-j}{k},$$ for $1 \leq j \leq m$, must satisfy the equations $$\label{one} (-1)^{m}{n \choose m}-(-1)^{m-1}{n \choose m-1}x_{1}-(-1)^{m-2}{n \choose m-2}x_{2}-\cdots-(-1)^{1}{n \choose 1}x_{m-1}-x_{m}=0,$$ and $$\label{two} (-1)^{k}{n \choose k}=(-1)^{k-1}{n \choose k-1}x_{1}+(-1)^{k-2}{n \choose k-2}x_{2}+\cdots+(-1)^{2}{n \choose 2}x_{k-2}+(-1)^{1}{n \choose 1}x_{k-1}.$$ Note that (\[one\]) is equivalent to $$\begin{aligned} 0 &=&\sum\limits_{j=0}^{m}(-1)^{j}{n \choose j}{n+m-2-j\choose m-j}\frac{k-(m-j)}{k}\\ &=&\sum\limits_{j=0}^{m}(-1)^{j}{n \choose j}{n+m-2-j \choose m-j}-\frac{1}{k}\sum\limits_{j=0}^{m-1}(-1)^{j}(m-j){n \choose j}{n+m-2-j \choose m-j}.\end{aligned}$$ To show (\[one\]), we will prove $$\label{first}\sum\limits_{j=0}^{m}(-1)^{j}{n \choose j}{n+m-2-j \choose m-j}=0,$$ and $$\label{second} \sum\limits_{j=0}^{m-1}(-1)^{j}(m-j){n \choose j}{n+m-2-j \choose m-j}=0.$$ Since $$(1+x)^{n}=\sum\limits_{j=0}^{n}{n \choose j}x^{j},$$ and $$(1+x)^{-n}=\sum\limits_{j=0}^{\infty}{-n \choose j}x^{j}=\sum\limits_{j=0}^{\infty}(-1)^{j}{n+j-1 \choose j}x^{j},$$ for $n \geq 0$, it follows that $$\label{forfirst} 1+x=(1+x)^{n}\times(1+x)^{-(n-1)}= \left (\sum\limits_{j=0}^{n}{n \choose j}x^{j} \right ) \left (\sum\limits_{j=0}^{\infty}(-1)^{j}{n+j-2 \choose j}x^{j} \right ).$$ One then observes that the coefficients of $x^m$ for $m \geq 0$ on the right side of (\[forfirst\]) are given by $$\sum\limits_{j=0}^{m}(-1)^{m-j}{n \choose j}{n+m-2-j \choose m-j}.$$ Comparing the coefficients of $x^m$ from both sides of (\[forfirst\]) for $m \geq 2$ now yields (\[first\]). Similarly, since $$-\frac{n-1}{(1+x)^{n}}=[(1+x)^{-(n-1)}]^{\prime}= \left [ \sum\limits_{j=0}^{\infty}(-1)^{j}{n+j-2 \choose j}x^{j} \right ]'=\sum\limits_{j=1}^{\infty}(-1)^{j}j{n+j-2 \choose j}x^{j-1},$$ we have $$\label{forsecond}1-n=(1+x)^{n}\times\frac{1-n}{(1+x)^{n}}=\left (\sum\limits_{j=0}^{n}{n \choose j}x^{j}\right ) \left (\sum\limits_{j=1}^{\infty}(-1)^{j}j{n+j-2 \choose j}x^{j-1}\right ).$$ The coefficients of $x^{m-1}$ for $m \geq1$ on the right side of (\[forsecond\]) are given by $$\sum \limits_{j=0}^{m-1}(-1)^{m-j}(m-j){n \choose j}{n+m-2-j \choose m-j},$$ and again, (\[second\]) readily follows from comparing the coefficients in (\[forsecond\]) for $m \geq 2$. Lastly, since $$\begin{aligned} 0&=&(-1)^{k}{n \choose k}-(-1)^{k-1}{n \choose k-1}x_{1}-(-1)^{k-2}{n \choose k-2}x_{2}-\cdots-(-1)^{1}{n \choose 1}x_{k-1}\\ &=&\sum\limits_{j=1}^{k}(-1)^{j}{n \choose j}{n+k-2-j\choose k-j}\frac{k-(k-j)}{k}\\ &=&\sum\limits_{j=1}^{k}(-1)^{j}{n \choose j}{n+k-2-j\choose k-j}-\frac{1}{k}\sum\limits_{j=1}^{k-1}(-1)^{j}(k-j){n \choose j}{n+k-2-j \choose k-j}\\ &=&\sum\limits_{j=0}^{k}(-1)^{j}{n \choose j}{n+k-2-j \choose k-j}-\frac{1}{k}\sum\limits_{j=0}^{k-1}(-1)^{j}(k-j){n \choose j}{n+k-2-j \choose k-j},\end{aligned}$$ ($\ref{two}$) holds based on what was done for ($\ref{one})$. \[x2\] Let $m, n \geq 2$ be positive integers. For each $1 \leq j \leq n+m-1$, set $$x_{j}=-{n+(j-2)\choose j}\frac{n+m-j}{n+m}.$$ Then, $$\left [ \begin{smallmatrix} (-1)^{1}{n \choose 1}&1&0&......&0&0&......&0&0&\\ (-1)^{2}{n \choose 2}&(-1)^{1}{n\choose 1}&1&......&0&0&......&0&0&\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots&\\ (-1)^{j}{n\choose j}&(-1)^{j-1}{n\choose j-1}&(-1)^{j-2}{n\choose j-2}&\cdots&0&0&......&0&0&\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots&\\ (-1)^{n-1}{n\choose n-1}&(-1)^{n-2}{n\choose n-2}&(-1)^{n-3}{n\choose n-3}&\cdots&0&0&......&0&0&\\ (-1)^{n}{n\choose n}&(-1)^{n-1}{n\choose n-1}&(-1)^{n-2}{n\choose n-2}&\cdots&1&0&......&0&0&\\ 0&(-1)^{n}{n\choose n}&(-1)^{n-1}{n\choose n-1}&\cdots&(-1)^{1}{n\choose 1}&1&\cdots&0&0&\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots&\\ 0&0&0&\cdots&0&0&\cdots&(-1)^{2}{n\choose 2}&(-1)^{1}{n\choose 1} \end{smallmatrix}\right ] \left [\begin{smallmatrix} x_{1}\\ x_{2}\\ \vdots\\ x_{j}\\ \vdots\\ x_{n-1}\\ x_{n}\\ x_{n+1}\\ \vdots\\ x_{n+m-1}\\ \end{smallmatrix}\right ] = \left [ \begin{smallmatrix} (-1)^{2}{n\choose 2}\\ (-1)^{3}{n\choose 3}\\ \vdots\\ (-1)^{j+1}{n\choose j+1}\\ \vdots\\ (-1)^{n}{n\choose n}\\ 0\\ 0\\ \vdots\\ 0\\ \end{smallmatrix}\right ].$$ The conclusion is equivalent to $${\linespread{0.5}\selectfont \left\{ \begin{array}{cc} {n \choose 2}=-{n \choose 1}x_{1}+x_{2}\\ \\ -{n \choose 3}={n \choose 2}x_{1}-{n \choose 1}x_{2}+x_{3}\\ \\ \vdots\\ \\ (-1)^{j+1}{n \choose j+1}=(-1)^{j}{n \choose j}x_{1}+(-1)^{j-1}{n \choose j-1}x_{2}+\cdots+(-1)^{1}{n \choose 1}x_{j}+x_{j+1}\\ \\ \vdots\\ \\ (-1)^{n}{n \choose n}=(-1)^{n-1}{n \choose n-1}x_{1}+(-1)^{n-2}{n \choose n-2}x_{2}+\cdots+(-1)^{1}{n \choose 1}x_{n-1}+x_{n}\\ \\ 0=(-1)^{n}{n \choose n}x_1+(-1)^{n-1}{n \choose {n-1}}x_2+\cdots +(-1)^{1}{n \choose 1}x_{n}+x_{1+n} \\ \vdots\\ \\ 0=(-1)^{n}{n \choose n}x_{i}+(-1)^{n-1}{n \choose n-1}x_{i+1}+\cdots+(-1)^{1}{n \choose 1}x_{i+n-1}+x_{i+n}\\ \\ \vdots\\ \\ 0=(-1)^{n}{n \choose n}x_{m}+(-1)^{n-1}{n \choose n-1}x_{m+1}+\cdots+(-1)^{1}{n \choose 1}x_{n+m-1}. \end{array} \right.}$$ If we set $$x_{1}=-{n+(1-2) \choose 1}\frac{n+m-1}{n+m},$$ then according to Lemma \[x1\] with $k=n$, $$x_{j}=-{n+(j-2)\choose j}\frac{n+m-j}{n+m},$$ for all $1 \leq j \leq n-1$. As in the previous lemma, we then show that $$x_{j}=-{n+(j-2)\choose j}\frac{n+m-j}{n+m},$$ defined for $1 \leq j \leq n+m-1$, satisfy the following three equations: First, $$\label{secondfirst} 0=(-1)^{n}{n \choose n}-(-1)^{n-1}{n \choose n-1}x_{1}-(-1)^{n-2}{n \choose n-2}x_{2}-\cdots-(-1)^{1}{n \choose 1}x_{n-1}-x_{n}.$$ Second, for $1 \leq i \leq m-1$, $$\label{secondsecond} 0=(-1)^{n}{n \choose n}x_{i}+(-1)^{n-1}{n \choose n-1}x_{i+1}+\cdots+(-1)^{1}{n \choose 1}x_{i+n-1}+x_{i+n},$$ and third, $$\label{secondthird} 0=(-1)^{n}{n \choose n}x_{m}+(-1)^{n-1}{n \choose n-1}x_{m+1}+\cdots+(-1)^{2}{n \choose 2}x_{n+m-2}+(-1)^{1}{n \choose 1}x_{n+m-1}.$$ For (\[secondfirst\]), we see that it is equivalent to $$\begin{aligned} 0 &=&\sum\limits_{j=0}^{n}(-1)^{j}{n \choose j}{2n-2-j \choose n-j}\frac{n+m-(n-j)}{n+m}\\ &=&\sum\limits_{j=0}^{n}(-1)^{j}{n \choose j}{2n-2-j \choose n-j}-\frac{1}{n+m}\sum\limits_{j=0}^{n}(-1)^{j}(n-j){n \choose j}{2n-2-j \choose n-j},\end{aligned}$$ while to prove (\[secondsecond\]), we show that for $1 \leq i \leq m-1$, $$\begin{aligned} 0 &=&-\sum\limits_{j=0}^{n}(-1)^{j}{n \choose j}{2n+i-2-j \choose n+i-j}\frac{n+m-(n+i-j)}{n+m}\\ &=&-\sum\limits_{j=0}^{n}(-1)^{j}{n \choose j}{2n+i-2-j \choose n+i-j}+\frac{1}{n+m}\sum\limits_{j=0}^{n}(-1)^{j}(n+i-j){n \choose j}{2n+i-2-j \choose n+i-j}.\end{aligned}$$ Finally, (\[secondthird\]) amounts to showing $$\begin{aligned} 0 &=&-\sum\limits_{j=1}^{n}(-1)^{j}{n \choose j}{2n+m-2-j \choose n+m-j}\frac{n+m-(n+m-j)}{n+m}\\ &=&-\sum\limits_{j=1}^{n}(-1)^{j}{n \choose j}{2n+m-2-j \choose n+m-j}+\frac{1}{n+m}\sum\limits_{j=1}^{n}(-1)^{j}(n+m-j){n \choose j}{2n+m-2-j \choose n+m-j}\\ &=&-\sum\limits_{j=0}^{n}(-1)^{j}{n \choose j}{2n+m-2-j \choose n+m-j}+\frac{1}{n+m}\sum\limits_{j=0}^{n}(-1)^{j}(n+m-j){n \choose j}{2n+m-2-j \choose n+m-j}.\end{aligned}$$ Note that once we prove that $$\label{endfirst} \sum\limits_{j=0}^{n}(-1)^{j}{n \choose j}{n+k-2-j \choose k-j}=0,$$ and that $$\label{endsecond} \sum\limits_{j=0}^{n}(-1)^{j}(k-j){n \choose j}{n+k-2-j \choose k-j}=0,$$ for $n \leq k \leq n+m$, the equations (\[secondfirst\]), (\[secondsecond\]), and (\[secondthird\]) will immediately follow. But it was already calculated in the previous lemma that for each $k \geq 0$, the term $$\sum\limits_{j=0}^{k}(-1)^{k-j}{n \choose j}{n+k-2-j \choose k-j}$$ represents the coefficient of $x^k$ in the expression $1+x$. Since $k \geq n \geq 2$ and ${n \choose k}=0$ for $k > n$, (\[endfirst\]) holds. One can show analogously that (\[endsecond\]) is true by using the fact that $$\sum \limits_{j=0}^{k-1}(-1)^{k-j}(k-j){n \choose j}{n+k-2-j \choose k-j}$$ is the coefficient of $x^{k-1}$ in the expression $1-n$ for $k \geq 1$. Proof of Theorem \[main\] ------------------------- The operator $T$ is of the form $$T \left (\sum_{j=0} ^{\infty} a_jz^j \right )=\sum_{j=0} ^{\infty} \frac{w_{j+1}}{w_j} a_{j+1}z^j,$$ and for the sake of simplicity, we will now set $$\lambda_j:=\frac{w_{j+1}}{w_j}.$$ Then, $$T=\begin{pmatrix} 0 & \sqrt{\lambda_0} & 0 & 0 & 0 & \cdots \\ 0 & 0 & \sqrt{\lambda_1} & 0 & 0 & \cdots \\ 0 & 0 & 0 & \sqrt{\lambda_2} & 0 & \cdots \\ \vdots & \vdots & \vdots & & \ddots\end{pmatrix},$$ and for every $m \geq 1$, $T^{*m}T^m$ is the diagonal matrix with the nonzero entry $$\prod_{j=k-1}^{m+k-2} \lambda_j,$$ in the $(m+k) \times (m+k)$ position for each positive integer $k$. If $T$ is a $1$-hypercontraction, that is, $$I-T^*T \geq 0,$$ then by looking at the entries of $I-T^*T$, we have $$1-\lambda_j \geq 0,$$ for every nonnegative integer $j$. This means that $\lambda_j \leq1={\frac{1+j}{1+j}}$ for every nonnegative integer $j$.\ If $T$ is a $2$-hypercontraction so that $$\sum_{j=0}^2 (-1)^j{2 \choose j}(T^*)^jT^j \geq 0,$$ then $$\begin{cases}1\geq0 \\ 1-{2 \choose 1}\lambda_0 \geq0\\ 1-{2 \choose 1}\lambda_1+{2 \choose 2}\lambda_1 \lambda_0 \geq0\\ \vdots\\ 1-{2 \choose 1}\lambda_k+{2 \choose 2} \ \lambda_k \lambda_{k-1}\geq0 \\ \vdots\\ \end{cases}.$$ From this, it is easy to see that $$\lambda_0 \leq {\frac{1+0}{2+0}} \text{ and }\lambda_1 \leq\frac{1}{2-\lambda_0}\ \leq \frac{1+1}{2+1}.$$ Now if we suppose that $$\lambda_{k-1} \leq {\frac{1+(k-1)}{2+(k-1)}},$$ then it follows that $$\lambda_k \leq \frac{1}{2-\lambda_{k-1}} \leq \frac{1+k}{2+k}.$$\ If $T$ is an $n$-hypercontraction for $n\geq3$, then $$\sum_{j=0}^n (-1)^j{n \choose j}(T^*)^jT^j \geq 0,$$ which equals $$\begin{cases} 1\geq0\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \\ 1-{n \choose 1}\lambda_0 \geq 0 \qquad\qquad\qquad\,\,\,\,\,\,\,\qquad\qquad\qquad\,\qquad\qquad\qquad\qquad (1)\\ 1-{n \choose 1}\lambda_1 +{n \choose 2}\lambda_1 \lambda_0 \geq0\qquad\qquad\qquad\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \qquad\qquad\qquad\,\,\,\,\, (2)\\ 1-{n \choose 1}\lambda_2+{n \choose 2}\lambda_2 \lambda_1-{n \choose 3}\lambda_2 \lambda_1 \lambda_0 \geq0\, \, \, \,\,\,\, \, \, \, \, \, \, \, \, \, \, \,\, \qquad \qquad \qquad \qquad (3)\\ \vdots\\ 1+\sum\limits_{j=1}^{k}(-1)^{j}{n \choose j}\lambda_{k-1}\lambda_{k-2}\cdots \lambda_{k-j} \geq0\,\,\,\,\,\,\,\qquad \qquad \qquad \qquad\qquad\,(k)\\ \vdots\\ 1+\sum\limits_{j=1}^{n}(-1)^{j}{n \choose j}\lambda_{n-1}\lambda_{n-2}\cdots \lambda_{n-j} \geq0\,\,\,\,\,\,\,\qquad \qquad \qquad \qquad\qquad(n)\\ 1+\sum\limits_{j=1}^{n}(-1)^{j}{n \choose j}\lambda_{n}\lambda_{n-1}\cdots \lambda_{n+1-j} \geq0\,\,\,\,\qquad\qquad \qquad \qquad\qquad(n+1)\\ \vdots\\ 1+\sum\limits_{j=1}^{n}(-1)^{j}{n \choose j}\lambda_{n+m-1}\lambda_{n+m-2}\cdots \lambda_{n+m-j} \geq0 \,\,\,\,\,\,\,\qquad \qquad \qquad(n+m)\\ \vdots \end{cases}$$ From inequality $(1)$, we get $$\lambda_0 \leq\frac{1}{n}=\frac{1+0}{n+0},$$ and we use it together with inequality $(2)$ to obtain $$\lambda_1 \leq\frac{1+1}{n+1}.$$ It is now the right time to resort to the lemmas that have been proved previously. Namely, by Lemma $1.2$, the $x_j=-{n+(j-2) \choose j}\frac{3-j}{3}$, for $1 \leq j \leq 2$, satisfy the equation $$\begin{bmatrix} -{n \choose 1}&1\\ {n \choose 2}&-{n \choose 1}\\ \end{bmatrix} \begin{bmatrix} \begin{array}{cc} x_{1}\\ x_{2}\\ \end{array} \end{bmatrix} = \begin{bmatrix} \begin{array}{cc} {n \choose 2}\\ -{n \choose 3}\\ \end{array} \end{bmatrix},$$ that is, $$\left\{ \begin{array}{cc} {n \choose 2}=-{n \choose 1}x_{1}+x_{2} \\ \\ -{n \choose 3}={n \choose 2}x_{1}-{n \choose 1}x_{2} \end{array}. \right.$$ Plugging this into inequality $(3)$, we have $0 \leq 1-{n \choose 1}\lambda_2+{n \choose 2}\lambda_2 \lambda_1-{n \choose 3}\lambda_2\lambda_1\lambda_0\\$\ $=1-{n \choose 1}\lambda_2+[-{n \choose 1}x_1+x_2 ]\lambda_2 \lambda_1+[{n \choose 2}x_1-{n \choose 1}x_2 ]\lambda_2 \lambda_1 \lambda_0\\$\ $=1-\lambda_2[{n \choose 1}+x_{1}]+x_{1}\lambda_2[1-{n \choose 1}\lambda_1+{n \choose 2}\lambda_1 \lambda_0 ]+x_{2}\lambda_2 \lambda_1 [1-{n \choose 1}\lambda_0].$ From the inequalities $(1)$ and $(2)$ of $(1.12)$, we obtain $$1-\left [{n \choose 1}+x_{1}\right ]\lambda_2 \geq 0,$$ by taking into account that for $1 \leq j \leq 2$, $$x_j <0.$$ Thus, $$\lambda_2 \leq\frac{1}{{n \choose 1}+x_{1}}=\frac{1+2}{n+2}.$$ In general, recall that Lemma \[x1\] states that for $2 \leq k \leq n$ and $x_{j}=-{n+(j-2) \choose j}\frac{k-j}{k}<0$, where $1 \leq j \leq k-1$, we have $$\linespread{0.5}\selectfont \left\{ \begin{array}{cc} {n \choose 2}=-{n \choose 1}x_{1}+x_{2} \\ \\ -{n \choose 3}={n \choose 2}x_{1}-{n \choose 1}x_{2}+x_{3}\\ \\ \vdots\\ \\ (-1)^{j}{n \choose j}=(-1)^{j-1}{n \choose j-1}x_{1}+(-1)^{j-2}{n \choose j-2}x_{2}+\cdots+(-1)^{1}{n \choose 1}x_{j-1}+x_{j}\\ \\ \vdots\\ \\ (-1)^{k-1}{n \choose k-1}=(-1)^{k-2}{n \choose k-2}x_{1}+(-1)^{k-3}{n \choose k-3}x_{2}+\cdots+(-1)^{1}{n \choose 1}x_{k-2}+x_{k-1}\\ \\ (-1)^{k}{n \choose k}=(-1)^{k-1}{n \choose k-1}x_{1}+(-1)^{k-2}{n \choose k-2}x_{2}+\cdots+(-1)^{2}{n \choose 2}x_{k-2}+(-1)^{1}{n \choose 1}x_{k-1}\\ \end{array}. \right.$$ Then from inequality $(k)$ of (1.12), we have Now based on the inequalities $(1)$ through $(k)$ of $(1.12)$, we have for every $1 \leq m \leq k$, $$1+\sum\limits_{j=1}^{m}(-1)^{j}{n \choose j}\lambda_{m-1}\lambda_{m-2}\cdots\lambda_{m-j}\geq0,$$ and therefore, using the fact that $x_j < 0$ for every $1 \leq j \leq k-1$, the inequality $$1-\left[{n \choose 1}+x_{1}\right]\lambda_{k-1}\geq0,$$ follows. Then for every $2 \leq k \leq n$, $$\lambda_{k-1}\leq\frac{1}{{n \choose 1}+x_{1}}=\frac{1+(k-1)}{n+(k-1)}.$$ Since it has been observed already that $\lambda_0 \leq \frac{1+0}{n+0}$, the inequality holds for all $1 \leq k \leq n$. For $n+1 \leq k$, we make use of Lemma 1.3 that states that for $$x_{j}=-{n+(j-2) \choose j}\frac{n+m-j}{n+m},$$ with $1 \leq j \leq n+m-1$ and $n, m \geq 2$, one has $${\linespread{0.5}\selectfont \left\{ \begin{array}{cc} {n \choose 2}=-{n \choose 1}x_{1}+x_{2}\\ \\ -{n \choose 3}={n \choose 2}x_{1}-{n \choose 1}x_{2}+x_{3}\\ \\ \vdots\\ \\ (-1)^{j+1}{n \choose j+1}=(-1)^{j}{n \choose j}x_{1}+(-1)^{j-1}{n \choose j-1}x_{2}+\cdots+(-1)^{1}{n \choose 1}x_{j}+x_{j+1}\\ \\ \vdots\\ \\ (-1)^{n}{n \choose n}=(-1)^{n-1}{n \choose n-1}x_{1}+(-1)^{n-2}{n \choose n-2}x_{2}+\cdots+(-1)^{1}{n \choose 1}x_{n-1}+x_{n}\\ \\ 0=(-1)^{n}{n \choose n}x_1+(-1)^{n-1}{n \choose {n-1}}x_2+\cdots +(-1)^{1}{n \choose 1}x_{n}+x_{1+n} \\ \vdots\\ \\ 0=(-1)^{n}{n \choose n}x_{i}+(-1)^{n-1}{n \choose n-1}x_{i+1}+\cdots+(-1)^{1}{n \choose 1}x_{i+n-1}+x_{i+n}\\ \\ \vdots\\ \\ 0=(-1)^{n}{n \choose n}x_{m}+(-1)^{n-1}{n \choose n-1}x_{m+1}+\cdots+(-1)^{1}{n \choose 1}x_{n+m-1} \end{array}. \right.}$$ Now, by inequality $(n+m)$ in $(1.12),$ we have $\footnotesize{\begin{array}{llll} \,&1+\sum\limits_{j=1}^{n}(-1)^{j}{n \choose j}\lambda_{n+m-1}\lambda_{n+m-2}\cdots\lambda_{n+m-j}\\ =&1-{n \choose 1}\lambda_{n+m-1}+ \sum\limits_{j=2}^{n}\left [\sum\limits_{l=0}^{j-1}(-1)^{l}{n \choose l}x_{j-l}\right ]\lambda_{n+m-1}\lambda_{n+m-2}\cdots \lambda_{n+m-j}\\ \,&+\sum\limits_{j=1}^{m-1}\left [\sum\limits_{l=0}^{n}(-1)^{l}{n \choose l}x_{n+j-l}\right ]\lambda_{n+m-1}\lambda_{n+m-2}\cdots\lambda_{m-j}+ \left[\sum\limits_{j=1}^{n}(-1)^{j}{n \choose j}x_{n+m-j}\right]\prod\limits_{l=1}^{n+m}\lambda_{l-1}\\ =&1-\left[{n \choose 1}+x_{1}\right]\lambda_{n+m-1}\\ &+\, \sum\limits_{j=1}^{m} \left(\left[1+\sum\limits_{l=1}^{n}(-1)^{l}{n \choose l}\lambda_{n+m-j-1}\lambda_{n+m-j-2}\cdots \lambda_{n+m-j-l}\right]x_{j}\lambda_{n+m-1} \cdots\lambda_{n+m-j}\right)\\ &+ \sum\limits_{j=m+1}^{n+m-1}\left( \left[1+\sum\limits_{l=1}^{n+m-j}(-1)^{l}{n \choose l}\lambda_{n+m-j-1}\cdots\lambda_{n+m-j-l}\right]x_{j}\lambda_{n+m-1} \cdots\lambda_{n+m-j}\right)\\ \geq&0. \end{array}}$ Again, the inequalities in $(1.12)$ give for all $1 \leq k \leq n$, $$1+\sum\limits_{j=1}^{k}(-1)^{j}{n \choose j}\lambda_{k-1}\lambda_{k-2}\cdots\lambda_{k-j}\geq0,$$ and for all $l \geq 0$, $$1+\sum\limits_{j=1}^{n}(-1)^{j}{n \choose j}\lambda_{n+l-1}\lambda_{n+l-2}\cdots\lambda_{n+l-j}\geq0.$$ Since $$x_{j}=-{n+(j-2) \choose j}\frac{n+m-j}{n+m}<0,$$ for $1 \leq j \leq n+m-1$, we conclude that $$1-\left[{n \choose 1}+x_{1} \right]\lambda_{n+m-1} \geq 0.$$ Thus, $$\lambda_{n+m-1}\leq\frac{1}{{n \choose 1}+x_{1}}=\frac{1+(n+m-1)}{n+(n+m-1)},$$ for every $m \geq 2$, and we then have $$\lambda_j =\frac{w_{j+1}}{w_j} \leq \frac{1+j}{n+j},$$ for every nonnegative integer $j$. Theorem \[main\] readily yields the following results. Recall that a *subnormal* operator is an operator with a normal extension. A weighted backward shift operator cannot be subnormal. It is known that an operator is an $n$-hypercontraction for all $n$ if and only if it is a subnormal contraction ([@Agler2]). Let $T$ be the backward shift operator on one of the spaces $H^2_w$ with weight sequence $\left \{\sqrt{\frac{w_{j+1}}{w_j}} \right \}_{j=0}^{\infty}$ and let it be subnormal. Since subnormality is preserved under the scalar multiplication operation, we can assume without generality that $\|T\| \leq 1$. Then by Theorem \[main\], for any fixed integer $j \geq 0$, we have for every integer $n \geq 1$, $\frac{w_{j+1}}{w_j} \leq \frac{1+j}{n+j}$. Since $\lim\limits_{n\rightarrow \infty} {\frac{1+j}{n+j}}=0,$ for every integer $j \geq 0$, $$\limsup_j \frac{w_{j+1}}{w_j}=0,$$ which is a contradiction to $\liminf_j |w_j|^{\frac{1}{j}}=1$. Next, let us recall how given an integer $n \geq 1$, the Hilbert space $\mathcal{M}_n$ of functions on the unit disk $\mathbb{D}$ is defined: $$\mathcal{M}_n= \{ f= \sum_{k=0}^{\infty} \hat{f}(k)z^k: \sum_{k=0}^{\infty} |\hat{f}(k)|^2 \frac{1}{{n+k-1 \choose k}} < \infty \}.$$ Using the proof of Theorem \[main\], one can also show that the backward shift operator $S^*_n$ on $\mathcal{M}_n$ is “almost" $n$-isometric”. Set $\mathcal{M}^1_n=\bigvee \{z^m: m \geq 1\}\subset \mathcal{M}_n$ and denote by $\mathcal{P}_n^1$ the orthogonal projection from $\mathcal{M}_n $ to $\mathcal{M}^1_n$. Then $$\mathcal{P}_n^1 \left (\sum_{j=0}^n (-1)^j{n \choose j}(S_n)^j(S^*_n)^j \right )\bigg \vert_{\mathcal{M}^1_n}=0.$$ In addition, we construct in the next corollary a weighted space whose backward shift operator satisfies an inequality involving curvatures with respect to the operator $S^*_n$ on $\mathcal{M}_n$. This inequality looks almost the same as the one that appears in the similarity criteria but one can no longer say anything about subharmonicity. The following well-known result by A. L. Shields that helps determine when two weighted shift operators are similar will be used in one part of the proof. \[SH\] Let $T_1$ and $T_2$ be unilateral shifts with weight sequences $\{\lambda_j\}_{j=0}^{\infty}$ and $\{\tilde{\lambda}_j\}_{j=0}^{\infty}$, respectively. Then $T_1$ and $T_2$ are similar if and only if there exist positive constants $C_1$ and $C_2$ such that $$0<C_1\leq \Big |\frac{\lambda_k\cdots \lambda_j}{\tilde{\lambda}_k\cdots \tilde{\lambda}_j} \Big | \leq C_2,$$ for all $k\leq j$. \[maincor\] For each operator $S^*_n$ on $\mathcal{M}_n$, there exist a weighted backward shift operator $T$ that is not an $n$-hypercontraction and a positive, bounded, real-analytic function $\psi$ defined on the unit disk $\mathbb{D}$ such that $$\partial\bar{\partial} \psi(w)=\mathcal{K}_{S^*_n}(w)-\mathcal{K}_T(w),$$ for every $w \in \mathbb{D}$. Moreover, $T$ is not similar to $S^*_n$. We will define our backward shift operator $T$ on some weighted space $$\mathcal{H}=\left \{ \sum_{j=0}^{\infty} a_jz^j: \sum_{j=0}^{\infty} |a_j|^2w_j < \infty \right \}.$$ The operator $S^*_n$ is an $n$-hypercontraction and using the reproducing kernel for the space $\mathcal{M}_n$, we have $$\mathcal{K}_{S^*_n}(w)=-\partial\overline{\partial}\log\frac{1}{(1-|w|^{2})^{n}}.$$ If we write $\mathcal{K}_T$ as $$\mathcal{K}_{T}(w)=-\partial\overline{\partial}\log k_{{w}}(w),$$ where $k_{{w}}(z)=\sum\limits_{j=0}^{\infty}\frac{\overline{w}^jz^j}{w_j}$ denotes the reproducing kernel of $\mathcal{H}$, then $$\mathcal{K}_{S^*_n}(w)-\mathcal{K}_{T}(w)=\partial\overline{\partial}\log\left [k_{{w}}(w){(1-|w|^{2})^{n}}\right ].$$ Hence, in order to prove that a positive, bounded, real-analytic function $\psi$ exists, we have to show that $k_{{w}}(w){(1-|w|^{2})^{n}}$ is bounded above and below by positive constants. We first consider the sequence $${\widetilde w}_{j}:=\frac{j!(n-1)!}{(n+j-1)!},$$ that appears in the following familiar expansion for $(1-x)^{-n}$: $$(1-x)^{-n}=\sum_{j=0}^{\infty}{ {n+j-1}\choose{j}} x^j.$$ We now construct the sequence ${w}_j$ for the space $\mathcal{H}$. Let $$w_{j}= \begin{cases} l{\widetilde w}_{j},\qquad j=N_{i}+l, \text{ for }1 \leq l \leq i,\\ l{\widetilde w}_{j},\qquad j=N_{i}+2i-l, \text{ for }1 \leq l \leq i,\\ {\widetilde w}_j,\qquad\qquad otherwise, \end{cases}$$ where the sequence $\{N_i\}_{i \geq 1}$ consists of positive integers $N_i > n-2$ with $$N_i+2i < N_{i+1}.$$ More details on the $N_i$ will be given later. Then, since $\frac{1}{w_{j}}-\frac{(n+j-1)!}{j!(n-1)} \neq 0$ only for $j=N_i+p$, where $2 \leq p \leq 2i-2$, we have $$\begin{aligned} k_{w}(w)&=&\sum\limits_{j=0}^{\infty}\frac{1}{w_{j}}(|w|^{2})^{j}\\ &=&\frac{1}{(1-|w|^{2})^{n}}+\sum\limits_{j=2}^{\infty}\left[\frac{1}{w_{j}}-\frac{(n+j-1)!}{j!(n-1)!}\right ](|w|^{2})^{j}\\ &=&\frac{1}{(1-|w|^{2})^{n}}+\sum\limits_{i=2}^{\infty} \sum\limits_{j=N_i+2}^{N_i+2i-2}\left[\frac{1}{w_{j}}-\frac{(n+j-1)!}{j!(n-1)!}\right ](|w|^{2})^{j}\\ &=:&\frac{1}{(1-|w|^2)^n}+\sum\limits_{i=2}^{\infty} g_i(w),\end{aligned}$$ where, $$\begin{aligned} |g_{i}(w)|=& \left | \sum\limits_{j=2}^{i}\left[\frac{1}{w_{N_{i}+j}}-\frac{(n+N_{i}+j-1)!}{(N_{i}+j)!(n-1)!}\right]|w|^{2N_{i}+2j}+\sum\limits_{j=2}^{i-1}\left[\frac{1}{w_{N_{i}+2i-j}}-\frac{(n+N_{i}+2i-j-1)!}{(N_{i}+2i-j)!(n-1)!}\right]|w|^{2N_{i}+4i-2j} \right | \\ =&\frac{(n+N_{i}-1)!}{N_{i}!(n-1)!}|w|^{2N_{i}}\left | \sum\limits_{j=2}^{i}\frac{(n+N_{i}+j-1)!N_{i}!}{(n+N_{i}-1)!(N_{i}+j)!}\left (\frac{1}{j}-1 \right)|w|^{2j}+\sum\limits_{j=2}^{i-1}\frac{(n+N_{i}+2i-j-1)!N_{i}!}{(n+N_{i}-1)!(N_{i}+2i-j)!}\left (\frac{1}{j}-1 \right)|w|^{4i-2j}\right |\\ \leq&\frac{(n+N_{i}-1)!}{N_{i}!(n-1)!}|w|^{2N_{i}}\left |\sum\limits_{j=2}^{i}2^{i}\left(\frac{1}{j}-1\right)|w|^{2j}+\sum\limits_{j=2}^{i-1}2^{i}\left (\frac{1}{j}-1\right)|w|^{4i-2j}\right |.\end{aligned}$$ Next, set $$M_i:=\sup \limits_{|w|<1} \left |\sum\limits_{j=2}^{i}2^{i}\left(\frac{1}{j}-1\right)|w|^{2j}+\sum\limits_{j=2}^{i-1}2^{i}\left (\frac{1}{j}-1\right)|w|^{4i-2j}\right |,$$ a constant that depends only on $i$ and not on the $N_i$. By direct calculation, one easily sees that $$M_i<i2^{i+1}.$$ We then have $$k_{w}(w)(1-|w|)^{n}=1+\sum\limits_{i=2}^{\infty}g_{i}(w)(1-|w|^{2})^{n},$$ and $$|g_{i}(w)|(1-|w|^{2})^{n}\leq M_{i}\frac{(n+N_{i}-1)!}{N_{i}!(n-1)!}(|w|^{2})^{N_{i}}(1-|w|^{2})^{n}.$$ Now, for $x \in \mathbb{D}$, if we let $$f(x):=x^{N_{i}}(1-x)^{n},$$ then $$f'(x)=N_{i}x^{N_{i}-1}(1-x)^{n}-nx^{N_{i}}(1-x)^{n-1}=x^{N_{i}-1}(1-x)^{n-1}\left[N_{i}(1-x)-nx \right].$$ Since the function $f(x)$ attains a maximum of $\left (\frac{N_{i}}{N_{i}+n} \right)^{N_{i}}\left (\frac{n}{N_{i}+n} \right)^{n}$ at $x=\frac{N_{i}}{N_{i}+n},$ $$|g_{i}(w)|(1-|w|^{2})^{n}\leq M_{i}\frac{(n+N_{i}-1)!}{N_{i}!(n-1)!}\left (\frac{N_{i}}{N_{i}+n} \right )^{N_{i}}\left (\frac{n}{N_{i}+n} \right )^{n} \leq M_{i}\frac{n^{n}}{(n-1)!(N_{i}+n)}.$$ Now if we choose $$N_{i}>max \left [\frac{2^{i+2}M_{i}n^{n}}{(n-1)!}-n, n-2 \right],$$ then $$|g_{i}(w)|(1-|w|^{2})^{n}\leq M_{i}\frac{n^{n}}{(n-1)!(N_{i}+n)}<\frac{1}{2^{i+2}}.$$ Notice that since $M_i<i2^{i+1},$ one could have chosen $N_i=\frac{i2^{2i+3}n^n}{(n-1)!}.$ Furthermore, it can be shown that $N_i+2i<N_{i+1}.$ Thus, we have that $$\frac{7}{8} < k_{w}(w)(1-|w|^{2})^{n} <\frac{9}{8},$$ and therefore, $k_{w}(w)(1-|w|^2)^n$ is indeed bounded by positive constants. To show that $T$ is not an $n$-hypercontraction, we note the existence of some $n_{0}=N_{j}+j-1$ such that $${\frac{{w}_{n_{0}+1}}{{w}_{n_{0}}}}={\frac{j \widetilde{w}_{N_{j}+j}}{(j-1) \widetilde{w}_{N_{j}+j-1}}}={\frac{j}{j-1}}{\frac{N_{j}+j}{n+N_{j}+j-1}}>{\frac{1+n_0}{n+n_0}},$$ and apply Theorem \[main\]. Lastly, to show that $T$ and $S^*$ are not similar, we choose $n_0=N_{j}+j-1$ as in the previous case to get $$\frac{\prod\limits_{k=0}^{n_0}{\frac{\widetilde{w}_{k+1}}{\widetilde{w}_{k}}}}{\prod\limits_{k=0}^{n_0}{\frac{{w}_{k+1}}{{w}_{k}}}} =\frac{\prod\limits_{k=0}^{N_{j}+j-1}{\frac{\widetilde{w}_{{k}+1}}{\widetilde{w}_{k}}}}{\prod\limits_{k=0}^{N_{j}+j-1}{\frac{{w}_{k+1}}{{w}_{k}}}} ={\frac{\widetilde{w}_{N_{j}+j}}{{w}_{N_{j}+j}}}={\frac{j{w}_{N_{j}+j}}{w_{N_{j}+j}}}={j}\rightarrow+\infty,$$ as $j\rightarrow+\infty$, and the conclusion follows from Lemma \[SH\]. Trace of curvature and similarity of reducible operators in $B_m(\Omega)$. ============================================================================ In this section, we give a simple example to show that the $n$-hypercontraction assumption is needed to determine the similarity of operators in $B_m(\Omega)$ in terms of the trace of the curvatures as was claimed in [@HJK]. We first introduce some definitions and mention some results about strongly irreducible operators. We assume throughout the section that $T \in \mathcal{L}(\mathcal{H})$. $T$ is said to be strongly irreducible (denoted str-irred.) if there is no nontrivial idempotent in the commutant ${\mathcal A}^{\prime}(T)$ of $T$, that is, $T$ cannot be written as $$T=T_1\stackrel{\cdot}{+}T_2,$$ for some $T_i\in \mathcal{L}({\mathcal H}_i)$, where $1 \leq i \leq 2$, and ${\mathcal H}={\mathcal H}_1\stackrel{.}{+}{\mathcal H}_2.$ Let $n < \infty$. A set ${\mathcal P}=\{P_i\}_{i=1}^{n}$ of idempotents in ${\mathcal L}({\mathcal H})$ is called a unit finite decomposition of $T$ if $1.\,\, P_i{\in} {\mathcal A}^{\prime}(T)$ for all $1{\leq}i{\leq}n$; $2.\,\, P_iP_j={\delta}_{ij}P_i$ for all $1{\leq}i, j{\leq}n,$ where ${\delta}_{ij}=\left\{ \begin{array}{cc} 1 & i=j\\ 0 & i \neq j \end{array} \right. $; and $3. \,\, \sum\limits_{i=1}^{n}P_i=I_{\mathcal H},$ where $I_{\mathcal H}$ denotes the identity operator on $\mathcal H$. If, in addition, $4. \,\, P_i$ is a minimal idempotent in ${\mathcal A}^{\prime}(T)$, that is , $T|_{{\operatorname{ran}}P_i} \text{ is str-irred. for}$ $1 \leq i \leq n,$ then ${\mathcal P}$ is said to be a unit finite strong irreducible decomposition of $T$ and we call the cardinality of ${\mathcal P}$ the strong irreducible cardinality of $T$. It is clear that an operator $T$ has a unit finite strong irreducible decomposition if and only if it can be expressed as the direct sum of finitely many strongly irreducible operators. \[UD\] Let ${\mathcal P}=\{P_i\}_{i=1}^{m}$ and ${\mathcal Q}=\{Q_i\}_{i=1}^{n}$ be two unit finite strong irreducible decompositions of $T$. We say that $T$ has a unique strong irreducible decomposition up to similarity if $m=n$ and there exist an invertible operator $X \in {\mathcal A}^{\prime}(T)$ and a permutation ${\Pi}$ of the set $(1, 2, \cdots, n)$ such that $XQ_{{\Pi}(i)}X^{-1}=P_i$ for all $1{\leq}i{\leq}n.$ The work of the first author, in collaboration with X. Guo and C. Jiang, shows how this concept is related to Cowen-Douglas operators. \[JGJ\] Let $T$ be a Cowen-Douglas operator and set $$T^{(n)}:=\bigoplus\limits^{n}_{i=1}T.$$ Then $ T^{(n)}$ has a unique strong irreducible decomposition up to similarity. Denote by $M_{k}({\mathcal A}^{\prime}(T))$ the collection of all $k\times k$ matrices with entries from the commutant ${\mathcal A}^{\prime}(T)$ of an operator $T$. Let $$M_{\infty}({\mathcal A}^{\prime}(T))=\bigcup\limits^{\infty}_{k=1}M_{k}({\mathcal A}^{\prime}(T)),$$ and let $\mbox{\rm Proj}(M_k({\mathcal A}^{\prime}(T)))$ be the algebraic equivalence classes of idempotents in $M_{\infty}({\mathcal A}^{\prime}(T))$. Set $\bigvee ({\mathcal A}^{\prime}(T))=\mbox{\rm Proj}(M_{\infty}({\mathcal A}^{\prime}(T))).$ If $p$ and $q$ are idempotents in $\bigvee ({\mathcal A}^{\prime}(T))$, then we will say that $p{\sim}_{st}q$ if $p{\oplus}r$ and $q{\oplus}r$ are algebraically equivalent for some idempotent $r \in \bigvee ({\mathcal A}^{\prime}(T))$. The relation ${\sim}_{st}$ is known as stable equivalence. The $K_0$-group of ${\mathcal A}^{\prime}(T)$, denoted $K_0({\mathcal A}^{\prime}(T))$, is defined to be the Grothendieck group of $\bigvee ({\mathcal A}^{\prime}(T))$. Now recall that for $\alpha \geq 1$, $\mathcal{M}_{\alpha}$ is a Hilbert space with reproducing kernel given by $K_{\alpha}(z, w)=\frac{1}{(1-\bar{w}z)^{\alpha}}$ and with the backward shift operator $S^*_{\alpha}$. Lemma \[SH\] above shows that the backward shift operators on two different spaces cannot be similar. \[ab\] $S^*_{\alpha}$ and $S^*_{\beta}$ are similar if and only if $\alpha=\beta.$ \[CFJ\]Let $T=\bigoplus\limits_{k=1}^{l}S_{\alpha_k}^*$ and $\widetilde{T}=\bigoplus\limits_{m=1}^{s}S_{\beta_m}^*$, where $\alpha_k, \beta_m \geq 1$, and $\alpha_k\neq \alpha_{k^{\prime}}$ and $\beta_{m}\neq \beta_{m^{\prime}}$, for $k \neq k'$ and $m \neq m'$. Then $T$ is similar to $\widetilde{T}$ if and only if $l=s$ and there exists a permutation ${\Pi}$ of the set $(1, 2, \cdots, l)$ such that for every $k\leq l$, $\alpha_k=\beta_{\Pi(k)}$. It suffices to prove one implication and therefore, we let $T$ and $\widetilde{T}$ be similar. Without loss of generality, we assume that $l < s$. It is well-known that $S^*_{\alpha_k}\in B_1(\mathbb{D})$ and that it is strongly irreducible in $\mathcal{L}({\mathcal M}_{\alpha_k})$. If we let ${\mathcal H}=\bigoplus\limits_{k=1}^l{\mathcal M}_{\alpha_k}$, then $T|_{Ran I_{{\mathcal H}_k}}=S^*_{\alpha_k}$. Analogous results hold for the operator $\widetilde{T}$. Moreover, since $T\in B_l(\mathbb{D})$ and $\widetilde{T}\in B_s(\mathbb{D})$, $$T\oplus \widetilde{T}= \bigoplus\limits_{k=1}^lS^*_{\alpha_k}\oplus \bigoplus\limits_{m=1}^{s}S_{\beta_m}^* \in B_{l+s}(\mathbb{D}).$$ By Theorem \[JGJ\], for any positive integer $n$, both $T^{(n)}\oplus \widetilde{T}^{(n)}$ and $T^{(2n)}$ have unique strong irreducible decompositions up to similarity. Moreover, we know from Lemma \[ab\] that $S^*_{\alpha_k}$ and $S^*_{\alpha_k'}$ are not similar for $k \neq k'$. The same is true for $S^*_{\beta_m}$. Now, the results in [@CFJ] show that $$K_0 \left ({\mathcal A}^{\prime} \left (\bigoplus\limits_{j=1}^t T_j \right ) \right) \cong \mathbb{Z}^{t},$$ where the $T_j$ are strongly irreducible Cowen-Douglas operators such that no two of them are similar to each other. Since each $S^*_{\alpha_k}$ is a strongly irreducible Cowen-Douglas operator, we then have $$K_0({\mathcal A}^{\prime}(T^{(2)}))=K_0(M_{2}(\mathbb{C})\otimes {\mathcal A}^{\prime}(T))= K_0({\mathcal A}^{\prime}(T)) \cong \mathbb{Z}^{l}.$$ Notice that if $T\oplus \widetilde{T}$ is similar to $T^{(2)}$, then $$K_0({\mathcal A}^{\prime}(T\oplus \widetilde{T}))=K_0({\mathcal A}^{\prime}(T^{(2)})) \cong \mathbb{Z}^{l}.$$ On the other hand, if there exists a $\beta_m$ such that $S^*_{\beta_m}$ is not similar to any $S^*_{\alpha_k}$ in $T\oplus \widetilde{T}= \bigoplus\limits_{k=1}^lS^*_{\alpha_k}\oplus \bigoplus\limits_{m=1}^{s}S_{\beta_m}^*,$ then one can find a positive number $l^{\prime}>l$ such that $$K_0({\mathcal A}^{\prime}(T\oplus \widetilde{T})) \cong \mathbb{Z}^{l^{\prime}}.$$ This is a contradiction. The following example shows that the $n$-hypercontractivity assumption cannot be dispensed with in determining similarity. A more general example can be constructed in the same way. For every $w \in \mathbb{D}$, $$\text{trace }\mathcal{K}_{S^*_1\oplus S^*_3}(w)=-\frac{2}{(1-|w|^2)^2}=\text{trace }\mathcal{K}_{S^*_2\oplus S^*_2}(w).$$ But by Proposition \[CFJ\], we know that $S^*_1\oplus S^*_3$ is similar to $S^*_2\bigoplus S^*_2$ if and only if both $S^*_1$ and $S^*_3$ are similar to $S^*_2$, which is a contradiction. In fact, since $S^*_1$ is not a 2-hypercontraction, $S^*_1 \oplus S^*_3$ cannot be a 2-hypercontraction, either. [99]{} J. Agler, *The Arveson extension theorem and coanalytic models,* [Integr. Equat. Op. Thy. ]{} $\mathbf{5}$ (1982), 608–631. J. Agler, *Hypercontractions and subnormality,* [J. Operator Theory,]{} $\mathbf{13}$ (1985), 203–217. Y. Cao, J. Fang, and C. Jiang, *$K_0$-group of Banach algebras and decomposition of strongly irreducible operators,* J. Operator Theory, $\mathbf{48}$ (2002), 235–253. , *Complex geometry and operator theory*, Acta Math. $\mathbf{141}$ (1978), 187–261. R. G. Douglas, H. Kwon, and S. Treil, *Similarity of n-hypercontractions and backward Bergman shifts,* J. Lond. Math. Soc. $\mathbf{88}$ (2013), No. 2, 637–648. Y. Hou, K. Ji, and H. Kwon, *The trace of the curvature determines similarity.*, Studia Math. $\mathbf{236}$ (2017), No. 2, 193–200. K. Ji and J. Sarkar, *Similarity of quotient Hilbert modules in the Cowen-Douglas class,* accepted to European J. Math. , *$K$-group and similarity classification of operators,* J. Funct. Anal. $\mathbf{225}$ (2005), 167–192. *Similarity of operators and geometry of eigenvector bundles,* Publ. Mat. $\mathbf{53}$ (2009), 417–438. *Curvature condition for non-contractions does not imply similarity to the backward shift,* Integr. Equat. Op. Thy. $\mathbf{66}$ (2010), 529–538. *Weighted shift operators and analytic function theory,* Math. Surveys, No. $\mathbf{13}$ Amer. Math. Soc., Providence, R.I., 1974, 49-128. [^1]: The first author is supported by National Natural Science Foundation of China (Grant No. 11831006)
--- abstract: 'We derive a novel variational expectation maximization approach based on truncated variational distributions. Truncated distributions are proportional to exact posteriors within a subset of a discrete state space and equal zero otherwise. The novel variational approach is realized by first generalizing the standard variational EM framework to include variational distributions with exact (‘hard’) zeros. A fully variational treatment of truncated distributions then allows for deriving novel and mathematically grounded results, which in turn can be used to formulate novel efficient algorithms to optimize the parameters of probabilistic generative models. We find the free energies which correspond to truncated distributions to be given by concise and efficiently computable expressions, while update equations for model parameters (M-steps) remain in their standard form. Furthermore, we obtain generic expressions for expectation values w.r.t. truncated distributions. Based on these observations, we show how efficient and easily applicable meta-algorithms can be formulated that guarantee a monotonic increase of the free energy. Example applications of the here derived framework provide novel theoretical results and learning procedures for latent variable models as well as mixture models including procedures to tightly couple sampling and variational optimization approaches. Furthermore, by considering a special case of truncated variational distributions, we can cleanly and fully embed the well-known ‘hard EM’ approaches into the variational EM framework, and we show that ‘hard EM’ (for models with discrete latents) provably optimizes a lower free energy bound of the data log-likelihood.' author: - |  \ Jörg Lücke\ Machine Learning Lab\ Carl von Ossietzky Universität Oldenburg\ 26111 Oldenburg, Germany  \ title: Truncated Variational Expectation Maximization --- Introduction {#SecIntro} ============ The application of expectation maximization [EM; @DempsterEtAl1977] is a standard approach to optimize the parameters of probabilistic data models. The EM meta-algorithm hereby seeks parameters that optimize the data likelihood given the data model and given a set of data points. Data models are typically defined based on directed acyclic graphs, which describe the data generation process using probabilistic descriptions of sets of hidden and observed variables and their interactions. EM approaches for most non-trivial such generative data models are intractable, however, and tractable approximations to EM are, therefore, very wide-spread. EM approximations range from sampling-based approximations of expectation values and related non-parameteric approaches [e.g. @GhahramaniJordan1995], over maximum a-posterior or ‘hard EM’ approaches [e.g. @JuangRabiner1990; @CeleuxGovaert1992; @OlshausenField1996; @LeeEtAl2007; @MairalEtAl2010], Laplace approximations [e.g. @KassSteffey1989; @FristonEtAl2007] to variational EM approaches [@SaulEtAl1996; @NealHinton1998; @JordanEtAl1999; @OpperWinther2005; @Seeger2008 and many more]. Instead of aiming at a direct maximization of the data likelihood, variational EM seeks to maximize a lower-bound of the likelihood: the variational free energy. Variational free energies can be formulated such that their optimization becomes tractable by avoiding the summation or integration over intractably large hidden state spaces. Since the framework of variational free energy approximations has first been explicitly introduced in Machine Learning [e.g. @SaulEtAl1996; @NealHinton1998; @JordanEtAl1999], variational approximations for the EM meta-algorithm have been widely applied and were generalized in many different ways. Variational EM is now routinely used to train latent variable models (or multiple-causes) models, to train time-series models or to train complex graphical models including models for deep unsupervised learning [see, e.g @Jaakkola2001; @Bishop2006; @Murphy2012; @PatelEtAl2016 for overviews]. Very prominent examples of variational EM are based on factored variational distributions [@JordanEtAl1999] or Gaussian variational distributions [e.g. @OpperWinther2005; @Seeger2008; @OpperArchambeau2009]. Truncated distributions were introduced later than factored or Gaussian approaches [@LuckeSahani2008; @LuckeEggert2010], and they used instead of a variational loop a sparsity assumption [@LuckeSahani2008] or preselection of latent states [@LuckeEggert2010; @SheikhEtAl2014; @DaiLucke2014; @SheltonEtAl2017]. Among approaches which are not considered variational are sampling based approximations [e.g. @ZhouetAl2009], or approaches which use just one state (the one with the approximate maximal posterior value) for the optimization of model parameters [@JuangRabiner1990; @CeleuxGovaert1992; @OlshausenField1996; @AllahverdyanGalstyan2011; @OordEtAl2014]. The latter is commonly referred to as ‘MAP training’ [@OlshausenField1996; @AllahverdyanGalstyan2011], ‘hard EM’ [@PoonDomingos2011; @AllahverdyanGalstyan2011; @OordEtAl2014], ‘zero temperature EM’ [@TurnerSahani2011], as ‘classification EM’ [@CeleuxGovaert1992] for mixture models, or as ‘Viterbi training’ for hidden Markov Models [@JuangRabiner1990; @CohenSmith2010; @AllahverdyanGalstyan2011]. In deep learning, ‘hard EM’ was used, for instance, for generative formulations of convolutional neural networks [@PatelEtAl2016], deep Gaussian Mixture Models [@OordEtAl2014], or Sum-Product Networks [@PoonDomingos2011]. Note in this respect that also variational approximations (primarily factored ones) have been considered for (deep) graphical models, and deep models have been generalized to fully Bayesian settings [e.g. @Attias2000; @Jaakkola2001; @BealGhahramani2003]. For the purposes of this study, we will introduce a novel variational EM approach assuming one set of observed and one set of hidden variables (without distinguishing subsets of these latents). Such a setup may suggest a bipartite graphical models with no further structure, i.e., a data model in which all hidden variables have the same form of influence on the observed variables [compare, e.g., @NealHinton1998]. Such models are presumably best suited to follow the introduction of the basic ideas of the novel approach and to highlight its elementary properties. However, as we will not make assumptions about the set of hidden variables, we here stress that the derived results apply for any directed graphical model with discrete latents, i.e., application to more intricate models including time-series models or deep directed models are straight-forward. The same does not apply for generalizations to fully Bayesian settings, which would require a major (and potentially very interesting) future research effort. This paper summarizes the main result about the novel truncated variational EM (TV-EM) approach in Sec.\[SecSummary\]. The reader interested in applying the approach will find the required information there and a partial and explicit form of the algorithm at the end of Sec.\[SecOptimization\]. In Sec.\[SecVarApproaches\], we first introduce standard variational EM (and Gaussian and factored variational distributions). The introduction of the standard framework serves (A) for highlighting the differences between truncated approaches and standard approaches, and (B) will be required to point out where we have generalized to non-standard variational distributions. Sec.\[SecTVEM\] formally shows that truncated distributions can be treated as fully variational distributions by generalizing the standard free energy framework to include distributions with exact zeros. Sec.\[SecOptimization\] then derives the theoretical results which are used to formulate TV-EM as a meta-algorithm. Sec.\[SecApplications\] presents example applications of the novel variational framework including its application to embed ‘hard EM’ into the free energy framework. We conclude by discussing the results in Sec.\[SecDiscussion\]. Truncated Variational EM – Summary of the Algorithm {#SecSummary} =================================================== The theoretical results of Secs.\[SecTVEM\] and \[SecOptimization\] will allow for the formulation of novel variational EM algorithms applicable to generative models with discrete latents, and for identifying existing algorithms as variational approaches. All derivations of these later sections are required to obtain the final results but the final result can be applied without detailed knowledge of the derivations’ details. We therefore summarize the main result in the following, and present the required derivations and more details and variants later. Problem Description and Notation -------------------------------- First we describe the problem addressed and the notation used. Following the framework of Expectation Maximization [EM; @DempsterEtAl1977], our aim is to maximize the data likelihood defined by a set of $N$ data points, , where we model the data distribution by a probabilistic generative model whose distribution $p({\vec{y}}\,|\,\Theta)$ is parameterized by the model parameters $\Theta$. We assume generative models with discrete hidden variables ${\vec{s}}$ and (discrete or continuous) observed variables ${\vec{y}}$ such that the modeled data distribution is given by: $$\begin{aligned} p({\vec{y}}\,|\,\Theta) &=& \sum_{{\vec{s}}} p({\vec{y}},{\vec{s}}\,|\,\Theta), \label{EqnGenModle}\end{aligned}$$ where $\sum_{{\vec{s}}}$ goes over all possible values of ${\vec{s}}$. The data log-likelihood is then given by: $$\begin{aligned} {{\cal L}}(\Theta) &=& \log\left(p({\vec{y}}^{(1)},\ldots,{\vec{y}}^{(N)}\,|\,\Theta)\right)\ = \ \sum_{n=1}^N\log\left(\sum_{{\vec{s}}} p({\vec{y}^{\,(n)}},{\vec{s}}\,|\,\Theta)\right). \label{EqnLikelihood}\end{aligned}$$ Expectation Maximization [EM; @DempsterEtAl1977] is a very popular meta-algorithm that allows for optimizing the likelihood by iterating an E- and an M-step. In its most elementary form, the E-step consists of computing the posterior probability $p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)$ for each data point ${\vec{y}^{\,(n)}}$ and latent state ${\vec{s}}$, while the M-step updates the parameters $\Theta$ to increase the likelihood. The basic EM algorithm was generalized in a number of contributions to provide justification for incremental or online versions and, more importantly for this contribution, to provide the theoretical foundation of variational EM approximations [see, e.g., @Hathaway1986; @SaulEtAl1996; @NealHinton1998]. Truncated Variational EM ------------------------ The basic idea of truncated EM is the use of truncated posterior distributions as approximations to the full posteriors [e.g. @LuckeEggert2010; @HennigesEtAl2014; @DaiLucke2014; @SheikhEtAl2014; @SheltonEtAl2017]. A truncated approximate posterior distribution is given by: $$\begin{aligned} \label{EqnQMain} \hspace{-4mm} {q^{(n)}}({\vec{s}})\hspace{-2mm} &:=& \hspace{-2mm}{q^{(n)}}({\vec{s}};{{\cal K}},\Theta)=\frac{{\displaystyle}\phantom{\int}p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\phantom{\int}} {\hspace{-2mm}{\displaystyle}\sum_{{{\vec{s}^{\,\prime}}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}p({{\vec{s}^{\,\prime}}}\,|\,{\vec{y}^{\,(n)}},\Theta)}\,\delta({\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}})=\frac{{\displaystyle}\phantom{\int}p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\phantom{\int}} {\hspace{-2mm}{\displaystyle}\sum_{{{\vec{s}^{\,\prime}}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}p({{\vec{s}^{\,\prime}}},{\vec{y}^{\,(n)}}\,|\,\Theta)}\,\delta({\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}), $$ where $\delta({\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}})$ is an indicator function, i.e., $\delta({\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}})=1$ if ${\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ and zero otherwise. Fig.\[FigTVEM\] shows an illustration of a truncated posterior approximation. The set ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ contains a finite number of hidden states ${\vec{s}}$. There is one such set for each data point $n$, and we will denote with ${{\cal K}}$ the collection of all sets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$, i.e., ${{\cal K}}={{\cal K}}^{(1:N)}=({{\cal K}}^{(1)},\ldots,{{\cal K}}^{(N)})$ (we will use the ‘colon’ notation to denote a range of indices throughout this paper). The expectation values w.r.t.truncated distributions are given by: $${\left\langle{}g({\vec{s}})\right\rangle}_{{q^{(n)}}}\,=\,{\left\langle{}g({\vec{s}})\right\rangle}_{{q^{(n)}}({\vec{s}};{{\cal K}},\Theta)}\,=\,\frac{{\displaystyle}\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}} p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\ g({\vec{s}}) }{{\displaystyle}\sum_{{{\vec{s}^{\,\prime}}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}p({{\vec{s}^{\,\prime}}},{\vec{y}^{\,(n)}}\,|\,\Theta)}\,, \label{EqnSuffStatMain}$$ where $g({\vec{s}})$ can be any (well-behaved) function over latents ${\vec{s}}$. For sufficiently small sets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$, the expectation values (\[EqnSuffStatMain\]) are computationally tractable if the joint distribution $p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)$ of a probabilistic generative model is efficiently computable. As for most directed graphical models the joint is indeed computational tractable, we will assume such tractability for the paper (unless stated otherwise). The variational distributions (\[EqnQMain\]) define, similar to other types of variational distributions, a free energy, which is given by: $$\begin{aligned} \label{EqnFreeEnergyTVEMOrgA} {{\cal F}}({{\cal K}},{\Theta^{\mathrm{old}}},\Theta)\,=\, \sum_{n=1}^{N} \bigg[ \sum_{{\vec{s}}}\ {q^{(n)}}({\vec{s}};{{\cal K}},{\Theta^{\mathrm{old}}})\ \log\left( p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\right) \bigg] + H(q({\vec{s}};{{\cal K}},{\Theta^{\mathrm{old}}}))\,, $$ where $H(q)$ is an entropy term in which $\Theta$ is held fixed at ${\Theta^{\mathrm{old}}}$. For ${\Theta^{\mathrm{old}}}=\Theta$ the free energy can be shown (see Sec.\[SecOptimization\], Prop.3) to take on a simplified form given by: $$\begin{aligned} \label{EqnFreeEnergyTVEMOrgB} {{\cal F}}({{\cal K}},\Theta) = {{\cal F}}({{\cal K}},\Theta,\Theta) &=& {\displaystyle}\sum_{n=1}^{N}\log\big({\hspace{-1.5mm}}\sum_{\ {\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}} p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\,\big)\,.$$ The free energy (\[EqnFreeEnergyTVEMOrgB\]) we will refer to as [*simplified*]{} truncated free energy or just truncated free energy. The truncated free energy is a lower bound of the log-likelihood (\[EqnLikelihood\]) and it is provably monotonically increased by the following procedure: $$\begin{aligned} & {{{\cal K}}^{\mathrm{new}}}= { \underset{{{\cal K}}}{\mathrm{argmax}} }\,\big\{{{\cal F}}({{\cal K}},{\Theta^{\mathrm{old}}})\big\} && \hspace{0mm}\text{TV-E-step} \label{EqnTVEStep}\\ &{\Theta^{\mathrm{new}}}= { \underset{\Theta}{\mathrm{argmax}} }\,\big\{{{\cal F}}({{{\cal K}}^{\mathrm{new}}},{\Theta^{\mathrm{old}}},\Theta)\big\} && \hspace{0mm}\text{TV-M-step} \label{EqnTVMStep}\\ &{\Theta^{\mathrm{old}}}= {\Theta^{\mathrm{new}}}&& \hspace{0mm}\text{}\label{EqnTVThird} $$ We will refer to one iteration of Eqns.\[EqnTVEStep\] to \[EqnTVThird\] as a [*truncated variational EM*]{} (TV-EM) iteration. The repetition of TV-EM iterations until convergence of $\Theta$ monotonically increases the lower free energy bound (\[EqnFreeEnergyTVEMOrgB\]) of the likelihood to at least local optima. The optimization of ${{\cal F}}({{\cal K}},\Theta)$ w.r.t.${{\cal K}}$ has hereby to be taken as optimization for sets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ with limited size. Small ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ of constrained size will ensure computational tractability as well as non-trivial solutions of (\[EqnTVEStep\]). The ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ can hereby be thought of as being all constraint to the same constant size, $|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|=const$ for all $n$, although the results which we will derive in this study will allow for other size constraints. For sufficiently small state sets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$, the TV-E-step (\[EqnTVEStep\]) is a constraint discrete optimization of a computationally tractable function, Eqn.\[EqnFreeEnergyTVEMOrgB\]. Furthermore, regarding the TV-M-step, any closed-form or gradient updates of $\Theta$ derived using (\[EqnTVMStep\]) are in this case computationally tractable because the expectation values w.r.t. ${q^{(n)}}({\vec{s}};{{\cal K}},\Theta)$ are tractable according to Eqn.\[EqnSuffStatMain\]. For many standard generative models (e.g., sparse coding models, mixture models, hidden Markov models etc.) the M-steps are well-known and often derivable in closed-form. The TV-M-step (\[EqnTVMStep\]) warrants that such M-step equations remain unchanged if TV-EM is applied – only the expectation values (which the M-steps depend on) have to be replaced by Eqn.\[EqnSuffStatMain\]. Like for other variational EM approximations, the variational E- and M-steps (Eqns.\[EqnTVEStep\] and \[EqnTVMStep\]), can also be changed to just partially optimize the free energy, i.e., to increase instead of maximize the free energy. For partial TV-E- and/or TV-M-steps the guarantee that the free energy monotonically increases does continue to hold (see Eqns.\[EqnTVEMOptStepsParialFinal\]). Eqns.\[EqnSuffStatMain\] to \[EqnTVThird\] sufficiently summarize the TV-EM meta-algorithm such that it can directly be applied to a generative model with discrete latents. Sec.\[SecExplicitForm\] reiterates the algorithm in a very explicit form which shows that the algorithm is formulated solely in terms of joint $p({\vec{y}},{\vec{s}}\,|\,\Theta)$ given by the generative model. In Sec.\[SecApplications\] we will discuss different realizations of concrete TV-EM algorithms. The reader interested in applying TV-EM may directly be referred to these sections. We will now proceed and derive TV-EM (Eqns.\[EqnTVEStep\] to \[EqnTVThird\] with Eqn.\[EqnFreeEnergyTVEMOrgB\]) step by step. All of the following derivations will be necessary to prove the properties of the TV-EM algorithm, and none of the Eqns.\[EqnFreeEnergyTVEMOrgB\] to \[EqnTVThird\] will turn out to be trivial. This includes Eqns.\[EqnTVMStep\] and \[EqnTVThird\] although they may seem straight-forward at first sight. Related Work: Variational Approaches to Expectation Maximization {#SecVarApproaches} ================================================================ Before we formally introduce and derive truncated variational EM as a novel type of variational EM, we first discuss related work by first reviewing variational approximations in general, and by discussing their most wide-spread special cases and some recent developments. While most of this section reviews well-known as well as some recent results, we will later show that some of the central derivations which allow for using prominent variational approaches (such as Gaussian variational or factored variational approaches) can and have to be generalized for truncated distributions. Furthermore, reviews of the well-known variational approaches will facilitate the introduction of truncated variational EM and will allow to contrast its properties with these and other related approaches. The Variational Free Energy Formulation {#SecVEM} --------------------------------------- Following the establishment of EM as a standard tool for likelihood maximization [@DempsterEtAl1977], its basic form was generalized to provide justification for incremental or online versions and, more importantly, to provide the theoretical foundation of variational EM approximations [see, e.g., @Hathaway1986; @SaulEtAl1996; @NealHinton1998]. As it will be of importance further below, we here briefly recapitulate the derivation of the free energy in its standard textbook form [see, e.g., @SaulEtAl1996; @Murphy2012; @Barber2012]. For this recall Jensen’s inequality, which can for our purposes be denoted as follows: Let $f$, $g$ and $q$ be real-valued functions such that $g({\vec{s}}),f({\vec{s}}),q({\vec{s}})\in{\mathbbm{R}}$ and let us denote the functions domain (the set of all possible states ${\vec{s}}$) by $\Omega$. For all ${\vec{s}}\in\Omega$ let $q({\vec{s}})$ be a non-negative function that sums to one, i.e., $\sum_{{\vec{s}}\in\Omega}\,q({\vec{s}})=1$. Then for any [*concave*]{} function $f$ the following inequality holds: $$\begin{aligned} f\big(\sum_{{\vec{s}}}\,q({\vec{s}})\,g({\vec{s}})\big)\,\geq\, \sum_{{\vec{s}}}\,q({\vec{s}})\,f(g({\vec{s}}))\,. \label{EqnJensen}\end{aligned}$$ Note that $\sum_{{\vec{s}}}$ in (\[EqnJensen\]) again denotes a summation over all states ${\vec{s}}$. Any distribution $q({\vec{s}})$ on $\Omega$ satisfies the conditions on $q$ for Jensen’s inequality. If we additionally demand $q({\vec{s}})$ to be [*strictly*]{} positive, i.e. $q({\vec{s}})>0$ for all ${\vec{s}}$, we can apply Jensen’s inequality to the data likelihood (\[EqnLikelihood\]) because the logarithm is a concave function. We obtain: $$\begin{aligned} {{\cal L}}(\Theta) &=& \sum_{n=1}^{N}\ \log\big(\sum_{{\vec{s}}} p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big)\ =\ \sum_{n=1}^{N}\ \log\big( \sum_{{\vec{s}}}\,{q^{(n)}}({\vec{s}})\,\frac{1}{{q^{(n)}}({\vec{s}})}\,p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big)\phantom{\int^f_g}\nonumber\\ \,&\geq&\,\, \sum_{n=1}^{N}\ \sum_{{\vec{s}}}\,{q^{(n)}}({\vec{s}})\, \log\big( \frac{1}{{q^{(n)}}({\vec{s}})}\,p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta) \big)\phantom{\int^f_g}\nonumber\\ \,&=&\,\, \sum_{n=1}^{N}\Big( \sum_{{\vec{s}}}\,{q^{(n)}}({\vec{s}})\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big)\,-\,\sum_{{\vec{s}}}\,{q^{(n)}}({\vec{s}})\,\log\big({q^{(n)}}({\vec{s}})\big)\Big)\,.\phantom{\int^f_g} \label{EqnFreeEnergyDeri}\end{aligned}$$ The crucial novel entity that emerges in this derivation is a lower bound of the likelihood which is termed the (variational) [*free energy*]{} [compare @SaulJordan1996; @NealHinton1998; @JordanEtAl1999; @MacKay2003]: $$\begin{aligned} \label{EqnFreeEnergy} {{\cal L}}(\Theta) \,\geq\, {{\cal F}}(q,\Theta)\,:=\, \sum_{n=1}^{N} \Big( \sum_{{\vec{s}}}\ {q^{(n)}}({\vec{s}})\ \log\big( p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big) \Big) + H(q)\,, $$ where $H(q) = - \sum_n\sum_{{\vec{s}}} {q^{(n)}}({\vec{s}}) \log({q^{(n)}}({\vec{s}}))$ is a function (the Shannon entropy) that depends on the distribution $q({\vec{s}})=q^{(1:N)}({\vec{s}})=(q^{(1)}({\vec{s}}),\ldots,q^{(N)}({\vec{s}}))$ but not on the model parameters $\Theta$. Given a data point ${\vec{y}^{\,(n)}}$, the distribution $q^{(n)}({\vec{s}})$ is called [*variational distribution*]{}. Also the collection of all distributions $q({\vec{s}})$ , with $q({\vec{s}})=(q^{(1)}({\vec{s}}),\ldots,q^{(N)}({\vec{s}}))$, is referred to as variational distribution. The name ‘free energy’ was inherited from statistical physics where approximations to the Helmholtz free energy of a physical system take a very similar mathematical form [see, e.g., @MacKay2003]. A basic result for variational EM [e.g. @JordanEtAl1999; @NealHinton1998; @MacKay2003; @Murphy2012] is that the difference between the likelihood [(\[EqnLikelihood\])]{} and the free energy [(\[EqnFreeEnergy\])]{} is given by the Kullback-Leibler divergence between the variational distributions and the posterior distributions: $$\begin{aligned} \label{EqnDKLStandard} {{\cal L}}(\Theta)-{{\cal F}}(q,\Theta)\ =\ \sum_{n}{D_{\mathrm{KL}}\big({q^{(n)}}({\vec{s}}),p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\big)}\ \geq\ 0, $$ where ${D_{\mathrm{KL}}\big(q,p\big)}$ denotes the Kullback-Leibler (KL) divergence. Using [(\[EqnDKLStandard\])]{} we observe that the free energy ${{\cal F}}(q,\Theta)$ is maximized if we choose (for all $n$) the posterior distribution as variational distribution: $$\begin{aligned} \label{EqnFreeEnergyExactEM} {q^{(n)}}({\vec{s}}) &:=& p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\ \ \ \Rightarrow\ \ \ {{\cal L}}(\Theta)={{\cal F}}(q,\Theta)\,, $$ which is the subject of Lemma 1 by @NealHinton1998. In the case of generative models with tractable posterior, Eqn.\[EqnFreeEnergyExactEM\] is used to maximize the likelihood [(\[EqnLikelihood\])]{} by iteratively maximizing the free energy [(\[EqnFreeEnergy\])]{} instead. Optimizing the latter is easier, as derivatives of the free energy can be taken while the parameters $\Theta$ of the posterior distribution are held fixed. After maximizing ${{\cal F}}(q,\Theta)$ w.r.t. $\Theta$ (the M-step), we can set ${q^{(n)}}({\vec{s}})=p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},{\Theta^{\mathrm{old}}})$ with ${\Theta^{\mathrm{old}}}=\Theta$ because this choice maximizes the free energy according to (\[EqnFreeEnergyExactEM\]). Reiterating these well-know results will be of importance for truncated distributions later on. Choosing ${q^{(n)}}({\vec{s}})$ equal to the full posterior is sometimes referred to as exact EM. However, the computation of posterior distributions represents the crucial computational intractability for most generative data models: typically neither the posteriors nor expectation values w.r.t. them are computationally tractable as they require summations over all possible states ${\vec{s}}$. The Variational Approximation of EM ----------------------------------- Overcoming the problem of computational intractability while maintaining an as good as possible approximation to the full posterior is the main motivation for variational approximations (and other approximation schemes). For variational EM the basic idea is to maximize the free energy ${{\cal F}}(q,\Theta)$ in [(\[EqnFreeEnergy\])]{} for a constrained class of variation distributions $q$. If we can find variational distributions $q$ such that\  \ (A) optimization of ${{\cal F}}(q,\Theta)$ is tractable, and\ (B) such that the lower bound ${{\cal F}}(q,\Theta)$ becomes as similar (as tight) as possible to ${{\cal L}}(\Theta)$,\  \ then a tractable approximate optimization of ${{\cal L}}(\Theta)$ is obtained. By definition of the free energy, almost no restrictions are imposed on the choice of the distributions $q$, such that they can, in principle, be chosen to take any functional form and to be dependent on any set of parameters. To fulfill requirement (A) some choice for a functional form of $q$ has to be made, however. As a consequence, the distributions ${q^{(n)}}({\vec{s}})$ become equipped with additional parameters and these parameters are then optimized to make ${{\cal F}}(q,\Theta)$ as tight as possible (requirement B). The variational distributions are then typically denoted by ${q^{(n)}}({\vec{s}},\Lambda)$ with the additional parameters $\Lambda$ being referred to as [*variational parameters*]{}. Having chosen a variational distribution, the free energy is often taken to depend on $\Lambda$ rather than on the variational distributions themselves, i.e., ${{\cal F}}(\Lambda,\Theta)$. Without being more specific about the choice of variational distributions and parameters, the iterative procedure to optimize the free energy may then be abstractly denoted by: $$\begin{aligned} \label{EqnFreeEnergyVEM} \mathrm{Opt\ 1:} & & & {\Lambda^{\mathrm{new}}}= { \underset{\Lambda}{\mathrm{argmax}} }{{\cal F}}(\Lambda,{\Theta^{\mathrm{old}}}) & & \mbox{while holding ${\Theta^{\mathrm{old}}}$ fixed} & \mbox{(V-E-step)}\\ \mathrm{Opt\ 2:} & & & {\Theta^{\mathrm{new}}}={ \underset{\Theta}{\mathrm{argmax}} }{{\cal F}}({\Lambda^{\mathrm{new}}},\Theta) & & \mbox{while holding ${\Lambda^{\mathrm{new}}}$ fixed} & \mbox{(V-M-step)} \\ \mathrm{} & & & {\Theta^{\mathrm{old}}}= {\Theta^{\mathrm{new}}}$$ The two optimization steps are repeated until the parameters $\Theta$ have sufficiently converged. In principle, many possible choices of variational distributions that fulfill requirements (A) and (B) are conceivable; and any choice would give rise to a variational EM procedure. Still, the application of variational EM has been dominated by two standard types of variational distributions: Gaussian variational distributions and factored variational distributions. Gaussian and Factored Variational EM ------------------------------------ As one of the most basic distributions is the Gaussian distribution, a natural choice for variational distributions (for continuous latents) is a multi-variate Gaussian: $$\begin{aligned} \label{EqnFreeEnergyGVEM} {q^{(n)}}({\vec{s}}) &:=& {q^{(n)}}({\vec{s}};\Lambda)\ =\ {q^{(n)}}({\vec{s}};\mu^{(n)},\Sigma^{(n)})\ =\ {{\cal N}}({\vec{s}};\mu^{(n)},\Sigma^{(n)})\,, $$ where $\Lambda^{(n)}=(\mu^{(n)},\Sigma^{(n)})$ and where $\Lambda=(\Lambda^{(1)},\ldots,\Lambda^{(N)})$ is the set of all variational parameters (one mean and one covariance matrix per data point). The Gaussian variational approach approximates each posterior distribution by a Gaussian distribution. The optimization of the variational parameters in (\[EqnFreeEnergyGVEM\]) results in update equations for mean and variance that maximize the free energy and minimize the KL-divergence between true posteriors and the variational Gaussians. Gaussian distributions are especially well suited for data models with mono-modal posteriors, and are consequently popular, e.g., for optimization of sparse linear models [e.g. @OpperWinther2005; @Seeger2008; @OpperArchambeau2009]. Gaussians can capture data correlations and jointly optimize mean and variance in the KL-divergence sense. The second (and more predominant) standard variant of variational EM builds up on the choice of variational distributions that factor over sets of hidden variables. Most commonly the choice is a fully factored distribution: $$\begin{aligned} \label{EqnFreeEnergyFVEM} {q^{(n)}}({\vec{s}}) &:=& {q^{(n)}}({\vec{s}};\Lambda)\ =\ \prod_{h=1}^{H}q_h(s_h;\vec{\lambda}_h^{(n)}) $$ where $\vec{\lambda}_h^{(n)}$ are parameters associated with one hidden variable $h$ for one data point $n$, and where $\Lambda$ is the collection of all these parameters (all $h$ and $n$ combinations). Using factored distributions in combination with specific choices for the factors $q_h(s_h;\vec{\lambda}_h^{(n)})$ then results in computationally tractable optimizations. For the usual generative models the factors are often chosen to be identical to the prior distributions of the individual hidden variables [e.g. @JordanEtAl1999; @HaftEtAl2004]. Because of a mathematical analogy to variational free energy approximations in statistical physics (which initially motivated variational EM), fully factored variational approaches (\[EqnFreeEnergyFVEM\]) are also frequently termed [*mean-field*]{} approximations. Further Variational Optimization Approaches ------------------------------------------- Any variational distributions that contain fully factored or Gaussian distributions as special cases, can potentially improve on these standard distributions because they can make the free energy a tighter lower-bound. Generalizations of fully factored approaches can be defined by allowing for dependencies between small sets of variables (doubles, triples etc) resulting in [*partially factored*]{} approaches. Such approaches can potentially capture more complex posterior interdependencies (correlations and higher-order dependencies), and they are therefore also termed [*structured variational approaches*]{} or [*structured mean-field*]{} to highlight their close relation to fully factored approaches [compare @SaulJordan1996; @MacKay2003; @Bouchard2009; @Murphy2012]. As factorized variational distributions (including structured ones) make potentially harmful assumptions [@IlinValpola2003; @MacKay2003; @TurnerSahani2011; @SheikhEtAl2014] alternative distributions have continuously been investigated. Examples are ‘normalizing flow’ approaches for continuous latents [@RezendeMohamed2015], or approaches that expand mean-field variational approaches hierarchically to include dependencies [@RanganathEtAl2015], or copula-based approaches [@TranEtAl2015]. Such approaches use specific transformations of distributions to allow for modeling complex dependencies among latent variables for improved posterior approximations. We will briefly discuss the relation of these approaches to truncated variational EM in the context of ‘black box’ optimization in Sec.7. Further work which generated recent attention [@HernandezEtAl2016; @RanganathEtAl2016] considers generalizations of the original likelihood and free energy objectives (Eqns.\[EqnLikelihood\] and \[EqnFreeEnergyTVEMOrgA\]). Also related in this context is work on variational approaches using stochastic variational inference [@HoffmanEtAl2013], where auxiliary distributions for Markov chains are defined and used to approximate true posteriors. Generalizations of free energy objectives and stochastic variational inference can both be considered complementary lines of research to the results discussed in this work. Truncated Variational Distributions {#SecTVEM} =================================== The introduction of Gaussian and factored variational distributions now provides the ground for the introduction of a novel class of variational distributions. In contrast to the prominent examples of variational EM, we will here neither assume monomodal variational distributions (like Gaussian variational EM) nor independent factors (like mean-field approaches). Furthermore, we will not choose a specific analytic function such as Gaussians or products of elementary distributions. Instead, we use the posterior distribution itself to define variational distributions. More precisely, we define the variational distributions to be proportional to the full posteriors. However, for any given data point ${\vec{y}^{\,(n)}}$, the proportionality will be constrained to a subspace ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ which will allow for computationally tractable procedures. States ${\vec{s}}$ not in ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ are assumed to have zero probability (‘hard’ zeros). Formally, such truncated distributions ${q^{(n)}}({\vec{s}})={q^{(n)}}({\vec{s}};{{\cal K}},\Theta)$ are given by Eqn.\[EqnQMain\]. Truncated distributions have been suggested previously and have successfully been applied to a number of elementary as well as more complex generative data models [@LuckeEggert2010; @DaiEtAl2013; @DaiLucke2014; @SheikhEtAl2014; @SheikhLucke2016; @SheltonEtAl2017]. Instead of using a variational optimization of approximation parameters similar to Gaussian or mean-field approaches, truncated approximations have, so far, used preselection mechanisms to reduce the number of states evaluated for a truncated approximation [@LuckeEggert2010]. While truncated EM was shown to be very efficient in practice [@SheikhLucke2016; @HughesSudderth2016; @SheltonEtAl2017], no fully variational treatment was provided, and no convergence guarantees and free energy results as they will be derived in this work, were given. We will later see, however, that preselection based truncated EM can be closely related to the fully variational framework developed here. Having defined the variational distribution of (\[EqnQMain\]), we can now seek to derive the corresponding free energy. However, by considering the standard derivation (\[EqnFreeEnergyDeri\]) which shows that the free energy is a lower bound of the likelihood, recall that we required the values of ${q^{(n)}}({\vec{s}})$ to be [*strictly*]{} positive, ${q^{(n)}}({\vec{s}})>0$ for all ${\vec{s}}$. In order to embed truncated distributions (\[EqnQMain\]) into the free energy framework, we therefore first have to generalize the free energy formalism to be applicable without the constraint of strict positivity. Such a generalization, which (to the knowledge of the authors) has not been investigated before, is required for our purposes but may be of more general use. As a fist step, we show that the following holds:  \  \ [**Proposition 1**]{}\ Let ${q^{(n)}}({\vec{s}})$ be variational distributions defined on a set of states $\Omega$ (with values not necessarily greater zero), then a free energy function ${{\cal F}}(q,\Theta)$ exists and is given by: $$\begin{aligned} \label{EqnPropOne} {{\cal F}}(q,\Theta)\,:=\, \sum_{n=1}^{N} \Big( \sum_{{\vec{s}}}\ {q^{(n)}}({\vec{s}})\ \log\big( p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big) \Big) + H(q)\,. $$  \ [**Proof**]{}\ First let us (as was also done for Eqn.\[EqnPropOne\]) drop the notation of variational parameters to increase readability. Then observe that the standard derivation (\[EqnFreeEnergyDeri\]) shows that the proposition is true if ${q^{(n)}}({\vec{s}})>0$ for all ${\vec{s}}\in\Omega$ and all $n$. To show that (\[EqnPropOne\]) is true for any distribution, consider the case of variational distributions ${q^{(n)}}({\vec{s}})$ for which ${q^{(n)}}({\vec{s}})>0$ is true only in proper subsets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ of the state space $\Omega$, and equal to zero for all ${\vec{s}}\not\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$. For a given data point $n$ a distribution is either ${q^{(n)}}({\vec{s}})>0$ for all ${\vec{s}}$ or there exists a proper subset ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$. In the latter case, it is trivial to define such a ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$, and as ${q^{(n)}}({\vec{s}})$ is a distribution (i.e., sums to one), this implies that ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ is not empty. For the main part of the proof we will now assume that for all $n$ there exists a proper subset ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ of $\Omega$. A general $q$ may consist of distributions ${q^{(n)}}({\vec{s}})$ that are strictly positive for some $n$ and have proper subsets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ for the other $n$. However, addressing these mixed cases will turn out to be straight-forward, and we will come back to this general case at the end of the proof. Given a distribution ${q^{(n)}}({\vec{s}})$ with set ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$, let us define an auxiliary function ${\tilde{q}^{(n)}}({\vec{s}})$ as follows: $${\tilde{q}^{(n)}}({\vec{s}}) = \left\{ \begin{array}{ll} {q^{(n)}}({\vec{s}})\,-\,{{\epsilon}_n}^- & \mbox{for all}\ {\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}\\ {q^{(n)}}({\vec{s}})\,+\,{{\epsilon}_n}^+ & \mbox{for all}\ {\vec{s}}\not\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}\label{EqnQt} \end{array} \right. $$ Let ${{\epsilon}_n}^-$ be greater zero but smaller than any value of ${q^{(n)}}({\vec{s}})$ in ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$, i.e., $$\begin{aligned} 0 \,<\,{{\epsilon}_n}^- \,<\, \min_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\{{q^{(n)}}({\vec{s}})\}\phantom{iiiii}\mbox{and let}\phantom{iiiii}{{\epsilon}_n}^+ \,:=\, \frac{|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|}{|\Omega|\,-\,|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|}\,{{\epsilon}_n}^-\,, \label{EqnEpsMinus} \label{EqnEpsPlus}\end{aligned}$$ where $|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|$ is the number of states in ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ and where $|\Omega|$ is the number of all possible states. As we have defined ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ to contain only values ${q^{(n)}}({\vec{s}})>0$, the minimum in ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ is greater zero, i.e., we can always find an ${{\epsilon}_n}^-$ satisfying (\[EqnEpsMinus\]). Consequently, ${{\epsilon}_n}^+$ is also greater zero and finite as we demanded the ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ to be proper subsets of $\Omega$ ($|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|<|\Omega|$). With definitions (\[EqnEpsMinus\]) observe that ${\tilde{q}^{(n)}}({\vec{s}})>0$ for all ${\vec{s}}$ and that $\sum_{{\vec{s}}}{\tilde{q}}({\vec{s}})=1$, i.e., ${\tilde{q}^{(n)}}({\vec{s}})$ is a distribution on $\Omega$ which satisfies (other than ${q^{(n)}}({\vec{s}})$) the requirement for the original derivation of the variational free energy (\[EqnFreeEnergyDeri\]). We can therefore use the free energy definition for ${\tilde{q}}({\vec{s}})=\big({\tilde{q}}^{(1)}({\vec{s}}),\ldots,{\tilde{q}}^{(N)}({\vec{s}})\big)$ and then consider the limit to small ${{\epsilon}_n}^-$. For this, we insert the definition of ${\tilde{q}^{(n)}}({\vec{s}})$ into the free energy and find: $$\begin{array}{rcll} {\displaystyle}&&{\hspace{-8mm}}{}{{\cal L}}(\Theta) \,\geq\,{\displaystyle}{{\cal F}}({\tilde{q}},\Theta)\,\phantom{\int^f_g}\nonumber\\ &&{\hspace{-8mm}}{}={\displaystyle}\,\sum_n\Big( \sum_{{\vec{s}}}\,{\tilde{q}^{(n)}}({\vec{s}})\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big)\,-\,\sum_{{\vec{s}}}\,{\tilde{q}^{(n)}}({\vec{s}})\,\log\big({\tilde{q}^{(n)}}({\vec{s}})\big) \Big)\,\phantom{\int^f_g}\nonumber\\ &&{\hspace{-8mm}}{}={\displaystyle}\,\sum_n\Big( \sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\,\big( {q^{(n)}}({\vec{s}})-{{\epsilon}_n}^-\big) \,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big)\,-\,\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\,\big( {q^{(n)}}({\vec{s}})-{{\epsilon}_n}^-\big)\,\log\big({q^{(n)}}({\vec{s}})-{{\epsilon}_n}^-\big)\phantom{\int^f_g} \\ &&{\hspace{-8mm}}{}{\displaystyle}\, \phantom{=}+\,\sum_{{\vec{s}}\not\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\,\big( {q^{(n)}}({\vec{s}})+{{\epsilon}_n}^+ \big)\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big)\,-\,\sum_{{\vec{s}}\not\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\,\big( {q^{(n)}}({\vec{s}})+{{\epsilon}_n}^+ \big)\,\log\big({q^{(n)}}({\vec{s}})+{{\epsilon}_n}^+\big) \Big)\phantom{\int^f_g} \\ &&{\hspace{-8mm}}{}={\displaystyle}\,\sum_n\Big( \sum_{{\vec{s}}}\,{q^{(n)}}({\vec{s}})\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big) \,-\, {{\epsilon}_n}^-\,\,\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big) + {{\epsilon}_n}^+\,\,\sum_{{\vec{s}}\not\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big)\phantom{\int^f_g}\\ &&{\hspace{-8mm}}{}{\displaystyle}\, \phantom{=}-\,\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\,\big( {q^{(n)}}({\vec{s}})-{{\epsilon}_n}^-\big)\,\log\big({q^{(n)}}({\vec{s}})-{{\epsilon}_n}^-\big) -{{\epsilon}_n}^+ \,\log({{\epsilon}_n}^+)\,\sum_{{\vec{s}}\not\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\,1\phantom{\int^f_g} \Big) \\ \end{array}$$ $$\begin{array}{rcll} {\displaystyle}&&{\hspace{-8mm}}{}={\displaystyle}\,\sum_n\Big( \sum_{{\vec{s}}}{q^{(n)}}({\vec{s}})\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big) - {{\epsilon}_n}^-\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big) + {{\epsilon}_n}^-\,\frac{|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|}{|\Omega|-|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|}\sum_{{\vec{s}}\not\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big)\phantom{\int^f_g} \\ &&{\hspace{-8mm}}{}{\displaystyle}\, \phantom{=}-\, \sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\,\big( {q^{(n)}}({\vec{s}})-{{\epsilon}_n}^-\big)\,\log\big({q^{(n)}}({\vec{s}})-{{\epsilon}_n}^-\big) - {{\epsilon}_n}^- \,|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|\,\log\big(\frac{|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|}{|\Omega|-|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|}\big) - |{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|\,{{\epsilon}_n}^- \,\log({{\epsilon}_n}^-) \Big) \phantom{\int^f_g} \\ &&{\hspace{-8mm}}{}={\displaystyle}\,\sum_n\Big( \sum_{{\vec{s}}}\,{q^{(n)}}({\vec{s}})\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big) - \sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\,\big( {q^{(n)}}({\vec{s}})-{{\epsilon}_n}^-\big)\,\log\big( {q^{(n)}}({\vec{s}})-{{\epsilon}_n}^- \big) - |{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|\,{{\epsilon}_n}^- \,\log({{\epsilon}_n}^-) \phantom{\int^f_g} \\ &&{\hspace{-8mm}}{}{\displaystyle}\, \phantom{=}-\, {{\epsilon}_n}^-\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big) + {{\epsilon}_n}^-\frac{|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|}{|\Omega|-|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|}\,\,\sum_{{\vec{s}}\not\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big) - {{\epsilon}_n}^- \,|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|\,\log\big(\frac{|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|}{|\Omega|-|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|}\big) \Big) \phantom{\int^f_g} \end{array}$$ This expression applies for any ${{\epsilon}_n}^-$ satisfying (\[EqnEpsMinus\]) and therefore also if we replace: $$\begin{aligned} {{\epsilon}_n}^- \,=\, {\epsilon}\,<\, \min_{n^\prime=1,\ldots,N}\{{\epsilon}_{n^\prime}^-\}, $$ such that we obtain: $$\begin{array}{rcll} {\displaystyle}&&{\hspace{-8mm}}{}\hspace{-4mm} {{\cal F}}({\tilde{q}},\Theta)={\displaystyle}\,\sum_n\Big( \sum_{{\vec{s}}}\,{q^{(n)}}({\vec{s}})\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big)\,-\, \sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\,\big( {q^{(n)}}({\vec{s}})-{\epsilon}\big)\,\log\big( {q^{(n)}}({\vec{s}})-{\epsilon}\big) - |{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|\,{\epsilon}\,\log({\epsilon}) \hspace{-9mm}\phantom{\int^f_g} \\ &&{\hspace{-8mm}}{}\hspace{-4mm} {\displaystyle}\, \phantom{=}-\, {\epsilon}\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big) + {\epsilon}\,\frac{|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|}{|\Omega|-|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|}\,\,\sum_{{\vec{s}}\not\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big) - {\epsilon}\,|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|\,\log\big(\frac{|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|}{|\Omega|-|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|}\big) \Big) \hspace{-9mm}\phantom{\int^f_g} \end{array} \label{EqnFFInter}$$ Let us now consider infinitesimally small ${\epsilon}>0$, i.e., let us consider the limit when ${\epsilon}\rightarrow{}0$. First, observe that in this case all summands of the last line of ${{\cal F}}({\tilde{q}},\Theta)$ in (\[EqnFFInter\]) trivially converge to zero. Then observe that the second summand of ${{\cal F}}({\tilde{q}},\Theta)$ converges to $$\begin{aligned} \lim_{{\epsilon}\rightarrow{}0} \Big( \sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\,\big( {q^{(n)}}({\vec{s}})\,-\,{\epsilon}\big)\,\log\big( {q^{(n)}}({\vec{s}})\,-\,{\epsilon}\big) \Big) \,=\, \sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\,{q^{(n)}}({\vec{s}})\,\log\big( {q^{(n)}}({\vec{s}}) \big) $$ as ${q^{(n)}}({\vec{s}})$ is finite and greater zero for all ${\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$. Finally, the third summand in (\[EqnFFInter\]) can be observed to converge (following l’Hôpital) to zero: $$\begin{aligned} \lim_{{\epsilon}\rightarrow{}0} \big( {\epsilon}\,\log({\epsilon}) \big) \,=\, \lim_{{\epsilon}\rightarrow{}0} \big( \frac{\log({\epsilon})}{\frac{1}{{\epsilon}}} \big) \,=\, \lim_{{\epsilon}\rightarrow{}0} \big( -\frac{\frac{1}{{\epsilon}}}{\frac{1}{{\epsilon}^2}} \big) \,=\, -\,\lim_{{\epsilon}\rightarrow{}0} \big( {\epsilon}\big) \,=\,0\,. \label{EqnConvSecond}\end{aligned}$$ Thus, the limit ${\epsilon}\rightarrow{}0$ exists and is given by: $${\displaystyle}\hspace{1mm} {{\cal F}}(q,\Theta)\,=\,\lim_{{\epsilon}\rightarrow{}0} {{\cal F}}({\tilde{q}},\Theta) \,=\,{\displaystyle}\,\sum_n \sum_{{\vec{s}}}\,{q^{(n)}}({\vec{s}})\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big)-\sum_n\sum_{{\vec{s}}}\, {q^{(n)}}({\vec{s}})\,\log\big( {q^{(n)}}({\vec{s}}) \big), \hspace{-5mm}\phantom{\int^f_g} \normalsize \label{EqnFFLimitRes}$$ where we have used (because of Eqn. \[EqnConvSecond\]) the convention ${q^{(n)}}({\vec{s}})\,\log\big( {q^{(n)}}({\vec{s}}) \big) = 0$ for all ${q^{(n)}}({\vec{s}})=0$. We will use this convention (which is also commonly used for the KL-divergence) throughout the paper. Eqn.\[EqnFFLimitRes\] shows that Proposition1 holds for any $q$ with distributions ${q^{(n)}}({\vec{s}})$ with proper subsets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ of $\Omega$. To finally show that Proposition1 holds for general $q$ (with discrete ${\vec{s}})$, consider the mixed case that $q$ contains strictly positive distributions ${q^{(n)}}$ for some $n$ and distributions ${q^{(n)}}$ with exact zeros for the other $n$’s. Let us define the set $J\subseteq\{1,\ldots,N\}$ to contain those $n$ with ${q^{(n)}}$ being strictly positive, and the complement ${\overline{J}}\subseteq\{1,\ldots,N\}$ to contain those $n$ with ${q^{(n)}}$ that are not strictly positive. Using $J$ and ${\overline{J}}$ we then define an auxiliary function ${\tilde{q}}$ by using ${\tilde{q}^{(n)}}({\vec{s}})$ of Eqn.\[EqnQt\] only for $n\in{\overline{J}}$ and by setting ${\tilde{q}^{(n)}}({\vec{s}})={q^{(n)}}({\vec{s}})$ for all ${\vec{s}}$ for all $n\in{}J$. The functions ${\tilde{q}^{(n)}}$ are then again strictly positive distributions on $\Omega$ for all $n$ and we can again apply the standard result [(\[EqnFreeEnergyDeri\])]{}:\ $$\begin{array}{rcll} {\displaystyle}{\hspace{-8mm}}{}{{\cal L}}(\Theta) \,\geq\,{\displaystyle}{{\cal F}}({\tilde{q}},\Theta)\phantom{ii} &&{\hspace{-8mm}}{}={\displaystyle}\,\phantom{iiiii}\sum_{n}\Big( \sum_{{\vec{s}}}\,{\tilde{q}^{(n)}}({\vec{s}})\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big)\,-\,\sum_{{\vec{s}}}\,{\tilde{q}^{(n)}}({\vec{s}})\,\log\big({\tilde{q}^{(n)}}({\vec{s}})\big) \Big)\,\phantom{\int^f_g}\nonumber\\ &&{\hspace{-8mm}}{}={\displaystyle}\phantom{\phantom{ii}+\,}\sum_{n\in{}J}\Big( \sum_{{\vec{s}}}\,{q^{(n)}}({\vec{s}})\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big)\,-\,\sum_{{\vec{s}}}\,{q^{(n)}}({\vec{s}})\,\log\big({q^{(n)}}({\vec{s}})\big) \Big)\,\phantom{\int^f_g}\nonumber\\ &&{\hspace{-8mm}}{}\phantom{iiiiii}+{\displaystyle}\,\sum_{n\in{\overline{J}}}\Big( \sum_{{\vec{s}}}\,{\tilde{q}^{(n)}}({\vec{s}})\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big)\,-\,\sum_{{\vec{s}}}\,{\tilde{q}^{(n)}}({\vec{s}})\,\log\big({\tilde{q}^{(n)}}({\vec{s}})\big) \Big)\,\phantom{\int^f_g}\nonumber\\ \end{array}$$ Now we apply the same arguments as above but only to the sum over all $n\in{}{\overline{J}}$, for which everything remains as for the derivation above. Eqn.\[EqnFFLimitRes\] therefore applies if we only consider sums over ${\overline{J}}$. Thus in the limit ${\epsilon}\rightarrow{}0$ we obtain in the mixed case: $$\begin{aligned} {\displaystyle}{{\cal F}}(q,\Theta) &=& \lim_{{\epsilon}\rightarrow{}0}\big( {{\cal F}}({\tilde{q}},\Theta) \big) \nonumber\\ &=&{\displaystyle}\sum_{n\in{}J}\Big( \sum_{{\vec{s}}}\,{q^{(n)}}({\vec{s}})\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big)\,-\,\sum_{{\vec{s}}}\,{q^{(n)}}({\vec{s}})\,\log\big({q^{(n)}}({\vec{s}})\big) \Big)\,\phantom{\int^f_g}\nonumber\\ &&{\hspace{-8mm}}{}\phantom{iiiiii}+{\displaystyle}\,\lim_{{\epsilon}\rightarrow{}0}\Big( \sum_{n\in{\overline{J}}}\Big( \sum_{{\vec{s}}}\,{\tilde{q}^{(n)}}({\vec{s}})\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big)\,-\,\sum_{{\vec{s}}}\,{\tilde{q}^{(n)}}({\vec{s}})\,\log\big({\tilde{q}^{(n)}}({\vec{s}})\big) \Big) \Big) \,\phantom{\int^f_g}\nonumber\\ &=&{\displaystyle}\sum_{n}\Big( \sum_{{\vec{s}}}\,{q^{(n)}}({\vec{s}})\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big)\,-\,\sum_{{\vec{s}}}\,{q^{(n)}}({\vec{s}})\,\log\big({q^{(n)}}({\vec{s}})\big) \Big)\,\phantom{\int^f_g}\nonumber \label{EqnFFLimitResMixed}\end{aligned}$$ which finally shows that Proposition 1 holds for any $q$ with any distributions ${q^{(n)}}$ on $\Omega$.\ [$\square$]{}\  \ As the free energy (\[EqnPropOne\]) for general $q$ was obtained as a limit, we have to show that it remains a lower bound of ${{\cal L}}(\Theta)$ also in this limit. Using the KL-divergence result (\[EqnDKLStandard\]), this can however be shown using a similar approach as for the proof above.  \ [**Proposition 2**]{}\ Let ${q^{(n)}}({\vec{s}})$ be variational distributions over a set of states $\Omega$ (with values not necessarily greater zero), then the corresponding free energy (\[EqnPropOne\]) is a lower bound of the likelihood, and the difference between likelihood and free energy is given by the sum over KL-divergences: $$\begin{aligned} \label{EqnPropTwo} {{\cal L}}(\Theta)-{{\cal F}}(q,\Theta)\ =\ \sum_{n}{D_{\mathrm{KL}}\big({q^{(n)}}({\vec{s}}),p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\big)}\ \geq\ 0. $$ [**Proof**]{}\ Given distributions ${q^{(n)}}({\vec{s}})$ let us consider the same auxiliary distributions ${\tilde{q}^{(n)}}({\vec{s}})$ as for the proof of Prop,1 (see Eqn.\[EqnQt\]). As ${\tilde{q}^{(n)}}({\vec{s}})$ are strictly positive distributions, Eqn.\[EqnDKLStandard\] applies:$$\begin{aligned} {{\cal L}}(\Theta)-{{\cal F}}({\tilde{q}},\Theta)\ =\ \sum_{n}{D_{\mathrm{KL}}\big({\tilde{q}^{(n)}}({\vec{s}}),p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\big)}\ \geq\ 0, $$ for any collection of ${{\epsilon}_n}^-$ satisfying (\[EqnEpsMinus\]). If we now consider a sequence ${\epsilon}_k=1/k$, we know that for any $n$ there exists a finite $K$ such that for all $k>K$ applies that ${{\epsilon}_n}^-={\epsilon}_k=1/k$ fulfills condition (\[EqnEpsMinus\]). If we now set ${{\epsilon}_n}^-={\epsilon}_k=1/k$ for all $n$, we know because of finitely many data points $n$ that there also exists a finite $K$ such that condition (\[EqnEpsMinus\]) is fulfilled for all $k>K$. If we now define a sequence of distributions ${\tilde{q}}_k({\vec{s}})=({\tilde{q}}^{(1)}_k,\ldots,{\tilde{q}}^{(N)}_k)$ by choosing ${{\epsilon}_n}^-={\epsilon}_k=1/k$ for all $n$, we know that for all $k>K$ applies: $$\begin{aligned} D_k\ =\ {{\cal L}}(\Theta)-{{\cal F}}({\tilde{q}}_k,\Theta)\ =\ \sum_{n}{D_{\mathrm{KL}}\big({\tilde{q}^{(n)}}_k({\vec{s}}),p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\big)}\ \geq\ 0. $$ The sequence $D_k$ is hence a sequence in the interval $[0,\infty)$. As the limit $\lim_{k\rightarrow\infty}{{\cal F}}({\tilde{q}}_k,\Theta)$ is finite, $D_k$ converges to a finite value within $[0,\infty)$ (which is left-closed). If $q$ contains some strictly positive ${q^{(n)}}$, we only use ${{\epsilon}_n}^-={\epsilon}_k=1/k$ for all $n\in{\overline{J}}$. Finally, by using Proposition 1, the limit $\lim_{k\rightarrow\infty}D_k$ is given by: $$\begin{array}{rcll} {\displaystyle}&&\hspace{-8mm} 0\,\leq\,{\displaystyle}\lim_{k\rightarrow\infty}D_k \,=\,{\displaystyle}{{\cal L}}(\Theta)-\lim_{k\rightarrow\infty}{{\cal F}}({\tilde{q}}_k,\Theta)\,\phantom{\int^f_g}\nonumber\\ &&\hspace{-6mm} {\displaystyle}\,=\,{{\cal L}}(\Theta)\,-\,{{\cal F}}(q,\Theta) \phantom{\int^f_g}\nonumber\\ &&\hspace{-6mm} {\displaystyle}\,=\,{{\cal L}}(\Theta)\,-\,\sum_n \sum_{{\vec{s}}}\,{q^{(n)}}({\vec{s}})\,\log\big(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big)\,+\, \sum_n\sum_{{\vec{s}}}\, {q^{(n)}}({\vec{s}})\,\log\big( {q^{(n)}}({\vec{s}}) \big) \phantom{\int^f_g}\nonumber\\ &&\hspace{-6mm} {\displaystyle}\,=\,{{\cal L}}(\Theta) -\sum_n \underbrace{\sum_{{\vec{s}}}\,{q^{(n)}}({\vec{s}})}_{1}\,\log\big(p({\vec{y}^{\,(n)}}\,|\,\Theta)\big)-\sum_n \sum_{{\vec{s}}}\,{q^{(n)}}({\vec{s}})\,\log\big(p({\vec{s}}\,|\,{\vec{y}},\Theta)\big) + \sum_n\sum_{{\vec{s}}}\, {q^{(n)}}({\vec{s}})\,\log\big( {q^{(n)}}({\vec{s}}) \big) \phantom{\int^f_g}\nonumber\\ &&\hspace{-6mm} {\displaystyle}\,=\,-\sum_n \Big( \sum_{{\vec{s}}}\,{q^{(n)}}({\vec{s}})\,\log\big(p({\vec{s}}\,|\,{\vec{y}},\Theta)\big) \,-\, \sum_{{\vec{s}}}\, {q^{(n)}}({\vec{s}})\,\log\big( {q^{(n)}}({\vec{s}}) \big) \Big) \phantom{\int^f_g}\nonumber\\ &&\hspace{-6mm} {\displaystyle}\,=\,\sum_n {D_{\mathrm{KL}}\big({q^{(n)}}({\vec{s}}),p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\big)}\,,\phantom{\int^f_g} \end{array}$$ where the last part follows the lines of the standard derivation for the difference ${{\cal L}}(\Theta)\,-\,{{\cal F}}(q,\Theta)$. Note that we again used the convention ${q^{(n)}}({\vec{s}})\,\log\big( {q^{(n)}}({\vec{s}}) \big) = 0$ for all ${q^{(n)}}({\vec{s}})=0$.\ [$\square$]{}\  \ Taken together, Propositions 1 and 2 mean that we can generalize the standard free energy framework to any variational distribution – the requirement of strictly positive distributions can be dropped. As a consequence, we can use the (generalized) free energy framework also for the truncated variational distributions in (\[EqnQMain\]). We can now insert the specific truncated distributions (\[EqnQMain\]) into the free energy (\[EqnPropOne\]). As for non-variational EM (\[EqnFreeEnergyExactEM\]) with exact posterior as variational distributions, ${q^{(n)}}({\vec{s}})\,=\,p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},{\hat{\Theta}})$, we will distinguish between parameters ${\hat{\Theta}}$ of the variational distribution and the parameters $\Theta$ of the generative data model. Like for the M-step of exact EM, this allows for taking derivatives of the log-joint $\log(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta))$ while the variational distributions can be treated as constant (we will come back to this point further below). Inserting the truncated distributions ${q^{(n)}}({\vec{s}};{{\cal K}},{\hat{\Theta}})$ into the free energy (\[EqnPropOne\]) then yields: $$\begin{aligned} \label{EqnFreeEnergyTVEMOrg} {{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)\,=\, \sum_{n=1}^{N} \bigg[ \sum_{{\vec{s}}}\ {q^{(n)}}({\vec{s}};{{\cal K}},{\hat{\Theta}})\ \log\left( p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\right) \bigg] + H(q({\vec{s}};{{\cal K}},{\hat{\Theta}}))\,.$$ The free energy now depends on [*three*]{} sets of parameters, ${{\cal K}}$, ${\hat{\Theta}}$ and $\Theta$. The Shannon-entropy $H(q({\vec{s}};{{\cal K}},{\hat{\Theta}}))$ is independent of the parameters $\Theta$. Optimization of Truncated Variational Free Energies {#SecOptimization} =================================================== The variational parameters of ${{\cal K}}$ and ${\hat{\Theta}}$ that we aim at optimizing are different from the typical variational parameters, e.g., different from those of factored variational approaches or of Gaussian approximations. For each $n$, the set ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ contains discrete points in latent space; and the parameters ${\hat{\Theta}}$ are of the same type as those of the generative model but with potentially different values. As the (generalized) variational framework of Sec.\[SecTVEM\] does not require strict positivity from the variational distributions, the truncated distributions ${q^{(n)}}({\vec{s}};{{\cal K}},{\hat{\Theta}})$ can now be treated within a variational free energy framework. Following the free energy approach we aim at optimizing the free energy (\[EqnFreeEnergyTVEMOrg\]) instead of directly optimizing the likelihood. We will do so by optimizing ${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$ step-by-step w.r.t. its three sets of parameters: $$\begin{array}{lcllll} \label{EqnTVEMOptSteps} \mathrm{Opt\ 1:} & & {{{\cal K}}^{\mathrm{new}}}&= \hspace{1mm}{ \underset{{{\cal K}}}{\mathrm{argmax}} }\big\{{{{\cal F}}}({{\cal K}},{{\hat{\Theta}}^{\mathrm{old}}},{\Theta^{\mathrm{old}}})\big\} & & \mbox{while holding ${{\hat{\Theta}}^{\mathrm{old}}}$ and ${\Theta^{\mathrm{old}}}$ fixed}\hspace{-6mm}\phantom{\int^f_g}\\[2mm] \mathrm{Opt\ 2:} & & {\Theta^{\mathrm{new}}}&=\hspace{1mm}{ \underset{\Theta}{\mathrm{argmax}} }\big\{{{{\cal F}}}({{{\cal K}}^{\mathrm{new}}},{{\hat{\Theta}}^{\mathrm{old}}},\Theta)\big\} & & \mbox{while holding ${{{\cal K}}^{\mathrm{new}}}$ and ${{\hat{\Theta}}^{\mathrm{old}}}$ fixed}\hspace{-6mm}\phantom{\int^f_g}\\[2mm] \mathrm{Opt\ 3:} & & {{\hat{\Theta}}^{\mathrm{new}}}&= \hspace{1mm}{ \underset{{\hat{\Theta}}}{\mathrm{argmax}} }\big\{{{{\cal F}}}({{{\cal K}}^{\mathrm{new}}},{\hat{\Theta}},{\Theta^{\mathrm{new}}})\big\} & & \mbox{while holding ${{{\cal K}}^{\mathrm{new}}}$ and ${\Theta^{\mathrm{new}}}$ fixed}\hspace{-6mm}\phantom{\int^f_g}\\[6mm] \mathrm{\ \ \ \ \ \,set} & & {{\hat{\Theta}}^{\mathrm{old}}}&\hspace{0mm}=\hspace{2mm} {{\hat{\Theta}}^{\mathrm{new}}}\mbox{\ and\ } {\Theta^{\mathrm{old}}}\hspace{2mm}=\hspace{2mm}{\Theta^{\mathrm{new}}}& &\hspace{-9mm}\mbox{and start-over with Opt 1} \end{array}$$ The order of the updates is chosen for later convenience. Each of the three optimization steps by definition increases the free energy ${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$ w.r.t. one of its arguments. Opt 2 which updates the model parameters $\Theta$ corresponds to the M-step. Opt 1 and Opt 3 optimize the two sets of variational parameters ${{\cal K}}$ and ${\hat{\Theta}}$, respectively, and correspond to the E-step for truncated variational distributions. One iteration of Eqns.\[EqnTVEMOptSteps\] will be referred to as TV-EM iteration (as in introduced in Sec.\[SecSummary\]), and by definition the free energy is monotonically increased. The optimization steps (\[EqnTVEMOptSteps\]) of TV-EM are formal definitions. In order to be applicable in practice, a more concrete procedure for each of the three optimization steps is required. The Truncated Free Energy ------------------------- Instead of investigating and applying the three optimization steps individually, we will carefully analyze each optimization in a theoretically grounded way. For this purpose, let us first introduce a free energy defined by setting the values of the variational parameters ${\hat{\Theta}}$ equal to the model parameters $\Theta$: $$\label{EqnFreeEnergyTVEM} \hspace{-0mm}\ {{{\cal F}}}({{\cal K}},\Theta)\,:=\,{{{\cal F}}}({{\cal K}},\Theta,\Theta)\,=\, \sum_{n=1}^{N} \bigg[ \sum_{{\vec{s}}}\ {q^{(n)}}({\vec{s}};{{\cal K}},\Theta)\ \log\left( p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\right) \bigg] + H({q^{(n)}}({\vec{s}};{{\cal K}},\Theta))\,. $$ Given the definition of the truncated variational distribution ${q^{(n)}}({\vec{s}};{{\cal K}},\Theta)$ in (\[EqnQMain\]), it can then be shown that the free energy ${{{\cal F}}}({{\cal K}},\Theta)$ can be decisively simplified as follows:\  \  \ [**Proposition 3**]{}\ Given a generative model defined by the joint distribution $p({\vec{s}},{\vec{y}}\,|\,\Theta)$. If ${{{\cal F}}}({{\cal K}},\Theta)$ is the free energy defined by (\[EqnFreeEnergyTVEM\]) with truncated distributions given by (\[EqnQMain\]), then it follows that $$\begin{aligned} {{{\cal F}}}({{\cal K}},\Theta) &=& \sum_{n=1}^{N}\ \log\big(\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}} p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\ \big). \label{EqnTruncatedF} $$  \ [**Proof**]{}\ Following Propositions 1 and 2, ${{{\cal F}}}({{\cal K}},\Theta)$ is a lower bound of ${{\cal L}}(\Theta)$, which satisfies:$$\begin{aligned} {\textstyle}{{\cal L}}(\Theta)-{{{\cal F}}}({{\cal K}},\Theta)=\sum_{n}{D_{\mathrm{KL}}\big({q^{(n)}}({\vec{s}};{{\cal K}},\Theta),p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\big)} $$ For notational purposes let us introduce the normalizer ${Z^{(n)}}=\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)$ such that: $$\begin{aligned} {q^{(n)}}({\vec{s}};{{\cal K}},\Theta) &=& \frac{1}{{Z^{(n)}}}\,p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\,\delta({\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}) $$ From the above it follows that: $$\begin{aligned} \lefteqn{{{{\cal F}}}({{\cal K}},\Theta)}\nonumber\\ &=& {{\cal L}}(\Theta)\,-\,\sum_{n}{D_{\mathrm{KL}}\big({q^{(n)}}({\vec{s}};{{\cal K}},\Theta),p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\big)}\nonumber\\ &=& \sum_{n}\log(p({\vec{y}^{\,(n)}}\,|\,\Theta))\,+\,\sum_{n}\sum_{{\vec{s}}} {q^{(n)}}({\vec{s}};{{\cal K}},\Theta) \log\big(\frac{p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)}{{q^{(n)}}({\vec{s}};{{\cal K}},\Theta)}\big)\nonumber\\ &=& \sum_{n}\log(p({\vec{y}^{\,(n)}}\,|\,\Theta))\,+\,\sum_{n}\sum_{{\vec{s}}} \frac{1}{{Z^{(n)}}}p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\,\delta({\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}) \log\big(\frac{p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)}{\frac{1}{{Z^{(n)}}}p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\,\delta({\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}))}\big)\nonumber\\ &=& \sum_{n}\log(p({\vec{y}^{\,(n)}}\,|\,\Theta))\,+\,\sum_{n}\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}} \frac{1}{{Z^{(n)}}}\,p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta) \log\big(\frac{p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)}{\frac{1}{{Z^{(n)}}}p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)}\big)\nonumber\\ &=& \sum_{n}\log(p({\vec{y}^{\,(n)}}\,|\,\Theta))\,+\,\sum_{n}\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}} \frac{1}{{Z^{(n)}}}\,p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta) \log\big({Z^{(n)}}\big). $$ Again we used the convention ${q^{(n)}}({\vec{s}})\,\log\big( {q^{(n)}}({\vec{s}}) \big) = 0$ for all ${q^{(n)}}({\vec{s}})=0$. Observing the intermediate result above, note that ${Z^{(n)}}$ is independent of ${\vec{s}}$. We can therefore continue as follows: $$\begin{aligned} {{{\cal F}}}({{\cal K}},\Theta) &=& \sum_{n}\log(p({\vec{y}^{\,(n)}}\,|\,\Theta))\,+\,\sum_{n}\log\big({Z^{(n)}}\big) \frac{\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)}{{Z^{(n)}}}\nonumber\\ &=& \sum_{n}\log(p({\vec{y}^{\,(n)}}\,|\,\Theta))\,+\,\sum_{n}\log\big({Z^{(n)}}\big)\nonumber\\ &=& \sum_{n}\log(p({\vec{y}^{\,(n)}}\,|\,\Theta))\,+\,\sum_{n}\log\big(\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\big)\nonumber\\ &=& \sum_{n}\log(p({\vec{y}^{\,(n)}}\,|\,\Theta))\,+\,\sum_{n}\log\big(\frac{\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)}{p({\vec{y}^{\,(n)}}\,|\,\Theta)}\big)\nonumber\\ &=& \sum_{n}\log\big(\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}} p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\ \big)\nonumber $$ \ [$\square$]{}\  \ Considering Eqn.\[EqnTruncatedF\] we instantly observe that ${{{\cal F}}}({{\cal K}},\Theta)$ is computationally tractable if ${{\cal K}}$ is sufficiently small. Also the fact that ${{\cal F}}({{\cal K}},\Theta)$ lower-bounds the log-likelihood can instantly be observed. Importantly, however, Prop.3 shows that Eqn.\[EqnTruncatedF\] is a variational free energy which corresponds to the truncated variational distributions (\[EqnQMain\]). Furthermore, Prop.3 directly relates (\[EqnTruncatedF\]) to the free energy (\[EqnFreeEnergyTVEMOrg\]). As ${{{\cal F}}}({{\cal K}},\Theta)$ is a special case of ${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$ both free energies are lower bounds of the likelihood ${{\cal L}}(\Theta)$. Furthermore, it can be shown that ${{\cal F}}({{\cal K}},{\hat{\Theta}},\Theta)$ can be obtained as a variational lower bound of ${{{\cal F}}}({{\cal K}},\Theta)$ and that the following holds.\  \ [**Proposition 4**]{}\ Given a generative model defined by the joint $p({\vec{s}},{\vec{y}}\,|\,\Theta)$, let ${{\cal L}}(\Theta)$ be the likelihood in Eqn.\[EqnLikelihood\] and let ${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$ and ${{{\cal F}}}({{\cal K}},\Theta)$ be the free energies defined by Eqn.\[EqnFreeEnergyTVEMOrg\] and \[EqnTruncatedF\], respectively. Then, for all values of ${{\cal K}}$, ${\hat{\Theta}}$, and $\Theta$ the following applies: $$\begin{aligned} {{\cal L}}(\Theta)\ \geq\ {{{\cal F}}}({{\cal K}},\Theta)\ \geq\ {{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)\,. $$ [**Proof**]{}\ ${{{\cal F}}}({{\cal K}},\Theta)$ is a special case of ${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$ by definition (Eqn.\[EqnFreeEnergyTVEM\]), and as such a lower bound of ${{\cal L}}(\Theta)$. To show that ${{{\cal F}}}({{\cal K}},\Theta)\ \geq\ {{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$, we use Proposition 3 and apply Jensen’s inequality: $$\begin{aligned} {{{\cal F}}}({{\cal K}},\Theta) &=& \sum_{n}\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\ \log\left( p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\right)\\ &=& \sum_{n}\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\ \log \left( {\hat{q}^{(n)}}({\vec{s}}) \frac{p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)}{{\hat{q}^{(n)}}({\vec{s}})} \right)\\ &\geq{}& \sum_{n}\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\ {\hat{q}^{(n)}}({\vec{s}}) \ \log \left( \frac{p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)}{{\hat{q}^{(n)}}({\vec{s}})} \right)\label{EqnFProof} $$ if $\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}{\hat{q}^{(n)}}({\vec{s}})=1$ and ${\hat{q}^{(n)}}({\vec{s}})\geq{}0$ for all ${\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$. We now define: $$\begin{aligned} {\hat{q}^{(n)}}({\vec{s}}) &=& \frac{p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},{\hat{\Theta}})}{\sum_{{\vec{s}}'\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}p({\vec{s}}'\,|\,{\vec{y}^{\,(n)}},{\hat{\Theta}})}\,. $$ The choice of ${\hat{q}^{(n)}}({\vec{s}})$ fulfills the conditions for Jensen’s inequality but note that it is not a probability density on the whole state space $\Omega$ of ${\vec{s}}$. Inserting ${\hat{q}^{(n)}}({\vec{s}})$ into (\[EqnFProof\]) we obtain: $$\begin{aligned} {{{\cal F}}}({{\cal K}},\Theta) &\geq& \sum_{n}\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\ \frac{p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},{\hat{\Theta}})}{\sum_{{\vec{s}}'\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}p({\vec{s}}'\,|\,{\vec{y}^{\,(n)}},{\hat{\Theta}})} \log\left( \frac{p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)}{\frac{p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},{\hat{\Theta}})}{\sum_{{\vec{s}}'\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}p({\vec{s}}'\,|\,{\vec{y}^{\,(n)}},{\hat{\Theta}})}}\right)\\ &=& \sum_{n}\sum_{{\vec{s}}}\ {q^{(n)}}({\vec{s}};{{\cal K}},{\hat{\Theta}})\ \log\left( \frac{p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)}{{q^{(n)}}({\vec{s}};{{\cal K}},{\hat{\Theta}})}\right)\\ &=& \sum_{n}\sum_{{\vec{s}}}\ {q^{(n)}}({\vec{s}};{{\cal K}},{\hat{\Theta}})\ \log\left(p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\right)\,+\,H({q^{(n)}}({\vec{s}};{{\cal K}},{\hat{\Theta}}))\\ &=& {{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)\,, $$ where ${q^{(n)}}({\vec{s}};{{\cal K}},{\hat{\Theta}})$ is the truncated variational distribution in (\[EqnQMain\]) and where ${{\cal F}}({{\cal K}},{\hat{\Theta}},\Theta)$ is the corresponding free energy in (\[EqnFreeEnergyTVEMOrg\]).\ [$\square$]{}\  \ Having established that ${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$ is a lower bound of ${{{\cal F}}}({{\cal K}},\Theta)$ for all ${\hat{\Theta}}$, the following applies for the differences between ${{\cal L}}(\Theta)$, ${{{\cal F}}}({{\cal K}},\Theta)$, and ${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$:\  \ [**Corollary 1**]{}\ All as above. $$\begin{aligned} {{\cal L}}(\Theta)-{{{\cal F}}}({{\cal K}},\Theta) {\hspace{-2mm}}{}&=&{\hspace{-2mm}}{} \sum_n{D_{\mathrm{KL}}\big({q^{(n)}}({\vec{s}};{{\cal K}},\Theta),p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\big)}\ \geq\ 0\,,\nonumber\\ {{{\cal F}}}({{\cal K}},\Theta)-{{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta){\hspace{-2mm}}{}&=&{\hspace{-2mm}}{} \sum_n{D_{\mathrm{KL}}\big({q^{(n)}}({\vec{s}};{{\cal K}},{\hat{\Theta}}),p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\big)}\,-\,\sum_n{D_{\mathrm{KL}}\big({q^{(n)}}({\vec{s}};{{\cal K}},\Theta),p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\big)}\geq\ 0\,,\nonumber\\ {{\cal L}}(\Theta)-{{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta) {\hspace{-2mm}}{}&=&{\hspace{-2mm}}{} \sum_n{D_{\mathrm{KL}}\big({q^{(n)}}({\vec{s}};{{\cal K}},{\hat{\Theta}}),p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\big)}\geq\ 0\,.\nonumber $$ [**Proof**]{}\ The first and the last equation are a direct consequence of Propositions 2 and 4 and of the fact that ${{{\cal F}}}({{\cal K}},\Theta)$ and ${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$ are both variational lower bounds of ${{\cal L}}(\Theta)$. The second equation is obtained by taking the difference of the first and the last equation. The fact that the difference of the KL-divergences of the second equation is greater zero follows from Proposition 4.\ [$\square$]{}\  \ By considering Corollary 1, we can now solve the last optimization step (Opt 3) in (\[EqnTVEMOptSteps\]) analytically. Indeed, it can be shown that ${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$ is optimized w.r.t. ${\hat{\Theta}}$ if the values of the variational parameters ${\hat{\Theta}}$ are set equal to model parameters $\Theta$:\  \ [**Proposition 5**]{}\ If ${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$ is the truncated free energy of Eqn.\[EqnFreeEnergyTVEMOrg\], then it applies for fixed ${{\cal K}}$ and $\Theta$ that $$\begin{aligned} { \underset{{\hat{\Theta}}}{\mathrm{argmax}} }\big\{ {{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta) \big\} &=& \Theta\,.\label{EqnThirdOpt} $$ [**Proof**]{}\ Let us first re-express ${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$ using Corollary 1: $$\begin{aligned} {{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta) &=& {{{\cal F}}}({{\cal K}},\Theta)\,+\,\sum_n{D_{\mathrm{KL}}\big({q^{(n)}}({\vec{s}};{{\cal K}},\Theta),p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\big)}\,-\,\sum_n{D_{\mathrm{KL}}\big({q^{(n)}}({\vec{s}};{{\cal K}},{\hat{\Theta}}),p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\big)}\nonumber\\ &=& {{{\cal F}}}({{\cal K}},\Theta)\,+\,\sum_n{D_{\mathrm{KL}}\big({q^{(n)}}(\Theta),{p^{(n)}}(\Theta)\big)}\,-\,\sum_n{D_{\mathrm{KL}}\big({q^{(n)}}({\hat{\Theta}}),{p^{(n)}}(\Theta)\big)},\normalsize \label{EqnFreeEnergyProofTemp}\end{aligned}$$ where the last line abbreviates the distributions for readability. As only the last summand depends on ${\hat{\Theta}}$, ${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$ is maximized if $\sum_n{D_{\mathrm{KL}}\big({q^{(n)}}({\hat{\Theta}}),{p^{(n)}}(\Theta)\big)}$ is minimized. Now we also know from Corollary 1 that $\sum_n{D_{\mathrm{KL}}\big({q^{(n)}}({\hat{\Theta}}),{p^{(n)}}(\Theta)\big)}\geq{}\sum_n{D_{\mathrm{KL}}\big({q^{(n)}}(\Theta),{p^{(n)}}(\Theta)\big)}$, where only the left-hand-side depends on ${\hat{\Theta}}$. Hence, the most minimal value for $\sum_n{D_{\mathrm{KL}}\big({q^{(n)}}({\hat{\Theta}}),{p^{(n)}}(\Theta)\big)}$ achievable is $$\begin{aligned} \sum_n{D_{\mathrm{KL}}\big({q^{(n)}}({\hat{\Theta}}),{p^{(n)}}(\Theta)\big)}\,=\,\sum_n{D_{\mathrm{KL}}\big({q^{(n)}}(\Theta),{p^{(n)}}(\Theta)\big)}\,. $$ By choosing ${\hat{\Theta}}=\Theta$ we can indeed satisfy the equality, and therefore know that takes on a global minimum for this choice. This global minimum then implies (because of Eqn.\[EqnFreeEnergyProofTemp\]) a global maximum of ${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$ w.r.t.${\hat{\Theta}}$.\ [$\square$]{}\  \ Note that there can potentially be other global maxima, e.g., due to permutations of the parameters without effect on ${q^{(n)}}({\vec{s}};{{\cal K}},{\hat{\Theta}})$. Using the KL-divergences of Corollary 1 makes it salient that it is sufficient to equate ${q^{(n)}}({\vec{s}};{{\cal K}},{\hat{\Theta}})$ and ${q^{(n)}}({\vec{s}};{{\cal K}},\Theta)$, which is a weaker condition than equating ${\hat{\Theta}}$ and $\Theta$. To show that ${\hat{\Theta}}=\Theta$ is maximizing ${{\cal F}}({{\cal K}},{\hat{\Theta}},\Theta)$ we can, alternatively, also use Proposition 4 directly, i.e., ${{{\cal F}}}({{\cal K}},\Theta)\geq{}{{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$. Either way, we can conclude that ${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$ is maximized for ${\hat{\Theta}}=\Theta$, in which case ${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$ becomes equal to ${{{\cal F}}}({{\cal K}},\Theta)$. Using Proposition 5 solves the third optimization step of Eqns.\[EqnTVEMOptSteps\], and one TV-EM iteration reduces to two optimizations: by applying (\[EqnThirdOpt\]) the third optimization (Opt 3 of Eqns.\[EqnTVEMOptSteps\]), is simply given by ${{\hat{\Theta}}^{\mathrm{new}}}={\Theta^{\mathrm{new}}}$. After combining this update with the last line of Eqns.\[EqnTVEMOptSteps\] we obtain: $$\label{EqnOptFinalPre} \begin{array}{lcllll} \mathrm{Opt\ 1:} & & {{{\cal K}}^{\mathrm{new}}}&= \hspace{1mm}{ \underset{{{\cal K}}}{\mathrm{argmax}} }\big\{{{{\cal F}}}({{\cal K}},{{\hat{\Theta}}^{\mathrm{old}}},{\Theta^{\mathrm{old}}})\big\} & & \mbox{}\hspace{-6mm}\phantom{\int^f_g}\\ \mathrm{Opt\ 2:} & & {\Theta^{\mathrm{new}}}&=\hspace{1mm}{ \underset{\Theta}{\mathrm{argmax}} }\big\{{{{\cal F}}}({{{\cal K}}^{\mathrm{new}}},{{\hat{\Theta}}^{\mathrm{old}}},\Theta)\big\} & & \mbox{}\hspace{-6mm}\phantom{\int^f_g}\\[3mm] \mathrm{} & & {{\hat{\Theta}}^{\mathrm{old}}}&\hspace{0mm}=\hspace{2mm} {\Theta^{\mathrm{new}}}\mbox{\ \ \ and\ \ \ } {\Theta^{\mathrm{old}}}\hspace{2mm}=\hspace{2mm}{\Theta^{\mathrm{new}}}\end{array} $$ As the variational parameters ${{\hat{\Theta}}^{\mathrm{old}}}$ and the model parameters ${\Theta^{\mathrm{old}}}$ are now both set to the same values in the last line of (\[EqnOptFinalPre\]), we can now for Opt1 use (without loss of generality) the simplified form of free energy ${{\cal F}}({{\cal K}},\Theta)$ as given by Eqn.\[EqnTruncatedF\]. As a consequence, we can replace the two resets of the parameters $\Theta$ in the last line by the single reset ${\Theta^{\mathrm{old}}}={\Theta^{\mathrm{new}}}$ and finally obtain the TV-EM formulation of Eqns.\[EqnTVEStep\] to \[EqnTVThird\] as stated in the beginning. To recapitulate, we have thus finally proven that iterating the TV-EM steps of Eqns.\[EqnTVEStep\] to \[EqnTVThird\] monotonically increases the free energy (\[EqnFreeEnergyTVEMOrgA\]), which is a lower bound of the log-likelihood (\[EqnLikelihood\]). The proof follows from the updates (\[EqnTVEMOptSteps\]), which monotonically increase the free energy by definition. Using Propositions 1 to 5 and Corollary 1, Eqns.\[EqnTVEStep\] to \[EqnTVThird\] result in the same updates as (\[EqnTVEMOptSteps\]) but represent a strong simplification. Note, in this respect, that it is important for the TV-M-step (\[EqnTVMStep\]) to be of the same form as for exact EM and other types of variational EM approximations because this form means that any M-step equations derived for any previously considered generative model can be reused for truncated variational EM. Maintaining the standard M-step update is non-trivial for truncated distributions and required the application of the theoretical results derived above: First, we needed to start with a three-stage optimization for ${{\cal F}}({{\cal K}},{\hat{\Theta}},\Theta)$. A direct optimization of the simplified free energy ${{\cal F}}({{\cal K}},\Theta)$ in a two-stage procedure would change the M-step to a non-standard form. This is because the truncated distributions do also depend on $\Theta$ (and derivatives w.r.t. to all $\Theta$’s would be required). By treating ${\hat{\Theta}}$ as variational parameters, derivatives exclusively w.r.t. the log-joint probability of the generative model can be taken, and this results, e.g., in the well-known closed-form updates of Gaussian Mixture Models, Hidden Markov Models, Factor Analysis etc. All these M-step results and similar such results for many other models can thus directly be used for TV-EM. Second, when we take derivatives w.r.t. $\Theta$ in the TV-M-step, it may feel straight-forward to hold the parameters $\Theta$ of the variational distributions fixed at their old values ${\Theta^{\mathrm{old}}}$; and to only afterwards set ${\Theta^{\mathrm{old}}}={\Theta^{\mathrm{new}}}$ as in (\[EqnTVThird\]). We are very used to this procedure for exact EM. Note, however, that for exact EM, it is required to prove that such a procedure never decreases the free energy. A possible such proof for exact EM would use Eqn.\[EqnDKLStandard\] and full posteriors $p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},{\hat{\Theta}})$ as variational distributions (using ${\hat{\Theta}}$ as variational parameters). The KL-divergence ${D_{\mathrm{KL}}\big(p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},{\hat{\Theta}}),p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\big)}$ can then be set to zero by choosing ${\hat{\Theta}}=\Theta$, which according to Eqn.\[EqnDKLStandard\] globally maximize the free energy [also see Lemma 1 of @NealHinton1998]. For general variational distributions with ${\hat{\Theta}}$ as variational parameters, the same does [*not*]{} apply. For truncated variational distributions, it is Prop.5 which shows that we can proceed with truncated distributions in the same way as we are used to for full posteriors in exact EM. Indeed, Prop.5 contains Lemma 1 of @NealHinton1998 as a special case: If we set all ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ to contain all states ${\vec{s}}$, i.e. ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}=\Omega$ for all $n$, then the variational distributions ${q^{(n)}}({\vec{s}};{{\cal K}},\Theta)$ in (\[EqnQMain\]) become equal to the full posteriors. Consequently, the free energy ${{\cal F}}({{\cal K}},{\hat{\Theta}},\Theta)$ becomes equal to the standard free energy for exact EM, and Prop.5 then shows that this free energy is maximized if we set ${\Theta^{\mathrm{old}}}$ of the posteriors $p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},{\Theta^{\mathrm{old}}})$ equal to the $\Theta$ obtained in the M-step. Also note that for ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}=\Omega$, the TV-EM algorithm reduces to exact EM as the TV-E-step (\[EqnTVEStep\]) becomes trivial and as the truncated distributions (\[EqnQMain\]) become equal to the exact posteriors. Partial Truncated E- and M-Steps {#SecPartialTVEM} -------------------------------- So far, we have considered with Eqns.\[EqnTVEMOptSteps\] full maximizations of truncated free energies, for which we derived the TV-EM algorithm given by Eqns.\[EqnTVEStep\] to \[EqnTVThird\]. However, for many generative models such full maximizations are analytically and/or computationally intractable. In order to also address these important cases, we here apply Props. 1 to 5 to partial TV-E- and partial TV-M-steps (which is analogous but not equal to the full maximization). Let us start with a three-stage optimization as before but instead we now consider a partial TV-EM procedure: $$\begin{array}{lcllll} \label{EqnTVEMOptStepsParial} \mathrm{Opt\ 1:} & & \mbox{choose ${{{\cal K}}^{\mathrm{new}}}$ such that\phantom{iiiiii}} {{{\cal F}}}({{{\cal K}}^{\mathrm{new}}},{{\hat{\Theta}}^{\mathrm{old}}},{\Theta^{\mathrm{old}}}) &\geq\hspace{2mm} {{{\cal F}}}({{{\cal K}}^{\mathrm{old}}},\ {{\hat{\Theta}}^{\mathrm{old}}},{\Theta^{\mathrm{old}}})\hspace{-6mm}\phantom{\int^f_g}\\ \mathrm{Opt\ 2:} & & \mbox{choose ${\Theta^{\mathrm{new}}}$ such that\phantom{iiiiii}} {{{\cal F}}}({{{\cal K}}^{\mathrm{new}}},{{\hat{\Theta}}^{\mathrm{old}}},{\Theta^{\mathrm{new}}}) &\geq\hspace{2mm} {{{\cal F}}}({{{\cal K}}^{\mathrm{new}}},{{\hat{\Theta}}^{\mathrm{old}}},{\Theta^{\mathrm{old}}})\hspace{-6mm}\phantom{\int^f_g}\\ \mathrm{Opt\ 3:} & & {{\hat{\Theta}}^{\mathrm{new}}}= \hspace{1mm}{ \underset{{\hat{\Theta}}}{\mathrm{argmax}} }\big\{{{{\cal F}}}({{{\cal K}}^{\mathrm{new}}},{\hat{\Theta}},{\Theta^{\mathrm{new}}})\big\} \hspace{-6mm}\phantom{\int^f_g}\\[2mm] \mathrm{} & & {{{\cal K}}^{\mathrm{old}}}\hspace{0mm}=\hspace{2mm} {{{\cal K}}^{\mathrm{new}}}, \mbox{\ \ } {\Theta^{\mathrm{old}}}\hspace{2mm}=\hspace{2mm}{\Theta^{\mathrm{new}}}, \mbox{\ \ } {{\hat{\Theta}}^{\mathrm{old}}}\hspace{0mm}=\hspace{2mm} {{\hat{\Theta}}^{\mathrm{new}}}\end{array}$$ An iteration using (\[EqnTVEMOptStepsParial\]) monotonically increases the free energy ${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$ as each individual optimization by definition never decreases ${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$. Opt 1 and Opt 2 are now partial optimizations (${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$ is increased, not maximized), while we maintain for Opt 3 a full maximization. By applying Prop.5 we can now, as before, replace the maximization of Opt 3 by setting ${{\hat{\Theta}}^{\mathrm{new}}}={\Theta^{\mathrm{new}}}$. Also as before, we then obtain by combining the analytical solution of Opt 3 with the last line of (\[EqnTVEMOptStepsParial\]) a two-stage optimization procedure: $$\begin{array}{lcllll} \label{EqnTVEMOptStepsParialTemp} \mathrm{Opt\ 1:} & & \mbox{choose ${{{\cal K}}^{\mathrm{new}}}$ such that\phantom{iiiiii}} {{{\cal F}}}({{{\cal K}}^{\mathrm{new}}},{{\hat{\Theta}}^{\mathrm{old}}},{\Theta^{\mathrm{old}}}) &\geq\hspace{2mm} {{{\cal F}}}({{{\cal K}}^{\mathrm{old}}},\ {{\hat{\Theta}}^{\mathrm{old}}},{\Theta^{\mathrm{old}}})\hspace{-6mm}\phantom{\int^f_g}\\ \mathrm{Opt\ 2:} & & \mbox{choose ${\Theta^{\mathrm{new}}}$ such that\phantom{iiiiii}} {{{\cal F}}}({{{\cal K}}^{\mathrm{new}}},{{\hat{\Theta}}^{\mathrm{old}}},{\Theta^{\mathrm{new}}}) &\geq\hspace{2mm} {{{\cal F}}}({{{\cal K}}^{\mathrm{new}}},{{\hat{\Theta}}^{\mathrm{old}}},{\Theta^{\mathrm{old}}})\hspace{-6mm}\phantom{\int^f_g}\\ \mathrm{} & & {{{\cal K}}^{\mathrm{old}}}\hspace{0mm}=\hspace{2mm} {{{\cal K}}^{\mathrm{new}}}, \mbox{\ \ } {\Theta^{\mathrm{old}}}\hspace{2mm}=\hspace{2mm}{\Theta^{\mathrm{new}}}, \mbox{\ \ } {{\hat{\Theta}}^{\mathrm{old}}}\hspace{0mm}=\hspace{2mm} {\Theta^{\mathrm{new}}}\end{array}$$ As ${\Theta^{\mathrm{old}}}$ and ${{\hat{\Theta}}^{\mathrm{old}}}$ are now set to the same values, we can replace the free energy ${{{\cal F}}}({{\cal K}},{\hat{\Theta}},\Theta)$ of Opt 1 in (\[EqnTVEMOptStepsParialTemp\]) by the simplified free energy (\[EqnTruncatedF\]) derived for Proposition 3. By further simplifying the last line of (\[EqnTVEMOptStepsParialTemp\]) we finally arrive at: $$\begin{array}{lcllll} \label{EqnTVEMOptStepsParialFinal} \mathrm{Opt\ 1:} & & \mbox{choose ${{{\cal K}}^{\mathrm{new}}}$ such that\phantom{iiiiii}} \phantom{iiiiiik}{{{\cal F}}}({{{\cal K}}^{\mathrm{new}}},{\Theta^{\mathrm{old}}}) &\geq\hspace{2mm} {{{\cal F}}}({{{\cal K}}^{\mathrm{old}}},\ {\Theta^{\mathrm{old}}})\hspace{-6mm}\phantom{\int^f_g}\\ \mathrm{Opt\ 2:} & & \mbox{choose ${\Theta^{\mathrm{new}}}$ such that\phantom{iiiiii}} {{{\cal F}}}({{{\cal K}}^{\mathrm{new}}},{\Theta^{\mathrm{old}}},{\Theta^{\mathrm{new}}}) &\geq\hspace{2mm} {{{\cal F}}}({{{\cal K}}^{\mathrm{new}}},{\Theta^{\mathrm{old}}},{\Theta^{\mathrm{old}}})\hspace{-6mm}\phantom{\int^f_g}\\ \mathrm{} & & {{{\cal K}}^{\mathrm{old}}}\hspace{0mm}=\hspace{2mm} {{{\cal K}}^{\mathrm{new}}}, \mbox{\ \ } {\Theta^{\mathrm{old}}}\hspace{2mm}=\hspace{2mm}{\Theta^{\mathrm{new}}}\end{array}$$ Eqns.\[EqnTVEMOptStepsParialFinal\] will be referred to as a partial TV-EM iteration. Like non-partial TV-EM (i.e., Eqns.\[EqnTVEStep\] to \[EqnTVThird\]), a partial TV-EM step monotonically increases the truncated free energy (\[EqnTruncatedF\]). While the derivation of partial TV-EM used the same theoretical results as non-partial TV-EM, note that a main difference is that the variational parameters ${{\cal K}}$ have to be memorized across partial TV-EM iterations. A full optimization does not necessarily require such a memorization. Furthermore, we require initial values of ${{\cal K}}$ for partial TV-EM. Finally, observe that we can, in the same way as above, define TV-EM algorithms with only the TV-E-step being a partial optimization or with only the TV-M-step being a partial optimization. Explicit Form {#SecExplicitForm} ------------- Before we consider applications of the theoretical results for TV-EM, let us formulate the algorithm given in the previous section more explicitly. Obtaining an explicit form for the TV-E-step is straight-forward by just inserting the simplified truncated free energy (\[EqnTruncatedF\]) into the first optimization of Eqns.\[EqnTVEMOptStepsParialFinal\] (see further below). Regarding the TV-M-step, consider the second optimization of Eqns.\[EqnTVEMOptStepsParialFinal\] and let us insert the (non-simplified) truncated free energy (\[EqnFreeEnergyTVEMOrg\]). After noting that (as usual for variational approaches) the entropy term is not relevant for the optimization of model parameters $\Theta$, the relevant function to optimize is given by: $$\begin{aligned} \label{EqnQFunction} {\hspace{-3mm}}{\hspace{-3mm}}Q(\Theta){\hspace{-1.5mm}}&=&{\hspace{-1.5mm}}\sum_{n=1}^{N} \sum_{{\vec{s}}}\ {q^{(n)}}({\vec{s}};{{\cal K}},{\Theta^{\mathrm{old}}})\ \log\left( p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\right) = \sum_{n=1}^{N}{\left\langle{}\log\big( p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\big)\right\rangle}_{{q^{(n)}}({\vec{s}};{{\cal K}},{\Theta^{\mathrm{old}}})} $$ If we now insert Eqn.\[EqnSuffStatMain\] for the expectation value w.r.t.${q^{(n)}}({\vec{s}};{{\cal K}},{\Theta^{\mathrm{old}}})$ in (\[EqnQFunction\]), we obtain (together with the TV-E-step) an explicit form of one TV-EM iteration given by: $$\begin{array}{llcll} \label{EqnTVEMExplicit} \hspace{-3mm}\mbox{\bf TV-E-step:} \\ \mbox{\ \ change ${{\cal K}}$ from ${{{\cal K}}_{\mathrm{old}}}$ to ${{{\cal K}}_{\mathrm{new}}}$ such that} & {\displaystyle}\sum_{n}\sum_{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}\ \log\left( p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,{\Theta^{\mathrm{old}}})\right)\phantom{i} \mbox{increases.} \hspace{-6mm}\phantom{\int^f_g}\\[9mm] \hspace{-3mm}\mbox{\bf TV-M-step:}\\[-7mm] \mbox{\ \ change $\Theta$ from ${\Theta^{\mathrm{old}}}$ to ${\Theta^{\mathrm{new}}}$ such that} & {\scriptstyle {\displaystyle}\sum_{n}\ \frac{{\displaystyle}\sum_{{\vec{s}}\in{{{\cal K}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}_{\mathrm{new}}}} p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,{\Theta^{\mathrm{old}}})\log\left( p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\right) }{{\displaystyle}\sum_{{\vec{s}^{\,\prime}}\in{{{\cal K}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}_{\mathrm{new}}}}p({\vec{s}^{\,\prime}},{\vec{y}^{\,(n)}}\,|\,{\Theta^{\mathrm{old}}})}}\phantom{i} \mbox{increases.} \\[8mm] \ \\ \hspace{-3mm}\mbox{\bf Reset:} & {{{\cal K}}_{\mathrm{old}}}\hspace{0mm}=\hspace{2mm} {{{\cal K}}_{\mathrm{new}}}, \mbox{\ \ } {\Theta^{\mathrm{old}}}\hspace{2mm}=\hspace{2mm}{\Theta^{\mathrm{new}}}\hspace{-9mm}\vspace{0mm} \end{array}$$ The form (\[EqnTVEMExplicit\]) of one TV-EM iteration makes the following explicit: (A) The procedure is fully defined by the joint probability of the considered generative model; and (B) all entities that have to be computed are computationally tractable given sufficiently small sets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ and efficiently computable joint probabilities. Eqns.\[EqnTVEMExplicit\] also highlight the very concise form of the procedure. If TV-EM is applied to a given generative model, the M-step typically relies on derivatives of the log-joint probability (be it either to derive closed-form update equations or gradient equations for more intricate models). The E-step will ultimately reduce to a pair-wise comparison of joint probabilities, which can be realized efficiently (we will provide more details of such procedures in the following). Applications of Theoretical Results {#SecApplications} =================================== Our theoretical results and the TV-EM meta-algorithm can now be applied to provide novel theoretical insights, and to point to novel ways to develop learning algorithms for generative models. We will consider three applications of our results: First, we will consider multiple-cause or latent variable generative models, i.e., models for which the values of multiple latent variables combine to generate the data. Second, we consider applications to mixture models, i.e., models for which an observation is always generated by one latent variable. Finally, in the third application we investigate the relation between TV-EM and ‘hard EM’, which is a very widely applied and in practice very successful form of paramter optimization in generative models. TV-EM for Multiple-Cause Models {#SecSampling} ------------------------------- Multiple-cause models or latent variable models are generative models in which a set of latent variables (latent causes) combine to generate an observation. Common examples are sparse coding models [e.g. @OlshausenField1996], noisy-OR Bayes nets [e.g. @SingliarHauskrecht2006] or sigmoid believe networks [@SaulEtAl1996; @JordanEtAl1999]. The interaction of multiple hidden variables typically gives rise to large latent spaces because of the combinatorics of individual latents. For any larger models, full posteriors over the latent space are not computationally tractable anymore, and variational EM is a standard technique to address such intractabilities. For our application to multiple-cause models let us consider partial TV-EM (Eqns. \[EqnTVEMOptStepsParialFinal\]). Similar to exact EM, the M-step of TV-EM requires the computation of derivatives of the log-joint probability of the considered generative model (compare Eqns.\[EqnTVEMExplicit\]). Computing any such derivations is standard except of the expectation values w.r.t. ${q^{(n)}}({\vec{s}};{{\cal K}},\Theta)$ which are for TV-EM computed using (\[EqnSuffStatMain\]). The crucial difference to previous variational approaches is hence the truncated variational E-step. Instead of solving fixed-point equations for the variational parameters as, e.g., for mean-field approaches, we have to find variational parameters that take the form of finite sets of hidden states. We will term these states [*variational states*]{}. A (partial) TV-E-step can now be implemented by suggesting new variational states ${\tilde{{{\cal K}}}}$ and to compare the free energy (\[EqnTruncatedF\]) of these new states with the free energy of the old variational states. The set ${{\cal K}}$ is then replaced by a new set ${\tilde{{{\cal K}}}}$ if the free energy increases. The efficiency of this procedure will, of course, depend crucially on the way how new variational states are suggested or how new sets ${{\cal K}}$ are defined. Before we consider more concrete examples of TV-E-steps, let us formulate the above described procedure (which directly follows form Eqns. \[EqnTVEMOptStepsParialFinal\]) as a meta-algorithm (see Alg.1). init model parameters ${\Theta^{\mathrm{old}}}$;[\ ]{}init variational states ${{{\cal K}}^{\mathrm{old}}}$;[\ ]{} The inner loop of Alg.1 repeatedly changes the states in ${{\cal K}}$ and its if-clause warrants that ${{{\cal F}}}({{\cal K}},\Theta)$ is increased w.r.t.${{\cal K}}$. The successive optimization of ${{{\cal F}}}({{\cal K}},\Theta)$ w.r.t.$\Theta$ can be accomplished using standard M-step update equations with expectation values estimated by (\[EqnSuffStatMain\]). Together with setting ${{{\cal K}}^{\mathrm{old}}}= {{{\cal K}}^{\mathrm{new}}}$ and ${\Theta^{\mathrm{old}}}= {\Theta^{\mathrm{new}}}$, one iteration thus realizes one partial TV-EM step. The outer loop of Alg.1 iterates over individual TV-EM steps such that Alg.1 provably monotonically increases the free energy ${{{\cal F}}}({{\cal K}},\Theta)$. As it is the case for the parameters $\Theta$ of the outer loop, we can terminate the inner loop (the TV-E-step) after one iteration (if the free energy increased) or once there are no or no significant changes of ${{\cal K}}$ observed anymore, or at any intermediate step. Considering Alg.1, the computation and comparison of free energies may seem computationally demanding. However, thanks to the specific functional form of the simplified free energy (\[EqnTruncatedF\]), it is sufficient to pair-wise compare the joint probabilities instead of the free energies. That is, for a given data point ${\vec{y}^{\,(n)}}$ and any newly suggested state, it is sufficient to compare the joint probability of the new state with those of the variational states in ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$:  \ [**Proposition 6**]{}\ Consider the application of TV-EM to a generative model given by the joint $p({\vec{s}},{\vec{y}}\,|\,\Theta)$ and let ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ be the set of variational states for a data point ${\vec{y}^{\,(n)}}$. If we now replace a state ${{\vec{s}}^{\,\mathrm{old}}}$ in ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ by a new state ${{\vec{s}}^{\,\mathrm{new}}}$ so far not in ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ then the free energy ${{\cal F}}({{\cal K}},\Theta)$ of (\[EqnTruncatedF\]) is increased if and only if $$\label{EqnCriterion} p({{\vec{s}}^{\,\mathrm{new}}},{\vec{y}^{\,(n)}}\,|\,\Theta)\ >\ p({{\vec{s}}^{\,\mathrm{old}}},{\vec{y}^{\,(n)}}\,|\,\Theta). $$ [**Proof**]{}\ For a given data point $n$ let us consider ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ and let ${{{\cal K}}_{\mathrm{new}}}^{(n)}$ be the set defined by replacing the latent state ${{\vec{s}}^{\,\mathrm{old}}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ with ${{\vec{s}}^{\,\mathrm{new}}}\not\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$, i.e., ${{{\cal K}}_{\mathrm{new}}}^{(n)}={{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}\backslash \{{{\vec{s}}^{\,\mathrm{old}}}\} \cup \{{{\vec{s}}^{\,\mathrm{new}}}\}$. Let us further define ${{{\cal K}}^{\mathrm{new}}}$ by replacing ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ with the set ${{{\cal K}}_{\mathrm{new}}}^{(n)}$, i.e., ${{{\cal K}}^{\mathrm{new}}}=\{{{\cal K}}^{(1)},\ldots,{{\cal K}}^{(n-1)},{{{\cal K}}_{\mathrm{new}}}^{(n)}, {{\cal K}}^{(n+1)},\ldots,{{\cal K}}^{(N)}$. Then it follows: $ \begin{array}{lrcl} & {{\cal F}}({{{\cal K}}^{\mathrm{new}}},\Theta) &>& {{\cal F}}({{\cal K}},\Theta)\\[2mm] \Leftrightarrow{} & \hspace{-0mm}{\displaystyle}\sum_{\underset{m\neq{}n}{m=1}}^{N}\log\big({\hspace{-1.5mm}}\sum_{\ {\vec{s}}\in{{\cal K}}^{(n)}} p({\vec{s}},{\vec{y}}^{(n)}\,|\,\Theta)\,\big)\,+\,\log\big({\hspace{-1.5mm}}\sum_{\ {\vec{s}}\in{{{\cal K}}_{\mathrm{new}}}^{(n)}} p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\,\big)\hspace{-45mm}\\[2mm] && \hspace{-50mm} >\ \ {\displaystyle}\sum_{\underset{m\neq{}n}{m=1}}^{N}\log\big({\hspace{-1.5mm}}\sum_{\ {\vec{s}}\in{{\cal K}}^{(n)}} p({\vec{s}},{\vec{y}}^{(n)}\,|\,\Theta)\,\big)\,+\,\log\big({\hspace{-1.5mm}}\sum_{\ {\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}} p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\,\big)\hspace{-40mm}\\[8mm] \Leftrightarrow{}& {\displaystyle}\log\big({\hspace{-1.5mm}}\sum_{\ {\vec{s}}\in{{{\cal K}}_{\mathrm{new}}}^{(n)}} p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\,\big) &>& {\displaystyle}\log\big({\hspace{-1.5mm}}\sum_{\ {\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}} p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\,\big)\\[4mm] \Leftrightarrow{}& {\displaystyle}\sum_{\underset{{\vec{s}}\neq{}{{\vec{s}}^{\,\mathrm{new}}}}{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}} p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\,+\,p({{\vec{s}}^{\,\mathrm{new}}},{\vec{y}^{\,(n)}}\,|\,\Theta) &>& {\displaystyle}\sum_{\underset{{\vec{s}}\neq{}{{\vec{s}}^{\,\mathrm{new}}}}{{\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}}} p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\,+\,p({{\vec{s}}^{\,\mathrm{old}}},{\vec{y}^{\,(n)}}\,|\,\Theta)\\[4mm] \Leftrightarrow{}& {\displaystyle}p({{\vec{s}}^{\,\mathrm{new}}},{\vec{y}^{\,(n)}}\,|\,\Theta) &>& p({{\vec{s}}^{\,\mathrm{old}}},{\vec{y}^{\,(n)}}\,|\,\Theta) \end{array} $\ [$\square$]{}\ The criterion (\[EqnCriterion\]) can not be derived for arbitrary functions $f({\vec{s}},{\vec{y}^{\,(n)}})$ nor does it become obvious by considering the original definition of ${{\cal F}}({{\cal K}},\Theta)$ in Eqn.\[EqnFreeEnergyTVEM\]. Only thanks to the simplified form of ${{\cal F}}({{\cal K}},\Theta)$ derived for Prop.3, the proof of Prop.6 becomes a straight-forward derivation (and the proof could be regarded as a formal, technical verification of what may have been seen directly by considering the specific functional form of Eqn.\[EqnTruncatedF\]). By virtue of Prop.6, the free energy increase for the inner loop of Alg.1 can be ensured if we, for a new state ${{\vec{s}}^{\,\mathrm{new}}}$, find one state in ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ with a lower joint probability than for ${{\vec{s}}^{\,\mathrm{new}}}$. Also large numbers of new states, generated in parallel, can be compared in a bunch to the states in ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$. The set of states that increases the free energy most, can then be obtained through efficient partial sorting [e.g. @BlumEtAl1973]. Alternatively, efficient data structures such as heaps or soft heaps [@Chazelle2000] to store the states of ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ according to their joint probabilities can be used. Any new set of states can then efficiently be compared with the states in ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ using inequality (\[EqnCriterion\]). Given efficient updates of sets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ using Prop.6 and the methods discussed above, the efficiency of the whole TV-E-step remains to depend on efficiently finding new states ${\tilde{{{\cal K}}}}$ that are indeed sufficiently effective in increasing the free energy.  \ [**Blind search.**]{} The easiest way to suggest new states ${\tilde{{{\cal K}}}}$ in Alg.1 is blind search. Such a search could be realized by randomly (e.g.uniformly) sampling new states of the latent state, and then to use Prop.6 to compare these sampled states to those in ${{{\cal K}}^{\mathrm{old}}}$. Alternatively, one could use random (blind) variations of the old states to generate new states. Applying Prop.6 would then realize a basic stochastic gradient ascent procedure which improves the free energy. However, especially for large hidden spaces the probability of newly generated states to increase the free energy will be small – any blind search procedure will thus presumably be inefficient in general.  \ [**Deterministic construction.**]{} Instead of blindly and randomly searching new sets ${{\cal K}}$ in Alg.1, an alternative would be to deterministically construct newly suggested sets ${\tilde{{{\cal K}}}}$. Such a construction could use procedures already developed previously [@LuckeEggert2010; @SheltonEtAl2011; @DaiEtAl2013; @DaiLucke2014], and Alg.1 would combine these constructions with the theoretical results derived for TV-EM. For instance, most previous work used oracle (or selection) functions to construct sets ${\tilde{{{\cal K}}}}$. A selection function could take the form of a scalar product [@SheltonEtAl2011], of approximations or upper-bounds of marginal probabilities [@LuckeEggert2010], it could be hand-crafted for specific (possibly relatively complex) generative models [@DaiEtAl2013; @DaiLucke2014], or a selection function which is itself learned from data could be used [@SheltonEtAl2017]. Typically, a selection function is first used to determine for each data point ${\vec{y}^{\,(n)}}$ a set $I^{(n)}$ of the most relevant latent variables (the other variables are assumed with $s_h=0$ to not contribute to the generation of ${\vec{y}^{\,(n)}}$). Given the set $I^{(n)}$, the set of states ${\tilde{{{\cal K}}}}^{(n)}$ can then be constructed, for instance, by assuming a sparse combinatorics of the relevant latent variables: $$\label{EqnKKtildeN} {\textstyle}{\tilde{{{\cal K}}}}^{(n)}\, =\, \{{\vec{s}}\ |\ \sum_h{}s_h\leq{}\gamma\ \mbox{and}\ \forall{}h \not\in{}I^{(n)}: s_h=0 \}\,,$$ where $\gamma$ parameterizes the considered sparsity level. For more details and for a visualization of such a constructed set ${\tilde{{{\cal K}}}}^{(n)}$ see, e.g., [@LuckeEggert2010 Fig.2] or [@SheikhEtAl2014]. The sets ${\tilde{{{\cal K}}}}^{(n)}$ were then directly used for the estimation of expectation values (\[EqnSuffStatMain\]). Instead of this previous direct use of ${\tilde{{{\cal K}}}}^{(n)}$ to define truncated posteriors, we can based on the results obtained here, use ${\tilde{{{\cal K}}}}^{(n)}$ as newly suggested states of Alg.1, and combine its elements with the old states ${{{\cal K}}_{\mathrm{old}}}^{(n)}$ to maximally increase the free energy. Such a procedure would profit from well methods to construct sets ${\tilde{{{\cal K}}}}$ for the different generative models of previous work. On the downside, any new generative model would require the definition of new construction procedures, and such constructions often make use of [*ad hoc*]{} assumptions such as sparsity [@HennigesEtAl2010; @DaiEtAl2013; @HennigesEtAl2014]. On the other hand, novel approaches to automate the construction of sets ${\tilde{{{\cal K}}}}$ [see @SheltonEtAl2017] can directly be applied in this context. In any case, deterministic construction closely links TV-EM to a series of previous pre-selection based EM approximations [@LuckeEggert2010] which have motivated this work initially. Importantly, these previous approaches can now be interpreted as TV-EM with an [*estimated*]{}, one-step partial TV-E-step. ‘Estimated’ because any direct definition of the states to define an approximation does not guarantee the free energy to increase; and the previous procedures are ‘one-step’ because the variational loop of Alg.1 is replaced by one construction step.  \ [**Combination with sampling.**]{} A further alternative to blind search would be a stochastic search for new states ${\tilde{{{\cal K}}}}$ in Alg.1 using knowledge given by the generative model. Already using the prior distribution of the generative model under consideration would be more efficient than to blindly sample hidden states, e.g., using a uniform distribution. Samples from the prior would lie in areas of the latent space where at least the average over all posteriors has a high probability mass. The space of high prior mass can still be very large, however, and for any given data point, large posterior mass may be located in areas of the latent space very different from areas of high [*average*]{} posterior mass. Procedures that generate ${\tilde{{{\cal K}}}}$ by sampling new variational states in a data-driven way promise to be much more efficient. For instance, if samples for newly suggested ${\tilde{{{\cal K}}}}^{(n)}$ are drawn from the posterior distribution $p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)$, their joint probabilities can be expected to be relatively high (and with them the free energy ${{{\cal F}}}({{\cal K}},\Theta)$). As the relative values of the joints are the crucial criterion to find good sets ${\tilde{{{\cal K}}}}^{(n)}$, the common normalizer $p({\vec{y}}\,|\,\Theta)$ is not relevant. This observation together with the existence of a well established research field on efficient procedures to generate posterior samples, would represent the advantages of such an approach. Potential disadvantages are that the highest posterior states will quickly be represented by the sets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ and that many new samples of $p({\vec{s}}\,|\,{\vec{y}},\Theta)$ may therefore already be contained in ${{\cal K}}$. Furthermore, posterior sampling is known to be challenging in high-dimensional latent spaces especially for discrete latents. Many of these challenges may potentially carry over to TV-EM if samples from the posterior are used. Preliminary work by @LuckeEtAl2017 investigates such sampling procedures using TV-EM for two concrete models, Binary Sparse Coding [BSC; @HaftEtAl2004; @HennigesEtAl2010] and sigmoid belief networks [@SaulEtAl1996; @JordanEtAl1999]. The approach emphasizes scalability and autonomous ‘black-box’ optimization procedures. Both is achieved using a combination of prior sampling and approximate marginal sampling, as both these sampling procedures can be defined without additional derivations. TV-EM for Mixture Models {#SecMixtureModels} ------------------------ Mixture models can be regarded as complementary to multiple-cause generative models. In their different versions, they are among the most widely applied generative data models and very successful in many tasks of image or sound processing as well as for general pattern analysis tasks [e.g. @McLachlanPeel2004; @Duda2007; @ZoranWeiss2011; @PoveyEtAl2011]. In contrast to multiple-cause models, any observed variable ${\vec{y}}$ is in mixture models assumed to be generated by exactly one cause, i.e., one class. In their most standard version, a discrete hidden variable $c$ is taken to represent one of $C$ causes or [*clusters*]{} and to generate data via a noise distribution $p({\vec{y}}\,|\,c,\Theta)$. The data distribution assumed by a mixture model is thus given by: $$\label{EqnMixtureModel} p({\vec{y}}\,|\,\Theta) \ =\ \sum_{c=1}^{C} \pi_c p({\vec{y}}\,|\,c,\Theta)\ \ \ \mbox{with}\ \ \sum_{c=1}^{C}\pi_c=1, $$ where the prior parameters $\pi_c=p(c\,|\,\Theta)\in[0,1]$ are model parameters commonly referred to as mixing proportions. As the hidden variable is discrete, TV-EM can directly be applied. We here only change the notation slightly to be more consistent with the conventional mixture model notation, i.e., we replace ${\vec{s}}$ for the hidden variable by the integer $c\in\{1,\ldots,C\}$. The sets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ then consequently contain subsets of class indices, ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}\subseteq\{1,\ldots,C\}$. Mixture models may not be considered as the typical application domain of variational EM procedures, but they were used as example applications already early on. @NealHinton1998, for instance, motivated the application of variational EM by its increased efficiency for mixture models, and we will see below that the same motivation applies for the application of TV-EM. For this example we consider TV-EM with a full E-step (Eqns.\[EqnTVEStep\] to \[EqnTVThird\]). Based on Eqn.\[EqnTVEStep\], the TV-E-step for mixture models corresponds to finding those states $c$ that globally maximize the simplified free energy ${{\cal F}}({{\cal K}},\Theta)$ (\[EqnTruncatedF\]) of Prop.3 w.r.t.${{\cal K}}$. Let us again constrain all sets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ to be of the same size, i.e., $|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|=C'\leq{}C$ in this case. Because of the form of the free energy in Eqn.\[EqnTruncatedF\], we can now show that it is sufficient and efficient to pair-wise compare all joint probabilities in order to find optimal ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$.  \ [**Proposition 7**]{}\ Consider the application of TV-EM to a mixture model given by (\[EqnMixtureModel\]) with $C$ clusters, and let ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ (with $|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|=C'$) be the set of variational states for a data point ${\vec{y}^{\,(n)}}$. Then the free energy is maximized in the TV-E-step (\[EqnTVEStep\]) if for all $n=1,\ldots,N$ the sets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ contain the $C'$ clusters with the largest joint probabilities, i.e., if $$\begin{aligned} \label{EqnLargestJoints} \mbox{for all}\ \ c\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}\ \mbox{ and }\ c'\not\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}:\ \ p(c,{\vec{y}^{\,(n)}}\,|\,\Theta) \geq p(c',{\vec{y}^{\,(n)}}\,|\,\Theta). $$ Such a maximum can be found using ${{\cal O}}(NC)$ comparisons of joint probabilities.  \ [**Proof**]{}\ Let us consider the set ${{\cal K}}=({{\cal K}}^{(1)},\ldots,{{\cal K}}^{(N)})$ for which each ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ fulfills criterion (\[EqnLargestJoints\]). If we now replace for a specific but arbitrary $n$ an arbitrary $c\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ by a $c'\not\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ then by our definition of ${{\cal K}}$: $ \begin{array}{lrcl} p(c',{\vec{y}^{\,(n)}}\,|\,\Theta) \leq p(c,{\vec{y}^{\,(n)}}\,|\,\Theta). \end{array} $ According to Prop.6, the free energy is then decreased or remains constant. As for any $n$ a change of any cluster $c\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ results in a decreased or constant free energy, the set ${{\cal K}}$ must represent a global maximum of ${{\cal F}}({{\cal K}},\Theta)$ (no better set can be found).\ Regarding the complexity of finding the maximum, let ${\cal C}^{(n)}$ be a list of all joint probabilities $p(c,{\vec{y}^{\,(n)}}\,|\,\Theta)$ for a fixed data point ${\vec{y}^{\,(n)}}$. All such lists ${\cal C}^{(n)}$ are of size $C$. Let us first suppose that all elements in ${\cal C}^{(n)}$ have different values. Finding the set ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ which fulfills (\[EqnLargestJoints\]) is then the problem of finding the $|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|$ largest elements in a list of $C$ elements. This partial sorting problem is according to [@BlumEtAl1973] solvable using ${{\cal O}}(C)$ comparisons of the elements. In case of two or more identical elements in ${\cal C}^{(n)}$, the same partial sorting procedure returns a list (i.e., a set ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$) which also fulfills $(\ref{EqnLargestJoints})$ (but there may now be more than one such ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ satisfying the criterion). By repeating the procedure $N$ times (once for each data point $n$), we can define a set ${{\cal K}}=({{\cal K}}^{(1)},\ldots,{{\cal K}}^{(N)})$ for which each ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ fulfills (\[EqnLargestJoints\]). The set ${{\cal K}}$ is therefore (A) computable using ${{\cal O}}(NC)$ comparisons of joint probabilities, and (B) it maximizes the free energy because all its ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ satisfy $(\ref{EqnLargestJoints})$.\ [$\square$]{}\ Prop.7 defines a concrete deterministic and efficient procedure applicable to any mixture model of the form (\[EqnMixtureModel\]). In practice, the procedure requires to actually compute all the joint probabilities first (at least up to a common factor) in order to realize the required comparison. The computational demand for computing the joints is ${{\cal O}}(NC)$ times the computations required for the evaluation of each joint (which is usually proportional to the number data space dimensions $D$), e.g., ${{\cal O}}(NCD)$ for Gaussian Mixture Models (GMMs). As mixture models have latent state spaces of size linear in $C$, it may not be considered surprising that TV-EM is applicable using ${{\cal O}}(NC)$ comparisons of joint probabilities. After all, an exact E-step of standard EM, e.g., for GMMs, also only requires ${{\cal O}}(NCD)$ computations. However, TV-EM can reduce the required computations for the M-step because it uses exact zeros, i.e., M-steps with expectation values (\[EqnSuffStatMain\]) can be shown to be less complex. The price to pay for this reduction is that the formal optimization problem of the TV-E-step (Eqn.\[EqnTVEStep\]) considers an optimization problem on a space larger than the state space. For the here (and throughout the paper) assumed equally sized sets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ (here $|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|=C'$), the number of all possible subsets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}\subseteq\{1,\ldots,C\}$ is $\left(\begin{array}{c}C\\C'\end{array}\right)$. Prop.7 ensures that we do not have to exhaustively visit all these subsets but can find the maximizing set for each $n$ efficiently. Again, this efficiency result is ultimately due to the simplified form of the free energy (\[EqnTruncatedF\]). init model parameters ${\Theta^{\mathrm{old}}}$;[\ ]{} The application of TV-EM to mixture models (as summarized by Alg.2) now allows for interpreting earlier applications to mixture models within the derived free energy framework. Truncated approximations were previously applied to mixture models, e.g., to GMMs in work by @SheltonEtAl2014 and later by @HughesSudderth2016. The cluster finding procedure used by @SheltonEtAl2014 can in the light of this study be recognized as an estimated TV-E-step (using Gaussian Processes to construct the sets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$), while the constrained likelihood optimization for exponential family mixtures as used by @HughesSudderth2016 can be recognized as a full TV-E-step. For Poisson mixtures, @ForsterLucke2017 directly applied the TV-EM algorithm (Alg.2) suggested by Prop.7. For all the above applications, our theoretical results show that the free energy (\[EqnTruncatedF\]) is the underlying objective function which is maximized. For the algorithms [@HughesSudderth2016; @ForsterLucke2017] the TV-EM application to mixture models, furthermore, warrants that the free energy is provably monotonically increased, which follows from Prop.5 and has not been shown previously. Furthermore, our results apply for [*any*]{} mixture model of the form (\[EqnMixtureModel\]) and for Alg.2 as well as for corresponding partial EM versions. The main motivation and focus of the previous truncated approximations for mixture models [@HughesSudderth2016; @ForsterLucke2017] was the increase of efficiency. The source for the reduction of computational efforts were hereby the hard zeros introduced by truncated posteriors, which significantly reduced the required number of numerical operations in the M-step. The work by @HughesSudderth2016 focuses on this M-step complexity reduction, and they empirically find that the whole EM optimization only requires about half the operations compared to exact EM. Also, @ForsterLucke2017 focus on the complexity reduction provided by the M-step and observe a similar efficiency increase for Poisson mixtures. Notably, TV-EM does not negatively effect the final likelihood values that were reported in these studies. On the contrary, faster convergence and higher final likelihoods for different datasets were observed empirically [@HughesSudderth2016; @ForsterLucke2017; @LuckeForster2017]. This is due to TV-EM avoiding local likelihood optima more efficiently than exact EM – an effect that has also been observed for sparse coding models and (preselection-based) truncated approximations [@ExarchakisEtAl2012]. Except of reducing the M-step complexity by TV-EM [@HughesSudderth2016; @ForsterLucke2017], the here derived results point to a further possibility for complexity reduction. In deriving partial TV-EM (Eqns.\[EqnTVEMOptStepsParialFinal\]) we have shown that the free energy (\[EqnTruncatedF\]) also monotonically increases for partial TV-E-steps. A full maximization is, hence, not required to obtain an algorithm that provably increases (\[EqnTruncatedF\]). As efficient criteria to verify increased free energies are available, very efficient partial E-steps were investigated in work parallel to this study. @ForsterLucke2017b have thus shown that clustering algorithms in which each EM iteration scales sublinearly with $C$ can be derived. Numerical experiments then show that the free energy (and likelihood) objective is still efficiently increased, which provides evidence for clustering being scalable sublinearly with $C$ [for details see @ForsterLucke2017b]. TV-EM and ‘hard EM’ {#SecHardEM} ------------------- A very wide-spread approach to optimize parameters of a given generative model is ‘hard EM’ also known as zero-temperature EM, MAP approximation, Viterbi training, classification EM, etc (see introduction). As the name suggests, ‘hard EM’ is typically introduced as an EM-like algorithm in which the computation of the full posterior in the E-step is replaced by the computation of the state ${\vec{s}}$ with maximum a-posterior (MAP) probability. In the M-step, the model parameters are then updated by only considering this maximum a-posterior state. Alg.\[AlgHardEM\] shows a standard form of the ‘hard EM’ algorithm. [r]{}[0.66]{}  \ init model parameters ${\Theta^{\mathrm{old}}}$;\  \ ‘Hard EM’ is often regarded as an [*ad hoc*]{} procedure, which replaces a computationally intractable full posterior in the E-step by a maximization. Such a maximization is easier because it can be reformulated as the maximization of a computationally tractable objective, the joint $p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)$ (as the normalizer $p({\vec{y}^{\,(n)}}\,|\,\Theta)$ does not depend on ${\vec{s}}$). ‘Hard EM’ can, however, also be derived from annealed versions of EM. Annealed EM is a procedure usually introduced in order to avoid local optima [e.g. @UedaNakano1998; @Sahani1999]. A temperature parameter is introduced which forces the probability values of the posterior to become more equal. While annealed EM algorithms are obtained for high temperatures, ‘hard EM’ is obtained if instead the limit to zero temperature is considered (see Appendix for details). In any case, ‘hard EM’ remains rather an heuristic approach; it requires to take a limit (temperature zero), and it implicitly assumes that states that maximize the posteriors can actually be found. By considering Alg.\[AlgHardEM\], ‘hard EM’ can be formulated by replacing the posterior by a $\delta$-function centered at the MAP state (often termed Dirac-$\delta$ in the continuous and Kronecker-$\delta$ in the discrete case). So far, $\delta$-functions have merely been considered in this context as a way to explicitly formulate the function which replaces the full posterior in the MAP approximation. Following the introduction of truncated posteriors for TV-EM, and in virtue of Props.1 and 2, we can now treat the $\delta$-functions fully variationally. For this, we consider the states for which the $\delta$-functions are non-zero as variational parameters of truncated distributions. Such a formulation then corresponds to a TV-EM algorithm with sets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ which each contain just one element, i.e. ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}=\{{\vec{s}^{\,(n)}}\}$ for all $n$. More precisely, we can for this boundary case of TV-EM show the following:  \ [**Proposition 8**]{}\ Consider a generative model $p({\vec{s}},{\vec{y}}\,|\,\Theta)$ with discrete latents ${\vec{s}}$. Then ‘hard EM’ for this model (Alg.\[AlgHardEM\]) is equivalent to a TV-EM algorithm which uses sets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ with just one element each.\ [**Proof**]{}\ Note that all results derived for truncated variational distributions, so far, apply for arbitrary (non-empty) sets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$. For a TV-EM algorithm with sets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ that contain just one element each, we can denote these elements by ${\vec{s}^{\,(n)}}$, i.e., ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}=\{{\vec{s}^{\,(n)}}\}$. Let us first consider the simplified truncated free energy (\[EqnTruncatedF\]) derived in Prop.3. For ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}=\{{\vec{s}^{\,(n)}}\}$ we then obtain: $${{\cal F}}({{\cal K}},{\Theta^{\mathrm{old}}}) = {\displaystyle}\sum_{n=1}^{N}\log\big({\hspace{-1.5mm}}\sum_{\ {\vec{s}}\in{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}} p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,{\Theta^{\mathrm{old}}})\,\big) = {\displaystyle}\sum_{n=1}^{N}\log\big( p({\vec{s}^{\,(n)}},{\vec{y}^{\,(n)}}\,|\,{\Theta^{\mathrm{old}}})\,\big)\,. $$ The maximum of ${{\cal F}}({{\cal K}},{\Theta^{\mathrm{old}}})$ w.r.t. ${{\cal K}}=({{\cal K}}^{(1)},\ldots,{{\cal K}}^{(N)})$ can be found by individually maximizing each summand $\log\big( p({\vec{s}^{\,(n)}},{\vec{y}^{\,(n)}}\,|\,{\Theta^{\mathrm{old}}})$ w.r.t.${\vec{s}^{\,(n)}}$. The states ${\vec{s}^{\,(n)}}$ that maximize the summands are then the same as those computed in the hard E-step of Alg.\[AlgHardEM\] because: $$\begin{aligned} {\vec{s}^{\,(n)}}&=& { \underset{{\vec{s}}}{\mathrm{argmax}} }\big\{ \log\big( p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,{\Theta^{\mathrm{old}}}) \big\}\ =\ { \underset{{\vec{s}}}{\mathrm{argmax}} }\big\{ p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},{\Theta^{\mathrm{old}}}) \big\}\,. $$ The new set ${{{\cal K}}^{\mathrm{new}}}$ which is computed by the TV-E-step (\[EqnTVEStep\]) is consequently given by ${{{\cal K}}^{\mathrm{new}}}=(\{{\vec{s}}^{(1)}\},\ldots,\{{\vec{s}}^{(N)}\})$. In the TV-M-step (\[EqnTVMStep\]) the truncated free energy ${{\cal F}}({{{\cal K}}^{\mathrm{new}}},{\Theta^{\mathrm{old}}},\Theta)$ is then optimized w.r.t. $\Theta$. When ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}=\{{\vec{s}^{\,(n)}}\}$, a truncated distribution ${q^{(n)}}({\vec{s}};{{{\cal K}}^{\mathrm{new}}},{\Theta^{\mathrm{old}}})$ is unequal zero only for the state ${\vec{s}}={\vec{s}^{\,(n)}}$, i.e., ${q^{(n)}}({\vec{s}};{{{\cal K}}^{\mathrm{new}}},{\Theta^{\mathrm{old}}})=\delta({\vec{s}}={\vec{s}^{\,(n)}})$. As, additionally, the entropy term of the free energy vanishes for such ${q^{(n)}}$, the free energy ${{\cal F}}({{\cal K}},{\Theta^{\mathrm{old}}},\Theta)$ reduces to: $$\begin{aligned} {{\cal F}}({{{\cal K}}^{\mathrm{new}}},{\Theta^{\mathrm{old}}},\Theta) &=& \sum_{n=1}^{N} \bigg[ \sum_{{\vec{s}}}\ {q^{(n)}}({\vec{s}};{{{\cal K}}^{\mathrm{new}}},{\Theta^{\mathrm{old}}})\ \log\left( p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\right) \bigg] + H(q({\vec{s}};{{{\cal K}}^{\mathrm{new}}},{\Theta^{\mathrm{old}}}))\nonumber\\ &=& \sum_{n=1}^{N} \bigg[ \sum_{{\vec{s}}}\ \delta({\vec{s}}={\vec{s}^{\,(n)}})\ \log\left( p({\vec{s}},{\vec{y}^{\,(n)}}\,|\,\Theta)\right) \ =\ \sum_{n=1}^{N} \log\Big( p({\vec{s}^{\,(n)}},{\vec{y}^{\,(n)}}\,|\,\Theta)\Big).\nonumber $$ Maximization of the free energy ${{\cal F}}({{\cal K}},{\Theta^{\mathrm{old}}},\Theta)$ w.r.t.$\Theta$ is thus equivalent to the ‘hard’ M-step of Alg.\[AlgHardEM\]. As the ${\vec{s}^{\,(n)}}$ used in the hard M-step are precisely those computed in the ‘hard’ E-step, TV-EM with just one element for each ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ is equivalent to ‘hard EM’ (Alg.\[AlgHardEM\]).\ [$\square$]{}\  \ In practice, the maximization in the hard E-step is often difficult to accomplish, such that states ${\vec{s}}$ are computed that only approximately maximize the posteriors $p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)$. Such a partial ‘hard EM’ approach can then be shown to correspond to TV-EM with a partial E-step (Eqns.\[EqnTVEMOptStepsParialFinal\]), and the proof follows along the same line as the proof for Prop.8. The equivalence of ‘hard EM’ and TV-EM with one state per ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ (as provided by Prop.8) applies for any generative model with discrete latents. Based on this equivalence we can instantly conclude that ‘hard EM’ optimizes a truncated free energy.  \ [**Corollary 2**]{}\ Consider a generative model $p({\vec{s}},{\vec{y}}\,|\,\Theta)$ with discrete latents ${\vec{s}}$. Then ‘hard EM’ monotonically increases a lower free energy bound of the log-likelihood given by: $$\begin{aligned} \label{EqnFreeEnergyHard} {{\cal F}}({\vec{s}}^{\,(1:N)},\Theta) = \sum_{n=1}^{N}\log\Big( p({\vec{s}^{\,(n)}},{\vec{y}^{\,(n)}}\,|\,\Theta)\Big) \leq {{\cal L}}(\Theta), $$ where ${\vec{s}^{\,(n)}}$ are the states computed in the ‘hard’ E-step and the $\Theta$ are the parameters computed by the hard M-step of Alg.\[AlgHardEM\]. The free energy (\[EqnFreeEnergyHard\]) is also monotonically increased if each ‘hard’ E-step in Alg.\[AlgHardEM\] just monotonically increases (instead of maximizes) the posteriors, i.e.if ${\vec{s}^{\,(n)}}$ are found such that for all $n$ applies:\ $$\begin{aligned} \label{EqnHardEMPartial} p({\vec{s}^{\,(n)}}\,|\,{\vec{y}^{\,(n)}},{\Theta^{\mathrm{old}}})\geq{}p({\vec{s}^{\,(n)}}_{\mathrm{old}}\,|\,{\vec{y}^{\,(n)}},{\Theta^{\mathrm{old}}}), $$ where ${\vec{s}^{\,(n)}}_{\mathrm{old}}$ are the states found in the previous ‘hard’ E-step. After each ‘hard EM’ iteration or partial ‘hard EM’ iteration, the difference between log-likelihood (\[EqnLikelihood\]) and free energy (\[EqnFreeEnergyHard\]) is given by: $$\begin{aligned} \label{EqnDiffHardEM} {{\cal L}}(\Theta) - {{\cal F}}({\vec{s}}^{(1:N)},\Theta) = - \sum_{n=1}^N \log\big(p({\vec{s}^{\,(n)}}\,|\,{\vec{y}^{\,(n)}},\Theta)\big). $$  \ [**Proof**]{}\ According to Prop.8, ‘hard’ EM is equivalent to TV-EM (with sets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}=\{{\vec{s}^{\,(n)}}\}$), which implies that results of Props.1 to 5 are applicable to ‘hard’ EM, including a guaranteed monotonic increase of the simplified truncated free energy (\[EqnTruncatedF\]). For ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}=\{{\vec{s}^{\,(n)}}\}$ the truncated free energy ${{\cal F}}({{\cal K}},\Theta)$ is given by (\[EqnFreeEnergyHard\]), where we replaced ${{\cal K}}=(\{{\vec{s}}^{(1)}\},\ldots,\{{\vec{s}}^{(N)}\})$ by ${\vec{s}}^{(1:N)}$, which proves the first claim. According to the results for partial TV-EM (Sec.\[SecPartialTVEM\]), the free energy (\[EqnFreeEnergyHard\]) also monotonically increases for a [*partial*]{} TV-E-step (Optimization 1 of Eqns.\[EqnTVEMOptStepsParialFinal\]). For ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}=\{{\vec{s}^{\,(n)}}\}$, the free energy (\[EqnFreeEnergyHard\]) is monotonically increased if for all $n$ applies $p({\vec{s}^{\,(n)}},{\vec{y}^{\,(n)}}\,|\,{\Theta^{\mathrm{old}}})\geq{}p({\vec{s}^{\,(n)}}_{\mathrm{old}},{\vec{y}^{\,(n)}}\,|\,{\Theta^{\mathrm{old}}})$ (compare Prop.6). As the data points are constant, this condition is equivalent to condition (\[EqnHardEMPartial\]) which proves the claim, i.e., it is sufficient to monotonically increase $p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},{\Theta^{\mathrm{old}}})$ for all $n$ given the current parameters ${\Theta^{\mathrm{old}}}$, and starting from the previous MAP states ${\vec{s}^{\,(n)}}_{\mathrm{old}}$. According to Props.1 to 3, ${{\cal F}}({\vec{s}}^{(1:N)},\Theta)$ is, as a special case of ${{\cal F}}({\vec{s}}^{(1:N)},{\Theta^{\mathrm{old}}},\Theta)$, a lower bound of the log-likelihood. Furthermore, the difference ${{\cal L}}(\Theta) - {{\cal F}}({\vec{s}}^{(1:N)},\Theta)$ is given by the KL-divergence $\sum_n{D_{\mathrm{KL}}\big({q^{(n)}}({\vec{s}};{{\cal K}},\Theta),p({\vec{s}}\,|\,{\vec{y}^{\,(n)}},\Theta)\big)}$ (Corollary 1). As the truncated variational distributions are here given by ${q^{(n)}}({\vec{s}};{{{\cal K}}^{\mathrm{new}}},{\Theta^{\mathrm{old}}})=\delta({\vec{s}}={\vec{s}^{\,(n)}})$, we obtain (\[EqnDiffHardEM\]). The relation between ${{\cal L}}(\Theta) - {{\cal F}}({\vec{s}}^{(1:N)},\Theta)$ and the KL-divergence also applies for partial TV-EM and consequently for partial ‘hard EM’ as a special case.\ [$\square$]{}\  \ Algorithmically, Prop.8 does not represent much novelty: optimizations by ‘hard EM’ or (non-partial) TV-EM with $|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|=1$ are the same. Importantly, however, Prop.8 and Corollary 2 for the first time fully embed the frequently used ‘hard EM’ approaches into the theoretical framework of variational free energy optimization. Importantly, this embedding includes the in practice frequently used ‘hard EM’ algorithms which use a partial E-step as a full maximization is computationally more expensive or even NP-hard [see, e.g., @CohenSmith2010]. In these cases, Corollary 2 provides the directly applicable result of monotonically increasing free energies, and it provides the theoretical justification for memorizing previous (approximate) MAP states as starting values for the next MAP optimization. The embedding of ‘hard EM’ into the free energy framework is made possible, first, by Prop.1 and 2, and, second, by the further theoretical results specific to truncated distributions (Props.3 to 5) which allow to interpret the MAP states ${\vec{s}^{\,(n)}}$ as variational parameters. Note that annealed EM (see Appendix) does not provide such an embedding. Using a rigorous treatment similar to the one for truncated distributions considered in this work, results similar to Props.4 and 5 may be derivable, and for the limit to zero temperature, results similar to those of Props.1 and 2 would have to be derived. Considering the free energy (\[EqnFreeEnergyHard\]), note that objective functions optimized by ‘hard EM’ have already previously been discussed. However, they were usually defined as a tool to interpret the heuristically introduced ‘hard EM’ algorithm[^1]. Instead, we have here shown that ‘hard EM’ and its optimization objective can be derived in the same mathematically grounded way as other variational EM procedures such as mean-field or Gaussian variational EM. That is, we considered a constrained set of functions for the approximation of full posteriors, and then canonically derived a variational EM algorithm and a provably increasing free energy objective. In addition to embedding ‘hard EM’ into the free energy framework, Prop.8 shows that ‘hard EM’ is a special case of TV-EM. We can, therefore, now also interpret TV-EM as a generalization of ‘hard EM’. TV-EM algorithms using sets ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ with more than one element do maintain ‘hard’ (i.e., exact) zeros but they do allow for more than one hidden state with non-zero probability. Such TV-EM generalization of ‘hard EM’ are notably non-trivial. For ‘hard EM’ we do not necessarily require Prop.5 in order to show that the free energy monotonically increases. This is because the free energy and its simplified version given by Prop.3 coincide for one element per ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$. In this case TV-EM (i.e., ‘hard EM’) becomes a straight-forward coordinate-wise ascent approach w.r.t. to this objective. However, any generalization to more than one state with non-zero probability changes the free energy objective ${{\cal F}}({{\cal K}},\Theta)$ to the more general form (\[EqnTruncatedF\]) given by Prop.3. Taking derivatives of ${{\cal F}}({{\cal K}},\Theta)$ in (\[EqnTruncatedF\]) w.r.t.$\Theta$ is now different from taking derivatives of the free energy ${{\cal F}}({{\cal K}},{\Theta^{\mathrm{old}}},\Theta)$ in (\[EqnFreeEnergyTVEMOrg\]). If the standard M-step equations for a given generative model are maintained (which optimize ${{\cal F}}({{\cal K}},{\Theta^{\mathrm{old}}},\Theta)$) then we require Prop.5 to show that TV-EM indeed monotonically increases the free energy. For mixture models, truncated approximations which maintain two, three or other low numbers of states have in empirical studies been found to work very well in practice. Such ‘almost hard EM’ approaches have been applied to invariant multiple-causes models [@DaiLucke2012b; @DaiEtAl2013], invariant mixture models [@DaiLucke2014], standard Gaussian mixture models [@SheltonEtAl2014; @HughesSudderth2016; @LuckeForster2017], Poisson mixtures [@ForsterLucke2017] and topic models [@HughesSudderth2016]. Already maintaining [*very*]{} low numbers of states per data point was in many experiments shown to efficiently and effectively recover ground-truth parameters, e.g., for GMMs [@SheltonEtAl2014; @HughesSudderth2016], and to very efficiently improve the likelihood objective [@HughesSudderth2016; @ForsterLucke2017; @LuckeForster2017; @ForsterLucke2017b]. Earlier applications of truncated approximations [e.g. @DaiEtAl2013; @DaiLucke2014; @SheltonEtAl2014] do not use the theoretical framework derived here but their E-steps can (according to Sec.\[SecSampling\]) be interpreted as estimated TV-E-steps. Discussion {#SecDiscussion} ========== We have defined and analyzed a novel variational approximation of expectation maximization (EM). Our approach is based on truncated a-posteriori distributions with latent states as variational parameters. Our first set of results (Props. 1 and 2) generalize the variational free energy approach as introduced, e.g., by @SaulEtAl1996 [@NealHinton1998; @JordanEtAl1999] by including discrete variational distributions with exact zeros. While this generalization is required in order to study truncated variational distributions, Props.1 and 2 also apply for any other discrete distributions with exact (‘hard’) zeros, i.e., these results are not restricted to the specific truncated distributions of Eqn.\[EqnQMain\]. As such, Props. 1 and 2 generalize the standard text book derivation of variational free energies [e.g. @Bishop2006; @Murphy2012; @Barber2012], i.e., the standard additional demand of $q({\vec{s}})>0$ can be dropped for the discrete case. Props. 3, 4, 5 and Corollary 1 then represent results specific to variational distributions in the form of truncated posteriors (Eqn.\[EqnQMain\]). Props. 6, 7 and 8 and Corollary 2, finally, represent example applications of the theoretical results. #### Relation to Preselection-Based Truncated Approximations. Previous algorithms based on truncated approximations of expectation values [@LuckeSahani2008; @LuckeEggert2010; @HennigesEtAl2014; @DaiLucke2014; @SheikhEtAl2014; @SheikhLucke2016; @SheltonEtAl2017] have motivated this work. These contributions have directly approximated expectation values by exploiting sparsity of latent activities [@LuckeSahani2008] and by additionally using a preselection procedure in what was termed Expectation Truncation [ET; @LuckeEggert2010]. Based on the theoretical framework derived in this work and Sec.\[SecSampling\], all previous selection-based approaches can be considered as approximations of TV-EM. [*Expectation Truncation*]{} can thus be embedded into the framework of variational approaches. While estimated E-steps using ET may potentially [*decrease*]{} the truncated free energy of Prop.3, the general effectiveness and efficiency of previous ET applications may be taken as evidence for the efficiency and effectiveness of truncated approximations in general. Such efficiency and effectiveness can now be generalized and theoretical guarantees for tractable free energies are available. Furthermore, and maybe most importantly, TV-EM can now provide (A) a straight-forward generalization and applicability to very advanced (including deep) data models, and (B) it avoids any additional effort to define model specific selection functions (compare discussion of ‘black box’ procedures below). In one aspects, ET provides a result that has not been addressed here, however: it can be shown that preselection allows for the definition of smaller generative models defined per data point, and that optimizing parameters of such smaller models approximately optimizes the parameters of the original larger generative model. This result, valid for a large class of generative models (but not for all), has been used in select-and-sample approaches [@SheltonEtAl2012; @SheltonEtAl2015], was formally proven in [@SheikhLucke2016], and carries over to recent work using selection functions that are themselves learned from data [@SheltonEtAl2017]. #### Relation to mean-field and ‘sparse EM’. Truncated variational EM is a variational approximation for models with discrete latents. Gaussian variational EM is not applicable to discrete hidden variables. The main standard class of variational approaches for comparison with TV-EM is therefore given by factored variational (i.e.[*mean-field*]{}) approaches [@SaulJordan1996; @JordanEtAl1999]. Compared to fully factored approaches (Eqn.\[EqnFreeEnergyFVEM\]), a main difference to TV-EM is that truncated approaches do not assume posterior independence. Assuming independence (i.e., neglecting explaining-away effects) has been observed to negatively impact likelihood optimization for different types of generative models [@IlinValpola2003; @MacKay2003; @TurnerSahani2011; @SheikhEtAl2014]. To address such potentially harmful consequences of posterior independence, partly factored approaches (sometimes called structured variational or [*structured mean-field*]{} approaches) have been studied [@SaulJordan1996; @MacKay2003; @Bouchard2009]. Any deviation from fully factored approaches does, however, often go along with increased analytical and computational effort, and may even provide only limited improvements compared to mean-field [e.g. @TurnerSahani2011 for a discussion]. In comparison, TV-EM does not assume independence. Computational tractability is instead achieved by approximating posteriors by considering only small subsets of the latent space. Other than mean-field variational approximations, ‘sparse EM’ is another (less frequently applied) alternative which was discussed early on in the very influential work by @NealHinton1998. Notably, @NealHinton1998 never explicitly mentioned [*factored*]{} variational distributions in their work. Instead the paper discussed with ‘sparse EM’ a variational approach that shares more similarities with TV-EM than with mean-field. In their ‘sparse algorithm’, @NealHinton1998 suggested a variational distribution defined on a subset of the state space which only contains ‘plausible’ values of the latents (their Sec.5). The probabilities outside of this set were ‘frozen’ to values of an earlier iteration but updated once in a while during learning. The procedure there described is not efficient, e.g. for latent variable models with large state spaces, because ‘sparse EM’ still requires an occasional evaluation of [*all*]{} latent states. As @NealHinton1998 discuss an application to a mixture model, this shortcoming is not very relevant for their paper. In contrast to ‘sparse EM’, truncated approximations assume exact zeros and can thus realize approximations without ever having to evaluate all hidden states while they are still able to find subsets of the latent space with high posterior mass. However, despite these differences, the results obtained for TV-EM in this work may be regarded as connecting back to an initial (and never followed up) train of thoughts expressed in the work by @NealHinton1998. #### Relation to ‘hard EM’. A further, very influential class of approximate EM alogrithms is ‘hard EM’ (alias ‘zero temperature EM’, ‘classification EM’ or EM using MAP approximations). ‘Hard EM’ approaches have been used for many types of data models including deep models, and they were often observed to work very well in practice. Because of their wide-spread use, e.g., in domains such as sparse coding [@OlshausenField1996; @MairalEtAl2010], compressive sensing [@Donoho2006; @Baraniuk2007] but also for relating generative modeling and deep learning [@PatelEtAl2016], ’hard EM’ may even be considered more wide-spread than any conventional or novel variational EM approach. As shown in Sec.\[SecHardEM\], we here identified ‘hard EM’ as a TV-EM algorithm with sets of variational parameters ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ just containing one state (Prop.8 and Corollary 2). ‘Hard EM’ (including versions with partial posterior optimization) can consequently be cleanly embedded into the variational EM framework and monotonically increasing free energies can be provided. Furthermore, TV-EM provides concrete procedures to generalize any ‘hard EM’ algorithm to algorithms with multiple ‘winning’ states. For instance, for training of deep networks with ‘hard EM’ [e.g. @PoonDomingos2011; @OordEtAl2014; @PatelEtAl2016], generalizations with more than one non-zero state maybe considered very interesting [especially considering the effectiveness of such approaches for mixture models; e.g. @HughesSudderth2016; @ForsterLucke2017]. Similarly, time-series models such as Hidden Markov Models (HMMs) and their many variants are often trained using ‘hard EM’ [@JuangRabiner1990; @CohenSmith2010; @AllahverdyanGalstyan2011]. TV-EM generalizations would thus provide promising future generalizations especially when noting that algorithms estimating multiple winning states are sometimes readily available [e.g. @Foreman1992; @HuangEtAl2012]. Again, TV-EM may also serve to interpret earlier combinations of ‘hard EM’ and standard EM for HMMs [e.g. @AllahverdyanGalstyan2011; @SpitkovskyEtAl2011] on the ground of a variational EM framework. But also for long-standing standard tasks such as clustering, the here obtained results are of direct theoretical and practical relevance. By applying TV-EM to a special case of GMMs (isotropic and equally weighted Gaussians), @LuckeForster2017 have shown, for instance, that the optimization of cluster centers decouples from the optimization of cluster variance for ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$ with just one element (while the same is not true for $|{{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}|>1$). The optimization of cluster centers is then observed to be equivalent to $k$-means. The equivalence is notably obtained without the requirement of taking the limit to zero cluster variances [which is the standard textbook procedure to relate GMMs and $k$-means, e.g., @MacKay2003; @Barber2012]. Furthermore, TV-EM for GMMs provides a free energy objective which is provably (and in this case strictly) increased by $k$-means. The objective bounds the GMM log-likelihood from below and is a function of the $k$-means objective, i.e., of the quantization error [see @LuckeForster2017 for details]. Finally, note that Prop.8 and Corollary 2 apply in general for any generative model, which (A) implies that TV-EM generalizations of any previous ‘hard EM’ approach are possible, and which (B) suggests that free energy results as for $k$-means and GMMs [@LuckeForster2017] may be obtained also for other generative models with discrete latents. #### Relation to Sampling. While standard variational EM procedures are typically regarded as being deterministic, TV-EM shares many properties with stochastic (i.e., sampling based) EM approaches. Like sampling approaches, TV-EM approximates probabilities (in our case posteriors) by a set of states in hidden space, and these states are then used to compute expectation values. Indeed, one option to realize concrete TV-EM optimizations is to suggest new states for ${{\cal K}}$ by sampling from appropriate distributions (Sec.\[SecSampling\]). TV-EM also shares with sampling that the accuracy of the approximation is only limited by computational demand. In the limit of infinite computational resources both sampling and TV-EM converge to EM with exact E-steps. The same can not be said about standard variational approaches like mean-field or Gaussian. TV-EM distinguishes itself from sampling, however, by being a variational approach that optimizes a free energy using variational distributions. While truncated approximations [*can*]{} make use of sampling as part of their optimization, sampling is just one option and other procedures like selection functions or other deterministic procedure (compare Sec.\[SecMixtureModels\]) can be applied. Also the distributions that are used to suggest states for ${{\cal K}}$ are not limited to posterior distributions as also other (potentially easier to use) distributions can be sufficiently efficient in providing samples which increase the truncated free energy. While these are all points of difference, the fact that both TV-EM and sampling are approximating posteriors based on finite sets of hidden states, makes TV-EM the variational approximation that is most closely related to sampling. Also other combinations of variational approaches and sampling have been investigated (see discussion of ‘black box’ approaches below) but none directly treats samples as variational parameters. #### Autonomous and General Purpose (‘Black Box’) Inference and Learning. Other than providing a general procedure to develop a learning algorithm for a specific generative model of interest, TV-EM is also of relevance for the field of autonomous machine learning, which recently attracted a lot of attention [@RezendeMohamed2015; @TranEtAl2015; @RanganathEtAl2015; @HernandezEtAl2016]. The goal of this field of study is to provide procedures that minimize expert intervention in the generation and application of learning and inference algorithms. Typically, user intervention is required for a number of steps in the process of developing a concrete learning algorithm for a given model. Both standard variational approaches and standard sampling have to overcome different analytical and practical computational challenges. A factored variational EM approach, for instance, first has to choose a specific form for the variational distributions, and then requires a potentially significant effort to derive update equations for their variational parameters. Instead, TV-EM does not require additional analytical steps for variational E-step equations, expectation values are computed directly based on the variational states. Also in case of sampling, deriving, e.g., an efficient sampler requires additional and potentially highly non-trivial analytical work. On the other hand, the optimization of ${{\cal K}}$ for TV-EM may require additional efforts. For previous truncated approximations, the construction of ${{\cal K}}$ using preselection was model dependent (see Sec\[SecSampling\]). However, given the novel results of this work, previous model specific constructions can be replaced by a general purpose optimization of ${{\cal K}}$. If such an optimization is provided, then TV-E-steps are obtained which just use the joint probability of the generative model under consideration. For the M-step, a similar model independence can be achieved, e.g., by using automatic differentiation techniques, which are continuously further developed. The explicit form of TV-EM in Sec.\[SecExplicitForm\] makes all the requirements of the algorithms very explicit and salient. The formulation in terms of joint probabilities (\[EqnTVEMExplicit\]) also relates TV-EM to research on unnormalized statistical models [e.g. @GutmannHyvarinen2012 and references therein]. The basic idea of the approach, e.g., by @GutmannHyvarinen2012, is the use of classifiers to discriminate between observed and generated data, which is not used by TV-EM. Further differences are our focus on discrete latents, and our requirement for computationally tractable joint probabilities. This latter requirement was for classifier-based training dropped in later extensions, e.g., by @GutmannCorander2016. On the other hand, classifier-based training [also compare @GoodfellowEtAl2014] usually requires the definition of comparison metrics in observed space which is not used by TV-EM. By considering one TV-EM iteration in its explicit form (\[EqnTVEMExplicit\]), also note that TV-EM is very different, e.g., (A) from @RezendeMohamed2015 who use normalizing flows and consider continuous latents, (B) from @RanganathEtAl2015 who successively apply mean-field approach to realize latent dependencies, or (C) from work by @TranEtAl2015 based on copulas. Further related studies are work by @SalimansEtAl2015 who use Markov Chain Monte Carlo (MCMC) samplers and treat the samples as auxiliary variables within a standard variational free energy framework as well as studies using stochastic variational inference [@HoffmanEtAl2013; @HoffmanBlei2015], where auxiliary distributions for Markov chains are defined and used to approximate true posteriors. Moreover, @GuEtAl2015 use variational distributions as proposal distributions to realize flexible and efficient MCMC sampling. All these approaches are more indirect than the direct treatment of latent states as variational parameters as done by TV-EM. @SalimansEtAl2015, for instance, also make a number of choices to define appropriate MCMC samplers (and they focus on models with continuous latents), @HoffmanBlei2015 do require sampling to estimate analytical intractabilities for their variational lower bound, and @GuEtAl2015 use variational distributions as a means to improve approximations by of MCMC sampling. For TV-EM, sampling is one option to vary the variational parameters, the procedure is by definition tractable for sufficiently small ${{{{\cal K}}}^{\hspace{-0.8ex}\phantom{1}^{(n)}}}$, and lower bounds are provably monotonically increased. On the other hand, TV-EM is constrained to discrete latents, and (as a novel approach) the performance for many concrete applications (which has been demonstrated for the other approaches discussed) remains to be explored. In any case, the very active research on such and other very recent methods for a ‘black-box’ optimization framework highlight (A) the requirement for powerful and efficient approaches for advanced data models, and (B) the short-comings of previous mean-field or Gaussian approximations. The use of collections of hidden states within variational approaches, e.g., as done by @SalimansEtAl2015 [@HoffmanBlei2015; @GuEtAl2015] or with truncated approaches [@LuckeEggert2010; @SheltonEtAl2011; @SheikhEtAl2014; @SheikhLucke2016], seems to be a promising general strategy in this respect. #### Outlook and Conclusion. The theoretical framework of free energy optimization [e.g @SaulEtAl1996; @NealHinton1998; @JordanEtAl1999] has been and is of exceptional significance for the development of Machine Learning algorithms. In this work we extend the framework to include variational distributions with ‘hard’ zeros and latent states as variational parameters. The theoretical results derived from these initial assumptions provided us with concise and easily applicable variational EM algorithms as well as concise and tractable forms of free energies. The derived results, consequently, (A) allow for developing novel algorithms, and (B) allow for embedding recent approaches as well as very established approaches into the framework of free energy optimization. Examples for the development of novel algorithms are combinations of variational EM with sampling or preselection methods (Sec.\[SecSampling\]), or novel algorithms for mixture models (Sec.\[SecMixtureModels\]). Approaches that the here derived results embed into a variational free energy framework are relatively recent algorithms based pm constrained likelihood optimizations by @HughesSudderth2016, and earlier truncated approximations [@LuckeSahani2008; @LuckeEggert2010; @HennigesEtAl2010; @DaiLucke2014; @SheikhEtAl2014]. Furthermore, the very popular ‘hard EM’ approaches can cleanly be embedded into the general free energy framework (Sec.\[SecHardEM\]). This embedding deepens the insights into the functioning and capabilities of ‘hard EM’, and it allows for its generalizations (Prop.8, Corollary 2). Still, Secs.\[SecSampling\], \[SecMixtureModels\] and \[SecHardEM\] are example applications of the main theoretical results. Further applications may include future derivations of algorithms other than Alg.1 and 2, or the embedding of further approaches into the free energy framework. The relation between k-means and Gaussian mixtures as studied by @LuckeForster2017 may serve as an example for such an application. Future examples may include generalizations of Viterbi training for HMMs [@JuangRabiner1990; @CohenSmith2010; @AllahverdyanGalstyan2011] or applications to train deep networks [@PoonDomingos2011; @OordEtAl2014; @PatelEtAl2016]. Appendix. ‘Hard EM’ and Annealed EM {#appendix.-hard-em-and-annealed-em .unnumbered} =================================== ‘Hard EM’ can be obtained from exact EM by considering annealed versions of EM [compare @Sahani1999; @MacKay2001]. For this we use the original posteriors $p({\vec{s}}\,|\,{\vec{y}},\Theta)$ to define annealed posteriors $p_{T}({\vec{s}}\,|\,{\vec{y}},\Theta)$ as follows [also compare @UedaNakano1998; @GhahramaniHinton2000; @MandtEtAl2016]: First we define a non-negative energy $E({\vec{s}},{\vec{y}};\Theta)$ given by: $$\label{EqnEnergy} E({\vec{s}},{\vec{y}};\Theta) = -\log\big(p({\vec{y}},{\vec{s}}\,|\,\Theta)\big)\,. $$ We then define the annealed posteriors using the Boltzmann distribution: $$\label{EqnAnnealedEM} p_{\small{T}}({\vec{s}}\,|\,{\vec{y}},\Theta) = \frac{1}{Z_T({\vec{y}};\Theta)}\exp\big(-T\,E({\vec{s}},{\vec{y}};\Theta)\big)\,=\, \frac{\exp\big(-T\,E({\vec{s}},{\vec{y}};\Theta)\big)}{\sum_{{\vec{s}}'}\exp\big(-T\,E({\vec{s}}',{\vec{y}};\Theta)\big)}\,, $$ where $T>0$ is a ’temperature’ parameter (often $\beta=\frac{1}{T}$ is used but for our purposes we remain with $T$). For temperatures $T>1$, the values of annealed posteriors become increasingly similar, and such posteriors are used for annealed EM. For $T=1$, we obtain the original posteriors. Finally, in the limit of $T\rightarrow{}0$ we obtain: $$\begin{aligned} \label{EqnZeroTemperatureEM} \lim_{T\rightarrow{}0} p_{T}({\vec{s}^{\,(n)}}\,|\,{\vec{y}^{\,(n)}},\Theta) &=&\lim_{T\rightarrow{}0} \frac{\exp\big(-T\,E({\vec{s}},{\vec{y}};\Theta)\big)}{\sum_{{\vec{s}}'}\exp\big(-T\,E({\vec{s}}',{\vec{y}};\Theta)\big)}\\[2mm] &=&\left\{\begin{array}{cl} 1 &\mbox{if}\ \ \ {\vec{s}}= { \underset{{\vec{s}}'}{\mathrm{argmin}} }\big\{E({\vec{s}}',{\vec{y}};\Theta)\big\}\\ 0 &\mbox{otherwise} \end{array}\right. \,=\,\left\{\begin{array}{cl} 1 &\mbox{if}\ \ \ {\vec{s}}= { \underset{{\vec{s}}'}{\mathrm{argmax}} }\big\{p({\vec{s}}'\,|\,{\vec{y}},\Theta)\big\}\\ 0 &\mbox{otherwise} \end{array}\right.\nonumber $$ As for this limit for $p_{T}({\vec{s}^{\,(n)}}\,|\,{\vec{y}^{\,(n)}},\Theta)$ coincides with maximum a-posteriori (MAP) estimates, it is sometimes referred to as ‘zero-temperature EM’ [e.g. @TurnerSahani2011]. Allahverdyan, A. and Galstyan, A. (2011). 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--- abstract: 'This paper investigates the evolution of strategic play where players drawn from a finite well-mixed population are offered the opportunity to play in a public goods game. All players accept the offer. However, due to the possibility of unforeseen circumstances, each player has a fixed probability of being unable to participate in the game, unlike similar models which assume voluntary participation. We first study how prescribed stochastic opting-out affects cooperation in finite populations. Moreover, in the model, cooperation is favored by natural selection over both neutral drift and defection if return on investment exceeds a threshold value defined solely by the population size, game size, and a player’s probability of opting-out. Ultimately, increasing the probability that each player is unable to fulfill her promise of participating in the public goods game facilitates natural selection of cooperators. We also use adaptive dynamics to study the coevolution of cooperation and opting-out behavior. However, given rare mutations minutely different from the original population, an analysis based on adaptive dynamics suggests that the over time the population will tend towards complete defection and non-participation, and subsequently, from there, participating cooperators will stand a chance to emerge by neutral drift. Nevertheless, increasing the probability of non-participation decreases the rate at which the population tends towards defection when participating. Our work sheds light on understanding how stochastic opting-out emerges in the first place and its role in the evolution of cooperation.' address: - 'Department of Mathematics, Michigan State University, 619 Red Cedar Road, C212 Wells Hall, East Lansing, MI 48824, USA' - 'Department of Mathematics, Dartmouth College, 27 N Main St., 6188 Kemeny Hall, Hanover, NH 03755, USA' - 'Department of Biomedical Data Science, Geisel School of Medicine at Dartmouth, Lebanon, NH 03756, USA' author: - Alexander G Ginsberg - Feng Fu title: 'Evolution of Cooperation in Public Goods Games with Stochastic Opting-Out' --- pairwise comparison ,adaptive dynamics ,finite populations ,social dilemmas ,evolutionary dynamics Introduction ============ Cooperation is everywhere. (See Axelrod (1984), Hölldobler and Wilson (2009), Traulsen and Nowak (2006), and Trivers (1971)). Bacteria cooperate. For example, bacteria cooperate in biofilm production, where bacteria go so far as to use quorum sensing to determine when there are enough cooperators that contributing to the biofilm is worthwhile (Nadell 2008). Ants cooperate, building vast anthills where members of a colony live together. Birds cooperate, sounding an alarm when predators are nearby. Moreover, humans cooperate. Indeed, whenever we contribute to a joint hunting effort, bring food to a potluck, or work together to combat climate change, we are cooperating. Why, though, do we see cooperation in all walks of life? How does cooperation evolve? Researchers have dedicated significant effort in the past twenty years towards studying the evolution of cooperation. (See Antal et al. (2009), Boyd et al. (2010), Hauert et al. (2002a), Hauert et al. (2002b), Nowak (2006b), and Priklopil et al. (2017), as examples).\ \ In particular, one common type of social interaction in which cooperation frequently arises and which has recently attracted attention by researchers is the public goods game (PGG). (See Hauert et al. (2002a), Hauert et al. (2002b), and Pacheco et al. (2015)). In a public goods game, cooperators contribute to a common pool which all participants of the game then share equally. In fact, in all of the instances of cooperation mentioned in the preceding paragraph, organisms contribute to a public good. In the case of bacteria, the public good is biofilm production. For ants, the good is the anthill. For birds, the good is the knowledge that a predator is nearby and hence that they should be careful. Lastly, for the party-goers, the good is the food at the potluck.\ \ However, whenever cooperators contribute to a common pool, there are free-riders, who benefit from the common pool without contributing. Game theorists frequently refer to such free-riders as *defectors*. These defectors cause the participants of the game to receive a smaller share of the common pool–a smaller payoff–than the social optimum where every player cooperates. In fact, regardless of what each other player does, a defector always earns a larger payoff because the defector does not have to contribute to the common pool, making defection the *dominant strategy*. Game theorists refer to a situation in which the dominant strategy is not socially optimal as a *social dilemma*. Consequentially, if each player were rational but unaware of the strategies of the other players, each player would choose to defect, and each player would receive no payoff.\ \ In reality, even though in any particular PGG defectors will outperform cooperators, averaging over all games, it may be the case that cooperators actually outperform defectors. Such a situation is an example of Simpson’s paradox (Hauert et al., 2002a). Additionally, there are many ways in which a tweak to the PGG may promote cooperation [@Battiston_NJP17; @Szolnoki_RSI15; @Szolnoki_PRSB15]. For instance, kin selection (Antal et al., 2009)(Nowak, 2006b), punishment of defectors (Boyd et al., 2010), signaling (Pacheco et al. 2015), and optional participation (Hauert et al. (2002a) and Hauert et al. (2002b)), and combinations of the preceding methods (Sigmund et al. 2010)(Hauert et al., 2008) have been used to promote cooperation. However, in the literature, a small but realistic tweak to the public goods game has yet to be addressed. Specifically, even if there is no punishment of defectors or if players cannot opt-out, due to unforeseen circumstances, at times players simply cannot participate in the PGG. For instance, an individual traveling to a hunting party may come across a flooded road and be forced to turn back. Or, an individual going to an international conference on climate change may suddenly become too ill to travel. It is even possible that on a whim, an individual may decide to engage in some activity other than the game. As a result, players participate in the public goods game stochastically, unable to participate independently of whether or not the player plans to cooperate or defect.\ \ We investigate such public goods games with stochastic non-participation. Ultimately, we add a fully analyzed stochastic model to the literature, improving the understanding of the evolution of cooperation. Moreover, our model demonstrates that a tweak even more slight than others in the current literature, can facilitate cooperation. We conclude with an analysis of adaptive dynamics for simplified 2 person PGGs in finite populations. In such an analysis, we demonstrate that increasing the probability of non-participation temporarily slows the rate at which the population tends to defection when participating given rare mutations only minutely different from the original population. The Model ========= We consider a well-mixed finite population of $n$ individuals, and suppose that frequently $N\leq n$ randomly selected individuals receive the opportunity to participate in a public goods game (PGG). In the PGG, individuals can choose to cooperate, investing 1 unit into a common pool, as in Hauert et al.(2002a) and Hauert et al. (2002b). Some force then multiplies the 1 unit each cooperator invests by some factor $N>r>1$ and thus for each unit invested by a cooperator, the common pool increases by $r$ units. At the end of the game, each PGG participant obtains an equal share of the common pool. However, the individuals who do not choose to cooperate choose to defect, receiving a share from the common pool without contributing. To simplify the model, we assume that individuals determine their strategies before the PGG has begun, ignoring group composition, as in Hauert et al. (2002a) and Hauert et al. (2002b).\ \ As stated, the preceding model leads to domination by defectors for all games where the multiplier $r$ is smaller than the game size $N$ and the game is thus a social dilemma. To promote cooperation, we assume that due to unforeseen circumstances each player has a fixed probability $\alpha$ of being unable to participate in the PGG, instead obtaining a fixed benefit or loss $\sigma$.\ \ Furthermore, our model needs a method by which the population may change its composition of players cooperating or defecting. We take pairwise comparison as such a method, where occasionally two individuals are randomly selected. One individual will update his or her strategy by comparing his or her success to the other individual. We let the probability $p$ that the updating individual adopts the strategy of the other individual be proportional to the expected payoff difference between individuals of the two strategies. Specifically, we let the probability of changing strategies be given by the Fermi function, as in Traulsen (2007) and Pacheco (2015): $$p=(1+\exp[-\gamma(\pi_{com}-\pi_{up})])^{-1}, %%Checked!$$ where $\pi_{com}$ represents the expected payoff of individuals playing the strategy of the individual selected for comparison, $\pi_{up}$ represents the expected payoff of individuals playing the strategy of the individual selected for updating, and $\gamma\geq 0$ represents a selection pressure, and corresponds to an inverse temperature (Traulsen et al., 2007). \[Model Schematic of Stochastic Opting-out\] ![Model Schematic of Stochastic Opting-out. Cooperators (blue) and defectors (red) are represented by dots. A fixed number of players are randomly drawn from the population to participate in a PGG, represented by the small tan rectangular area. While most players are able to make it to the game, some are not. Players then return to the general populace, where no game is occurring.](GameModel.pdf "fig:"){width="\textwidth"} \ \ When it comes to Adaptive Dynamics in finite populations, for simplicity, we assume game size is 2. Furthermore, applying adaptive dynamics to the problem as done in Imhof and Nowak (2010), we assume that a single mutant who plays a strategy similar to that to the original population invades the original population. Specifically, we suppose that every player plays a strategy in the strategy space $(\beta,\alpha)$, where $\beta$ is the probability that the player cooperates if he or she plays, and $\alpha$ is the probability that due to unforeseen circumstances the player cannot play. We let the original population be composed solely of players with strategy $(\beta,\alpha)$, and we suppose that the population is invaded by a single player with strategy $(\beta_1, \alpha_1)$. Then, we let $\beta_1 \rightarrow \beta$ and $\alpha_1 \rightarrow \alpha$. As in Imhof and Nowak (2010), we also assume rare mutation. That is, we assume sufficient time passes between mutations that either fixation, or extinction, of the mutant type occurs. Results ======= Pairwise Invasion Based on Fixation Probability ----------------------------------------------- To proceed with the analysis of the stochastic model, we must calculate the expected payoffs for cooperators and defectors, $\pi_c$ and $\pi_d$, respectively. To calculate $\pi_d$, we use the method presented by Hauert et al. (2002b). First, we observe that in a game with $S$ players, defectors receive a benefit $rn_c/S$, where $n_c$ is the number of cooperators in the game, if $S>1$. However, if $S=1$, that player must be a loner, and will receive payoff $\sigma$. Then, noting that any player does not play with probability $\alpha$ and plays with probability $1-\alpha$, and letting $x_c$ be the proportion of cooperators in the population, $$\pi_d=\alpha\sigma+(1-\alpha)[rx_c[1-(1-\alpha^N)/(1-\alpha)]+\alpha^{N-1}\sigma].$$ We defer the details to Appendix A. Employing a similar method, $$\pi_c=\pi_d+r/(n-1)[\alpha(1-\alpha^{N-1})]+(1-\alpha)[-1+(1-r)\alpha^{N-1}+(r/N)(1-\alpha^N)/(1-\alpha)].$$ We defer the details to Appendix B. Hence, $$\pi_c-\pi_d=r/(n-1)[\alpha(1-\alpha^{N-1})]+(1-\alpha)[-1+(1-r)\alpha^{N-1}+(r/N)(1-\alpha^N)/(1-\alpha)],$$ a constant. Then, inputting $\pi_c-\pi_d$ into (1), the probability that a cooperator becomes a defector given that a cooperator is selected for updating and a defector is selected for comparison is $$p_{cd}=(1+\exp[\gamma(r/(n-1)[\alpha(1-\alpha^{N-1})]+(1-\alpha)[-1+(1-r)\alpha^{N-1}+(r/N)(1-\alpha^N)/(1-\alpha)])])^{-1},$$ which is constant regardless of the number of cooperators. Thus the probability that the number of cooperators decreases by one in one iteration of the pairwise comparison model is $$p_{cd}i(N-i)/[N(N-1)].$$ Likewise, the probability that a defector becomes a cooperator given that the defector is selected for updating and the cooperator is selected for comparison is $$p_{dc}=(1+\exp[-\gamma(r/(n-1)[\alpha(1-\alpha^{N-1})]+(1-\alpha)[-1+(1-r)\alpha^{N-1}+(r/N)(1-\alpha^N)/(1-\alpha)])])^{-1},$$ also a constant. Hence, the probability that the number of cooperators increases by one in one iteration of the pairwise comparison model is $$p_{dc}i(N-i)/[N(N-1)].$$ Of course, though, if the number of cooperators, $i$, is 0 or n, the probabilities that a cooperator will change to a defector and that a defector will change to a cooperator are both zero, and the number of cooperators remains at 0 or $n$. That is, $i=0$ and $i=n$ are absorption states in the model.\ \ Moreover, now knowing $p_{cd}$ and $p_{dc}$, and noting that $p_{cd}+p_{dc}=1$, we can calculate the transition matrix $P$ for the Markov chain in which pairwise selection is iterated repeatedly. However, as the transition matrix itself is not vital for our analysis, we defer discussion of the transition matrix to Appendix C. On the other hand, the fixation probability of cooperation, that is, the probability that given i cooperators in a population of defectors that every individual will become a cooperator, *is* vital. Following the procedure outlined by Nowak (2006a), we demonstrate that the fixation probability of cooperation given $i\geq 1$ cooperators, $x_i$, is $$x_i=(1+\Sigma_{j=1}^{i-1} \Pi_{k=1}^jp_{cd}/p_{dc})/(1+\Sigma_{j=1}^{n-1} \Pi_{k=1}^jp_{cd}/p_{dc}),$$ where $i=1$ implies the numerator is 1, where we denote $p_{cd}/p_{dc}$ by\ $G(\alpha,\gamma,N,n,r)$, and $$G(\alpha,\gamma,N,n,r)=(1 +\exp(-\gamma(\pi_c-\pi_d)))/(1 + \exp(\gamma(\pi_c-\pi_d))).$$ Notably, G is constant over $i$. Hence, we may expand the numerator and denominator of $x_i$ as geometric series. So, if $G\neq 1$, $$x_i=(1-G^i)/(1-G^n).$$ However, $G=1$ implies that $p_{cd}=p_{dc}=1/2$, which implies neutral drift. We assume for now that $G\neq 1$. Then, observing that $p_{dc}/p_{cd}=G^{-1}$, the fixation probability of defection given i defectors is simply $x_i$ with $G$ replaced by $G^{-1}$: $$y_i=[G^{n-i}-G^n]/[1-G^n].$$ Hence, the fixation probability given $i$ cooperators is $$y_{n-i}=[G^i-G^n]/[1-G^n].$$ Thus, the probability of fixation of cooperators or defectors given i defectors is $$x_i+y_{n-i}=1.$$ Consequentially, the system always reaches an absorption state.\ \ Furthermore, now knowing the probabilities of fixation of cooperation given $i$ cooperators, $x_i$, and of defection given $i$ defectors, $y_i$, we can calculate the strategy favored by natural selection. Moreover, as in Nowak (2006a), natural selection favors cooperation over defection if and only if $x_1>y_1$. Likewise, natural selection favors defection over cooperation if and only if $y_1>x_1$ (Nowak, 2006a). Additionally, natural selection favors cooperation over neutral drift if and only if $x_1>1/n$ = the probability of fixation given natural drift (Nowak, 2006a). Likewise, natural selection favors defection over neutral drift if and only if $y_1>1/n$ (Nowak, 2006a). In fact, $$x_1>1/n \Leftrightarrow G<1.$$ We defer the proof to appendix D. Also, $G=1$, implies neutral drift, since $G=1$ implies $p_{cd}=p_{dc}=1/2$. Since $G\neq1$ implies either $p_{cd}>p_{dc}$ or vice-versa, there is neutral drift if and only if $G=1$. Thus, $x_1<1/n$ if and only if $G>1$. Hence, natural selection favors cooperation over neutral drift if and only if $G<1$, favors neither cooperation nor neutral drift one over the other if and only if $G=1$, and favors neutral drift over cooperation if and only if $G>1$. On the other hand, $$G<1 \Rightarrow y_1<1/n,$$ and $$G>1 \Rightarrow y_1>1/n.$$ We defer proofs of the two preceding assertions to Appendix D. Additionally if $G=1$, then there is neutral drift, as we demonstrated above, so $y_1=1/n$. Thus, if $G>1$, $y_1>1/n>x_1$; if $G=1$, then $y_1=1/n=x_1$; and otherwise, i.e $0<G<1$, $y_1<1/n<x_1$. Additionally, noting that if the model is not experiencing neutral drift, then recalling that $(1 +\exp(-\gamma(\pi_c-\pi_d)))/(1 + \exp(\gamma(\pi_c-\pi_d)))$, clearly $$G>1 \Leftrightarrow \pi_c-\pi_d<0.$$ Likewise, $$G<1 \Leftrightarrow \pi_c-\pi_d>0.$$ Also, $G=1$ if and only if $\pi_c-\pi_d=0$. Thus, there are three possibilities:\ \ 1) Natural selection favors cooperation over neutral drift, and neutral drift over defection ($\pi_c-\pi_d>0$), or\ \ 2) Natural selection favors neither cooperation nor neutral nor defection one over the other, ($\pi_c-\pi_d=0$), or\ \ 3) Natural selection favors defection over neutral drift, and neutral drift over cooperation ($\pi_c-\pi_d<0$).\ \ Thus, the sign of $\pi_c-\pi_d$, a function of the probability that a given player opts out $\alpha$, the game size $N$, the population size $n$, and the return on investments by cooperators, $r$, exclusively determines which strategies, cooperation or defection, natural selection favors one over the another and whether or not natural selection favors each strategy over neutral drift. \ \ This has profound implications. Primarily, there exists a minimum value of $r$, $R$, for given $N$ and $\alpha$ such that $r>R$ implies that $\pi_c-\pi_d>0$, $r<R$ implies that $\pi_c-\pi_d<0$, and $r=R$ implies that $\pi_c-\pi_d=0$. Indeed, recalling that $\pi_c-\pi_d=r/(n-1)[\alpha(1-\alpha^{N-1})]+(1-\alpha)[-1+(1-r)\alpha^{N-1}+(r/N)(1-\alpha^N)/(1-\alpha)]$, and noting that by the lemma in Appendix D, $[1-\alpha^N]/[N(1-\alpha)]-\alpha^{N-1} >0$, and thus that $[\alpha(1-\alpha^{N-1})]/[(n-1)(1-\alpha)]+[1-\alpha^N]/[N(1-\alpha)]-\alpha^{N-1}>0$, it follows that $$\begin{aligned} \pi_c-\pi_d>&0 \Leftrightarrow\\ r>&\dfrac{1-\alpha^{N-1}}{{[\alpha(1-\alpha^{N-1})]/[(n-1)(1-\alpha)]+[1-\alpha^N]/[N(1-\alpha)]-\alpha^{N-1}}}=R(\alpha), %%Checked\end{aligned}$$ where $R(\alpha)$ is defined for \[0,1). Simplifying, $$R(\alpha)=N\dfrac{1-\alpha-\alpha^{N-1}+\alpha^N}{1+\alpha N/(n-1)-\alpha^{N-1}N+\alpha^N(N-1-N/(n-1))},$$ demonstrating that on \[0,1), $R$ is also the quotient of two polynomials of degree $N$, and hence is continuous. By analogous proofs, $\pi_c-\pi_d=0$ if and only if $r=R(\alpha)$ and $\pi_c-\pi_d<0$ if and only if $r<R(\alpha)$. Additionally, on \[0,1), $$R(\alpha)<N,$$ (we defer the proof to Appendix D), so it is always possible to choose $r$ such that natural selection favors cooperation. Moreover, as proven in Appendix E, $R(\alpha)$ is strictly decreasing on \[0,1). Thus, given investment $r=R$, there is a threshold $\alpha_0$ such that $\alpha>\alpha_0$ implies natural selection favors cooperation. This threshold is analogous to the threshold on the proportion of individuals who choose to opt-opt suggested by Hauert et al. (2002b), which deals with an infinite rather than finite population and with planned rather than unplanned non-participation. Moreover, this threshold is the value of $\alpha$ satisfying $r=R(\alpha)$. Thus, as $\alpha$ increases, the requirements on $r$ such that natural selection favors cooperation become less and less stringent. In other words, increasing the probability for players to be unable to participate facilitates cooperation. Adaptive Dynamics in Finite Populations --------------------------------------- Considering that increasing the probability of non-participation facilitates cooperation, it may be surprising that the adaptive dynamics for the two player game discussed in *The Model* indicates that natural selection will push individuals to always defect when participating or to never participate. To see why, we consider a population consisting of two types of players, type one and type two, defined by their strategies $(\beta_1, \alpha_1)$ and $(\beta_2,\alpha_2)$, respectively. Otherwise maintaining the notation used in section 3.1, the expected payoff for players of type 1 is $$\begin{aligned} \begin{split} \pi_1=&\dfrac{n - i}{n - 1}(1 - \alpha_1)(1 - \alpha_2)(r\beta_2/2 + r\beta_1/2- \beta_1) + \dfrac{i - 1}{n - 1}(1 - \alpha_1)^2\beta_1(r - 1) +\\& \sigma\alpha_1 + \sigma(1 - \alpha_1)(\dfrac{n - i}{n - 1}\alpha_2 + \dfrac{i - 1}{n - 1}\alpha_1). \end{split}\end{aligned}$$ We defer the derivation of the expected payoff for players of type 1 to Appendix F. Moreover, since the game is symmetric, the expected payoff for players of type 2 may be determined simply by replacing the number of players of type 1, i, with the number of players of type 2, n-i, and by switching subscripts. Specifically, the expected payoff for players of type 2 is $$\begin{split} \pi_2=&\dfrac{i}{n - 1}(1 - \alpha_2)(1 - \alpha_1)(r\beta_1/2 + r\beta_2/2- \beta_2) + \dfrac{n-i- 1}{n - 1}(1 - \alpha_2)^2\beta_2(r - 1)+\\& \sigma\alpha_2 + \sigma(1 - \alpha_2)(\dfrac{i}{n - 1}\alpha_1 + \dfrac{n-i - 1}{n - 1}\alpha_2). \end{split}$$ Continuing to use the pairwise comparison model, the probability that a player of type 1 will adopt the strategy of a player of type 2 given that the player of type 1 updates and the player of type 2 compares is $$p_{1\rightarrow 2}=(1+\exp[-\gamma(\pi_2-\pi_1)])^{-1},$$ where $\gamma$ is a selection pressure, just as in section 3.1. Similarly, the analogous probability for players of type 2 is $$p_{2\rightarrow 1}=(1+\exp[-\gamma(\pi_1-\pi_2)])^{-1}.$$ Then, again following the method proposed by Nowak (2006a), the fixation probability of a player of type 1 given i players of type 1 in a population of players of type 2 is $$x_i=(1+\Sigma_{j=1}^{i-1} \Pi_{k=1}^jp_{1\rightarrow 2}/p_{2 \rightarrow 1})/(1+\Sigma_{j=1}^{n-1} \Pi_{k=1}^jp_{1\rightarrow 2}/p_{2\rightarrow 1}).$$ For the remainder of this section, we will assume players of type 2 compose the invaded population, and hence we will drop the subscripts on $\alpha_2$ and $\beta_2$.\ \ To investigate the adaptive dynamics, consider $\vec{f}(\beta,\alpha)=\lim_{(\alpha_1,\beta_1)\rightarrow(\alpha,\beta)}(\partial x_1/\partial \alpha_1,\partial x_1/\partial \beta_1)$. The direction given by $\vec{f}$ for $(\alpha,\beta)$, plotting f as a vector field, is the direction in the strategy space which maximizes the fixation probability of the invading mutant population given one invading mutant, $x_1$. Following the directions which maximize $x_1$ in the strategy space starting at an initial $(\alpha,\beta)$, that is, following the streamlines of $\vec{f}$, indicates the most likely path in the strategy space that a population will take as mutants with similar strategies eventually fixate in the population, as suggested by Imhof and Nowak (2010). Moreover, applying the StreamPlot function of Mathematica to the model for various combinations of $r$ and $\sigma$ in a population of size $n$ indicates that the probability an individual cooperates will decrease and that increasing $r$ or decreasing $\sigma$ will facilitate participation. Notably, Mathematica demonstrates that for $\gamma=1$ $$\lim_{(\alpha_1,\beta_1)\rightarrow(\alpha,\beta)}(\partial x_1/\partial \alpha_1)=(\alpha-1)(n-2)((r-1)\beta-\sigma)/(2n),$$ and $$\lim_{(\alpha_1,\beta_1)\rightarrow(\alpha,\beta)}(\partial x_1/\partial \beta_1)=(1-\alpha)^2(2-2n-2r+nr)/(4n).$$ Unfortunately, the problem proves too complicated to calculate a closed-form solution of $\vec{f}$ for every $\gamma$. Nevertheless, we conjecture that for any $\gamma$, $$\lim_{(\alpha_1,\beta_1)\rightarrow(\alpha,\beta)}(\partial x_1/\partial \alpha_1)=(\alpha-1)\gamma(n-2)((r-1)\beta-\sigma)/(2n),$$ and $$\lim_{(\alpha_1,\beta_1)\rightarrow(\alpha,\beta)}(\partial x_1/\partial \beta_1)=(1-\alpha)^2\gamma(2-2n-2r+nr)/(4n).$$ (30) and (31) imply (28) and (29), respectively, and hold for a variety of other test values of $n$, $r$, $\sigma$, and $\gamma$. In particular, (30) and (31) both hold if $\beta=0$. ![Coevolution of Cooperation and Stochastic Opting-out. Shown are the adaptive dynamics using the stream plot function of Mathematica in a finite population of size $n = 5$ for selection strength $\gamma=1$ and various values of return on investment, $r$, and payoff for non-participants $\sigma$. Following the arrows leads to the most likely path the population will take in the strategy space. Observe that as $r$ increases or $\sigma$ decreases the population tends to participation at a faster rate or the population tends to non-participation at a slower rate, where the rate is indicated by the slope of the arrows. Also, the population always tends to defection when participating, and otherwise the population appears to tend to non-participation. However, increasing $\alpha$ decreases the rate at which individuals in the population tend to complete defection.](AdaptiveDynamicsDiagram.pdf "fig:"){width="\textwidth"}\[\] \ \ Furthermore, if (31) is indeed true, then $\lim_{(\alpha_1,\beta_1)\rightarrow(\alpha,\beta)}(\partial x_1/\partial \beta_1)$ would be independent of $\beta$ and $\sigma$. Moreover, if $n>2$, $\alpha_2<1$, $\gamma>0$, and (31) is valid, simple algebraic manipulation yields $$\lim_{(\alpha_1,\beta_1)\rightarrow(\alpha,\beta)}(\partial x_1/\partial \beta_1)>0 \Leftrightarrow r>1+n/(n-2)>2.$$ Likewise, $\lim_{(\alpha_1,\beta_1)\rightarrow(\alpha,\beta)}(\partial x_1/\partial \beta_1)=0$ if and only if $r=1+n/(n-2)$, and $\lim_{(\alpha_1,\beta_1)\rightarrow(\alpha,\beta)}(\partial x_1/\partial \beta_1)<0$ if and only if $r<1+n/(n-2)$. On the other hand, if $\gamma=0$, or $\alpha_2=1$, then $\lim_{(\alpha_1,\beta_1)\rightarrow(\alpha,\beta)}(\partial x_1/\partial \beta_1)=0$, and if $n=2$, $\lim_{(\alpha_1,\beta_1)\rightarrow(\alpha,\beta)}(\partial x_1/\partial \beta_1)=(-1/4)(1-\alpha)^2\gamma$. Hence, for all games which are social dilemmas, i.e $r<2$, and even for some games which are not social dilemmas, the presence of rare and minute mutations leads each individual towards defection when participating as long as $\alpha<1$ and $\gamma>0$.\ \ Moreover, if (31) holds, then cooperation is never stable if initially $\alpha \nrightarrow 1$, and each individual in the population tends towards always defecting, i.e, $\beta\rightarrow 0$. Thus, the sign of the right-hand-side of (30) becomes the sign of $\sigma$ when $n>2$ and 0 otherwise, (although if $n=2$ the right-hand side of (30) is *always* 0). Hence, if $\sigma<0$, each individual in the population tends towards always defecting and always participating if (30) is valid. If $\sigma=0$ or $n=0$, (30) and (31) demonstrate that defection with some degree of participation is stable, but cooperation is not. Instead, if $\sigma>0$, (30) and (31) show that $\alpha \rightarrow 1$ anyways. On the other hand, if initially $\alpha \rightarrow 1$, $\alpha$ remains near $1$ by (30) and (31). However, if $\alpha \rightarrow 1$, nobody participates (everyone gets the same payoff $\sigma$) and thus neutral drift allows the establishment of cooperation along the edge $\alpha =1$. Also, if (31) is valid, increasing $\alpha$ increases $\lim_{(\alpha_1,\beta_1)\rightarrow(\alpha,\beta)}(\partial x_1/\partial \beta_1)$, i.e increasing $\alpha$ decreases the rate at which individuals in the population tend towards complete defection. Moreover, from Eq. (31), we can obtain a possible rest point $\beta^* = \sigma/(r-1)$ on the edge of $\alpha =1$, as long as the value of $\beta^*\in (0,1)$. This can be confirmed in Fig. 3(b2) and Fig. 3(c2). Conclusion ========== Notably, for games where every player participates, i.e $\alpha=0 $, the threshold return on investment, $R(0)$, above which natural selection favors cooperation is the game size, $N$. Hence, if the game size is reduced by a factor $(1-\alpha)$, where $0<\alpha<1$, the threshold value on investment is $N(1-\alpha)$. Moreover, if instead $\alpha$ is the probability that any given player does not participate, the law of large numbers suggests that for very large game sizes, the number of people participating in the game will be $N(1-\alpha)$. If the game size is very large but is small with respect to the population size, this is exactly the threshold value $R(\alpha)$ for return on investment above which natural selection favors cooperation. Hence, for very large games which are small with respect to the population size, reducing $\alpha$ appears to be the sole factor which facilitates cooperation. However, if population size and game size are both very large but population size is no longer arbitrarily large with respect to game size, there is a second factor at work. In the equation for the threshold, $R(\alpha)$, this second factor arises from the term $((n-1)/N)\alpha(1-\alpha^{N-1})/[1-\alpha]$. This factor, always positive, reduces the threshold R at every positive value of $\alpha$, thereby facilitating cooperation. Furthermore, if the game size is small but population size is still very large population, $R \rightarrow 2$ as $\alpha\rightarrow 1$. Also considering that $R$ is strictly decreasing implies that the threshold curve $R(\alpha)$ also satisfies $R(\alpha)>N(1-\alpha)$ for large $\alpha$, so there appears to be some other factor which resists cooperation in small groups. Specifically, this factor occurs at least in part because for $0<<\alpha<1$, games become rare and the vast majority of games become two player games, where natural selection favors cooperation if and only if the return on investment by cooperators is larger than 2.\ \ Despite increasing $\alpha$ always reducing the threshold value for cooperation, the adaptive dynamics suggest that in the presence of minute and rare mutations individuals in the population always tend towards always defecting or never participating. Nevertheless, the adaptive dynamics also indicate that the rate at which the population tends to defection is slower for larger values of $\alpha$. So, while assuming that players have a fixed probability of non-participation in the adaptive dynamics does not make cooperation stable in the presence of rare and minute mutations, by decreasing the rate at which the population tends to defection, it essentially increases the time which the population spends at higher levels of cooperation. Also, cooperation emerges on the edge $\alpha=1$. Ultimately, either in the presence of rare and minute mutations or not, assuming players are unable to participate with a fixed probability facilitates cooperation. Acknowledgments {#acknowledgments .unnumbered} =============== The authors would like to thank the National Science Foundation and Dartmouth College for funding the REU program at which the research was conducted. In particular, Alexander G Ginsberg would also like to thank professor Feng Fu, professor Anne Gelb, Tracy Moloney, and Amy Powell, all of Dartmouth College, for personally overseeing the program. Lastly, he would like to thank professors Ignacio Uriarte-Tuero, George Pappas, Tsvetanka Tsendova, and Teena Gerhardt, all of Michigan State University, for making sure he attended a program that fit his needs. Feng Fu acknowledges generous support from the Dartmouth Startup Fund, the Walter and Constance Burke Research Initiation Award, the NIH (grant no. C16A12652-A10712), and DARPA (grant no. D17PC00002-002) Derivation of $\pi_d$ ===================== We define the probability that an event E occurs be denoted by P(E), and let the probability that E occurs given a second event F occurs be denoted by $P(E|F)$. Then, $$\begin{aligned} \pi_d=&\alpha\sigma+ \Sigma P(n_c \cap S \cap plays)*payoff\\ %%Checked! =&\alpha\sigma+ \Sigma P(plays)P(S|plays)P(n_c |S \cap plays)*payoff\\%%Checked! =&\alpha\sigma+(1-\alpha)\Sigma P(S|plays)P(n_c |S \cap plays)*payoff\\%%Checked! =&\alpha\sigma+(1-\alpha)[\Sigma_{S=2}^{N}P(S|plays)\Sigma_{n_c=0}^{S-1}P(n_c|S\cap plays)rn_c/S+P(S=1|plays)\sigma]\\ %%Checked! =&\alpha\sigma+(1-\alpha)[\Sigma_{S=2}^{N}P(S|plays)/S \Sigma_{n_c=0}^{S-1}P(n_c|S\cap plays)rn_c+P(S=1|plays)\sigma]\\ %%Checked! \begin{split} =&\alpha\sigma+(1-\alpha)[r\Sigma_{S=2}^{N}(1/S){N-1 \choose S-1}(1-\alpha)^{S-1}\alpha^{N-S}\Sigma_{n_c=0}^{S-1}{S-1\choose n_c}x_c^{n_c}*\\&(1-x_c)^{S-1-n_c}n_c+\alpha^{N-1}\sigma]\\ %%x_c=#cooperators/(n-1)!!!!! =&\alpha\sigma+(1-\alpha)[r\Sigma_{S=2}^{N}{N-1 \choose S-1}(1-\alpha)^{S-1}\alpha^{N-S}x_c(S-1)/S\Sigma_{n_c=1}^{S-1}{S-2\choose n_c-1}x_c^{n_c-1}*\\&(1-x_c)^{S-1-n_c}+\alpha^{N-1}\sigma]\\ =&\alpha\sigma+(1-\alpha)[r\Sigma_{S=2}^{N}{N-1 \choose S-1}(1-\alpha)^{S-1}\alpha^{N-S}x_c(S-1)/S\Sigma_{k=0}^{S-2}{S-2\choose k}x_c^{k}*\\&(1-x_c)^{S-2-k}+ \alpha^{N-1}\sigma]\\ =&\alpha\sigma+(1-\alpha)[r\Sigma_{S=2}^{N}{N-1 \choose S-1}(1-\alpha)^{S-1}\alpha^{N-S}x_c(S-1)/S+\alpha^{N-1}\sigma]\\ =&\alpha\sigma+(1-\alpha)[rx_c(\Sigma_{S=2}^{N}{N-1 \choose S-1}(1-\alpha)^{S-1}\alpha^{N-S}- \Sigma_{S=2}^{N}{N-1 \choose S-1}(1-\alpha)^{S-1}*\\&\alpha^{N-S}/S)+\alpha^{N-1}\sigma]\\ =&\alpha\sigma+(1-\alpha)[rx_c(\Sigma_{k=1}^{N-1}{N-1 \choose k}(1-\alpha)^k\alpha^{N-k-1}-(1/N)\Sigma_{S=2}^{N}{N \choose S}(1-\alpha)^{S-1}*\\&\alpha^{N-S})+\alpha^{N-1}\sigma]\\ =&\alpha\sigma+(1-\alpha)[rx_c((1-\alpha^{N-1})-1/(N(1-\alpha))\Sigma_{S=2}^{N}{N \choose S}(1-\alpha)^{S}\alpha^{N-S})+\alpha^{N-1}\sigma]\\ =&\alpha\sigma+(1-\alpha)[rx_c((1-\alpha^{N-1})-[1-N(1-\alpha)\alpha^{N-1}-\alpha^N]/[N(1-\alpha)]+\alpha^{N-1}\sigma]\\ =&\alpha\sigma+(1-\alpha)[rx_c(N-N\alpha-[1-\alpha^N])/(N[1-\alpha])+\alpha^{N-1}\sigma]\\ =&\alpha\sigma+(1-\alpha)[rx_c[1-(1-\alpha^N)/(1-\alpha)]+\alpha^{N-1}\sigma]. \end{split}\end{aligned}$$ We have verified via Mathematica and via Hauert et al. (2002b) that $$\begin{split} &\Sigma_{S=2}^{N}P(S|plays)\Sigma_{n_c=0}^{N-1}P(n_c|S\cap plays)rn_c/S+P(S=1|plays)\sigma=\\& rx_c[1-(1-\alpha^N)/(1-\alpha)]+\alpha^{N-1}\sigma. \end{split}$$ Derivation of $\pi_c$ ===================== $$\begin{aligned} \begin{split} \pi_c=&\alpha\sigma+ \Sigma P(n_c \cap S \cap plays)*payoff\\ %%Checked! =&\alpha\sigma+(1-\alpha)\Sigma P(S|plays)P(n_c |S \cap plays)*payoff\\ %%Checked! =&\alpha\sigma+(1-\alpha)[\Sigma_{S=2}^{N}P(S|plays)\Sigma_{n_c=0}^{S-1}P(n_c|S\cap plays)((r/S)(n_c+1)-1)+\\&P(S=1|plays)\sigma]\\ %% Checked! =&\alpha\sigma+(1-\alpha)[\Sigma_{S=2}^{N}P(S|plays)\Sigma_{n_c=0}^{S-1}P(n_c|S\cap plays)rn_c/S+\\&\alpha^{N-1}\sigma+\Sigma_{S=2}^{N}P(S|plays)\Sigma_{n_c=0}^{S-1}P(n_c|S\cap plays)(r/S-1)]. %%Checked! \end{split}\end{aligned}$$ Noting that $$\alpha\sigma+(1-\alpha)[\Sigma_{S=2}^{N}P(S|plays)\Sigma_{n_c=0}^{S-1}P(n_c|S\cap plays)rn_c/S+\alpha^{N-1}\sigma]=\pi_d, %%Checked$$ where $x_c$ is replaced by $x_c-1/(n-1)$, it follows that $$\begin{aligned} \begin{split} \pi_c=&\pi_d+r/(n-1)[\alpha(1-\alpha^{N-1})]+(1-\alpha)\Sigma_{S=2}^{N}P(S|plays)\Sigma_{n_c=0}^{S-1}P(n_c|S\cap plays)*\\&(r/S-1)\\%%Checked! =&\pi_d+r/(n-1)[\alpha(1-\alpha^{N-1})]+(1-\alpha)\Sigma_{S=2}^{N}P(S|plays)(r/S-1)\\ =&\pi_d+r/(n-1)[\alpha(1-\alpha^{N-1})]+(1-\alpha)[r\Sigma_{S=2}^{N}(1/S){N-1 \choose S-1}(1-\alpha)^{S-1}\alpha^{N-S}-\\& \Sigma_{S=2}^{N}{N-1 \choose S-1}(1-\alpha)^{S-1}\alpha^{N-S}]\\ =&\pi_d+r/(n-1)[\alpha(1-\alpha^{N-1})]+(1-\alpha)[(r/N)\Sigma_{S=2}^{N}{N \choose S}(1-\alpha)^{S-1}\alpha^{N-S}-\\& \Sigma_{k=1}^{N-1}{N-1 \choose k}(1-\alpha)^k\alpha^{N-k-1}]\\ =&\pi_d+r/(n-1)[\alpha(1-\alpha^{N-1})]+(1-\alpha)[(r/[N(1-\alpha)])\Sigma_{S=2}^{N}{N \choose S}(1-\alpha)^S\alpha^{N-S}-\\&(1-\alpha^{N-1})] \end{split}\end{aligned}$$ $$\begin{aligned} \begin{split} =&\pi_d+r/(n-1)[\alpha(1-\alpha^{N-1})]+(1-\alpha)[r/[N(1-\alpha)](1-N(1-\alpha)\alpha^{N-1}-\alpha^N)-\\&1+\alpha^{N-1}] \end{split}\end{aligned}$$ $$\begin{aligned} \begin{split} =&\pi_d+r/(n-1)[\alpha(1-\alpha^{N-1})]+(1-\alpha)[-1-r\alpha^{N-1}+\alpha^{N-1}+(r/N)(1-\alpha^N)/\\&(1-\alpha)] \end{split}\end{aligned}$$ $$\begin{aligned} \begin{split} =&\pi_d+r/(n-1)[\alpha(1-\alpha^{N-1})]+(1-\alpha)[-1+(1-r)\alpha^{N-1}+(r/N)(1-\alpha^N)/\\&(1-\alpha)].%%Checked! \end{split}\end{aligned}$$ Again, we have verified via Mathematica and via Hauert et al. (2002b) that $$\begin{aligned} &\Sigma_{S=2}^{N}P(S|plays)\Sigma_{n_c=0}^{S-1}P(n_c|S\cap plays)(r/S-1)=\\&-1+(1-r)\alpha^{N-1}+(r/N)(1-\alpha^N)/(1-\alpha).%% Checked!\end{aligned}$$ Transition Matrix ================= We define $P$ be the transition matrix for the Markov chain formed by repeatedly iterating pairwise comparison. Then, $P_{i,i-1}=p_{cd}i(n-i)/[n(n-1)]$, and $P_{i,i+1}=p_{dc}i(n-i)/[n(n-1)]$, for i=2, 3, ..., n-1. Since the only other transition from i cooperators per iteration is the absence of transition, $P_{i,i}=1-P_{i,i-1}-P_{i,i+1}$, and the remaining entries in the $i^{th}$ row are 0. Also considering that $i=0$ cooperators and $i=n$ cooperators are absorbing states, it follows that P is the tridiagonal (n+1)x(n+1) matrix $$\begin{bmatrix} 1 & 0 & 0 & 0 & \hdots &0 \\ P_{2,1} & P_{2,2}& P_{2,3} & 0& \hdots & 0 \\ 0 & P_{3,2}&P_{3,3} &P_{3,4} & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots &0\\ 0 & \hdots& 0 & P_{n-1,n-2} & P_{n-1,n-1}& P_{n-1,n}\\ 0 & \hdots & 0 & 0 & 0 & 1 \end{bmatrix}.$$ Fortunately, the calculation $P^k$ as $k\rightarrow\infty$ is relatively straightforward. Indeed, the calculated the fixation probabilities $x_i$ in (11), and $y_{n-i}$ in (13), represent, respectively, the last and first entries in the $i^{th}$ row of $\lim_{n\rightarrow\infty}P^n$. Also considering that the entries in any given row of $P^n$ must sum to 1 as $P^n$ is a stochastic matrix, and that $x_i+y_{n-i}=1$, it follows that $$\lim_{n\rightarrow\infty}P^n= \begin{bmatrix} 1 & 0 & \hdots & 0 & 0\\ x_1 & 0 &\hdots & 0 & y_{n-1}\\ x_2 & 0 &\hdots & 0 & y_{n-2}\\ \vdots & \vdots & \vdots & \vdots & \vdots\\ x_{n-1} & 0 & \hdots & 0 & y_{1}\\ 0 & 0 & \hdots & 0 & 1 \end{bmatrix}.$$ Thus, $\lim_{n\rightarrow\infty}XP^n$ converges to a vector of the form $(a, 0, ..., 0,b)$. Namely, the set of vectors of the form (a, 0, ..., 0, b) is the set of eigenvectors of $\lim_{n\rightarrow\infty}P^n$, which in turn is the set of eigenvectors of P with eigenvalue 1. Moreover, if $X=(Prob(i=0), Prob(i=1), ..., Prob(i=n))$, then $\lim_{n\rightarrow\infty}XP^n$ converges to a vector of the form $(\alpha, 0, ..., 0, \beta)$, where $\alpha+\beta=1$. Since the set of vectors of the form $(\alpha, 0, ..., 0, \beta)$ with $\alpha+\beta=1$ is the set of stochastic eigenvectors of $P$ with eigenvalue 1, it follows that depending on the initial probability vector for the system, $X=(Prob(i=0), Prob(i=1), ..., Prob(i=n))$, the system can potentially converge to any stochastic eigenvector. Inequalities ============ Proof of (15) ------------- $$\begin{aligned} &x_1>1/n \Leftrightarrow \\ &[1-G]/[1-G^n]>1/n \Leftrightarrow \\ &[1-G^n]/[1-G]<n \Leftrightarrow \\ &\Sigma_{k=0}^{n-1}G^k<\Sigma_{k=0}^{n-1}1 \Leftrightarrow\\ &G<1. \square\end{aligned}$$ Proof of (16) ------------- $$\begin{aligned} &y_1<1/n \Leftrightarrow\\ &[G^{n-1}-G^n]/[{1-G^n}]<1/n \Leftrightarrow\\ &[1-G]/[1-G^n]<1/(nG^{n-1}) \Leftrightarrow\\ &[1-G^n]/[1-G]>nG^{n-1} \Leftrightarrow\\ &(1/n)\Sigma_{k=0}^{n-1}G^k>G^{n-1} \Leftrightarrow\\ &(1/n)\Sigma_{k=0}^{n-1}G^k>(G^{(n-1)(n)/2})^{2/n} \Leftrightarrow \\ &(1/n)\Sigma_{k=0}^{n-1}G^k>((\Pi_{k=0}^{n-1}G^k)^{1/n})^2. \end{aligned}$$ Moreover, if $G<1$, then $(\Pi_{k=0}^{n-1}G^k)^{\dfrac{1}{n}}>((\Pi_{k=0}^{n-1}G^k)^{\dfrac{1}{n}})^2$. Hence, if $G<1$, applying the arithmetic-mean-geometric-mean (AM-GM) inequality demonstrates that $$(1/n)\Sigma_{k=0}^{n-1}G^k>((\Pi_{k=0}^{n-1}G^k)^{1/n})^2.$$ Thus, $G<1$ implies that $y_1<1/n$. $\square$ Proof of (17) ------------- If $G>1$ and $n>1$, note that $$\begin{aligned} &y_1>1/n \Leftrightarrow\\ &[G^{n-1}-G^n]/[1-G^n]>1/n \Leftrightarrow\\ &\dfrac{G-1}{G-1/G^{n-1}}>1/n\Leftrightarrow\\ &G-G^{1-n}<nG-n.\end{aligned}$$ Then, observe that $$\dfrac{d^2}{d^2G} (G-G^{1-n})=-n(n-1)G^{-n-1}<0,$$ for $n>1$. Thus, $$\dfrac{d}{dG} (G-G^{1-n})=1-(1-n)G^{-n}$$ is decreasing whereas $$\dfrac{d}{dG}(nG-n)=n$$ is constant. Also considering that $$\dfrac{d}{dG} (G-G^{1-n})|_{G\rightarrow 1}=n=\dfrac{d}{dG}(nG-n)|_{G\rightarrow 1},$$ it follows that $$\dfrac{d}{dG} (G-G^{1-n})<\dfrac{d}{dG}(nG-n),$$ for $n>1$. Since it is also true that as $G\rightarrow 1$, $G-G^{n-1} \rightarrow 0$ and $nG-n \rightarrow 0$, $$G-G^{1-n}<nG-n.$$ Therefore, if $G>1$, $y_1>1/N$. $\square$ Lemma: $1-\alpha^{N-1}N+\alpha^N(N-1)>0$ ---------------------------------------- For $\alpha=0$, $1-\alpha^{N-1}N+\alpha^N(N-1)=1$. Then, if $\alpha\in (0,1)$ and $N>1$, $$\begin{aligned} 1-\alpha^{N-1}N+\alpha^N(N-1)>&0 \Leftrightarrow\\ [1-\alpha^N]/[N(1-\alpha)]-\alpha^{N-1}>&0 \Leftrightarrow\\ (1/N)(\Sigma_{k=0}^{N-1}\alpha^k)-\alpha^{N-1}>&0 \Leftrightarrow\\ (1/N)(\Sigma_{k=0}^{N-1}\alpha^k)>&(\alpha^{(N-1)N/2})^{\dfrac{2}{N}} \Leftrightarrow\\ (1/N)(\Sigma_{k=0}^{N-1}\alpha^k)>&((\Pi_{k=0}^{N-1}\alpha^k)^{1/N})^2.\end{aligned}$$ However, since $((\Pi_{k=0}^{N-1}\alpha^k)^{1/N})^2=\alpha^{N-1}<1$, $$((\Pi_{k=0}^{N-1}\alpha^k)^{1/N})^2<((\Pi_{k=0}^{N-1}\alpha^k)^{1/N}),$$ and since by the AM-GM inequality, $$(1/N)(\Sigma_{k=0}^{k=N-1}\alpha^k)>((\Pi_{k=0}^{N-1}\alpha^k)^{1/N})$$ D.28 must be valid. $\square$. Proof of (22) ------------- If $N>1$, $$\begin{aligned} &N>\dfrac{1-\alpha^{N-1}}{{[1-\alpha^N]/[N(1-\alpha)]-\alpha^{N-1}}} \Leftrightarrow \\ &[1-\alpha^N]/[1-\alpha]-N\alpha^{N-1}>1-\alpha^{N-1} \Leftrightarrow \\ &\Sigma_{k=0}^{N-1}\alpha^k-(N-1)\alpha^{N-1}>1 \Leftrightarrow \\ &(1/(N-1))\Sigma_{k=1}^{N-1}\alpha^k>\alpha^{N-1}.\end{aligned}$$ However, by the AM-GM inequality, $$(1/(N-1))\Sigma_{k=1}^{N-1}\alpha^k>\sqrt{\alpha^N}.$$ D.32 is true if and only if the denominator of D.31 is positive. This is true by the lemma. Also considering that $0<\alpha<1$, and so for $N>2$ $$\begin{aligned} 1>&\alpha^{N-2} \Rightarrow \\ \alpha^N>&\alpha^{2N-2} \Rightarrow \\ \sqrt{\alpha^N}>&\alpha^{N-1},\end{aligned}$$ it follows that (D.34) is always true. Since, $$[\alpha/(n-1)][1-\alpha^{N-1}]/[1-\alpha]=\alpha/(n-1)\Sigma_{k=0}^{N-2}\alpha^k>0$$ for $\alpha\in (0,1)$ and $N\geq 2$, $$\dfrac{1-\alpha^{N-1}}{{[1-\alpha^N]/[N(1-\alpha)]-\alpha^{N-1}}}> R(\alpha).$$ Therefore, $N>R$ for $N>2$. However, if $N=2$, $$\dfrac{1-\alpha^{N-1}}{{[1-\alpha^N]/[N(1-\alpha)]-\alpha^{N-1}}}=2.$$ Applying (D.40) to (D.41), for $\alpha\in (0,1)$ and $N\geq 2$, $N>R(\alpha)$. $\square$ Proof that as $n/N \rightarrow 0$, $R(\alpha)>N(1-\alpha)$ ---------------------------------------------------------- As $n/N \rightarrow 0$, $R(\alpha)\rightarrow N\dfrac{1-\alpha-\alpha^{N-1}+\alpha^N}{1-\alpha^{N-1}N+\alpha^N(N-1)}$. Hence, $$\begin{aligned} N(1-\alpha)<&N\dfrac{1-\alpha-\alpha^{N-1}+\alpha^N}{1-\alpha^{N-1}N+\alpha^N(N-1)} \Leftrightarrow\\ (1-\alpha)(1-\alpha^{N-1}N+\alpha^N(N-1))<&1-\alpha-\alpha^{N-1}+\alpha^N \Leftrightarrow\\ 1-\alpha^{N-1}N+\alpha^N(N-1)<&1-\alpha^{N-1} \Leftrightarrow\\ \alpha^N(N-1)<&\alpha^{N-1}(N-1) \Leftrightarrow \\ \alpha^N<\alpha^{N-1},\end{aligned}$$ which is true for $\alpha\in(0,1)$. D.43 holds if and only if the denominator of the right-hand-side of D.42 is positive. This is true by the lemma. $\square$ Proof that $R(\alpha)$ Is Strictly Decreasing on $[0,1)$ ======================================================== Let $$F(\alpha)=r([1-\alpha^N]/[N(1-\alpha)]-\alpha^{N-1})-(1-\alpha^{N-1}).$$ As shown in Hauert et al. (2002b), $F$ on $(0,1)$ has no root for $r \leq 2$. The preceding result does not hold, though, if $N=2$. We address the case for which $N=2$ at the end of the following proof. For now, we suppose $N>2$. Then, for every $r>2$ there exists exactly one $\alpha$ such that $F=0$. We consider $$Q(\alpha)=\dfrac{1-\alpha^{N-1}}{[1-\alpha^N]/[N(1-\alpha)]-\alpha^{N-1}}.$$ Q gives the values of r given $\alpha$ for which F is zero. Hence, Q is injective where it is defined. Since $[1-\alpha^N]/[N(1-\alpha)]-\alpha^{N-1}$ is positive on $[0,1)$ by the lemma in Appendix D, Q is defined and thus injective on (0,1). Thus, Q is either strictly decreasing or strictly increasing on (0,1). However, $$\lim_{\alpha\rightarrow 0} Q(\alpha) = N,$$ and $$\lim_{\alpha\rightarrow 1} Q(\alpha)=2,$$ applying l’Hôpital’s rule twice. Since Q is continuous on (0,1) there exist $\delta_1<1/2$ and $\delta_2<1/2$ such that for $\alpha \in (0, 0+\delta_1)$ and for $\alpha \in (1-\delta_2 ,1)$, $|Q(\alpha)-N|<1/3$ and $|Q(\alpha)-2|<1/3$, respectively. Choosing arbitrary $c_1 \in (0,0+\delta_1)$ and $c_2 \in (1-\delta_2,1)$, it follows that for $N>2$, $Q(c_1)>Q(c_2)$ and $c_1<c_2$. Hence, Q must be strictly decreasing on $(0,1)$. Moreover, $Q(0)=N$. Also considering that $Q<N$ on $(0,1)$, as shown in the proof of (22), Q is strictly decreasing on $[0,1)$. Then, we let the numerator of Q be $$S(\alpha)=1-\alpha^{N-1},$$ and note that S is strictly decreasing but positive on $[0,1)$. Next, we let the denominator of Q be $$T(\alpha)=[1-\alpha^N]/[N(1-\alpha)]-\alpha^{N-1},$$ which is positive in \[0,1) by the lemma in Appendix D. Lastly we let $$U(\alpha)=[\alpha/(n-1)][1-\alpha^{N-1}]/[1-\alpha],$$ which is non-negative on \[0,1) since it is the product of three non-negative terms. Also, $[1-\alpha^{N-1}]/[1-\alpha]=\Sigma_{k=0}^{N-2}\alpha^k$ for $N\geq2$, a strictly increasing function of $\alpha$ for $\alpha\geq0$ if $N>2$ and constant if $N=2$. Since $\alpha/(n-1)$ is strictly increasing, it follows that $U(\alpha)$ is also strictly increasing on $[0,\infty)$ for $N \geq 2$. Next, note that $$R(\alpha)=S(\alpha)/[T(\alpha)+U(\alpha)],$$ and consider any $\alpha_1, \alpha_2 \in [0,1)$ such that $\alpha_1<\alpha_2$. Then, $$\begin{aligned} R(\alpha_1)&>R(\alpha_2) \Leftrightarrow \\ S(\alpha_1)/[T(\alpha_1)+U(\alpha_1)]&>S(\alpha_2)/[T(\alpha_2)+U(\alpha_2)] \Leftrightarrow \\ S(\alpha_1)T(\alpha_2)+S(\alpha_1)U(\alpha_2)&>S(\alpha_ 2)T(\alpha_1)+S(\alpha_2)U(\alpha_1).\end{aligned}$$ However, since $S$ is strictly decreasing, $S(\alpha_1)>S(\alpha_2)$. Also considering that since $U$ is strictly increasing, $U(\alpha_2)>U(\alpha_1)$, and that $S$ is positive and $U$ is non-negative with a zero only at $\alpha=0$, it follows that $$S(\alpha_1)U(\alpha_2)> S(\alpha_2)U(\alpha_1).$$ Furthermore, since Q is strictly decreasing and T is positive, $$S(\alpha_1)T(\alpha_2)>S(\alpha_2)T(\alpha_1).$$ (E.12) and (E.13) together imply that E.11 is valid. Thus, R is strictly decreasing on $[0,1)$ for $N>2$. However, if $N=2$, then the only change from the above proof is that Q is constant rather than strictly decreasing. Then, (E.12) still holds, and we replace (E.13) by $$S(\alpha_1)T(\alpha_2)=S(\alpha_2)T(\alpha_1).$$ Thus, (E.11) still holds. Hence, R is strictly decreasing on $[0,1)$ for $N\geq 2$. $\square$ Derivation of $\pi_1$ (equation 24) =================================== The payoff matrix for a two person public goods game in which cooperators invest 1 unit which is then multiplied by r and distributed equally among all players is $$\begin{array}{c|cc} & c & d \\ \hline c& r-1 & r/2-1 \\ d& r/2 & 0 \end{array}$$ Then, we suppose that there are i players of type 1 in a population of n individuals, and that the remaining individuals are of type 2. We let a player of type 1 be one of the players invited to play in the two person PGG and call that player “player A”. Next, we let $A_c$, $A_d$, $A_n$, and $A_n^c$ represent the events where player A cooperates, defects, does not participate, and participates, respectively. We suppose “player B” is the other individual invited to play. We let $B_c$, $B_d$, $B_n$ be the events where player B cooperates, defects, and does not participate, respectively. Lastly, we let $E_1$ and $E_2$ be the events where Player B is of type 1 and of type 2, respectively. Denoting the intersection of any two events F and G by FG, and the probability that an event F occurs by p(F), $$\begin{aligned} \begin{split} \pi_1=&(r-1)[p(A_cE_1B_c)+p(A_cE_2B_c)]+(r/2-1)[p(A_cE_1B_d)+p(A_cE_2B_d)]\\&+r/2[p(A_dE_1B_c)+p(A_dE_2B_c)]+\sigma[p(A_n)+p(A_n^cE_1B_n)+p(A_n^cE_2B_n)]\\ %%Checked =&(r-1)[p(A_c)p(E_1|A_c)p(B_c|A_cE_1)+p(A_c)p(E_2|A_c)p(B_c|A_cE_2)]\\&+(r/2-1)[p(A_c)p(E_1|A_c)p(B_d|A_cE_1)+p(A_c)p(E_2|A_c)p(B_d|A_cE_2)]\\&+r/2[p(A_d)p(E_1|A_d)p(B_c|A_dE_1)+p(A_d)p(E_2|A_d)p(B_c|A_dE_2)]\\&+\sigma[p(A_n)+p(A_n^c)p(E_1|A_n^c)p(B_n|A_n^cE_1)+p(A_n^c)p(E_2|A_n^c)p(B_n|A_n^cE_2)]\\ %%Checked =&(r-1)\beta_1(1-\alpha_1)[p(E_1|A_c)\beta_1(1-\alpha_1)+p(E_2|A_c)\beta_2(1-\alpha_2)]\\&+(r/2-1)\beta_1(1-\alpha_1)[p(E_1|A_c)(1-\beta_1)(1-\alpha_1)+p(E_2|A_c)(1-\beta_2)(1-\alpha_2)]\\&+r/2(1-\beta_1)(1-\alpha_1)[p(E_1|A_d)\beta_1(1-\alpha_1)+p(E_2|A_d)\beta_2(1-\alpha_2)]\\&+\sigma[\alpha_1+(1-\alpha_1)[p(E_1|A_n^c)\alpha_1+p(E_2|A_n^c)\alpha_2]\\ %%Checked =&(r-1)\beta_1(1-\alpha_1)[\dfrac{i-1}{n-1}\beta_1(1-\alpha_1)+\dfrac{n-i}{n-1}\beta_2(1-\alpha_2)]\\&+(r/2-1)\beta_1(1-\alpha_1)[\dfrac{i-1}{n-1}(1-\beta_1)(1-\alpha_1)+\dfrac{n-i}{n-1}(1-\beta_2)(1-\alpha_2)]\\&+r/2(1-\beta_1)(1-\alpha_1)[\dfrac{i-1}{n-1}\beta_1(1-\alpha_1)+\dfrac{n-i}{n-1}\beta_2(1-\alpha_2)]\\&+\sigma[\alpha_1+(1-\alpha_1)[\dfrac{i-1}{n-1}\alpha_1+\dfrac{n-i}{n-1}\alpha_2]\\ %%Checked =&\dfrac{n - i}{n - 1}(1 - \alpha_1)(1 - \alpha_2)(r\beta_2/2 + r\beta_1/2- \beta_1) + \dfrac{i - 1}{n - 1}(1 - \alpha_1)^2\beta_1(r - 1)\\& + \sigma\alpha_1 + \sigma(1 - \alpha_1)(\dfrac{n - i}{n - 1}\alpha_2 + \dfrac{i - 1}{n - 1}\alpha_1) \end{split} %%Checked\end{aligned}$$ Justification of Approximations =============================== Approximation for $R(\alpha)$ as $N\rightarrow \infty$, $\dfrac{N}{n-1}=c$ -------------------------------------------------------------------------- We let $c$ be a real number in \[0,1\]. As $N\rightarrow\infty$, $\alpha^{N-1}N$, $\alpha^N(N-1-N/(n-1))$, $\alpha^{N-1}$, and $\alpha^N$ $\rightarrow 0$ as long as $\alpha\nrightarrow 1$. Hence, for $\alpha\nrightarrow 1$, $$\begin{aligned} R(\alpha)=&N\dfrac{1-\alpha-\alpha^{N-1}+\alpha^N}{1+\alpha N/(n-1)-\alpha^{N-1}N+\alpha^N(N-1-N/(n-1))}\\ \approx & N(1-\alpha)/(1+c\alpha).\end{aligned}$$ However, applying l’Hôpital’s rule yields $\lim_{\alpha \rightarrow 1}R(\alpha)=0$, which is $\lim_{\alpha\rightarrow 1}N(1-\alpha)/(1+1/2\alpha)$. $\square$ Approximation for $R(\alpha)$ for $n>>N>>0$ ------------------------------------------- As $n \rightarrow \infty$, $N \rightarrow \infty$, $N/n \rightarrow 0$, for $alpha \nrightarrow 1$, $$\begin{aligned} R(\alpha)=&N\dfrac{1-\alpha-\alpha^{N-1}+\alpha^N}{1+\alpha N/(n-1)-\alpha^{N-1}N+\alpha^N(N-1-N/(n-1))}\\ \approx & N(1-\alpha).\end{aligned}$$ However, as in the preceding proof, applying l’Hôpital’s rule yields $\lim_{\alpha \rightarrow 1}R(\alpha)=0$, which is $\lim_{\alpha\rightarrow 1}N(1-\alpha)$. $\square$\ \ **References** [00]{} Antal, T., Ohtsuki, H., Wakeley, J., Taylor, P.D. and Nowak, M.A., 2009. 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--- abstract: 'An improved understanding of nonperturbative QCD can be obtained by the recently developed soft color interaction models. Their essence is the variation of color string-field topologies, giving a unified description of final states in high energy interactions, [e.g.]{}, diffractive and nondiffractive events in $ep$ and [$p\bar{p}$]{}. Here we present a detailed study of such models (the soft color interaction model and the generalized area law model) applied to [$p\bar{p}$]{}, considering also the general problem of the underlying event including beam particle remnants. With models tuned to HERA $ep$ data, we find a good description also of Tevatron data on production of $W ,$ beauty and jets in diffractive events defined either by leading antiprotons or by one or two rapidity gaps in the forward or backward regions. We also give predictions for diffractive $J/\psi$ production where the soft exchange mechanism produces both a gap and a color singlet $c\bar{c}$ state in the same event. This soft color interaction approach is also compared with Pomeron-based models for diffraction, and some possibilities to experimentally discriminate between these different approaches are discussed.' address: | $^a$ High Energy Physics, Uppsala University, Box 535, S-751 21 Uppsala, Sweden\ $^b$ Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, D-22603 Hamburg, Germany author: - 'R. Enberg$^a$, G. Ingelman$^{ab}$, N. Tîmneanu$^a$' title: |  \  \  \ Soft Color Interactions and Diffractive Hard Scattering\ at the Fermilab Tevatron --- DESY 01-076\ June 2001 Introduction {#sec-intro} ============ A major unsolved problem in particle physics is to understand strong interaction processes with a small (‘soft’) momentum transfer. The most striking illustration of this is the confinement of quarks and gluons in hadrons and the related hadronization process giving the observable hadronic final states in high energy collisions. In terms of Quantum Chromodynamics (QCD), small momentum transfers have a large coupling $\alpha_s$ such that a perturbative expansion in terms proportional to powers of $\alpha_s^n$ does not work. This is in contrast to processes with a ‘hard’ scale, [i.e.]{}, a large momentum transfer, where $\alpha_s$ is small and perturbative QCD (pQCD) on the level of quarks and gluons works well. To gain understanding of soft, nonperturbative QCD (non-pQCD) it is therefore advantageous to first consider soft effects in hard scattering events, since the hard scale gives a firm ground in terms of a parton level process which is calculable in pQCD. This hard-soft interplay is the basis for the topical research field of diffractive hard scattering [@StCroix]. Diffractive events can be characterized by having a rapidity gap, [i.e.]{}, a region in rapidity (or polar angle) without any particles. Another definition is to require a leading particle carrying most of the beam particle momentum ($x_F\gtrsim 0.9$), which is kinematically related to a rapidity gap. These rapidity gaps in the forward or backward rapidity regions, connect to the soft part of the event and therefore nonperturbative effects on a long space-time scale are certainly important. The central rapidity gaps between high-$p_\perp$ jets, observed at the Tevatron [@j-g-j], may be of a different kind since the hard momentum transfer is across the gap. This gap phenomenon will therefore not be considered here, but is studied separately [@singlet-exchange]. Diffractive scattering has traditionally been explained in the Regge framework by the exchange of a Pomeron [@GoulianosReview]. For processes with a hard scale, a parton structure of the Pomeron may be considered [@IS]. With the Pomeron flux given by Regge phenomenology, the HERA data on diffractive deep inelastic scattering can be well described by fitting parton density functions in the Pomeron [@HERA-pomeron; @HERA-F2D]. However, applying exactly the same model for $p\bar{p}$ gives a too large cross section for diffractive hard processes. Compared to the Tevatron data in Table \[tab-gapratios\], such a Pomeron model gives about a factor six too large rates of $W$ and dijets with one gap and two orders of magnitude too large rates of dijets with two gaps [@Alvero]. This is related to the failure of the factorization theorem for hard diffractive hadron-hadron scattering, although it holds in diffractive deep inelastic scattering (DIS) [@Collins-Factorization]. It is also an indication of a non-universality problem of the Pomeron model, which may be related to the Pomeron flux. Since this flux specifies the leading particle spectrum, it is interesting to note that the new Tevatron data [@CDF-AP] with a leading antiproton show a similar problem of the Pomeron model. These problems of the Pomeron approach are further discussed in Section \[sec-pomeron\]. In order to better understand nonperturbative dynamics and to provide a unified description of all final states, the soft color interaction (SCI) model [@SCI; @unified] and the generalized area law (GAL) model [@GAL] were developed. These are added to Monte Carlo generators ([<span style="font-variant:small-caps;">lepto</span>]{}[@Lepto] for $ep$ and [<span style="font-variant:small-caps;">pythia</span>]{}[@Pythia] for $p\bar{p}$) which simulate the interaction dynamics and provide a complete final state of observable particles, such that an experimental approach can be taken to classify events depending on the characteristics of the final state: [e.g.]{}, gaps or no-gaps, leading protons or neutrons, etc. The basic assumption of the models is that variations in the topology of the confining color force fields (strings [@lund]) lead to different hadronic final states after hadronization. The pQCD interaction gives a set of partons with a specific color order. However, this order may change owing to soft, nonperturbative interactions. The details of our models for such interactions are described in Section \[sec-model\]. One may at first think that this approach is some kind of model for the Pomeron. To the extent that the term ‘Pomeron’ is associated with the Regge approach, this is not the case since nothing from the Regge formalism is being used or referred to. The soft color interaction models also give quite different results when applied to diffractive hard scattering at the Tevatron. An overall summary of the relative rates of various diffractive hard processes is given in Table \[tab-gapratios\], which shows that this approach can account for several different gap phenomena (taking the uncertainty in models and data into account). The details of this and other results are presented and discussed in Sections \[sec-single\], \[sec-dpe\] and \[sec-jpsi\]. ----------------------- ------------ ---------------- ------------------- ----- ----- -- Observable $\sqrt{s}$ \[GeV\] Experiment Observed SCI GAL $W$ - gap 1800 CDF [@CDF-W] $1.15 \pm 0.55$ 1.2 0.8 $Z$ - gap 1800 — — 1.0 0.5 $b\bar b$ - gap 1800 CDF [@CDF-B] $0.62 \pm 0.25$ 0.7 1.4 $J/\psi$ - gap 1800 — — 1-2 1-2 $jj$ - gap 1800 CDF [@CDF-JJ] $0.75 \pm 0.10$ 0.7 0.6 $jj$ - gap 1800   DØ[@D0-JJ] $0.65 \pm 0.04$ 0.7 0.6 $jj$ - gap   630   DØ[@D0-JJ] $1.19 \pm 0.08$ 0.9 1.2 gap - $jj$ -gap 1800 CDF [@CDF-DP] $ 0.26 \pm 0.06 $ 0.2 0.1 $\bar{p}$ - $jj$ -gap 1800 CDF [@CDF-DPE] $0.80 \pm 0.26$ 0.5 0.4 ----------------------- ------------ ---------------- ------------------- ----- ----- -- : Ratios diffractive/inclusive for hard scattering processes in [$p\bar{p}$]{} collisions at the Tevatron, showing experimental results from CDF and D0 compared to the SCI and GAL soft color exchange models.[]{data-label="tab-gapratios"} \ \ \ As opposed to the standard Pomeron approach, the SCI and GAL models can describe diffractive events both at HERA and at the Tevatron. This is not achieved by introducing several free parameters. On the contrary, the models have essentially only [*one*]{} new parameter to account for the unknown nonperturbative dynamics. This parameter is determined from the HERA data on the diffractive structure function $F_2^D$ [@HERA-F2D] and then used with the same computer code implemented in [<span style="font-variant:small-caps;">pythia</span>]{} to simulate [$p\bar{p}$]{} at the Tevatron. The SCI and GAL models are very general in that they are able to describe a large set of different data. This does not only refer to diffraction, but also various nondiffractive observables. Particularly noteworthy is that the SCI model reproduces the observed rate of high-$p_\perp$ charmonium and bottomonium at the Tevatron [@SCI-onium], which is factors of 10 larger than the predictions based on the color singlet model in conventional pQCD. Although the SCI and GAL models are too simple and have too weak theoretical content to provide a satisfactory understanding, their general applicability and success in describing different kinds of observables show that different phenomena may have a common explanation. They represent a new approach which, together with others mentioned in the Conclusions, may lead us towards a proper understanding of nonperturbative QCD. Pomeron problems {#sec-pomeron} ================ The inability to describe both HERA and $p\bar{p}$ collider data on hard diffraction is a problem for the Pomeron model. It shows that the ‘standard’ Pomeron flux factor [@DLpomeron], $$\label{eq:pomeron-flux} f_{{I\!\!P}/p}(x_{{I\!\!P}},t)=\frac{9\beta_0^2}{4\pi^2} \left( \frac{1}{x_{{I\!\!P}}}\right) ^{2\alpha_{{I\!\!P}}(t)-1} \left[ F_1(t)\right]^2$$ and Pomeron parton densities, $f_{i/{I\!\!P}}(x,Q^2)$, cannot be used universally. This flux is found to give a much larger cross section for inclusive single diffraction than measured at $p\bar{p}$ colliders, although it works well for lower energy data. This is due to the increase of the flux as the minimum $x_{{I\!\!P}\, min}=M^2_{X\, min}/s$ gets smaller with increasing cms energy $\sqrt{s}$. To avoid this unphysical increase, a Pomeron flux ‘renormalization’ has been proposed [@fluxrenorm] by enforcing that the integral of the flux saturates at unity (by dividing by the integral whenever it is larger than unity). This prescription not only gives the correct inclusive single diffractive cross section at collider energies, but it also makes the HERA and Tevatron data on hard diffraction compatible with the Pomeron hard scattering model. The model result for HERA is not affected, but at the higher energy of the Tevatron the Pomeron flux is reduced such that the data are essentially reproduced. In another proposal [@Erhan-Schlein] based on an analysis of single diffraction cross sections, the Pomeron flux is reduced at small $x_{{I\!\!P}}$ through an $x_{{I\!\!P}}$- and $t$-dependent damping factor. The pros and cons of these two approaches to modify the Pomeron flux have been debated. It has recently been shown [@CFL-Wproduction] that the Tevatron data on diffractive $W$ production can be reproduced if a harder Pomeron flux is introduced together with a Pomeron intercept higher than the value extracted from HERA data. These changes from the conventional Pomeron model illustrate the problem of having a universally applicable Pomeron model. In a proposed new phenomenological approach [@Dino-new] the structure of the Pomeron is derived from that of the parent proton such that the gap probability is obtained from the soft parton density at $x_{{I\!\!P}}$. Some general features of diffractive DIS are obtained, but a more detailed confrontation with data remains to be performed. A difference between diffraction in $ep$ and $p\bar{p}$ is the possibility for coherent Pomeron interactions in the latter [@CollinsFrankfurtStrikman]. In the incoherent interaction only one parton from the Pomeron participates and any others are spectators. However, in the Pomeron-proton interaction with ${I\!\!P}=gg$ both gluons may take part in the hard interaction giving a coherent interaction. For example, in the ${I\!\!P}p$ hard scattering subprocess $gg\to q\bar{q}$, the second gluon from the Pomeron may couple to the gluon from the proton. Such diagrams cancel when summing over all final states for the inclusive hard scattering cross section (the factorization theorem). For gap events, however, the sum is not over all final states and the cancellation fails leading to factorization breaking for these coherent interactions where the whole Pomeron momentum goes into the hard scattering system. This coherent interaction cannot occur in the same way in deep inelastic scattering (DIS) since the Pomeron interacts with a particle without colored constituents. This difference between $ep$ and $p\bar{p}$ means that there should be no complete universality of parton densities in the Pomeron. The difference between diffractive hard scattering at HERA and the Tevatron can be described in terms of an overall suppression factor or gap ‘survival probability’, due to extra soft rescattering effects in $p\bar{p}$, estimated using an eikonal model [@Khoze]. Although modified Pomeron models may describe the rapidity gap events reasonably well, there is no satisfactory understanding of the Pomeron and its interaction mechanisms. On the contrary, there are conceptual and theoretical problems with this framework. The Pomeron is not a real state, but only a virtual exchanged spacelike object. The concept of a structure function is then not well defined and, in particular, it is unclear whether a momentum sum rule should apply. In fact, the factorization into a Pomeron flux and a Pomeron structure function cannot be uniquely defined since only the product is an observable quantity [@Landshoff-Paris]. It may therefore be improper to regard the Pomeron as being ‘emitted’ by the proton, having QCD evolution as a separate entity and being ‘decoupled’ from the proton during and after the hard scattering. Since the Pomeron-proton interaction is soft, its time scale is long compared to the short space-time scale of the hard interaction. It is therefore natural to expect soft interactions between the Pomeron system and the proton both before and after the snapshot of the high-$Q^2$ probe provided by the hard scattering. The Pomeron can then not be viewed as decoupled from the proton and, in particular, is not a separate part of the QCD evolution in the proton. Large efforts have been made to understand the Pomeron as a two-gluon system or a gluon ladder in pQCD. By going to the soft limit one may then hope to gain understanding of non-pQCD and, perhaps, establish a connection between pQCD in the small-$x$ limit and Regge theory. More explicitly, diffractive DIS has been considered in terms of models based on two-gluon exchange in pQCD, see [e.g.]{}[@pQCD-pomeron]. The basic idea is to take two gluons in a color singlet state from the proton and couple them to the $q\bar{q}$ system from the virtual photon. With higher orders included the diagrams and calculations become quite involved. Nevertheless, these approaches can be made to describe the main features of the diffractive DIS data. Although this illustrates the possibilities of the pQCD approach to the Pomeron, one is still forced to include nonperturbative modeling to connect the two gluons in a soft vertex to the proton which goes beyond the conventional use of parton densities. Thus, even if one can gain understanding by working as far as possible in pQCD, one cannot escape the fundamental problem of understanding non-pQCD. Models for soft color interactions {#sec-model} ================================== Given these practical and conceptual problems of the Regge-based Pomeron model and the impossibility to cover all important aspects by a pQCD treatment, new approaches should be investigated. We are here exploring new ideas to model non-pQCD interactions, which avoid the concept of a Pomeron and provide a single simple model that describes all final states, with or without rapidity gaps. The starting point is that the hadronic final state is produced through the hadronization of partons emerging from a hard scattering process which can be well described by pQCD. The basic new idea is that there may be additional soft color interactions at a scale below the cutoff $Q_0^2$ for the perturbative treatment. Obviously, interactions will not disappear below this cutoff. On the contrary, they will be abundant due to the large coupling [$\alpha_s$]{} at small scales. The question is rather how to describe these interactions properly. Here, we introduce soft color interactions which do not change the dynamics of the hard scattering, but change the color topology of the state such that another hadronic final state emerges after hadronization. This topology can be described in terms of color triplet strings and the standard Lund model [@lund] can be used for a well established treatment of the hadronization of any given string configuration. We have tried two different ways to model the soft exchange of color-anticolor representing nonperturbative gluon exchange. The soft color interaction model is formulated in a parton basis with color exchanges between the partons emerging from the hard scattering process (including remnants of initial hadrons). The generalized area law model is instead formulated in a string basis, since strings are here assumed to be the proper states for soft exchanges that may not resolve partons. In spite of this difference, the models have a very similar structure and may be regarded as variations on the same general theme. The SCI and GAL models are constructed as subroutines added to the Monte Carlo event generators [<span style="font-variant:small-caps;">lepto</span>]{}[@Lepto] for $ep$ and [<span style="font-variant:small-caps;">pythia</span>]{}[@Pythia] for $p\bar{p}$. This gives powerful tools for detailed investigations of the models and their ability to reproduce experimental data. Since the soft non-pQCD processes cannot alter the hard perturbative scattering processes, the latter should be kept unchanged in the models. Therefore, the hard parton level interactions are treated in the normal way using standard hard scattering matrix elements (electroweak or QCD) plus initial and final state parton showers based on the DGLAP leading logarithm evolution equations [@DGLAP] to simulate higher order pQCD processes. Thus, the set of partons, including those in beam hadron remnants, are generated as in conventional $ep$ and [$p\bar{p}$]{} hard scattering processes. The SCI and GAL models are then added as an extra intermediate step before the hadronization is performed using the Lund Monte Carlo [jetset]{} [@Pythia]. In this section we first describe these two models in some detail and then discuss other aspects of soft interactions which are common for both models and must be considered in a complete Monte Carlo model. The SCI model {#sec-SCI} ------------- The soft color interaction (SCI) model [@SCI; @unified] is applied to the parton state emerging from the hard scattering. It gives the possibility for each pair of these color charged partons to make a soft interaction. One may here include all possible pairs of partons or require that one parton belongs to the remnant. In the latter case, one may view this as the perturbatively produced quarks and gluons interacting softly with the color medium of the proton as they propagate through it. The soft interaction changes only the color but not the momentum and may be viewed as soft nonperturbative gluon exchange. This should be a natural part of the process in which bare perturbative partons are dressed into nonperturbative ones and the formation of the confining color flux tube in between them. This necessarily involves some, not yet understood nonperturbative interactions which the model attempts to describe. Being a nonperturbative process, the exchange probability cannot presently be calculated and is therefore described by a phenomenological parameter $P$. The number of soft exchanges will vary event-by-event and change the color topology such that, in some cases, color singlet subsystems arise separated in rapidity, as illustrated Fig. \[DIS-SCI\] where, [e.g.]{}, a color exchange between the perturbatively produced quark and the quark in the remnant has taken place. Color exchanges between the perturbatively produced partons and the partons in the proton remnant (representing the color field of the proton) are of particular importance for the gap formation. It should be emphasized, however, that the model is quite general giving rise to events both with and without rapidity gaps. Since DIS is a simpler and cleaner process than [$p\bar{p}$]{} collisions, the model was first developed for DIS and successfully tested against diffractive DIS data from HERA [@SCI; @unified; @sce-heramc]. The rate and main properties of the gap events are qualitatively reproduced. The rate of gap events depends on the parameter $P$, but the dependence is not strong giving a stable model with $P\simeq 0.2$–0.5. This color exchange probability is the only new parameter in the model. Other parameters belong to the conventional [<span style="font-variant:small-caps;">lepto</span>]{} model [@Lepto] and have their usual values. The rate and size of gaps do, however, depend on the amount of parton emission. In particular, more initial state parton shower emissions will tend to populate the forward rapidity region and prevent gap formation [@unified]. The gap events show the properties characteristic of diffraction. The exponential $t$-dependence arises in the model from the intrinsic transverse momentum (Fermi motion) of the interacting parton which is balanced by the proton remnant system. This remnant gives rise to leading protons with a peak at large fractional momentum $x_F$, as well as proton dissociation. The salient features of the measured diffractive structure function are also reproduced [@sce-heramc]. The behavior of the data on $F_2^D(\beta,Q^2)$ is in the SCI model understood as normal pQCD evolution in the proton. The rise with $\ln{Q^2}$ also at larger $\beta$ is simply the normal behavior at the small momentum fraction $x=\beta x_{{I\!\!P}}$ of the parton in the proton. Here, $x_{{I\!\!P}} =\frac{Q^2+M_X^2-t}{Q^2+W^2-m_p^2} \approx \frac{x(Q^2+M_X^2)}{Q^2}$ is only an extra variable related to the gap size or $M_X$, which does not require a Pomeron interpretation. The flat $\beta$-dependence of $x_{{I\!\!P}} F_2^D=\frac{x}{\beta} F_2^D$ is due to the factor $x$ compensating the well-known increase at small-$x$ of the proton structure function $F_2$. This Monte Carlo model gives a general description of DIS, with and without gaps. In fact, it can give a fair account of such ‘orthogonal’ observables as rapidity gaps and the large forward $E_T$ flow [@unified]. Diffractive events are in this model defined through the topology of the final state, in terms of rapidity gaps or leading protons just as in experiments. There is no particular theoretical mechanism or description in a separate model, like Pomeron exchange, that defines what is labeled as diffraction. This provides a smooth transition between diffractive gap events and nondiffractive (no-gap) events [@SCI-forward]. In addition, leading neutrons are also obtained in fair agreement with recent experimental measurements [@leading-pn]. In a conventional Regge-based approach, Pomeron exchange would be used to get diffraction, pion exchange added to get leading neutrons and still other exchanges added to get a smooth transition to normal DIS. The SCI model demonstrates that a simpler theoretical description can be obtained. The GAL model {#sec-GAL} ------------- The generalized area law (GAL) model [@GAL] for color string re-interactions is a model for soft color exchanges which is similar in spirit to the SCI model. Whereas the SCI model is formulated as soft exchanges between the partons emerging from the hard scattering process, the GAL model is formulated in terms of interactions between the strings connecting these partons. Soft color exchanges between strings change the color topology resulting in another string configuration, as illustrated in Fig. \[DIS-SCI\]. The probability for two strings to interact is in the GAL model obtained as a generalization of the area law suppression $e^{-bA}$ with the area $A$ swept out by a string in energy-momentum space. The model uses the measure $A_{ij}=(p_i+p_j)^2-(m_i+m_j)^2$ for the piece of string between two partons $i$ and $j$. This results in the probability $P=P_0[1-\exp{(-b\Delta A)}]$ depending on the change $\Delta A$ of the areas spanned by the strings in the two alternative configurations of the strings, [i.e.]{}, with or without the topology-changing soft color exchange. The exponential factor favors making ‘shorter’ strings, [e.g.]{}, events with gaps, whereas making ‘longer’ strings is suppressed. The fixed probability for soft color exchange in SCI is thus in GAL replaced by a dynamically varying one. There is only one new parameter in the GAL model, [i.e.]{}, $P_0$ instead of $P$ in SCI. $b$ is one of the usual hadronization parameters in the Lund model [@lund], but its value must be retuned when changing the string configuration. Since the GAL model is formulated in terms of strings, it should be applicable to all interactions producing strings, [i.e.]{}, also to hadronic final states in $e^+e^-$. The parameter values used in the GAL model were obtained [@GAL] by making a simultaneous tuning to the diffractive structure function in DIS and the charged particle multiplicity distribution and momentum distribution for $\pi^{\pm}$ in $e^+e^-$ annihilation at the $Z^0$-resonance. This resulted in $P_0=0.1$, $b=0.45$ GeV$^{-2}$ and $Q_0=2$ GeV, where $Q_0$ is the cutoff for initial and final state parton showers. It is not possible to have the [[<span style="font-variant:small-caps;">jetset</span>]{}]{} default cutoff $Q_0=1$ GeV in the parton showers and simultaneously reproduce the multiplicity distribution. One might worry that the obtained cutoff is relatively large compared to the default value. However, it is not obvious that perturbation theory should be valid for so small scales when more exclusive final states are considered. Therefore, $Q_0$ can be seen as as a free parameter describing the boundary below which it is more fruitful to describe the fragmentation process in terms of strings instead of perturbative partons. With this parameter tuning the GAL model gives very similar results [@GAL] for the final state in $e^+e^-\to Z^0\to hadrons$ as default [<span style="font-variant:small-caps;">jetset</span>]{}. This concerns multiplicity distributions, momentum distributions and string effects. Also the conventional rapidity gap behavior is obtained, [i.e.]{}, an exponentially falling distribution with increasing size $\Delta y$ of the largest rapidity gap in the event. Applying the GAL model to DIS at HERA [@sce-heramc] gives a quite good description of the diffractive structure function $F_2^{D(3)}(x_{{I\!\!P}},\beta,Q^2)$ observed by H1. The details at low $Q^2$ is actually better reproduced with GAL than with SCI. The GAL model cures the problem the SCI model has in producing somewhat too many soft hadrons in inclusive DIS, but results in too low transverse energy flow in the forward region. These effects are related to events where the string after SCI goes back-and-forth producing a zig-zag shape, [i.e.]{}, a longer string, giving more but softer hadrons after hadronization. Conversely, the GAL model suppresses topologies with long strings. Remnants and soft underlying event {#sec-remnant} ---------------------------------- To obtain a complete model for the production of the observable hadronic final state there are further issues of nonperturbative dynamics that have to be considered. These include not only the hadronization process itself, but also the treatment of remnants of the colliding hadrons and possible additional dynamics in order to achieve a decent description of the soft underlying event, [i.e.]{}, underlying the hard scattering part of the event. Here, we essentially use the standard models developed for the family of Lund Monte Carlo programs, but with some modifications and further developments as will be described in this subsection. The standard Lund hadronization model [@lund] as implemented in [<span style="font-variant:small-caps;">jetset</span>]{}[@Pythia] is used for the formation of hadrons from color triplet string fields. However, the final state will depend on how the strings are stretched between partons, as exemplified by the SCI and GAL models above. Similarly, the resulting string system will depend on how the hadron remnants are treated and if additional strings are formed, [e.g.]{}, to produce additional hadronic activity in the underlying event. The remnant system is the initial (anti)proton ‘minus’ the parton entering the hard scattering process, [i.e.]{}, the hard $2\to 2$ scattering given by matrix elements combined with parton showers. The initial parton carries a momentum fraction $x_0$ of the beam proton as given by the parton density distributions $f_i(x_0,Q^2_0)$ at the scale $Q_0^2$ where the initial state parton shower is terminated in its backwards evolution simulation. This leaves the fraction $1-x_0$ for the proton remnant system. The initial parton can be either a valence quark, a sea quark or a gluon. In case a valence quark is removed from the initial proton, the remnant is a diquark with an anti-triplet color charge that defines the endpoint of a triplet string. If the initial parton is a gluon, the remnant contains all three valence quarks in a color octet state which is split into a color triplet quark and a color anti-triplet diquark that form the end-points on two triplet strings. Here, the quark and diquark share the remnant momentum in the fractions $\chi$ and $1-\chi$, respectively, as given by parametrizations of $\cal P(\chi)$ in [<span style="font-variant:small-caps;">pythia</span>]{} and to be further discussed below. In case a sea quark is removed from the initial proton, the remnant system is more complex, containing all three valence quarks plus the partner of the interacting sea quark in order to conserve quantum numbers. Here, a more elaborate sea quark treatment (SQT) has been introduced [@Lepto; @unified]. The interacting quark, with flavor and momentum $x_0$ obtained from the initial state parton shower evolution, is taken as a valence or sea quark based on the relative sizes of the corresponding parton distributions $q_{val}(x_0,Q_0^2)$ and $q_{sea}(x_0,Q_0^2)$. In case of a sea quark, the left-over partner is given an explicit momentum. Here, we have tried two possibilities to model this unknown dynamics. In the first (SQT1), the longitudinal momentum fraction is given by the Altarelli-Parisi splitting function $P(g\to q\bar{q})$, [i.e.]{}, the pQCD initial state parton shower routine is used to model a $g\to q\bar{q}$ process which is strictly speaking below the original parton shower cutoff. As an alternative (SQT2) the sea quark partner is assigned a longitudinal momentum chosen from the corresponding sea quark momentum distribution in the proton. In both cases the transverse momentum is chosen from the same Gaussian used for the primordial transverse (Fermi) momentum. These two methods give similar results, but differ in some details as will be discussed below. The sea quark partner and the three valence quarks, which are split into a quark and a diquark as described, define the dynamics of the remnant system. These three color (anti)triplet objects in the remnants are then end-points on strings, implying additional string topology possibilities. Since the sea quark partner has only a small transverse momentum, it affects in particular the very forward part of the final hadronic state. Therefore, it is of interest for the formation of rapidity gaps studied here. A related issue is the treatment of a color singlet system (string) with small invariant mass. The Lund hadronization model is constructed for large mass strings, but can be applied to systems of invariant mass which is as low as the sum of the end-point parton masses plus an additional $\sim 1$ GeV. When the string mass is so small that only one or two hadrons can be formed, normal string hadronization is not applicable since energy-momentum constraints and resonance phenomena demand special treatment. This is instead achieved through the new routines ([lsmall]{} in [<span style="font-variant:small-caps;">lepto</span>]{} and [pysmall]{} in [<span style="font-variant:small-caps;">pythia</span>]{}). Of particular importance for the investigations in this paper is the formation of a single leading proton (or antiproton) giving the diffractive signature. The mapping of a string with a continuous mass distribution onto a particular on-shell hadron with fixed mass, requires a shuffling of energy-momentum to another string system in order to conserve energy-momentum in the event [@Pythia]. By transferring the required energy-momentum to/from another parton which is as far away as possible in phase space, the relative disturbance on the four-vectors is kept minimal and typically of order tens of MeV, [i.e.]{}, small even on the hadronization momentum transfer scale. Starting with the hard scattering processes (matrix elements and parton showers) and adding this remnant treatment followed by Lund string model hadronization results in a Monte Carlo event generator producing a complete hadronic final state. The resulting hadronic activity is, however, too small compared to collider data [@Multiple]. The observed multiplicities are larger, with the multiplicity distribution extending in a longer tail to large multiplicities. Furthermore, the number of particles per unit rapidity is larger and gives a higher rapidity plateau or ‘pedestal’ below high-[$p_{\perp}$]{} jets than obtained in the model. This additional activity in the underlying event is related to soft QCD processes and is therefore difficult to describe in a theoretically satisfactory way. In [<span style="font-variant:small-caps;">pythia</span>]{} this additional activity in the underlying event is achieved by a model for multiple interactions (MI) [@Multiple; @Pythia]. This is constructed based on multiple parton-parton scatterings described by the QCD $2\to 2$ matrix elements. At small momentum transfers this cross section becomes large, even larger than the [$p\bar{p}$]{} total cross section which is interpreted as having more than one such parton-parton scattering in the same event. These scatterings can sometimes be hard enough to contribute to the rate of low-[$p_{\perp}$]{} jets and minijets, but dominantly they have too small [$p_{\perp}$]{} to give observable jet structure. These small-[$p_{\perp}$]{} partons will stretch additional strings that produce more hadrons over large rapidity regions and thereby contribute substantially to the underlying event. The cross section for these multiple scatterings diverge when the scattered parton $p_\perp \to 0$. This is avoided by some (arbitrary) regularization or a cutoff on [$p_{\perp}$]{}, which will be the main regulator of the amount of multiple scatterings that are generated. In the default version of the MI model in [<span style="font-variant:small-caps;">pythia</span>]{} 5.7 a sharp cutoff $p_{\perp}^{\mathrm{min}}=1.4$ GeV is used, although more complicated alternatives are available as options [@Pythia]. Using this MI model, data on multiplicities, rapidity distributions and pedestal effects at the S[$p\bar{p}$]{}S ($\sqrt{s}=540$ and $630$ GeV) [@Underlying] can be reasonably described [@Multiple]. Measurements of this kind have only recently been made at the Tevatron and the model has not yet been tested or tuned at the energy of interest in our study. Although the MI model is based on pQCD parton-parton scattering, in this context the model is used to emulate soft nonperturbative effects. The soft color exchange models are also introduced to account for soft effects on the hadronic final state. The SCI model, in particular, can give zig-zag shaped strings which produce a larger number of hadrons per unit rapidity, [i.e.]{}, more activity in the underlying event. There is therefore a risk of ‘double counting’ the soft effects and producing too much underlying soft activity if the SCI/GAL model and the MI model are simply added. With the SCI/GAL model tuned to data on rapidity gaps, we therefore lower the amount of multiple interactions by increasing the $p_\perp^{\mathrm{min}}$ parameter. This means that the pQCD-based MI model is not pushed to generate the softest dynamics, which is instead treated by the soft exchange models. We have studied this issue in some detail by looking at jet profiles, rapidity plateaus and charged particle multiplicities obtained by running [<span style="font-variant:small-caps;">pythia</span>]{} with SCI/GAL added and the default MI model. Keeping the default value of $p_\perp^{\mathrm{min}}$ gives, as expected, too much underlying event activity, whereas increasing to $p_\perp^{\mathrm{min}}=2.5$ GeV for SCI and to $p_\perp^{\mathrm{min}}=2.0$ GeV for GAL, one obtains essentially the same results as default [<span style="font-variant:small-caps;">pythia</span>]{}, and thereby reproduce data equally well. The lower value for the GAL model reflects the fact that longer strings are suppressed, and therefore GAL contributes less to the underlying event activity than SCI. We note that in the recently released version 6 of [<span style="font-variant:small-caps;">pythia</span>]{} [@Pythia6], the MI cutoff has been made energy dependent giving the value $p_\perp^{\mathrm{min}}=2.1$ GeV at the Tevatron, [i.e.]{}, closer to our values and indicating that the GAL model adds very little activity to the underlying event. Our $p_\perp^{\mathrm{min}}$ values have also been obtained by comparing with the diffractive data studied in this paper, but this will be discussed further below. The sensitivity of our results to variations in these details of the modeling of the remnant and the underlying event has been investigated and is discussed below in connection with the comparison of our models and the available data. Single diffractive hard scattering {#sec-single} ================================== Before discussing the details of how the SCI and GAL models apply in different single diffractive hard scattering processes in the following subsections, we first discuss some general aspects. Single diffractive scattering is characterized by a large rapidity gap in the forward or backward hemisphere of a [$p\bar{p}$]{} collision. The occurrence of rapidity gaps is very strongly affected by soft effects, as demonstrated in Fig. \[plotmaxgapsize\] for the case of diffractive $W$ production. At the parton level, arising from the hard processes described by matrix elements and parton showers, there can be large regions of phase space where no partons have been emitted and thereby no strong suppression of the probability for large rapidity gaps. The partons are, however, connected by color force fields which through hadronization produce hadrons which fill these gaps in the final state. Thus, applying hadronization using the standard Lund string model, causes the drastic transition from the dashed to the dash-dotted curve in Fig. \[plotmaxgapsize\] such that large rapidity gaps in the final state of hadrons become exponentially suppressed. An extreme case is provided by the peak in the parton level curve, which arises from events where the $W$ is produced by valence quark annihilation without parton radiation resulting in a huge rapidity separation between the two remnant systems (diquarks). Hadronization of the color string between these remnants produce hadrons in the full rapidity range, leaving no trace of the parton level gap. This very strong effect of hadronization implies that modifications of the modeling of the poorly known nonperturbative QCD processes can have substantial effects. Applying the SCI model of last section, leads to an increased probability for large rapidity gaps (full curve in Fig. \[plotmaxgapsize\]) at the hadron level, but still far below the parton level result. This difference relative to default [<span style="font-variant:small-caps;">pythia</span>]{} may at first seem small, but for large gaps it is exactly what is needed to describe data as will be discussed in detail below. One may worry that there is no flat region, [i.e.]{}, where the probability does not decrease with increasing gap size, which is sometimes taken as a characteristic for diffraction. This is due to the kinematical restriction on high-$x_F$ leading protons imposed by the large $W$ mass, as verified in the Monte Carlo by lowering $m_W$ resulting in the expected diffractive behavior shown by the dotted curve in Fig. \[plotmaxgapsize\]. \ Diffractive events can be defined experimentally in two different ways: by a rapidity gap or by a leading (anti)proton. (Given the symmetry between proton and antiproton beams at the Tevatron, we usually mean either proton or antiproton when speaking of a leading proton.) The two methods are related, since kinematics requires an event with a leading proton to also have a gap. This has been explicitly investigated with our Monte Carlo model resulting in Fig. \[gapsize\]. Events with a very large gap do typically have a leading proton. At Tevatron energies, however, gaps of substantial size are kinematically allowed also for protons with not so high $x_F$ as shown by the nontrivial correlations in Fig. \[gapsize\]b and c. This calls for some caution when comparing results based on these two definitions of diffractive events. Irrespectively of this warning, comparing Fig. \[gapsize\]b and c shows the effect of the SCI model to produce more events with large-$x_F$ protons and large gaps. When the leading proton is at a low $x_F$ there may be another leading system of small invariant mass, in particular a large-$x_F$ neutron. In addition to the events included in Fig. \[gapsize\]b and c, there is a substantial amount of gap events without a proton, but with other leading particles. Such events, which are natural products of the Monte Carlo model, must be included when using a gap definition of diffraction. The diffractive rates obtained with a gap definition are therefore usually larger than those obtained with a leading proton definition. The experimental results on diffractive hard scattering processes have mainly been presented as relative rates, [i.e.]{}, the cross section for a diffractive process divided by the total cross section for the same hard process. We denote this diffractive ratio by $R_{\mathrm{hard}}$, where ‘hard’ stands for the relevant hard subprocess. The first experimental analyses, [e.g.]{}[@CDF-W; @CDF-B; @CDF-JJ], used a gap definition of diffraction. This is, however, essentially equivalent to requiring a leading proton with $x_F>0.9$, such that the diffractive ratio $R_{\mathrm{hard}}$ can be expressed as $$R_{\mathrm{hard}} = \frac{1}{\sigma_{\mathrm{hard}}^{\mathrm{tot}}} \int_{{x_F}_{\mathrm{min}}}^1 dx_F \, \frac{d\sigma_{\mathrm{hard}}}{dx_F}. \label{RW}$$ where ${x_F}_{\mathrm{min}}$ is the minimum leading proton $x_F$ for an event to be considered diffractive. The values of $R_{\mathrm{hard}}$ in Table \[tab-gapratios\] were obtained with ${x_F}_{\mathrm{min}}=0.9$, corresponding to the experimental analyses. This is also in accordance with the conventional definition of diffraction in the Regge approach and comparisons with simulations of Pomeron exchange at $x_{{I\!\!P}}=1-x_F < 0.1$ were made, leading to the problems discussed in Section \[sec-pomeron\]. The variation of $R_{\mathrm{hard}}$ with ${x_F}_{\mathrm{min}}$ in the models will be discussed below. Some more recent CDF analyses [@CDF-AP; @CDF-DPE] could define diffraction through leading antiprotons observed in Roman pot detectors. This provided additional information, on $x_{{I\!\!P}}$ and the momentum fraction $x$ of the struck parton in the incoming antiproton, making the results less inclusive. This gives additional handles to test the models, as will be discussed below. Our results presented below were obtained by Monte Carlo simulations using [<span style="font-variant:small-caps;">pythia</span>]{} version 5.7. As a reference, called ‘default’, we use standard [<span style="font-variant:small-caps;">pythia</span>]{} with all parameters and switches at their default values. The parton distributions CTEQ3L [@CTEQ3] were used for the simulations with default [<span style="font-variant:small-caps;">pythia</span>]{} and with the SCI model, and CTEQ4L [@CTEQ4] were used with GAL. There is a slight variation of the results depending on this choice, see Table \[tab-RW\] and the discussion in section \[Wprod\]. The SCI and GAL models are simulated using added subroutines as described in Section \[sec-model\]. This includes the improved procedures for beam particle remnants, with the treatment of sea quark interactions and small mass string systems. In order to compare with the Pomeron model, we have also included results from simulations using the [<span style="font-variant:small-caps;">pompyt</span>]{} program (version 2.6) [@Pompyt]. Here, the Donnachie-Landshoff (DL) [@DLpomeron] Pomeron flux and the Gehrmann-Stirling (GS) [@GSpomeron] Pomeron structure functions were used. The GS parametrizations have two variants, referred to as model I and II. In short, model I describes the Pomeron as a hadronic system of quarks and gluons. Model II has, apart from this ‘resolved’ component, also a ‘direct’ component with a photon-Pomeron coupling. Both models have been tuned to describe HERA data. We have mainly used model I, as this describes the Tevatron data better, but we have also tested model II. Still other parametrizations of the Pomeron structure function are available, but using them will not change the results in an essential way. After having defined our models and described the general framework, we can now turn to the specific diffractive hard scattering processes. Diffractive W production {#Wprod} ------------------------ Diffractive $W$ production has been experimentally observed at the Tevatron by the CDF collaboration at a relative rate $R_W^{\mathrm{CDF}}=(1.15\pm 0.55)\%$ [@CDF-W]. Only leptonic decays of the $W$’s are considered here, since they are easier to reconstruct due to a lower background. The interpretation of diffractive $W$ production in the soft color exchange models is illustrated in Fig. \[fig:Wsci\]. In order to have a leading proton, it is necessary to have a gluon-initiated process, [i.e.]{}, taking a gluon from a beam (anti)proton. The color octet charge of the remnant can then be neutralized by a soft gluon exchange between this remnant and some other color charge in the event. This may be described in a parton basis as in the SCI model or in a string basis as in the GAL model. In any case, this gives the possibility to produce a small mass leading system, [e.g.]{}, a single proton, separated by a rapidity gap to the central system containing the $W$. In order to produce a leading proton, a parton with not too large energy-momentum fraction $x$ from one beam proton will interact with a parton from the other beam particle. Because of the small energy loss ($x$) from the leading proton and the large mass of the $W$, the parton from the other beam hadron will have to be quite energetic and is therefore typically a valence quark. This also implies that the $W$ predominantly emerges in the hemisphere opposite to that of the gap or the leading proton. These effects in our Monte Carlo simulation produce the same correlations of rapidities and $W$ charge as observed by CDF and used in the experimental analysis. In Pomeron models, on the other hand, $W$ production can be described, as originally proposed and calculated in [@Bruni+I], by the processes in Fig. \[pomW\]. As discussed above, one folds a Pomeron flux from one of the initial hadrons with a hard Pomeron-proton collision using parton densities in the Pomeron. Since the charge-rapidity correlations are essentially of kinematical origin, they also appear in this model. The main results of our $W$ simulations are shown in Fig. \[plotW\]b which shows that the SCI and GAL models reproduce the rate of diffractive $W$ as observed by CDF, whereas the Pomeron model result is far above (about a factor six) and standard [<span style="font-variant:small-caps;">pythia</span>]{} is much below the measured value. Here one should remember that the SCI and GAL models are [*not*]{} adjusted to these data, but have an absolute normalization which is fixed by the rate of rapidity gaps in DIS at HERA, as discussed in Section \[sec-model\]. This ability to reproduce these CDF data is related to the increased rate of high-$x_F$ protons as shown in Fig. \[plotW\]a. The Pomeron model, which is only applicable for $x_F\gtrsim 0.9$, overshoots the Tevatron diffractive $W$ rate if taken directly over from its tuning to diffractive HERA data. As discussed in Section \[sec-pomeron\], this problem can be cured by introducing some essential modification of the Pomeron model. Since the Pomeron model only applies in a limited $x_F$ range, the curve in Fig. \[plotW\]a cannot be normalized to unit area and is instead normalized based on its absolute cross in relation to the other models. Concerning this Pomeron model curve, one should note that it is quite flat. The basic $1/(1-x_F)$ dependence in the Pomeron model is here strongly distorted by the kinematical suppression for $x_F\to 1$ imposed by the $W$ mass. This implies that the cross section for diffractive $W$, as opposed to inclusive single diffraction, is quite sensitive to the cutoff $x_{F\, min}$. \ \ As pointed out, the rate of diffractive events may depend on whether they are defined in terms of a gap or a leading proton. This CDF result is based on the observation of a gap, but is essentially equivalent to requiring a leading proton with $x_F>0.9$. The mild, essentially linear variation of the SCI and GAL model results with $x_{F\, min}$ shown in Fig. \[plotW\]b, demonstrate that the exact $x_{F\, min}$ value is not crucial; in particular in view of the presently rather large error bars on the experimental ratio $R_W$. It is now interesting to investigate how variations in the models affect the results. To start with, we find that there is almost no discernible difference between the results from the two variants of the SCI model, [i.e.]{}, the one which allows color reconnections between any pair of partons and the one which requires one of the interacting partons to be in the remnant. This is because practically all rapidity gaps come from reconnections involving a parton in the remnant, representing the color background field. Color exchanges between the more centrally produced partons from the hard scattering do not give rise to large gaps between the central and the leading systems. Consequently, in all simulations in this paper, the standard version of SCI is taken as the one involving at least one parton in the remnant. One may ask how the diffractive ratio depends on the soft color exchange probability $P$ for the SCI model or $P_0$ for the GAL model. Qualitatively, if $P$ is large there will be more color reconnections, increasing the rate of gap events. However, an increasing number of color exchanges may also destroy gaps, through the possibility of reconnecting strings ‘across’ an already formed gap. This behavior is indeed found in the simulation and shown in Fig. \[plotW\]c obtained using the SCI model. As can be seen, there is only a quite weak dependence on $P$ as long as it does not approach its limiting values 0 or 1. In accordance with earlier studies [@unified], we take $P=0.5$ as our value for the SCI probability. For the GAL model, we use the original value $P_0=0.1$ [@GAL] as discussed in Section \[sec-GAL\]. The improved model for sea quark treatment, which assigns some dynamics to the sea quark partner in the case of scattering off a sea quark in the proton, should be of relevance. The reason is that a sea quark may be viewed as coming from $g\to q\bar{q}$ and thereby be like a gluon-induced process giving diffractive $W$ production as discussed above. As described in Section \[sec-remnant\], there are two variants of this sea quark treatment, SQT1 and SQT2. Using one or the other, or neglecting this sea quark treatment, gives somewhat different diffractive $W$ rates as shown in Table \[tab-RW\], but the results are all within the experimental error. It could be argued that SQT2 is more correct since it uses sea quark parton distributions to assign momenta, while SQT1 uses pQCD parton splitting functions in the nonperturbative region. Together with the fact that SQT2 gives slightly better agreement with data, this is the preferred version that we use as standard. The multiple interaction model discussed in Section \[sec-remnant\] has an important influence on the results. It is clear that in Pomeron models, additional parton-parton scatterings in an event would destroy any gaps, and therefore the existence of a gap would signify that there were no such extra scatterings in the event. In contrast, multiple interactions do not exclude gaps in the soft color interaction models. In fact, the gap ratios shown here for the SCI and GAL models include multiple interactions. The gap rate does depend on the amount of multiple interactions, but switching them off only leads to the somewhat increased gap rate shown in Table \[tab-RW\] which is still consistent with the observed $R_W$. Therefore, at this stage of accuracy, the multiple interactions do not present a problem. They must, of course, be included at some level in order to reproduce various characteristics of the underlying event. As discussed above, we have found a slightly increased value of the basic transverse momentum cutoff parameter for these additional parton-parton scatterings to avoid double counting of soft phenomena. Model $R_W$ (%) ------------------------------------------------------------------ ------------- [**SCI**]{} incl. SQT2, MI [**1.2**]{} ” changing to SQT1 1.7 ” switching off SQT 0.9 ” switching off MI 1.7 ” switching to CTEQ4L 1.0 [**GAL**]{} [**0.8**]{} ” switching to CTEQ3L 1.0 [**Pompyt**]{} GS I [**7.2**]{} ” GS II 11.6 Default [<span style="font-variant:small-caps;">pythia</span>]{} 0.1 : Ratio $R_W$ of diffractive $W$ production obtained from different variations of the models: sea quark treatment (SQT), multiple interactions (MI), parametrization of parton densities in the proton (CTEQ) and in the Pomeron (GS). Results from standard version models are shown in boldface.[]{data-label="tab-RW"} Finally, we have checked the dependence on the choice of parton distribution parametrizations, and we find that the diffractive ratios are slightly smaller (about 15–20%) with CTEQ4L than with CTEQ3L. These variations of the SCI and GAL models result in changes of the diffractive ratios (Table \[tab-RW\]) which illustrate the uncertainty of the models. We note that these variations are all within the errors of the present experimental results. In contrast to the soft color exchange models it is, as already emphasized, not possible to take the Pomeron model directly from HERA and use it to reproduce the Tevatron data. The best result is achieved using model I for the parton densities in the Pomeron, which results in a diffractive $W$ ratio which is six times too large, whereas model II gives a rate about ten times too large (see Table \[tab-RW\]). Other parametrizations of the Pomeron parton densities exist, but using them will not essentially change this disagreement with Tevatron data which is also compatible with other investigations [@Alvero]. We find, however, that some general characteristics such as the $\eta$ distributions of particles in an event, are the same for [<span style="font-variant:small-caps;">pythia</span>]{} with SCI and for [<span style="font-variant:small-caps;">pompyt</span>]{}. At this point, having examined variations of the models, we make an important observation: the measurement of diffractive $W$ production was only made for the leptonic decay channel $W \to e \nu$. When the $W$ instead decays to quarks, these quarks must also be included in the soft color interactions since, given the short $W$ lifetime, they are produced in a very small space-time region embedded in the color background field of the colliding hadrons. This gives the possibility that reconnections with these decay quarks rearrange the color structure of the event and destroy rapidity gaps. Therefore, the probability for a diffractive event can be lower for hadronic than for leptonic $W$ decays. This effect could be seen as an [*apparent*]{} change in the branching ratios of $W$ decays, so that in a diffractive sample of events there will be a higher branching ratio to leptons and a lower branching ratio to hadrons than what is observed in the total, inclusive sample. In Pomeron models on the other hand, no such effect should be present since the hard scattering is independent of the gap-formation process. This has been confirmed by simulations with [<span style="font-variant:small-caps;">pompyt</span>]{}. The real branching ratios for $W$ are $B(W \to l\nu) = 32.2 \%$ and $B(W \to q\bar{q}') = 67.8 \%$, and thus $B(W \to l\nu) / B(W \to q\bar{q}') = 0.475$. Now, using the SCI model, but with both the leptonic and the hadronic decay channels of the $W$ included, we find $$\left. \frac{B^{SCI}(W \to l\nu)}{B^{SCI}(W \to q\bar{q}')} \right|_{\mathrm{diffractive}} = \frac{39 \%}{61 \%} = 0.63 > 0.475. \label{BR}$$ Thus there are indeed different apparent branching ratios in the biased diffractive $W$ sample. This is also reflected in the diffractive ratio $R_W$, which drops from 1.2 to 1.0 when including hadronic $W$ decays. Naively we would expect the same effect in the GAL model, but this is not observed in our simulations. The reason is that for reconnections with the decay products of the $W$ the price in terms of increased string area is too large. The quarks from the $W$ decay will form a separate color singlet system, which is central in rapidity. Reconnecting this string with a string from a more noncentral parton will typically mean an increase in area, which is strongly suppressed in the model. Therefore we do not observe any shifted apparent branching ratios in the GAL model, only in the SCI model. The CDF paper [@CDF-W] also contains a study of the jet structure of diffractive $W$ production. Only 8 out of 34 diffractive events were observed to have a jet giving the ratio 24% , but the relative error is large because of the low statistics. This fraction was used to estimate the quark and gluon content of the Pomeron, and it was found that the measurement was consistent with a quark dominated Pomeron (although the measured value of $R_W$ favors a gluonic Pomeron). An SCI model interpretation is also quite in order, since we have verified that it can reproduce this measured rate of jets in diffractive $W$ events. Here $W$ production with pQCD corrections in terms of next-to-leading order tree level matrix elements and parton showers was employed, however, the description turns out to be equally good using only LO matrix elements and parton showers. Before moving on to other processes, we will briefly consider diffractive $Z$ production, as this should be qualitatively similar to the $W$ case. This has not been observed experimentally yet since the cross section and branching ratio to leptons are both smaller for $Z$ than for $W$. We predict diffractive ratios $R_Z$ that are smaller than the corresponding $R_W$ (see Table \[tab-gapratios\]): we get $(R_Z/R_W)_{\mathrm{SCI}}=0.83$ and $(R_Z/R_W)_{\mathrm{GAL}}=0.64$. This difference is essentially accounted for by the mass difference; it takes more energy to produce a $Z$, so there will be less energy available for the leading proton, which will on average have a lower $x_F$. Thus $R_Z$ will be lower than $R_W$. We have checked this by a simulation where the $Z$ mass was set equal to the $W$ mass, resulting in a ratio consistent with unity for SCI. We find similar results in the Pomeron model, as expected based on general kinematical mass effects. In the GAL model, however, the suppression of $Z$ compared to $W$ is larger. Simulating with the GAL model and the $Z$ mass changed to $m_W$, we get $(R_Z^\prime/R_W)_{\mathrm{GAL}}=0.8$. Hence the larger mass is not the whole reason. The difference between the SCI model and the GAL model is larger for $Z$ than it is for $W$. This gives an indication that the dependence on the hard scale is different between the two models, as will be discussed in more detail later. To summarize this subsection, we have demonstrated that the SCI and GAL models can indeed reproduce experimental data on diffractive $W$ production, while the Pomeron model cannot without modifications. We have also studied some variations of the models, and found that the ‘best model’ is the same model as the one used to reproduce diffractive HERA data, namely, SCI or GAL together with the new model for sea quark treatment (SQT2), but here also with the multiple interaction model necessary for [[$p\bar{p}$]{}]{} collisions. We have also pointed out some differences between the SCI, GAL and Pomeron models, which could be used to experimentally discriminate between them. An interesting such observable is the phenomenon of different apparent $W$ branching ratios in diffractive events. Diffractive beauty production ----------------------------- CDF has also measured diffractive [$b\bar{b}$]{} production in terms of open beauty in events with rapidity gaps, defined in the same way as in the $W$ case. The resulting ratio of diffractive beauty production is $R_{b\bar b}=(0.62\pm 0.25)\%$ [@CDF-B]. In contrast to $W$ production, the description of heavy quark production needs to include also higher order diagrams. In leading order (LO) pQCD heavy-quark production occurs through $gg\to b\bar{b}$ and $q\bar{q}\to b\bar{b}$, Fig. \[bbdiag\]ab. However, higher order processes involving gluon splitting $g\to b\bar{b}$ are important. For example, the process $gg\to gb\bar{b}$ illustrated in Fig. \[bbdiag\]c gives a large contribution because it is an $\alpha_s$ correction to the large cross section for gluon scattering ($gg\to gg$). Matrix elements with explicit heavy-quark mass are available up to next-to-leading order (NLO), but still higher orders may contribute at collider energies. These can only be taken into account through the parton shower (PS) approach which, although being approximate, has the advantage of resumming leading logarithms to all orders. We therefore investigate beauty production both in leading order and in higher orders (HO) using [<span style="font-variant:small-caps;">pythia</span>]{}. The LO matrix elements include the $b$-quark mass $m_b=4.5$ GeV. The higher orders are obtained through $g\to b\bar{b}$ in the parton showers added to all LO $2\to 2$ QCD processes, except those producing $b\bar{b}$. The LO and HO contributions can then be added with their respective cross section weights. The higher orders are tree level diagrams, whereas virtual corrections are not taken into account in this approximation. \[bb\_ptslope\] The diffractive ratios obtained in this way are listed in Table \[tab-gapratios\] and are plotted as functions of ${x_F}_{\mathrm{min}}$ in Fig. \[bb\_R\]. The separate LO and HO contributions in the figure show that the LO gives a larger gap ratio, but the HO gives a larger contribution to the total cross section. In contrast to $W$ production, GAL here gives a larger gap ratio than SCI and is not in very good agreement with the experimental value. The SCI model gives excellent agreement as usual, whereas the Pomeron model is a factor 15 too large compared to the measurement. Here we again note that the GAL model has a different energy dependence, larger ratios for smaller hard scales and smaller ratios for larger hard scales, as compared to the SCI model. This was already seen for $Z$ production and we here anticipate the results from Sections \[sec-dij\] and \[sec-jpsi\] and observe that the same holds for diffractive dijets and $J/\psi$. The experimental observation of $B$ mesons is based on electrons from their decay. One requires these electrons to have a transverse momentum larger than $p_{\perp \mathrm{min}}^e=9.5$ GeV. This is an important point, since we find that the diffractive ratio $R_{b\bar b}$ depends on the value of $p_{\perp \mathrm{min}}^e$, as shown in Fig. \[bb\_ptslope\]. The three SCI curves shown (LO, HO, and total) all have the same slopes. The Pomeron curve also has the same slope, but as the absolute normalization is a factor 15 too large, it has been correspondingly rescaled in the figure. The GAL curve is at the same level as the SCI model for small $p_{\perp \mathrm{min}}^e$, but its slope is smaller such that it overshoots the experimental data point. This different slope of the GAL model is again a manifestation of its different scale dependence. This dependence on $p_{\perp \mathrm{min}}^e$, which is effectively a requirement on the transverse momentum of the $b(\bar b)$ quark, can arise from an interplay of several effects. First, a higher $p_\perp$ requires larger momentum fractions taken from the colliding protons, which means less energy left for leading protons. Second, with higher $p_\perp$ the incoming and outgoing partons radiate more, thus filling gaps. It is not a priori clear how the underlying event affects this, but we have found that multiple interactions do not change the slope of the curves, only the normalization. Given these effects, one should realize that the measured diffractive beauty ratio might be biased towards a lower value given the requirement of a high-$p_\perp$ electron. Diffractive dijet production {#sec-dij} ---------------------------- The process originally considered when introducing the concept of diffractive hard scattering was jet production in high energy hadronic interactions [@IS]. The transverse momentum ($p_\perp$) or transverse energy ($E_T$) of the jets provides the hard scale necessary for the study of diffraction based on a firm underlying parton picture. The experimental discovery by UA8 of hard scattering phenomena in diffractive scattering was also in terms of events with a leading proton and high-$p_\perp$ jets at the CERN [$p\bar{p}$]{} collider [@UA8-1]. Additional UA8 data [@UA8-2] gave important results, which were mainly interpreted in terms of the Pomeron model resulting in hard parton density distributions in the Pomeron. Diffractive dijet production has also been observed by the CDF and D[Ø]{} experiments at the Tevatron. Initially, CDF observed [@CDF-JJ] events with high transverse energy jets ($E_T > 20$ GeV) and a gap in the rapidity region opposite to the dijets in $p\bar p$ collisions at $\sqrt s = 1800$ GeV, while D[Ø]{} has reported [@D0-JJ] observation of events with a similar topology ($E_T > 12$ GeV and $E_T > 15$ GeV) at the two center of mass energies $\sqrt{s}=630$ and 1800 GeV, respectively. The analyses are quite analogous to that for diffractive $W$ discussed above, with the observed gap equivalent to a leading proton with $x_F>0.9$. This kind of events occurs naturally in the soft color exchange models as illustrated in Fig. \[pp-dijets\]. Applying the SCI and GAL models to jet production in [<span style="font-variant:small-caps;">pythia</span>]{}, described by leading order QCD $2\to 2$ scattering processes with parton showers added for higher orders, results in a good description of the observed diffractive dijet ratios $R_{jj}$, as shown in Table \[tab-gapratios\] and Fig. \[figjj\]. We emphasize that it is exactly the same SCI and GAL models as used for diffractive $W$ and $b\bar{b}$ above, only the hard subprocess has been changed. We have investigated the dependence of the results on the reconnection probability $P$, $p_\perp^{\mathrm{min}}$ in the multiple interaction model, different aspects of the sea quark treatment, and arrived at the same conclusions as for the $W$ case in Section \[Wprod\]. The other models cannot reproduce the measured $R_{jj}$; default [<span style="font-variant:small-caps;">pythia</span>]{} is far below data (Fig. \[figjj\]) and the Pomeron model is above (not shown explicitly). CDF has recently presented a new sample of diffractive dijet events, where the signature of diffraction is a leading antiproton observed in Roman pot detectors [@CDF-AP]. The reported results are based on events with antiprotons in the range $0.905< x_F < 0.965$ and two jets with $E_T > 7$ GeV. Since this offers a new testing ground for the models, we have investigated the production of dijet events with a leading antiproton and compared the results of the models with the observed CDF data. We note that CDF uses the variable $\xi$ to denote the antiproton fractional momentum loss, which is related by $x_F=1-\xi$ to the variable $x_F$ consistently used in this paper. In Fig. \[eta\] we compare characteristic features of the dijet systems in our models and in data. The data show that the $E_T$ distribution of the diffractive sample falls steeper than that of the nondiffractive sample. This behavior is present in both the SCI and GAL models, although the exact shape is not very well reproduced. This may be related to a mismatch between our jet reconstruction procedure and the experimental one, or [<span style="font-variant:small-caps;">pythia</span>]{} being limited to leading order matrix elements without next-to-leading order corrections for the basic jet cross section. The rapidity distribution of the jets is in the diffractive sample shifted into the hemisphere opposite to the leading antiproton, a characteristic which is well described by both models, see Fig. \[eta\]b for the case of SCI. CDF has furthermore extracted the ratio of diffractive to nondiffractive dijet events as a function of the momentum fraction $x$ of struck parton in the antiproton. This $x$ can be evaluated from the transverse energy and rapidity of the jets using the relation $$x=\frac{1}{\sqrt{s}} \sum_{j=1}^{\mathrm{2\; or\; 3}} E_T^j ~ e^{-\eta^j} \label{x-antip}$$ where the sum includes the two leading jets, plus a third jet if it has $E_T > 5$ GeV. In Fig. \[xf\]a we compare their data with the results from the models. The Pomeron model overshoots the data by an order of magnitude, while default [<span style="font-variant:small-caps;">pythia</span>]{} is too low by a similar factor. The soft color exchange models give a fairly correct description, reproducing the overall behavior and giving the correct total ratio. Going into finer details, we note that as $x_F$ approaches unity ($x_F>0.965$), the slope of this ratio with $x$ becomes more steep in the models (as seen in Fig. \[xf\]a, where this contribution is included in the full curve). This behavior seems not to be quite in accord with CDF results, which indicate a constant slope as $x_F$ varies [@CDF-AP]. This dependence in the model is mainly due to the details of the remnant treatment, which affect the steepness of the ratio. The measurement of the leading antiproton provides a test of exactly how the beam particle remnant is handled in the model. In order to explore this we have investigated the effects of the alternative remnant handling procedures available in [<span style="font-variant:small-caps;">pythia</span>]{}. Since diffractive events arise dominantly in the SCI and GAL models from gluon-induced processes, the remnant typically contains the three valence quarks. As described in Section \[sec-remnant\], this remnant is split into a quark and a diquark taking energy-momentum fractions $\chi$ and $1-\chi$, respectively. The probability distribution ${\cal P}(\chi)$ cannot be deduced from first principles, but is given by some parametrization. As our standard choice we use ${\cal P}(\chi) \sim (1-\chi)$, giving in the mean one third of the remnant energy-momentum to the quark and two thirds to the diquark. We have also tried other parametrizations, in particular the parton distribution-like form ${\cal P}(\chi) \sim \chi ^{-1} (1-\chi)^3$. The antiproton $x_F$ spectra obtained are shown in Fig. \[xf\]b. The SCI and GAL results are quite similar, but both depend significantly on this remnant treatment. Of course, the main effect in Fig. \[xf\]b is the large increase of antiprotons at large $x_F$ when going from default [<span style="font-variant:small-caps;">pythia</span>]{} to the SCI or GAL model resulting in an overall description of the diffractive rates. The finer details of the diffractive events will, however, depend on the details in the modeling of the remnant. Summarizing the investigation of diffractive dijets, the soft exchange models do a very good job in reproducing the overall ratios of diffractive to nondiffractive dijet production. They also give a good agreement with the kinematical distributions observed for this type of events. However, some detailed results depend on the treatment of the proton remnant in the Monte Carlo. The new diffractive Tevatron data based on a leading antiproton provide additional tests of the models. DPE – ‘Double leading Proton Events’ {#sec-dpe} ==================================== Related to single diffraction are events with [*two*]{} leading protons with associated gaps. These protons are at the opposite extremes in phase space, [i.e.]{}, at $x_F\to +1$ and $x_F\to -1$, and their associated gaps are in the forward and backward rapidity regions, respectively. In the Regge framework these events are described by a process where the two beam protons each emit a Pomeron. These Pomerons then interact, producing a central system which is separated in rapidity from the two quasi-elastically scattered beam protons. This class of events has therefore been called double Pomeron exchange (DPE). This nomenclature is, however, based on an interpretation in a certain model and it would be better to classify them independently of any model and only based on their experimental signature. In order to keep the well established abbreviation DPE, we propose to call them ‘Double leading Proton Events’. These DPE events occur naturally in the soft color interaction models, where the final color string topology may also produce two rapidity gaps as illustrated in Fig. \[pp-dijets\]c. With one single mechanism for soft color exchanges, different final states will emerge and can be classified in the same way as experimentally observed events: no-gap events, single diffractive events with one gap or a leading proton, or DPE events with two gaps or two leading protons. It is therefore straightforward to extract such events from the Monte Carlo simulations based on the SCI and GAL models. Both CDF [@CDF-DP] and D[Ø]{} [@D0-DP] have observed such DPE events having a dijet system in the central region. They were first identified by two rapidity gaps, one in the forward and one in the backward region. The ratio of two-gap events to one-gap events observed by CDF is well reproduced by the SCI model, as can be seen in Table \[tab-gapratios\]. Although D[Ø]{} has not made such a ratio available, the expectation from the models would be of the same magnitude ($\sim 0.2 \%$). Recently, CDF has reported DPE dijet events defined by a leading antiproton and a rapidity gap on the opposite proton side [@CDF-DPE]. In the data set of single diffractive dijet events with leading antiproton, they have observed a subset with a rapidity gap on the outgoing proton side at a rate given in Table \[tab-dpe\]. By studying the kinematical correlations between a leading particle and the associated gap, CDF describes the DPE events in terms of a leading proton with $0.97<x_F<0.99$, although no such proton is actually observed. $\tilde{R}^{DPE}_{SD}\;\;\; [\% ]$  $\sigma^{DPE}\;\;\; [nb]$ ---------------- ------------------------------------- --------------------------- CDF [@CDF-DPE] $0.80 \pm 0.26$ $43.6\pm 4.4 \pm 21.6 $ SCI $0.54\; \pm 0.05$ $5$ – $25$  GAL $0.44\; \pm 0.05$ $6$ – $40$  : Rates of DPE dijet events in data compared to SCI and GAL models; relative to single diffractive dijet events and absolute cross section.[]{data-label="tab-dpe"} \ \ \ Table \[tab-dpe\] also contains the results of the SCI and GAL models. Applying the leading proton condition strictly results in too low cross sections, but when instead using the more generous gap definition the models reproduce the measured cross section within the errors. This difference between the two approaches illustrates our warning above that leading particle and gap definitions need not be exactly equivalent. In particular, experimental smearing effects may become important when approaching the phase space limit $x_F\to 1$. The absolute cross section is more sensitive to details in the model, such as the remnant treatment and the previously mentioned lack of NLO corrections in [<span style="font-variant:small-caps;">pythia</span>]{} may also play a role. With the uncertainties in both data and models in mind, one may conclude that the models give essentially the correct cross section for DPE events. This discussion illustrates the difficulty to exactly reproduce data in a Monte Carlo model which is ambitious enough to attempt to describe the detailed dynamics of nonperturbative QCD processes. This problem is accentuated for DPE events, where the gaps and leading particles in both the forward and backward region mean a stronger dependence on the details of the remnant treatment. Using the remnant splitting ${\cal P}(\chi) \sim (1-\chi)$ based on simple counting rules, the ratio of DPE to SD events and of SD to ND events gets closer to the measured values than other options for ${\cal P}(\chi)$ provided in [<span style="font-variant:small-caps;">pythia</span>]{}. The $x$-dependence of these ratios are shown in Fig. \[plot\_r\_dpe\]. The curve for DPE/SD is obtained with the same leading proton requirement as CDF derived from the observed rapidity gap. The SD/ND ratio differs from the one in Fig. \[xf\]a by being calculated per unit $x_F^{\bar p}$, which not only changes the normalization but also the slope. The main features of the data are described by the SCI model, but there are discrepancies related to the mentioned problems of the remnant treatment. The main result in Fig. \[plot\_r\_dpe\] is, however, the breakdown of diffractive factorization, which is quantified by the ratio of SD/ND to DPE/SD (=$0.19\pm 0.07$) being so clearly different from unity [@CDF-DPE]. This important result also emerges from the models. After this discussion of the rates of DPE events, we turn to some of their internal properties. Fig. \[dpe\_jets\] shows some essentials of the jets in DPE events compared to inclusive and single diffractive events. Higher jet multiplicities are clearly suppressed in DPE events compared to the inclusive sample. The slopes of the jet-$E_T$ distributions have a tendency to increase from nondiffractive to single diffractive to DPE events. This can be understood by the limitations on the energy in the hard scattering subsystem due to leading particle effects. The rapidity distribution, which is symmetric around zero for nondiffractive events, is shifted when gap or leading proton conditions are applied on either side. All these features are qualitatively reproduced by the SCI and GAL models. Some discrepancies can, however, be found in the details. The $E_T$ distributions in the models seem to have somewhat too small slopes and higher jet multiplicities are not sufficiently suppressed in DPE events. These deficiencies may be due to a mismatch between data and model regarding the jet reconstruction or the lack of NLO corrections in the hard scattering matrix elements used in [<span style="font-variant:small-caps;">pythia</span>]{}. We have shown in this section how soft color exchange models go beyond their original purpose and explain more than just single diffraction; thus giving a natural description of diffractive events with two gaps or corresponding leading particles. The two leading particles imply an increased sensitivity to the remnant treatment, providing possibilities to test and improve the details of the Monte Carlo model. Diffractive $J/\psi$ production {#sec-jpsi} =============================== In the last section it was shown that the soft color interactions can produce two rapidity gaps in the same event and thereby provide a description of DPE. In this section we will demonstrate an even more striking effect where the soft color interactions give rise to two different phenomena in the same event, namely both a rapidity gap and turning a color octet [$c\bar{c}$]{} pair into a singlet giving a $J/\psi$. The results of our models are predictions to be tested against the data that should appear soon given the very recent observation by CDF of such diffractive $J/\psi$ events. It will be a highly nontrivial result if both the gap formation and the $J/\psi$ production can be well explained with one and the same model for non-pQCD dynamics. To start with, let us leave diffraction aside and concentrate on the $J/\psi$ production. The main point here is that the soft color interaction, [e.g.]{}, seen as a soft color-anticolor gluon exchange, can change the color charge of a [$c\bar{c}$]{} pair. A sizable fraction of the large cross section for pQCD production of color octet [$c\bar{c}$]{} pairs can then be turned into color singlet [$c\bar{c}$]{}. These will form onium states when their invariant mass is below the threshold for open charm production. It is a remarkable fact [@SCI-onium] that exactly the same SCI model that was used above, reproduces the observed cross sections of high-$p_\perp$ charmonium and bottomonium in [$p\bar{p}$]{} at the Tevatron. Since these cross sections are factors of ten larger than the prediction of conventional pQCD in terms of the color singlet model, where the [$c\bar{c}$]{} is produced in a singlet state, they need a radically new explanation. The production of charmonium states in fixed target hadronic interactions at different energies can also be described by these kinds of soft color interaction models, as demonstrated in [@Cristiano]. Furthermore, elastic and inelastic photoproduction of $J/\psi$ at HERA has been investigated from the perspective of soft color exchanges [@johan-quarkonium]. Although SCI and GAL show good agreement with data for the energy dependence of the cross section, the normalization is uncertain since these models are based on leading order matrix elements. The results are, therefore, sensitive to the choice of factorization and renormalization scale, and in the elastic case, the treatment of the proton remnant. Given this success of soft color interaction models to describe inclusive heavy quarkonium production, we now turn to diffractive $J/\psi$ production at the Tevatron. The predictions of the SCI and GAL models are shown in Fig. \[diffJpsi\]. The ratio of diffractive to nondiffractive $J/\psi$ events is in the range 1–2%, depending on $p_{\bot}$ and $\eta$ of the $J/\psi$. This predicted ratio seems to be in agreement with the recent preliminary CDF result [@Goulianos] of $(0.64 \pm 0.12)/{\cal A}$, where ${\cal A} \sim 0.4$ is an estimated rapidity gap acceptance. These diffractive events are experimentally defined as events with a rapidity gap and we have performed the analysis similarly to the aforementioned hard processes with a rapidity gap. For production of $c\bar c$ with appropriate invariant mass to form $J/\psi$, we found that higher order contributions are very important, which was also demonstrated in [@SCI-onium]. The leading order production through $gg \rightarrow c\bar{c}$ and $q\bar{q} \rightarrow c\bar{c}$ (Fig. \[bbdiag\] a,b) are included through massive matrix elements, while the higher order tree level contributions are taken into account approximately through the parton shower approach (main contribution in Fig. \[bbdiag\]c). LO and HO give the same ratio of diffractive to nondiffractive $J/\psi$ when considered independently, but the HO mechanism gives a higher absolute cross section and therefore dominates the diffractive $J/\psi$ events. To conclude, we find that the ratio of diffractive to nondiffractive $J/\psi$ predicted in the SCI and GAL models seems to be in agreement with expectations based on recent preliminary experimental results. This shows that the same soft color interaction mechanism can be used to describe both gap formation and quarkonium production, even occurring in the same event! Conclusions {#sec-conclusions} =========== A proper understanding of nonperturbative QCD has not yet been possible based on rigorous theory. The development of phenomenological models is therefore a useful approach. By considering soft effects in hard scattering events one can have a firm basis in terms of a parton level process which can be calculated in perturbation theory. Below the cutoff for the perturbative treatment, further interactions occur abundantly because of the large coupling $\alpha_s$ at small scales. The problem is then to model these soft interactions properly. The soft interactions can have large effects on the hadronic final state. This was demonstrated in Fig. \[plotmaxgapsize\], where frequently occurring large rapidity gaps on the parton level were filled through the hadronization process resulting in a strong, exponential suppression of large gaps at the hadron level. Conventional hadronization models, like the Lund string model, have a substantial theoretical input and describe very well many aspects of the hadronic final states. Nevertheless, they are still not derived from fundamental QCD theory, but are of phenomenological character and depend on which data have been considered when constructing them. The models may therefore need the introduction of new aspects or new dynamics as other data or new observations are considered. The soft color interaction approach investigated in this paper is an example of such new dynamics. We have argued that these interactions are a natural part of the process in which bare perturbative partons are dressed into nonperturbative ones and of the formation of color flux tubes between them. In the SCI model this may be viewed as the perturbatively produced partons interacting softly with the color medium of the proton as they propagate through it. Interactions of a color charge with a color background field is a more general problem which has been investigated using other theoretical approaches and received increasing interest in recent years. Examples of effects considered are large $K$-factors in Drell-Yan processes and synchrotron radiation of soft photons [@Nachtmann] as well as diffractive DIS in a semiclassical model [@Buchmueller]. The new approach to diffraction in [@Dino-new] may also be possible to interpret in a soft color interaction scenario. Our phenomenological approach is formulated in terms of the SCI and GAL models which are added to the well-known Monte Carlo programs [<span style="font-variant:small-caps;">lepto</span>]{} and [<span style="font-variant:small-caps;">pythia</span>]{}. A new stage of soft color interactions is introduced after the conventional perturbative processes, described by matrix elements and parton showers, but before applying the standard Lund string hadronization model. The SCI model is formulated in a parton basis, with soft color exchange between quarks and gluons, whereas the GAL model is formulated in a string basis, with soft color exchange between strings. In both cases, this causes a change of the color string topology of the event such that another hadronic final state will result after hadronization. These fluctuations will sometimes result in a region where no string is stretched giving a rapidity gap after hadronization. In both models there is only one new parameter, giving the probability for such color exchanges. The value of this parameter is chosen such that the rate of diffractive rapidity gap events observed in DIS at HERA is reproduced. The main result of this paper is that the same soft color interaction models, using the same value for this single new parameter, give a good description of the single diffractive hard scattering phenomena observed at the Tevatron: $W$, dijets and beauty mesons. Also the observed rate of double leading proton events (DPE), conventionally interpreted as double Pomeron exchange, is well reproduced by the SCI and GAL models. Here, the same soft color interaction mechanism produces two leading protons with associated rapidity gaps in the same event and it is a nontrivial result that the correct rate of DPE events are produced. Another, even more striking effect of two observables in the same event being explained with the soft color interaction mechanism is diffractive $J/\psi$ production. Here, both a rapidity gap is produced and a color octet [$c\bar{c}$]{}pair is turned into a color singlet such that a charmonium state can be produced. As a result we have predicted a rate of diffractive $J/\psi$ production which seems to be in good agreement with the recent preliminary CDF result. Data on inclusive charmonium and bottomonium production (without gap requirements) are also reproduced, as demonstrated in [@SCI-onium] for the case of high-$p_\perp$ $J/\psi$, $\psi '$ and $\Upsilon$ at the Tevatron and in [@Cristiano] for $J/\psi$ and $\psi '$ production at fixed target energies. Diffractive events at the Tevatron were first obtained based on the observation of rapidity gaps. CDF has also obtained samples defined by measured leading antiprotons in their Roman pot detectors. Compared to the gap definition, this gives consistent results on diffractive rates, but provides additional information. We have used this to test details of the models, in particular the treatment of the hadron remnant which is poorly constrained from data. Here, one has to address issues like the treatment of a complex remnant containing several partons and the hadronization of systems with small invariant mass. Comparing the different diffractive hard scattering processes we find a general tendency that their ratio to the corresponding nondiffractive processes decreases with increasing scale ($m_{J/\psi}$, $m_{\perp \, b}$, $p_{\perp\, jet}$, $m_W$, $m_Z$) of the hard process. This behavior arises naturally in the models due to two effects. The first is the simple kinematical correlation that an increased hard scale requires a larger momentum fraction $x$ of the incoming parton, leaving less to the hadron remnant and thereby a reduced probability for a leading proton with large $x_F$. The second effect is more pQCD parton radiation which can populate rapidity regions such that no gap is formed. This decrease of the diffractive ratio with increasing hard scale is somewhat stronger in the GAL model than in the SCI model. This is related to the larger cutoff for parton showers in GAL, leaving less room for radiation at lower hard scales in particular. Furthermore, the interaction probability in GAL depends on the invariant masses of parton pairs, making string reconnections from high-$p_\perp$ partons more likely than from low-$p_\perp$ ones. The experimental measurements do not yet have high enough precision to provide any clear conclusions on this scale dependence. We have also compared the results of the Pomeron model to our models and to data. With Pomeron parton density parametrizations obtained from diffractive DIS at HERA, the Pomeron model gives diffractive rates at the Tevatron that are clearly too large. The problems of the Pomeron approach have been discussed together with possible modifications, [e.g.]{} of the Pomeron flux, to obtain the correct diffractive rates. There are, however, other more detailed observables that may be used to discriminate between the models. An example was here presented in terms of the SCI model giving different apparent branching ratios of the $W$ in the diffractive sample. A $q\bar{q}$ from the $W$ decay will take part in the soft color interactions and affect the probability for gap formation, whereas leptonic $W$ decays will not have this effect. This means that the sample of $W$ events with a gap requirement becomes biased to having more leptonic $W$ decays. So far, diffractively produced $W$’s have only been reconstructed through their leptonic decays. Future measurements of hadronic $W$ decays in diffractive events are required to explore this difference of the models regarding apparent branching ratios. We note that this effect will not be present for diffractive $J/\psi$ or beauty mesons, since their life times are long enough that their decay products will be produced outside the color background field of the primary interaction. New data from Run II at the Tevatron with increased luminosity can give valuable new information and higher precision diffractive data. 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--- abstract: '[**Abstract:**]{} The generalized Tsallis statistics produces a distribution function appropriate to describe the interior solar plasma, thought as a stellar polytrope, showing a tail depleted respect to the Maxwell-Boltzmann distribution and reduces to zero at energies greater than about $20 \, k_{_B} T$. The Tsallis statistics can theoretically support the distribution suggested in the past by Clayton and collaborators, which shows also a depleted tail, to explain the solar neutrino counting rate.' address: | Dipartimento di Fisica and INFM- Politecnico di Torino - Corso Duca degli Abruzzi 24, 10129 Torino, Italy\ Istituto Nazionale di Fisica Nucleare, Sezioni di Cagliari e di Torino author: - 'G. Kaniadakis, A. Lavagno and P. Quarati [^1]' title: Generalized Statistics and Solar Neutrinos --- By imposing a variational principle one can conveniently generalize both Boltzmann-Gibbs statistics and standard thermodynamics [@tsa; @cura]. The lack of adequacy of the Boltzmann entropy is related to the breakdown of Boltzmann-Gibbs statistics for systems with long-range interactions [@pla; @nob] such as, among others, the interior solar plasma, where the particles are exposed to many-body collisions and the energy available to a particular pair of particles is not defined [@cla2]. Even for a non perfect plasma, the ideal gas approximation is usually considered a good one. The Maxwell-Boltzmann (MB) distribution is believed to be highly correct and the many-body physics involved does not determine sensible deviations from the standard statistics used [@kip]. However, non-Maxewellian (flattened) distributions in plasmas heated by inverse bremsstrahlung (collisional) absorption of sufficiently strong electromagnetic fields have been predicted and recently measured [@mat; @erd; @zito]; in $D$ - $^3 He$ fusion plasma experiments, a nonthermal component of the ion distribution and the possible consequences on the nuclear rates have been investigated [@fis; @lap]. Among the distributions that deviate from the MB one, we pose our attention on distributions coming from generalized statistics. We give a very brief outline of the Tsallis generalization of thermodynamics and statistical physics [@tsa; @cura], suitable for describing systems with long-range interactions [@pla; @nob].\ If we have a system with $W$ microscopic states, each with probability $f_i \ge 0$ normalized as $$\sum_{i=1}^{W} f_i=1\ \ ,$$ then the entropy is given by $$S_q=\frac{k}{q-1} \sum_{i=1}^W f_i (1-f_i^{q-1})\ \ ,$$ where $k$ and $q$ are constants. In the limit as $q$ approaches unity, the well known expression $S_{1}=-k \sum_i f_i \log f_i$ is recovered and we can fix $k=k_{_B}$ (the Boltzmann constant), i.e. the Tsallis statistics reduces to the standard one as $q\rightarrow 1$. We have shown a generalization of the Tsallis statistics by using a kinetic approach, based on the Fokker-Planck equation, and we have given to the Tsallis parameter $q$ the meaning of a measure of the deviation from constant behavior of the diffusion coefficient $D(v)$, a quadratic function of the velocity $v$ [@ka1; @ka2; @ka3]. The Tsallis generalized statistics is actually widely used in many different physical problems (we send the reader to Ref.s [@bar; @levy] where many references on the different applications are quoted). One of the first problems where Tsallis statistics has been applied is that of stellar polytropes [@pla]. Nobre and Tsallis [@nob] and other authors have recently discussed the physical needs for departure from Boltzmann-Gibbs statistical mechanics and thermodynamics for gravitational like systems [@bog] and anomalous diffusion [@levy; @zan] (this last subject could be of great interest in the study of solar core, due to the well known problem of the diffusion of light elements like $Li$ and $Be$ [@ric]). Plastino and Plastino [@pla] have sought help from Tsallis entropy to find sensible distribution functions for stellar polytropes while that of MB gives unphysical distribution functions.\ They found a range of variability of $q$ from a relation between the polytropic index $n$ and $q$ deduced comparing two different but equivalent expressions of the distribution. We obtain a different relation between $n$ and $q$ because we impose a special constraint on the solar gravitational potential.\ The internal structure of the sun can be considered polytropic with index $n$ ranging between the value 3/2 ($\gamma$=5/3) and 5 ($\gamma$=6/5)[@kip] ($\gamma$ is the adiabatic parameter). In the interior regions where hydrogen ionization is changing rapidly, $\gamma$ is taken to be very close to unity ($n=\infty$, $q=1$), therefore in these circumstances the MB distribution holds. We want to apply generalized statistics to derive a distribution function for the interior solar plasma, relevant in calculating the nuclear fusion reaction rates responsible for the neutrino flux emitted by the sun [@bah2; @gal]. The following arguments support the program of this work. It is well known that one of the main problems with solar physics is related to the detected neutrinos arriving from the sun. All the different experiments have confirmed a deficit in the flux relative to the predictions of standard theory of nuclear physics [@ric; @gal]. In particular, the neutrino flux from the reactions involving $^7 Be$ and $^8 B$, mainly due to collisions at energies higher than the effective energy of $pp$ reactions (4.58 $k_{_B} T$), is much lower than that predicted by the standard models, which uses MB distributions [@ric]. In the recent past we have tried to contribute to the solution of the solar neutrino problem deriving, as steady state solutions of the Fokker-Planck equation, statistical distributions that differ from the MB distribution for a depleted tail at high energies [@ka1; @ka3]. The distribution $f$ is also the solution of a Boltzmann equation without collisions (at equilibrium the distributions obtained with and without the collision term coincide).\ Non-Maxwellian distribution can give thermonuclear reaction rates $r=<\sigma v>$ smaller than the standard ones [@lap], allowing a reduced flux of neutrinos from the sun interior, particularly at energies above few $k_{_B} T$. Very recently we have shown that the Tsallis distribution is a particular case of the family of distributions we derived by means of the Fokker-Planck equation kinetic approach [@ka2; @ka3]. A well defined statistics can be derived for any particular pair of values (M,N) indicating the degree of the polynomials, in the velocity variable, used to describe the drift $J$ and diffusion $D$ coefficients. The Tsallis classical and quantum statistics are related to the pair (0,1) (in this paper we do not derive the drift and diffusion coefficients of the solar core, this will be investigated elsewhere, rather we use straight way the generalized Tsallis statistics). In addition to this, let us recall that Clayton and collaborators [@cla2; @cla3] suggested, on phenomenological grounds, that neutrino counting rate will be much reduced if the high energy tail of the MB distribution of relative energies is depleted, the depletion being described by the factor $\exp \{-\delta \, (E/k_{_B}T)^2\}$ with a suggested value $\delta \approx 0.01$.\ The distribution that we have derived in Ref.[@ka1; @ka2], of the Tsallis type, can take into account this behavior, certainly due to the long-range gravitational interaction. The Tsallis parameter $q$ can be related to the Clayton parameter $\delta=(1-q)/2$; our constraints impose a value of $\delta$ slightly different from the value argued by Clayton. We obtain that the value of $\delta$ must satisfy the range of variability $0.02 \le \delta \le 0.05$. To show the analogies of our approach with the distribution suggested by Clayton, we introduce, by means of an expansion, an approximated expression of the Tsallis distribution which is correct in the range of energies of interest here.\ The Tsallis distribution function reduces to zero at $E \approx (10\div 25) \, k_{_B}T$ depending on the value of $\delta$; therefore, high energy collisions are greatly reduced or absent and the neutrino flux from $^7 Be$ and $^8 B$ reactions is smaller than foreseen by standard theories, the importance of the neutrino flux from $pp$ reactions is consequently increased. In this work we show that the suggestion of Clayton and collaborators is well founded on theoretical grounds (long-range gravitational interaction), it is equivalent to using the distribution we have derived elsewhere [@ka1; @ka2; @ka3] and that this distribution is well motivated within the Tsallis statistics. The neutrino counting rate, compared to the standard predictions, can be explained using the above prescriptions. A complete treatment of this subject can be carried on only within a complete solar model code [@ich; @cas; @tur], it will be done and reported elsewhere. Following Tsallis [@tsa; @cura] Plastino and Plastino [@pla] and Boghosian [@bog], the distribution function $f$ of stellar polytropes is $$f\propto \left [ 1+(q-1)(\alpha+\beta\epsilon) \right ] ^{1/(1-q)} \ \ ,$$ where $q$ is the Tsallis parameter and $\epsilon=E+\Phi(r)$, where $E$ is the c.m. kinetic energy, $\Phi(r)$ is the gravitational potential, $\alpha=-\beta\mu$, where $\mu$ is the chemical potential, $\beta=1/(k_{_B}T)$. The distribution (3) becomes the MB distribution $\exp[-(\alpha+\beta\epsilon)]$ as $q$ goes to one. Let us define the relative gravitational potential $\Psi$ $$\Psi=-\Phi+\Phi_0 \ \ ,$$ where $\Phi_0$ is a constant to be chosen such that $\Psi$ vanishes at the edge of the system [@kip]. The following relation between the relative potential $\Psi$ and the density $\rho$ holds $$\rho^{\gamma-1}=\frac{\gamma-1}{{\cal K}\gamma} \Psi \ \ ,$$ where ${\cal K}={\cal P}/\rho^{\gamma}$, ${\cal P}$ is the pressure and $\gamma$ is the adiabatic parameter.\ The polytropic index $n$ is defined by $$\gamma=1+\frac{1}{n} \ \ .$$ We introduce the relative energy ${\cal E}=\Psi-E$, the distribution $f$ can be written as $$f\propto \left [ 1+(q-1)(\alpha+\beta\Phi_0)-(q-1)\beta{\cal E} \right ]^{1/(1-q)} \ \ .$$ The quantity $\alpha$ can be chosen, without loosing generality, in such a way that [@bog] $$1+(q-1)(\alpha+\beta\Phi_0)=0 \ \ .$$ We want to compare the Tsallis statistics $f$ to the distribution introduced by Clayton et al. [@cla3] which is equal to the MB distribution $C \, \exp(-\beta E)$ times a correction factor $$f=C \, e^{-\beta E} \, e^{\varphi(E)}=e^{\beta_0-\beta_1 E-\beta_2 E^2} \ \ ,$$ where we set $\beta_1=\beta+B$ ($B$ is a constant).\ We observe that the Tsallis distribution can be written in a classical form $f\propto e^{-\hat\epsilon}$ where $$-\hat\epsilon=\frac{1}{1-q} \left\{\log [1+(q-1)(\alpha+\beta\epsilon)] \right\} \ \ .$$ The function $\hat\epsilon$ may be interpreted as a generalized energy that takes into account many body collective interactions [@lav].\ The logarithmic term in Eq.(10) can be expanded in powers of $(q-1)(\alpha+\beta\epsilon)$ with the conditions $\mid (q-1)(\alpha+\beta\epsilon)\mid \le 1$ and $(q-1)(\alpha+\beta\epsilon) \ne -1$.\ We impose that $$\alpha+\beta\Phi=0 \ \ ,$$ to allow that $\beta_1=\beta$, as required for physical reasons in the approximated distribution by Clayton [@cla2; @cla3].\ Infact, $B$ does not have any effect and can be ignored because we maintain the solar luminosity at its known value and the increase of the central temperature $T$ to counteract the effect of $B$ on the power of the sun raises the $^8 B$ neutrino flux back to the value it had at $B=0$.\ The special condition (11) or $\alpha=\beta(\Psi-\Phi_0)$, which is a constraint characteristic of the solar core, shows that $\alpha$ is a function of $\Psi$ or $({\cal E}+E)$, therefore the comparison between two different but equivalent expressions of $f$ done to derive a relation between $n$ and $q$ in Ref.s[@pla; @bog] is not allowed in this case. Finally, we find that the parameters $\beta_0$, $\beta_1$, $\beta_2$ are: $\beta_0=\log C$, $\beta_1=\beta$ and $\beta_2=(1-q) \beta^2 /2=\delta\beta^2$.\ The above constraints (8) and (11) imply $$1+(q-1)\beta\Psi=0 \ \ ,$$ and from Eq.s (5), (6) and (12) we obtain $$q=1-\frac{\tau}{n+1} \ \ {\rm or} \ \ \ \delta \, (n+1)=\frac{\tau}{2} \ \ ,$$ with $\tau=k_{_B} T \, \rho/\cal P$. This relation, which links $n$ and $q$, differs from the relation given by Plastino and Plastino [@pla] or from the one given by Boghosian [@bog] because of the constraint (11) valid for the solar core. In Ref.[@kip] it is shown that, if we select $n=3$, the following values of $\rho$, $\cal P$ and $T$ of the solar core can be derived: the proton density $\rho=53.47$ gr/cm$^3=0.32 \, 10^{-13}$ protons/fm$^3$, the pressure ${\cal P}=0.77$ $10^{-16}$ MeV/fm$^{-3}$ and the temperature $k_{_B}T=1.034$ $10^{-3}$ MeV which is slightly lower than the temperature fixed by other models ($1.29$ $10^{-3}$ MeV). These last two figures are determined supposing a particular central composition; changing the composition by increasing the $He$ concentration a greater temperature can be reached.\ With this selection of values we obtain $\tau=0.43$ and $\delta=0.05$. In the solar core the value of the different physical quantities of interest in this work have been reported by Ichimaru [@ich] to be: $k_{_B}T=1.29 \,\, 10^{-3}$ MeV, $\beta=0.77 \,\, 10^3$ MeV$^{-1}$, $\rho=56.2$ gr/cm$^{3}$, which is the 36% of 156 gr/cm$^3$, for the free protons and ${\cal P}=2.12 \, \, 10^{-16}$ MeV/fm$^3$.\ By inserting these values into Eq.(13) we obtain (with $n=3$) $\tau=0.2$ and $\delta (n+1)=0.1$; then we can fix the central composition to maintain the magnitude of $\tau$ to be $0.2$ and select different couples of values of $n$ and $\delta$, e.g.: $n=3/2$ and $\delta=0.04$, $n=3$ and $\delta=0.025$, $n=5$ and $\delta=0.017$. In conclusion, we may expect that the appropriate value of $\delta$ be in the range between $0.02$ and $0.05$. The value proposed by Clayton ($\delta=0.01$) gives (when $\tau=0.2$) $n=9$ which is unphysical, because greater than $5$. The approximated distribution function with $\delta=0.02$ is $$f_{approx}=C^{'} \, e^{-0.774 \, E_{\rm keV}-0.012 \, E^2_{\rm keV}} \ \ ,$$ where $C^{'}$ is the normalization constant. It is easy to verify that the correct Tsallis distribution reduces to zero at $E=k_{_B}T/(2\delta)$ forbidding the existence of ions with energies greater than $10\div 25$ times the sun temperature. We wish to recall that the Eq.(14) represents a Druyvenstein distribution [@dru].\ All quantities relevant in solar physics and solar neutrino emission can be expressed and evaluated as functions of the parameter $\delta$. We report here, for instance, the thermonuclear reaction rate corrected respect to the MB rate $r_{_{MB}}$ by the depletion factor and calculated in Ref.[@cla3] $$r=r_{_{MB}}\, (1+\frac{15}{4} \delta-\frac{7}{3}\delta \frac{E_0}{k_{_B}T}+\cdots)\, e^{-\Delta} \ \ ,$$ where $E_0$ is the most effective energy ($E_0=4.5$ $k_{_B}T$ for $pp$ reactions) and $\Delta$ is a function of $E_0$, $ k_{_B}T $ and $\delta$. The neutrino counting can be reduced sensibly above few $ k_{_B}T$ with some changes in the solar model parameters. These can be compensated by small changes in the initial $He$ concentration. Most of this reduction has come at the expenses of the $^7 Be$ and $^8 B$ neutrino fluxes. Counting rates within the solar model used by Clayton and collaborators can be extrapolated from the curves reported in Fig.2 of their work [@cla3]. A value of $\delta$ different from zero makes a star more luminous and reduces the rate of energy production at a given temperature. The solar core contracts to higher temperature. As shown by Clayton et al. the increase of temperature at given solar luminosity does not increase the neutrino fluxes that decrease with $\delta$. We do not discuss further, in this work, the measured results and the predictions of the neutrino fluxes; we leave complete and more definitive discussion after the proposed distribution will be tested within the available solar models [@bah2; @cas; @tur; @cla4]. Of course, the results reported in this work depend on the values of the parameters of the solar core one takes as input. We can expect that the trend of definitive results will be on the line of the present description. We hope that the content of this work could be useful to the operating and proposed solar neutrino and underground nuclear astrophysics experiments [@arp1; @arp2]. We thank B. Boghosian, D. Clayton, A. Erdas, G. Fiorentini, M. Lissia and C. Tsallis for critical reading of the manuscript, comments and discussions. C. 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--- abstract: 'We demonstrate a $p$-wave optical Feshbach resonance (OFR) using purely long-range molecular states of a fermionic isotope of ytterbium $^{171}$Yb, following the proposition made by K. Goyal *et al.* \[Phys. Rev. A [**82**]{}, 062704 (2010)\]. The $p$-wave OFR is clearly observed as a modification of a photoassociation rate for atomic ensembles at about 5 $\mu $K. A scattering phase shift variation of $\delta \eta=0.022$ rad is observed with an atom loss rate coefficient $K=28.0\times$10$^{-12}$ cm$^3$/s.' author: - Rekishu Yamazaki - Shintaro Taie - Seiji Sugawa - Katsunari Enomoto - Yoshiro Takahashi title: 'Observation of a $p$-wave Optical Feshbach Resonance' --- Ultracold atoms have recently been utilized as a versatile test bench for the study of many-body physics. In particular, the capability to tune a wide range of interatomic interaction with fine controllability offered by magnetic Feshbach resonances (MFRs) [@Kohler2006; @Chin2010] is so powerful and has enabled numerous novel observations, including a Mott insulator of fermionic atoms in an optical lattice [@Jordens2008], Bose-Einstein condensation(BEC)-Bardeen-Cooper-Schriefer(BCS) crossover [@Regal2004; @Zwierlein2004; @Zwierlein2005], and Efimov trimers[@Kraemer2006]. For these studies, alkali atoms equipped with the MFR have been the main work horse. ![(color online) Relevant molecular states and adiabatic potentials for the $p$-wave OFR experiment. Tuning and monitor lasers ($\lambda=555.8$ nm) are tuned near PA resonances at $-355.4$ MHz and $-212.4$ MHz from the $^1S_0$+$^3P_1$ asymptote. The excited states are previously observed PLR molecular states arising from the hyperfine interaction. A part of the wavefunction of the $p$-wave scattering state $F_{\text{scat}}$ as well as the wavefunction of the state for the monitoring $\phi_{\text{mon}}$ are shown. Note that the energy scale of the ground state potential is enlarged by a factor of hundred compared with the excited state potential to show the centrifugal barrier of approximately 44 $\mu$K in height.[]{data-label="fig1"}](fig1.eps "fig:"){width="8cm"}\ ![image](fig2.eps){width="17.7cm"}\ An alternative approach, using an optical transition to artificially form a Feshbach resonance, *optical Feshbach resonance* (OFR) was proposed [@Fedichev1996] and successfully demonstrated using alkali atoms [@Fatemi2000; @Theis2004; @Wu2012]. These experiments were accompanied with a rather large two-body inelastic atom loss due to the photoassociation (PA). This loss can be mitigated, however, using a narrow intercombination transition in the alkaline-earth-like atoms, as suggested by Ciuryło *et al.* [@Ciurylo2005]. This narrow line OFR has been demonstrated experimentally in thermal gases and condensates of ytterbium(Yb) [@Enomoto2008; @Yamazaki2010], which showed about an order of magnitude suppression of the atom loss as compared to the case for the alkali atoms. Later experiments using strontium atoms also showed similar advantages [@Blatt2011; @Yan2012]. Besides the suppression of an atom loss, the optical manner to control the interatomic interaction, including the optically controlled MFR [@Bauer2009], introduces new possibilities in quantum gas manipulation such as a fast temporal manipulation and fine spatial resolution, which have been successfully demonstrated in Ref.[@Yamazaki2010]. Arbitrary control of interatomic interactions among different kinds of atom pairs in a mixture of gases would be also an interesting possibility. These technical advancements can broaden the spectrum of experiments that can be performed with quantum gases. In all of the previous experiments, the effect of the OFR has been examined only on the $s$-wave interaction. In this paper, we extend the ability of the OFR to control interatomic interaction of higher-partial waves [@Deb2009; @Goyal2010]. We successfully demonstrate the $p$-wave OFR effect in fermionic $^{171}$Yb atoms in the vicinity of the purely long-range (PLR) molecular resonance , the use of which has been suggested by Goyal *et al.* [@Goyal2010]. An intriguing application of a $p$-wave OFR would be the study on the $p$-orbital bands of spinless fermions trapped in an optical lattice [@Goyal2010; @Hauke2011]. Such $p$-orbital physics in optical lattices has been discussed in Refs. [@Wu2008; @Wu2008a]. The radial part of the energy-normalized ground-state $p$-wave scattering wavefunction has an asymptotic form $F_{\text{scat}}(r,k)=(2\mu/\pi\hbar^2k)^{1/2}\sin(kr-\pi/2+\eta)$ at a long interatomic distance $r$, where $\eta$ is the scattering phase shift, $\mu$ is the reduced mass of the scattering atom pair, and $k$ is the wave number of the atom pair, approximated as $k=\sqrt{2\mu k_B T }/\hbar$ with $T$ the temperature of the atomic sample, $k_B$ the Boltzmann constant, and $\hbar$ the Planck constant divided by $2\pi$. The OFR effect is induced by the laser excitation to a molecular state, which alters the ground state scattering wavefunction. Using the semi-analytic formalism by Bohn and Julienne [@Bohn1999], a general form of the scattering matrix element $S=\exp(2i\eta)$ for the colliding ground state atoms modified by the OFR laser excitation is given by $$S=\exp(2i\eta_0)\frac{\Delta-i\left(\Gamma-\gamma\right)/2}{\Delta+i\left(\Gamma+\gamma\right)/2},$$ where $\eta_0$ is the scattering phase shift associated with the unperturbed scattering wavefunction. $\gamma$ is the radiative decay rate of the excited molecular state, and $\Delta$ is the detuning of the OFR laser with respect to the molecular PA resonance. $\Gamma$ is the laser-induced width given by $$\label{Gamma} \Gamma=\frac{\pi}{2}\left(\frac{I}{I_{sat}}\right)\hbar\gamma_a^2f_{FC},$$ where $I$ is the OFR laser intensity, $I_{sat}=0.14$ mW/cm$^2$ and $\gamma_a/2\pi=182$ kHz are the saturation intensity and linewidth for the atomic $^3P_1$ state, respectively. $f_{FC}=|\langle \phi_{\text{e}} | \mathbf{d} \cdot \mathbf{\epsilon}_L | F_{\text{scat}}\rangle|^2/2d_A^2$ is the Franck-Condon factor including the rotational correction, where $\mathbf{d}$, $d_A$, and $\mathbf{\epsilon}_L$ are the dipole moment operator, atomic dipole moment and the laser polarization, respectively. $\phi_{\text{e}}$ and $F_{\text{scat}}$ are the wavefunction of the excited state and scattering state, including the spin and rotational degree of freedom. The variation of the scattering properties by the OFR effect can be determined from the $S$ matrix. The loss rate coefficient $K$ and scattering phase shift $\eta$ can be derived from $S$-matrix as, $$\begin{aligned} % \nonumber to remove numbering (before each equation) K &=& \frac{\pi\hbar}{\mu k}(1-|S|^2)=\frac{\pi\hbar}{\mu k}\frac{\Gamma\gamma}{\Delta^2+\frac{(\Gamma+\gamma)^2}{4}} \label{K} \\ \eta &=& \arg(S)/2. \label{eta}\end{aligned}$$ For the demonstration of the $p$-wave OFR effect, we tune the OFR laser near to the previously observed PLR molecular state arising from the hyperfine interaction [@Enomoto2008b]. Relevant molecular states and adiabatic potentials are denoted in Fig. \[fig1\]. The PLR molecular states have the inner turning point at $\sim$2.5 nm and have small Franck-Condon overlap with tightly bound states, possibly suppressing the inelastic atom loss. In order to determine the variation of the scattering phase shift in a systematic manner, we use the same technique previously demonstrated for the measurement of the $s$-wave OFR effect [@Fatemi2000; @Enomoto2008]. Along with the strong OFR laser (tuning laser) which tunes the scattering phase shift, we apply a much weaker laser (monitor laser) tuned right on another PA resonance. PA resonances at $-212.4$ MHz and $-355.4$ MHz are associated with $T_{\text{e}}=3$ PLR states which are optically accessible from the $p$-wave scattering states, and are used for the monitor and tuning lasers, respectively, where $T_{\text{e}}$ is the total angular momentum of the molecular states. Condon radius of the tuning and monitor states are $R_c=7.22$ and 5.96 nm, respectively. In the previous demonstration of the $s$-wave OFR, the tuning laser varies the $s$-wave scattering phase shift, effectively translating the scattering wavefunction in the radial direction. This translation results in the variation of the Franck-Condon factor for the monitor PA transition, which enables to observe the variation of the phase shift $\eta$ in terms of the monitor PA rate variation. For the current study of the $p$-wave OFR, however, we should be careful due to the existence of the centrifugal barrier as discussed later in details. The experimental setup is nearly the same as our previous works on the PA experiments [@Tojo2006; @Kitagawa2008]. A typical atom number obtained after the evaporation is $1.7 \times 10^5$ at 5.6 $\mu$K. After the preparation of the cold atom sample with mixed spin, the tuning and monitor lasers with $\lambda=556$ nm are turned on to perform the OFR experiment. The two beams are combined and the polarization is cleaned with a polarizer to the same polarization before sent into a polarization maintaining fiber for the experiment. Linearly polarized beams are focused at the atomic cloud with the beam waist of 70 $\mu$m. The tuning laser power is kept at 525 $\mu$W, which is the maximum power obtainable with the current setup, while monitor laser power is adjusted to 113 $\mu$W and 37 $\mu$W for $T=7.2$ $\mu$K and 4.5 $\mu$K, respectively. Two beams are sent in together with the pulse duration of 30 ms. The monitor laser frequency is fixed to the peak of the PA resonance at $-212.4$ MHz, while the tuning laser is scanned around the PA resonance at $-355.4$ MHz. The atom loss induced by the two lasers is monitored by taking the absorption image of the sample. We perform the measurements at a nearly zero magnetic field. The residual magnetic field is estimated to be below 50 mG, which results in the Zeeman splitting of the excited state sublevels to less than 2 kHz, well within the linewidth of the excited state. The observed atom loss spectra are shown in Fig. \[fig2\] for (a) $T=7.2$ $\mu$K and (b) 4.5 $\mu$K. The $p$-wave centrifugal barrier for Yb atoms is approximately 920 kHz (44 $\mu$K), and we were unable to observe clear PA spectra below 3.5 $\mu$K, due to lower starting atom number from the large loss in the evaporative cooling and PA rate suppression at lower temperature. With the tuning laser only, the spectra are symmetric as expected for the ordinary PA resonance. When the monitor laser is turned on, the additional atom loss due to the monitor laser PA is introduced, which is observed for the data at off-resonance in Fig. 2(a) and 2(b). The OFR effect manifests itself around the tuning laser PA resonance, where the dispersive-shaped spectra are observed. This behavior suggests that the Franck-Condon factor between the scattering wave state and the monitor state is altered dispersively around the tuning PA resonance, with the effect diminishing at a lower temperature. ![(color online) Calculated wavefunction overlap integral, $\langle\phi_{\text{mon}}|F_{\text{scat}}\rangle$, for the monitor state ($-212.4$ MHz) and the scattering state as a function of the phase shift $\eta$, for (a) a wide range and (b) near the background phase shift $\eta_0=0.127$ at $T=5.6$ $\mu$K. The overlap integral is normalized with respect to the overlap integral at $\eta_0$, denoted as $\langle\phi_{\text{mon}}|F_{\text{scat}}\rangle_0$. Dots are the numerical results with lines showing the guide to the eye. Large overlap integral can be seen near $\eta=1.7$ and $-1.4$, representing the location of the shape resonances. Near $\eta_0$ the variation of the overlap integral is quite linear with respect to $\eta$, with zero at $\eta'=0.116$. The shaded region shows the range where the variation of $\eta$ is observed at a current study.[]{data-label="fig3"}](fig3.eps "fig:"){width="8cm"}\ For the quantitative analysis of the OFR spectra with the monitor laser on, it is important to analyze the overlap integral between the scattering wavefunction of the ground state $F_{\text{scat}}$ and that of the molecular bound state $\phi_{\text{mon}}$ used for the monitoring PA. We performed numerical calculation of the bound and scattering wavefunction using Numerov method with a potential provided by van der Waals potential with potential constant C$_6$ = 1931.7 a.u. and the centrifugal term. The monitor-laser-induced atom loss rate coefficient $K_{\text{mon}}$ is proportional to $\Gamma_{\text{mon}}$ when the laser is tuned on-resonance ($\Delta_{\text{mon}}=0$) with the weak excitation $\Gamma_{\text{mon}} \ll \gamma$, as understood from Eq. \[K\]. As shown in Fig. 1, the wavefunction of the molecular bound state $\phi_{\text{mon}}$ is mostly inside the centrifugal barrier ($r<$7.0 nm) of the $p$-wave scattering state. The calculated wavefunction overlap integral, $\langle\phi_{\text{mon}}|F_{\text{scat}}\rangle$ as a function of the scattering phase shift $\eta$ is shown in Fig. \[fig3\](a). The overlap integral varies sinusoidally with respect to $\eta$. The background scattering phase shift calculated for $^{171}$Yb is $\eta_0=0.127$ at $T=5.6$ $\mu$K. As shown in Fig. \[fig3\](b), the variation of the overlap integral is quite linear in the vicinity of $\eta_0$, with zero overlap at $\eta'=0.116$. Therefore, we expect the form of the monitor-laser-induced atom loss rate coefficient $K_{\text{mon}}=K_0(\eta-\eta')^2$ for the current experiment. ![ Calculated variation of the scattering wavefunction $F_{\text{scat}}$ for various $\eta$. A large modification of the wavefunction inside the centrifugal barrier ($r<7.0$ nm) can be seen within a small variation of $\eta$. []{data-label="fig4"}](fig4.eps "fig:"){width="7cm"}\ The data with the monitor laser is fitted using an inelastic rate equation $\dot{n}=-2K_{\text{total}}n^2$, where $n$ is the atom density, with a PA rate coefficient $K_{\text{total}}=K_{\text{tun}}+K_{\text{mon}}$. We fit the atom loss spectra with fitting parameters $\Gamma_{\text{tun}}$ and $K_0$, with $\gamma/2\pi=364$ kHz used for the molecular state radiative decay rate, and the results are shown in solid lines in Fig. \[fig2\](a) and \[fig2\](b). To include the effect of the collision at different energies, thermal averaging is included in the fit, assuming Maxwell-Boltzmann velocity distribution of the atoms for the given temperature. The calculated results fit the data remarkably well. Although the fit slightly overestimates the atom loss near the resonance, the overall fitting quality including the reproducibility of the asymmetric spectra is satisfiable. We obtained the best fit with $\Gamma_{\text{tun}}/2\pi=16.5$ and 16.2 kHz, where the maximum scattering phase shift $\eta=0.149(2)$ and 0.144(3) radian and corresponding atom loss rate coefficient $K_{\text{tun}}=28.0\times 10^{-12}$ and $17.2\times 10^{-12}$ cm$^3$/s are calculated, for $T=7.2$ and 4.5 $\mu$K, respectively. From $\Gamma_{\text{tun}}$ obtained from the fit, we also calculate the atom loss spectra with a loss rate coefficient $K_{\text{total}}=K_{\text{tun}}$ for the data without the monitor laser and results are shown in dotted lines in the same figure. The loss spectra are well reproduced with the fitted parameter. The calculated variation of $\eta$ from the obtained value of $\Gamma_{\text{tun}}$ at $T=7.2$ $\mu$K is shown in Fig. \[fig2\](c). The phase shift variation $\delta\eta=\eta_{max}-\eta_0=0.022$ radian is observed. On the blue side of the resonance, the OFR induced $\eta$ crosses the $\eta_{min}=0.116$, where we expect the Franck-Condon factor for the monitor PA to be zero. The atom loss curve with the tune and monitor laser (solid line) tangentially connect to the atom loss with the tuning laser only (dotted line) at this region. From all the data including that for a different temperature, we calculated the average optical volume[@Goyal2010] $V_{\text{opt}}=\Gamma /2k^3\gamma =27(6)$ nm$^3$ at the tuning laser intensity $I=1$ W/cm$^2$ for this transition which is lower than the previously calculated value 39.2 nm$^3$ at $I=1$ W/cm$^2$, which is the average value over different projections of the partial-wave angular momentum and the excitation photon helicity [@Goyal2010]. The uncertainties in the measurement are mainly from the PA lasers and atom sample characterizations. The uncertainty in the laser intensity determination is the dominant source of error, which includes the power fluctuation, beam size uncertainty, and power meter calibration error, resulting in total of 23% uncertainty. The atom number fluctuation is approximately 5 %, while the temperature variation is limited to about 3 %. While the values obtained in experiment and theory show a discrepancy, a typical $p$-wave scattering volume (as in van der Waals length scale in the $s$-wave scattering) is on the order of (100 $a_0$)$^3　\sim 150$ nm$^3$, where $a_0$ is Bohr radius. The observed difference between the experiment and theory is much smaller compared to this typical scale size. One may wonder why the change of the monitor PA rate is rather strong, showing strong asymmetric atom loss spectra, while the phase shift variation $\delta \eta$ is not significantly large. In Fig. \[fig4\], we show calculated scattering wavefunctions for $\eta$ ranging from 0.097 to 0.171 radian. Despite the small variation in $\eta$, a drastic change in the shape of the wavefunction can be observed inside the centrifugal barrier ($r<7.0$ nm). At $\eta_0$, the scattering wavefunction is away from the shape resonance condition, and the penetration inside the centrifugal barrier is small and thus the amplitude of the wavefunction inside the centrifugal barrier is small. When the OFR causes a change of the phase shift $\eta$, even if it is not so large, a large modification of the wavefunction can take place inside to alter the Franck-Condon factor with respect to $\phi_{\text{mon}}$. By choosing the molecular state which is located well inside the centrifugal barrier, the monitor PA signal amplifies the small shift of $\eta$, resulting in a large spectral modification. In conclusion, we successfully observed the $p$-wave OFR effect in the vicinity of the previously reported PRL molecular state. The scattering phase shift variation of $\delta\eta=0.022$ radian at about 5 $\mu $K is observed. In order to realize a larger phase shift $\eta$ at a low temperature, it is beneficial to use an atomic species which has a $p$-wave shape resonance, which results in an enhancement of $\Gamma_{\text{tun}}$ through the large Frack-Condon factor. The $^{173}$Yb homonuclear pair has a $p$-wave shape resonance at the collision energy of tens of micro Kelvin [@Kitagawa2008], and thus it would be a good candidate for an efficient $p$-wave OFR. A future investigation will also include a use of ultra-narrow transitions to $^3P_{0,2}$ states in alkaline-earth-like atoms to further suppress the atom loss and heating of the sample. We acknowledge useful discussions with B. Deb, I. Deutsch, I. Reichenbach, and P. Zhang. This work is supported by the Grant-in-Aid for Scientific Research of JSPS (No. 18204035, No. 21102005C01, and No. 21104513A03 Quantum Cybernetics), GCOE Program The Next Generation of Physics, Spun from Universality and Emergence from MEXT of Japan, and World-Leading Innovative R&D on Science and Technology (FIRST). S. T. and S. S. acknowledge support from JSPS. [26]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , ****, (). , , , , ****, (). , , , , , ****, (). , , , ****, (). , , , , , , ****, (). , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , ****, (). , , , ****, (). , , , , , , , ****, (). , ****, (). , , , ****, (). , , , , ****, (). , , , , ****, (). , , , , , , , ****, (). , , , , , (). , , , , , ****, (). , ****, (). , , , ****, (). , , , , , , ****, (). , ****, (). , ****, (). , ****, (). , , , , ****, (). , , , , , , , ****, (). , , , , , , , ****, ().
--- abstract: | By $BMO_{\textrm{o}}(\mathbb{R})$ we denote the space consisting of all those odd and bounded mean oscillation functions on $\mathbb{R}$. In this paper we characterize the functions in $BMO_{\textrm{o}}(\mathbb{R})$ with bounded support as those ones that can be written as a sum of a bounded function on $(0,\infty )$ plus the balayage of a Carleson measure on $(0,\infty )\times (0,\infty)$ with respect to the Poisson semigroup associated with the Bessel operator $$B_\lambda :=-x^{-\lambda }\frac{d}{dx}x^{2\lambda }\frac{d}{dx}x^{-\lambda },\quad \lambda >0.$$ This result can be seen as an extension to Bessel setting of a classical result due to Carleson. address: - 'Víctor Almeida, Jorge J. Betancor, Juan C. Fariña, Lourdes Rodríguez-Mesa Departamento de Análisis Matemático, Universidad de La Laguna, Campus de Anchieta, Avda. Astrofísico Sánchez, s/n, 38721 La Laguna (Sta. Cruz de Tenerife), Spain' - 'Alejandro J. Castro Department of Mathematics, Nazarbayev University, 010000 Astana, Kazakhstan' author: - 'V. Almeida' - 'J. J. Betancor' - 'A. J. Castro' - 'J. C. Fariña' - 'L. Rodríguez-Mesa' title: BMO functions and balayage of Carleson measures in the Bessel setting --- [^1] Introduction ============ In this paper we extend [@Ca Theorem 2] (See also [@Wi Theorem A]) due to Carleson to Bessel settings. A measurable function $f$ on ${\mathbb{R}^n}$ is said to have bounded mean oscillation, in short $f \in BMO({\mathbb{R}^n})$, when there exists $C>0$ such that, for every cube $Q \subset {\mathbb{R}^n}$ with sides parallel to the coordinate axes, $$\frac{1}{|Q|} \int_Q |f(y)-f_Q| dy \leq C,$$ where $$f_Q := \frac{1}{|Q|} \int_Q f(y)dy.$$ It is defined, for every $f \in BMO({\mathbb{R}^n})$, $$\|f\|_{BMO({\mathbb{R}^n})} := \sup_{Q} \frac{1}{|Q|} \int_Q |f(y)-f_Q| dy,$$ where the supremum is taken over all the cubes $Q \subset {\mathbb{R}^n}$ with sides parallel to the coordinate axis. It is clear that $\| \cdot \|_{BMO({\mathbb{R}^n})}$ is a norm when two functions $f$ and $g$ in $BMO({\mathbb{R}^n})$ are identified provided that the difference $f-g$ is constant in ${\mathbb{R}^n}$. $BMO({\mathbb{R}^n})$ is also called the John-Nirenberg space ([@JN]). The $BMO({\mathbb{R}^n})$ space is closely connected with the so called Carleson measure in ${\mathbb{R}}_+^{n+1}:={\mathbb{R}^n}\times (0,\infty)$. If $Q$ is a cube in ${\mathbb{R}^n}$, the Carleson box is given by $\widehat{Q}:=Q \times (0,|Q|)$, where $|Q|$ denotes the Lebesgue measure of $Q$. A Borel measure $\mu$ on ${\mathbb{R}}_+^{n+1}$ is said to be a Carleson measure, in short $\mu \in \mathcal{C}({\mathbb{R}}_+^{n+1})$, when there exists $C>0$ such that, for every cube $Q \subset {\mathbb{R}^n}$ with sides parallel to the coordinate axes, $$\frac{|\mu|(\widehat{Q})}{|Q|} \leq C,$$ where $|\mu|$ denotes the total variation measure of $\mu$. If $\mu \in \mathcal{C}({\mathbb{R}}_+^{n+1})$ it can be defined the norm $$\|\mu\|_{\mathcal{C}({\mathbb{R}}_+^{n+1})} := \sup_{Q} \frac{|\mu|(\widehat{Q})}{|Q|},$$ where the supremum is taken over all cubes $Q \subset {\mathbb{R}^n}$ with sides parallel to the coordinate axes. The classical Poisson semigroup $\{P_t\}_{t>0}$ generated by $-\sqrt{- \Delta}$, where $\Delta$ represents the Laplace operator $\Delta := \sum_{i=1}^n \partial_{x_i}^2$ in ${\mathbb{R}^n}$, is defined for every $f \in L^p({\mathbb{R}^n})$, $1 \leq p \leq \infty$, by $$P_t(f)(x) := c_n \int_{{\mathbb{R}^n}} P_t(x-y) f(y) dy, \quad x \in {\mathbb{R}^n}, \ t>0,$$ where $c_n:=\Gamma((n+1)/2)/\pi^{(n+1)/2}$. Here the Poisson kernel is $$P_t(x) := \frac{t}{(|x|^2+t^2)^{(n+1)/2}}, \quad x \in {\mathbb{R}^n}, \ t>0.$$ If $f \in BMO({\mathbb{R}^n})$, then ([@St p. 141]) $$\label{eq1} \int_{{\mathbb{R}^n}} \frac{|f(y)|}{(1+|y|)^{n+1}} dy < \infty,$$ and $P_t(|f|)(x) < \infty$, for every $x \in {\mathbb{R}^n}$ and $t>0$. It is well known ([@St p. 159 and p. 165]) that a function $f \in L^1({\mathbb{R}^n},(1+|y|)^{-n-1} dy)$ is in $BMO({\mathbb{R}^n})$ if, and only if, the measure $\mu_f$ on ${\mathbb{R}}_+^{n+1}$ defined by $$d\mu_f(x,t) := | t \partial_t P_t(f)(x) |^2 \frac{dx dt}{t}, \quad (x,t) \in {\mathbb{R}}_+^{n+1},$$ is a Carleson measure. If $\mu$ is a positive measure on ${\mathbb{R}}^{n+1}_+$ the balayage $S_{\mu,P}$ with respect to the Poisson semigroup $\{P_t\}_{t>0}$ is defined by $$S_{\mu,P}(x) := \int_{{\mathbb{R}}_+^{n+1}} P_t(x-y) d\mu(y,t), \quad x \in {\mathbb{R}^n}.$$ Carleson ([@Ca Theorem 2]) (see also [@Wi Theorem A]) proved that a function $f$ with compact support is in $BMO({\mathbb{R}^n})$ if, and only if, there exist $g \in L^\infty({\mathbb{R}^n})$ and a Carleson measure $\mu$ on ${\mathbb{R}}_+^{n+1}$ such that $f=g+S_{\mu,P}$ and $$\|f\|_{BMO({\mathbb{R}^n})} \sim \|g\|_{L^\infty({\mathbb{R}^n})} + \|\mu\|_{\mathcal{C}({\mathbb{R}}_+^{n+1})}.$$ Actually, this result was established for uniparametric families $\{K_t\}_{t>0}$ being the Poisson semigroup $\{P_t\}_{t>0}$ a special case. An extension of [@Ca Theorem 2] to spaces of homogeneous type was proved by Uchiyama ([@Uc]). The proofs of the mentioned results in [@Ca] and [@Uc] (see also [@GJ]) are based on an iterative argument. Other proof was presented in [@Wi]. Here, we will adapt Wilson’s ideas to our setting. Recently, Chen, Duong, Li, Song and Yan ([@CDLSY]) have established a version of Carleson’s result ([@Ca Theorem 2]) where the Laplace operator $\Delta$ is replaced by the Schrödinger operator $\mathcal{L}_V:=-\Delta + V$, where the nonnegative potential $V$ belongs to the reverse Hölder class $B_q$ for some $q \geq n$. Definitions and main properties about $BMO$ spaces associated with $\mathcal{L}_V$ can be encountered in [@DYZ] and [@DGMTZ] (see also [@Sh]). Harmonic analysis associated with Bessel operators was initiated by Muckenhoupt and Stein ([@MS]). They considered the Bessel operators $$\Delta_\lambda := -x^{-2\lambda} \frac{d}{dx} x^{2\lambda} \frac{d}{dx}, \quad \lambda >0,$$ and studied $L^p$-boundedness properties of maximal operators associated with Poisson semigroups defined by $\Delta_\lambda$ and Riesz transforms in this setting. Recently, harmonic analysis related to Bessel operators has raised interest again (see [@BCS], [@DLOWY], [@DLWY], [@LW], [@NS1], [@NS2], [@Vi] and [@YY], among others). We consider the Bessel type operator on $(0,\infty)$ $$B_\lambda := - x^{-\lambda} \frac{d}{dx} x^{2\lambda} \frac{d}{dx} x^{-\lambda} = - \frac{d^2}{dx^2} + \frac{\lambda (\lambda-1)}{x^2}, \quad \lambda >0.$$ Note that the potential $V_\lambda(x):=\lambda(\lambda-1)/x^2$, $x \in (0,\infty)$, does not satisfy any reverse Hölder property and it has a singularity at $x=0$. Then, $B_\lambda$ is not included in the class of Schrödinger operators considered in [@CDLSY] and [@DGMTZ]. Assume that $\lambda>0$. According to [@MS §16] the Poisson semigroup $\{P_t^\lambda\}_{t>0}$ associated with the Bessel operator $B_\lambda$ is defined as follows $$P_t^\lambda(f)(x) := \int_0^\infty P_t^\lambda(x,y)f(y) dy, \quad x,t \in (0,\infty),$$ for every $f \in L^p(0,\infty)$, $1 \leq p \leq \infty$. Here, the Poisson kernel $P_t^\lambda(x,y)$ is given by $$P_t^\lambda(x,y) := \frac{2\lambda}{\pi} (xy)^\lambda t \int_0^\pi \frac{(\sin \theta)^{2\lambda-1}}{[(x-y)^2+t^2+2xy(1-\cos \theta)]^{\lambda+1}} d\theta, \quad x,y,t \in (0,\infty).$$ For every $1 \leq p \leq \infty$, the family $\{P_t^\lambda\}_{t>0}$ is bounded on $L^p(0,\infty)$. Note that $\{P_t^\lambda\}_{t>0}$ is not Markovian, that is, $P_t^\lambda(1) \neq 1$. Indeed, according to [@BSt Lemma 2.2 and Remark 2.5] the function $v(x,t):=P_t^\lambda(1)(x)$ satisfies $$(\partial_t^2 - B_{\lambda,x})v(x,t)=0, \quad x,t \in (0,\infty),$$ but clearly $$(\partial_t^2 - B_{\lambda,x})1 = -\frac{\lambda (\lambda-1)}{x^2}, \quad x,t \in (0,\infty).$$ We also remark that the function ${\mathfrak f}_\lambda(x)=x^\lambda$, $x \in (0,\infty)$, does not belong to $L^p(0,\infty)$, for any $1 \leq p \leq \infty$. However, $P_t^\lambda({\mathfrak f}_\lambda)={\mathfrak f}_\lambda$, $t>0$ (see, [@BCaFR p. 455]). We denote by $P_*^\lambda$ the maximal operator defined by $\{P_t^\lambda\}_{t>0}$, that is, $$P_*^\lambda(f) := \sup_{t>0} |P_t^\lambda(f)|, \quad f \in L^p(0,\infty), \quad 1 \leq p \leq \infty.$$ $P_*^\lambda$ is bounded from $L^p(0,\infty)$ into itself when $1<p \leq \infty$ and from $L^1(0,\infty)$ into $L^{1,\infty}(0,\infty)$ ([@BSt Theorem 2.4 and Remark 2.5]). The Hardy space $H^1_\lambda(0,\infty)$ associated to the operator $B_\lambda$ was studied in [@BDT]. It is said that a function $f \in L^1(0,\infty)$ is in $H^1_\lambda(0,\infty)$ when $P_*^\lambda(f) \in L^1(0,\infty)$. On $H^1_\lambda(0,\infty)$ it is considered the norm $\|\cdot \|_{H^1_\lambda(0,\infty)}$ given by $$\|f \|_{H^1_\lambda(0,\infty)} := \|f \|_{L^1(0,\infty)} + \|P_*^\lambda(f) \|_{L^1(0,\infty)}, \quad f \in H^1_\lambda(0,\infty).$$ The dual space of $H^1_\lambda(0,\infty)$ can be characterized as a $BMO$-type space. A function $f \in L^1(0,a)$, for every $a>0$, is in $BMO_{\textrm{o}}(\mathbb{R})$ when there exists $C>0$ such that - for every bounded interval $I \subset (0,\infty)$, $$\frac{1}{|I|} \int_I |f(y)-f_I| dy \leq C,$$ - for every $a \in (0,\infty)$, $$\frac{1}{a} \int_0^a |f(y)| dy \leq C.$$ On $BMO_{\textrm{o}}(\mathbb{R})$ the norm $\|\cdot \|_{BMO_{\textrm{o}}(\mathbb{R})}$ is defined by $$\| f \|_{BMO_{\textrm{o}}(\mathbb{R})} := \inf \{C>0 \text{ : {\it a}) and {\it b}) hold}\}.$$ The space $BMO_{\textrm{o}}(\mathbb{R})$ can be characterized as that one consisting on all the functions $f$ defined on $(0,\infty)$ such that the odd extension $f_{\textrm{o}}$ of $f$ to ${\mathbb{R}}$ is in $BMO({\mathbb{R}})$ ([@BCFR1 p.465]). This property, that justifies the notation $BMO_{\textrm{o}}(\mathbb{R})$ for our space, will be very useful in the sequel. The space $BMO_{\textrm{o}}(\mathbb{R})$ coincides, in the usual way, with the dual space of $H^1_\lambda(0,\infty)$ (see [@BCFR1 p. 466]). We say that a Borel measure $\mu$ on $(0,\infty)\times (0,\infty )$ is a Carleson measure on $(0,\infty)\times (0,\infty )$ when there exists $C>0$ such that, for every bounded interval $I \subset (0,\infty)$, $$\frac{|\mu|(\widehat{I})}{|I|} \leq C.$$ Here, as above, $|\mu|$ represents the total variation measure of $\mu$, $|I|$ denotes the length of the interval $I$ and $\widehat{I}:=I \times (0,|I|)$. If $\mu$ is a Carleson measure on $(0,\infty)\times (0,\infty )$ we define $$\|\mu\|_{\mathcal{C}} := \sup_{I} \frac{|\mu|(\widehat{I})}{|I|},$$ where the supremum is taken over all bounded intervals $I \subset (0,\infty)$. Next result shows the connection between $BMO_{\textrm{o}}(\mathbb{R})$ and the Carleson measures on $(0,\infty)\times (0,\infty )$ by using Poisson semigroups $\{P_t^\lambda\}_{t>0}$. \[Th1.1\] [([@BCFR1 Theorem 1.1])]{} Let $\lambda >0$. Suppose that $f \in L^1(0,a)$, for every $a>0$. Then, $f \in BMO_{\textrm{o}}(\mathbb{R})$ if, and only if, $f \in L^1((0,\infty),(1+x)^{-2} dx)$ and the measure $\mu_f$ on $(0,\infty)\times (0,\infty )$ defined by $$\mu_f(x,t) := | t \partial_t P_t^\lambda(f)(x) |^2 \frac{dxdt}{t}, \quad x,t \in (0,\infty),$$ is Carleson. Moreover, the quantities $\|f\|_{BMO_{\textrm{o}}(\mathbb{R})}^2$ and $\|\mu_f\|_{\mathcal{C}}$ are equivalent. In our $B_\lambda$-Bessel setting we consider the gradient $\nabla_{\lambda} :=(\partial_t,D_{\lambda ,x})$, where $D_{\lambda ,x}:=x^\lambda \partial_x x^{-\lambda}$ . \[Th1.2\] [([@BCaFR Theorem 1])]{} Let $\lambda >1$. Assume that $u$ is a function defined in ${\mathbb{R}}\times (0,\infty)$ such that $x^{-\lambda} u(x,t) \in C^\infty({\mathbb{R}}\times (0,\infty))$ and it is even in the $x$-variable. Suppose also that $(\partial_t^2 - B_\lambda)u=0$, on $(0,\infty)\times (0,\infty )$. Then, the following assertions are equivalent. - There exists $f \in BMO_{\textrm{o}}(\mathbb{R})$ such that $u(x,t)=P_t^{\lambda}(f)(x)$, $x,t \in (0,\infty)$. - The measure $\mu_\lambda$ on $(0,\infty)\times (0,\infty )$ defined by $$d\mu_\lambda(x,t) := | t \nabla_\lambda u(x,t) |^2 \frac{dxdt}{t}, \quad x,t \in (0,\infty),$$ is Carleson. Moreover, the quantities $\|f\|_{BMO_{\textrm{o}}(\mathbb{R})}^2$ and $\|\mu_\lambda\|_{\mathcal{C}}$ are equivalent. The main result of this paper is the following, which can be seen as a version of the Carleson’s result in [@Ca Theorem 2] (see also [@Wi Theorem A]) in our Bessel setting. \[Th1.3\] Let $\lambda >0$. - If $\mu$ is a Carleson measure on $(0,\infty)\times (0,\infty )$, the balayage of $\mu$ with respect to the Poisson semigroup $\{P_t^\lambda\}_{t>0}$ associated with $B_\lambda$ defined by $$S_{\mu,P^\lambda}(x) := \int_0^\infty\int_0^\infty P_t^\lambda(x,y) d\mu(y,t), \quad x \in (0,\infty),$$ is in $BMO_{\textrm{o}}(\mathbb{R})$ and $$\|S_{\mu,P^\lambda}\|_{BMO_{\textrm{o}}(\mathbb{R})} \leq C \|\mu\|_{\mathcal{C}}.$$ Here $C>0$ does not depend on $\mu$. - Let $f \in BMO_{\textrm{o}}(\mathbb{R})$ such that $f=0$ on $(a,\infty)$, for some $a>0$. Then, there exist $g \in L^\infty(0,\infty)$ and a Carleson measure $\mu$ on $(0,\infty)\times (0,\infty )$ such that $f = g + S_{\mu,P^\lambda}$ and $$\|g\|_{L^\infty(0,\infty)} + \|\mu\|_{\mathcal{C}} \leq C \|f\|_{BMO_{\textrm{o}}(\mathbb{R})},$$ where $C>0$ does not depend on $f$. In order to prove this theorem we are going to adapt the procedure developed by Wilson ([@Wi]) to our Bessel setting. The heat semigroup $\{W_t^\lambda\}_{t>0}$ associated to the Bessel operator $B_\lambda$ is defined, for every $f \in L^p(0,\infty)$, $1 \leq p \leq \infty$, by $$W_t^\lambda(f)(x) := \int_0^\infty W_t^\lambda(x,y)f(y)dy, \quad x \in (0,\infty),$$ where the heat kernel is given by $$W_t^\lambda(x,y) := \frac{\sqrt{xy}}{2t} I_{\lambda-1/2}\Big( \frac{xy}{2t}\Big) e^{-(x^2+y^2)/(4t)}, \quad x,y,t \in (0,\infty).$$ Here, $I_\nu$ denotes the modified Bessel function of the first kind and order $\nu$. If $\mu$ is a Borel measure on $(0,\infty)\times (0,\infty )$ we define the balayage $S_{\mu,W^\lambda}$ of $\mu$ with respect to $\{W_t^\lambda\}_{t>0}$ in the natural way. The well known subordination formula connects Bessel Poisson and heat semigroups as follows. For every $f \in L^p(0,\infty)$, $1 \leq p \leq \infty$, $$P_t^\lambda(f)(x) = \frac{1}{\sqrt{\pi}} \int_0^\infty \frac{e^{-u}}{\sqrt{u}} W^\lambda_{t^2/(4u)}(f)(x) du, \quad x,t \in (0,\infty).$$ By using this equality from Theorem \[Th1.3\] we can immediately deduce the following property (see [@CDLSY proof of Theorem 3.5]). \[Cor1.1\] Let $\lambda >0$ and $f \in BMO_{\textrm{o}}(\mathbb{R})$ such that $f=0$ on $(a,\infty)$, for some $a>0$. Then, there exist $g \in L^\infty(0,\infty)$ and a Carleson measure $\mu$ on $(0,\infty)\times (0,\infty )$ such that $f = g + S_{\mu,W^\lambda}$ and $$\|g\|_{ L^\infty(0,\infty)} + \|\mu\|_{\mathcal{C}} \leq C \|f\|_{BMO_{\textrm{o}}(\mathbb{R})}.$$ Here $C>0$ does not depend on $f$. This paper is organized as follows. In Section \[Sect2\], we present some properties of the Poisson kernel and Poisson semigroups for Bessel operators that will be useful in the sequel. In Section \[Sect3\] we prove new properties of the space $BMO_{\textrm{o}}(\mathbb{R})$ that are needed to establish Theorem \[Th1.3\]. The proof of Theorem \[Th1.3\] is presented in Section \[Sect4\]. Throughout this paper by $C$ we always denote a positive constant that is not necessarily the same in each occurrence. Also, we always consider $\lambda>0$. Some useful properties of Bessel Poisson semigroups {#Sect2} =================================================== As it was mentioned in the introduction, according to [@MS §16] the $B_\lambda$-Poisson kernel is given by $$\label{D1} P_t^\lambda(x,y) := \frac{2\lambda}{\pi} t (xy)^\lambda \int_0^\pi \frac{(\sin \theta)^{2\lambda-1}}{[(x-y)^2+t^2+2xy(1-\cos \theta)]^{\lambda+1}} d\theta, \quad x,y,t \in (0,\infty).$$ From it is straightforward that $$\label{D2} 0 \leq P_t^\lambda(x,y) \leq C \frac{t (xy)^\lambda}{((x-y)^2+t^2)^{\lambda + 1}}, \quad x,y,t \in (0,\infty).$$ Also, by [@MS (b) p. 86] we get $$\label{D3} P_t^\lambda(x,y) \leq C \frac{t}{(x-y)^2+t^2}, \quad x,y,t \in (0,\infty).$$ We also need estimations for the derivatives of $P_t^\lambda(x,y)$. \[D4\] Let $\lambda>0$. Then, for every $t,x,y \in (0,\infty)$, $$|\partial_t P_t^\lambda(x,y)|+|D_{\lambda ,x}P_t^\lambda(x,y)| \leq \frac{C}{t} P_t^\lambda(x,y), \qquad$$ We have that $$\begin{aligned} | \partial_t P_t^\lambda(x,y) | & = \Big| \frac{2\lambda}{\pi} (xy)^\lambda \Big\{ \int_0^\pi \frac{(\sin \theta)^{2\lambda-1}}{[(x-y)^2+t^2+2xy(1-\cos \theta)]^{\lambda+1}} d\theta \nonumber \\ & \qquad \qquad \quad - 2(\lambda +1)t^2 \int_0^\pi \frac{(\sin \theta)^{2\lambda-1}}{[(x-y)^2+t^2+2xy(1-\cos \theta)]^{\lambda+2}} d\theta\Big\} \Big| \nonumber \\ & \leq \frac{C}{t} P_t^\lambda(x,y).\end{aligned}$$ On the other hand, we can write $$\begin{aligned} D_{\lambda,x} P_t^\lambda(x,y) & = -\frac{4\lambda(\lambda+1)}{\pi} t(xy)^\lambda \int_0^\pi \frac{((x-y)+y(1-\cos \theta))(\sin \theta )^{2\lambda -1}}{[(x-y)^2+t^2+2xy(1-\cos \theta)]^{\lambda+2}} d\theta,\end{aligned}$$ and also, $$\begin{aligned} D_{\lambda,x}P_t^\lambda(x,y) & = -\frac{4\lambda(\lambda+1)}{\pi} t(xy)^\lambda \int_0^\pi \frac{((x-y)\cos \theta +x(1-\cos \theta))(\sin \theta)^{2\lambda-1}}{((x-y)^2+t^2+2xy(1-\cos \theta))^{\lambda+2}} d\theta.\end{aligned}$$ Then, $$\Big| D_{\lambda,x} P_t^\lambda(x,y) \Big|\leq Ct(xy)^\lambda \int_0^\pi \frac{(|x-y|+\min\{x,y\}(1-\cos \theta))(\sin \theta)^{2\lambda-1}}{[(x-y)^2+t^2+2xy(1-\cos \theta)]^{\lambda+2}} d\theta .$$ Since $$\frac{|x-y|+\min\{x,y\}(1-\cos \theta)}{(x-y)^2+t^2+2xy(1-\cos \theta)}\leq \frac{C}{t}\Big(1+\frac{\min \{x,y\}}{\sqrt{xy}}\Big)\leq \frac{C}{t},\quad x,y,t\in (0,\infty )\mbox{ and }\theta \in [0,\pi],$$ we conclude that $$\begin{aligned} \Big| D_{\lambda,x} P_t^\lambda(x,y) \Big| & \leq \frac{C}{t} P_t^\lambda(x,y).\end{aligned}$$ The Hankel transform $h_\lambda(f)$ of $f \in L^1(0,\infty)$ is defined by $$h_\lambda(f)(x):= \int_0^\infty \sqrt{xy} J_{\lambda-1/2}(xy) f(y) dy, \quad x \in (0,\infty),$$ where $J_\nu$ denotes the Bessel function of the first kind and order $\nu$. Since the function $\sqrt{z} J_{\lambda-1/2}(z)$ is bounded on $(0,\infty)$, it is clear that $$\| h_\lambda(f) \|_{L^\infty(0,\infty)} \leq C \|f\|_{L^1(0,\infty)}, \quad f \in L^1(0,\infty).$$ The Hankel transform $h_\lambda$ can be extended from $L^1(0,\infty) \cap L^2(0,\infty)$ to $L^2(0,\infty)$ as an isometry in $L^2(0,\infty)$ ([@Tit p. 473 (1)]). The Bessel Poisson kernel can be written in the following way ([@MS (16.1’)]) $$P_t^\lambda(x,y) = \int_0^\infty e^{-tz} \sqrt{xz} J_{\lambda-1/2}(xz) \sqrt{yz} J_{\lambda-1/2}(yz) \, dz, \quad x,y,t\in (0,\infty).$$ Then, we have that $$P_t^\lambda(x,y) = h_\lambda \Big( e^{-tz} \sqrt{xz} J_{\lambda-1/2}(xz) \Big)(y), \quad x,y,t\in (0,\infty),$$ and $$\label{A.4} \partial_t P_t^\lambda (x,y) = - h_\lambda\Big(z e^{-tz}\sqrt{xz} J_{\lambda-1/2}(xz) \Big)(y), \quad x,y,t\in (0,\infty).$$ We can also obtain that if $f\in L^2(0,\infty)$, $$\label{D7} P_t^\lambda (f) = h_\lambda\Big(e^{-tz}h_\lambda(f)(z)\Big), \quad t \in (0,\infty),$$ and $$\label{A.3} \partial_t P_t^\lambda (f) = - h_\lambda\Big(z e^{-tz}h_\lambda(f)(z)\Big), \quad t \in (0,\infty).$$ Indeed, since the function $\sqrt{z} J_\lambda(z)$ is bounded on $(0,\infty)$, we get $$\begin{aligned} & \int_0^\infty |\sqrt{yz} J_{\lambda-1/2}(yz) e^{-tz}h_\lambda(f)(z) | \, dz \leq C \int_0^\infty e^{-tz} |h_\lambda(f)(z) |\, dz \\ & \qquad \qquad \leq C \Big(\int_0^\infty e^{-2tz} \, dz \Big)^{1/2} \|h_\lambda(f)\|_{L^2(0,\infty)} \leq \frac{C}{t^{1/2}} \|f\|_{L^2(0,\infty)} < \infty, \quad y, t \in (0,\infty),\end{aligned}$$ which allows us to establish . In analogous way the differentiation under the integral sign in can be justified. Also, since (see [@Le (5.3.5)]), $$\partial_y [(yz)^{-\nu} J_\nu(yz)] = -z(yz)^{-\nu} J_{\nu+1}(yz), \quad y,z \in (0,\infty),$$ it follows that $$\begin{aligned} \label{DyKernel} D_{\lambda ,y}[P_t^\lambda (x,y)] & = y^\lambda \partial_y \int_0^\infty (yz)^{-\lambda + 1/2} J_{\lambda-1/2}(yz) z^\lambda e^{-tz} \sqrt{xz} J_{\lambda-1/2}(xz) \, dz \nonumber\\ & =- \int_0^\infty \sqrt{yz} J_{\lambda+1/2}(yz) z e^{-tz} \sqrt{xz} J_{\lambda-1/2}(xz) \, dz \nonumber\\ & = -h_{\lambda +1}\Big( z e^{-tz} \sqrt{xz} J_{\lambda-1/2}(xz) \Big) (y), \quad x,y, t \in (0,\infty),\end{aligned}$$ and, for $f\in L^2(0,\infty )$, $$\label{DyPoisson} D_{\lambda ,y}[ P_t^\lambda(f)(y)] = -h_{\lambda +1}\Big( z e^{-tz} h_\lambda(f)(z) \Big) (y), \quad y,t \in (0,\infty).$$ Differentiation under the integral sign can be justified as above. On the other hand, since $h_\lambda(f) \in L^2(0,\infty)$, the dominated convergence theorem implies that $$\lim_{t \to 0^+}tz e^{-tz} h_\lambda(f)(z) = \lim_{t \to +\infty}tz e^{-tz} h_\lambda(f)(z) =\lim_{t \to +\infty}e^{-tz}h_\lambda(f)(z)=0, \quad \text{ in } L^2(0,\infty),$$ and $$\lim_{t \to 0^+}e^{-tz}h_\lambda(f)(z)=h_\lambda(f)(z),\quad \text{ in } L^2(0,\infty).$$ Then, from , and the $L^2$-boundedness of $h_\lambda$ we get that $$\label{D8} \lim_{t\to 0^+}t \partial_t (P_{t}^\lambda(f)(z)) = \lim_{t\to +\infty}t \partial_t (P_{t}^\lambda(f)(z))=\lim_{t\to +\infty} P_{t}^\lambda(f)(z)=0, \quad \text{ in } L^2(0,\infty),$$ and $$\label{A.4.1} \lim_{t\to 0^+}P_t^\lambda(f)(z)=f, \quad \text{ in } L^2(0,\infty).$$ Our next objective is to prove the following lemma. Let $f\in L^2(0,\infty)$. Then, $$\label{A.2} f(x) = 2 \lim_{{\varepsilon}\to 0^+} \int_{{\varepsilon}}^{1/{\varepsilon}} \int_0^\infty t \nabla_{\lambda,y} (P_t^\lambda (x,y)) \cdot \nabla_{\lambda,y}(P_t^\lambda (f)(y)) \, dy \, dt,$$ where the equality is understood in $L^2(0,\infty)$ and also in a distributional sense. By using and , Plancherel inequality for $h_\lambda$ leads to $$\begin{aligned} \int_0^\infty \partial_t P_t^\lambda (x,y) \, \partial_t P_t^\lambda(f)(y) dy & = \int_0^\infty h_\lambda\Big(z e^{-tz}\sqrt{xz} J_{\lambda-1/2}(xz) \Big)(y) \, h_\lambda\Big(z e^{-tz}h_\lambda(f)(z)\Big)(y) dy \\ & = \int_0^\infty z^2 e^{-2tz}\sqrt{xz} J_{\lambda-1/2}(xz)h_\lambda(f)(z) dz \\ & = \frac{1}{4} \partial_t^2 [P_{2t}^\lambda(f)(x)], \quad x, t \in (0,\infty).\end{aligned}$$ In analogous way from , and Plancherel equality for $h_{\lambda + 1}$ we obtain $$\begin{aligned} & \int_0^\infty D_{\lambda ,y}[P_t^\lambda (x,y)] \, D_{\lambda ,y}[P_t^\lambda(f)(y)] \, dy \\ & \qquad \qquad = \int_0^\infty h_{\lambda+1}\Big(z e^{-tz}\sqrt{xz} J_{\lambda-1/2}(xz) \Big)(y) \, h_{\lambda+1}\Big(z e^{-tz}h_\lambda(f)(z)\Big)(y) dy \\ & \qquad \qquad = \int_0^\infty z^2 e^{-2tz} h_\lambda(f)(z) \sqrt{xz} J_{\lambda-1/2}(xz) dz \\ & \qquad \qquad = \frac{1}{4} \partial_t^2 [P_{2t}^\lambda(f)(x)], \quad x, t \in (0,\infty).\end{aligned}$$ By partial integration we obtain $$\begin{aligned} &\int_{{\varepsilon}}^{1/{\varepsilon}} \int_0^\infty t \nabla_{\lambda,y} (P_t^\lambda (x,y)) \cdot \nabla_{\lambda,y}(P_t^\lambda (f)(y))\, dy \, dt = \frac{1}{2} \int_{{\varepsilon}}^{1/{\varepsilon}} t \partial_t^2 [P_{2t}^\lambda(f)(x)] \, dt \\ & \qquad \qquad = \frac{1}{2} \Big\{ t \partial_t (P_{2t}^\lambda(f)(x)) \Big]_{t={\varepsilon}}^{t=1/{\varepsilon}} - P_{2t}^\lambda(f)(x)\Big]_{t={\varepsilon}}^{t=1/{\varepsilon}} \Big\}, \quad x \in (0,\infty) \text{ and } 0<{\varepsilon}<1.\end{aligned}$$ We conclude from and that $$\lim_{{\varepsilon}\to 0^+} \int_{{\varepsilon}}^{1/{\varepsilon}} \int_0^\infty t \nabla_{\lambda,y} (P_t^\lambda (x,y)) \cdot \nabla_{\lambda,y}(P_t^\lambda (f)(y)) \, dy \, dt = \frac{f(x)}{2}, \quad \text{in } L^2(0,\infty),$$ and then, also, in a distributional sense. $BMO$ spaces associated with Bessel operators {#Sect3} ============================================= In this section we establish some properties for the functions in the space $BMO_{\textrm{o}}(\mathbb{R})$ that will be useful in the proof of Theorem \[Th1.3\]. As it was mentioned in the introduction, in [@BDT] Hardy spaces associated with Bessel operators $B_\lambda $ were introduced by using maximal operators. A function $f\in L^1(0,\infty )$ is in the Hardy space $H_\lambda ^1(0,\infty )$ provided that the maximal function $P_*^\lambda (f) \in L^1 (0,\infty )$. Here, $P_*^\lambda $ is defined by $$P_*^\lambda (f) :=\sup_{t>0}|P_t^\lambda (f)|,\quad f\in L^1(0,\infty ).$$ According to [@BDT Theorem 1.10 and Proposition 3.8] $H_\lambda^1(0,\infty )$ can be also defined by using the maximal operator associated to the heat semigroup $\{W_t^\lambda \}_{t>0}$ generated by $-B_\lambda$. The area integral defined by the Poisson semigroup $\{P_t^\lambda\}_{t>0}$, $g_\lambda (f)$ of $f\in L^1 (0,\infty )$ is defined by $$g_\lambda (f)(x) :=\Big(\int_{\Gamma _+(x)}|t\partial _tP_t^\lambda (f)(y)|^2\frac{dtdy}{t^2}\Big)^{1/2},\quad x\in (0,\infty ),$$ where $\Gamma _+(x):=\{(y,t)\in (0,\infty )\times (0,\infty ): |x-y|<t\}$, $x\in (0,\infty )$. In [@BCFR1 Proposition 4.1] it was proved that $f\in L^1(0,\infty )$ is in $H_\lambda^1(0,\infty )$ if and only if $g_\lambda (f)\in L^1(0,\infty )$. Actually, the space $H_\lambda ^1(0,\infty )$ does not depend on $\lambda $ because, according to [@BDT Theorem 1.10] and [@Fri Theorem 2.1], a function $f\in L^1 (0,\infty )$ is in $H_\lambda ^1(0,\infty )$ when and only when the odd extension $f_{\textrm{o}}$ of $f$ to $\mathbb{R}$ is in the classical Hardy space $H^1(\mathbb{R})$. Other characterizations for the space $H_\lambda ^1(0,\infty )$ can be found in [@BDLWY] (even in the multiparametric case). The dual space of $H_\lambda ^1(0,\infty )$ is the space $BMO_{\textrm{o}}$ ([@BCFR1 p. 466]). By using duality and the description of $H_\lambda ^1(0,\infty )$ in terms of $g_\lambda$ we deduce a new characterization of $BMO_{\textrm{o}}(\mathbb{R})$. According to (\[D3\]) we have that $P_t^\lambda (f)(x)<\infty$, for every $x,t\in (0,\infty )$, provided that $f$ is a complex measurable function on $(0,\infty )$ such that $$\int_0^\infty \frac{|f(x)|}{(1+x)^2}dx<\infty .$$ We say that a function $f\in L^1((0,\infty ),(1+x)^{-2}dx)$ is in $BMO(P^\lambda )$ when $$\|f\|_{BMO(P^\lambda )}:=\sup \frac{1}{|I|}\int_I|f(x)-P_{|I|}^\lambda (f)(x)|dx<\infty,$$ where the supremum is taken over all bounded intervals $I$ in $(0,\infty )$. We now characterize $BMO(P^\lambda )$ as the dual space of $H^1_\lambda (0,\infty )$. In order to do this we consider the odd-atoms introduced in [@Fri]. A measurable function $\mathfrak a$ on $(0,\infty )$ is an odd-atom when it satisfies one of the following properties: $(a)$ $\mathfrak a=\frac{1}{\delta}\chi _{(0,\delta)}$, for some $\delta >0$. Here $\chi _{(0,\delta)}$ denotes the characteristic function of $(0,\delta)$, for every $\delta >0$. $(b)$ There exists a bounded interval $I\subset (0,\infty)$ such that ${\mathop{\mathrm{supp}}}\mathfrak a\subset I$, $\int_I\mathfrak a(x)dx=0$ and $\|\mathfrak a\|_\infty \leq |I|^{-1}$. We say that a function $f\in L^1(0,\infty )$ is in $H_{\textrm{o}, at}^1(0,\infty )$ when, for every $j\in\mathbb{N}$, there exist $\lambda _j>0$ and an odd-atom $\mathfrak a_j$ such that $f=\sum_{j\in \mathbb{N}}\lambda _j\mathfrak a_j$, in $L^1(0,\infty)$, and $\sum_{j\in \mathbb{N}}\lambda _j<\infty$. We define, for every $f\in H_{\textrm{o}, at}^1(0,\infty )$, $$\|f\|_{H_{\textrm{o}, at}^1(0,\infty )} :=\inf\sum_{j\in \mathbb{N}}\lambda _j,$$ where the infimum is taken over all the sequences $\{\lambda_j\}_{j\in \mathbb{N}}\subset (0,\infty )$ such that $\sum_{j\in\mathbb{N}}\lambda _j<\infty$ and $f=\sum_{j\in \mathbb{N}}\lambda_j\mathfrak a_j$, in $L^1(0,\infty )$, where $\mathfrak a_j$ is an odd-atom, for every $j\in \mathbb{N}$. According to [@BDT Proposition 3.7] we have that $H_{\textrm{o}, at}^1(0,\infty )=H_\lambda ^1(0,\infty )$ algebraic and topologically. Note that this equality implies that $\mathcal{A}=\mbox{span}\{\mbox{odd atoms}\}$ is a dense subspace of $H_\lambda ^1(0,\infty )$. We now characterize $BMO(P^\lambda )$ as the dual space of $H_\lambda ^1(0,\infty )$. \[Prop2.1\] Let $\lambda >0$. - Let $f\in BMO(P^\lambda )$. We define the functional $T_f$ on $\mathcal{A}$ by $$T_f(b) :=\int_0^\infty f(x)b(x)dx,\quad b\in \mathcal{A}.$$ Then, $T_f$ can be extended to $H_\lambda ^1(0,\infty )$ as a bounded operator from $H_\lambda ^1(0,\infty )$ into $\mathbb{C}$. Furthermore, $$\|T_f\|_{(H_\lambda ^1(0,\infty ))'}\leq C\|f\|_{BMO(P^\lambda)},$$ where $C>0$ does not depend on $f$. - There exists $C>0$ such that, for every $T\in (H_\lambda ^1(0,\infty ))'$, there exists $f\in BMO(P^\lambda )$ such that $T=T_f$ on $\mathcal{A}$ and $$\|f\|_{BMO(P^\lambda )}\leq C\|T\|_{(H_\lambda ^1(0,\infty ))'}.$$ Since $f\in BMO(P^\lambda )$, we can affirm that the measure $\rho _f$ on $(0,\infty )\times (0,\infty )$ defined by $$d\rho _f(x,t) :=|t(\partial _tP_t^\lambda )(i_d -P_t^\lambda )f(x)|^2\frac{dxdt}{t},$$ where $i_d$ represents the identity operator, is Carleson and $$\|\rho _f\|_{\mathcal{C}}\leq C\|f\|^2_{BMO(P^\lambda )},$$ with $C>0$. In order to prove this assertion, we can proceed as in the proof of [@DY Lemma 4.6]. Indeed, it is sufficient to see that, there exists $C>0$ such that, for every bounded interval $I\subset (0,\infty )$ we have that $$\label{2.1} \int_{\widehat{I}}|t(\partial_tP_t^\lambda) (i_d-P_t^\lambda )(i_d-P_{|I|}^\lambda)f(x)|^2\frac{dxdt}{t}\leq C|I|\|f\|_{BMO(P^\lambda )}^2,$$ and $$\label{2.2} \int_{\widehat{I}}|t(\partial_tP_t^\lambda )(i_d-P_t^\lambda )P_{|I|}^\lambda (f)(x)|^2\frac{dxdt}{t}\leq C|I|\|f\|_{BMO(P^\lambda )}^2.$$ We consider the Littlewood-Paley type function $G_\lambda$ defined by $$G_\lambda (g)(x) :=\Big(\int_0^\infty |t(\partial _tP_t^\lambda )(i_d-P_t^\lambda )g(x)|^2\frac{dt}{t}\Big)^{1/2},\quad x\in (0,\infty ).$$ Let $g\in L^2(0,\infty )$. According to , and the fact that $h_\lambda ^2=i_d$, we can write $$t(\partial _tP_t^\lambda )(i_d-P_t^\lambda )g=-h_\lambda (tze^{-tz}(1-e^{-tz})h_\lambda (g)),\quad t>0.$$ Then, since $h_\lambda $ is a bounded operator from $L^2(0,\infty )$ into itself, we get $$\begin{aligned} \|G_\lambda (g)\|_2 &=\Big(\int_0^\infty \int_0^\infty |h_\lambda (tze^{-tz}(1-e^{-tz})h_\lambda (g))(x)|^2dx\frac{dt}{t}\Big)^{1/2}\\ &\leq C\Big(\int_0^\infty \int_0^\infty |tze^{-tz}(1-e^{-tz})|^2\frac{dt}{t}|h_\lambda (g)(z)|^2dz\Big)^{1/2}\\ &\leq C\|g\|_2.\end{aligned}$$ Hence, the sublinear operator $G_\lambda$ is bounded from $L^2(0,\infty )$ into itself. Let $I$ be a bounded interval in $(0,\infty )$. We now decompose the function $(i_d-P_{|I|}^\lambda )f$ as follows: $$(i_d-P_{|I|}^\lambda )f=\chi _{2I}(i_d-P_{|I|}^\lambda )f+\chi _{(0,\infty )\setminus 2I}(i_d-P_{|I|}^\lambda )f=:g_1+g_2.$$ The arguments in [@DY p. 956] (see also [@DY2]) allow us to obtain $$\label{2.3} \int_{\widehat{I}}|t(\partial _tP_t^\lambda )(i_d-P_t^\lambda )(g_1)(x)|^2\frac{dxdt}{t}\leq \|G_\lambda (g_1)\|_2^2\leq C\|g_1\|_2^2\leq C|I|\|f\|_{BMO(P^\lambda )}^2,$$ and also, by using Lemma \[D4\] and , that $$|t(\partial _tP_t^\lambda )(i_d-P_t^\lambda )(g_2)(x)|\leq C\frac{t}{|I|}\|f\|_{BMO(P^\lambda )}.$$ Then, we get $$\label{2.4} \int_{\widehat{I}}|t(\partial _tP_t^\lambda )(i_d-P_t^\lambda )(g_2)(x)|^2\frac{dxdt}{t}\leq C|I|\|f\|_{BMO(P^\lambda )}^2.$$ Inequality follows now from and . According again to Lemma \[D4\], and by proceeding as in the bottom of [@DY p. 956] we obtain $$|t(\partial _tP_t^\lambda )(i_d-P_t^\lambda )P_{|I|}^\lambda (f)(x)|\leq C\frac{t}{|I|}\|f\|_{BMO(P^\lambda )},$$ and can be established. If $F$ is a measurable function on $(0,\infty )\times (0,\infty )$ we define (see [@BCFR1 p. 488]) $$\Phi (F)(x) :=\sup_{I\subset (0,\infty ), I \;{\rm bounded},x\in I} \Big(\frac{1}{|I|}\int_0^{|I|}\int_I|F(y,t)|^2\frac{dydt}{t}\Big)^{1/2},\quad x\in (0,\infty ),$$ and $$\Psi (F)(x) :=\Big(\int_{\Gamma_+(x)}|F(y,t)|^2\frac{dydt}{t^2}\Big)^{1/2},\quad x\in (0,\infty ).$$ Suppose that $b\in \mathcal{A}$ and consider $F(x,t):=t(\partial _tP_t^\lambda )(i_d-P_t^\lambda )f$ and $G(x,t):=t\partial _tP_t^\lambda (b)(x)$, $x,t\in (0, \infty )$. Since $\rho _f $ is a Carleson measure we have that $\Phi (F)\in L^\infty (0,\infty )$ and $$\|\Phi (F)\|_\infty \leq C\|\rho _f\|_{\mathcal{C}}^{1/2}.$$ On the other hand, from [@BCFR1 Proposition 4.1], we have that $\Psi (G)\in L^1(0,\infty )$ and $\|\Psi(G)\|_1\leq C\|b\|_{H_\lambda ^1(0,\infty )}$. Then, according to [@BCFR1 Proposition 4.3] we get $$\begin{aligned} \label{2.5} &\int_0^\infty \int_0^\infty |t (\partial _tP_t^\lambda )(i_d-P_t^\lambda )f(y)||t\partial _tP_t^\lambda (b)(y)|\frac{dydt}{t}\nonumber\\ & \qquad \leq C\int_0^\infty \Phi (F)(y)\Psi (G)(y)dy \leq C\|f\|_{BMO(P^\lambda )}\|b\|_{H^1_\lambda (0,\infty )}.\end{aligned}$$ It follows that $$\int_0^\infty \int_0^\infty t(\partial _tP_t^\lambda )(i_d-P_t^\lambda )f(y)t\partial _tP_t^\lambda (b)(y)\frac{dydt}{t}=\lim_{\varepsilon \rightarrow 0^+,N\rightarrow \infty }H(\varepsilon ,N),$$ where, for every $0<\varepsilon <N<\infty$, $$H(\varepsilon ,N) :=\int_\varepsilon ^N \int_0^\infty t(\partial _tP_t^\lambda )(i_d-P_t^\lambda )f(y)t\partial _tP_t^\lambda (b)(y)\frac{dydt}{t}.$$ We can write $$t(\partial _tP_t^\lambda )(i_d-P_t^\lambda )=t\partial _tP_t^\lambda -\frac{1}{2}t\partial _tP_{2t}^\lambda ,\quad t>0.$$ Since $f\in L^1((0,\infty ), (1+x)^{-2}dx)$ and $b\in \mathcal{A}$, by using [@BCFR1 (4.8)] we deduce that $$\begin{aligned} &\int_0^\infty t(\partial _tP_t^\lambda )(i_d-P_t^\lambda )f(y)\partial _tP_t^\lambda (b)(y)dy\\ &\qquad \qquad =\int_0^\infty f(z)\int_0^\infty t\Big(\partial _tP_t^\lambda (y,z)-\frac{1}{2}\partial _tP_{2t}^\lambda (y,z)\Big)\partial _tP_t^\lambda (b)(y)dydz,\quad t>0,\end{aligned}$$ and, for every $0<\varepsilon<N<\infty$, $$H(\varepsilon ,N)=\int_0^\infty f(z)\int_\varepsilon ^N\int_0^\infty t\Big(\partial _tP_t^\lambda (y,z)-\frac{1}{2}\partial _tP_{2t}^\lambda (y,z)\Big)\partial _tP_t^\lambda (b)(y)dydtdz.$$ According to [@BCFR1 (4.15)] we obtain $$\lim_{\varepsilon \rightarrow 0^+, N\rightarrow \infty }\int_\varepsilon ^N\int_0^\infty t\partial _tP_t^\lambda (y,z)\partial _tP_t^\lambda (b)(y)dydt=\frac{b(z)}{4},\quad \mbox{ in }L^2(0,\infty ).$$ In a similar way we can see that $$\lim_{\varepsilon \rightarrow 0^+, N\rightarrow \infty }\int_\varepsilon ^N\int_0^\infty t\partial _tP_{2t}^\lambda (y,z)\partial _tP_t^\lambda (b)(y)dydt=\frac{2b(z)}{9},\quad \mbox{ in }L^2(0,\infty ).$$ Then, $$\lim_{\varepsilon \rightarrow 0^+, N\rightarrow \infty }\int_\varepsilon ^N\int_0^\infty t\Big(\partial _tP_t^\lambda (y,z)-\frac{1}{2}\partial _tP_{2t}^\lambda (y,z)\Big)\partial _tP_t^\lambda (b)(y)dydt=\frac{5b(z)}{36},$$ in $L^2(0,\infty )$. By using now dominated convergence theorem as in [@BCFR1 p. 492] we conclude that $$\label{2.6} \int_0^\infty \int_0^\infty t\partial _t(P_t^\lambda )(i_d-P_t^\lambda )f(x)\partial _tP_t^\lambda (b)(x)\frac{dxdt}{t}=\frac{5}{36}\int_0^\infty f(x)b(x)dx.$$ By combining and we get $$|T_f(b)|=\Big|\int_0^\infty f(x)b(x)dx\Big| \leq C\|f\|_{BMO(P^\lambda )}\|b\|_{H_\lambda ^1(0,\infty )}.$$ Assume that $T\in (H_\lambda ^1(0,\infty ))'$. There exists $f\in BMO_{\textrm{o}}(\mathbb{R})$ such that $Tg=T_fg$, for every $g\in \mathcal{A}$, and $\|f\|_{BMO_{\textrm{o}}(\mathbb{R})}\leq C\|T\|_{(H_\lambda ^1(0,\infty ))'}$ ([@BCFR1 p. 466] and [@BDT Theorem 1.10]). We are going to see that $f\in BMO(P^\lambda )$. Let $I$ be a bounded interval in $(0,\infty )$. We can write $$\begin{aligned} \frac{1}{|I|}\int_I|f(x)-P_{|I|}^\lambda (f)(x)|dx&\leq \frac{1}{|I|}\int_I|f(x)-f_I|dx+\frac{1}{|I|}\int_I|P_{|I|}^\lambda (f-f_I)(x)|dx\\ &\quad +\frac{|f_I|}{|I|}\int_I|P_{|I|}^\lambda (1)(x)-1|dx=:J_1+J_2+J_3.\end{aligned}$$ Since $f\in BMO_{\textrm{o}}(\mathbb{R})$, $J_1\leq \|f\|_{BMO_{\textrm{o}}(\mathbb{R})}$. According to we have that $$\begin{aligned} \label{3.6.1} |P_{|I|}^\lambda (f-f_I)(x)| &\leq C\int_0^\infty \frac{|I|}{|x-y|^2+|I|^2}|f(y)-f_I|dy \nonumber\\ & \leq C \Big(\int_{(0,\infty )\cap (x-|I|/2,x+|I|/2)}+\sum_{k\in \mathbb{N}}\int_{(0,\infty )\cap (B(x,2^k|I|)\setminus B(x,2^{k-1}|I|)}\Big)\frac{|I||f(y)-f_I|}{(x-y)^2+|I|^2}dy \nonumber\\ & \leq C\Big(\frac{1}{|I|}\int_{(0,\infty )\cap (2I)}|f(y)-f_I|dy+\sum_{k\in \mathbb{N}}\frac{1}{2^{2k}|I|}\int_{(0,\infty )\cap 2^{k+2}I}|f(y)-f_I|dy\Big) \nonumber\\ & \leq C\|f\|_{BMO_{\textrm{o}}(\mathbb{R})}\Big(1+\sum_{k\in \mathbb{N}}\frac{k}{2^k}\Big)\leq C\|f\|_{BMO_{\textrm{o}}(\mathbb{R})},\quad x\in I.\end{aligned}$$ Then, $J_2\leq C\|f\|_{BMO_{\textrm{o}}(\mathbb{R})}$. In order to estimate $J_3$ we distinguish two cases. We consider firstly that $x_I\leq |I|$. According to (\[D3\]) we get $$\label{dif1} |P_{|I|}^\lambda (1)(x)-1|\leq C\int_0^\infty \frac{|I|}{(x-y)^2+|I|^2}dy +1\leq C,\quad x\in (0,\infty ).$$ Then $$\label{J3} J_3\leq C|f_I|\leq \frac{C}{|I|}\int_I|f(y)|dy\leq C\frac{x_I+|I|}{|I|(x_I+|I|)}\int_0^{x_I+|I|}|f(y)|dy\leq C\|f\|_{BMO_{\textrm{o}}(\mathbb{R})}.$$ Suppose now that $I=(x_I-|I|/2, x_I+|I|/2)$, where $x_I>|I|$. We write $$\begin{aligned} P_{|I|}^\lambda (1)(x)-1 &=\int_0^\infty P_{|I|}^\lambda (x,y)dy-\frac{1}{\pi}\int_{-\infty }^{+\infty }\frac{|I|}{(x-y)^2+|I|^2}dy\\ &=\Big(\int_0^{x/2} +\int_{2x}^\infty \Big) P_{|I|}^\lambda (x,y)dy -\frac{1}{\pi }\Big(\int_{-\infty }^{x/2}+\int_{2x}^\infty \Big)\frac{|I|}{(x-y)^2+|I|^2}dy\\ &\quad +\int_{x/2}^{2x}\Big(P_{|I|}^\lambda (x,y)-\frac{1}{\pi }\frac{|I|}{(x-y)^2+|I|^2}\Big)dy\\ &=:\sum_{i=1}^3R_i(x),\quad x\in (0,\infty ).\end{aligned}$$ By using we get $$\begin{aligned} |R_1(x)+R_2(x)|& \leq C\left\{\Big(\int_0^{x/2}+\int _{2x}^\infty \Big)\frac{|I|}{(x-y)^2+|I|^2}dy+\int_0^\infty \frac{|I|}{(x+y)^2+|I|^2}dy \right\}\\ &\leq C|I|\left\{\Big(\int_0^{x/2}+\int_{2x}^\infty\Big)\frac{1}{(x-y)^2+|I|^2}dy +\int_{x/2}^{2x} \frac{1}{(x+y)^2+|I|^2}dy\right\}\\ &\leq C|I|\Big(\int_0^{x/2}\frac{dy}{x^2}+\int_{x/2}^\infty \frac{dy}{y^2}\Big) \leq C\frac{|I|}{x}, \quad x\in (0,\infty ).\end{aligned}$$ To analyze $R_3(x)$, $x\in (0,\infty )$, we write $$\begin{aligned} P_t^\lambda (x,y) &=\frac{2\lambda t(xy)^\lambda }{\pi }\Big(\int_0^{\pi /2}+\int_{\pi /2}^\pi \Big) \frac{(\sin \theta )^{2\lambda -1}}{((x-y)^2+t^2+2xy(1-\cos \theta ))^{\lambda +1}}d\theta\\ &=:P_t^{\lambda ,1}(x,y)+P_t^{\lambda ,2}(x,y),\quad x,y,t\in (0,\infty ),\end{aligned}$$ and $$\begin{aligned} P_t^{\lambda ,1}(x,y) &=\frac{2\lambda t(xy)^\lambda }{\pi }\Big\{\int_0^{\pi /2}\frac{(\sin \theta )^{2\lambda -1}-\theta ^{2\lambda -1}}{((x-y)^2+t^2+2xy(1-\cos \theta))^{\lambda +1}}d\theta \\ &\quad +\int_0^{\pi /2}\Big(\frac{\theta ^{2\lambda -1}}{((x-y)^2+t^2+2xy(1-\cos \theta))^{\lambda +1}}-\frac{\theta ^{2\lambda -1}}{((x-y)^2+t^2+xy\theta^2)^{\lambda +1}}\Big)d\theta \\ &\quad +\int_0^\infty \frac{\theta ^{2\lambda -1}}{((x-y)^2+t^2+xy\theta ^2)^{\lambda +1}}d\theta -\int_{\pi /2}^\infty \frac{\theta ^{2\lambda -1}}{((x-y)^2+t^2+xy\theta ^2)^{\lambda +1}}d\theta \Big\}\\ &=:\frac{2\lambda t(xy)^\lambda }{\pi }\sum_{i=1}^4I_i(x,y,t), \quad x,y,t\in (0,\infty ).\end{aligned}$$ We observe that $$\frac{2\lambda t(xy)^\lambda }{\pi }I_3(x,y,t)=\frac{1}{\pi}\frac{t}{(x-y)^2+t^2}=P_t(x-y),\quad x,y,t\in (0,\infty ),$$ and then, we obtain, for each $x,y,t\in (0,\infty )$, $$\label{decomposition} P_t^\lambda (x,y)-P_t(x-y)=\frac{2\lambda t(xy)^\lambda }{\pi }(I_1(x,y,t)+I_2(x,y,t)+I_4(x,y,t))+P_t^{\lambda ,2}(x,y).$$ We have that, $$|P_t^{\lambda,2}(x,y)|\leq C\frac{t(xy)^\lambda }{(x^2+y^2+t^2)^{\lambda +1}}\leq C\frac{t}{x^2},\quad x,y,t\in (0,\infty).$$ By using mean value theorem we get $$\begin{aligned} |I_1(x,y,t)|&\leq C\int_0^{\pi /2}\frac{\theta ^{2\lambda +1}}{((x-y)^2+t^2+xy\theta ^2)^{\lambda +1}}d\theta\leq C\int_0^{\pi /2}\frac{\theta ^{2\lambda +1}}{(|x-y|+t+x\theta )^{2\lambda +2}}d\theta\\ &\leq \frac{C}{x^{2\lambda +3/2}|x-y|^{1/2}},\quad 0<\frac{x}{2}<y<2x,\;t>0,\end{aligned}$$ and $$\begin{aligned} |I_2(x,y,t)|&\leq C\int_0^{\pi /2}\frac{xy\theta ^{2\lambda +3}}{((x-y)^2+t^2+xy\theta ^2)^{\lambda +2}}d\theta \leq C\int_0^{\pi /2}\frac{x^2\theta ^{2\lambda +3}}{(|x-y|+t+x\theta )^{2\lambda +4}}d\theta \\ &\leq \frac{C}{x^{2\lambda +3/2}|x-y|^{1/2}},\quad 0<\frac{x}{2}<y<2x,\;t>0.\end{aligned}$$ Also, for every $t>0$ and $0<\frac{x}{2}<y<2x$, we can write $$\begin{aligned} |I_4(x,y,t)|&=\frac{(xy)^{-\lambda}}{(x-y)^2+t^2}\int_{\frac{\pi}{2}\sqrt{xy/((x-y)^2+t^2)}}^\infty \frac{u^{2\lambda -1}}{(1+u^2)^{\lambda +1}}du\\ &\leq C\frac{x^{-2\lambda}}{(x-y)^2+t^2} \int_{\frac{\pi}{2}\sqrt{xy/((x-y)^2+t^2)}}^\infty \frac{du}{u^3}\leq \frac{C}{x^{2\lambda +2}}.\end{aligned}$$ From and by putting together the above estimates we get $$\left|P_t^\lambda (x,y)-P_t(x-y)\right| \leq Ct\Big(\frac{1}{x^{3/2}|x-y|^{1/2}}+\frac{1}{x^2}\Big),\quad 0<\frac{x}{2}<y<2x,\;t>0.$$ It follows that $$|R_3(x)| \leq C|I|\int_{x/2}^{2x}\Big(\frac{1}{x^{3/2}|x-y|^{1/2}}+\frac{1}{x^2}\Big)dy \leq C\frac{|I|}{x},\quad x\in (0,\infty ).$$ We obtain that $$\label{dif2} |P_{|I|}^\lambda (1)(x)-1|\leq C\frac{|I|}{x},\quad x\in (0,\infty ).$$ Since $x_I>|I|$, then $x_I-|I|/2>2^{-1}x_I$, and we get $$\begin{aligned} \label{J3b} J_3&\leq C|f_I|\int_I\frac{1}{x}dx\leq \frac{C}{|I|}\int_I|f(y)|dy\frac{|I|}{x_I-|I|/2}\nonumber\\ &\leq C\frac{x_I+|I|}{(x_I-|I|/2)(x_I+|I|)}\int_0^{x_I+|I|}|f(y)|dy\leq C\|f\|_{BMO_{\textrm{o}}(\mathbb{R})}.\end{aligned}$$ We conclude that $$\frac{1}{|I|}\int_I|f(x)-P_{|I|}^\lambda (f)(x)|dx\leq C\|f\|_{BMO_{\textrm{o}}(\mathbb{R})}.$$ Thus we prove that $f\in BMO(P^\lambda )$ and that $$\|f\|_{BMO(P^\lambda )}\leq C\|f\|_{BMO_{\textrm{o}}(\mathbb{R})}\leq C\|T\|_{(H_1^\lambda (0,\infty ))'}.$$ From Proposition \[Prop2.1\] and since $BMO_{\textrm{o}}(\mathbb{R})$ is the dual space of $H_\lambda ^1(0,\infty )$ ([@BCFR1 p. 466]), we can deduce the equality of $BMO(P^\lambda )$ and $BMO_{\textrm{o}}(\mathbb{R})$. \[cor2.1\] Let $\lambda >0$. Then, $BMO(P^\lambda )=BMO_{\textrm{o}}(\mathbb{R})$ algebraic and topologically. The following property will be very useful in the sequel. \[Prop2.2\] Let $\lambda >0$. There exists $C>0$ such that, for every $f\in BMO_{\textrm{o}}(\mathbb{R})$, $$|t\partial _tP_t^\lambda (f)(x)|+|tD_{\lambda ,x}P_t^\lambda (f)(x)|\leq C\|f\|_{BMO_{\textrm{o}}(\mathbb{R})},\quad x,t\in (0,\infty ).$$ Let $f\in BMO_{\textrm{o}}(\mathbb{R})$ and consider the odd extension $f_{\rm{o}}$ of $f$ to $\mathbb{R}$. We have that $f_{\rm{o}}\in BMO(\mathbb{R})$ and $$\begin{aligned} P_t(f_{\rm o})(x)&=\int_0^\infty (P_t(x-y)-P_t(x+y))f(y)dy,\quad x,t\in (0,\infty ).\end{aligned}$$ Since $f_{\rm o}\in BMO(\mathbb{R})$ and $\|f_{\rm o}\|_{BMO(\mathbb{R})}\leq C\|f\|_{BMO_{\rm o}(\mathbb{R})}$, from [@Wi (2), p. 22] (see also [@Ga]) we deduce that $$|t\partial _tP_t(f_{\rm o})(x)|+|t\partial_xP_t(f_{\rm o})(x)| \leq C\|f\|_{BMO_{\rm o}(\mathbb{R})},\quad t,x\in (0,\infty ).$$ Also, we can write $$\begin{aligned} \label{BesselClassical} t\partial _tP_t^\lambda (f)(x) &=t\partial _tP_t(f_{\rm o})(x)+\int_0^\infty t\partial _t(P_t^\lambda (x,y)-P_t(x-y)+P_t(x+y))f(y)dy\nonumber\\ &=t\partial _tP_t(f_{\rm o})(x) +\Big(\int_0^{x/2}+\int_{2x}^\infty\Big)t\partial _tP_t^\lambda (x,y)f(y)dy \nonumber\\ &\quad +\Big(\int_0^{x/2}+\int_{2x}^\infty\Big)t\partial _t(P_t(x+y)-P_t(x-y))f(y)dy\nonumber\\ &\quad +\int_{x/2}^{2x}t\partial _tP_t(x+y)f(y)dy+\int_{x/2}^{2x}t\partial _t(P_t^\lambda (x,y)-P_t(x-y))f(y)dy\nonumber\\ &=t\partial _tP_t(f_{\rm o})(x)+\sum_{i=1}^4 J_i(x,t),\quad x,t\in (0,\infty ).\end{aligned}$$ By using Lemma \[D4\] and (\[D2\]) we deduce that $$\begin{aligned} \label{J1} |J_1(x,t)| &\leq C\Big(\int_0^{x/2}+\int_{2x}^\infty \Big) \frac{t(xy)^\lambda |f(y)|}{((x-y)^2+t^2)^{\lambda +1}}dy\nonumber\\ &\leq C\Big(\int_0^{x/2}\frac{tx^{2\lambda}|f(y)|}{(x^2+t^2)^{\lambda +1}}dy+\int_{2x}^\infty \frac{t(xy)^\lambda|f(y)|}{(y^2+t^2)^{\lambda +1}}dy\Big)\nonumber\\ &\leq C\Big(\frac{1}{x}\int_0^x|f(y)|dy+x^\lambda\int_{2x}^\infty \frac{|f(y)|}{y^{\lambda +1}}dy\Big) \nonumber\\ &\leq C\Big(\|f\|_{BMO_{\textrm{o}}(\mathbb{R})}+x^\lambda\sum_{k=1}^\infty \int_{2^kx}^{2^{k+1}x}\frac{|f(y)|}{y^{\lambda +1}}dy\Big)\nonumber\\ &\leq C\Big(\|f\|_{BMO_{\textrm{o}}(\mathbb{R})}+x^\lambda \sum_{k=1}^\infty \frac{1}{(2^kx)^{\lambda +1}}\int_0^{2^{k+1}x}|f(y)|dy\Big)\nonumber\\ &\leq C\Big(\|f\|_{BMO_{\textrm{o}}(\mathbb{R})}+\sum_{k=1}^\infty 2^{-k\lambda }\frac{1}{2^{k+1}x}\int_0^{2^{k+1}x}|f(y)|dy\Big)\nonumber\\ &\leq C\|f\|_{BMO_{\rm{o}}(\mathbb{R})},\quad x,t\in (0,\infty ). \end{aligned}$$ On the other hand, $$\begin{aligned} |J_2(x,t)| &=\frac{1}{\pi}\Big|\Big(\int_0^{x/2}+\int_{2x}^\infty\Big)t\partial _t\Big(\frac{4xyt}{((x-y)^2+t^2)((x+y)^2+t^2)}\Big)f(y)dy\Big|\\ &=\frac{1}{\pi}\Big|\Big(\int_0^{x/2}+\int_{2x}^\infty\Big)\frac{4xyt}{((x-y)^2+t^2)((x+y)^2+t^2)} \\ & \qquad \qquad \qquad \qquad -\frac{8xyt^3((x+y)^2+(x-y)^2+2t^2)}{((x-y)^2+t^2)^2((x+y)^2+t^2)^2}f(y)dy\Big|\\ &\leq C\Big(\int_0^{x/2}+\int_{2x}^\infty\Big)\frac{xyt}{((x-y)^2+t^2)((x+y)^2+t^2)}|f(y)|dy\\ &\leq C\Big(\frac{tx^2}{(x^2+t^2)^2}\int_0^x|f(y)|dy+tx\int_{2x}^\infty \frac{y|f(y)|}{(y^2+t^2)^2}dy\Big)\\ &\leq C\Big(\frac{1}{x}\int_0^x|f(y)|dy+x\int_{2x}^\infty \frac{|f(y)|}{y^2}dy\Big) \leq C\|f\|_{BMO_{\rm{o}}(\mathbb{R})},\quad x,t\in (0,\infty ).\end{aligned}$$ The last inequality is obtained by proceeding as in for $\lambda =1$. Also, we have that $$|J_3(x,t)| =\frac{1}{\pi}\Big|\int_{x/2}^{2x}\Big(\frac{t}{(x+y)^2+t^2}-\frac{2t^3}{((x+y)^2+t^2)^2}\Big)f(y)dy\Big| \leq \frac{C}{x}\int_{x/2}^{2x}|f(y)|dy \leq C\|f\|_{BMO_{\rm o}(\mathbb{R})}.$$ Finally, in order to estimate $J_4(x,t)$, $x,t\in (0,\infty )$, we consider and write, for each $x,y,t\in (0,\infty )$, $$\label{diferencia} t\partial _t(P_t^\lambda (x,y)-P_t(x-y))=\frac{2\lambda }{\pi}(xy)^\lambda\sum_{i=1,2,4}t\partial _t(tI_i(x,y,t))+t\partial _tP_t^{\lambda ,2}(x,y).$$ By using the mean value theorem we get $$\begin{aligned} |t\partial _t(tI_1(x,y,t))| &=\Big|tI_1(x,y,t)-2(\lambda +1)t^3\int_0^{\pi /2}\frac{(\sin \theta )^{2\lambda -1}-\theta ^{2\lambda -1}}{((x-y)^2+t^2+2xy(1-\cos \theta ))^{\lambda +2}}d\theta\Big|\\ &\leq Ct\int_0^{\pi /2}\frac{|(\sin \theta )^{2\lambda -1}-\theta ^{2\lambda -1}|}{((x-y)^2+t^2+2xy(1-\cos \theta ))^{\lambda +1}}d\theta\\ &\leq Ct\int_0^{\pi /2}\frac{\theta ^{2\lambda +1}}{(|x-y|+t+x\theta )^{2\lambda +2}}d\theta \leq \frac{C}{x^{2\lambda +1}},\quad 0<\frac{x}{2}<y<2x,\;t>0, \end{aligned}$$ and $$\begin{aligned} |t\partial _t(tI_2(x,y,t))| &=\Big|tI_2(x,y,t)-2(\lambda +1)t^3\int_0^{\pi /2}\Big[\frac{\theta ^{2\lambda -1}}{((x-y)^2+t^2+2xy(1-\cos \theta ))^{\lambda +2}} \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad - \frac{\theta ^{2\lambda -1}}{((x-y)^2+t^2+xy\theta ^2)^{\lambda +2}}\Big]d\theta \Big| \\ & \leq Cxy\Big(\int_0^{\pi /2}\frac{t\theta ^{2\lambda +3}}{((x-y)^2+t^2+xy\theta ^2)^{\lambda +2}}d\theta +\int_0^{\pi /2}\frac{t^3\theta ^{2\lambda +3}}{((x-y)^2+t^2+xy\theta ^2)^{\lambda +3}}d\theta \Big)\\ & \leq Ctx^2\int_0^{\pi /2}\frac{\theta ^{2\lambda +3}}{(|x-y|+t+x\theta )^{2\lambda +4}}d\theta \leq \frac{C}{x^{2\lambda +1}}, \quad 0<\frac{x}{2}<y<2x,\;t>0.\end{aligned}$$ Also, we obtain, when $0<x/2<y<2x$ and $t>0$, $$\begin{aligned} |t\partial _t(tI_4(x,y,t))| &=\Big|tI_4(x,y,t)-2(\lambda +1)t^3\int_{\pi /2}^\infty \frac{\theta ^{2\lambda -1}}{((x-y)^2+t^2+xy\theta ^2)^{\lambda +2}}d\theta \Big|\\ &\leq Ct\int_{\pi /2}^\infty \frac{\theta ^{2\lambda -1}}{((x-y)^2+t^2+xy\theta ^2)^{\lambda +1}}d\theta\leq \frac{C}{x^{2\lambda +1}}\int_{\pi /2}^\infty \frac{d\theta}{\theta ^2}\leq \frac{C}{x^{2\lambda +1}}.\end{aligned}$$ Finally, we can write $$\begin{aligned} |t\partial _t(P_t^{\lambda ,2}(x,y))| &=\Big|P_t^{\lambda ,2}(x,y)-\frac{4\lambda (\lambda +1)}{\pi}t^3 (xy)^\lambda \int_{\pi /2}^\pi\frac{(\sin \theta )^{2\lambda -1}}{((x-y)^2+t^2+2xy(1-\cos \theta))^{\lambda +2}}d\theta \Big|\\ &\leq Ctx^{2\lambda }\int_{\pi /2}^\pi \frac{\theta ^{2\lambda -1}}{(x+y+t)^{2\lambda +2}}d\theta\leq \frac{C}{x}, \quad 0<\frac{x}{2}<y<2x,\;t>0..\end{aligned}$$ From and by combining the above estimates, it follows that $$J_4(x,t)\leq \frac{C}{x}\int_0^{2x}|f(y)|dy\leq C\|f\|_{BMO_{\rm o}(\mathbb{R})},\quad x,t\in (0,\infty ).$$ Equality allows us to conclude that $$|t\partial _tP_t^\lambda (f)(x)|\leq C\|f\|_{BMO_{\rm o}(\mathbb{R})},\quad x,t \in (0,\infty ).$$ We are going to see now that $$|tD_{\lambda ,x}P_t^\lambda(f)(x)|\leq C\|f\|_{BMO_{\rm o}(\mathbb{R})},\quad x,t\in (0,\infty ).$$ We use a decomposition similar to and write $$\begin{aligned} \label{BesselClassicalD} tD_{\lambda ,x}P_t^\lambda (f)(x) &=t\partial _xP_t(f_{\rm o})(x)+t\int_0^\infty [D_{\lambda ,x}(P_t^\lambda (x,y))-\partial _x(P_t(x-y)+P_t(x+y))]f(y)dy\nonumber\\ &=t\partial _xP_t(f_{\rm o})(x) +\Big(\int_0^{x/2}+\int_{2x}^\infty\Big)tD_{\lambda ,x}(P_t^\lambda (x,y))f(y)dy\nonumber\\ &\quad +\Big(\int_0^{x/2}+\int_{2x}^\infty\Big)t\partial _x(P_t(x+y)-P_t(x-y))f(y)dy\nonumber\\ &\quad +\int_{x/2}^{2x}t\partial _xP_t(x+y)f(y)dy+\int_{x/2}^{2x}t[D_{\lambda ,x}(P_t^\lambda (x,y))-\partial _x(P_t(x-y))]f(y)dy\nonumber\\ &=:t\partial _xP_t(f_{\rm o})(x)+\sum_{i=1}^4 H_i(x,t),\quad x,t\in (0,\infty ).\end{aligned}$$ By using Lemma \[D4\] and (\[D2\]) in the same way as in we obtain $$|H_1(x,t)|\leq C\|f\|_{BMO_{\rm o}(\mathbb{R})},\quad x,t\in (0,\infty ).$$ On the other hand, $$\begin{aligned} |H_2(x,t)| &=\frac{1}{\pi}\Big|\Big(\int_0^{x/2}+\int_{2x}^\infty\Big)\partial _x\Big(\frac{4xyt^2 }{((x-y)^2+t^2)((x+y)^2+t^2)}\Big)f(y)dy\Big|\\ &\hspace{-1cm}=\frac{1}{\pi}\Big|\Big(\int_0^{x/2}+\int_{2x}^\infty\Big)\frac{4yt^2}{((x-y)^2+t^2)((x+y)^2+t^2)}\\ &\quad -4xyt^2\frac{2(x-y)((x+y)^2+t^2)+2(x+y)((x-y)^2+t^2)}{((x-y)^2+t^2)^2((x+y)^2+t^2)^2}f(y)dy\Big|\\ &\hspace{-1cm}\leq C\Big(\int_0^{x/2}+\int_{2x}^\infty\Big)\frac{yt^2}{((x-y)^2+t^2)((x+y)^2+t^2)}\Big(1+\frac{x|x-y|}{(|x-y|+t)^2}+ \frac{x(x+y)}{(x+y+t)^2}\Big)|f(y)|dy\\ &\hspace{-1cm}\leq C\Big(\frac{xt^2}{(x^2+t^2)^2}\int_0^x|f(y)|dy+\int_{2x}^\infty \frac{yt^2|f(y)|}{(y^2+t^2)(x+y+t)^2}dy\Big)\\ &\hspace{-1cm} \leq C\Big(\frac{1}{x}\int_0^x|f(y)|dy+\int_{2x}^\infty \frac{t|f(y)|}{(x+y+t)^2}dy\Big)\\ &\hspace{-1cm} \leq C\Big(\|f\|_{BMO_{\rm{o}}(\mathbb{R})} + \Big(\int_{2x}^{2x+t}+\sum_{k=0}^\infty \int_{2x+2^kt}^{2x+2^{k+1}t}\Big)\frac{t}{(x+y+t)^2}|f(y)|dy\Big)\\ &\hspace{-1cm} \leq C\Big(\|f\|_{BMO_{\rm{o}}(\mathbb{R})}+\frac{1}{2x+t}\int_0^{2x+t}|f(y)|dy +\sum_{k=0}^\infty 2^{-k}\frac{1}{2x+2^{k+1}t}\int_0^{2x+2^{k+1}t}|f(y)|dy\Big)\\ &\hspace{-1cm} \leq C\|f\|_{BMO_{\rm o}(\mathbb{R})},\quad x,t\in (0,\infty ).\end{aligned}$$ Also, it follows that $$|H_3(x,t)| =\frac{2}{\pi}\Big|\int_{x/2}^{2x}\frac{(x+y)t^2}{((x+y)^2+t^2)^2}f(y)dy\Big| \leq \frac{C}{x}\int_0^{2x}|f(y)|dy\leq C\|f\|_{BMO_{\rm o}(\mathbb{R})}.$$ We deal now with $H_4(x,t)$, $x,t\in (0,\infty )$. From we have that $$\begin{aligned} \label{Dlambda} D_{\lambda ,x}P_t^\lambda (x,y)-\partial _xP_t(x-y)&=\partial _x[P_t^\lambda (x,y)-P_t(x-y)]-\frac{\lambda}{x}P_t^\lambda (x,y)\nonumber\\ &\hspace{-2cm}=\frac{2\lambda}{\pi}\sum_{i=1,2,4}t\partial_x[(xy)^\lambda I_i(x,y,t)]+\partial_xP_t^{\lambda , 2}(x,y)-\frac{\lambda}{x}P_t^\lambda (x,y),\quad x,y,t\in (0,\infty ).\end{aligned}$$ Again by using the mean value theorem we get $$\begin{aligned} |t\partial _x[(xy)^\lambda I_1(x,y,t)]| &=\Big| \frac{\lambda t}{x} (xy)^\lambda I_1(x,y,t)\\ & \qquad -2(\lambda +1)t(xy)^\lambda\int_0^{\pi /2}\frac{[(\sin \theta )^{2\lambda -1}-\theta ^{2\lambda -1}][(x-y)+y(1-\cos \theta)]}{((x-y)^2+t^2+2xy(1-\cos \theta ))^{\lambda +2}}d\theta\Big|\\ &\leq Ctx^{2\lambda} \Big(\frac{1}{x}\int_0^{\pi /2}\frac{\theta ^{2\lambda +1}}{(|x-y|+t+x\theta )^{2\lambda +2}}d\theta+\int_0^{\pi /2}\frac{\theta ^{2\lambda +1}(|x-y|+y\theta ^2)}{(|x-y|+t+x\theta )^{2\lambda +4}}d\theta\Big)\\ &\leq \frac{C}{tx}\Big(\int_0^{\pi /2}\theta d\theta +\int_0^{\pi/2}(1+\theta )d\theta \Big) \leq \frac{C}{tx},\quad 0<\frac{x}{2}<y<2x,\;t>0, \end{aligned}$$ and $$\begin{aligned} |t\partial _x[(xy)^\lambda I_2(x,y,t)]| &=\Big|\frac{\lambda t}{x} (xy)^\lambda I_2(x,y,t)\\ &\quad -2(\lambda +1)t(xy)^\lambda\int_0^{\pi /2}\theta ^{2\lambda -1}\Big[\frac{(x-y)+y(1-\cos \theta)}{((x-y)^2+t^2+2xy(1-\cos \theta ))^{\lambda +2}} \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad - \frac{(x-y)+y\theta ^2}{((x-y)^2+t^2+xy\theta ^2)^{\lambda +2}} \Big]d\theta\Big| \\ &\leq Ct(xy)^\lambda \Big\{\frac{1}{x}\int_0^{\pi /2}\frac{\theta ^{2\lambda +3}xy}{(|x-y|+t+x\theta )^{2\lambda +4}}d\theta \\ &\quad +\int_0^{\pi /2}\theta ^{2\lambda +3}\Big(\frac{y}{(|x-y|+t+x\theta )^{2\lambda +4}}+\frac{xy(|x-y|+y\theta^2 )}{(|x-y|+t+x\theta )^{2\lambda +6}}\Big)d\theta \Big\}\\ &\leq C\Big(tx^{2\lambda +1}\int_0^{\pi /2}\frac{\theta ^{2\lambda +2}}{(|x-y|+t+x\theta )^{2\lambda +4}}d\theta \Big) \leq \frac{C}{tx},\quad 0<\frac{x}{2}<y<2x,\;t>0.\end{aligned}$$ Now, we write $$\begin{aligned} |t\partial _x((xy)^\lambda I_4(x,y,t))|&=\Big|\frac{\lambda t}{x} (xy)^\lambda I_4(x,y,t)\\ &\quad -2(\lambda +1)t(xy)^\lambda \int_{\pi /2}^\infty\frac{\theta ^{2\lambda -1}(x-y+y\theta ^2)}{((x-y)^2+t^2+xy\theta ^2)^{\lambda +2}}d\theta \Big|\\ &\leq Ctx^{2\lambda -1}\int_{\pi /2}^\infty \frac{\theta ^{2\lambda -1}}{(|x-y|+t+x\theta )^{2\lambda +2}}d\theta \\ &=C\frac{t}{x(|x-y|+t)^2}\int_{\frac{\pi x}{2(|x-y|+t)}}^\infty \frac{u^{2\lambda -1}}{(1+u)^{2\lambda +2}}du\leq \frac{C}{tx}, \quad 0<\frac{x}{2}<y<2x,\;t>0.\end{aligned}$$ Finally, it is clear from that $$\frac{1}{x}|P_t^\lambda (x,y)|\leq \frac{C}{tx},\quad x,t\in (0,\infty ),$$ and, also, $$\begin{aligned} |\partial _x(P_t^{\lambda ,2}(x,y))| &=\Big|\frac{\lambda }{x}P_t^{\lambda ,2}(x,y)-\frac{4\lambda(\lambda +1)}{\pi}t(xy)^\lambda \int_{\pi /2}^\pi\frac{(\sin \theta )^{2\lambda -1}[x-y+y(1-\cos \theta)]}{((x-y)^2+t^2+2xy(1-\cos \theta))^{\lambda +2}}d\theta \Big|\\ &\leq Ctx^{2\lambda -1}\int_{\pi /2}^\pi \frac{\theta ^{2\lambda -1}}{(x+y+t)^{2\lambda +2}}d\theta\leq \frac{C}{tx}, \quad 0<\frac{x}{2}<y<2x,\;t>0.\end{aligned}$$ By combining the above estimates and taking into account it follows that $$H_4(x,t)\leq \frac{C}{x}\int_0^{2x}|f(y)|dy\leq C\|f\|_{BMO_{\rm o}(\mathbb{R})},\quad x,t\in (0,\infty ).$$ Then, from we conclude that $$|D_{\lambda ,x}P_t^\lambda (f)(x)|\leq C\|f\|_{BMO_{\rm o}(\mathbb{R})},\quad x,t\in (0,\infty ).$$ Proof of Theorem \[Th1.3\] {#Sect4} ========================== Suppose that $\mu$ is a Carleson measure on $(0,\infty)\times (0,\infty )$. According to Corollary \[cor2.1\] in order to see that $S_{\mu,P^\lambda}\in BMO_{\rm o}(\mathbb R)$ it is sufficient to see that there exists $C>0$ such that, for every bounded interval $I\subset (0,\infty)$, $$\label{C.1} \frac{1}{|I|}\int_I|S_{\mu,P^\lambda}(x)-P_{|I|}^\lambda(S_{\mu,P^\lambda})(x)|dx\leq C\|\mu\|_{\mathcal C}.$$ We proceed as in the proof of [@CDLSY Proposition 2.5]. Let $I$ be a bounded interval in $(0,\infty)$. We can write $$\begin{aligned} & \int_I |S_{\mu,P^\lambda}(x)-P_{|I|}^\lambda(S_{\mu,P^\lambda})(x)|dx \\ & \qquad \leq C \int_I\int_{(0,\infty)^2}|P_t^\lambda(x,y)-P_{t+|I|}^\lambda(x,y)|d\mu(y,t)dx \\ & \qquad \leq\Big(\int_I\int_{\widehat{2I}}+\int_I\int_{(0,\infty)^2\setminus \widehat{2I}}\Big)|P_t^\lambda(x,y)-P_{t+|I|}^\lambda(x,y)|d\mu(y,t)dx \\ & \qquad =:I_1+I_2.\end{aligned}$$ According to (\[D3\]), since $\mu$ is a Carleson measure on $(0,\infty)\times (0,\infty )$, we get $$\begin{aligned} I_1\leq C & \int_{\widehat{2I}}\int_I\Big(\frac{t}{(x-y)^2+t^2}+ \frac{t+|I|}{(x-y)^2+(t+|I|)^2}\Big)dxd\mu(y,t) \leq C\mu(\widehat{2I})\leq C|I|\|\mu\|_{\mathcal C}.\end{aligned}$$ Also by Lemmma \[D4\] and (\[D3\]), we obtain $$\begin{aligned} I_2 &\leq C \int_I\int_{(0,\infty)^2\setminus \widehat{2I}}\int_0^{|I|}|\partial_sP_{t+s}^\lambda(x,y)|dsd\mu(y,t)dx \\ & \leq C\sum_{k=1}^\infty\int_I\int_0^{|I|}\int_{\widehat{2^{k+1}I}\setminus\widehat{2^{k}I}}\frac{1}{(x-y)^2+(s+t)^2}d\mu(y,t)dsdx \\ & \leq C\sum_{k=1}^\infty\mu(\widehat{2^{k+1}I})\frac{1}{(2^{k+1}|I|)^2}|I|^2\leq C|I|\|\mu\|_{\mathcal C}.\end{aligned}$$ Thus, (\[C.1\]) is proved. We will use the procedure developed by Wilson ([@Wi]) (see also [@CDLSY]). We need to make modifications and to justify each step in our setting. Let $Q$ be a bounded interval in $(0,\infty )$. In what follows we consider right-open intervals and denote by $x_Q$ the center of $Q$, and by $t_Q$ the length of $Q$. Assume that $f\in BMO_{\rm o}(\mathbb{R})$ with $\mbox{supp }f\subset (0,1)$. We consider $u(x,t):=P_t^\lambda (f)(x)$, $x,t\in (0,\infty )$, and take $Q_0:=[0,2)$. In what follows we consider right-open intervals. We now construct the $k$-th generation of subintervals of $Q_0$ as follows. By $A$ we denote a positive constant that will be fixed later. The 0-th generation is defined by $G_0:=\{Q_0\}$. For every $k \in {\mathbb{N}}$, the $(k+1)$-th generation $G_{k+1}$ is defined recursively as follows. A dyadic interval $Q\subset Q_0$ is in $G_{k+1}$ when - there exists $Q_1\in G_k$ such that $Q\subset Q_1$, - $Q$ is a maximal dyadic with respect to the property $$|x_Q^{-\lambda }u(x_Q,t_Q)-x_{Q_1}^{-\lambda }u(x_{Q_1},t_{Q_1})|>Ax_Q^{-\lambda}.$$ Note that the properties of the dyadic intervals and the maximal property ($b$) imply that, if $k\in \mathbb{N}$ and $Q_1,Q_2\in G_k$, then $Q_1=Q_2$ or $Q_1\cap Q_2=\emptyset$. For every $k\in \mathbb{N}$ and $Q\in G_k$ we define the set $$\Sigma_Q :=\widehat{Q}\setminus \bigcup_{Q'\subset Q, \, Q'\in G_{k+1}}\widehat{Q'}.$$ In the following figure where a possible $\Sigma _Q$ is represented, the dark grey squares are the Carleson boxes of those cubes $Q'\subset Q$ that belong to $G_{k+1}$. (2.2,0) – (2.2,2); at (2.5,1) [$|Q|$]{}; at (1,-0.3) [$Q$]{}; (0,0.5)–(0.5,0.5)–(0.5,0)–(0.75,0)–(0.75,0.25)–(0.5,0.25)–(1,0.25)–(1,0)–(1.125,0)–(1.125,0.125)–(1.25,0.125)–(1.25,0)–(1.5,0)–(1.5,0.125)–(1.625,0.125)–(1.625,0)–(2,0)–(2,2)–(0,2); (0,0) rectangle (0.5,0.5); (0.75,0) rectangle (1,0.25); (1.125,0) rectangle (1.25,0.125); (1.5,0) rectangle (1.625,0.125); at (1,1.2)[$\Sigma _Q$]{}; (0,0) – (0,2) ; (0,2) – (2,2); (0,0) – (2,0); (2,0) – (2,2); The set $\Sigma_Q$, $Q\in G_k$, $k\in \mathbb{N}$, can be written in a different and useful way. For every interval $J$ we define $T(J)$ as follows $$T(J) :=\Big\{(x,t): x\in J\mbox{ and } \frac{\ell (J)}{2}\leq t< \ell (J)\Big\}.$$ It is clear that, for every dyadic interval $S\subset (0,\infty)$, we have that $$\widehat{S} =\bigcup_{J\subset S, \, J \;{\rm dyadic }}T(J).$$ Then, for every $k\in \mathbb{N}$ and $Q\in G_k$, we have that $$\Sigma_Q=\bigcup _{J\in \mathcal{A}(Q)}T(J),$$ where $\mathcal{A}(Q):=\{J\mbox{ dyadic}: J\subseteq Q, J\cap S^{\rm c}\not=\emptyset, \mbox{ for every }S\in G_{k+1}\}$. Now, take $k\in \mathbb{N}$ and $Q\in G_k$. We are going to see that $$\label{3.1} |x^{-\lambda }u(x,t)-x_Q^{-\lambda }u(x_Q,t_Q)|\leq Cx^{-\lambda }(A+\|f\|_{BMO_{\rm o}(\mathbb{R})}),\quad (x,t)\in \Sigma _Q.$$ Here $C>0$ does not depend on $k$ or $Q$. Suppose that $(x,t)\in \Sigma _Q$. There exists $J\in \mathcal{A}(Q)$ such that $(x,t)\in T(J)$. According to the definition of $G_{k+1}$ and since $x\leq 2x_J$ we get $$|x^{-\lambda }_Qu(x_Q,t_Q)-x_J^{-\lambda }u(x_J,t_J)|\leq Ax_J^{-\lambda }\leq 2^\lambda Ax^{-\lambda }.$$ On the other hand, for some $z$ in the segment joining $x$ and $x_J$ and for some $s$ in the segment joining $t$ and $t_J$, we have that $$x^{-\lambda }u(x,t)-x_J^{-\lambda}u(x_J,t_J)=\partial _z(z^{-\lambda }u(z,t))(x_J-x)+\partial _s(x_J^{-\lambda }u(x_J,s))(t_J-t).$$ Since $x\leq 2x_J$, from Proposition \[Prop2.2\] it follows that $$|x^{-\lambda}u(x,t)-x_J^{-\lambda}u(x_J,t_J)|\leq Cx^{-\lambda }\|f\|_{BMO_{\rm o}(\mathbb{R})},$$ and is checked. Next, we show that $$\label{3.2} \sum_{J\subset Q, \, J\in G_{k+1}}|J|\leq \frac{C}{A}|Q| \, \|f\|_{BMO_{\rm o}(\mathbb{R})}.$$ For that, we write $$\sum_{J\subset Q, J\in G_{k+1}}|J|\leq \frac{1}{A}\sum_{J\subset Q, J\in G_{k+1}}|J| \, x_J^\lambda |x_Q^{-\lambda}u(x_Q,t_Q)-x_J^{-\lambda }u(x_J,t_J)|,$$ and use the following decomposition for every $J\in G_{k+1}$, $J\subset Q$, $$\begin{aligned} x_Q^{-\lambda}u(x_Q,t_Q)-x_J^{-\lambda }u(x_J,t_J)&=x_Q^{-\lambda }P_{t_Q}^\lambda (f-f_Q)(x_Q)-x_J^{-\lambda}P_{t_J}^\lambda (f-f_J)(x_J)\\ &\quad +x_Q^{-\lambda}f_Q[P_{t_J}^\lambda (1)(x_Q)-1]- x_J^{-\lambda }f_J[P_{t_J}^\lambda (1)(x_J)-1]\\ &\quad +[x_Q^{-\lambda }f_Q-x_J^{-\lambda }f_J]\\ &=:\sum_{i=1}^5H_i (J) .\end{aligned}$$ Let $J\in G_{k+1}$, $J\subset Q$. According to and since $x_J\leq 2x_Q$ we obtain $$x_J^\lambda |H_1(J)+H_2(J)|\leq C\Big[\Big(\frac{x_J}{x_Q}\Big)^\lambda +1\Big]\|f\|_{BMO_{\rm o}(\mathbb{R})}\leq C\|f\|_{BMO_{\rm o}(\mathbb{R})}.$$ By using and and proceeding as in and in we also get $$x_J^\lambda |H_3(J)+H_4(J)|\leq C(|f_Q||P_{t_Q}^\lambda (1)(x_Q)-1|+|f_J||P_{t_J}^\lambda (1)(x_J)-1|)\leq C\|f\|_{BMO_{\rm o}(\mathbb{R})}.$$ We now study $H_5$. We can write $$\begin{aligned} x_J^\lambda H_5&\leq \Big|\Big[\Big(\frac{x_J}{x_Q}\Big)^{\lambda }-1\Big]f_Q\Big|+|f_Q-f_J|.\end{aligned}$$ If $x_Q\leq t_Q$, then, as in , $$\Big|\Big[\Big(\frac{x_J}{x_Q}\Big)^{\lambda }-1\Big]f_Q\Big|\leq C|f_Q|\leq C\|f\|_{BMO_{\rm o}(\mathbb{R})}.$$ In the case that $x_Q>t_Q$, since $J\subset Q$, it follows that $x_J/x_Q\subset (1/2,3/2)$, and then, by applying the mean value theorem we get $$\Big|\Big[\Big(\frac{x_J}{x_Q}\Big)^{\lambda }-1\Big]f_Q\Big|\leq C\frac{|x_Q-x_J|}{x_Q}|f_Q|\leq C\frac{t_Q}{x_Q}|f_Q|\leq C\frac{x_Q+t_Q}{x_Q}\|f\|_{BMO_{\rm o}(\mathbb{R})}\leq C\|f\|_{BMO_{\rm o}(\mathbb{R})}.$$ By combining the above estimates we conclude that $$\begin{aligned} \sum_{J\subset Q, J\in G_{k+1}}|J|& \leq \frac{C}{A}\sum_{J\subset Q, J\in G_{k+1}}|J|(\|f\|_{BMO_{\rm o}(\mathbb{R})}+|f_Q-f_J|)\\ &\leq \frac{C}{A}\Big(|Q|\|f\|_{BMO_{\rm o}(\mathbb{R})}+\sum_{J\subset Q, J\in G_{k+1}}\int_J|f(y)-f_Q|dy\Big)\\ &\leq \frac{C}{A}\Big(|Q|\|f\|_{BMO_{\rm o}(\mathbb{R})}+\int_Q|f(y)-f_Q|dy\Big)\leq \frac{C}{A}|Q|\|f\|_{BMO_{\rm o}(\mathbb{R})},\end{aligned}$$ and is established. By choosing $A:=2C(1+\|f\|_{BMO_{\rm o}(\mathbb{R})})$ we obtain $$\label{mitad} \sum_{J\subset Q, \, J\in G_{k+1}}|J|\leq \frac{|Q|}{2},$$ for every $Q\in G_k$, $k\in \mathbb{N}$. Another helpful property is the following. Let $0 \leq a < b < \infty$ and $0<c<d<\infty$. For every $\alpha \in \mathbb{R}$, we have that $$\begin{aligned} \label{A.5} 2\int_c^d\int_a^bt\nabla_{\lambda,y}(P_t^\lambda(x,y))\cdot \nabla_{\lambda,y}(P_t^\lambda(f)(y)-\alpha y^\lambda)dydt& \nonumber \\ &\hspace{-7cm} =\int_a^b \Big[t\partial_t(P_t^\lambda(x,y))(P_t^\lambda(f)(y)-\alpha y^\lambda )+tP_t^\lambda(x,y)\partial_t(P_t^\lambda(f)(y)) \nonumber\\ & \hspace{-7cm} \quad \quad - P_t^\lambda(x,y)(P_t^\lambda(f)(y)-\alpha y^\lambda)\Big]_{t=c}^{t=d} dy\nonumber\\ & \hspace{-7cm}\quad + \int_c^dt\Big[P_t^\lambda(x,y)D_{\lambda ,y}(P_t^\lambda(f)(y))+D_{\lambda ,y}(P_t^\lambda(x,y)) (P_t^\lambda(f)(y)-\alpha y^\lambda)\Big]_{y=a}^{y=b} dt\nonumber\\ &\hspace{-7cm} :=\int_a^b \Big[H_\alpha (x,y,t)\Big]_{t=c}^{t=d}dy+\int_c^d\Big[V_\alpha (x,y,t)\Big]_{y=a}^{y=b}dt,\quad x\in (0,\infty ). \end{aligned}$$ Indeed, by integrating by parts we get, for all $x \in (0,\infty)$, $$\begin{aligned} & \int_{c}^{d} \int_a^b t D_{\lambda,y} (P_t^\lambda (x,y))D_{\lambda,y} (P_t^\lambda(f)(y))dy \,dt \\ & \qquad \qquad = \frac{1}{2} \Big\{ \int_{c}^{d} \Big[t P_t^\lambda (x,y) \, D_{\lambda,y} (P_t^\lambda(f)(y)) + t D_{\lambda,y} (P_t^\lambda (x,y)) \, P_t^\lambda(f)(y) \Big]_{y=a}^{y=b} \, dt \\ & \qquad \qquad \qquad + \int_{c}^{d} \int_a^b t \Big( P_t^\lambda (x,y) \, B_{\lambda,y} (P_t^\lambda(f)(y)) + B_{\lambda,y}(P_t^\lambda (x,y)) \, P_t^\lambda(f)(y) \Big) \, dy \, dt \Big\}.\end{aligned}$$ Also we have that $$\begin{aligned} & \int_{c}^{d} \int_a^b t \partial_t (P_t^\lambda (x,y)) \, \partial_t (P_t^\lambda(f)(y)) \, dy \, dt \\ & \qquad = \frac{1}{2} \int_{c}^{d} \int_a^b t \Big\{ \partial_t^2 [P_t^\lambda (x,y) P_t^\lambda(f)(y)] - \partial_t^2 (P_t^\lambda (x,y)) P_t^\lambda(f)(y) \\ & \qquad \qquad - P_t^\lambda (x,y) \partial_t^2 (P_t^\lambda(f)(y)) \Big\} \, dy \, dt, \quad x \in (0,\infty).\end{aligned}$$ By [@BSt Lemma 2.2 and (2.12)] $$(\partial_t^2-B_{\lambda,y})P_t^\lambda(f)(y)=0 \quad \text{and} \quad (\partial_t^2-B_{\lambda,y})P_t^\lambda(x,y)=0, \quad t,x,y\in (0,\infty),$$ we obtain, for $x\in (0,\infty)$, $$\begin{aligned} & \int_c^d\int_a^bt\nabla_{\lambda,y}(P_t^\lambda(x,y))\cdot \nabla_{\lambda,y}(P_t^\lambda(f)(y))dydt \\ & \qquad =\frac{1}{2}\Big( \int_a^b\int_c^d t\partial_t ^2[P_t^\lambda(x,y)P_t^\lambda(f)(y)]dtdy \\ & \qquad \qquad+ \int_c^d\left[tP_t^\lambda(x,y)D_{\lambda,y} (P_t^\lambda(f)(y))+tD_{\lambda,y}(P_t^\lambda(x,y)) P_t^\lambda(f)(y)\right]_{y=a}^{y=b} dt\Big) \\ & \qquad =\frac{1}{2}\Big( \int_a^b\left[ t\partial_t[P_t^\lambda(x,y)P_t^\lambda(f)(y)]- P_t^\lambda(x,y)P_t^\lambda(f)(y)\right]_{t=c}^{t=d}dy \\ & \qquad \qquad+ \int_c^d\left[tP_t^\lambda(x,y)D_{\lambda,y} (P_t^\lambda(f)(y))+tD_{\lambda,y}(P_t^\lambda(x,y)) P_t^\lambda(f)(y)\right]_{y=a}^{y=b} dt\Big) \\ & \qquad =\frac{1}{2}\Big( \int_a^b\left[ t\partial_t(P_t^\lambda(x,y))P_t^\lambda(f)(y)+tP_t^\lambda(x,y)\partial_t(P_t^\lambda(f)(y))- P_t^\lambda(x,y)P_t^\lambda(f)(y)\right]_{t=c}^{t=d}dy \\ & \qquad \qquad+ \int_c^d\left[tP_t^\lambda(x,y)D_{\lambda,y} (P_t^\lambda(f)(y))+tD_{\lambda,y}(P_t^\lambda(x,y)) P_t^\lambda(f)(y)\right]_{y=a}^{y=b} dt\Big).\end{aligned}$$ Then, by taking into account that $\nabla_{\lambda ,y}(\alpha y^\lambda)=(0,0)$, for every $\alpha\in\mathbb R$, we obtain . In order to prove Theorem \[Th1.3\], ([*[ii]{}*]{}), by using (\[A.2\]), we can write $$f(x) = 2 \lim_{n \to \infty} \int_{2^{-n}}^{2^{n}} \int_0^\infty t \nabla_{\lambda,y} (P_t^\lambda (x,y)) \cdot \nabla_ {\lambda,y}(P_t^\lambda (f)(y)) \, dy \, dt, \quad \text{in } L^2(0,\infty).$$ For every $n \in {\mathbb{N}}$ we define the sets $$\begin{aligned} U_n&=[0,2)\times (2^{-n}, 2),\\ W_n&=[0,\infty) \times (2^{-n},2^n)\setminus U_n,\end{aligned}$$ and, for every $k \in {\mathbb{N}}$ and $Q \in G_k$, $\Sigma_{Q,n}:= \Sigma_Q \cap U_n.$ Note that if $k,n\in \mathbb{N}$ and $k >n$, then $\Sigma_{Q,n} = \emptyset$, for each $Q \in G_k$. (0,0.25) rectangle (1.5,1.5); at (0.75,0.9)[$U_n$]{}; (0,1.5) rectangle (1.5,2.3); (1.5,0.25) rectangle (3,2.3); at (2.1,1.3)[$W_n$]{}; (-0.2,0) – (3,0); (0,-0.2) – (0,2.5) ; (0,1.5) – (1.5,1.5); (-0.05,1.5) – (0.05,1.5); at (-0.3,1.5) [$2$]{}; (1.5,0) – (1.5,1.5); (1.5,-0.05) – (1.5,0.05); at (1.5,-0.3) [$2$]{}; (0,2.3) – (3,2.3); (-0.05,2.3) – (0.05,2.3); at (-0.3,2.3) [$2^{n}$]{}; (0,0.25) – (3,0.25); (-0.05,0.25) – (0.05,0.25); at (-0.3,0.25) [$2^{-n}$]{}; (-0.6,0) – (2.5,0); (2.2,0) – (2.2,2); at (2.5,1) [$|Q|$]{}; at (1,-0.3) [$Q$]{}; (0,0.5)–(0.5,0.5)–(0.5,0.25)–(1.5,0.25)–(2,0.25)–(2,2)–(0,2)–(0,0.5); (0,0) rectangle (0.5,0.5); (0.75,0) rectangle (1,0.25); (1.125,0) rectangle (1.25,0.125); (1.5,0) rectangle (1.625,0.125); at (1,1.2)[$\Sigma _{Q,n}$]{}; (0,0) – (0,2) ; (0,2) – (2,2); (2,0) – (2,2); (-0.4,0.25) – (2.5,0.25); (-0.45,0.25) – (-0.35,0.25); at (-0.7,0.25) [$2^{-n}$]{}; \[fig:regions\] We obtain, for each $n\in \mathbb{N}$, $$\begin{aligned} 2\int_{2^{-n}}^{2^{n}} \int_0^\infty t \nabla_{\lambda,y} (P_t^\lambda (x,y)) \, \cdot \, \nabla_{\lambda,y} (P_t^\lambda (f)(y)) \, dy \, dt &\\ &\hspace{-4cm} =2\Big(\int_{U_n}+\int_{W_n}\Big)t \nabla_{\lambda,y} (P_t^\lambda (x,y)) \, \cdot \, \nabla_{\lambda,y} (P_t^\lambda (f)(y)) \, dy \, dt\\ & \hspace{-4cm}=2 \Big(\int_{\bigcup_{Q\in \cup_{k=0}^n G_k}\Sigma_{Q,n}} + \int_{W_n} \Big) t\nabla_{\lambda,y} (P_t^\lambda (x,y)) \, \cdot \, \nabla_{\lambda,y} (P_t^\lambda (f)(y)) \, dy \, dt \\ & \hspace{-4cm}=: G_{1,n}(x) + G_{2,n}(x), \quad x \in (0,\infty).\end{aligned}$$ Our objective is to establish that, there exists an increasing sequence $\{n_i\}_{i\in \mathbb{N}}$ of nonnegative integers such that $$\label{O1} \lim_{i\rightarrow \infty}G_{1,n_i}(x)=g_1(x)+S_{\sigma _1, P^\lambda }(x),\quad \mbox{ a.e. }x\in (0,\infty),$$ and $$\label{O2} \lim_{i\rightarrow \infty}G_{2,n_i}(x)=g_2(x)+S_{\sigma _2, P^\lambda }(x),\quad \mbox{ a.e. }x\in (0,\infty),$$ for certain $g_1$ and $g_2\in L^\infty(0,\infty)$ and $\sigma _1$ and $\sigma _2$ Carleson measures on $(0,\infty )\times (0,\infty )$ such that $$\|g_1\|_\infty +\|g_2\|_\infty +\|\sigma _1\|_\mathcal{C}+\|\sigma _2\|_\mathcal{C}\leq C(A+\|f\|_{BMO_{\rm o}(\mathbb{R})}),$$ and thus we can conclude our result. Let $n\in \mathbb{N}$. First we deal with the function $G_{1,n}$. We can write $$G_{1,n}(x)=\sum_{Q\in \bigcup_{k\in \mathbb{N}} G_k}2\int_{\Sigma _{Q,n}} t\nabla_{\lambda,y} (P_t^\lambda (x,y)) \, \cdot \, \nabla_{\lambda,y} [P_t^\lambda (f)(y)-c_Qy^\lambda] \, dy \, dt,$$ where $c_Q:=y^{-\lambda} P_t^\lambda(f)(y)_{|_{(y,t)=(x_Q,t_Q)}}$. Let $k\in \mathbb{N}$ and $Q\in G_k$. By taking into account (\[A.5\]) it follows that the integral $$\int_{\Sigma_{Q,n}}t\nabla_{\lambda,y}(P_t^\lambda(x,y))\cdot\nabla_{\lambda,y}[P_t^\lambda(f)(y)-c_Qy^\lambda]dydt$$ reduces to an integral over the boundary $\partial\Sigma_{Q,n}$ of $\Sigma_{Q,n}$. We decompose this boundary in vertical and horizontal segments as follows. Let us denote by ${\mathcal V}_{Q,n}$ the set of vertical segments in $\partial\Sigma_{Q,n}\cap ([0,2]\times [2^{-n},2])$, by ${\mathcal H}_{Q,n}$ the set constituted by all horizontal segments in $\partial\Sigma_{Q,n}\cap ([0,2]\times (2^{-n},2])$ and those ones in $\partial\Sigma_{Q,n}\cap ([0,2]\times \{2^{-n}\})$ which belong to the boundary of some $Q'\subset Q$, $Q'\in G_{k+1}$ with $|Q'|=2^{-n}$ and finally we consider ${\mathcal H}_{Q,n}^0$ the set of all horizontal segments in $\partial\Sigma_{Q,n}\cap ([0,2]\times \{2^{-n}\})$ that are not in ${\mathcal H}_{Q,n}$. Indeed we can write $$\begin{aligned} {\mathcal H}_{Q,n}&=\bigcup_{I\in \mathbb{I}_{Q,n}}(I\times \{|I|\}),\\ {\mathcal H}_{Q,n}^0&=\bigcup_{J\in \mathbb{I}_{Q,n}^0}(J\times \{2^{-n}\}),\\ {\mathcal V}_{Q,n}&=\bigcup_{K\in \mathbb{K}_{Q,n}}(\{a_K\}\times K),\end{aligned}$$ where, when $k\leq n$, $\mathbb{I}_{Q,n}$ is the set constituted by $Q$ and all intervals $I\subset Q$, $I\in G_{k+1}$ with $|I|\geq 2^{-n}$, $\mathbb{I}_{Q,n}^0$ contains the maximal dyadic intervals $J\subset Q\setminus \{I\subset Q, I\in G_{k+1},|I|\geq 2^{-n}\}$, $\mathbb{K}_{Q,n}$ is a finite set of dyadic intervals in $[2^{-n},2]$, and $a_K\in [0,2]$, for every $K\in \mathbb{K}_{Q,n}$. When $k>n$, we consider $\mathbb{I}_{Q,n}=\mathbb{I}_{Q,n}^0=\mathbb{K}_{Q,n}=\emptyset$. According to we have that $$\begin{aligned} \label{G1} G_{1,n}(x)&=-\sum_{Q\in \bigcup_{k\in \mathbb{N}} G_k}\sum_{J\in \mathbb{I}_{Q,n}^0}\int_JH_{c_Q}(x,y,t)_{|t=2^{-n}}dy\nonumber\\ &\quad +\sum_{Q\in \bigcup_{k\in \mathbb{N}}G_k}\left(\sum_{I\in \mathbb I_{Q,n}}\varepsilon_I\int_I H_{c_Q}(x,y,t)_{|t=|I|}dy +\sum_{K\in \mathbb K_{Q,n}}\varepsilon _K \int_KV_{c_Q}(x,y,t)_{|y=a_K}dt\right)\nonumber\\ & :=\mathfrak{g}_{1,n}(x)+\mathfrak{g}_{2,n}(x),\quad x\in (0,\infty ).\end{aligned}$$ Here $\varepsilon_J=\pm 1$, $J\in \mathbb{I}_{Q,n}\cup\mathbb{K}_{Q,n}$. Next we show that $$\label{A.6} \lim_{n\rightarrow \infty}\mathfrak{g}_{1,n}(x)=\sum_{Q\in\cup_{k\in\mathbb N}G_k}(f(x)-c_Qx^\lambda)\chi_{\partial\Sigma_Q\cap([0,2]\times\{0\})}(x) =:g_1(x),$$ in $L^2(0,\infty )$. We can write, for every $n\in \mathbb{N}$, $$\begin{aligned} \mathfrak{g}_{1,n}(x)&=- \int_0^\infty\sum_{Q\in \cup_{\in \mathbb{N}}G_k}\chi_{\mathbb{I}_{Q,n}^0}(y)H_{c_Q}(x,y,t)_{|t=2^{-n}}dy\\ &=\int_0^\infty \Big(P_t^\lambda (x,y)\sum_{Q\in \cup_{k\in \mathbb{N}}G_k}\chi_{\mathbb{I}_{Q,n}^0}(y)[P_t^\lambda (f)(y)-c_Qy^\lambda ]\Big)_{|t=2^{-n}}dy\\ &\quad -\int_0^\infty \Big(t\partial _tP_t^\lambda (x,y)\sum_{Q\in \cup_{k\in \mathbb{N}}G_k}\chi_{\mathbb{I}_{Q,n}^0}(y)[P_t^\lambda (f)(y)-c_Qy^\lambda ]\Big)_{|t=2^{-n}}dy\\ &\quad -\int_0^\infty \Big(P_t^\lambda (x,y)\sum_{Q\in \cup_{k\in \mathbb{N}}G_k}\chi_{\mathbb{I}_{Q,n}^0}(y)t\partial _t(P_t^\lambda (f)(y))\Big)_{|t=2^{-n}}dy,\quad x\in (0,\infty ).\end{aligned}$$ According to (\[3.1\]), for $k\in\mathbb N$ and $Q\in G_k,$ $$\label{A.7*} |P_t^\lambda(f)(y)-c_Qy^\lambda |=y^\lambda |y^{-\lambda}u(y,t)-x_Q^{-\lambda }u(x_Q,t_Q)|\leq C(A+\|f\|_{BMO_{\rm o}(\mathbb R)}),\quad(y,t)\in\Sigma_Q.$$ By using (\[A.4.1\]), since $f\in L^2(0,\infty)$, $$\lim_{t\rightarrow 0^+}P_t^\lambda(f)(y)=f(y),\;\;\;\mbox{a.e.}\;\;y\in (0,\infty).$$ Then it follows that $$\label{A.7} |f(y)-c_Qy^\lambda |\leq C(A+\|f\|_{BMO_{\rm o}(\mathbb R)}),\;\;\;\mbox{a.e.}\;\;y\in \partial\Sigma_Q\cap([0,2]\times\{0\}).$$ We observe also that $${\mathop{\mathrm{supp}}}\Big(\sum_{Q\in \cup_{\in \mathbb{N}}G_k}\chi_{\mathbb{I}_{Q,n}^0}(y)[P_t^\lambda (f)(y)-c_Qy^\lambda ]_{|t=2^{-n}}-g_1(y)\Big)\subset [0,2],\quad n\in\mathbb N ,$$ and $$\label{converg} \lim_{n\rightarrow \infty}\sum_{Q\in \cup_{k\in \mathbb{N}}G_k}\chi_{\mathbb{I}_{Q,n}^0}(y)[P_t^\lambda (f)(y)-c_Qy^\lambda ]_{|t=2^{-n}}=g_1(y),\quad \mbox{ a.e. }y\in (0,\infty ),$$ (actually, we can assure that is true for all $y\in Q_0$ which is not a dyadic number). By the dominated convergence theorem, we get in $L^2(0,\infty)$. According to and we have, for each $n\in \mathbb{N}$, $$\begin{aligned} \mathfrak{g}_{1,n}(x)-g_1(x)&=h_\lambda \Big(e^{-tz}h_\lambda \Big(\sum_{Q\in \cup_{\in \mathbb{N}}G_k}\chi_{\mathbb{I}_{Q,n}^0}(y)[P_t^\lambda (f)(y)-c_Qy^\lambda ]-g_1(y)\Big)(z)_{|t=2^{-n}}\Big)(x)\\ &\quad +h_\lambda ((e^{-tz}-1)_{|t=2^{-n}}h_\lambda (g_1)(z))(x) \\ &\quad +h_\lambda \Big(tze^{-tz}h_\lambda\Big(\sum_{Q\in \cup_{k\in \mathbb{N}}G_k}\chi_{\mathbb{I}_{Q,n}^0}(y)[P_t^\lambda (f)(y)-c_Qy^\lambda ]-g_1(y)\Big)(z)_{|t=2^{-n}}\Big)(x)\\ &\quad -(t\partial _tP_t^\lambda (g_1)(x))_{|t=2^{-n}}\\ &\quad -h_\lambda\Big(e^{-tz}h_\lambda \Big(\sum_{Q\in \cup_{k\in \mathbb{N}}G_k}\chi_{\mathbb{I}_{Q,n}^0}(y)t\partial _t(P_t^\lambda (f)(y))\Big)(z)_{|t=2^{-n}}\Big)(x),\quad x\in (0,\infty ).\end{aligned}$$ Then, by taking into account the $L^2$-boundedness of $h_\lambda$ we get $$\begin{aligned} \|\mathfrak{g}_{1,n}-g_1\|_2&\leq C\Big(\Big\|\sum_{Q\in \cup_{k\in \mathbb{N}}G_k}\chi_{\mathbb{I}_{Q,n}^0}(y)[P_t^\lambda (f)(y)-c_Qy^\lambda ]_{|t=2^{-n}}-g_1(y) \Big\|_2\\ &\quad +\|(e^{-tz}-1)_{|t=2^{-n}}h_\lambda (g_1)\|_2+\|(t\partial _tP_t^\lambda (g_1))_{|t=2^{-n}}\|_2\\ &\quad +\|(t\partial_t P_t^\lambda (f))_{t=2^{-n}}\|_2\Big),\quad n\in \mathbb{N}.\end{aligned}$$ From , and the dominated convergence theorem we obtain . Note that by we have that $\|g_1\|_{\infty}\leq C(A+\|f\|_{BMO_{\rm o}(\mathbb R)})$. Now, let us show that there exists a Carleson measure $\sigma_1$ on $(0,\infty )^2$ such that $$\label{sigma1} \lim_{n\rightarrow \infty}\mathfrak{g}_{2,n}(x)=S_{\sigma_1,P^\lambda}(x),\quad x\in (0,\infty ).$$ For that, we write, for each $n\in \mathbb{N}$, $$\begin{aligned} \mathfrak{g}_{2,n}(x)&=\sum_{Q\in \bigcup_{k\in \mathbb{N}}G_k}\left(\sum_{I\in \mathcal{H}_{Q,n}}\int_I P_t^\lambda(x,y)M_{Q,1}(y,t)dy_I +\sum_{J\in \mathcal{V}_{Q,n}}\int_JP_t^\lambda (x,y)L_{Q,1}(y,t)dt_J\right)\\ &\quad +\sum_{Q\in \bigcup_{k\in \mathbb{N}}G_k}\left(\sum_{I\in \mathcal{H}_{Q,n}}\int_I M_{Q,2}(x,y,t)dy_I +\sum_{J\in \mathcal{V}_{Q,n}}\int_JL_{Q,2}(x,y,t)dt_J\right)\\ &:=F_{1,n}(x)+F_{2,n}(x),\quad x\in (0,\infty ),\end{aligned}$$ where, for every $Q\in \cup_{k\in \mathbb{N}}G_k$, $$\begin{aligned} \label{M} &M_{Q,1}(y,t) :=t\partial_tP_t^\lambda(f)(y)-[P_t^\lambda(f)(y)-c_Qy^\lambda],\\ &L_{Q,1}(y,t) :=tD_{\lambda,y}P_t^\lambda(f)(y),\nonumber\\ &M_{Q,2}(x,y,t) :=t\partial_t(P_t^\lambda(x,y))[P_t^\lambda(f)(y)-c_Qy^\lambda],\nonumber\end{aligned}$$ and $$L_{Q,2}(x,y,t) :=tD_{\lambda,y}P_t^\lambda(x,y)[P_t^\lambda(f)(y)-c_Qy^\lambda].$$ Let us consider $${\mathcal H}_{Q} :=\{\mbox{horizontal segments in } \partial\Sigma_{Q}\cap ([0,2]\times(0,2])\},$$ and $${\mathcal V}_{Q} :=\{\mbox{vertical segments in } \partial\Sigma_{Q}\}.$$ By and according to [@Ga p. 346] the measures $$\nu :=\sum_{Q\in \cup_{k\in\mathbb{N}} G_k}\Big(\sum_{I\in {\mathcal H}_{Q}}dy_I+\sum_{J\in {\mathcal V}_{Q}}dt_J\Big),$$ and $$\nu_n :=\sum_{Q\in \cup_{k\in\mathbb{N}}G_k}\Big(\sum_{I\in {\mathcal H}_{Q,n}}dy_I+\sum_{J\in {\mathcal V}_{Q,n}}dt_J\Big),\quad n\in \mathbb{N},$$ are Carleson measures. Moreover, we can write $$d\nu_n(y,t)=k_n(y,t)d\alpha (y,t) \mbox{ and }d\nu(y,t)=k(y,t)d\alpha (y,t),$$ for certain positive measure $\alpha$ and nonnegative functions $k_n$ and $k$ such that $k_n\uparrow k$, as $n\rightarrow \infty$, pointwisely. Then, $\|\nu_n\|_{\mathcal C}\leq \|\nu\|_{\mathcal C}$, $n\in \mathbb{N}$, and by the monotone convergence theorem, $$\lim_{n\rightarrow \infty}S_{\nu _n,P^\lambda }(x)=S_{\nu, P^\lambda}(x),\quad x\in (0,\infty ).$$ We now define $$\mu_n(y,t):= \sum_{Q\in \cup_{k\in \mathbb{N}} G_k} \Big(\sum_{I\in{\mathcal H}_{Q,n}}M_{Q,1}(y,t) dy_I +\sum_{J\in{\mathcal V}_{Q,n}}L_{Q,1}(y,t)dt_J\Big), \quad n\in\mathbb N,$$ and $$\mu(y,t):= \sum_{Q\in \cup_{k\in \mathbb{N}} G_k} \Big(\sum_{I\in{\mathcal H}_{Q}}M_{Q,1}(y,t) dy_I + \sum_{J\in{\mathcal V}_{Q}}L_{Q,1}(y,t)dt_J\Big).$$ From Proposition \[Prop2.2\] and it follows that, for every $Q\in \bigcup_{k=0}^\infty G_k$, $$|M_{Q,1}(y,t)|\leq C(A+\|f\|_{BMO_{\rm o}(\mathbb R)}), \quad (y,t)\in \Sigma _Q,$$ and $$|L_{Q,1}(y,t)|\leq C\|f\|_{BMO_{\rm o}(\mathbb R)},\quad y,t\in (0,\infty).$$ Then the measures $\mu$ and $\mu_n$, $n\in\mathbb N$, are Carleson measures, satisfying that $$\|\mu_n\|_{\mathcal{C}}\leq C(A+\|f\|_{BMO_{\rm o}(\mathbb R)}),\;\;\;n\in\mathbb N,$$ $$\|\mu\|_{\mathcal{C}}\leq C(A+\|f\|_{BMO_{\rm o}(\mathbb R)}),$$ and, by Theorem \[Th1.3\], $(i)$ and the dominated convergence theorem $$\lim_{n\rightarrow\infty}S_{\mu_n, P^\lambda }(x)=S_{\mu, P^\lambda }(x),\quad x\in (0,\infty ).$$ Note that, for every $n\in\mathbb N$, $F_{1,n}(x)=S_{\mu_n,P^\lambda}(x)$, $x\in (0,\infty )$. Then, $$\lim_{n\rightarrow\infty}F_{1,n}(x)=S_{\mu,P^\lambda}(x),\quad x\in (0,\infty).$$ Now we study $F_{2,n}$, $n\in \mathbb{N}$. By Lemma \[D4\] and (\[D3\]) we get $$|t\partial_tP_t^\lambda(x,y)|+|tD_{\lambda,y}P_t^\lambda(x,y)|\leq C\frac{t}{(x-y)^2+t^2},\quad x,y,t\in (0,\infty).$$ Then, by taking into account (\[A.7\*\]) and proceeding as in [@Wi p. 25 and 26] (see also [@CDLSY p. 2088]), we get, for every $n\in\mathbb N$, a Carleson measure $\rho_n$ such that $F_{2,n}=S_{\rho_n,P^\lambda}$, and a Carleson measure $\rho$, such that $$\sum_{Q\in\bigcup_{k\in \mathbb{N}}G_k}\left(\sum_{I\in \mathcal{H}_Q}\int_I M_{Q,2}(x,y,t)dy_I +\sum_{J\in \mathcal{V}_Q}\int_JL_{Q,2}(x,y,t)dt_J\right)=S_{\rho ,P^\lambda}(x),\quad x\in (0,\infty ).$$ We have that, $$\lim_{n\rightarrow\infty}S_{\rho_n,P^\lambda}(x)=S_{\rho,P^\lambda}(x),\quad x\in (0,\infty ),$$ and $\|\rho \|_{\mathcal{C}}\leq C(A+\|f\|_{BMO_{\rm o}(\mathbb{R})})$. Then we obtain for $\sigma_1=\mu+\rho$. From , and we can find an increasing sequence $\{n_i\}_{i\in \mathbb{N}}$ of nonnegative integers such that $$\lim_{i\rightarrow \infty}G_{1,n_i}(x)=g_1(x)+S_{\mu +\rho,P^\lambda }(x),\quad \mbox{ a.e. }x\in (0,\infty ).$$ Note that $\|g_1\|_\infty +\|\sigma _1\|_{\mathcal{C}}\leq C(A+\|f\|_{BMO_{\rm o}(\mathbb{R})})$, so is thus established. We now deal with $G_{2,n}$, $n\in \mathbb{N}$. Let $M>2$. We define $$W_{n,M}:=\Big\{(x,t)\in W_n: x\in (0,M)\Big\}, \quad n\in\mathbb N.$$ By (\[A.5\]) we have that $$\begin{aligned} G_{2,n,M}(x) & :=2\int_{W_{n,M}}t\nabla_{\lambda,y}(P_t^\lambda(x,y))\cdot\nabla_{\lambda,y}[P_t^\lambda(f)(y)-c_0y^\lambda]dydt \\ &=\int_0^MH_{c_0}(x,y,t)_{|t=2^n}dy-\int_2^MH_{c_0}(x,y,t)_{|t=2^{-n}} dy-\int_0^2 H_{c_0}(x,y,t)_{|t=2}dy\\ & \quad -\int_2^{2^n}V_{c_0}(x,y,t)_{|y=0} dt -\int_{2^{-n}}^2V_{c_0}(x,y,t)_{|y=2} dt+\int_{2^{-n}}^{2^n}V_{c_0}(x,y,t)_{|y=M} dt \\ & =:\sum_{i=1}^6I_{i,n,M}(x),\;\;\;x\in (0,\infty)\;\mbox{and}\;n\in\mathbb N,\end{aligned}$$ where $c_0:=x_{Q_0}^{-\lambda}u(x_{Q_0},t_{Q_0})$. Observe that, actually $I_{3,n,M}$ is independent of $n$ and $M$ and $I_{4,n,M}$ and $I_{5,n,M}$ do not depend on $M$. First, we note that $I_{4,n,M}(x)=0$, $n\in \mathbb{N}$. Indeed, by Lemma \[D4\] and it follows that $$\lim_{y\rightarrow 0^+}P_t^\lambda(x,y)=\lim_{y\rightarrow 0^+}D_{\lambda ,y}P_t^\lambda (x,y)=0,\quad x,t\in (0,\infty ).$$ Then, by taking into account Proposition \[Prop2.2\] and for $Q=Q_0$ it follows that $V_{c_0}(x,y,t)_{|y=0}=0$, $x,t\in (0,\infty )$. On the other hand, we have that $$\begin{aligned} I_{3,n,M}(x) & = -\int_0^2( P_t^\lambda(x,y)M_{Q_0,1}(y,t))_{|t=2} dy -\int_0^2M_{Q_0,2}(y,t)_{|t=2} dy\\ &:=I_3^1(x)+I_3^2(x),\quad x\in (0,\infty ).\end{aligned}$$ Here $M_{Q_0,1}$ and $M_{Q_0,2}$ are as in with $Q=Q_0$. By considering Proposition \[Prop2.2\] and we get a Carleson measure $\alpha_3^1$ such that $I_3^1(x)=S_{\alpha _3^1,P^\lambda }(x)$, $x\in (0,\infty )$ and $\|\alpha_3^1\|_{\mathcal C}\leq C(A+\|f\|_{BMO_{\rm o}(\mathbb R)})$. Also, from Lemma \[D4\], and we deduce as above (see [@Wi p. 25 and 26]) that $I_3^{2}(x)=S_{\alpha_3^2,P^\lambda}(x)$, $x\in (0,\infty )$, for some Carleson measure $\alpha_3^2$ such that $\|\alpha_3^2\|_{\mathcal C}\leq C(A+\|f\|_{BMO_{\rm o}(\mathbb R)})$. Then there exists a Carleson measure $\alpha_3=\alpha_3^1+\alpha_3^2$ on $(0,\infty )^2$ such that $I_{3,n,M}(x)=S_{\alpha_3,P^\lambda }(x)$, $x\in (0,\infty )$. In a similar way we can see that, for every $n\in \mathbb{N}$, there exists a Carleson measure $\alpha_{5,n}$ such that $I_{5,n,M}=S_{\alpha_{5,n},P^\lambda}$ and $\|\alpha_{5,n}\|_{\mathcal C}\leq C(A+\|f\|_{BMO_{\rm o}(\mathbb R)})$. And, as above, by Theorem \[Th1.3\] and the dominated convergence theorem, there exists $\alpha_5\in \mathcal C$ such that $S_{\alpha_{5,n},P^\lambda}(x)\rightarrow S_{\alpha_5,P^\lambda}(x)$, as $n\rightarrow\infty$, for a.e. $x\in (0,\infty)$. We have also that, for every $n\in \mathbb{N}$, $$\lim_{M\rightarrow \infty}I_{6,n,M}(x)=0,\quad x\in (0,\infty ).$$ It is sufficient to note that by Lemma \[D4\] and it follows that $$\begin{aligned} |I_{6,n,M}(x)| & \leq C M^\lambda\int_{2^{-n}}^{2^n}\frac{t(xM)^\lambda }{((x-M)^2+t^2)^{\lambda +1}}\Big(1+\frac{1}{t^{2\lambda +2}}\int_0^1|f(z)|dz\Big)dt\\ &\leq C\frac{x^\lambda M^{2\lambda }}{(|x-M|+2^{-n})^{2\lambda +1}}\int_{2^{-n}}^{2^n}\Big(1+\frac{1}{t^{2\lambda +2}}\Big)dt\leq \frac{C_{n,x}M^{2\lambda }}{(|x-M|+2^{-n})^{2\lambda +1}},\quad x\in (0,\infty),\end{aligned}$$ for every $n\in \mathbb{N}$ and for certain $C_{n,x}>0$. On the other hand, for each $n\in \mathbb{N}$, $$\label{I_1} \lim_{M\rightarrow \infty}I_{1,n,M}(x)=\int_0^\infty H_{c_0}(x,y,t)_{|t=2^n}dy,\quad x\in (0,\infty ),$$ and $$\label{I_2} \lim_{M\rightarrow \infty}I_{2,n,M}(x)=\int_2^\infty H_{c_0}(x,y,t)_{|t=2^{-n}}dy,\quad x\in (0,\infty ).$$ It is sufficient to show that for each $n\in \mathbb{N}$, the integrals in the right side of and are absolutely convergent for every $x\in (0,\infty)$. Indeed, from Lemma \[D4\] and (\[D2\]) and Proposition \[Prop2.2\], we get $$|H_{c_0}(x,y,t)|\leq C\frac{t(xy)^\lambda }{((x-y)^2+t^2)^{\lambda +1}}(|P_t^\lambda (f)(y)|+y^\lambda +1),\quad x,y,t\in (0,\infty ).$$ Since $f\in L^2(0,\infty)$, also $P_t^\lambda(f)\in L^2(0,\infty)$ and, by Hölder’s inequality we deduce that $$\begin{aligned} \int_0^\infty |H_{c_0}(x,y,t)|dy&\leq Ctx^\lambda \int_0^\infty\frac{y^\lambda}{((x-y)^2+t^2)^{\lambda +1}}(|P_t^\lambda(f)(y)|+y^\lambda +1)dy \\ & \leq Ctx^\lambda\Big\{\Big(\int_0^\infty\frac{y^{2\lambda}}{((x-y)^2+t^2)^{2\lambda +2}}dy\Big)^{1/2}\|P_t^\lambda(f)\|_2\\ & \qquad \qquad + \int_0^\infty \frac{y^\lambda+y^{2\lambda}}{((x-y)^2+t)^{\lambda +1}}dy\Big\}<\infty,\;\;x,t\in (0,\infty).\end{aligned}$$ We conclude that, for every $n\in \mathbb{N}$, $$\begin{aligned} G_{2,n}(x) &= \int_0^\infty H_{c_0}(x,y,t)_{|t=2^n}dy-\int_2^\infty H_{c_0}(x,y,t)_{|t=2^{-n}}dy\\ &\quad +S_{\alpha _3,P^\lambda}(x)+S_{\alpha_{5,n},P^\lambda}(x),\quad x\in (0,\infty).\end{aligned}$$ In order to finish the proof we are going to show that $$\label{H1} \lim_{n\rightarrow \infty}\int_0^\infty H_{c_0}(x,y,t)_{|t=2^n}dy=c_0x^\lambda ,\quad x\in (0,\infty ),$$ and $$\label{H2} \lim_{i\rightarrow \infty}\int_2^\infty H_{c_0}(x,y,t)_{|t=2^{-n_i}}=c_0x^\lambda \chi _{(2,\infty)}(x),\quad \mbox{ a.e. }x\in (0,\infty ),$$ for certain increasing sequence $\{n_i\}_{i\in \mathbb{N}}$ of nonnegative integers. Thus we obtain that $$\lim_{i\rightarrow \infty }G_{2,n_i}(x)=c_0x^\lambda \chi _{(0,2)}(x)+S_{\alpha_3+\alpha_5,P^\lambda}(x),\quad \mbox{ a.e. }x\in (0,\infty ),$$ and we get with $g_2(x)=c_0x^\lambda \chi _{(0,2)}(x)$, $x\in (0,\infty)$, and $\sigma_2=\alpha_3+\alpha_5$. Since $P_t^\lambda (y^\lambda )(x)=x^\lambda$, $x,t\in (0,\infty )$ ([@BCaFR p. 455]), $$\int_0^\infty \partial_t P_t^\lambda(x,y)y^\lambda dy=\partial_t\int_0^\infty P_t^\lambda(x,y)y^\lambda dy=0, \quad x,t\in (0,\infty).$$ The derivation under the integral sign is justified because by Lemma \[D4\] and it follows that $$\int_0^\infty |\partial_t P_t^\lambda(x,y)y^\lambda| dy\leq C\int_0^\infty\frac{x^\lambda y^{2\lambda}} {((x-y)^2+t^2)^{\lambda +1}}dy <\infty,\quad x,t\in (0,\infty).$$ Then, we can write $$\begin{aligned} \int_0^\infty H_{c_0}(x,y,t)dy&=\int_0^\infty (t\partial_ t(P_t^\lambda(x,y))P_t^\lambda(f)(y)+tP_t^\lambda(x,y)\partial_tP_t^\lambda(f)(y)-P_t^\lambda(x,y)P_t^\lambda(f)(y))dy\\ &\quad +c_0x^\lambda ,\quad x,t\in (0,\infty ). \end{aligned}$$ From Lemma \[D4\] and we have that $$\begin{aligned} \Big|\int_0^\infty (t\partial_ t(P_t^\lambda(x,y))P_t^\lambda(f)(y)+tP_t^\lambda(x,y)\partial_tP_t^\lambda(f)(y)-P_t^\lambda(x,y)P_t^\lambda(f)(y))dy\Big| & \\ &\hspace{-9cm} \leq C \int_0^\infty \frac{t}{(x-y)^2+t^2}\int_0^1\frac{t}{(y-z)^2+t^2}|f(z)|dzdy \\ &\hspace{-9cm} \leq \frac{C}{t}\int_0^\infty \frac{t}{(x-y)^2+t^2}\int_0^1|f(z)|dzdy\\ &\hspace{-9cm} \leq C\frac{\|f\|_{BMO_{\rm o}(\mathbb{R})}}{t}\Big(\int_0^{2x}\frac{dy}{t}+\int_{2x}^\infty \frac{t}{(y+t)^2}dy\Big)\\ &\hspace{-9cm} \leq \frac{C}{t}\Big(\frac{x}{t}+\frac{t}{x+t}\Big),\quad x, t\in (0, \infty).\end{aligned}$$ Note that, for each $x\in (0,\infty )$, the last term tends to zero as $t\rightarrow \infty$, and consequently we get . On the other hand, we have that $$\begin{aligned} \int_2^\infty H_{c_0}(x,y,t)dy&=\int_2^\infty (t\partial_ t(P_t^\lambda(x,y))P_t^\lambda(f)(y)+tP_t^\lambda(x,y)\partial_tP_t^\lambda(f)(y)-P_t^\lambda(x,y)P_t^\lambda(f)(y))dy\\ &\quad -c_0t\partial _t\int_2^\infty P_t^\lambda (x,y)y^\lambda dy+c_0\int_2^\infty P_t^\lambda (x,y)y^\lambda dy\\ &=\int_2^\infty (t\partial_ t(P_t^\lambda(x,y))P_t^\lambda(f)(y)+tP_t^\lambda(x,y)\partial_tP_t^\lambda(f)(y)-P_t^\lambda(x,y)P_t^\lambda(f)(y))dy\\ &\quad +c_0t\partial_tP_t^\lambda (y^\lambda \chi _{(0,2)}(y))(x)+c_0x^\lambda -c_0P_t^\lambda (y^\lambda \chi _{(0,2)}(y))(x),\quad x,t\in (0,\infty ). \end{aligned}$$ As above, by using Lemma \[D4\] and it follows that $$\begin{aligned} \Big|\int_2^\infty (t\partial_ t(P_t^\lambda(x,y))P_t^\lambda(f)(y)+tP_t^\lambda(x,y)\partial_tP_t^\lambda(f)(y)-P_t^\lambda(x,y)P_t^\lambda(f)(y))dy\Big| & \\ &\hspace{-9cm} \leq C \int_2^\infty P_t^\lambda (x,y)\int_0^1\frac{t(yz)^\lambda}{((y-z)^2+t^2)^{\lambda +1}}|f(z)|dzdy \\ &\hspace{-9cm} \leq Ct\int_2^\infty P_t^\lambda (x,y)y^\lambda \int_0^1\frac{|f(z)|}{(1+t^2)^{\lambda +1}}dzdy\\ &\hspace{-9cm} \leq C\|f\|_{BMO_{\rm o}(\mathbb{R})}t\int_0^\infty P_t^\lambda (x,y)y^\lambda dy\\ &\hspace{-9cm} \leq Cx^\lambda t,\quad x, t\in (0, \infty).\end{aligned}$$ Observe that, for each $x\in (0,\infty )$, the last term tends to zero as $t\rightarrow 0$. 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--- abstract: | We study anomalous scaling and multiscaling of two-dimensional crack profiles in the random fuse model using both periodic and open boundary conditions. Our large scale and extensively sampled numerical results reveal the importance of crack branching and coalescence of microcracks, which induce jumps in the solid-on-solid crack profiles. Removal of overhangs (jumps) in the crack profiles eliminates the multiscaling observed in earlier studies and reduces anomalous scaling. We find that the probability density distribution $p(\Delta h(\ell))$ of the height differences $\Delta h(\ell) = [h(x+\ell) - h(x)]$ of the crack profile obtained after removing the jumps in the profiles has the scaling form $p(\Delta h(\ell)) = \langle\Delta h^2(\ell)\rangle^{-1/2} ~f\left(\frac{\Delta h(\ell)}{\langle\Delta h^2(\ell)\rangle^{1/2} }\right)$, and follows a Gaussian distribution even for small bin sizes $\ell$. The anomalous scaling can be summarized with the scaling relation $\left[\frac{\langle\Delta h^2(\ell)\rangle^{1/2}}{\langle\Delta h^2(L/2)\rangle^{1/2}}\right]^{1/\zeta_{loc}} + \frac{(\ell-L/2)^2}{(L/2)^2} = 1$, where $\langle\Delta h^2(L/2)\rangle^{1/2} \sim L^{\zeta}$. author: - 'Phani K.V.V. Nukala' - Stefano Zapperi - 'Mikko J. Alava' - Sran Šimunović title: Anomalous Roughness of Fracture Surfaces in 2D Fuse Models --- Introduction ============ For over two decades, scaling of fracture surfaces has been a well studied, yet a controversial issue [@breakdown; @alava06]. Experiments on several materials under different loading conditions have shown that the fracture surface is self-affine [@man], which implies that if the in plane length scales of a fracture surface are scaled by a factor $\lambda$ then the out of plane length scales (height) of the fracture surface scales by $\lambda^\zeta$, where $\zeta$ is the roughness exponent. Many experiments on several materials including metals [@metals], glass [@glass], rocks [@rocks] and ceramics [@cera] have tested the scaling in three dimensions. The scaling regime in some cases has been quite impressive, spanning five decades in metallic alloys [@bouch]. It is an interesting question as to whether the self-affinity measured so often can be replaced by more complicated scenarios and whether in any particular setup and geometry the exponents are universal as in the line-depinning scenario [@ponson06ijf; @ponson06]. In two dimensions, recent studies have debated the picture of simple self-affinity. In other words, for a two-dimensional crack profile $h(x)$ one can look at various statistical measures including the dependence of the roughness on sample/system size and the scaling of various moments of $h(x)$. There is a discussion on whether the two-dimensional fracture surfaces would exhibit self-affine or multi-affine scaling [@procaccia; @jstat2; @bakke07; @santucci07]. Ref. [@procaccia] argues that a crack line $h(x)$ in two-dimensions is not self-affine; instead, it exhibits a much complicated multi-affine (or multiscaling) structure, with a non-constant scaling exponent $\zeta_q$ for the $q$-th order correlation function $C_q(\ell) = \langle |h(x+\ell)-h(x)|^q\rangle^{1/q} \sim \ell^{\zeta_q}$. In analogy to kinetic roughening of surfaces, it has been argued that fracture surfaces exhibit anomalous scaling [@anomalous]: the [*global*]{} exponent describing the scaling of the crack width with the sample size is larger than the local exponent measured on a single sample [@exp-ano; @exp-ano2]. This means that the typical slope of $h(x)$ develops an algebraic dependence on the system size $L$, and it is necessary to introduce two roughness exponents a global one ($\zeta$) and a local one ($\zeta_{loc}$) whose difference measures the $L$-dependent extra lengthscale. In two dimensions, the available experimental results, mainly obtained for paper samples, indicate a roughness exponent in the range $\zeta \simeq 0.6-0.7$ [@jstat2; @santucci07; @kertesz93; @engoy94; @salminen03; @rosti01]. However, one should note that apparently one can measure for various ordinary, industrial papers values that are significantly higher than $\zeta=0.7$ [@menezessobrinho05]. The reasons for these discrepancies are not clear. It has also been noted that the roughness exponent is dependent on the crack velocity: at the onset of fast crack propagation the exponent makes a small jump from its value when the crack still grows in a stable fashion [@vanel]. The theoretical understanding of the origin and universality of crack surface roughness is often investigated by discrete lattice (fuse, central-force, and beam) models. In these models the elastic medium is described by a network of discrete elements such as fuses, springs and beams with random failure thresholds. In the simplest approximation of a scalar displacement, one recovers the random fuse model (RFM) where a lattice of fuses with random threshold are subject to an increasing external voltage [@deArcangelis85]. Using two-dimensional RFM, the estimated crack surface roughness exponents are: $\zeta = 0.7\pm0.07$ [@hansen91b], $\zeta_{loc}=2/3$ [@sep-00], and $\zeta = 0.74\pm0.02$ [@bakke]. Recently, using large system sizes (up to $L = 1024$) with extensive sample averaging, we found that the crack roughness exhibits anomalous scaling [@zns05]. The local and global roughness exponents estimated using two different lattice topologies are: $\zeta_{loc} = 0.72\pm0.02$ and $\zeta = 0.84\pm0.03$. Anomalous scaling has been noted in the 3D numerical simulations as well [@nukalapre3D]. The origins of anomalous scaling of fracture surfaces is not yet clear although recent studies [@jstat2; @santucci07; @bakke07] suggest that the origin of multiscaling and anomalous scaling in numerical simulations may be due to the existence of overhangs (jumps) in the crack profile. In this paper, we further quantify the influence of these overhangs in the crack profiles on multi-affine scaling and anomalous scaling of crack roughness exponents. In particular, the questions we would like to address in this article are the following: (i) whether anomalous scaling of roughness observed in numerical simulations is a result of these overhangs (or jumps) in the crack profiles, and (ii) whether removing the jumps in the crack profiles completely eliminates multiscaling. This should then imply a constant scaling exponent $\zeta_{loc}$ such that the $q$-th order correlation function $C_q(\ell) = \langle |h(x+\ell)-h(x)|^q\rangle^{1/q} \sim \ell^{\zeta_{loc}}$. It should be noted that Gaussian distribution for $p(\Delta h(\ell))$ has been noted in Refs. [@salminen03; @jstat2; @santucci07; @bakke07] only above a characteristic scale where self-affine scaling of crack surfaces is observed. In this study, we would like to further investigate whether removing these jumps in the crack profiles extends the validity of Gaussian probability density distribution $p(\Delta h(\ell))$ of the height differences $\Delta h(\ell) = [h(x+\ell) - h(x)]$ of the crack profile to even smaller window sizes $\ell$. We also discuss the cases of open (OBC) and periodic boundary conditions (PBC), since the presence of the former might have an effect on whether “anomalous scaling” exists. For the PBC case, we show that the crack profiles can be collapsed to a “semi-circle law”, a scaling ansatz followed by many stochastic processes that return to the origin [@baldassarri03]. The rest of the article consists of three sections: first we introduce the numerical details. In Section III, we go through all the numerical results, and finally Section IV presents the conclusions. Model ===== We consider numerical simulations using two-dimensional random fuse model (RFM), where a lattice of fuses with random threshold are subject to an increasing external voltage [@deArcangelis85]. The lattice system we consider is a triangular lattice of linear size $L$ with a central notch of length $a_0$ (unnotched specimens imply $a_0 = 0$). All of the lattice bonds have the same conductance, but the bond breaking thresholds, $t$, are randomly distributed based on a thresholds probability distribution, $p(t)$. The burning of a fuse occurs irreversibly, whenever the electrical current in the fuse exceeds the breaking threshold current value, $t$, of the fuse. Periodic boundary conditions are imposed in the horizontal directions ($x$ direction) to simulate an infinite system and a constant voltage difference, $V$, is applied between the top and the bottom of the lattice system bus bars. A power-law thresholds distribution $p(t)$ is used by assigning $t = X^D$, where $X \in [0,1]$ is a uniform random variable with density $p_X(X) = 1$ and $D$ represents a quantitative measure of disorder. The larger $D$ is, the stronger the disorder. This results in $t$ values between 0 and 1, with a cumulative distribution $P(t) = t^{1/D}$. The average breaking threshold is $<t> = 1/(D+1)$, and the probability that a fuse will have breaking threshold less than the average breaking threshold $<t>$ is $P(<t>) = (1/(D+1))^{1/D}$. That is, the larger the $D$ is, the smaller the average breaking threshold and the larger the probability that a randomly chosen bond will have breaking threshold smaller than the average breaking threshold. Numerically, a unit voltage difference, $V = 1$, is set between the bus bars (in the $y$ direction) and the Kirchhoff equations are solved to determine the current flowing in each of the fuses. Subsequently, for each fuse $j$, the ratio between the current $i_j$ and the breaking threshold $t_j$ is evaluated, and the bond $j_c$ having the largest value, $\mbox{max}_j \frac{i_j}{t_j}$, is irreversibly removed (burnt). The current is redistributed instantaneously after a fuse is burnt implying that the current relaxation in the lattice system is much faster than the breaking of a fuse. Each time a fuse is burnt, it is necessary to re-calculate the current redistribution in the lattice to determine the subsequent breaking of a bond. The process of breaking of a bond, one at a time, is repeated until the lattice system falls apart. Using the algorithm proposed in Ref. [@nukalajpamg1], we have performed numerical simulation of fracture up to system sizes $L = 512$ for unnotched samples and up to $L = 320$ for notched samples. Our simulations cover an extensive parametric space of ($L$, $D$ and $a_0$) given by: $L = \{64, 128, 192, 256, 320, 512\}$; $D = \{0.3, 0.4, 0.5, 0.6, 0.75, 1.0\}$; and $a_0/L = \{0, 1/32, 1/16, 3/32, 1/8, 3/16, 1/4, 5/16, 3/8\}$. A minimum of 200 realizations have been performed for each case, but for many cases 2000 realizations have been used to reduce the statistical error. Crack Roughness =============== Crack width ----------- Once the sample has failed, we identify the final crack, which typically displays dangling ends and overhangs (see Fig. \[fig:crack\]). We remove them and obtain a single valued crack line $h_x$, where the values of $x \in [0,L]$. For self-affine cracks, the local width, $w(l)\equiv \langle \sum_x (h_x- (1/l)\sum_X h_X)^2 \rangle^{1/2}$, where the sums are restricted to regions of length $l$ and the average is over different realizations, scales as $w(l) \sim l^\zeta$ for $l \ll L$ and saturates to a value $W=w(L) \sim L^\zeta$ corresponding to the global width. The power spectrum $S(k)\equiv \langle \hat{h}_k \hat{h}_{-k} \rangle/L$, where $\hat{h}_k \equiv \sum_x h_x \exp i(2\pi xk/L)$, decays as $S(k) \sim k^{-(2\zeta+1)}$. When anomalous scaling is present [@anomalous; @exp-ano; @exp-ano2], the exponent describing the system size dependence of the surface differs from the local exponent measured for a fixed system size $L$. In particular, the local width scales as $w(\ell) \sim \ell^{\zeta_{loc}}L^{\zeta-\zeta_{loc}}$, so that the global roughness $W$ scales as $L^\zeta$ with $\zeta>\zeta_{loc}$. Consequently, the power spectrum scales as $S(k) \sim k^{-(2\zeta_{loc}+1)}L^{2(\zeta-\zeta_{loc})}$. ![(Color online) A typical crack in a fuse lattice system of size $L \times L$ with $L = 512$. This crack, identified as (a) in the figure has dangling ends, which are removed to obtain a single valued crack profile $h(x)$, identified as (b) in the figure. This final crack $h(x)$ possesses finite jumps that arise due to the solid-on-solid projection to obtain a single-valued fracture surface. The inset shows a zoomed portion of the crack.[]{data-label="fig:crack"}](crack_prof_102.eps){width="8cm"} Figure \[fig:widthD\]a presents the scaling of local and global crack widths in systems with different disorder values and an initial relative notch size of $a_0/L = 1/16$. The slopes of the curves presented in Fig. \[fig:widthD\]a suggest that a local roughness exponent $\zeta_{loc} = 0.71$ that is independent of the disorder. The global roughness exponent is estimated to be $\zeta = 0.87$, and differs considerably from the local roughness exponent $\zeta_{loc}$. The collapse of the data in Fig. \[fig:widthD\]b clearly demonstrates that crack widths follow such an anomalous scaling law. Notice that we have scaled away the amplitudes of the roughness for all the different $D$ to achieve the maximal data collapse to illustrate the universality. In the range of $D$ considered here the amplitudes vary by about 20 %. The inset in Fig. \[fig:widthD\]b reports the data collapse of the power spectra based on anomalous scaling for different disorder values. This collapse of the data once again suggests that local roughness is independent of disorder. A fit of the power law decay of the spectrum yields a local roughness exponent of $\zeta_{loc}=0.74$. This result is in close agreement with the real space estimate and we can attribute the differences to the bias associated to the methods employed [@sch-95]. ![(Color online) (a) Scaling of local and global widths $w(l)$ and $W$ of the crack for different system sizes $L = \{64,128,256,320\}$, disorder values $D$ and a fixed $a_0/L = 1/16$ value (top). The local crack width exponent $\zeta_{loc} = 0.71$ is independent of disorder and differs considerably from the global crack width exponent $\zeta = 0.87$. (b) Collapse of the crack width data using the anomalous scaling law (bottom). $L_c = (L-a_0)$ is the effective length of the crack profile. Collapse of the data including a disorder dependent prefactor $A(D)$ for a given disorder value implies that local and global roughness exponents are independent of disorder. The inset shows collapse of power spectrum $S(k)$ using the anomalous scaling law with $\zeta_{loc} = 0.71$ and $\zeta = 0.87$. The slope in the inset defines the local exponent via $-(2\zeta_{loc}+1) = -2.48$. (a)-(b) present a total of 20 data sets.[]{data-label="fig:widthD"}](halfseg_rough_mov_L320_L256_L128_L64.eps "fig:"){width="8cm"} ![(Color online) (a) Scaling of local and global widths $w(l)$ and $W$ of the crack for different system sizes $L = \{64,128,256,320\}$, disorder values $D$ and a fixed $a_0/L = 1/16$ value (top). The local crack width exponent $\zeta_{loc} = 0.71$ is independent of disorder and differs considerably from the global crack width exponent $\zeta = 0.87$. (b) Collapse of the crack width data using the anomalous scaling law (bottom). $L_c = (L-a_0)$ is the effective length of the crack profile. Collapse of the data including a disorder dependent prefactor $A(D)$ for a given disorder value implies that local and global roughness exponents are independent of disorder. The inset shows collapse of power spectrum $S(k)$ using the anomalous scaling law with $\zeta_{loc} = 0.71$ and $\zeta = 0.87$. The slope in the inset defines the local exponent via $-(2\zeta_{loc}+1) = -2.48$. (a)-(b) present a total of 20 data sets.[]{data-label="fig:widthD"}](halfseg_ALLL_inset_AD.eps "fig:"){width="8cm"} Anomalous Scaling ----------------- The scaling properties of the crack profiles $h(x)$ can also be studied using the probability density distribution $p(\Delta h(\ell))$ of the height differences $\Delta h(\ell) = [h(x+\ell) - h(x)]$ of the crack profile between any two points on the reference line ($x$-axis) separated by a distance $\ell$. Assuming the self-affine property of the crack profiles implies that the probability density distribution $p(\Delta h(\ell))$ follows the relation $$\begin{aligned} p(\Delta h(\ell)) & \sim & \langle\Delta h^2(\ell) \rangle^{-1/2} ~f\left(\frac{\Delta h(\ell)}{\langle\Delta h^2(\ell)\rangle^{1/2} }\right) \label{pdelt}\end{aligned}$$ where $\langle\Delta h^2(\ell)\rangle^{1/2}$ denotes the width of the height difference $\Delta h(\ell)$ over a length scale $\ell$. Since for PBC the periodicity in crack profiles is analogous to return-to-origin excursions arising in stochastic processes, we propose the following ansatz for the local width $\langle\Delta h^2(\ell)\rangle^{1/2}$ in height differences $\Delta h(\ell)$ $$\begin{aligned} \langle\Delta h^2 (\ell)\rangle^{1/2} & = & \langle\Delta h^2(L/2)\rangle^{1/2} ~\phi\left(\frac{\ell}{L/2}\right) \label{dhell}\end{aligned}$$ with $\langle\Delta h^2(L/2)\rangle^{1/2} = L^\zeta$. The function $\phi\left(\frac{\ell}{L/2}\right)$ is symmetric about $\ell = L/2$ and is constrained such that $\phi\left(\frac{\ell}{L/2}\right) = 0$ at $\ell = 0$ and $\ell = L$, and $\phi\left(\frac{\ell}{L/2}\right) = 1$ at $\ell = L/2$. Based on these conditions, a scaling ansatz of the form $$\begin{aligned} \left[\frac{\langle\Delta h^2(\ell)\rangle^{1/2}} {\langle\Delta h^2(L/2)\rangle^{1/2}}\right]^{1/\zeta_{loc}} + \frac{(\ell-L/2)^2}{(L/2)^2} & = & 1 \label{eq3}\end{aligned}$$ similar to stochastic excursions or bridges can be proposed for $\langle\Delta h^2(\ell)\rangle^{1/2}$, which implies a functional form $$\begin{aligned} \phi\left (\frac{\ell}{L/2}\right) & = & \left[1 - \left(\frac{(\ell - L/2)}{L/2}\right)^2\right]^{\zeta_{loc}} \label{dhell1}\end{aligned}$$ for $\phi\left(\frac{\ell}{L/2}\right)$ that is satisfied to a good approximation by our numerical results. This scaling ansatz implies anomalous scaling when $\zeta_{loc} \neq \zeta$. Upon further simplication, Eq. (\[dhell1\]) results in $$\begin{aligned} \phi\left(\frac{\ell}{L/2}\right) & = & 4^{\zeta_{loc}} \left(\frac{\ell}{L}\right)^{\zeta_{loc}} \left(1 - \frac{\ell}{L}\right)^{\zeta_{loc}} \label{dhell12}\end{aligned}$$ which along with $\langle\Delta h^2(L/2)\rangle^{1/2} = L^\zeta$ and Eq. (\[dhell\]) illustrates how anomalous scaling appears in the scaling of local widths $\langle\Delta h^2(\ell)\rangle^{1/2}$, and how local and global roughness exponents $\zeta_{loc}$ and $\zeta$ can be computed based on numerical results. Figure \[fig:pbc\_multi\] presents the scaling of $\langle\Delta h^2(\ell)\rangle^{1/2}$ based on the above ansatz (Eq. (\[eq3\])). The collapse of the $\langle\Delta h^2(\ell)\rangle^{1/2}/\langle\Delta h^2(L/2)\rangle^{1/2}$ data for different system sizes $L$ and window sizes $\ell$ onto a scaling form given by Eq. (\[eq3\]) with $\zeta_{loc} = 0.64$ can be clearly seen in Figs. \[fig:pbc\_multi\](a)-(c). In particular, Figs. \[fig:pbc\_multi\](a)-(c) present the data for unnotched and notched samples with varying amounts of disorder $D$ and relative crack sizes $a_0/L$. The collapse of the data for varying amounts of disorder ($0.3 \le D \le 1$) and relative crack sizes ($0 \le a_0/L \le 3/8$) can be clearly seen in these figures and suggests that local roughness exponent $\zeta_{loc}$ is independent of disorder, at least for the disorder ranges considered here. It is an interesting question as to why the $\zeta_{loc}$ from collapsing the average crack profiles does not agree with the value from the local width, demonstrated in Fig. \[fig:widthD\]. Figures \[fig:width\](a)-(b) present the scaling of $\langle\Delta h^2(L/2)\rangle^{1/2}$ for various notched and unnotched samples with varying amounts of disorder and relative crack sizes. The data presented in these figures shows that $\langle\Delta h^2(L/2)\rangle^{1/2} \sim L^\zeta$ with $\zeta = 0.87$ in agreement with the previously given value for the global width exponent. In Figure \[fig:width\](b) one can note that there is a $a_0/L$-dependent amplitude and the data follow the 0.87-exponent at fixed $a_0/L$. Since there exists a significant difference between the global and local roughness exponents ($\zeta$ and $\zeta_{loc}$, respectively), here again we can conclude that crack profiles obtained using the fuse models exhibit anomalous roughness scaling. ------------------------------------------------ ---------------------------------------- --------------------------------------------- -- ![image](stddh_uncrack_D1_00.eps){width="6cm"} ![image](stddh_varD0.eps){width="6cm"} ![image](stddh_vara0_ALLL.eps){width="6cm"} ------------------------------------------------ ---------------------------------------- --------------------------------------------- -- ![(Color online) Scaling of $\langle\Delta h^2(L/2)\rangle^{1/2}$ with system size $L$. For notched samples, we use the effective length of the crack profile $L_c = L - a_0$. (a) Scaling of $\langle\Delta h^2(L/2)\rangle^{1/2}$ is shown for unnotched samples and for samples with a fixed relative notch size of $a_0/L = 1/16$ having varying amounts of disorder $D$. (b) Scaling of $\langle\Delta h^2(L_c/2)\rangle^{1/2}$ for samples with varying notch sizes and a fixed disorder of $D = 0.6$ (bottom).[]{data-label="fig:width"}](stddhmax_disorder.eps "fig:"){width="8cm"} ![(Color online) Scaling of $\langle\Delta h^2(L/2)\rangle^{1/2}$ with system size $L$. For notched samples, we use the effective length of the crack profile $L_c = L - a_0$. (a) Scaling of $\langle\Delta h^2(L/2)\rangle^{1/2}$ is shown for unnotched samples and for samples with a fixed relative notch size of $a_0/L = 1/16$ having varying amounts of disorder $D$. (b) Scaling of $\langle\Delta h^2(L_c/2)\rangle^{1/2}$ for samples with varying notch sizes and a fixed disorder of $D = 0.6$ (bottom).[]{data-label="fig:width"}](stddhmax_vara0_ALLL.eps "fig:"){width="8cm"} The questions that we would like to resolve in the following are whether this anomalous scaling and multi-affine scaling of crack surface roughness are a consequence of the jumps in the crack profiles induced by the crack overhangs (see Fig. \[fig:crack\]). As shown in Fig. \[fig:crack\_nojumps\], removal of jumps from an initially periodic crack profile $h(x)$ makes the resulting crack profile $h_{NP}(x)$ nonperiodic, where the subscript $NP$ refers to [*nonperiodicity*]{} of the profiles. A direct evaluation of the roughness exponent using these nonperiodic profiles can be made. However, such an evaluation of roughness exhibits finite size effects for window sizes $\ell > L/2$. Alternatively, the roughness of these resulting nonperiodic profiles can be evaluated by first subtracting a linear profile $h_{lin}(x) = \left[h_{NP}(0) + \frac{(h_{NP}(L)-h_{NP}(0))}{L} x\right]$ from the nonperiodic profile $h_{NP}(x)$, and then evaluating the roughness of the resulting periodic profile $h_{P}(x)$. In the following, we consider the scaling of $h_{P}(x)$. ![(Color online) Figure shows a typical single valued crack profile $h(x)$ with jumps based on solid-on-solid projection scheme (identified as (a)). Removing the jumps in the crack profile $h(x)$ makes it a non-periodic profile (identified as (b)). Subtracting a linear profile from this non-periodic profile results in a periodic profile (identified as (c)).[]{data-label="fig:crack_nojumps"}](crack_prof_pbc2nopbc_nojumps_102.eps){width="8cm"} Figure \[fig:pbc\_width\_nojumps\] presents the scaling of crack width $w(\ell)$ with window size $\ell$ for crack profiles without the jumps. The data presented in Fig. \[fig:pbc\_width\_nojumps\]a suggests that local and global roughness exponents ($\zeta_{loc} = 0.74$ and $\zeta = 0.80$) are not the same even after removing the jumps in the crack profiles, although the difference between these exponents is small. We have also investigated the power spectra $S(k)$ of the crack profiles without the jumps in the crack profiles (see Fig. \[fig:pbc\_width\_nojumps\]b). An excellent collapse of the data is obtained using the anomalous scaling law for power spectrum with $2(\zeta-\zeta_{loc}) = 0.1$ and $\zeta_{loc} = 0.74$. This result is consistent with the exponents measured using the crack widths as in Fig. \[fig:pbc\_width\_nojumps\]a. ![(Color online) Scaling of crack profiles without the jumps. (a) Scaling of crack width $w(\ell)$. The local and global roughness exponents appear to be different although the difference is considerably smaller than that with the jumps in the profiles. (b) Scaling of power spectra of crack profiles. An excellent collapse of the data is obtained using anomalous scaling law with $2(\zeta-\zeta_{loc}) = 0.1$ and $\zeta_{loc} = 0.74$. Removal of overhangs in the crack profiles does not appear to eliminate the anomalous scaling of crack roughness in fuse models.[]{data-label="fig:pbc_width_nojumps"}](rough_mov_nojumps.eps "fig:"){width="8cm"} ![(Color online) Scaling of crack profiles without the jumps. (a) Scaling of crack width $w(\ell)$. The local and global roughness exponents appear to be different although the difference is considerably smaller than that with the jumps in the profiles. (b) Scaling of power spectra of crack profiles. An excellent collapse of the data is obtained using anomalous scaling law with $2(\zeta-\zeta_{loc}) = 0.1$ and $\zeta_{loc} = 0.74$. Removal of overhangs in the crack profiles does not appear to eliminate the anomalous scaling of crack roughness in fuse models.[]{data-label="fig:pbc_width_nojumps"}](rough_ps_nojumps_anomalous.eps "fig:"){width="8cm"} In addition to the above two methods (crack width scaling and power spectrum method) used for estimating the local and global roughness exponents, we also used the scaling ansatz proposed in Eq. (\[eq3\]) to estimate the local and global roughness exponents. Since the difference between the local and global exponents is small, alternate ways of measuring these exponents provide a sense of reliability into these estimates. Figures \[fig:pbc\_multi\_nojumps\](a)-(c) present the scaling of crack profiles $h_{P}(x)$. The collapse of the $\langle\Delta h_P^2(\ell)\rangle^{1/2}/\langle\Delta h_P^2(L/2)\rangle^{1/2}$ data for different system sizes $L$ and window sizes $\ell$ onto a scaling form given by Eq. (\[eq3\]) with $\zeta_{loc} = 0.62$ can be clearly seen in Fig. \[fig:pbc\_multi\_nojumps\](a). This value is fairly close to the 0.64 quoted before. In addition, the collapse of the data presented in Fig. \[fig:pbc\_multi\_nojumps\](c) for $\langle\Delta h_P^q(\ell)\rangle^{1/q}/\langle\Delta h_P^q(L/2)\rangle^{1/q}$ demonstrates that multi-affine scaling of fracture surfaces arises because of overhangs (jumps) with certain statistics in the crack profile and removal of these jumps in the crack profiles completely eliminates multiscaling of fracture surfaces. The simple scaling (no multiaffinity) is also evident through the scaling of $\langle\Delta h_P^q(L/2)\rangle^{1/q}$ presented in Fig. \[fig:pbc\_multi\_nojumps\](b). The slopes of the data for moments $q = 1$ to $6$ of $\Delta h_P(L/2)$ are identical. An interesting observation to be made is that $\langle\Delta h_P^q(L/2)\rangle^{1/q} \sim L^\zeta$ with $\zeta = 0.80$ whereas the local roughness exponent as obtained from Figs. \[fig:pbc\_multi\_nojumps\](a) and (c) is $\zeta_{loc} = 0.62$. A similar behavior is observed even when the linearity in the profile is not subtracted: the scaling of these nonperiodic profiles $h_{NP}(x)$ is in agreement with that obtained for periodic profiles for window size $\ell \le L/2$ although finite size effects are observed when window sizes $\ell > L/2$ are considered. The difference in these exponents even after removing the jumps caused by overhangs in the crack profile indicates that anomalous scaling is present in two-dimensional fracture simulations using the fuse models and this anomalous scaling is not due to the jumps in the crack profiles. ![(Color online) (a) Scaling of $\langle\Delta h_{P}^2(\ell)\rangle^{1/2}$ with window size $\ell$ (top); (b) Scaling of $\langle\Delta h_P^2(L/2)\rangle^{1/2}$ with system size $L$ (middle); (c) Scaling of $\langle\Delta h_P^q(\ell)\rangle^{1/q}$ with window size $\ell$ for different moments $q$ of crack profiles without overhangs (bottom).[]{data-label="fig:pbc_multi_nojumps"}](rough_multi_nojumps.eps "fig:"){width="8cm"} ![(Color online) (a) Scaling of $\langle\Delta h_{P}^2(\ell)\rangle^{1/2}$ with window size $\ell$ (top); (b) Scaling of $\langle\Delta h_P^2(L/2)\rangle^{1/2}$ with system size $L$ (middle); (c) Scaling of $\langle\Delta h_P^q(\ell)\rangle^{1/q}$ with window size $\ell$ for different moments $q$ of crack profiles without overhangs (bottom).[]{data-label="fig:pbc_multi_nojumps"}](deltahqL2.eps "fig:"){width="8cm"} ![(Color online) (a) Scaling of $\langle\Delta h_{P}^2(\ell)\rangle^{1/2}$ with window size $\ell$ (top); (b) Scaling of $\langle\Delta h_P^2(L/2)\rangle^{1/2}$ with system size $L$ (middle); (c) Scaling of $\langle\Delta h_P^q(\ell)\rangle^{1/q}$ with window size $\ell$ for different moments $q$ of crack profiles without overhangs (bottom).[]{data-label="fig:pbc_multi_nojumps"}](height_multi_nojumps_512.eps "fig:"){width="8cm"} The case of open boundaries --------------------------- It is interesting to compare the PBC case with that of open boundary conditions. Figure \[fig:obc\_width\] presents the scaling of crack widths for fuse lattice simulations with open boundary conditions. The data in Fig. \[fig:obc\_width\]a indicates that local roughness exponent is $\zeta_{loc} = 0.75$. However, the data for different system sizes does not collapse, which is an indication of anomalous scaling. The inset in Fig. \[fig:obc\_width\]a shows that a simple L-dependent shifting of the data achieves a perfect collapse of the data with possible finite size deviations for window sizes $\ell$ approaching the system size. Figure \[fig:obc\_width\]b presents the scaling of $w(\ell)$ for crack profiles without the jumps. Even after removing the jumps from the crack profiles, the crack widths data does not collapse onto a single curve. This suggests that removal of overhangs in the crack profile does not eliminate this apparent anomalous scaling of crack roughness even for open boundary conditions. On the other hand, removing the jumps in the crack profiles once again completely eliminates the multiscaling. Figure \[fig:nopbc\_nomultiscaling\]a presents the scaling of $q$-th order correlation function $C_q(\ell) = \langle |h(x+\ell)-h(x)|^q\rangle^{1/q}$ measured using the original crack profiles. Multiscaling below a characteristic length scale can be clearly seen in Fig. \[fig:nopbc\_nomultiscaling\]a. The data in Fig. \[fig:nopbc\_nomultiscaling\]b represents the scaling of $q$-th order correlation function $C_q(\ell)$ measured after removing the jumps in the profiles. The Figure shows that the plots for different crack profile moments $q$ are parallel to one another, and thus the removal of jumps in the crack profiles eliminates multiscaling. A collapse of these plots is shown in the inset and the local roughness exponent is estimated to be $\zeta_{loc} = 0.72$, close to the PBC value. ![(Color online) Scaling of crack width $w(\ell)$ with window size $\ell$ for open boundary conditions. (a) Scaling of $w(\ell)$ for crack profiles with jumps. The inset presents the data shown in the main figure after a $L$-dependent shift is applied. A power law fit to the data estimates the local roughness exponent to be $\zeta_{loc} = 0.75$. (b) Scaling of $w(\ell)$ for crack profiles obtained after removing the jumps. plots in figure (b) indicate that even with the removal of overhangs in the crack profile does not eliminate this apparent anomalous scaling of crack roughness.[]{data-label="fig:obc_width"}](nopbc_mov_shifted_inset.eps "fig:"){width="8cm"} ![(Color online) Scaling of crack width $w(\ell)$ with window size $\ell$ for open boundary conditions. (a) Scaling of $w(\ell)$ for crack profiles with jumps. The inset presents the data shown in the main figure after a $L$-dependent shift is applied. A power law fit to the data estimates the local roughness exponent to be $\zeta_{loc} = 0.75$. (b) Scaling of $w(\ell)$ for crack profiles obtained after removing the jumps. plots in figure (b) indicate that even with the removal of overhangs in the crack profile does not eliminate this apparent anomalous scaling of crack roughness.[]{data-label="fig:obc_width"}](nopbc_nojumps_mov.eps "fig:"){width="8cm"} ![(Color online) Scaling of $q$-th order correlation function $C_q(\ell)$. The data presented is for a system of size $L = 320$ simulated with open boundary conditions. (a) $C_q(\ell)$ measured using original crak profiles with jumps. (b) $C_q(\ell)$ measured using crack profiles without jumps. Removal of overhangs in the crack profile eliminates apparent multiscaling. Inset shows that normalization of the data leads to collapse of the curves with a local roughness exponent $\zeta_{loc} = 0.72$.[]{data-label="fig:nopbc_nomultiscaling"}](height_multi_nopbc_320.eps "fig:"){width="8cm"} ![(Color online) Scaling of $q$-th order correlation function $C_q(\ell)$. The data presented is for a system of size $L = 320$ simulated with open boundary conditions. (a) $C_q(\ell)$ measured using original crak profiles with jumps. (b) $C_q(\ell)$ measured using crack profiles without jumps. Removal of overhangs in the crack profile eliminates apparent multiscaling. Inset shows that normalization of the data leads to collapse of the curves with a local roughness exponent $\zeta_{loc} = 0.72$.[]{data-label="fig:nopbc_nomultiscaling"}](height_multi_collapse_nopbc_nojumps_inset.eps "fig:"){width="8cm"} In the following, we finally investigate the probability density $p(\Delta h(\ell))$ of height differences $\Delta h(\ell)$. In Refs. [@jstat2; @santucci07], the $p(\Delta h(\ell))$ distribution is shown to follow a Gaussian distribution above a cutoff length scale and the deviations away from Gaussian distribution in the tails of the distribution have been attributed to finite jumps in the crack profiles. A self-affine scaling of $p(\Delta h(\ell))$ as given by Eq. (\[pdelt\]) implies that the cumulative distribution $P(\Delta h(\ell))$ scales as $P(\Delta h(\ell)) \sim P(\Delta h(\ell)/\langle\Delta h^2(\ell)\rangle^{1/2})$. Figure \[fig:pdeltah\](a) presents the raw data of cumulative probability distributions $P(\Delta h(\ell))$ of the height differences $\Delta h(\ell)$ on a normal or Gaussian paper for bin sizes $\ell \ll L$. As observed in Refs. [@jstat2; @santucci07], and in Ref. [@salminen03] Fig. \[fig:pdeltah\](a) shows large deviations away from Gaussian distribution for these small bin sizes. However, for moderate bin sizes, the distribution is Gaussian with deviations in the tails of the distribution beyond the $3\sigma = 3\langle\Delta h^2(\ell)\rangle^{1/2}$ limit (data not shown in Figure). Removing the jumps in the crack profiles however collapses the $P(\Delta h_{P}(\ell))$ distributions onto a straight line (see Fig. \[fig:pdeltah\](b)) indicating the adequacy of Gaussian distribution even for small window sizes $\ell$. Indeed, Fig. \[fig:pdeltah\](b) shows the collapse of the $P(\Delta h_{P}(\ell))$ data for a system size $L = 512$ with a variety of bin sizes $2 \le \ell \le L/2$. Removing the jumps in the profiles not only made the $P(\Delta h_{P}(\ell))$ distributions Gaussian even for small window sizes $\ell$ but also extended the validity of $P(\Delta h_{P}(\ell))$ Gaussian distribution for moderate bin sizes to a $4\sigma = 4\langle\Delta h_{P}^2(\ell)\rangle^{1/2}$ ($99.993\%$ confidence) limit. ![(Color online) Plots of cumulative probability distributions $P(\Delta h(\ell))$ of the height differences $\Delta h(\ell) = [h(x+\ell) - h(x)]$ of the crack profile $h(x)$ for various bin sizes $\ell$ on a normal paper. $\Phi^{-1}$ denotes inverse Gaussian. The collapse of the profiles onto a straight line with unit slope indicates that a Gaussian distribution is adequate to represent $P(\Delta h(\ell))$. (a) $P(\Delta h(\ell))$ distributions for $L = 512$ and $\ell \ll L$. Large deviation from Gaussian profiles is observed for these window sizes. (b) Removing the jumps in the profiles however collapses the $P(\Delta h(\ell))$ distributions onto a straight line indicating the adequacy of a Gaussian even for small window sizes $\ell$. The data is for $L = 512$ and $\ell = (2,4,8,16,32,64,96,128,160,256)$.[]{data-label="fig:pdeltah"}](pdeltah_unnotch_small_ell_512.eps "fig:"){width="8cm"} ![(Color online) Plots of cumulative probability distributions $P(\Delta h(\ell))$ of the height differences $\Delta h(\ell) = [h(x+\ell) - h(x)]$ of the crack profile $h(x)$ for various bin sizes $\ell$ on a normal paper. $\Phi^{-1}$ denotes inverse Gaussian. The collapse of the profiles onto a straight line with unit slope indicates that a Gaussian distribution is adequate to represent $P(\Delta h(\ell))$. (a) $P(\Delta h(\ell))$ distributions for $L = 512$ and $\ell \ll L$. Large deviation from Gaussian profiles is observed for these window sizes. (b) Removing the jumps in the profiles however collapses the $P(\Delta h(\ell))$ distributions onto a straight line indicating the adequacy of a Gaussian even for small window sizes $\ell$. The data is for $L = 512$ and $\ell = (2,4,8,16,32,64,96,128,160,256)$.[]{data-label="fig:pdeltah"}](pdeltah_nojumps_unnotch_Gaussian_512.eps "fig:"){width="8cm"} Discussion ========== In summary, we have here considered the nature of the roughness of the crack surfaces in the two-dimensional RFM. The results presented here indicate universality of local roughness exponent for both notched and unnotched samples with different disorders $D$ in the range $0.3 \le D \le 1.0$ and for different relative crack sizes $a_0/L$. This is true both for open and periodic boundary conditions. The results indicate that anomalous scaling of roughness is a generic feature of two-dimensional fracture in the fuse model. This is in contrast to e.g. the beam model [@newbeam], where the global and local exponents are equal. The difference of the global and local exponents arises due to an additional lengthscale, which scales as a power-law of the system size $L$. We further investigated whether anomalous scaling of roughness is an artifact of presence of large jumps in the tails of $p(\Delta h(\ell))$ distribution. To do this, we considered the $\Delta h(\ell)$ data that is only within $\pm 3\sigma$ range of mean of $p(\Delta h(\ell))$ distribution, and computed the corresponding $\langle\Delta h^2(\ell)\rangle^{1/2}$ for various window sizes $\ell$. However, the data even from these truncated $p(\Delta h(\ell))$ distributions showed anomalous scaling. We repeated our investigation with $\pm 2\sigma$ range as well, but with a similar result. This indicates that anomalous scaling of roughness is not due to the tails of $p(\Delta h(\ell))$ distribution and persists in the mean as a function of $L$. Our results provide a concrete proof that the apparent multi-scaling of crack profiles observed in Ref. [@procaccia] is an artifact of jumps in the crack profiles that are formed due to the solid-on-solid approximation used in extracting the crack profiles. The removal of these jumps from the crack profiles completely eliminates this apparent multi-scaling of crack profiles. Furthermore, removing these jumps in the crack profiles extends the validity of Gaussian probability density distribution $p(\Delta h(\ell))$ of the height differences to even smaller window sizes $\ell$ and to a range (of $4\sigma = 4\langle\Delta h_{P}^2(\ell)\rangle^{1/2}$) well beyond that observed in earlier studies. In conclusion, though the RFM is a “toy model” of (two-dimensional here) fracture, it still poses interesting issues and can be used to study questions that are also relevant for experiments. Our numerical results presented here raise three basic theoretical questions related to the morphology of two-dimensional RFM fracture surfaces that still remain to be answered. First, why does the extra lengthscale that leads to anomalous scaling have to be algebraic? Second, models explaining the dynamics in the final avalanche (unstable crack propagation) and the roughness exponent values would be theoretically interesting to develop. Third, how strong is the universality of the roughness exponents for disorders very different from the ones used here, in particular, to those leading to percolative damage or finite densities of infinitely strong fuses? 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--- abstract: 'Different phenomenological RG transformations based on scaling relations for the derivatives of the inverse correlation length and singular part of the free-energy density are considered. These transformations are tested on the 2D square Ising and Potts models as well as on the 3D simple-cubic Ising model. Variants of RG equations yielding more accurate results than Nightingale’s RG scheme are obtained. In the 2D case the finite-size equations which give the [*exact*]{} values of the critical point or the critical exponent are found.' address: - '*L. D. Landau Institute for Theoretical Physics of Russian Academy of Sciences, 142432 Chernogolovka, Moscow Region, Russia*' - '*Vasilsursk Laboratory, Radiophysical Research Institute, 606263 Vasilsursk, Nizhny Novgorod Region, Russia*' author: - 'M. A. Yurishchev' title: '**Improved Transfer-Matrix Schemes of Phenomenological Renormalization [^1]**' --- Introduction {#sec:Intro} ============ The phenomenological renormalization-group (RG) method [@Nigh7679] is a powerful tool for the investigation of critical phenomena. As it is known [@dSS81], phenomenological RG can be constructed by using not just the correlation length as it is done in Nigtingale’s approach [@Nigh7679], but using any other quantity with a power-law divergence at criticality. Binder [@B81] suggested a phenomenological renormalization scheme by using the order-parameter moments (cumulants) which are, on the one hand, related to the higher susceptibilities and, on the other hand, immediately suitable for the Monte Carlo simulations. Recently Itakura [@I96] extended Binder’s cumulant crossing method taking linear combination of several different reduced moments. In this report, I discuss various RG transformations which follow from general scaling functional equations. These equations are evaluated in terms of the eigenvalues and eigenvectors of the transfer matrices. By large transverse sizes of partly finite subsystems, all those transformations must yield the same results. However, for the small sizes which normally are used in practice, different RG equations lead to estimates with distinct accuracies. My aim is to find the best strategies of a phenomenological renormalization group method. This is especially important for 3D systems. Phenomenological RG Equations {#sec:PRGE} ============================= Let us write the finite-size scaling equations for the [*derivatives*]{} of the inverse correlation length $\kappa_L$ and the singular part of the reduced free-energy density $f^s_L$: $$\label{eq:kappa^mn} \kappa_L^{(m,n)}(t, h) = b^{my_t+ny_h-1}\kappa_{L/b}^{(m,n)}(t', h'),$$ $$\label{eq:f^smn} f_L^{s\,(m,n)}(t, h) = b^{my_t+ny_h-d}f_{L/b}^{s\,(m,n)}(t', h'),$$ where $z^{(m,n)}(x,y)=\partial^{m+n}z/\partial x^m\partial y^n$ ($z$ is $\kappa_L$ or $f_L^s$), $t=K-K_c$ is the deviation from critical coupling, $h$ is a normalized external field, $y_t$ and $y_h$ are, respectively, thermal and magnetic critical exponents of the system, $d$ is the space dimensionality, $L$ is a characteristic size of a subsystem and $b=L/L'$ is the rescaling factor. In the phenomenological approach proposed by Nightingale [@Nigh7679], eq.(\[eq:kappa\^mn\]) with $m=n=0$ is combined with the ordinary expression for the inverse correlation length $$\label{eq:kappa_L} \kappa_L=\ln(\lambda_1^{(L)}/\lambda_2^{(L)}),$$ in which $\lambda_1^{(L)}$ and $\lambda_2^{(L)}$ are the largest and second-largest eigenvalues, respectively, of the associated transfer matrix. In the absence of a symmetry breaking field, the critical coupling $K_c$ is estimated from the equation $$\label{eq:kappa} L\kappa_L(K_c)=(L-1)\kappa_{L-1}(K_c).$$ In writing this equation, one sets $L'=L-1$. Another possible way to produce a phenomenological renormalization group is obtained by using eq.(\[eq:f\^smn\]) with $m=n=0$. The fixed point is given by the relation $$\label{eq:fs} L^df_L^s(K_c)=(L-1)^df_{L-1}^s(K_c).$$ The dimensionless free-energy density, $f_L=f_\infty + f_L^s$, of a subsystem $L^{d-1}\times\infty$ is calculated by the formula $$\label{eq:f_L} f_L=L^{1-d}\ln\lambda_1^{(L)}$$ and the “background” $f_\infty$ is introduced as an extra parameter. Besides eqs.(\[eq:kappa\]) and (\[eq:fs\]), in this paper I also consider the following RG equations (resulting from the relations (\[eq:kappa\^mn\]) and (\[eq:f\^smn\])): $$\label{eq:chi4} \frac{\chi_L^{(4)}}{L^d\chi_L^2}\Big|_{K_c}= \frac{\chi_{L-1}^{(4)}}{(L-1)^d\chi_{L-1}^2}\Big|_{K_c},$$ where $\chi_L=\partial^2f_L/\partial h^2|_{h=0}=f_L^{s\,(0,2)}(K,0)$ is the zero-field susceptibility and $\chi_L^{(4)}=\partial^4f_L/\partial h^4|_{h=0}=f_L^{s\,(0,4)}(K,0)$ is a nonlinear susceptibility (eq.(\[eq:chi4\]) corresponds to Binder’s phenomenological renormalization group); $$\label{eq:kappa1} L^{2-d}(\kappa_L^{(1)})^2/\chi_L= (L-1)^{2-d}(\kappa_{L-1}^{(1)})^2/\chi_{L-1};$$ $$\label{eq:kappa2} L^{1-d}\kappa_L^{(2)}/\chi_L= (L-1)^{1-d}\kappa_{L-1}^{(2)}/\chi_{L-1};$$ $$\label{eq:kappa4} L^{1-2d}\kappa_L^{(4)}/\chi_L^2= (L-1)^{1-2d}\kappa_{L-1}^{(4)}/\chi_{L-1}^2.$$ Here, $\kappa_L^{(n)}=\partial^n\kappa_L/\partial h^n|_{h=0}=\kappa_L^{(0,n)}(K,0)$. Expressions for the derivatives of the inverse correlation length and the free energy with respect to $h$ in terms of eigenvalues and eigenvectors of the transfer matrix are available in [@Yu9497]. Results and discussions {#sec:RD} ======================= To represent numerical data in tables, eqs.(\[eq:kappa\]), (\[eq:fs\]), (\[eq:chi4\]), (\[eq:kappa1\]), (\[eq:kappa2\]) and (\[eq:kappa4\]) will be labeled by symbols “$\kappa$”, “$f^s$”, “$\chi^{(4)}/\chi^2$”, “$(\kappa^{(1)})^2/\chi$”, “$\kappa^{(2)}/\chi$” and “$\kappa^{(4)}/\chi^2$”, respectively. In table \[tab:2DI\], results for the critical coupling in the Ising model on a square lattice are given. The calculations were carried out for strips $L\times\infty$ with a periodic boundary condition in the transverse direction. The estimates are shown for the pairs $(L-1,L)$ with $L\leq5$. In the case of $(3,4)$ pairs, the errors are also given. The type of phenomenological RG equations which have been used are indicated in the first column of the table. In this model $K_c={1\over2}\ln(1+\sqrt2)$ and $f_\infty=2G/\pi + {1\over2}\ln2$ ($G$ is Catalan’s constant) [@O44]. ------------------------- --------- --------------------- --------- eq. $(2,3)$ $(3,4)$ $(4,5)$ \[2mm\] $(\kappa)$ 0.42236 0.43088 $(-2.23\%)$ 0.43595 $(\chi^{(4)}/\chi^2)$ 0.42593 0.43242 $(-1.88\%)$ 0.43672 $(\kappa^{(4)}/\chi^2)$ 0.42596 0.43243 $(-1.87\%)$ 0.43673 \[1mm\] $(f^{s})$ 0.44324 0.44168 $(+0.23\%)$ 0.44105 $(\kappa^{(2)}/\chi)$ 0.47420 0.45153 $(+2.64\%)$ 0.44626 ------------------------- --------- --------------------- --------- : Estimates of $K_c$ for the 2D sq Ising lattice; $K_c^{exact}=0.440\,686\ldots$[]{data-label="tab:2DI"} It is seen from table \[tab:2DI\] that the best lower bound is given by eq.(\[eq:kappa4\]). Slightly worse results are obtained by Binder’s phenomenological renormalization-group procedure. This approach which is normally implemented by Monte Carlo simulations was used in the transfer-matrix version in [@SD85]. Nightingale’s renormalization (first line in table 1) which is traditionally used by transfer-matrix calculations has only the third position in accuracy among the lower estimates. I also found the phenomenological RG equations leading to the upper bounds for $K_c$ (last two lines in table \[tab:2DI\]). Among these more accurate results are provided by eq.(\[eq:fs\]). The magnitude of the error in line four of table 1 is the least among all lower and upper estimates of $K_c$. Unfortunately, such approach requires a knowledge about the background $f_\infty$. Transfer-matrix eigenvalues for the Ising strips are known in analytical form [@O44]. Using this fact I considered the RG transformation (\[eq:f\^smn\]) which makes use of the first derivative with respect to $K$ $(m=1, n=0; h=0)$. For the fixed point this transformation gives $$\label{eq:us} L^{d-y_t}u_L^s(K_c)=(L-1)^{d-y_t}u_{L-1}^s(K_c),$$ where $u_L^s=u_L-u_\infty$ is the singular part of the reduced energy density, and $u_L=\partial f_L/\partial K$. Remarkably, the root of eq.(\[eq:us\]) is equal to the exact value of $K_c$ since $u_L^s(K_c)\equiv0$ for all $L$. In other words, all finite-size corrections to the background $u_\infty(=\sqrt2$ [@O44]) are zero. Moreover, in the 2D Ising model $\partial\kappa_L/\partial K$ at $K=K_c$ is also independed of $L$ and, therefore, eq.(\[eq:kappa\^mn\]) with $m=1$ and $n=0$ gives the exact value for the critical exponent: $\nu\equiv1/y_t=1$. The data obtained for the 3-state square Potts lattice are collected in table \[tab:2DP\]. For this model $K_c=\ln(1+\sqrt3)$ and $f_\infty=4G/3\pi+\ln(2\sqrt3)+{1\over3}\ln(2+\sqrt3)$ [@B82]. Inspecting table \[tab:2DP\], it is seen that eqs.(\[eq:fs\]) and (\[eq:kappa1\]) lead to more qualitative estimates than Nightingale’s approach. Again, the lowest absolute error is yielded by the phenomenological RG equation based on $f_L^s$. Numerical calculations on strips $L\times\infty$ show us that in the 2D $q$-state Potts model the finite-size corrections to the background energy, $u_\infty=1+1/\sqrt q$ [@B82], are also absent and, consequently, the equation $$\label{eq:u_L} u_L(K_c)=u_{L'}(K_c)$$ yields the exact value of $K_c$. Note that this equation has been derived earlier from other considerations [@W94]. ------------------------- --------- -------------------- --------- eq. $(2,3)$ $(3,4)$ $(4,5)$ \[2mm\] $(\kappa)$ 0.96248 0.98350 $(-2.1\%)$ 0.99467 $(\kappa^{(1)^2}/\chi)$ 0.99311 0.99920 $(-0.6\%)$ 1.00380 \[1mm\] $(f^{s})$ 1.00927 1.00667 $(+0.2\%)$ 1.00565 ------------------------- --------- -------------------- --------- : Estimates of $K_c$ for the 2D sq 3-state Potts lattice; $K_c^{exact}=1.005\,052\ldots$[]{data-label="tab:2DP"} Let us discuss now the results presented in table \[tab:3DI\] for the 3D Ising model on a simple-cubic lattice. For this model $K_c=0.221\,6544(3)$ [@TB96] and $f_\infty=0.777\,90(2)$ [@M89]. Renormalizations were done for the $L\times L\times\infty$ parallelepipeds with periodic boundary conditions in both transverse directions. As in the 2D case, the best lower values of $K_c$ are obtained from eq.(\[eq:kappa4\]). ------------------------- --------- --------------------- eq. $(2,3)$ $(3,4)$ \[2mm\] $(\kappa)$ 0.21340 0.21826 $(-1.53\%)$ $(\chi^{(4)}/\chi^2)$ 0.21823 0.22002 $(-0.74\%)$ $(\kappa^{(4)}/\chi^2)$ 0.21824 0.22006 $(-0.72\%)$ \[1mm\] $(f^{s})$ 0.22354 0.22236 $(+0.32\%)$ $(\kappa^{(2)}/\chi)$ 0.22658 0.22314 $(+0.67\%)$ ------------------------- --------- --------------------- : Estimates of $K_c$ for the 3D sc Ising lattice; $K_c^{exact}=0.221\,6544(6)$[]{data-label="tab:3DI"} In the 3D case the amplitudes of the finite-size corrections to the critical-point energy are not equal to zero. As a result, eq.(\[eq:u\_L\]) only yields an approximate value of $K_c$. The author thanks A. A. Belavin for stimulating discussions and valuable remarks. I am indebted also to M. I. Polikarpov for his encouragement. This work was supported by RFBR Grant Nos. 99-02-16472 and 99-02-26660. 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--- abstract: 'In a typical single molecule experiment, dynamics of an unfolded proteins is studied by determining the reconfiguration time using long-range F$\ddot{o}$rster resonance energy transfer where the reconfiguration time is the characteristic decay time of the position correlation between two residues of the protein. In this paper we theoretically calculate the reconfiguration time for a single flexible polymer in presence of active noise. The study suggests that though the MSD grows faster, the chain reconfiguration is always slower in presence of long-lived active noise with exponential temporal correlation. Similar behavior is observed for a worm like semi-flexible chain and a Zimm chain. However it is primarily the characteristic correlation time of the active noise and not the strength that controls the increase in the reconfiguration time. In a nutshell, such active noise makes the polymer to move faster but the correlation loss between the monomers becomes slower.' author: - 'Nairhita Samanta and Rajarshi Chakrabarti\*' title: Chain reconfiguration in active noise --- Introduction ============ Active processes giving rise to non-equilibrium fluctuations are ubiquitous in biological systems. This is notably distinct from the incessant motion exhibited by particles in any fluid known as Brownian motion which results from the constant collision of the particle with its surrounding solvent molecules [@Chandrasekhar]. However in biological systems active motion are driven by the chemical energy produced from the hydrolysis of adenosine triphosphate (ATP). For instance the motion of cytoskeleton inside the cells is controlled by the motor proteins which involves ATP hydrolysis [@Weitz]. Other examples would be cell membranes which are perpetually out of equilibrium through active processes [@Bassereau] and swimming bacteria which control the active transport of nutrients in aqueous medium [@Libchaber]. In a very new study it has been shown that the dynamics of DNA is also influenced by the processes dependent on the energy derived from ATP hydrolysis [@Mitchison]. A series of simulation studies have also been performed to investigate the looping dynamics in active system. Shin *et al* have recently shown that in presence of self-propelled particles the loop formation in polymer become faster due to increased diffusion [@metzler2015njp]. In another study it has been found that looping is also faster when the polymer itself is active, having a catalytic monomer. This catalytic monomer generates a concentration gradient prompting faster diffusion of the non-catalytic monomer resulting in rapid ring-closure [@snigdha2014]. Such studies are extremely important as loop formation in biopolymers is an essential process in protein folding, DNA replication etc. Experimentally there have been many attempts to study the dynamics of unfolded proteins mainly involving long-range F$\ddot{o}$rster resonance energy transfer (FRET) [@schuler2008; @schuler2012]. In this particular technique two residues of a protein are labelled with a donor and an acceptor using fluorescence probes to study the fluctuation of the distance between them (Fig. 1). This distance is temporally correlated with a characteristic decay time, referred as reconfiguration time ($\tau_{N0}$) which is determined by fitting the long time decay of the second order intensity correlation function [@makarov2003; @makarov2010]. To the best of our knowledge no such experimental study has been performed till date which will provide insights into the reconfiguration dynamics of a chain in an active medium. In this paper we theoretically analyze the dynamics of a single chain polymer in presence of active noise. By active noise we refer to a long-ranged temporal noise where the time correlation is independent of ambient temperature. In contrary to the recent simulation studies we find such long temporally correlated noise to result in a slower reconfiguration of a polymer chain, be it flexible or semi-flexible. Even in the presence of non-local hydrodynamics interactions in addition to the active noise, reconfiguration of the chain is slower. The paper is arranged as follows. In section $\bf{II}$ we have introduced the model for active noise, in section $\bf{III}$ the calculation methods are discussed. The results are presented in section $\bf{IV}$ and the paper is concluded in section $\bf{V}$. ![Schematics showing the end-to-end monomers of a protein labelled with the donor and the acceptor. The arrow depicts the distance ($R_{N0}(t)$) between the donor and acceptor monitored in the experiment.[]{data-label="fig:a0"}](1.jpg){width="55.00000%"} Model ===== For an one-dimensional Brownian particle, moving in a harmonic trap, the dynamics in the over-damped regime is best described by the Langevin equation [@doibook; @kawakatsubook] $$\xi\frac{dx(t)}{d{t}}=-k{x(t)}+{f}(t) \label{eq:langevin}$$ Where, $k$ is the force constant and $f(t)$ is the Gaussian random force with first and second moments $$\left<f(t)\right>=0, \left<f(t^{\prime})f(t^{\prime\prime})\right>=2 \xi k_B T \delta(t^{\prime}-t^{\prime\prime}) \label{eq:random-force}$$ Here the strength of the correlation depends on the ambient temperature $T$. Now when the system is subjected to an active noise of strength $f_A$ the equation of motion becomes [@ghosh2014], $$\xi\frac{dx(t)}{d{t}}=-k{x(t)}+{f}(t)+{f_A}(t) \label{eq:langevinact}$$ Here, $f_A$ is considered to be exponentially correlated with a characteristic decay time $\tau_A$ and Gaussian distribution with moments $$\left<f_A(t)\right>=0,\left\langle f_A(t^{\prime})f_A(t^{\prime\prime}) \right\rangle)=Ce^{-\frac{|t^{\prime}-t^{\prime\prime}|}{\tau_A}} \label{eq:active-force}$$ Importantly $C$, the strength of the active noise is independent of $T$ and can be related to probability of active force ($P(f_A)$) and the force $f_A$ acting on the particle as f $C \sim P(f_A)f_A^2$. Being independent of ambient temperature $T$ the active noise drives the system off the equilibrium and only in the infinite time limit a stationary state can be realized. In between, the system remains in a non-equilibrium state. Such a choice of noise correlation comes from earlier simulation studies on red-blood cell membrane fluctuations, where the force $f_A(t)$ originates from the non-equilibrium fluctuations of the motor proteins [@Shokef]. The position correlation function $\left\langle x(t)x(0)\right \rangle$ with the active noise is analytically trackable and has the following expression. Readers are referred to the appendix for the detailed derivation. $$\phi(t)=\left\langle x(t)x(0) \right\rangle=\frac{k_BT}{k}e^{-\frac{t}{\tau}}+\frac{C}{\xi^2}e^{-\frac{t}{\tau}}\left(\frac{1}{\frac{1}{\tau}\left(\frac{1}{\tau}+\frac{1}{\tau_A}\right)}+\frac{e^{\frac{t}{\tau}-\frac{t}{\tau_A}}-1}{\left(\frac{1}{\tau^2}-\frac{1}{\tau_A^2} \right)} \right) \label{eq:corrpart}$$ In the absence of active noise, $C \rightarrow 0$, the above expression is reduced to $$\lim_{C\to0}\phi(t)=\frac{k_BT}{k}e^{-\frac{t}{\tau}}$$ This is the time correlation function for an over-damped Brownian particle in harmonic potential in the presence of only thermal noise or the Ornstein-Uhlenbeck process and $\tau=\frac{\xi}{k}$ is the corresponding relaxation time [@sokolovbook]. It is obvious from Eq. (\[eq:corrpart\]) that although the correlation function is translationally invariant, it is not single exponential. However, this can be approximated as a single exponential with an effective relaxation time $\tau_{eff}$, $$\phi_{eff}(t)=\frac{k_BT_{0}}{k}e^{-\frac{t}{\tau_{eff}}}$$ with, $$\begin{aligned} \tau_{eff} & = \int\limits_{0}^{\infty} dt \frac{\phi(t)}{\phi(0)} \\ & = \tau \left[\frac{\left(k_BT\xi^2+C\tau\tau_Ak\right)\left(\tau+\tau_A\right)}{C\tau^2\tau_Ak+k_BT\xi^2\left(\tau+\tau_A\right)}\right] \end{aligned}$$ \ $\tau_{eff}$ is bound from the above and below with $\tau$ and $\tau+\tau_A$. In the limit, $T \rightarrow \infty$, $\tau_{eff}=\tau$, in other extreme $T \rightarrow 0$, $\tau_{eff}=\tau+\tau_A$. Similarly, as $C \rightarrow \infty$, the correlation decay also become slower with $\tau_{eff}=\tau+\tau_A$ and in the absence of noise when $C \rightarrow 0$, $\tau_{eff}=\tau$. Other than $\tau_{eff}$ another parameter is $T_{0}$ that defines the effective correlation function $\phi_{eff}(t)$. $T_0$ is related to the ambient temperature as follows, $k_BT_0=k_B\left(T+\frac{Ck\tau}{\xi^2\left(\frac{1}{\tau}+\frac{1}{\tau_A}\right)}\right)$.\ Thus, $k_BT_0$ defines a renormalized thermal energy, but only in the limit $t \rightarrow \infty$. This directly follows from the mean square displacement (MSD) of the particle. $$\left\langle\left(x(t)-x(0)\right)^2\right\rangle=2\left(\phi(0)-\phi(t)\right) \\ \\ = \frac{2k_BT}{k}\left(1-e^{-\frac{t}{\tau}}\right)+\frac{2C}{\xi^2\frac{1}{\tau}\left(\frac{1}{\tau}+\frac{1}{\tau_A}\right)}\left(1-e^{-\frac{t}{\tau}}\right)-\frac{2C\left(e^{-\frac{t}{\tau_A}}-e^{-\frac{t}{\tau}}\right)}{\xi^2\left(\frac{1}{\tau^2}-\frac{1}{\tau_A^2}\right)} \label{eq:msdpart}$$ This MSD grows with time $t$ and saturates as expected due to confinement. However, the initial growth of MSD is faster in presence of active noise ($C\neq0, \tau_A\neq0$). A detailed derivation of MSD is presented in the appendix. This model can be further extended to describe a many-body system such as a flexible or a semi-flexible polymer as long as the chain is assumed to have a Gaussian distribution. Rouse chain ----------- Rouse model is the simplest yet widely used model to describe a polymer with Gaussian statistics devoid of any hydrodynamics and excluded volume interaction. The equation of motion of $n^{th}$ monomer is given by [@doibook; @kawakatsubook] $$\xi\frac{\partial{R_n}(t)}{\partial{t}}=k\frac{\partial^2{R_{n}(t)}}{\partial{n^2}}+{f}(n,t) \label{eq:rouse-model}$$ Where, ${R_n}(t)$ is the position of the $n^{th}$ monomer at time $t$ and $n$ can vary from $0$ to $N$ for a polymer with $(N+1)$ monomers. The friction coefficient is denoted by $\xi$, which is proportional to solvent viscosity and $k$ is the spring constant which is related to the Kuhn length, $b$ as $k=\frac{3k_BT}{b^2}$ with ${f}(n,t)$ being the random force acting on $n^{th}$ monomer at time $t$ which denotes the collision between the monomer with its surrounding solvent molecules. $$\left<f(n,t)\right>=0, \left<f_{\alpha}(n,t^{\prime})f_{\beta}(m,t^{\prime\prime})\right>=2 \xi k_B T \delta_{\alpha\beta}\delta(n-m)\delta(t^{\prime}-t^{\prime\prime}) \label{eq:random-forcerouse}$$ As a simple extension of the above model one can consider a Rouse chain in presence of active noise $f_A(t)$. $$\xi\frac{\partial{R_n}(t)}{\partial{t}}=k\frac{\partial^2{R_{n}(t)}}{\partial{n^2}}+{f}(n,t)+f_A(n,t) \label{eq:rouse-model-act}$$ It is a very standard procedure to decouple the equation of motion of the monomers using normal modes having independent motions as follows, $R_{n}(t)={X_0} + 2 \sum\limits_{p=1}^\infty X_p(t)cos(\frac{p \pi n}{N})$ and as long as the noises $f(n,t)$ and $f_A(n,t)$ are uncorrelated Eq. (\[eq:rouse-model-act\]) converts to $$\xi_p \frac{d{X_p}(t)}{d{t}}=-k_p X_{p}(t)+f_p(t)+f_{A,p}(t) \label{eq:rouse-mode}$$ Where, $k_p=\frac{6k_BTp^2\pi^2}{Nb^2}$ and $\xi_p=2N\xi$. The relaxation time for the $p^{th}$ normal mode in absence of any active noise is $\tau_p=\frac{\xi_p}{k_p}=\frac{\tau_1}{p^2}$, where $\tau_1 =\frac{\xi N^2 b^2}{3 k_BT\pi^2}$ is known as Rouse time. $f_{p}(t)$ and $f_{A,p}(t)$ are random and active forces respectively which follow $$\left<f_{p\alpha}(t)\right>=0,\left<f_{p\alpha}(t^{\prime})f_{q\beta}(t^{\prime\prime})\right>=2 \xi_p k_B T \delta_{\alpha\beta}\delta_{pq}\delta(t^{\prime}-t^{\prime\prime}) \label{eq:randomforce_modes}$$ $$\left<f_{A,p\alpha}(t)\right>=0,\left\langle f_{A,p\alpha}(t^{\prime})f_{A,q\beta}(t^{\prime\prime}) \right\rangle)=2NC\delta_{\alpha\beta}\delta_{pq}e^{-\frac{|t^{\prime}-t^{\prime\prime}|}{\tau_A}} \label{eq:activeforce_modes}$$ The above equation (Eq. (\[eq:rouse-mode\])) is structurally the same as that of Eq. (\[eq:langevinact\]), the only difference being it is for the $p^{th}$ mode of a chain. It is obvious that each mode of the chain behaves as an over-damped Brownian particle in the presence of the active noise trapped in a harmonic well. Zimm chain ---------- When pre-averaged hydrodynamic interactions are considered under $\theta$ condition, the normal modes of the polymer behave very similarly as that of a Rouse chain and have the same structure [@doibook] $$\xi_p^Z \frac{d{X_p}(t)}{d{t}}=-k_p^Z X_{p}(t)+f_p(t)+f_{A,p}(t) \label{eq:zimm-mode}$$ with $\xi_{p}^Z=\xi\sqrt{\frac{\pi N p}{3}}$ where $k_p^Z=k_p$ and $\tau_p^Z=\frac{\xi_p^Z}{k_p^Z}$ and $\tau_1^Z=\frac{\xi N^{3/2}b^2}{6\sqrt{3}\pi^{3/2}k_BT}$ [@chakrabartiphysica1]. Wormlike chain -------------- The semi-flexible polymer is modeled as Kratky-Porod wormlike chain which is unstretchable and includes the effect of bending energy [@doibook; @Liverpool2003]. The equation of motion for a semi-flexible chain without incorporating the effects from hydrodynamic interactions is given by $$\xi\frac{\partial{R_n}(t)}{\partial{t}}=k\frac{\partial^2{R_{n}(t)}}{\partial{n^2}}-\kappa\frac{\partial^4{R_{n}(t)}}{\partial{n^4}}+{f}(n,t)+f_A(n,t) \label{eq:rousesemi-model}$$ In normal mode description semi-flexible chain is similar to a flexible chain except $k_p$ which has a fourth order dependence on the mode number $p$ unlike flexible chain. $$\xi_p^S \frac{d{X_p}(t)}{d{t}}=-k_p^S X_{p}(t)+f_p(t)+f_{A,p}(t) \label{eq:semi-mode}$$ Where, $k_p^S=\frac{6k_BTp^2\pi^2}{Nb^2}+\frac{2\kappa p^4\pi^4}{N^3b^3}$ and $\kappa$, bending rigidity is related to the persistence length $l_p$ of the polymer as follows $\kappa =k_BTl_p$. However, $\xi_{p}^S=\xi_{p}=2N\xi$ and $\tau_p^S=\frac{\xi_p^S}{k_p^S }$. Calculation methods =================== The time-correlation function for the normal modes has a very general structure for the flexible as well as the semi-flexible chain and it remains the same even when the hydrodynamic interactions are incorporated. The form of the expression is very similar to that of a single over-damped Brownian particle moving in a harmonic well in the presence of active noise $$\left\langle X_{p\alpha}(t)X_{q\beta}(0)\right\rangle=\frac{k_BT}{k_p}\delta_{\alpha\beta}\delta_{pq}e^{-\frac{t}{\tau_p}}+\frac{2 N C}{\xi_p^2}\delta_{\alpha\beta}\delta_{pq}e^{-\frac{t}{\tau_p}}\left(\frac{1}{\frac{1}{\tau_p}\left(\frac{1}{\tau_p}+\frac{1}{\tau_A}\right)}+\frac{e^{\frac{t}{\tau_p}-\frac{t}{\tau_A}}-1}{\left(\frac{1}{\tau_p^2}-\frac{1}{\tau_A^2} \right)} \right) \label{eq:corrgen}$$ To find the exact expression for the flexible, semi-flexible or the Zimm chain one just need to select the exact forms of $k_p$, $\xi_p$ and $\tau_p$ for a Rouse chain, $k_p^Z$, $\xi_p^Z$, $\tau_p^Z$ for a Zimm chain and $k_p^S$, $\xi_p^S$, $\tau_p^S$ for a semi-flexible chain. The time correlation function for the vector ($R_{N0}$) connecting the $N^{th}$ and the $0^{th}$ monomer can easily be calculated from the above expression, which is the summation over the correlation functions of all normal modes describing the polymer. $${\phi_{N0}}(t)=\left<{R}_{N0}(t).{R}_{N0}(0)\right> =16 \sum\limits_{p=odd}^\infty 3\left\langle X_{p}(t)X_{p}(0)\right\rangle \label{eq:normcorr}$$ The reconfiguration time ($\tau_{N0}$) corresponding to the fluctuation of the distance between the end-to-end monomers is theoretically calculated by taking a time integration of the corresponding normalized correlation function ($\Phi_{N0}(t)$) [@chakrabarti2014; @chakrabarti2015] $${\tau}_{N0}=\int\limits_{0}^\infty dt {\Phi}_{N0}(t) \label{eq:recon}$$ Where, ${\Phi}_{N0}(t)=\frac{\phi(t)}{\phi(0)}$ Similarly, the expression of the MSD of the vector ($R_{N0}$) can also be derived from the MSD of the normal modes [@toan; @chakrabarti2014]. This is again similar to that of a single particle. In a recent study, Ghosh *et al.* [@ghosh2014] have demonstrated how MSD of a semi-flexible chain grows in presence of such active noise. Higher the strength of the active noise, faster the growth. $$\begin{aligned} \left\langle\left(R_{N0}(t)-R_{N0}(0)\right)^2\right\rangle= 2\left(\phi_{N0}(0)-\phi_{N0}(t)\right) \\ \\ =16\sum_{p=odd}^{\infty} 3 \Bigg( \frac{2k_BT}{k_p}\left(1-e^{-\frac{t}{\tau_p}}\right)+& \frac{2C}{\xi_p^2\frac{1}{\tau_p}\left(\frac{1}{\tau_p}+\frac{1}{\tau_A}\right)}\left(1-e^{-\frac{t}{\tau_p}}\right)\\ &-\frac{2C\left(e^{-\frac{t}{\tau_A}}-e^{-\frac{t}{\tau_p}}\right)}{\xi_p^2\left(\frac{1}{\tau_p^2}-\frac{1}{\tau_A^2}\right)} \Bigg) \end{aligned} \label{eq:MSD}$$ Results and discussions ======================= In Fig. (\[fig:a\]) we show the normalized correlation function for Rouse chain in presence and absence of active noise which is calculated using the generalized expression given in Eq. (\[eq:corrgen\]). The parameters are chosen in consistence with the real values such as $N = 100$, $b=3.8\times10^{-10} m$, $k_B = 1.38\times10^{-23} JK^{-1}$, $T = 300 K$ and $\xi = 9.42\times10^{-12} kgs^{-1}$ which is in agreement with the viscosity of water. As mentioned earlier $C$ is the strength of the active noise and $C=\frac{P(f_A) f_A^2}{b}$. It has been experimentally observed that in biological systems motor proteins like myosin, kinesin exert force in the $\sim5-10 pN$ range [@Cell_Biology_book]. For our calculations we have considered $f_a = 10\times10^{-12} N$ and $P(f_A)=1$. For a fixed value of $C$ we have chosen two different values of $\tau_A$, such as $0.2\tau_1$ and $5.0\tau_1$ which are in the same order of magnitude of $\tau_1$. From the plot it can be seen that correlation decay is always slower in presence of an active noise even when the characteristic decay time of the active noise $\tau_A$ is very small and as $\tau_A$ increases the decay of $\Phi_{N0}$ becomes even slower. However, this correlation loss has very weak dependence on the strength of the active noise $C$. Changing the strength practically brings no difference in the correlation function. The log-log plot of reconfiguration time against chain length ($N$) is shown in Fig (\[fig:b\]) where the reconfiguration time is calculated using Eq. (\[eq:recon\]) and as expected, reconfiguration time increases as the temporal correlation loss of the active noise becomes slower. For, a $100$ monomer chain $\tau_{N0}$ increases $\sim 1.4$ times in the presence of active noise when $\tau_A=0.2\tau_1$, whereas it becomes $\sim 7$ times higher when the decay time of active noise $\tau_A=5\tau_1$. But, surprisingly the chain length dependence remains unchanged even in the presence of active noise. In all three cases $\nu=2$ where, $\tau_{N0}\sim N^{\nu}$. It is well known that the reconfiguration time is a summation of the relaxation times of each mode i.e. $\tau_{N0}=16 \sum\limits_{p=odd}^\infty 3\left\langle X_{p}(t)X_{p}(0)\right\rangle$, which has the analytically exact expression $$\tau_{N0}=16 \sum\limits_{p=odd}^{\infty}3{\tau}_p \left[\frac{\left(k_BT\xi_p^2+C{\tau}_p\tau_Ak_p\right)\left({\tau}_p+\tau_A\right)}{C{\tau}_p^2\tau_Ak_p+k_BT\xi_p^2\left({\tau}_p+\tau_A\right)}\right] \label{eq:recon_gen}$$ Where, the $N$ dependence comes through ${\tau}_p$, $\xi_p$ and $k_p$. A careful analysis of the preceding expression shows that if the active noise strength $C$ is very small i.e. $C \rightarrow 0$, the above expression reduces to $\tau_{N0}\simeq 16 \sum\limits_{p=odd}^{\infty}3{\tau}_p$ and since, $\tau_p \sim N^2$, the dependence of the reconfiguration time is also identical. Now what happens if $C$ becomes very large i.e. $C \rightarrow \infty$, $\tau_{N0}\simeq 16 \sum\limits_{p=odd}^{\infty}3{\tau}_p+\tilde{\tau}$, where $\tilde{\tau}$ is a constant. Even in this case the $N$ dependence comes only from $\tau_p$ and $\tau_{N0}\sim N^2$. In between these two extreme cases the active noise cause very small change in the $N$ dependence of the reconfiguration time which is reflected Fig (\[fig:b\]). ![Plot of $\Phi_{N0}(t)$ vs $t$ for Rouse chain[]{data-label="fig:a"}](2.jpg){width="80.00000%"} ![Log-log plot of reconfiguration time ($\tau_{N0}$) vs Chain length ($N$) for Rouse chain []{data-label="fig:b"}](3.jpg){width="80.00000%"} The same set of calculations have been performed for a flexible polymer including the pre-averaged hydrodynamic interaction under $\theta$ condition. The plot of normalized time-correlation function against time is shown in Fig. (\[fig:c\]) which shows a similar trend as that of Rouse chain, i.e. the correlation loss becomes slower whenever active noise is introduced to the system. Next, the chain length dependence of the reconfiguration time is determined for the Zimm chain from Fig. (\[fig:d\]), and it is found to be $\sim N^{1.5}$ which is in agreement to the previous work done by Chakrabarti [@chakrabartiphysica1]. In this case also the chain-length dependence of reconfiguration time does not differ in the presence of active noise. ![Plot of $\Phi_{N0}(t)$ vs $t$ for Zimm chain[]{data-label="fig:c"}](zimm1.jpg){width="80.00000%"} ![Log-log plot of reconfiguration time ($\tau_{N0}$) vs Chain length ($N$) for Zimm chain []{data-label="fig:d"}](zimm2.jpg){width="80.00000%"} Fig. (\[fig:e\]) shows the normalized time correlation function of the end-to-end vector for a semi-flexible chain. Here also the behavior of the correlation loss in presence of active noise is identical to Rouse and Zimm chain. The Kuhn length for semi-flexible has been considered to be $b = 50\times10^{-9} m$. This is roughly the Kuhn length of DNA which has a series of different Kuhn length depending upon the solvent condition [@Manning]. The persistence length of the semi-flexible chain has been considered to be half of the Kuhn length during the calculations. When chain length dependence of semi-flexible chain is determined it is found to be $\sim N^{2}$. This might seem surprising since $\tau_p^S$ has a fourth order dependence on $N$ in addition to the usual second order dependence, for which the dependence on chain length should be higher than that of Rouse model. However, if we take a look at the expression for $\tau_p^S$, since the value of $\kappa$ considered in the calculations is very small the contribution from $N^4$ is negligibly small and that is the reason even for semi-flexible chain the $\tau_{N0}\sim N^2$. ![Plot of $\Phi_{N0}(t)$ vs $t$ for semi-flexible chain[]{data-label="fig:e"}](semi1.jpg){width="80.00000%"} ![Log-log plot of reconfiguration time ($\tau_{N0}$) vs Chain length ($N$) for semi-flexible chain []{data-label="fig:f"}](semi2.jpg){width="80.00000%"} Conclusion ========== In this work we have looked into the effect of active noise in chain reconfiguration for a flexible polymer where the active noise is modelled with a long temporally correlated non-equilibrium force. It can be clearly seen that in the presence of active noise the chain reconfiguration becomes slower. However slowing down of the reconfiguration dynamics seems to be controlled by the correlation time $\tau_A$ rather than the strength of the correlation. Thus in a typical FRET like experiment, measurement of reconfiguration time of a protein should show slowing down in an environment with active noise. Similar behaviors are also observed when pre-averaged hydrodynamic interaction is considered. For a worm-like semi-flexible chain the trend remains the same. However, the dependence on chain-length of the reconfiguration time does not change in the presence of active noise. Keeping long story short, our study suggests that not always presence of active noise can guarantee faster reconfiguration of a polymer chain. In an environment, where a long temporal noise acts on the chain, FRET type measurement would show the chain to retain the correlation for longer time than in absence of such noise. Appendix ======== Correlation function -------------------- The equation of motion for a single over-damped Brownian particle trapped in harmonic potential in presence of active noise $\frac{d{{x}(t)}}{d{t}}+\frac{k}{\xi}{x}(t)=\frac{1}{\xi}\left({f}(t)+f_A(t)\right)$\ Multiplying the integrating factor $e^{\frac{k}{\xi}t}$ on both sides we get, $\left(\frac{d{{x}(t)}}{d{t}}+\frac{k}{\xi}{x}(t)\right)e^{\frac{k}{\xi}t}=\frac{1}{\xi}\left({f}(t)e^{\frac{k}{\xi}t}+f_A(t)e^{\frac{k}{\xi}t}\right)$\ Integrating boths side from $-\infty$ to $t$ (which means we assume the system to start evolving at infinite past). $\int\limits_{-\infty}^t\frac{d}{dt}\left(x(t^{\prime})e^{\frac{k}{\xi}t^{\prime}}\right)=\frac{1}{\xi}\int\limits_{-\infty}^t\left({f}(t^{\prime})e^{\frac{k}{\xi}t^{\prime}}+f_A(t^{\prime})e^{\frac{k}{\xi}t^{\prime}}\right)$ or, $x(t)=\frac{e^{-\frac{t}{\tau}}}{\xi}\int\limits_{-\infty}^t dt^{\prime}\left({f}(t^{\prime})e^{\frac{t^{\prime}}{\tau}}+f_A(t^{\prime})e^{\frac{t^{\prime}}{\tau}}\right)$ as, $\frac{\xi}{k}=\tau$ Since the thermal and the active noise are uncorrelated, they come separately as a summation in the correlation function and the position correlation function for the thermal noise has standard solution which is not shown here, $$\begin{split} \left\langle x(t)x(0) \right\rangle & = \frac{e^{-\frac{t}{\tau}}}{\xi^2}\int\limits_{-\infty}^t dt^{\prime}\int\limits_{-\infty}^0dt^{\prime\prime}\left(\left\langle f(t^{\prime})f(t^{\prime\prime}) \right\rangle + \left\langle f_A(t^{\prime})f_A(t^{\prime\prime}) \right\rangle\right) e^{\left(\frac{t^{\prime}+t^{\prime\prime}}{\tau}\right)}\\ & = \frac{k_BT}{k}e^{-\frac{t}{\tau}}+\frac{C}{\xi^2}e^{-\frac{t}{\tau}}\int\limits_{-\infty}^t dt^{\prime}\int\limits_{-\infty}^0dt^{\prime\prime}e^{\frac{t^{\prime}+t^{\prime\prime}}{\tau}} e^{-\frac{|t^{\prime}-t^{\prime\prime}|}{\tau_A}} \end{split}$$ The position correlation function for The time correlation function of the active noise involves a modulus of time, therefore the integration is split in two parts. One where $t^{\prime}>t^{\prime\prime}$ and another considering $t^{\prime}<t^{\prime\prime}$. ![The shaded region represents the range of time that has to be integrated for active noise to calculate the time correlation function.[]{data-label="fig:g"}](act_1.JPG){width="80.00000%"} $$\begin{aligned} \left\langle x(t)x(0) \right\rangle=&\frac{k_BT}{k}e^{-\frac{t}{\tau}}+\frac{C}{\xi^2}e^{-\frac{t}{\tau}} \Bigg( \int\limits_{t^{\prime\prime}=-\infty}^0 dt^{\prime\prime}\int\limits_{t^{\prime}=t^{\prime\prime}}^{0}dt^{\prime}e^{\left( \frac{t^{\prime}}{\tau}-\frac{t^{\prime}}{\tau_A} \right)} e^{\left( \frac{t^{\prime\prime}}{\tau}+\frac{t^{\prime\prime}}{\tau_A} \right)}+ \\ &\int\limits_{t^{\prime}=-\infty}^0 dt^{\prime}\int\limits_{t^{\prime\prime}=t^{\prime}}^{0}dt^{\prime\prime}e^{\left( \frac{t^{\prime\prime}}{\tau}-\frac{t^{\prime\prime}}{\tau_A} \right)} e^{\left( \frac{t^{\prime}}{\tau}+\frac{t^{\prime}}{\tau_A} \right)}+\int\limits_{t^{\prime\prime}=-\infty}^0 dt^{\prime\prime}\int\limits_{t^{\prime}=0}^{t}dt^{\prime}e^{\left( \frac{t^{\prime}}{\tau}-\frac{t^{\prime}}{\tau_A} \right)} e^{\left( \frac{t^{\prime\prime}}{\tau}+\frac{t^{\prime\prime}}{\tau_A} \right)}\Bigg) \\ &=\frac{k_BT}{k}e^{-\frac{t}{\tau}}+\frac{C}{\xi^2}e^{-\frac{t}{\tau}}\Bigg(2\int\limits_{t^{\prime\prime}=-\infty}^0 dt^{\prime\prime}\int\limits_{t^{\prime}=t^{\prime\prime}}^{0}dt^{\prime}e^{\left( \frac{t^{\prime}}{\tau}-\frac{t^{\prime}}{\tau_A} \right)} e^{\left( \frac{t^{\prime\prime}}{\tau}+\frac{t^{\prime\prime}}{\tau_A} \right)}+\\ & \int\limits_{t^{\prime\prime}=-\infty}^0 dt^{\prime\prime}\int\limits_{t^{\prime}=0}^{t}dt^{\prime}e^{\left( \frac{t^{\prime}}{\tau}-\frac{t^{\prime}}{\tau_A} \right)} e^{\left( \frac{t^{\prime\prime}}{\tau}+\frac{t^{\prime\prime}}{\tau_A} \right)}\Bigg)\\ &=\frac{k_BT}{k}e^{-\frac{t}{\tau}}+\frac{C}{\xi^2}e^{-\frac{t}{\tau}}\left(\frac{2}{\left(\frac{1}{\tau}-\frac{1}{\tau_A} \right)}\int\limits_{t^{\prime\prime}=-\infty}^0dt^{\prime\prime}\left(e^{\frac{2t^{\prime\prime}}{\tau}}-e^{\frac{t^{\prime\prime}}{\tau}+\frac{t^{\prime\prime}}{\tau_A}}\right)+\frac{1}{\left(\frac{1}{\tau}+\frac{1}{\tau_A} \right)}\frac{e^{\frac{t}{\tau}-\frac{t}{\tau_A}}-1}{\left(\frac{1}{\tau}-\frac{1}{\tau_A} \right)}\right) \end{aligned}$$ $$\left\langle x(t)x(0) \right\rangle=\frac{k_BT}{k}e^{-\frac{t}{\tau}}+\frac{C}{\xi^2}e^{-\frac{t}{\tau}}\left(\frac{1}{\frac{1}{\tau}\left(\frac{1}{\tau}+\frac{1}{\tau_A}\right)}+\frac{e^{\frac{t}{\tau}-\frac{t}{\tau_A}}-1}{\left(\frac{1}{\tau^2}-\frac{1}{\tau_A^2} \right)} \right) \label{eq:corrpart1}$$ MSD --- The mean-square displacement(MSD) of single particle in one-dimension is as follows, $\left\langle\left(x(t)-x(0)\right)^2\right\rangle$ $x(t)=\frac{e^{-\frac{t}{\tau}}}{\xi}\int\limits_{-\infty}^t dt^{\prime}\left({f}(t^{\prime})e^{\frac{t^{\prime}}{\tau}}+f_A(t^{\prime})e^{\frac{t^{\prime}}{\tau}}\right)$ $x(t)-x(0)=\frac{e^{-\frac{t}{\tau}}}{\xi}\int\limits_{-\infty}^t dt^{\prime}\left({f}(t^{\prime})e^{\frac{t^{\prime}}{\tau}}+f_A(t^{\prime})e^{\frac{t^{\prime}}{\tau}}\right)-\frac{1}{\xi}\int\limits_{-\infty}^0 dt^{\prime}\left({f}(t^{\prime})e^{\frac{t^{\prime}}{\tau}}+f_A(t^{\prime})e^{\frac{t^{\prime}}{\tau}}\right)$ $$\begin{aligned} \left\langle\left(x(t)-x(0)\right)^2\right\rangle=&\frac{e^{-\frac{2t}{\tau}}}{\xi^2}\int\limits_{-\infty}^t dt^{\prime}\int\limits_{-\infty}^t dt^{\prime\prime}\left(\left\langle f(t^{\prime})f(t^{\prime\prime}) \right\rangle + \left\langle f_A(t^{\prime})f_A(t^{\prime\prime}) \right\rangle\right) e^{\left(\frac{t^{\prime}+t^{\prime\prime}}{\tau}\right)}+\\ &\frac{1}{\xi^2}\int\limits_{-\infty}^0 dt^{\prime}\int\limits_{-\infty}^0 dt^{\prime\prime}\left(\left\langle f(t^{\prime})f(t^{\prime\prime}) \right\rangle + \left\langle f_A(t^{\prime})f_A(t^{\prime\prime}) \right\rangle\right) e^{\left(\frac{t^{\prime}+t^{\prime\prime}}{\tau}\right)}-\\ &2\frac{e^{-\frac{t}{\tau}}}{\xi^2}\int\limits_{-\infty}^t dt^{\prime}\int\limits_{-\infty}^0 dt^{\prime\prime}\left(\left\langle f(t^{\prime})f(t^{\prime\prime}) \right\rangle + \left\langle f_A(t^{\prime})f_A(t^{\prime\prime}) \right\rangle\right) e^{\left(\frac{t^{\prime}+t^{\prime\prime}}{\tau}\right)} \end{aligned}$$ Again the MSD for an over-damped Brownian particle in presence of harmonic potential is well-known which is not shown here in detail, $$\begin{aligned} \frac{e^{-\frac{2t}{\tau}}}{\xi^2} \int\limits_{-\infty}^t dt^{\prime}\int\limits_{-\infty}^t dt^{\prime\prime}\big(\left\langle f(t^{\prime})f(t^{\prime\prime}) \right\rangle & + \left\langle f_A(t^{\prime})f_A(t^{\prime\prime}) \right\rangle \big) e^{\left(\frac{t^{\prime}+t^{\prime\prime}}{\tau}\right)}=\\ &\frac{k_BT}{k}+\frac{C}{\xi^2}\Bigg(e^{-\frac{2t}{\tau}}\left(\frac{1}{\frac{1}{\tau}\left(\frac{1}{\tau}+\frac{1}{\tau_A}\right)}+2\frac{e^{\frac{t}{\tau}-\frac{t}{\tau_A}}-1}{\left(\frac{1}{\tau^2}-\frac{1}{\tau_A^2}\right)} \right)+\\ &\frac{1}{\frac{1}{\tau}\left(\frac{1}{\tau}+\frac{1}{\tau_A} \right)}\left(1-e^{-\frac{2t}{\tau}}\right)-\frac{2e^{-\frac{t}{\tau}}}{\left(\frac{1}{\tau^2}-\frac{1}{\tau_A^2} \right)}\left(e^{-\frac{t}{\tau_A}}-e^{-\frac{t}{\tau}}\right)\Bigg) \end{aligned}$$ $$\begin{aligned} \frac{1}{\xi^2}\int\limits_{-\infty}^0 dt^{\prime}\int\limits_{-\infty}^0 dt^{\prime\prime}\left(\left\langle f(t^{\prime})f(t^{\prime\prime}) \right\rangle + \left\langle f_A(t^{\prime})f_A(t^{\prime\prime}) \right\rangle\right) e^{\left(\frac{t^{\prime}+t^{\prime\prime}}{\tau}\right)}=\\ &\frac{k_BT}{k}+\frac{C}{\xi^2}\left(\frac{1}{\frac{1}{\tau}\left(\frac{1}{\tau}+\frac{1}{\tau_A}\right)}\right) \end{aligned}$$ $$\begin{aligned} 2\frac{e^{-\frac{t}{\tau}}}{\xi^2} \int\limits_{-\infty}^t dt^{\prime}\int\limits_{-\infty}^0 dt^{\prime\prime}\big(\left\langle f(t^{\prime})f(t^{\prime\prime}) \right\rangle & + \left\langle f_A(t^{\prime})f_A(t^{\prime\prime}) \right\rangle\big) e^{\left(\frac{t^{\prime}+t^{\prime\prime}}{\tau}\right)}=\\ &\frac{2k_BT}{k}e^{-\frac{t}{\tau}}+\frac{2C}{\xi^2}e^{-\frac{t}{\tau}}\left(\frac{1}{\frac{1}{\tau}\left(\frac{1}{\tau}+\frac{1}{\tau_A}\right)}+\frac{e^{\frac{t}{\tau}-\frac{t}{\tau_A}}-1}{\left(\frac{1}{\tau^2}-\frac{1}{\tau_A^2} \right)} \right) \end{aligned}$$ $$\left\langle\left(x(t)-x(0)\right)^2\right\rangle=\frac{2k_BT}{k}\left(1-e^{-\frac{t}{\tau}}\right)+\frac{2C}{\xi^2\frac{1}{\tau}\left(\frac{1}{\tau}+\frac{1}{\tau_A}\right)}\left(1-e^{-\frac{t}{\tau}}\right)-\frac{2C\left(e^{-\frac{t}{\tau_A}}-e^{-\frac{t}{\tau}}\right)}{\xi^2\left(\frac{1}{\tau^2}-\frac{1}{\tau_A^2}\right)} \label{eq:msdpart1}$$ [23]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url 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--- abstract: 'By employing a local two-fluid theory, we investigate an obliquely propagating electromagnetic instability in the lower hybrid frequency range driven by cross-field current or relative drifts between electrons and ions. The theory self-consistently takes into account local cross-field current and accompanying pressure gradients. It is found that the instability is caused by reactive coupling between the backward propagating whistler (fast) waves in the moving electron frame and the forward propagating sound (slow) waves in the ion frame when the relative drifts are large. The unstable waves we consider propagate obliquely to the unperturbed magnetic field and have mixed polarization with significant electromagnetic components. A physical picture of the instability emerges in the limit of large wavenumber characteristic of the local approximation. The primary positive feedback mechanism is based on reinforcement of initial electron density perturbations by compression of electron fluid via induced Lorentz force. The resultant waves are qualitatively consistent with the measured electromagnetic fluctuations in reconnecting current sheet in a laboratory plasma.' author: - 'Hantao Ji, Russell Kulsrud, William Fox, Masaaki Yamada' title: An Obliquely Propagating Electromagnetic Drift Instability in the Lower Hybrid Frequency Range --- Introduction ============ Current-driven instabilities with frequencies higher than ion cyclotron frequency ($\omega > \Omega_i$) or wavelengths shorter than ion skin depth ($k\lambda_i > 1; \lambda_i \equiv c/\omega_{pi}$) have been a popular subject for space and laboratory plasma research [see [*e.g.*]{} @gary93]. Recently, this topic has been revisited in the context of magnetic reconnection [see [*e.g.*]{} @biskampbook], where intense current density exists locally in the diffusion region. In particular, the Lower Hybrid Drift Instability [@krall71] (LHDI) driven by a density gradient has received considerable attention as a potential source of anomalous resistivity. When the LHDI propagates nearly perpendicular to the magnetic field it is purely electrotatic. Such waves have been observed at the low-$\beta$ edge of the current sheet in the laboratory [@carter02a], in numerical simulations [see [*e.g.*]{} @scholer03], and in space [@shinohara98; @bale02]. They are driven unstable by inverse Landau damping of the drifting electrons. However, these electrostatic modes are largely stabilized [@davidson77] inside the high-$\beta$ reconnection layer, where the magnetic field gradient is large and the $\nabla B $ drift of the electrons is in the wrong direction to amplify the waves. Further, it is observed that their amplitudes do not correlate with the fast reconnection in the Magnetic Reconnection Experiment or MRX [@carter02b]. By contrast, magnetic fluctuations up to the lower hybrid frequency range have been more recently detected [@ji04a] in this high-$\beta$ center of the current sheet in the MRX. These propagate obliquely to the magnetic field, and their amplitudes exhibit positive correlations with fast reconnection. A theoretical explanation for the origin of these magnetic fluctuations, other than the electrostatic perpendicularly propagating LHDI waves, is therefore in order. Earlier, motivated by observations of high frequency magnetic fluctuations in a magnetic shock experiment, Ross attempted [@ross70] the first theoretical exploration of such candidate obliquely propagating electromagnetic high-frequency waves driven by a relative drift between electrons and ions associated with local currents. Based on a two-fluid formalism in the electron frame, Ross showed that unstable waves propagating obliquely to the magnetic field are excited by reactive coupling between ion beam and whistler waves. Such an instability is generally known as the Modified Two Stream Instability [@mcbride72; @seiler76] (MTSI) since it is driven by a local current across a magnetic field unrelated to a diagmagnetic drift. Extensions to a full kinetic treatment of both ions and electrons were made for this instability [@lemons77; @wu83; @tsai84]. Unlike the perpendicular LHDI, the obliquely propagating MTSI persists in high-$\beta$ plasmas, where the critical values of relative drift for the instability are typically a few times the local Alfvén velocity, and possesses significant electromagnetic components. However, in most of these works, a finite pressure gradient self-consistent with the cross-field current was left out in the wave dynamics. This neglect throws doubt on the applicability of the MTSI to the MRX, where all the current is due to inhomogeneities. Recently, global eigenmode analyses [@daughton99; @yoon02; @daughton03] of the current driven instabilities have been carried out to take into account the effects of boundary conditions of a Harris current sheet [@harris62]. This followed earlier work on the same subject [@huba80]. It was found that for short wavelengths ($k\lambda_e \sim 1; \lambda_e \equiv c/\omega_{pe}$), the unstable modes concentrate at the low-$\beta$ edge, and they are predominantly electrostatic similar to the perpendicular propagating LHDI. In contrast, for relatively longer wavelengths ($k\sqrt{\lambda_e \lambda_i} \sim 1$), unstable modes with significant electromagnetic components develop in the center region. These are similar to the MTSI at high-$\beta$. For even longer wavelengths ($k\lambda_i \sim 1$), a drift kink instability [@daughton99] is known to exist but this has a slower growth rate at more realistic ion-electron mass ratios. More recently, these analyses have been further extended to non-Harris current sheets [@yoon04; @sitnov04]. When relative drift between electrons and ions is enhanced, the central region is clearly dominated by instabilities resembling the MTSI. The first numerical simulations of the MTSI have been carried out in a two-dimensional local model [@winske85], but focused on the electron heating. Particle simulations have also been carried out in three-dimensions to study stability of a Harris current sheet under various but limited conditions [@horiuchi99; @lapenta02; @daughton03; @scholer03; @shinohara04; @ricci04]. It was found that at first the LHDI like instabilities are active only at the low-$\beta$ edge, and modify the current profile which then leads to the long wavelength electromagnetic modes, such as drift kink instabilities or Kelvin-Helmholtz instabilities [@lapenta02]. Recent simulations using more realistic parameters (larger mass ratios with more particles) in larger dimensions indicate [@ricci04] that the MTSI-like modes also develop in the central region. While the characteristics of the observed waves in the MRX current sheet are generally consistent with these linear stability analyses and nonlinear simulation results, there has been yet no convincing physical explanation of the observed electromagnetic waves in the lower hybrid frequency range. Comparisons between MTSI and LHDI, the latter of which involves a self-consistent pressure gradient, were made based on local kinetic theories [@hsia79; @yoon94; @silveira02], but with a focus on nearly perpendicularly propagating waves. Extensions to larger propagation angles were also attempted earlier [@zhou83; @zhou91] but with few discussions on the underlying physics. Motivated by the observations in the MRX and these recent theoretical developments, we investigate this instability based on a local two-fluid formalism in this paper. Our analysis is of the MRX and includes the self-consistent pressure gradient with large propagation angles. A local treatment is justified if the wavelength is short ($k\lambda_i \gg 1$) and the growth rate is large ($\gamma \gg \Omega_i$), compared to the global eigenmode analyses extending throughout the current layer (see for example [@kulsrud67]). Our focus here is to reveal the underlying physics of the instability by using the simplest possible model rather than to carry out more involved calculations. We find that when the relative drifts are large, the instability is caused by a reactive coupling between the backward propagating whistler (fast) waves in the moving electron frame and the forward propagating sound (slow) waves in the ion frame. The unstable waves have a mixed electromagnetic character with both electrostatic and magnetic components. They propagate obliquely to the unperturbed magnetic field. The primary positive feedback mechanism for the instability is identified as reinforcement of initial electron density perturbations by an induced Lorentz force. The role this instability plays in magnetic reconnection, such as anomalous resistivity and heating, will be discussed in a forthcoming paper [@kulsrud04] that is based on quasi-linear theory (see also [@winske85; @basu92; @yoon93].) Theoretical Model ================= The basic features of our model are described in this section. Since our main objective is to understand physics of the underlying instability, we develop a theoretical model, which contains the essential ingredients for the instability, yet remains simple enough so that the feedback mechanism can be understood. In contrast to the past work, most of which is based on full kinetic theory, we find that we are able to use a simple two-fluid theory and still obtain reliable results. We show that most features of the instability can be revealed by this simple model. Method of the calculation ------------------------- We wish to treat the LHDI mode by an approach somewhat different from earlier approaches. Our basic assumption is that the drift velocity is produced by equilibrium gradients (LHDI) rather than an ion beam (MTSI). In the MRX the gradients are the origin of the relative drift velocity of the ions and electrons which is just the diamagnetic currents, so that the instability is an LHDI. However, since the LHDI has been usually treated as a nearly perpendicular propagating mode and we restrict ourselves in this paper to propagation at angles finitely different from $ 90 $ degrees, we refer to our instability as the obliquely propagating LHDI or more briefly the oblique LHDI. The reason we do not consider the LHDI near 90 degrees is that it has been shown to be stable in the central regions of the MRX and, as discussed in the introduction, we are interested in explaining the observed instabilities there. In fact, as shown by [@carter02b], the gradient of the magnetic field is large there and the $\nabla B$ drifts cause the resonant electrons to drift in the opposite direction than inferred from their current. The oblique instability we investigate is a non resonant one. We assume that the mode is at a large frequency compared to the ion cyclotron frequency, and the wave length is small compared to the ion gyration radius, so that the ions may be considered to be unmagnetized. We also assume that the frequency is small compared to the electron cyclotron frequency, $ \Omega_e $, and that the wave length is large compared to the electron gyration radius, $ \rho_e $, so that the electron can be treated by the drift kinetic theory. This theory is described in Appendix A1, but the upshot of it is that one expands the Vlasov equation in the small parameter $ \rho_e/ \lambda $ where $ \lambda $ is the perpendicular scale of the perturbation as well as the equilibrium. One solves the Vlasov equation to lowest order to obtain the zero order electron distribution function $ f_0 $ from which one can obtain the electron pressure tensor, $ {\bf P}_e $. Then one calculates the perpendicular velocity moment of the first order distribution function $ f_1 $, to find the perpendicular electron current. But this calculation is equivalent to taking the perpendicular electron fluid equation of motion with this pressure tensor. The parallel current is then obtained from the continuity equation, $ \nabla \cdot {\bf j}^e - \partial (n_{e1}e) /\partial t = 0 $. This procedure is totally equivalent to previous calculations giving identical results in the small $ \rho_e $ limit. It might be argued that one should consider waves with $ k_{\perp} \rho_e \sim 1$ since in previous work on the perpendicular LHDI the maximum growth occurs when $ k_{\perp} \rho_e \sim 1 $. However, for the oblique LHDI the maximum growth actually occurs when $ k_{\perp} \rho_e \ll 1$ and the mode becomes stable for $ k_{\perp} \rho_e $ that approaches unity. (The guiding center treatment is appropriate for inhomogeneous systems, since it makes no assumption about [*near*]{} homogeneity and avoids the complicating approximations concerning it that are usually made.) It turns out that it is not appropriate to treat the pressure tensor as anisotropic for the MRX experiments, in which the magnetic fluctuations are observed. This is because the electron-ion collision rate is comparable to the frequencies and growth rates of the mode, so it is just as accurate to take the pressure as isotropic. Further, it is also appropriate to assume that the plasma is isothermal, so that $ p= nT $ in general; $ p_0 = n_0(x) T $ in the equilibrium and $ p_1 = n_1 T $ is the perturbation. The fact that $ T $ is constant in the equilibrium over the region occupied by the mode is supported directly from observations. The fact that perturbations in the temperature are zero follows from the very large thermal conductivity along the lines, so that the thermal relaxation time is shorter than the perturbation growth time. With this assumption we can avoid the solution for $ f_0 $ and work entirely from the electron fluid equation, to determine the perpendicular electron currents. At this point we are in a position to solve for the ion and electron currents in terms of the electric fields. However, one further physical result, charge neutrality, allows us to further shorten the calculation. Since the Debye length is very small compared to even the electron gyration radius, we may assume to an excellent approximation that the perturbed electron density $ n_{e1} $ is equal to the perturbed ion density $ n_{i1} $ and this enable us to easily evaluate the relevant terms in the perturbed equation of motions of the electrons. (Of course if we had avoided this step and solved directly for the ion and electron currents separately and then substituted in Maxwell’s equations, charge neutrality would have followed automatically. Introducing charge neutrality earlier leads to considerable simplicity in the calculation and more physical insight.) To summarize our calculation: we first write down the equilibrium conditions. Then next we calculate the perturbed ion current and density from the unmagnetized ion dynamics. We then calculate the perturbed perpendicular electron current from the perpendicular equation of motion for the electrons. We can then find the parallel electron current from $ \nabla \cdot {\bf j} = \nabla \cdot ({\bf j}^i +{\bf j}^e) = 0 $. Knowing these currents, we then substitute them into Maxwell’s equations to find three independent relations for the wave electric fields. However, it turns out that one of the three Maxwell’s equations can be simplified to the electron force balance along the field line. Thus, this eliminates the needs to calculate the parallel electron current directly from $ \nabla \cdot {\bf j} = 0$, which is demanded by the charge neutrality condition. Equilibrium ----------- For definiteness, we assume the MRX equilibrium is a Harris equilibrium and study it in the ion frame. This seems the most physical frame in which to study the instability since it turns out to be essentially an unstable sound mode which is carreid by the ions. We concentrate our attention on a small region say about half way out from the center of the Harris sheet. In this frame as shown in Fig.1(a), there is an electric field $ E_0 $ balancing their pressure force, $T_i \partial n_0/\partial y$, in the $y$ direction: $$en_0E_0=T_i {\partial n_0 \over \partial y}. \label{i_balance}$$ The magnetic field, $B_0$, is chosen in the $z$ direction. A current is carried by electrons drifting in the $x$ direction with a speed $V_0$. Force balance of the electron fluid then is given by $$-en_0(E_0-V_0B_0)=T_e {\partial n_0 \over \partial y} \label{e_balance}$$ Eliminating ${\partial n_0 / \partial y}$ in Eqs.(\[i\_balance\],\[e\_balance\]), we have $$E_0={T_i \over T_e+T_i} V_0B_0. \label{e0}$$ If the plasma resistivity is finite, the electron current in the $x$ direction cannot be maintained without an electric field in the same direction, $E_{x0}$. We shall see later, however, that its effects on the wave dynamics are small as in the MRX. Dispersion Relation =================== All wave quantities are assumed to have a normal mode decomposition proportional to $$\exp[i({\bf k} \cdot {\bf x} -\omega t)]$$ with the wave vector ${\bf k}=(k_x,0,k_z)$ and the wave angular frequency $\omega$. Note that ${\bf k}$ here does not have a $y$ component. This assumption is justified in a local theory if wavelengths are much smaller than the current layer thickness in the $y$ direction. The governing equation between $\omega$ and ${\bf k}$, or the dispersion relation, follows from three independent equations that relate the three components of the wave electric field, $E_x$, $E_y$, and $E_z$. These can be derived from Ampere’s law and Faraday’s law, $${\bf k} \times ({\bf k} \times {\bf E}) = -i\omega \mu_0 {\bf j}, \label{AmpFaraday}$$ which leads to $$\begin{aligned} k_z^2 E_x - k_x k_z E_z & = & i\omega \mu_0 j_x \label{eq1}\\ k^2 E_y & = & i \omega \mu_0 j_y \label{eq2}\\ k_x^2 E_z - k_x k_z E_x & = & i\omega \mu_0 j_z. \label{eq3}\end{aligned}$$ Here $\mu_0$ is the vacuum magnetic permeability. Next, we separately consider ion and electron dynamics to express the above equations in terms of the electric field. Ion Dynamics ------------ We take the ions as unmagnetized and solve the kinetic equation for the perturbed distribution function assuming the equilibrium ion distribution function is Maxwellian with constant temperature, but variable density, $ d n_0/d y = \epsilon n_0 $. From Eq.(\[i\_balance\]), $ \epsilon = e E_0/T_i = 2 e E_0/M v_i^2 $. The solution of the ion Vlasov equation is carried out as an expansion to first order in $ \epsilon $. The result is most easily expressed in terms of the electric field components $ E_1 $ and $ E_3 $ defined in Fig.1(b), in which $ E_1 $ is the component parallel to $ {\bf k} $, and $ E_3 $ is the component perpendicular to it and in the $ x-z $ plane. The perturbed ion current can then be written (Appendix B1), $$\begin{aligned} {\bf j}^i = -i \frac{ n_0 e^2}{M} \frac{1}{k v_i} \left[ Z(\zeta) {\bf E} \right. & - & ( \zeta Z' + Z) ({\bf E} \cdot \hat{{\bf k} }) \hat{{\bf k} } \nonumber \\ & - &\left. i (\epsilon/k) (\zeta Z' + Z ) E_y \hat{{\bf k} } \right] \label{jip}\end{aligned}$$ and the perturbed ion density is $$n = i\frac{n_0 e}{M k^2 v_i^2} Z'(\zeta) \left( {\bf k} \cdot {\bf E} + i \epsilon E_y \right) \label{ni}$$ where $ \hat{{\bf k} } ={\bf k}/k $, $ \zeta = \omega /k v_i $, and $Z$ is the plasma dispersion function. We find that for the principal instabilities the phase velocity is somewhat larger than $ v_i $ so for convenience we first take the $ \zeta \gg 1 $ limit (the cold limit), determine the parameter range of instability. Then, in Appendix B3, we are able to employ a simple modification of the dispersion relation to extract the correct growth rate including the finite ion thermal effects. In the cold limit the ion current neglecting the $ \epsilon $ correction is obtained from the $ \zeta \gg 1 $ limit and is $${\bf j}^i \approx i {\omega_{pi}^2 \over \omega} \epsilon_0 {\bf E}, \label{ji}$$ where the ion plasma angular frequency $\omega_{pi} \equiv \sqrt{n_0e^2/M\epsilon_0}$ and $\epsilon_0$ is the vacuum susceptibility. In the same limit the perturbed ion density is $$n = i\frac{n_0 e}{M\omega^2}( {\bf k} \cdot {\bf E} +i \epsilon E_y ) \approx i{en_0 \over M\omega^2} ({\bf k} \cdot {\bf E}). \label{n}$$ The neglected $\epsilon$ term is much smaller than the other one since, for our local theory, we assume $k/\epsilon \gg 1$. Indeed, it is shown in Appendix B2 that the neglected term only has a small effect on the dispersion relation. Electron Dynamics ----------------- As we have shown in Appendix A1, the perpendicular electron current can be obtained from the first order force balance for the electron fluid, $${\bf j}^e \times {\bf B}_0 = en_0 {\bf V}_0 \times {\bf B} + en_0 {\bf E} + en {\bf E}_0 + T_e\nabla n + m n_0 {\partial {\bf U}_E \over \partial t} \label{je}$$ where $ {\bf U}_E = {\bf E} \times {\bf B_0} /B_0^2 $ and $m$ is the electron mass. As shown in Appendix A2, the electron inertial terms contribute a small effect to the distpersion relation and we can neglect them when determining the instability. The $y$ and $x$ components of Eq.(\[je\]), therefore, are given by $$\begin{aligned} -j_x^e B_0 & = & -en_0V_0B_z + en_0E_y + enE_0 \label{jex} \\ j_y^eB_0 & =& en_0 E_x + ik_xT_en \label{jey}\end{aligned}$$ respectively. Here, $B_z = k_x E_y/\omega$. Since $E_0$ and $n$ are given already by Eq.(\[e0\]) and Eq.(\[n\]), $j_x^e$ and $j_y^e$ can be expressed in terms of the electric field. We note on the righthand side of Eq.(\[jey\]) that there would be another term, $enE_{x0}$, where $E_{x0}$ is the unperturbed electric field. However, we will treat it as second order and balanced by quasi-linear terms [@kulsrud04]. In fact, the contribution from this term is small when compared with the last term if $k_x \gg eE_{x0}/T_e$ as is often satisfied in the MRX. The $z$-component of the electron current, $j_z^e$, is not determined by Eq.(\[je\]). It turns out, however, that it is unnecessary to explicitly calculate it in order to obtain the dispersion relation due to simplifications of the $z$-component of Maxwell’s equation, Eq.(\[eq3\]). This is because the electrons are so easily accelerated along the field line by the force, $F_z^e$, on the electron fluid where $$F_z^e = -n_0 e \left( E_z + V_0 B_x + i k_z {T_e \over e} {n \over n_0}\right).$$ The various terms in this force are separately large and must balance closely to avoid very large parallel electrons currents. In fact, taking $j_z^e= -n_0 e v_z^e = - i(e/m\omega) F_z^e$ and using Eq.(\[ji\]) for the ion current, we can write the $z$-component of Maxwell’s equation, Eq.(\[eq3\]), as $$\begin{aligned} && k_x^2 E_z - k_x k_z E_x = i \omega \mu_0 (j_z^i + j_z^e) \nonumber \\ && = -{\omega_{pi}^2\over c^2} E_z - {\omega_{pe}^2\over c^2} \left( E_z + V_0 B_x + i k_z {T_e \over e} {n \over n_0}\right). \label{eq3p}\end{aligned}$$ Since $\omega_{pi}^2/\omega_{pe}^2=m/M \ll 1 $ and $(k\lambda_e)^2 \ll 1$ to our interests here, the above equation simplifies to the one demanding the electron force balance in the $z$ direction, $$E_z + V_0 B_y + ik_z {T_e \over e} {n \over n_0} = 0, \label{eq3a}$$ where $B_y$ can be expressed in terms of the electric field using Faraday’s law, $$B_y = {k_z E_x - k_x E_z \over \omega}.$$ In Appendix A2, we show that the neglected terms have only a small effect on the dispersion relation. We note that, although unneeded for the dispersion relation, the $z$-component of the electron current, $j_z^e$, can be determined by $\nabla \cdot {\bf j} =0$. This is a consequence of the charge-neutrality condition, which is in turn enforced by Eq.(\[eq3a\]). It is interesting to note that if we allow the propagation angle approach to $90^\circ$, the parallel phase velocity can be comparable to the electron thermal velocity. In this case, we need to include a Landau term in Eq.(\[eq3a\]). Then, if $\beta_e \ll 1$, we would be able to recover the electrostatic perpendicular LHDI [@krall71]. However, since this electrostatic LHDI disappears at the high-$\beta$ of interest to us, we need not include the Landau term. Dispersion Relation ------------------- Substituting expressions of ${\bf j}^e$, ${\bf j}^i$, and $n$ \[Eqs.(\[ji\],\[jex\],\[jey\],\[n\])\] into Eqs.(\[eq1\],\[eq2\],\[eq3a\]) we obtain, after some algebra, the dispersion relation $$\left(\begin{array}{ccc}D_{xx} & D_{xy} & D_{xz} \\D_{yx} & D_{yy} & D_{yz} \\D_{zx} & D_{yz} & D_{zz} \end{array}\right) \left( \begin{array}{c} E_x \\ E_y \\ E_z \end{array} \right)=0, \label{disp}$$ where $$\begin{aligned} D_{xx}=& K^2\cos^2 \theta +1 -\displaystyle{\beta_i \over \beta_e +\beta_i} {KV \sin \theta \over \Omega} \nonumber \\ D_{xy}=& i(\Omega-KV\sin\theta) \nonumber \\ D_{xz}=& -K^2\sin\theta\cos\theta-\displaystyle{\beta_i \over \beta_e +\beta_i} \displaystyle{KV \cos\theta\over \Omega} \nonumber \\ D_{yx}=& -i\left(\Omega-\displaystyle{\beta_e\over 2}\displaystyle{K^2\sin^2\theta\over\Omega}\right) \nonumber \\ D_{yy}=& K^2+1 \nonumber\\ D_{yz}=& i\displaystyle{\beta_e\over 2}\displaystyle{K^2\sin\theta\cos\theta\over\Omega} \nonumber\\ D_{zx}=& KV\cos\theta -\displaystyle{\beta_e\over 2}\displaystyle{K^2\sin\theta\cos\theta\over\Omega} \nonumber\\ D_{zy}=& 0 \nonumber\\ D_{zz}=& \Omega-KV\sin\theta -\displaystyle{\beta_e\over 2} \displaystyle{K^2\cos^2\theta\over\Omega}. \nonumber\end{aligned}$$ Here the dimensionless parameters are defined by , K k[c\_[pi]{}]{}, V , \_e ,\ \_i , . \[parameter\] Here, $\omega_{ci}$ is the ion cyclotron angular frequency $eB_0/M$ and $V_A$ the Alfven speed $B_0/\sqrt{\mu_0Mn_0}$. The $ KV $ term in $ D_{{x x}} $ and in $ D_{xz} $ and the $ \beta_e $ terms all result from replacing the kinetic equation for the perturbed density $ n $ by it cold limit. The ’one’s in $ D_{x x} $ and $ D_{yy} $ are ion currents which are similarly appoximated. The resultant dispersion relation $\Omega(K)$ is a fourth order algebraic equation in $\Omega$ with 4 controlling parameters, $V$, $\beta_e$, $\beta_i$, and $\theta$, \^4 -2KV\^3\ -\^2\ +KV\ +K\^2=0. \[dispersion\] Wave Characteristics and Instability ==================================== Basic Wave Characteristics without Drift ---------------------------------------- The basic wave characteristics described by Eq.(\[dispersion\]) are summarized here for the case that there is no drift between ions and electrons. When $V=0$ and $\theta=0$, Eq.(\[dispersion\]) reduces to $$\left[\Omega^2-(K^2+1)^2\right]\left[\Omega^2-{\beta_e\over 2}K^2\right]=0$$ which represents four waves, as shown in Fig.2 for the case of $\beta_e=\beta_i=1$. Two waves are whistler waves, traditionally termed fast waves, while the other two waves are sound waves or slow waves. One of each waves propagates along the background magnetic field and the other propagates against. As expected, the whistler waves are largely transverse waves or electromagnetic waves since the electric field vectors are perpendicular to the propagation (${\bf k}$) direction $\phi \simeq 90^\circ$, where $\cos\phi \equiv {\bf k} \cdot {\bf E}/ (|{\bf k}||{\bf E}|)$. In contrast, the sound waves are largely longitudinal waves or electrostatic waves since $\phi \simeq 0$. The situation changes when $\theta$ and $\beta$ are varied. In Fig.3, the angles between ${\bf E}$ and ${\bf k}$, $\phi$, are shown for $V=0$ and a few cases of $\theta$ and $\beta$. It can be seen that when $\theta$ is larger, the whistler waves become less electromagnetic and more electrostatic while the sound waves become more electromagnetic and less electrostatic. This trend is stronger for larger values of $\beta$. An Oblique Electromagnetic Instability -------------------------------------- It is evident that the whistler waves are supported by fast electron dynamics while the sound waves are supported by slow ion dynamics. When there is no drift between these two fluids, all wave branches stay separate in the dispersion diagram as shown in Fig.2 for $\theta=0$. The situation is similar for more general cases of $\theta \neq 0$. If $V=0$, Eq.(\[dispersion\]) reduces to \^4 - \^2\ +[\_e2]{}K\^2(K\^2+1)\^2\^2=0, \[dispnov\] which represents four waves in the left panels in Fig.4 for the case of $\theta=60^\circ$ and $\beta_e=\beta_i=1$. It is seen that at this propagation angle, $\phi$ is $\sim 40^\circ$ for whistler waves and $\sim 0^\circ$ for sound waves. When there is a finite electron drift in the ion rest frame, the whistler waves are doppler-shifted so that each $\Omega$ from Eq.(\[dispnov\]) is increased by $KV\sin \theta$, shown as dotted curves in the top-right panel of Fig.4 for the case of $V=6$. In constrast, sound waves, unaffected by the drift, are shown as dotted straight lines. When the drift is large, some part of the backward propagating whistler waves branch can intercept with the forward propagating sound wave branch, resulting in instabilities through reactive couplings. The case of $V=6$ is shown in the right panels of Fig.4 and all other parameters are the same as in the left panels. It is seen that when $K < \sim 6$ or $K > \sim 16$, all four roots are real and thus all waves are stable. When $6 < K <16$, two of roots become complex conjugates as a result of coupling; one of them is damped and another growing (the growth rates are shown in the middle-right panel). The maximum growth rate is about 8 times of $\omega_{ci}$ at $K\simeq 11$. Since the polarization angle $\phi \simeq 15^\circ$, the unstable waves have significant electromagnetic components. Figure 5 shows the unstable region and contours of polarization angle in the $\theta-K$ plane for a few values of $V$. It is seen that the unstable waves are localized to small $K$ when $\theta$ is small and to large $K$ when $\theta$ is large. The unstable region expands and the growth rate increases with increasing $V$. The polarization angle $\phi$ ranges between $10^\circ$ to $25^\circ$, and is larger near the small $K$ and small $\theta$ corner. A Physical Picture ================== Further Simplification of Electron Dynamics ------------------------------------------- In order to understand the primary feedback mechanism of our instability we make further simplifications to the dispersion relation given by Eq.(\[disp\]). We first start by rotating the coordinate for ${\bf E}$ as shown in Fig.1(b): $(E_x,E_y,E_z)$ to $(E_1,E_2,E_3)$. $E_1$ is in the $k$ direction, representing the electrostatic component. $E_2$ is the same as $E_y$ and $E_3$ is another perpendicular component to ${\bf k}$, and both of these are electromagnetic components. Using the new bases, $(E_1,E_2,E_3)$, Eq.(\[disp\]) reduces to $$\left(\begin{array}{ccc}D_{11} & D_{12} & D_{13} \\D_{21} & D_{22} & D_{23} \\D_{31} & D_{32} & D_{33} \end{array}\right) \left( \begin{array}{c} E_1 \\ E_2 \\ E_3 \end{array} \right)=0, \label{dispprime}$$ where $$\begin{aligned} D_{11}=& \sin\theta-\displaystyle{\beta_i \over \beta_e+\beta_i} {KV \over \Omega} \nonumber \\ D_{12}=& i(\Omega-KV\sin\theta) \nonumber \\ D_{13}=& -(K^2+1)\cos\theta \nonumber \\ D_{21}=& -i \displaystyle{\sin\theta \over \Omega} \left(\Omega^2-\displaystyle{\beta_e\over 2}K^2\right) \nonumber \\ D_{22}=& K^2+1 \nonumber\\ D_{23}=& i\Omega\cos\theta \nonumber\\ D_{31}=& \displaystyle{\cos\theta \over \Omega} \left(\Omega^2-\displaystyle{\beta_e\over 2}K^2\right) \nonumber\\ D_{32}=& 0 \nonumber\\ D_{33}=& \Omega\sin\theta-KV. \nonumber\end{aligned}$$ Again the $ KV $ term in $ D_{11} $ and the $ \beta_e $ terms result form approximating the perturbed density, and the ’one’s in $ D_{13} $ and $ D_{22} $ from approximating the ion currents. Next we simplify these equations by taking the limit of large $\Omega$, $K$, and $V$ since this asymptotic limit will make the physical mechanism of the instability clear. The simplified matrix then reduces to $$\left(\begin{array}{ccc} - \displaystyle{{\beta_i \over \beta_e+\beta_i} {KV \over \Omega}} & -iKV\sin\theta & -K^2\cos\theta\\ -i\displaystyle{\sin\theta \over \Omega} \left(\Omega^2- \displaystyle{\beta_e \over 2}K^2\right) & K^2 & 0\\ \displaystyle{\cos\theta \over \Omega} \left(\Omega^2- \displaystyle{\beta_e \over 2}K^2\right) & 0 & -KV\end{array}\right). \label{dispred}$$ Each line of the above matrix equation represents the balance of the leading forces on the electron fluid along the three coordinate directions $ y, x, z, $ respectively. By referring back to Eq.(\[eq3\]) and Eq.(\[je\]) we can see that the force balance can be written $$\begin{array}{ccc} y: & -enE_0 - j_{0x} B_z - j_x B_0 = 0\\ x: & -en_0E_1\sin\theta -\partial p_e/\partial x+ j_y B_0 = 0\\ z: & -en_0E_1\cos\theta -\partial p_e/\partial z + j_{0x} B_y = 0 \end{array}, \label{forces}$$ where in the asymptotic limit the current ${\bf j}$ is all due to the electrons. Interestingly, the electrostatic force is balanced by the Lorentz force in all directions. In the $y$-direction, the unperturbed electrostatic field acting on the perturbed electron density is balanced by the Lorentz force, which consists of both magnetic pressure gradient, $-j_{0x}B_z $, and tension $-j_x B_0$ forces. By contrast, the perturbed electrostatic field is balanced by the magnetic tension, $j_{0x}B_y$, in the $z$-direction. The Case of $\theta$=0 ---------------------- We start with the simplest case, $\theta=0$, in which there are no perturbed forces in the $x$-direction. In the $y$-direction the perturbed magnetic pressure force is also zero since $B_z=k_xE_y/\omega=0$. Therefore, the electrostatic force, $-enE_0$, must be balanced by the magnetic tension force, $-j_xB_0$. Suppose that the electron density is perturbed in a way such that $n>0$ at the origin as illustrated in Fig.6(a) in the $y-z$ plane. Because ${\bf E}_0$ points in the positive $y$-direction, the perturbed electrostatic force on the electron fluid, $-enE_0$, points in the negative $y$-direction at the origin. Since it varies in $z$, this force bends the field line until its magnetic tension force $-j_xB_0$ balances the $-enE_0$ force. (Here the field-line bending can also be understood as a result of the perturbed $j_x$ due to changing the number of the charged carriers by the perturbed density $n$.) In the $z$-direction, there is now a component of the magnetic tension force towards the origin $ j_{0x} B_y $ due to the bent line, as illustrated in Fig.6(a). This force reduces or reverses the perturbed electrostatic force $ - e n_0 E_1$ produced by the electron density perturbation. In the [*latter*]{} case, the perturbed electrostatic force is directed away from the regions where $n>0$ and towards the regions where $n<0$. As a result the perturbed electric field, ${\bf E}_1$, must point from the regions where $n<0$ to the regions where $n>0$, such as the origin. To see that this leads to instability consider the ions which only see the electrostatic field $ {\bf E}_1$. This electrostatic field will force the ions to condense further at the origin increasing their density perturbation. By charge neutrality this will increase the initially assumed electron density perturbation and thus lead to instability. The Case of $\theta > 0$ ------------------------ We find that it is convenient to take the limit of $\beta_e=0$ for the discussion of this more general and complicated case. Here the feedback to initial perturbations through compression or decompression of the electron fluid along the $z$-direction is unaffected except for a reduced efficiency. However, there are perturbed forces in the $x$-direction. As before, we suppose an electron density perturbation $n>0$ at the origin. When the mode is unstable, the perturbed electrostatic force, which is parallel to ${\bf k}$, has an $x$-component, $-en_0E_1\sin\theta$, pointing away from the regions where $n>0$ towards the regions where $n<0$ also as before. This force on the electrons decompresses the magnetic field in the $n>0$ regions and compresses it in the $n<0$ regions. This is illustrated in Fig.6(b) in the $x-z$ plane. Because ${\bf k}$ makes a finite angle to ${\bf B}_0$, the magnetic field lines are distorted to have both a tension force and also a magnetic pressure force. Therefore, $B_z$ must be negative (decompressed) at the origin where $n>0$ and thus, the associated magnetic pressure force in the $y$-direction, $-j_{0x}B_z$, is directed towards positive $y$-direction. As a result, this force counters the initial electrostatic force, $-enE_0$, (which bends the field line) and thus, reduces the tendency towards instability. Both these stabilizing and destabilizing forces are included in the dispersion relation from Eq.(\[dispred\]), in which we restore $\beta_e$ to obtain $$\Omega^2={\beta_e\over 2} K^2 + {\beta_i \over \beta_e + \beta_i} {K^2V^2 \over V^2\sin^2\theta-K^2\cos^2\theta}.$$ Consider a given (large enough) $V$, it can be seen that instability occurs when $K$ exceeds some threshold values, and stability returns eventually in the limit of large $K$, consistent with Fig.4. Thus, if $\rho_e$ is small enough, the growth rate reaches its peak at a wavelength longer then $\rho_e$. However, it is clear from the above equation that, if $\beta_e=0$, the instability persists over all $K$ above its critical value (at least until some finite electron gyroradius effects become important.) From this, we can see that our calculation is essentially based on a two-fluid model, and it is not strictly a Hall MHD calculation since the ions are totally unmagnetized and one cannot set $\beta_e=0$ without losing the above physical contents. Our calculation is perhaps closer to a hybrid model [see @birn01] with kinetic ions and a massless electron fluid, but in three dimensions. We emphasize here that the background ion pressure gradient is essential for the instability in both $\theta=0$ and $\theta > 0$ cases because of the important role played by the associated equilibrium electric field, ${\bf E}_0$. Discussions and Conclusions =========================== In the MRX, it has been observed that the usual electrostatic LHDI, propagating perpendicularly to the magnetic field, is active only in the low-$\beta$ edge of the reconnection region, but not in the high-$\beta$ central region [@carter02a]. This is consistent with the theoretical prediction that the perpendicular LHDI is stable at the high-$\beta$ [@davidson77; @carter02b]. On the other hand, it has been found that, in the high-$\beta$ central region, obliquely propagating, electromagnetic waves in a similar frequency range are active, and their amplitude positively correlate with the reconnection rate [@ji04a]. Motivated by these observations, we have developed a simple two-fluid formalism to derive and analyze in detail an electromagnetic drift instability in the lower-hybrid frequency range. We term this the oblique LHDI. We show that the main features of the instability are consistent with fully electromagnetic kinetic calculations [@lemons77; @wu83; @tsai84]. We find that, contrary to the perpendicular LHDI, the oblique LHDI persists in high-$\beta$ plasmas. Further, the growth rate peaks at longer wavelength than electron gyroradius, justifying our assumption that the electrons are magnetized. The resultant waves have mixed polarization and significant electromagnetic components. The instability is caused by reactive coupling between the backward propagating whistler (fast) waves in the moving electron frame and the forward propagating sound (slow) waves in the ion frame, and occurs when the relative drifts are large. After further simplifications of the model, the primary positive feedback mechanism is identified as a reinforcement of initial electron density perturbations by compression of the electron fluid by an induced Lorentz force. Interestingly, the revealed mechanism of the instability requires close interactions between the electrostatic and electromagnetic forces. In contrast to most of previous theories on MTSI, our analysis also suggest that the self-consistent background-ion-pressure gradient is essential for the instability. A few comments on three-dimensional particle simulations are in order. In addition to the dimensionless parameters of Eq.(\[parameter\]), the mass ratio, $M/m$, is another important parameter. To make the simulations feasible, often $M/m$ is limited to a few hundreds. In contrast, our analysis based on the above simple local model is valid in the limit of large $M/m$ since ions are treated as unmagnetized. Small mass ratios used in simulations will limit available wavenumber window for the instability due to the condition of $\lambda_i^{-1} \ll k \ll \lambda_e^{-1}$. In addition, the limited grid size and resolution may not permit numerical treatment of the large oblique wavenumber range where our instability resides. Future numerical simulations with increasingly powerful computers may help to elucidate these effects more clearly especially with regard to nonlinear consequences in magnetic reconnection. Simulations of non-Harris current sheets, as attempted in the linear analyses [@yoon04; @sitnov04], may prove to be more physically meaningful since they may represent the reality more accurately. Many of the predicted features of unstable waves discussed in this paper are also qualitatively consistent with the observed magnetic fluctuations in the MRX [@ji04a], including their existence in the high-$\beta$ region, their frequency range, and their propagation direction with respect to the background magnetic field. In fact, the parameters we use in the calculation have been drawn directly from the MRX experiments, and they are valid throughout the bulk of the MRX current sheet. Also, the instability does indeed persist into the $\beta_e\gg 1$ regimes, but the physics of the instability is still uncertain in the region where the magnetic field nearly vanishes. One particular comment on their phase velocity is worth making. The experimentally measured phase velocity is of the same order as the relative drift velocity. Even given the large experimental uncertainties such as the measurement location and the unknown relative velocity between the ion frame and the laboratory frame, the measured phase velocities are considerably larger than our theoretical predictions. As seen in Fig.4, the unstable waves should have phase velocities on the order of the ion thermal speed. However, the theory presented here is limited to the case where $k_y=0$. The phase velocity may be substantially increased by incorporating a nonzero $k_y$. This is a subject for future work. Increasing the phase velocity to values much larger than ion thermal speed may also help mitigate another shortcoming of our analysis: the reduction of the growth rates by ion thermal effect. The role which this instability plays in magnetic reconnection, such as in the production of anomalous resistivity and its effect on heating, is discussed in a forthcoming paper [@kulsrud04] that is based on quasi-linear theory. Detailed Calculations of Electron Dynamics ========================================== Drift Kinetic Equation for Electrons ------------------------------------ Normally, the drift kinetic equation is developed for both electrons and ions, and is combined with Maxwell’s equations to achieve some important simplifications. This full formulation is described in a number of places, for example in [@kulsrud83]. However, if the ions are unmagnetized, as in this paper, the formulation is reduced to that of solving the electron Vlasov equation alone, as an expansion in $ \rho_e/\lambda $, and $ 1/\omega_{ce} t $ where $ \lambda $ is the length scale of the phenomena, and $ t $ is its time scale. We follow the procedure given in the handbook article. It is clear that the electronic charge can be used as a guide to the expansion and we use $ 1/e $ as the expansion parameter. The electron Vlasov equation is $$\label{A1} \frac{\partial f}{\partial t} +{\bf v} \cdot \nabla f - \frac{e}{m} \left( {\bf E} + {\bf v} \times {\bf B} \right) \cdot \nabla_{{\bf v}} f = 0.$$ We first carry out the expansion for the full distribution, (equilibrium $ f $ and perturbed $ \delta f $) and later carry out the expansion in the instability perturbation. The lowest order Vlasov equation is accordingly $$\label{A2} - \frac{e}{m} \left( {\bf E} + {\bf v} \times {\bf B} \right) \cdot \nabla_{{\bf v}} f_0 = 0.$$ We introduce the $ {\bf E} \times {\bf B} $ velocity by $${\bf {\bf U}_E} = \frac{{\bf E} \times {\bf B} }{B^2}$$ and carry out the transformation of the velocity at each point $ {\bf r} $, $${\bf v} ={\bf {\bf U}_E}({\bf r}) +{\bf v'} = {\bf U}_E +v_{\perp} \cos \phi \hat{{\bf x'}} + v_{\perp} \sin \phi \hat{{\bf y'}} + v_{\parallel } {\bf b}$$ where $ \hat{{\bf x'}} , \hat{{\bf y'}} $ and $ {\bf b} $ are local coordinates at each point $ {\bf r} $, and $ v_{\perp }, \phi $ and $ v_{\parallel} $ are cylindrical coordinates for $ {\bf v'} $. Then Eq.(\[A2\]) becomes $$\frac{e B}{m} \frac{\partial f_0}{\partial \phi } - \frac{e E_{\parallel }}{m} \frac{\partial f_0}{\partial v_{\parallel}} =0.$$ If $ E_{\parallel } $ is non zero, $ f_0 $ would be constant along a helical orbit in velocity space that extends to infinity, which is impossible. Thus, $ E_{\parallel} $ must vanish to lowest order and $ E_{\parallel} $ must be considered first order. Dropping the second term we see that $ f_0 $ is independent of $ \phi $ (gyrotropic) and, thus, a function only of $ t, {\bf r}, v_{\perp} , $ and $ v_{\parallel} $. Proceeding to next order in $ 1/e $ we get $$\label{A6} -\frac{eB}{m} \frac{\partial f_1}{\partial \phi} = \left(\frac{\partial f_0}{\partial t} + {\bf v} \cdot \nabla f_0 \right) - \frac{e}{m} E^1_{\parallel} \frac{\partial f_0}{\partial v_{\parallel} }$$ where the expression in parentheses must be transformed to $ t, {\bf r} ,v_{\perp}, v_{\parallel}, \phi $ coordinates. Equation (\[A6\]) can only be solved for $ f_1 $ if its average over $ \phi $ (which eliminates $ \partial f_1/\partial \phi $ ) vanishes. The result is $$\begin{aligned} \frac{\partial f_0}{\partial t} & + & ( {\bf U}_E + v_{\parallel} {\bf b} ) \cdot \nabla f_0 \nonumber \\ & - & \frac{v_{\perp} }{2} \left( \nabla \cdot {\bf U}_E - {\bf b \cdot \nabla U}_E \cdot {\bf b} + v_{\parallel} \nabla \cdot {\bf b} \right) \frac{\partial f_0}{\partial v_{\perp} } \nonumber\\ & + & \left( - {\bf b} \cdot \frac{D {\bf U}_E}{D t} \cdot {\bf b} +\frac{v_{\perp}^2}{2} (\nabla \cdot {\bf b } ) + \frac{e}{m} E_{\parallel} \right) \frac{\partial f_0}{\partial v_{\perp} } = 0 \nonumber \\ && \label{A7}\end{aligned}$$ where $D{\bf U}_E/Dt \equiv \partial {\bf U}_E /\partial t + ( {\bf U}_E+{\bf b}v_\parallel)\cdot \nabla {\bf U}_E$. (Note that the $ e E_\parallel $ term is zero order since $ E_{\parallel} $ is first order and $ e $ is minus first order.) In principle, $ f_0 $ can be solved for from this equation. For the case of the instability, $ f_0 $ can be written as $ f^0_0 + \delta f_0 $ where $ f_0 $ is a local Maxwellian, $ {\bf U}_E $ is a perturbation, and $ {\bf B}_0 = B_0 \hat{{\bf z} } $ to lowest order. The only equilibrium term that survives is the $ v_{\parallel} {\bf b} \cdot \nabla f_0^0 $ term so the only restriction on $ f^0_0 $ is that it be constant along the magnetic field. To get the electron current perpendicular to $ {\bf B} $ we need $ f_1 $, $$\label{A8} {\bf j}_{\perp}^e = e \int {\bf v}_{\perp} f_1 d \phi v_{\perp} d v_{\perp} d v_{\parallel} = e \int \frac{\partial {\bf v}_{\perp} }{\partial \phi } \frac{\partial f_1}{\partial \phi } d^3 {\bf v}$$ which can be obtained directly from Eq.(\[A6\]) by multiplying it by $ \partial {\bf v}_{\perp} /\partial \phi = - v_{\perp} \sin \phi \hat{{\bf x} } + v_{\perp} \cos \phi \hat{{\bf y} } $, dividing by $ B $ and inegrating over velocity space. In fact, we could just as well have multiplied Eq.(\[A6\]) by $ {\bf v}_{\perp} $ and integrated it to find $ {\bf j}_e \times {\bf B} $. Even simpler, we could have multiplied Eq.(\[A1\]) by $ {\bf B} $ integrated over velocity space and taken the perpendicular part of the result. This result would be the perpendicular part of $$\label{A9} n m \left( \frac{\partial {\bf v}_e}{\partial t } + {\bf v}_e \cdot \nabla {\bf v}_e \right) = {\bf j}^e \times {\bf B} - \nabla \cdot {\bf P}^e + n e {\bf E}$$ Here the stress tensor is zero order, and can be found from $ f_0 $ once we have solved Eq.(\[A7\]) for it. If we inspect Eq.(\[A9\]) we see that the inertia term and the $ \nabla \cdot {\bf P} $ are zeroth order, but the $ n e E $ term is minus first order in the $ 1/e $ expansion. Thus, $ {\bf j}^e $ has a minus first order part, $ n e $ time the $ {\bf E} \times {\bf B} $ drift, and zero order parts, essentially the diamagnetic and polarization currents. If the ions were magnetized, this minus first order current would be cancelled by the corresponding $ {\bf E} \times {\bf B} $ current of the ions, but this is no longer the case for unmagnetized ions. This procedure gives the perpendicular current of the electrons. The parallel current is given by the continuity condition $$\nabla \cdot {\bf j}^e + \frac{\partial }{\partial t} (n e) = 0$$ Again for finite $ n $ the $ ne $ term is minus first order. $ n_0 $ is given by the zero moment of $ f_0 $. However, $ n_1 $ is needed to give the finite parallel electron current, and for it we need the zero moment of $ f_1 $. This zero moment cannot be obtained from Eq.(\[A6\]), which only gives the $ \phi $ dependent part of $ f_1$, $\partial f_1/ \partial \phi $. To get the mean part it is necessary to go to next order in the $ 1/e $ expansion of the Vlasov equation. This has been done some time ago [@frieman66], and will yield $ n_1 $. This procedure is certainly possible to carry out in all detail as outlined above and is fairly easy for our perturbation problem. In fact if it is carried out in a velocity frame in which the equilibrium electric field is zero (the so-called Harris frame) the results turn out to be essentially identical to those calculated by [@yoon94] in the common limit of approximation, small gyration radius and small frequency compared to the electron cyclotron frequency. As stated in the text, we can avoid some of the calculation by taking the perturbed density from that of the ions by quasi neutrality. This also avoids going to next order in the Vlasov equation to find $ n_1 $. This assumption puts a constraint on $ E_{\parallel} $ in an early phase in the calculation rather than waiting for substitution in Maxwell’s equations to enforce it. In any event the drift kinetic approach is completely consistent with earlier calculations of LHDI. Electron Inertial Terms ----------------------- The first two rows of the matrix in Eq.(\[disp\]) represent $-\Omega/n_0e$ times the $y$ and $x$ components of Eq.(\[je\]). Their initial terms are $i(m/M) n_0e\Omega E_x$ and $-i(m/M) n_0e\Omega E_y$, respectively. Multiplying these by $(-\Omega/n_0e)$ we get for the first two rows of the matrix equation $$\begin{array}{ccc} D_{xx} - \displaystyle{\frac{m}{M}} \Omega^2 &D_{xy} &D_{xz} \\ D_{yx} & D_{yy} + \displaystyle{\frac{m}{M}} \Omega^2 & D_{yz} \end{array}$$ where the coefficients $D$ are given by Eq.(\[disp\]) as before. The last row represents $-\Omega c^2/\omega_{pe}^2$ times the last term in Eq.(\[eq3p\]). Bringing all the other terms to the right-hand side and multiplying these by $-c^2/\omega_{pe}^2$, we get $-(m/M)K^2\sin\theta\cos\theta E_x +(m/M) (1+K^2\sin^2\theta)E_z$. Multiplying these by $\Omega$ and adding the results to the last row of the matrix equation, we obtain $$D_{zx} -\displaystyle{\frac{m}{M}} K^2 \Omega \sin \theta \cos \theta \quad D_{zy} \quad D_{zz} +\displaystyle{\frac{m}{M}}\Omega(1+K^2 \sin^2 \theta).$$ Transforming to the $(E_1, E_2, E_3)$ components of the electric field, we have $$\begin{array}{ccc} D_{11} - \displaystyle{\frac{m}{M}}\Omega^2\sin\theta & D_{12} & D_{13} +i\displaystyle{\frac{m}{M}} \Omega^2 \cos \theta \\ D_{21} & D_{22} + \displaystyle{\frac{m}{M}} \Omega^2 & D_{23} \\ D_{31} + \displaystyle{\frac{m}{M}}\Omega \cos \theta & D_{32} & D_{33} + \displaystyle{\frac{m}{M}} \Omega \sin \theta (1+ K^2) \end{array}$$ and for the limit of large $ K , V $ and $ \Omega $, Eq.(\[dispred\]) becomes $$\left(\begin{array}{ccc} - \displaystyle{{\beta_i \over \beta_e+\beta_i} {KV \over \Omega}} - \displaystyle{\frac{m}{M}}\Omega^2\sin\theta & -iKV\sin\theta & -K^2\cos\theta +i\displaystyle{\frac{m}{M}} \Omega^2 \cos \theta \\ -i\displaystyle{\sin\theta \over \Omega} \left(\Omega^2- \displaystyle{\beta_e \over 2}K^2\right) & K^2 + \displaystyle{\frac{m}{M}} \Omega^2 & 0\\ \displaystyle{\cos\theta \over \Omega} \left(\Omega^2- \displaystyle{\beta_e \over 2}K^2\right) + \displaystyle{\frac{m}{M}}\Omega \cos \theta & 0 & -KV + \displaystyle{\frac{m}{M}} \Omega \sin \theta (1+ K^2) \end{array}\right).$$ If we regard $ K, V $, and $ \Omega $ as all of order $ K $, then we can we can see that the relative corrections are of order at most $ m/M $ except in the one-one and three-three elements where they are of order $\sim (m/M) K $. These corrections are all small and can be neglected as long as $ K \ll M/m $. Incidentally, the correction in the third line represents the extra parallel electron field needed to accelerate the electrons along the magnetic field to achieve charge neutrality. Its smallness indicates the ease with which the electrons are able to achieve charge neutrality. Detailed Calculations of Ion Dynamics ===================================== Perturbed Ion Current and Density --------------------------------- The expressions for the unmagnetized ion current and density given in Eqs. (\[jip\]) and (\[ni\]), which keep the equilibrium density gradient, as a first order correction are found from the perturbed ion distribution function with the same correction. The latter is obtained by iterating the perturbed ion Vlasov equation $$-i(\omega - {\bf k} \cdot {\bf v} ) f_1 + v_y \frac{\partial f_1}{\partial y} + \frac{e}{M} E_0 \frac{\partial f_1}{\partial v_y} + \frac{e}{M} {\bf E}_1 \cdot \frac{\partial f_0}{\partial {\bf v} } =0$$ The second and third terms are the correction terms. Therefore, drop them at first and solve for the uncorrected $ f_1 $ from the remaining equation, in the standard way. $$f_1 = \frac{-i n_0 e}{k M v_i^5} \frac{2 {\bf v} \cdot {\bf E}_1 }{(v_z - \omega/k )} \frac{e^{-v^2/v_i^2}}{\pi^{3/2}}$$ where $ v_i^2= 2 T_i/M $, and where without loss of generality we take the $ z $ axis along $ {\bf k} $. We see that $ \partial f_1/\partial y = \epsilon f_1 $ where $ \epsilon = (d n_0/d y)/n_0 $. Next, we insert this expression into the second and third terms of the full Vlasov equation and solve for the correction, $ \delta f $, to $ f_1 $ which satisfies $$-i(\omega - {\bf k} \cdot {\bf v} ) \delta f = - v_y \epsilon f_1 - \frac{e}{M} E_0 \frac{\partial f_1}{\partial v_y}$$ After some algebra we can express the zero and first moments of $ (f_1 + \delta f ) $ in terms of the plasma dispersion function, $ Z $, of $ \zeta = \omega /k v_i $ and, thus, obtain Eqs.(\[jip\]) and (\[ni\]) of the main text. Dispersion Relation with the Correction from Background Density Gradient ------------------------------------------------------------------------ In Eq.(\[n\]), a term proportional to the density gradient has been neglected in deriving the dispersion relation. It is straightforward to show that, by including this term, the dispersion matrix is given by $$\left(\begin{array}{ccc} D_{xx} & D_{xy} + 2i \displaystyle{{\beta_i \over (\beta_e+\beta_i)^2} {V^2 \over \Omega} } & D_{xz}\\ D_{yx} & D_{yy} + \displaystyle{{\beta_e \over \beta_e+\beta_i} {KV\sin \theta \over \Omega}} & D_{yz}\\ D_{zx} & D_{zy}+ i\displaystyle{{\beta_e \over \beta_e+\beta_i} {KV\cos \theta \over \Omega}} & D_{xz} \end{array}\right), \label{dispgrad}$$ where the coefficients $D$ are given by Eq.(\[disp\]). The resultant dispersion relation remains as a fourth order equation, and the added new terms only have a small effect on the solutions. In the right and middle panel of Fig.4, the growth rate by Eq.(\[dispgrad\]) is shown as the dotted line, which differs little from the solid line by Eq.(\[disp\]) especially in the large $K$ limit. (The dotted line indicating instability at very small $K$ has no physical significance since the local approximation becomes clearly questionable for such cases.) Growth Rates with Warm Ions --------------------------- The most important instabilities occur for very local perturbations with large $ K,V $ and $ \Omega $. We restrict the discussion of the thermal corrections to this case. The cold ion approximation involves using Eq.(\[n\]) for the ion density instead of Eq.(\[ni\]) and Eq.(\[ji\]) for the ion currents instead of Eq.(\[jip\]). In equation (18) the ion currents are negligible and only the one-one element and the $ \beta_e $ terms are proportion to the perturbed ion density. Thus the matrix of Eq.(\[dispred\]) with the corrected ion density is $$\left(\begin{array}{{ccc}} -\alpha \displaystyle{\frac{\beta_i }{\beta_e + \beta_i}} KV &-i KV \sin \theta & - K^2 \cos \theta \\ -i \sin \theta \left(\Omega^2 - \displaystyle{\frac{\beta_e}{2}} K^2 \alpha \right) &K^2 & 0 \\ \cos \theta \left(\Omega^2 - \displaystyle{\frac{\beta_e}{2}} K^2 \alpha \right) & 0 & -KV \\ \end{array}\right) \label{B1}$$ where $$\alpha = \frac{n_{true}}{n} = \zeta ^2 Z'(\zeta )$$ where $ \zeta = \omega /(k v_i) = \Omega/ (K \sqrt{\beta_i}) $. The dispersion relation from Eq.(\[B1\]) can thus be written $$\begin{aligned} - K^4V^2 \frac{\beta_i}{\beta_e + \beta_i } \alpha &+& K^2V^2 \sin^2 \theta\left( \Omega^2 - \alpha \frac{\beta_e}{2} K^2\right) \nonumber\\ &-& K^4 \cos^2 \theta \left( \Omega^2 - \alpha \frac{\beta_e}{2} K^2\right) =0.\end{aligned}$$ By dividing this equation by $ \alpha $ we see that $ \Omega^2/\alpha $ satisfies the same equation as $ \Omega_0^2 $, the approximate solution for the growth rate with cold ions. Thus we can write $$\label{correct} \frac{\zeta ^2}{\zeta ^2 Z'(\zeta) } = \frac{1}{Z'(\zeta) } = \zeta^2_0$$ where $ \zeta_0 ^2 = \Omega^2_0/(K^2 \beta_i). $ Thus from Eq.(\[correct\]) we plot ratio of the true $\zeta$ to the approximate $ \zeta _0 $ as a function of $\zeta_0$ in Fig.7. (Actually, the $ \Omega $’s are pure imaginary so we plot $\Gamma/(K \sqrt{\beta_i}) $’s where the $\Gamma $’s refer to approximate and exact normalized growth rates.) We see that there is indeed a difference of order unity between the approximate and exact values of $ \Omega $ or $ \zeta $. Since we see from Fig.4 that the peak $\Gamma_0/K \approx 1$, the true $ \Gamma/K \approx 0.5 $ when $ \beta_i =1$ and $\Gamma/K \approx 0.7$ when $\beta_i = 0.5 $. In spite of this reduction, we see that the oblique LHDI is still unstable. The authors are grateful to Mr. Y. Ren for his contributions to initial assessments of wave dispersion of high-$\beta$ plasmas. Drs. P. Yoon, A. T. Y. Lui, W. Daughton, and M. Sitnov are acknowledged for useful discussions. This work was jointly supported by DOE, NASA, and NSF. ![image](2005ja011188-f01_orig.eps){width="2.5truein"} ![image](2005ja011188-f02_orig.eps){width="3.0truein"} ![Angle between ${\bf E}$ and ${\bf k}$ for the cases of $\theta=0,45^\circ,85^\circ$ and $\beta_e(=\beta_i)=1$ and 10. Solid lines represent whistler waves and dotted lines represent sound waves.[]{data-label="esem"}](2005ja011188-f03_orig.eps){width="5.0truein"} ![image](2005ja011188-f04_orig.eps){width="5.5truein"} ![Unstable region where Im$(\Omega)>0$ (filled regions in top panels) and contours of polarization angle ($\phi$, bottom panels) in the $\theta-K$ plane for the cases of $V=3,6,10$ and $\beta_e=\beta_i=1$.[]{data-label="thetak"}](2005ja011188-f05_orig.eps){width="6truein"} ![image](2005ja011188-f06_orig.eps){width="5.5truein"} ![image](2005ja011188-f07_orig.eps){width="3.5truein"} [39]{} \[1\][\#1]{} urlstyle \[1\][doi:\#1]{} Bale, S., F. Mozer, and T. Phan (2002), Observation of lower hybrid drift instability in the diffusion region at a reconnecting magnetopause, *Geophys. Res. Lett.*, *29*, 2180. Basu, B., and B. Coppi (1992), Electromagnetic modes driven by streaming proton populations, *J. Geophys. Res.*, *97*, 17,033. Birn, J., et al. 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[**[On fields inspired with the polar $HSV-RGB$ theory of Colour]{} [^1]** ]{} [**Ján Haluška[^2]** ]{} Introduction ============ Traditional Colour theories --------------------------- There are more different sources of the theory of Colour which approach to the subject from different sides and are complementary in this sense. #### Computer graphics The HSV (Hue-Saturation-Value) theory is the most common representation of points in an RGB (Red-Green-Blue) color *technical model*. Computer graphics pioneers developed the HSV model in the 1970s for computer graphics applications (A. R. Smith in 1978, also in the same issue, A. Joblove and H. Greenberg). A HSV theory is used today in color pickers, in image editing software, and less commonly in image analysis and computer vision. A rather extensive explanation of the present State of Art in industry we can find in [@So]. #### Biophysics Th. Young and H. Helmholtz proposed a trichromatic theory. Their theory states that the human retina contains dispersed photo-sensitive clusters, where each of these clusters consists of three types of sensitive cones which peaks in *short* (420–440 nm), *middle* (530–540 nm), and long (560–580 nm) wavelengths. Weighting a total light power spectrum by the individual spectral sensitivities of the three types of cone cells gives three effective stimulus values; these three values make up a tristimulus specification of the objective color of the light spectrum. In Fig. \[birds\], there are schematic behaviours of these sensitive clusters of cells. #### Fine arts For the revelatory theories of Colour written by authors who are not mathematicians (artistic photographers, visual artists), c.f. [@Briggs; @Hirsch; @Hunt]. Terminology ----------- For terminology, basic and also advanced concepts about Colour, we refer to [@Stockman], Chapter 11 (Vol. III, Vision and vision optics; Chap. 11, Color vision mechanism). Comments to modelling of Colour ------------------------------- A reflected electro-magnetic vibration energy is filtered with the (human) tristimulus apparatus in the eye retina into three functions (also called the $RGB$ stimulus curves). Here is an information loss, because we see in the wavelength interval approximately 350 nm – 750 nm although there is some power output theoretically within the whole $[0, + \infty)$. Vibrations within other vibration intervals are partially perceived by other senses (hearing, touch). Also variuos animals have various intervals within they can see. The vision process continues in the brain where the obtained three stimuli curves are aggregated back. The result is a registry of Colour of the object. In our theory, this aggregation is a linear combination of three basic colours (poles) with the coefficients which are complex functions defined on the interval $[0, +\infty)$ (= all possible frequencies). In Fig. \[birds\], we see that realistic stimulus curves may have parts which are particularly in the negative (under the $x$-axis), i.e., the accession to the resulting Colour may happen also such that the parts of curves “absorbs” energy. Some practical aspects about $RGB$ in computer graphics we can find in [@Pascale]. ![A realistically shaped tristimulus reaction on a Colour[]{data-label="birds"}](%obrazky/ VisibleSpectrum-png.eps) ![A horse-shoe planar cut of the visible colour space; Black = non visible[]{data-label="cie1931"}](%obrazky/ cie1931-png.eps) ![A Colour wheel with colours of the equal energy (the upper surface)[]{data-label="wheelsatu"}](%obrazky/ HSV-jpg.eps) Outside a certain area of the domain, the colour aggregated from $R, B, G$ curves is not very exact because of distorted perceptions by human senses, see Fig. \[cie1931\]. We use an imagination of the Colour Hues in the Colour wheel. The reason is that there are colour hues which are not in the linear rainbow palette (e.g., Pink). We suppose that angles of the basic $RGB$ colours in the colour wheel correspond to the angles $0 \equiv 2 \pi$, $2 \pi/3$, $4\pi/3$, respectively, which is approximately true in reality. An incoming white sun light is Colours is decomposed into a sum of three $RGB$ curves. So it is also in our model. But abstracting from the natural perception, the number three for basic colours is not substantial. Our polar theory works for arbitrary natural $K\geq 3$. For a review, in the realm of animals, there are mono-chromatic Arctic mammals; most of mammals have sensibility only for two colours, they are dichromats; birds and insects are mostly tetrachromats. Concerning primates, the human vision is trichromatic. The record keeps the Mantis shrimp’s vision with $K=12$, cf. [@mantis]. There are also artificial colour schemes coming and used in the industry for some good reasons. E.g., we know the CMYK (cyan, magenta yellow, black), RGYB (red, green, yellow, blue) systems, and others. The point and interval characteristics of light ----------------------------------------------- Hue is a point characteristic (it is determined in a point), the saturation and brightness are “interval” characteristics, i.e., they are determined for an interval, not at a point (similarly as the notions concavity-convexity has no sense at a points). #### Hue of Colour is the wavelength within the visible-light spectrum at which the energy output from a source is greatest. This is shown as the peak of the sum of the three intensity curves in the accompanying graphs of intensity versus wavelength. In the illustrative examples in the pictures Fig.\[obr3\], Fig.\[obr2\], Fig.\[obr1\], all $3\times 3$ = nine colors there have the same Hue with a wavelength 500 nm, in the yellow-green portion of the spectrum. #### Saturation is an expression for the relative bandwidth of the visible output from a light source. In the diagram Fig.\[obr3\], the notion of saturation is represented relatively, by the steepness of the slopes of the curves. Here, the blue curve represents a color having the greatest saturation. As saturation increases, the color with the same Hue appears more “pure.” As saturation decreases, colors appear more “washed-out.” #### Brightness is a relative expression of the intensity of the energy output of a visible light source. It can be expressed as a total energy value (different for each of the curves in the diagram Fig. \[obr2\]), or as the amplitude at the wavelength where the intensity is greatest. Energy is imagined as the area under the curve. In the picture the blue curve has the lowest brightness. As we can see, Saturation and Brightness are generally non-comparable parameters of Colour, cf. Fig.\[obr1\]. One commonly supposes that all possible colours can be specified according to these three parameters and that Colors can be represented in terms of the $RGB$ components. Thus the whole information in the $RBG$ Colour theory is contained the three tristimulus curves. A concept of triangular coefficients mathematically reflects this idea. Semi-field of triangular coefficients {#coef} ------------------------------------- A *semi-field* $\mathbb{X}$ is a set equipped with an algebraic structure with binary operations of addition $(+)$ and multiplication $(\cdot)$, where $(\mathbb{X},+)$ is a commutative semi-group. $(X, \cdot, 1)$ is a multiplicative group with the unit $1$, and multiplication is distributive with respect to addition from both sides. For a review of semi-fields, c.f. [@Vechto]. A semi-field $\mathbb{X}$ is called a *semi-field with zero* if there exists an element $\mathbb{O}\in \mathbb{X}$ such that $\mathbf{x} \cdot \mathbb{O} = \mathbb{O}$ and $\mathbf{x} + \mathbb{O} = \mathbf{x}$ for every $\mathbf{x} \in \mathbb{X}$ and the distributivity of multiplication from both sides is preserved for the extended system. *Let $\mathbb{R}_{+,0} := (0,\infty) \cup \{0\} = [0, \infty)$ be a ray with all structures heredited from the real line $\mathbb{R}$. This is one of *trivial real semi-field with zero*.* We say that a set $$T = \{ q + \psi(f) \varepsilon \mid q \geq 0, \psi(f) \in \mathbb{R}^{[0, +\infty)}, f \in [0, + \infty) \}$$ is called to be the *set of all triangular coefficients*, where $\varepsilon$ is the *parabolic imaginary unit*, $|\varepsilon|=1, \varepsilon^2 = 0$, cf. [@Harkin]. If $[a + b(f) \varepsilon] \in T$ and $[A + B(f) \varepsilon] \in T$ are two triangular coefficients, we define the operations of addition, multiplication, and division as following. For every $ a, A \geq 0$; $b(f), B(f) \in \mathbb{R}^{[0, +\infty)}$, $f \in [0, + \infty)$, $$\begin{aligned} \label{tplus} [a+ b(f)\varepsilon] + [A + B(f)\varepsilon] := [a + A ] + [b(f)+B(f)]\varepsilon, \end{aligned}$$ $$\begin{aligned} \label{trnumbermulti} [a + b(f)\varepsilon] \cdot [A + B(f)\varepsilon] := a A + [a B(f) + A b(f)]\varepsilon, \end{aligned}$$ $$\begin{aligned} \label{trnumber2} A \neq 0 \implies \frac{a + b(f)\varepsilon}{A + B(f)\varepsilon} := \frac{a}{A} + \frac{A b(f) - aB(f)}{A^2}\varepsilon. \end{aligned}$$ For the better reading, triangular coefficients are written in square brackets in the sequel of the paper. The set $T$ with the operations defined in (\[tplus\]), (\[trnumbermulti\]), (\[trnumber2\]) is a semi-field. [[**Proof.**   ]{}]{}Saving the denotation of the previous definition we see that $[a+ b(f)\varepsilon] + [A + B(f)\varepsilon] \in T$, $[a + b(f)\varepsilon] \cdot [A + B(f)\varepsilon] \in T$, and for $A \neq 0$, $ \frac{a + b(f)\varepsilon}{A + B(f)\varepsilon} \in T$. The assertion is an obvious enlargement of the result for parabolic-complex numbers to functions defined on the non-negative real ray. [$\Box\;\;$ ]{} In our polar theory of Colour, a role of scalars will play elements of the semi-field $T$. For every function $B(f), f \in [0, + \infty)$, and by (\[trnumber2\]), $$A \neq 0 \implies \frac{1}{A + B(f)\varepsilon} = \frac{1}{A} + \frac{ - B(f)}{A^2} \varepsilon \in T.$$ ![Monotonicity of Saturation; Brightness and Hue are fixed[]{data-label="obr3"}](%obrazky/ obr3.eps) ![Monotonicity of Brightness; Saturation and Hue are fixed[]{data-label="obr2"}](%obrazky/ obr2.eps) ![Non-comparable Saturation and Brightness, Hue is fixed[]{data-label="obr1"}](%obrazky/ obr1.eps) The aim of the paper {#the-aim-of-the-paper .unnumbered} -------------------- *To model a Colour space as a three-polar field.* Mastering polar Colour spaces ============================= Poles and the definition of Colour space {#pooles} ---------------------------------------- The real line is two-polar and we are using the obvious polar operators, i.e., the signs $(+)$ and $(-)$. In this sense, we will also understand three signs (poles) $R, G, B$ in this paper. The poles $R, G, B$ could be understand also as the *generalized signs*. In fact, poles can be chosen as objects of various mathematical nature. In this paper, we use the following finite set of poles: $$\label{Examplesx} \mathbb{A} = \left\{ R= 1 + 0 \imath, G = -\frac{1}{2} + \frac{\sqrt{3}}{2} \imath, B= -\frac{1}{2} -\frac{\sqrt{3}}{2}\imath \right\},$$ a set of three vertices of an equilateral triangle in the elliptic complex plane, where $\imath$ is a usual (elliptic) complex unit, $\imath^2 = -1$, $|\imath| = 1$. Sometimes we will equivalently speak that poles are *polar operators*. Which properties are asked from poles in general? Various other objects of different nature can also be used as poles. E.g., functions, operators, the set $\mathbb{A}$ when replacing complex unit $\imath$ by the complex units $\varepsilon, \kappa$ of the parabolic ($\varepsilon^2=0$) or hyperbolic ($\kappa^2=1$) complex units, respectively, etc. In our case, we use the operators $R, G, B \in \mathbb{C}$ which 1. are applicable to “all objects” (similarly as signs plus and minus); 2. fulfil the condition $R + G + B = 0$ (presence of the *white point* of the colour space); 3. $R,G,B$ are different and non-collinear points; 4. a symmetry in some sense of the set $\mathbb{A}$ is desirable. All these terms will be precised below. Let us denote by $$\begin{array} {rcl}\triangle & := &\{ R[r+\rho(f)\varepsilon]+ G[g + \gamma(f)\varepsilon] + B[b + \beta(f)\varepsilon] \\ && \mid [r +\rho(f)\varepsilon] \in T, [g + \gamma(f)\varepsilon] \in T, [b + \beta(f)\varepsilon] \in T\},\end{array} \label{3polarity}$$ where $r, g, b \geq 0$; $\rho(f), \gamma(f), \beta(f) \in \mathbb{R}^{[0, +\infty)}, f \in [0, +\infty)$. The set $\triangle$ is called the *colour space*. An element of the colour space is called to be the *colour*. \(1) Although the functions $\rho(f), \gamma(f), \beta(f) \in \mathbb{R}^{[0, \infty)}$ are arbitrary in our paper, in the praxis they are supposed to have “good” properties, e.g., they are supposed to be unimodal, continuous, etc. (2) Elements of $\triangle$ are (mixed elliptic–parabolic) bicomplex numbers. For (elliptic–elliptic) bicomplex numbers, cf. [@Shapiro]. Achromatic part of Colour {#achroma} ------------------------- Every Colour is a composite of chromatic (pure colours) and achromatic (grey and noises) parts. Both parts of colour can be derived from the decomposition of the tristimulus sum into three individual curves. In this paper, the achromatic part of colour is derived from an element $[a + \alpha(f)] \in T$, where $a \in [0,\infty)$ determines a Hue of Grey.[^3] The function $\alpha(f) \in \mathbb{R}^{[0,\infty)}$ represents noise. We incorporate an achromatic part of Colour into our theory dealing with the following subsets (cuts) of the colour space $\triangle$: Let $[a + \alpha(f)\varepsilon] \in T$ and $[r + \rho(f)\varepsilon] \in T$, $[g + \gamma(f)\varepsilon] \in T$, $[b + \beta(f)\varepsilon] \in T$. Denote by $$\mathfrak{o}_{[a + \alpha(f)\varepsilon]} := R[a +\alpha(f)\varepsilon]+ G[a + \alpha(f)\varepsilon] + B[a + \alpha(f))\varepsilon],$$ $$\mathfrak{O} := \bigcup_{[a + \alpha(f)\varepsilon] \in T} \mathfrak{o}_{[a + \alpha(f)\varepsilon]} \subset \triangle,$$ and $$\mathfrak{s}_a := \{\mathbf{x} \in \triangle \mid \mathbf{x} = R[a +\rho(f)\varepsilon]+ G[a + \gamma(f)\varepsilon] + B[a + \beta(f)\varepsilon]\},$$ $$\mathfrak{S} := \bigcup_{a \in [0, \infty)} \mathfrak{s}_{a},$$ where $a\geq 0$ and $\alpha(f), \rho(f), \gamma(f), \beta(f)$ are arbitrary functions in $\mathbb{R}^{[0,\infty)}$. \[congruence\] Let $\mathbf{x}\ \in \triangle$. Let $\lambda \in \mathfrak{O}$. Then $$\mathbf{x} = \mathbf{x} + \lambda.$$ [[**Proof.**   ]{}]{}From definition of $\mathfrak{O}$ it follows that the triangular coefficients $[r + \rho(f)\varepsilon]$, $[g + \gamma(f)\varepsilon]$, $[b + \beta(f)\varepsilon]$ in $\mathbf{x} \in \triangle$ are ambiguous since for every arbitrary $[a + \alpha(f)\varepsilon] \in T$, there holds $$\begin{array}{rcl} \mathbf{x} & = \phantom{ + } & R [r + \rho(f)\varepsilon] + G [g + \gamma(f)\varepsilon] + B [b + \beta(f)\varepsilon] \\ & = \phantom{ + } & R\{[r + \rho(f)\varepsilon] + [a + \alpha(f)\varepsilon]\} \\ & \phantom { = } + & G\{[(g + \gamma(f)\varepsilon] +[a + \alpha(f)\varepsilon]\} \\ & \phantom { = } + & B\{[(b + \beta(f)\varepsilon] +[a + \alpha(f)\varepsilon]\} =\mathbf{x} + \lambda, \lambda \in \mathfrak{O}. \hskip1cm\Box \end{array}$$ In the view of this lemma we introduce the following notion. Let $X \in \mathbb{A}$. We say that colour $\mathbf{x} \in \triangle$ is $X$-*polarized* if it can be expressed in the form $\mathbf{x} = X [x +\xi(f) \varepsilon]$, $[x +\xi(f) \varepsilon] \in T$. The congruence given in Lemma \[congruence\] is known also as the *Cancellation law*, c.f. [@Gregor-Haluska1]). So, (this holds concerning all arithmetic operations in $\triangle$ we will define in the sequel of this paper). So, we operate with the accuracy of congruent triples of triangular coefficients. Physically Cancellation law means an ambiguity with respect to the achromatic parts of Colour. This is expressed with using of the phrase “with respect to Cancellation law”. For the sake of simplicity and without loss of precision, this phrase will be often omitted in the text. Arithmetic operations in $\triangle$ ==================================== Let us denote for the following sections: $$\begin{array}{rcl}\mathbf{x}& =& \{ R[r + \rho(f)\varepsilon] + G [g + \gamma(f)\varepsilon] + B [b + \beta(f)\varepsilon]\} \in \triangle, \\ \mathbf{y} & = & \{R [u + \sigma(f)\varepsilon] + G[v + \chi(f)\varepsilon] + B[t + \xi(f)\varepsilon]\} \in \triangle \end{array}$$ be two colours. Addition in $\triangle$ (*Mixing of Colours*) --------------------------------------------- We define: $$\begin{array}{rcl}\mathbf{x} \oplus \mathbf{y}& := & \phantom{+} R\{[r + \rho(f)\varepsilon] + [u + \sigma(f)\varepsilon]\} \\ & &+ G\{[g + \gamma(f)\varepsilon] + [v + \chi(f)\varepsilon] \} \\ & & + B\{[b+ \beta(f)\varepsilon] + [t + \xi(f)\varepsilon]\},\\ & = & \phantom{+} R[\{r + u\} + \{ \rho(f) + \sigma(f) \}\varepsilon] \\ & &+ G[\{g + v\} + \{ \gamma(f) + \chi(f)\}\varepsilon] \\ & & + B[\{b+ t\} + \{\beta(f) + \xi(f)\}\varepsilon]. \end{array}$$ Remind, that the result of operation of addition is with respect to Cancellation law, i.e., for every $\lambda_1, \lambda_2, \lambda_3 \in \mathfrak{O},$ $$\mathbf{x} \oplus \mathbf{y} = (\mathbf{x} + \lambda_1 )\oplus (\mathbf{y} + \lambda_2) = (\mathbf{x} \oplus \mathbf{y}) + \lambda_3.$$ Subtraction in $\triangle$ (*Inverse colours*) ---------------------------------------------- We define Subtraction in $\triangle$ as the addition of inverse elements, $$\mathbf{x} \ominus \mathbf{y} := \mathbf{x} \oplus (\ominus \mathbf{y}).$$ The inverse elements of the basic colours are defined as follows: $$\begin{array}{ccl} \ominus R [r + \rho(f)\varepsilon] & := &G [r + \rho(f)\varepsilon] + B[r + \rho(f)\varepsilon],\\ \ominus G [g + \gamma(f)\varepsilon] & := &R [g + \gamma(f)\varepsilon] + B[g + \gamma(f)\varepsilon],\\ \ominus B [b+ \beta(f)\varepsilon] & := &R [b+ \beta(f)\varepsilon] + G[b+ \beta(f)\varepsilon]. \end{array}$$ So, $$\begin{array} {rcl}\ominus\mathbf{x} & := & \phantom{ + } R \{ [g + \gamma(f)\varepsilon] + [b+ \beta(f)\varepsilon]\} \\ && + G\{[r + \rho(f)\varepsilon] + [b+ \beta(f)\varepsilon]\} \\ && + B\{ [r + \rho(f)\varepsilon] + [g + \gamma(f)\varepsilon] \}. \end{array}$$ And the subtraction is defined then as following: $$\begin{array}{ccl} \mathbf{x}\ominus \mathbf{y} &=& \phantom{\ominus} \{R[r + \rho(f)\varepsilon] + [g + \gamma(f)\varepsilon] G+ B[b+ \beta(f)\varepsilon]\} \\ & & \ominus \{R[u + \sigma(f)\varepsilon] + G[v + \chi(f)\varepsilon] + B[t + \xi(f)\varepsilon]\} \\ &&\\ & := & \phantom{\oplus} \{R[r + \rho(f)\varepsilon] + G[g + \gamma(f)\varepsilon] + B[b+ \beta(f)\varepsilon]\} \\ & & \oplus \{ G[u + \sigma(f)\varepsilon] + B[u + \sigma(f)\varepsilon]\} \\ & & \oplus \{ R[v + \chi(f)\varepsilon] +B[v + \chi(f)\varepsilon] \} \\ & & \oplus \{ R[t + \xi(f)\varepsilon] + G[t + \xi(f)\varepsilon]\} \\ && \\ & = & \phantom{ + }R\{ [r + \rho(f)\varepsilon] + [v + \chi(f)\varepsilon] +[t + \xi(f)\varepsilon] \} \\ & & + G\{[g + \gamma(f)\varepsilon] +[u + \sigma(f)\varepsilon] + [t + \xi(f)\varepsilon] \} \\ & & + B\{[b+ \beta(f)\varepsilon] +[u + \sigma(f)\varepsilon] + [v + \chi(f)\varepsilon] \}. \end{array}$$ We can replace polar operators $R,G,B$ with their *inverse operators* $$C := \ominus R =-1, M:= \ominus G = 1/2 - (\sqrt{3}/2))\imath, Y := \ominus B = 1/2 + (\sqrt{3}/2))\imath.$$ This way we obtain the $C,M,Y$ (cyan - magenta - yellow) colour scheme. These colour systems are mathematically equivalent, but to White should correspond Black as the inverse Colour. But Black does not physically exist in the electro-magnetic spectrum as Colour (in the $RGB$ scheme, Black means an absence of energy). Therefore in praxis (e.g. in the print industry), Black is artificially added to the $CMY$ system to have $CMYK$ system (the character $K$ is added as the abbreviation for Black). Subsuming the previous two sections, the following lemma holds: The triple $(\triangle, \oplus, \mathfrak{O})$ is an Abel additive group with respect to Cancellation law. Multiplication in $\triangle$ *New transformations of Colours* =============================================================== For simple mixing of two Colours, it is enough to deal with the additive group of Colours. However, there are theoretical transformations of Colours which can be called as multiplications according their mathematical properties. The author did not know about any appearance of these operations in the praxis. However, using a computer digitalization, this theory enables, we can artificially produce and explore these Colours. This section is about multiplication of Colours in the Colour space $\triangle$ and about division in the factorized Colour space $\triangle | \mathfrak{S}$ where $\mathfrak{S}$ is the ideal of singular elements for division of Colours, cf. Section \[greyideal\]. Cyclic compositions of polar operators in $\mathbb{C}$ {#squares} ------------------------------------------------------ It is easy to see that for the number 3 ($R, G, B$), there are possible six symmetric Latin squares which respectively yield 6 commutative operations (“multiplications”) $\otimes_i:\mathbb{A} \times \mathbb{A} \to \mathbb{A}, i= 1,2,\dots, 6$ of Colours. $$\begin{array}{c} \begin{array}{c|ccc} \otimes_1 &R&G&B\\ \hline R&R&G&B\\ G&G&B&R\\ B&B&R&G \end{array}, \hskip5mm \begin{array}{c|ccc} \otimes_2 &R&G&B\\ \hline R&G&B&R\\ G&B&R&G\\ B&R&G&B \end{array}, \hskip5mm \begin{array}{c|ccc} \otimes_3 &R&G&B\\ \hline R&B&R&G\\ G&R&G&B\\ B&G&B&R \end{array}, \\ \\ \begin{array}{c|ccc} \otimes_4 &R&G&B\\ \hline R&R&B&G\\ G&B&G&R\\ B&G&R&B \end{array}, \hskip5mm \begin{array}{c|ccc} \otimes_5 &R&G&B\\ \hline R&B&G&R\\ G&G&R&B\\ B&R&B&G \end{array}, \hskip5mm \begin{array}{c|ccc} \otimes_6 &R&G&B\\ \hline R&G&R&B\\ G&R&B&G\\ B&B&G&R \end{array}. \end{array}$$ In this paper, we will deal only with commutative operations $\otimes$ which have the property $\exists Y \in \mathbb{A}; \forall X \in \mathbb{A} \mid Y \otimes X = X$. Such are cases $\otimes_1$, $\otimes_2$, $\otimes_3$. For the reason of cyclic change, we will only deal with the Latin square $\otimes = \otimes_1$. So, let us define the compositions of poles $\otimes:\mathbb{A} \otimes \mathbb{A} \to \mathbb{A}$ with the following Latin square table. $$\begin{array}{c|ccc} \otimes &R&G&B\\ \hline R&R&G&B\\ G&G&B&R\\ B&B&R&G \end{array}$$ The operation of multiplication $\odot$ in $\triangle$ is defined in accord both with the table of the composition $\otimes $ and the multiplication in the semi field $T$. Namely, $$\mathbf{x} \odot \mathbf{y} = \left(\sum_{i=1,2,3} A ^{(i)}[x_i + \xi_i(f)\varepsilon]\right) \odot \left(\sum_{j =1,2,3} A ^{(j)}[y_j + \chi_j(f)\varepsilon]\right)$$ $$:= \sum_{i=1,2,3}\sum_{j=1,2,3}\left[ A ^{(i)}\otimes A ^{(j)} \right] \left\{[x_i + \xi_i(f)\varepsilon] \cdot [y_j + \chi_j(f)\varepsilon] \right\}$$ $$= \sum_{i=1,2,3}\sum_{j=1,2,3}\left[A ^{(i)}\otimes A ^{(j)}\right] [x_i y_j + \{x_i \chi_j(f) + y_j\xi_i(f)\}\varepsilon],\label{star}$$ where $A ^{(i)}, A ^{(j)} \in \mathbb{A}$, $i,j =1,2,3$; $\mathbf{x} = R [x_i + \xi_i(f)\varepsilon] + G[x_2 + \xi_2(f)\varepsilon] + B[x_3 +\xi_3(f)\varepsilon] \in \triangle$; $\mathbf{y} = R[y_1+\chi_1(f)\varepsilon] + G [y_2 + \chi_2(f)\varepsilon] + G [y_3 + \chi_3(f)\varepsilon] \in \triangle$. Note that multiplication in Equation (\[star\]) is parabolic complex. The proof of the following lemma is evident.   1. The result of operation of multiplication is with respect to Cancellation law, i.e., $$\mathbf{x \odot y} = (\mathbf{x} \oplus \lambda_1) \odot (\mathbf{y} \oplus \lambda_2) \oplus \lambda_3,$$ for every $\lambda_1, \lambda_2, \lambda_3 \in \mathfrak{O}$. 2. The element $$\mathbf{1} := R[1 + 0(f)\varepsilon] + G[0 + 0(f)\varepsilon] + B[0 +0(f)\varepsilon] \in \triangle$$ is an *unit element* for the operation of multiplication in $\triangle$ (and hence also $\mathbf{1} + \lambda$, $\lambda \in \mathfrak{O}$, with respect to the congruence given with Cancellation law). Conjugation in $\triangle$ *(Polarization of the light)* -------------------------------------------------------- To define an operation of division, we introduce an operation of *conjugation*. Conjugation physically means a polarization of light; according to the chosen Latin square $\otimes_1$, a projection will be done to the $R$-axis. *Let $\mathbf{x} = R[r + \rho(f)\varepsilon] + G[g + \gamma(f)\varepsilon] + B[b+ \beta(f)\varepsilon] \in \triangle$.* We say that an element $\overline{\mathbf{x}} \in \triangle$ is a *conjugation* of the element $\mathbf{x}$ if $$\overline{\mathbf{x}} := R[r + \rho(f)\varepsilon]+ G [b+ \beta(f)\varepsilon] + B [g + \gamma(f)\varepsilon] \in \triangle.$$ \[4.3\] Let $\mathbf{x} \in \triangle$ be as in previous definition, let $$\mathbf{y} = [R(u + \sigma(f)\varepsilon) + G(v + \chi(f)\varepsilon)] + B(t + \xi(f)\varepsilon) \in \triangle.$$ Then 1. $\overline{\overline{\mathbf{x}}} = \mathbf{x} \in \triangle$, 2. $\overline{\mathbf{x}\oplus \mathbf{y}} = \overline{\mathbf{x}} \oplus \overline{\mathbf{y}} \in \triangle,$ 3. $\overline{\mathbf{x}\odot \mathbf{y}} = \overline{\mathbf{x}} \odot \overline{\mathbf{y}} \in \triangle$, 4. $ \mathbf{y}\odot \overline{\mathbf{y}} = R \Theta$, where $$\begin{array}{cl} \Theta = & \Big[ \frac{ ( u - v)^2 + (v- t)^2 + (t - u)^2}{2} \\ & + \{(u-v)(\sigma(f) - \xi(f)) \\ & + (v-t)(\xi(f)- \chi(f)) + (t-u)(\chi(f) - \xi(f))\}\varepsilon \Big] \in T. \end{array}$$ [[**Proof.**   ]{}]{}The proofs of items 1., 2., 3. are exercises in algebra, we let them to the reader. We prove the last statement 4. We have: $$\begin{array}{rcl} \mathbf{y} \odot \overline {\mathbf{y}} & = & \phantom{ \odot } \{R[u + \sigma(f)\varepsilon] + G[v + \chi(f)\varepsilon] + B[t + \xi(f)\varepsilon]\} \\ && \odot \{R[u + \sigma(f)\varepsilon] + G[t + \xi(f)\varepsilon] + B[v + \chi(f)\varepsilon]\} = \end{array}$$ using the composition table of poles and the distributive law, we continue: $$= R\big\{[u + \sigma(f)\varepsilon]^2 + [v + \chi(f)\varepsilon]^2 + [t + \xi(f)\varepsilon]^2\big\}$$ $$+ G\big\{[u + \sigma(f)\varepsilon][v + \chi(f)\varepsilon] + [u + \sigma(f)\varepsilon][t + \xi(f)\varepsilon] + [v + \chi(f)\varepsilon] [t + \xi(f)\varepsilon]\big\}$$ $$+ B\big\{ [u + \sigma(f)\varepsilon][v + \chi(f)\varepsilon] + [u + \sigma(f)\varepsilon] [t + \xi(f)\varepsilon] + [v + \chi(f)\varepsilon][t + \xi(f)\varepsilon]\big\}=$$ By the definition of subtraction, $$=R \Big[ \{[u + \sigma(f)\varepsilon]^2 + [v + \chi(f)\varepsilon]^2 + [t + \xi(f)\varepsilon]^2\}$$ $$- \{[u + \sigma(f)\varepsilon][v + \chi(f)\varepsilon]$$ $$+ [u + \sigma(f)\varepsilon] [t + \xi(f)\varepsilon]$$ $$+ [v + \chi(f)\varepsilon] [t + \xi(f)\varepsilon]\} \Big] =$$ Now, we have to show that this is a $R$-polarized element. Indeed, we continue: $$\begin{array}{c} = R \Big\{[u + \sigma(f)\varepsilon]^2 + [v + \chi(f)\varepsilon]^2 + [t + \xi(f)\varepsilon] ^2 \\ - [u + \sigma(f)\varepsilon] [v + \chi(f)\varepsilon] \\ - [v + \chi(f)\varepsilon] [t + \xi(f)\varepsilon] \\ - [t + \xi(f)\varepsilon] [u + \sigma(f)\varepsilon]\Big\} =\label{sucet} \end{array}$$ $$= R \Big\{ \left[\frac{1}{2}[u + \sigma(f)\varepsilon] ^2 - [u + \sigma(f)\varepsilon] [v + \chi(f)\varepsilon] + \frac{1}{2}[v + \chi(f)\varepsilon] ^2\right]$$ $$+ \left[\frac{1}{2} [u + \sigma(f)\varepsilon] ^2 - [u + \sigma(f)\varepsilon] [t + \xi(f)\varepsilon]+ \frac{1}{2} [t + \xi(f)\varepsilon] ^2 \right]$$ $$+ \left[\frac{1}{2} [v + \chi(f)\varepsilon] ^2 - [v + \chi(f)\varepsilon] [t + \xi(f)\varepsilon] + \frac{1}{2} [t + \xi(f)\varepsilon] ^2 \right] \Big\}$$ $$\label{==}\begin{array} {cl} = \frac {R}{2} \{ & \Big[ [u + \sigma(f)\varepsilon] -[v + \chi(f)\varepsilon] \big]^2 \\ & + \big[ [u + \sigma(f)\varepsilon] - [t + \xi(f)\varepsilon] \big]^2 \\ & +\big[ [v + \chi(f)\varepsilon] - [t + \xi(f)\varepsilon] \big]^2 \Big\} \end{array}$$ $$\begin{array}{rl} = \frac{R}{2} & \Big\{ [\{u - v\}+ \{\sigma(f) - \chi(f)\}\varepsilon]^2 + [\{v - t\} + \{ \chi(f) - \xi(f)\}\varepsilon ]^2 \\ & + [\{t - u \} +\{\xi(f) - \sigma(f))\}\varepsilon ]^2 \Big\} \end {array}$$ $$\label{H}\begin{array}{cl} = R & \Big[\frac{( u - v)^2 + (v- t)^2 + (t - u)^2}{2} \\ & + \{(u-v)(\sigma(f) - \xi(f)) \\ & + (v-t)(\xi(f)- \chi(f)) + (t-u)(\chi(f) - \xi(f))\}\varepsilon \Big], \end{array}$$ Since $( u - v)^2 + (v- t)^2 + (t - u)^2 \geq 0$, then $\mathbf{y}\odot \overline{\mathbf{y}} = R \Theta$, where $\Theta \in T$. [$\Box\;\;$ ]{} The ideal $\mathfrak{S}$\[greyideal\] ------------------------------------- We proved that $\mathbf{y} \odot \overline{\mathbf{y}}$ is an $R$-polarized element in $\triangle$. But what about elements $\mathbf{y} \in \triangle$ such that $\mathbf{y} \odot \overline{\mathbf{y}} \in \mathfrak{O}$ and $\mathbf{y} \not\in \mathfrak{O}$? An element $\mathbf{y} \in \triangle$ such that $\mathbf{y} \odot \overline{\mathbf{y}} = \mathfrak{O}$ and $\mathbf{y} \neq \mathfrak{O}$ is called a *singular element*. The set of all singular elements (including elements of $\mathfrak{O}$) is denoted by $\mathfrak{H}$. \[5.5\] $$\mathfrak{H} = \mathfrak{S}.$$ [[**Proof.**   ]{}]{} Let $\mathbf{y} = R [u + \sigma(f)\varepsilon] + G [v + \chi(f)\varepsilon] + B [t + \xi(f)\varepsilon]\in \triangle$. From Equation (\[H\]) it follows that $\mathbf{y}\odot \overline{\mathbf{y}} \in \mathfrak{O}$ if and only if $u = v = t$ for arbitrary real functions $\sigma(f), \chi(f), \xi(f) \in \mathbb{R}^{[0, + \infty)}$, and every $u = v = t \geq 0$. [$\Box\;\;$ ]{} Since the result $t=u=v$ of the previous Lemma proof is symmetrical and the analogical result can be obtained using with the cyclic change for the Latin squares $(\otimes_1), \otimes_2, \otimes_3$, the ideal $\mathfrak{S}$ of all singular elements for divisions derived from matrices $\otimes_1, \otimes_2, \otimes_3$ is the same. $\mathfrak{S}$ is a two-sided ideal in the ring $(\triangle, \oplus, \odot, \mathfrak{O})$ with respect to operation of multiplication $\odot$ with respect to Cancellation law. [[**Proof.**   ]{}]{}We have to proof: $$\mathfrak{O} \varsubsetneqq \mathfrak{S} \varsubsetneqq \triangle,$$ $$\mathbf{x} \in \mathfrak{S}\ \& \ \mathbf{y} \in \mathfrak{S} \implies \mathbf{x} + \mathbf{y} \in \mathfrak{S},$$ and $$\mathbf{x} \in \mathfrak{S} \ \& \ \mathbf{y} \in \triangle \implies \mathbf{x} \odot \mathbf{y} \in \mathfrak{S}.$$ The first two assertions are evident. Prove the third assertion. Let $\mathbf{x} = R[a + \rho(f)\varepsilon] + G[a + \gamma(f)\varepsilon] + B[a+ \beta(f)\varepsilon] \in \mathfrak{S}$ and let $\mathbf{y} = R [u + \sigma(f)\varepsilon] + G [v + \chi(f)\varepsilon] + B [t + \xi(f)\varepsilon] \in \triangle$. We have: $$\mathbf{x} \odot \mathbf{y} = \{R[a + \rho(f)\varepsilon] + G[a + \gamma(f)\varepsilon] + B[a+ \beta(f)\varepsilon] \}$$ $$\odot \{R [u + \sigma(f)\varepsilon] + G [v + \chi(f)\varepsilon] + B [t + \xi(f)\varepsilon]\} =$$ since $\mathbf{x} \in \mathfrak{S}$, $$= \{R\rho(f)\varepsilon + G\gamma(f)\varepsilon + B\beta(f)\varepsilon \} \odot \{R [u + \sigma(f)\varepsilon] + G [v + \chi(f)\varepsilon] + B [t + \xi(f)\varepsilon]\} =$$ since $\varepsilon^2=0$, $$= \varepsilon \{ R\rho(f) + G\gamma(f) + B\beta(f) \} \odot \{R u + G v + B t \} =$$ using the composition table of poles, $$= \varepsilon \{ R [u\rho(f) + v \beta(f) + t \gamma(f)] + G[v\rho(f) + u\gamma(f) +t\beta(f)] +B[t\rho(f) + v\gamma(f) + u\beta(f)]\}.$$ We obtained an element in $\mathfrak{S}$. [$\Box\;\;$ ]{} Division in the factor space $\triangle | \mathfrak{S}$ -------------------------------------------------------- Let $\mathbf{x,y} \in \triangle$ and $\mathbf{y} \notin \mathfrak{S}$. The unit element $\mathbf {1} = R[1+0(f)\varepsilon] + G[0+0(f)\varepsilon] + B[0+0(f)\varepsilon]$ is a mathematical abstraction. In the real world, there is no tristimulus of this kind. However, using this theoretical object $\mathbf {1}$, we are able to divide real Colours. Division is defined as follows: $$\mathbf{x} \oslash \mathbf{y} :=(\mathbf{x} \odot \mathbf{1}) \oslash \mathbf{y} = \mathbf{x} \odot (\mathbf{1} \oslash \mathbf{y}).$$ Find the element $ \mathbf{1} \oslash \mathbf{y} \in \triangle$ if it exists. \[4.7\] Let $\mathbf{y} = R[u + \sigma(f)\varepsilon] + G[v + \chi(f)\varepsilon] + B[t + \xi(f)\varepsilon] \not\in \mathfrak{S}$. Then $$\mathbf{1} \oslash \mathbf{y} = \{ R [u + \sigma(f)\varepsilon] + G[t + \xi(f)\varepsilon] + B[v + \chi(f)\varepsilon]\} \cdot (2/\Theta),$$ where $\Theta$ is defined in (\[H\]). [[**Proof.**   ]{}]{}By Theorem \[4.3\], the item 4., and Lemma \[5.5\], $$\mathbf{1} \oslash \mathbf{y} = \frac{\overline{\mathbf{y}}} {\mathbf{y} \odot \overline{\mathbf{y}}}. \ \ \Box$$ From Theorem \[4.7\] it follows that we can divide every colour by another colour but elements in the ideal $\mathfrak{S}$. Compatibility of the additive and multiplicative structures of $\triangle$ -------------------------------------------------------------------------- To be complete in verifying of the axioms of the field, we bring the following more-less evident lemma. 1. $\mathbf{1} \not\in \mathfrak{S}$; 2. $\mathbf{1} \oslash \mathbf{1} = \mathbf{1}$; 3. Let $\lambda \in \mathfrak{O}$. Then $\mathbf{1} \neq \lambda$; 4. Let $\mathbf{x,y,z} \in \triangle$. Then $$\mathbf{x} \odot (\mathbf{y} \oplus \mathbf{z}) = (\mathbf{x} \odot \mathbf{y}) \oplus (\mathbf{y} \odot \mathbf{z}).$$ [[**Proof.**   ]{}]{}(1)(2) The first and second items are trivial. (3)Let $[a + \alpha(f)\varepsilon] \in T$. Then $$\mathbf{1} = R[(1+a) + \alpha(f)\varepsilon] + G[a + \alpha(f)\varepsilon] + B[a + \alpha(f)\varepsilon]$$ $$\neq R[a + \alpha(f)\varepsilon] + G[a + \alpha(f)\varepsilon] + G [a + \alpha(f)\varepsilon] \in \mathfrak{O}.$$ \(4) The fourth item (distributive law) is ensured by construction of the additive $\oplus$ and multiplicative $\odot$ operations in the set $\triangle$. The verification of this equation is elementary and we let it to the reader as an exercise. [$\Box\;\;$ ]{} Mathematical subsuming ====================== We collect mathematical results of the paper into the following theorem. Let $\mathbb{A}\subset \mathbb{C}$ be a set of three poles. Let $T$ be a semi-field of triangular coefficients, cf. Subsection \[coef\]. For a fixed Latin square $\otimes\in \{\otimes_1, \otimes_2, \otimes_3\}$, cf. Section \[squares\], the system $(\triangle$, $\oplus$, $\odot$, $\mathfrak{O}$, $\mathbf{1})$ (called the Colour space) is an commutative Abel ring (with respect to Cancellation law congruence) and particular division, c.f. Subsection \[pooles\]. There are two Abel groups: an additive group $(\triangle, \oplus, \mathfrak{O})$ (with respect to Cancellation law). The second group $(\triangle | \mathfrak{S}, \odot, \mathbf{1} )$ (with respect to Cancellation law) is a multiplicative group with the unit $\mathbf{1}$, where the set $\mathfrak{S}$, see Lemma \[5.5\], is an ideal, cf. Section \[achroma\]. The additive and multiplicative groups are linked together by the distributive law of addition with respect to the multiplication/division commutatively from both sides. The structure $(\triangle | \mathfrak{S}, \oplus, \odot, \mathfrak{S}, \mathbf{1})$ is an field (with respect to Cancellation law). Conclusions =========== We created and described a mathematical theory of Colour space which factorized with the ideal $\mathfrak{S}$ is a tripolar $RGB$ field with respect to Cancellation law. This theory has practical and theoretical application to everything where the phenomenon Colour is sofisticated on the $RGB$ language. For practical applications, the model needs to include a more thorough theory of the achromatic part of Colour and also it supposed to take into account some corrections implied from the technical equipment limitations and human sensory distortions. [100]{} D. Briggs, The Dimensions of Color, e-book, Art gallery of New South Wales; Julian Ashton Art Gallery - Sydney; National Art School - Sydney; copyrighted for years 2007-2013, www.huevaluechroma.com J. S. Golan, Some recent applications of semiring theory. Int. Conf. on Algebra in memory of Kostia Beidar, National Cheng Kung University, Tainan, March 6–12, 2005, pp. 18. T. Gregor, Three-polar space over the semi-field of double numbers, Tatra Mount. Math. Publ. 61(2014), 167-173. T. Gregor, J. Haluška, Lexicographical ordering and field operations in the complex plane. Stud. Mat. 41(2014), 123–133. A. A. Harkin– J. B. Harkin, Geometry of general complex numbers. Mathematics magazine, 77(2004), 118–129. R. Hirsch, Exploring Colour Photography: A Complete Guide. Laurence King Publishing, 2004. ISBN 1-85669-420-8. R. W. G. Hunt, The Reproduction of Colour (6th ed.). Chichester UK: Wiley, IS & T Series in Imaging Science and Technology. 2004. ISBN 0-470-02425-9. M.E. Luna-Elizarrarás, M. Shapiro, D. C. Struppa, A. Vajiac Schmid, Bicomplex Numbers and their Elementary Functions. CUBO A Mathematical Journal 02(2012), 61–80. E. Ružický, A. Ferko, Computer graphics and image processing (in Slovak). Sapientia, Bratislava 1995, ISBN 80-967180-2-9, 325 pp. + ix. D. Pascale, A review of RGB color spaces ... from $xyY$ to $R'G'B'$, The Babel Color Co., Montreal 2002. (revised 2003). R. M. Soneira, Display technology shoot-out: comparing CRT, LCD, plasma and DLP displays, 1990–2005, Parts: Overview, I., II., III., IV. http:/ / www.displaymate.com (Part II: Gray-Scale and Color Accuracy); copyrighted for years 1990–2005. A. Stockman – D. H. Brainard: Color vision mechanisms. In M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. MacDonald, V. Mahajan, and E. van Stryland (Eds.), The Optical Society of America Handbook of Optics, 3rd edition, Volume III: Vision and Vision Optics., McGraw Hill, New York 2010. H. H. Thoen–M. J. How–T. H. Chiou –J. Marshall, A different form of color vision in mantis shrimp, J. Science 343 (2014), 411–413. E. M. Vechtomov, A. V. Cheraneva, Semifields and their properties, Jour. of Math. Sciences, Vol. 163, no. 6, 2009; translated from, Fundamentaľnaya i prikladnaya matematika, Vol. 14, no. 5, pp.3-54, 2008 (in Russian). A learning community for photographers Cambridge Colours, e-tutorial on Colour photography, ©2015, www.cambridgeincolour.com . [^1]: *Mathematics Subject Classification (2000): *92C55, 12E30. RGB representation of colour, polar colour space, parabolic complex numbers HSV theory This paper was supported by Grant VEGA 2/0178/14 and by the Slovak-Ukrainian joint research project “Vector valued measures and integration in polarized vector spaces”.** [^2]: Ján Haluška, Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, 040 00 Košice, Slovakia, e-mail: jhaluska@saske.sk [^3]: The so called Value, which is a term overtaken from the HSV Colour theory; not very apt for a mathematical theory.
--- abstract: 'In nature or societies, the power-law is present ubiquitously, and then it is important to investigate the characteristics of power-laws in the recent era of big data. In this paper we prove the superposition of non-identical stochastic processes with power-laws converges in density to a unique stable distribution. This property can be used to explain the universality of stable laws such that the sums of the logarithmic return of non-identical stock price fluctuations follow stable distributions.' author: - Masaru Shintani - Ken Umeno date: 'Aug 22, 2017' title: | Super Generalized Central Limit Theorem\ –Limit distributions for sums of non-identical random variables with power-laws– --- [*Introduction—.*]{}There are a lot of data that follow the power-laws in the world. Examples of recent studies include, but are not limited to the financial market [@mandelbrot1997variation; @mantegna1994stochastic; @mantegna1995scaling; @gopikrishnan1998inverse; @gabaix2006institutional; @denys2016universality; @tanaka2016statistical], the distribution of people’s assets [@druagulescu2001exponential], the distribution of waiting times between earthquakes occurring [@bak2002unified] and the dependence of the number of wars on its intensity [@roberts1998fractality]. It is then important to investigate the general characteristics of power-laws. In particular, as for the data in the financial market, Mandelbrot [@mandelbrot1997variation] firstly argued that the distribution of the price fluctuations of cotton follows a stable law. Since the 1990’s, there has been a controversy as to whether the central limit theorem or the generalized central limit theorem (GCLT) [@Kolmogorov] as sums of power-law distributions can be applied to the data of the logarithmic return of stock price fluctuations. In particular, Mantegna and Stanley argued that the logarithmic return follows a stable distribution with the power-law index $\alpha<2$ [@mantegna1994stochastic; @mantegna1995scaling], and later they denied their own argument by introducing the cubic laws ($\alpha=3$) [@gopikrishnan1998inverse]. Even recently, some researchers [@gabaix2006institutional; @denys2016universality; @tanaka2016statistical] have argued whether a distribution of the logarithmic returns follows power-laws with $\alpha>2$ or stable laws with $\alpha<2$. On the other hand, it is necessary to prepare very large data sets to elucidate true tail behavior of distributions [@weron2001levy]. In this respect, the recent study [@tanaka2016statistical] showed that the large and high-frequency arrowhead data of the Tokyo stock exchange (TSE) support stable laws with $1<\alpha<2$. In this study, we show that the sums of the logarithmic return of multiple stock price fluctuations follows stable laws, and it can be described from a theoretical background. We will extend the GCLT to sums of independent non-identical stochastic processes. We call this Super Generalized Central Limit Theorem (SGCLT). [*Summary of stable distributions and the GCLT—.*]{}A probability density function $S(x;\alpha,\beta,\gamma,\mu)$ of random variable $X$ following a stable distribution [@nolan2003stable] is defined with its characteristic function $\phi(t)$ as: $$\begin{aligned} S(x;\alpha,\beta,\gamma,\mu)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\phi(t)e^{-ixt}{\rm d}x,\end{aligned}$$ where $\phi(t;\alpha,\beta,\gamma,\mu)$ is expressed as: $$\begin{aligned} &\phi(t)=\exp{\{i\mu t-\gamma^\alpha|t|^{\alpha}(1-i\beta {\rm sgn}(t)w(\alpha,t))\}}\\ & w(\alpha,t) =\left\{\begin{aligned} &\tan\left(\pi\alpha/2\right) & \text{if} &\ \ \alpha\neq 1\\ &-2/\pi\log|t| & \text{if} & \ \ \alpha = 1. \end{aligned}\right. \end{aligned}$$ The parameters $\alpha,\beta,\gamma$ and $\mu$ are real constants satisfying $0<\alpha\le 2$, $-1\le \beta\le 1$, $\gamma>0$, and denote the indices for power-law in stable distributions, the skewness, the scale parameter and the location, respectively. When $\alpha=2$ and $\beta=0$, the probability density function obeys a normal distribution. Note that explicit forms of stable distributions are not known for general parameters $\alpha$ and $\beta$ except for a few cases such as the Cauchy distribution ($\alpha=1,\beta=0$). A stable random variable satisfies the following property for the scale and the location parameters. A random variable $X$ follows $S(\alpha,\beta,\gamma,\mu)$, when $$\begin{aligned} \label{eq:scaleandshift} X\overset{d}{=} \left\{ \begin{aligned} &\gamma X_0+\mu &\ \text{if} & \ \ \alpha\neq 1\\ &\gamma X_0+\mu+\frac{2}{\pi}\beta\gamma\ln \gamma &\ \text{if}& \ \ \alpha=1, \end{aligned} \right.\end{aligned}$$ where $X_0=S(\alpha,\beta,1,0)$. When the random variables $X_j$ satisfy $X_j \sim S(x;\alpha,\beta_j,\gamma_j,0)$, the superposition $Z_n=(X_1+\cdots+ X_n)/n^{\frac{1}{\alpha}}$ of independent random variables $\{X_j\}_{j=1,\cdots,n}$ that have [*different*]{} parameters except for $\alpha$ is also in the stable distribution family as: $$\begin{aligned} \label{eq:superposition} Z_n \sim S(\alpha,\hat{\beta},\hat{\gamma},\hat{\mu}),\end{aligned}$$ where the parameters $\hat{\beta},\hat{\gamma}$ and $\hat{\mu}$ are expressed as: $$\begin{aligned} & \displaystyle \hat{\beta}=\frac{\sum_{j=1}^{n}\beta_j\gamma_j^\alpha}{\sum_{j=1}^{n}\gamma_j^\alpha},\hat{\gamma}=\left\{\frac{\sum_{j=1}^{n}\gamma_j^\alpha}{n}\right\}^{\frac{1}{\alpha}}\ \text{and}\\ & \hat{\mu}= \left\{ \begin{array}{lll} 0 & \text{if} & \alpha\neq 1\\ -\frac{2\ln n}{n\pi}\sum_{j=1}^n\beta_j\gamma_j & \text{if} & \alpha=1. \end{array}\right.\end{aligned}$$ We can prove this immediately by the use of the characteristic function for the sums of random variables expressed as the product of their characteristic functions: $$\begin{aligned} \phi (t;\alpha,\hat{\beta},\hat{\gamma},\hat{\mu})=\prod_{j=1}^n\phi \left(t/n^{\frac{1}{\alpha}};\alpha,\beta_j,\gamma_j,0 \right).\end{aligned}$$ We focus on the GCLT. Let $f$ of $x$ be a probability density function of a random variable $X$ for $0<\alpha<2$: $$\begin{aligned} \label{eq:GCLTcondition} f(x) \simeq\left\{ \begin{aligned} & c_{+}x^{-(\alpha+1)} &\text{for} &\ \ x\to \infty\\%$0<\alpha<2$\\ & c_{-}|x|^{-(\alpha+1)} &\text{for} &\ \ x\to -\infty,%,$0<\alpha<2$ \end{aligned}\right.\end{aligned}$$ with $c_+,c_- >0$ being real constants. Then, according to the GCLT [@Kolmogorov], the superposition of independent, identically distributed random variables $X_1,\cdots,X_n$ converges in density to a unique stable distribution $S(x;\alpha,\beta,\gamma,0)$ for $n \to \infty$, that is $$\begin{aligned} \label{eq:GCLT} \begin{split} Y_n&=\frac{\sum_{i=1}^{n}X_i-A_n}{n^{\frac{1}{\alpha}}} \xrightarrow{d} S(\alpha,\beta,\gamma,0) \ \text{for} \ n\to \infty,\\ A_n &= \left\{ \begin{aligned} & 0& \text{if} & \ \ 0<\alpha<1\\ & n^2 \Im \ln (\varphi_X\left(1/n\right))& \text{if} &\ \ \alpha=1\\ & n\mathbb{E}[X] & \text{if} & \ \ 1<\alpha<2, \end{aligned}\right. \end{split}\end{aligned}$$ where $\varphi_X$ is a characteristic function of $X$ as the expected value of $\exp(itX)$, $\mathbb{E}[X]$ is the expectation value of $X$, $\Im$ is an imaginary part of the argument, and parameters $\beta$ and $\gamma$ are expressed as: $$\begin{aligned} \beta&=&\frac{c_{+}-c_{-}}{c_{+}+c_{-}}, \ \gamma=\left\{\frac{\pi(c_{+}+c_{-})}{2\alpha\sin(\frac{\pi\alpha}{2})\Gamma(\alpha)}\right\}^{\frac{1}{\alpha}},\end{aligned}$$ with $\Gamma$ being the Gamma function. When $\alpha= 2$, we obtain $\mu=\int xf(x){\rm d}x$, $\sigma^2=\int x^2 f(x){\rm d}x$ and the superposition $Y_n$ of the independent, identically distributed random variables converges in density to a normal distribution: $$\begin{aligned} Y_n&=&\frac{\sum_{i=1}^{n}X_i-n\mu}{\sqrt{n}\sigma} \xrightarrow{d} \mathcal{N}(0,1), \ \text{for} \ n\to \infty.\end{aligned}$$ [*Our generalization—.*]{}We consider an extension of this existing theorem for sums of non-identical random variables. In what follows we assume that the random variables $\{X_i\}_{i=1,\cdots,n}$ satisfy the following two conditions. (Condition 1): The random variables $C_{+}>0$, $C_{-}>0$ obey respectively the distributions ${\rm P}_{c_+}(c)$, ${\rm P}_{c_-}(c)$, and satisfy $\mathbb{E}[C_+]<\infty$, $\mathbb{E}[C_-]<\infty$. (Condition 2): The probability distribution function $f_i(x)$ of the random variables $X_i$ satisfies in $0<\alpha<2$: $$\begin{aligned} \label{eq:SGCLTcondition} f_i(x) \simeq\left\{ \begin{aligned} & c_{+i}x^{-(\alpha+1)} &\text{for} &\ \ x\to \infty\\%$0<\alpha<2$\\ & c_{-i}|x|^{-(\alpha+1)} &\text{for} &\ \ x\to -\infty,%,$0<\alpha<2$ \end{aligned}\right.\end{aligned}$$ where $c_{+i}$ and $c_{-i}$ are samples obtained by $C_{+}$ and $C_{-}$. We emphasize that the probability distribution function may not be obtained even when we integrate $f_i(x)$ over $c_{+i}$ and $c_{-i}$. The main claim of this paper is the following generalization of GCLT: The following superposition $S_n$ of [*non-identical*]{} random variables with power-laws converges in density to a [*unique stable*]{} distribution $S(x;\alpha,\beta^*,\gamma^*,0)$ for $n \to \infty$, where $$\begin{aligned} \label{eq:SGCLT} \begin{aligned} S_n&=\frac{\sum_{i=1}^{n}X_i-A_n}{n^{\frac{1}{\alpha}}} \xrightarrow{d} S(x;\alpha,\beta^*,\gamma^*,0) \ \ \text{for} \ n\to \infty,\\ % A_n &= \left\{ % \begin{aligned} % & 0& \text{if} & \ \ 0<\alpha<1\\ % & n\sum_{i=1}^n \Im \ln (\varphi_i\left(1/n\right))& % \text{if} &\ \ \alpha=1\\ % & \sum_{i=1}^n E[X_i]& \text{if} & \ \ 1<\alpha<2, % \end{aligned}\right. A_n &= \left\{ \begin{array}{lll} 0& \text{if} & \ 0<\alpha<1\\ n\sum_{i=1}^n \Im \ln (\varphi_i\left(1/n\right))& \text{if} &\ \alpha=1\\ \sum_{i=1}^n \mathbb{E}[X_i]& \text{if} & \ 1<\alpha<2, \end{array}\right. \end{aligned}\end{aligned}$$ with $\varphi_i$ being a characteristic function of $X_i$ as the expected value of $\exp(itX_i)$, and parameters $\beta^*,\gamma^*,\beta_i,\gamma_i$ are expressed as: $$\begin{aligned} \beta^*&=&\frac{{\rm E}_{C_+,C_-}[\beta_i \gamma_i^\alpha]}{{\rm E}_{C_+,C_-}[\gamma_i^\alpha]}, \ \ \gamma^*=\left\{{\rm E}_{C_+,C_-}[\gamma_i^\alpha]\right\}^{\frac{1}{\alpha}},\\%\left(=E[\gamma_i]\right)\\ \beta_i&=&\frac{c_{+i}-c_{-i}}{c_{+i}+c_{-i}}, \ \ \gamma_i=\left\{\frac{\pi(c_{+i}+c_{-i})}{2\alpha\sin(\frac{\pi\alpha}{2})\Gamma(\alpha)}\right\}^{\frac{1}{\alpha}}.\end{aligned}$$ Here ${\rm E}_{C_+,C_-}[X]$ denotes the expectation value of $X$ with respect to random parameter distributions ${\rm P}_{c_{+}}$ and ${\rm P}_{c_{-}}$. [*Proof—.*]{}Although the following is not mathematically rigorous, we give the following intuitive proof. The probability distribution function of random variables $\{X_j\}_{j=1,\cdots,N}$ satisfying the Conditions 1-2 is expressed as: $$\begin{aligned} f_{j}(x)\simeq \left\{ \begin{aligned} &c_{+j}x^{-(\alpha+1)} &\text{for} &\ x\to +\infty\\ &c_{-j}|x|^{-(\alpha+1)} &\text{for} &\ x\to -\infty, \end{aligned}\right.\end{aligned}$$ where $c_{+j}>0$ and $c_{-j}>0$ satisfy $\mathbb{E}[C_+]>0$ and $\mathbb{E}[C_-]>0$. The superposition $S_N$ is then defined as: $$\begin{aligned} \begin{split} S_N&=\frac{\sum_{j=1}^{N}X_j-A_N}{N^{\frac{1}{\alpha}}},\\ % A_N &= \left\{ % \begin{aligned} % & 0& \text{if} & \ \ 0<\alpha<1\\ % & N \sum_{j=1}^N \Im \ln (\varphi_j\left(1/N\right)) & % \text{if} & \ \ \alpha=1\\ % & \sum_{j=1}^NE[X_j] & \text{if} & \ \ 1<\alpha<2, % \end{aligned}\right. A_N &= \left\{ \begin{array}{lll} 0& \text{if} & \ 0<\alpha<1\\ N \sum_{j=1}^N \Im \ln (\varphi_j\left(1/N\right)) & \text{if} & \ \alpha=1\\ \sum_{j=1}^N\mathbb{E}[X_j] & \text{if} & \ 1<\alpha<2, \end{array}\right. \end{split}\end{aligned}$$ where $\varphi_j$ is a characteristic function of $X_j$. On the other hand, let $N'$ be $M\times N$ with some $M$, and $\{X_{ij}\}_{i=1,\cdots,M,j=1,\cdots,N}$ be samples given by the same parent to $X_j$ for each $j$. Then $\{X_{ij}\}_{i=1,\cdots,M,j=1,\cdots,N}$ are independent, identically distributed for $i=1,\cdots,M$ at a fixed index $j$. Then, we define the superposition $S_{N'}$ as follows: $$\begin{aligned} \begin{aligned} &S_{N'} = \frac{\sum_{i=1}^M\sum_{j=1}^{N} X_{ij}-A_{N'}}{N'^{\frac{1}{\alpha}}},\\ A_{N'} &= \left\{ \begin{array}{lll} 0& \text{if} &\ 0<\alpha<1\\ M^2N \sum_{j=1}^N \left( \Im \ln (\varphi_{j}\left(1/(MN))\right)\right) & \text{if} &\ \alpha=1\\ M\sum_{j=1}^N\mathbb{E}[X_j] & \text{if} &\ 1<\alpha<2. \end{array}\right. \end{aligned}\end{aligned}$$ Here, we do not consider the convergence of $S_N$ in density for $N \to \infty$, but consider the superposition $S_{N'}$ for $N'\to \infty$, since the superposition $S_N$ will converge to the same limiting distribution of $S_{N'}$ if $S_N$ converges in density. We focus on the convergence in density of $S_{N'}$ for $M\to\infty$ and $N\to\infty$ as follows. About the previous $A_{N'}$ in $S_{N'}$, we express it as $A_{N'}=\sum_{j=1}^{N}A_{M_{j}}$ with the following $A_{M_j}(j=1,\cdots,N)$, $$\begin{aligned} A_{M_{j}} &=& \left\{ \begin{aligned} & 0& \text{if} & \ \ 0<\alpha<1\\ & M^2N \Im \ln (\varphi_{j}\left(1/MN\right)) & \text{if} & \ \ \alpha=1\\ & M \mathbb{E}[X_{j}] & \text{if} & \ \ 1<\alpha<2. \end{aligned}\right.\end{aligned}$$ Here, the superposition $S_{N'}$ is described as: $$\begin{aligned} S_{N'} &=& \frac{\sum_{i=1}^M\sum_{j=1}^{N} X_{ij}-A_{N'}}{N'^{\frac{1}{\alpha}}}\\ &=& \frac{\frac{\sum_{i=1}^MX_{i1}-A_{M_{1}}}{M^{\frac{1}{\alpha}}}+\cdots+\frac{\sum_{i=1}^MX_{iN}-A_{M_{N}}}{M^{\frac{1}{\alpha}}}}{N^{\frac{1}{\alpha}}}.\end{aligned}$$ When $\alpha\neq 1$, let $Y_{M_j}$ be the superposition $\left(\sum_{i=1}^MX_{ij}-A_{M_{j}}\right)/{M^{\frac{1}{\alpha}}}$. Then, $Y_{M_j}$ converges in density to $S(\alpha,\beta_j,\gamma_j,0)$ for $M \to \infty$ according to the GCLT , that is $$\begin{aligned} Y_{M_j}=\frac{\sum_{i=1}^MX_{ij}-A_{M_{j}}}{M^{\frac{1}{\alpha}}} \overset{d}{\to} S(\alpha,\beta_j,\gamma_j,0) \ \text{for} \ M\to \infty,\end{aligned}$$ where $\beta_j$ and $\gamma_j$ are $$\begin{aligned} \beta_j&=&\frac{c_{+j}-c_{-j}}{c_{+j}+c_{-j}}, \ \gamma_j=\left\{\frac{\pi(c_{+j}+c_{-j})}{2\alpha\sin(\frac{\pi\alpha}{2})\Gamma(\alpha)}\right\}^{\frac{1}{\alpha}}. \end{aligned}$$ Thus, with the stable property , we obtain the convergence of the superposition $S_{N'}$ as follows: $$\begin{aligned} S_{N'} &=& \frac{\sum_{j=1}^NY_{M_j}}{N^{\frac{1}{\alpha}}}\\ &\overset{d}{\to}& \frac{\sum_{j=1}^NY_{j}}{N^{\frac{1}{\alpha}}} \ \ \text{for} \ M\to \infty , (Y_j\sim S(\alpha,\beta_j,\gamma_j,0)) \\ &\overset{d}{\to}& S(x;\alpha,\beta^*,\gamma^*,0) \ \ \text{for} \ N\to\infty,\end{aligned}$$ where $\beta^*$ and $\gamma^*$ are: $$\begin{aligned} \beta^*&=&\lim_{N\to \infty}\frac{\sum_{j=1}^{N}\beta_j\gamma_j^\alpha}{\sum_{j=1}^{N}\gamma_j^\alpha}\\ &=& \lim_{N\to \infty}\frac{\frac{1}{N}\sum_{j=1}^{N}\beta_j\gamma_j^\alpha}{\frac{1}{N}\sum_{j=1}^{N}\gamma_j^\alpha} = \frac{{\rm E}_{C_+,C_-}[\beta_j\gamma_j^\alpha]}{{\rm E}_{C_+,C_-}[\gamma_j^\alpha]},\\ \gamma^* &=& \lim_{N\to \infty}\left\{\frac{\sum_{j=1}^{N}\gamma_j^\alpha}{N}\right\}^{\frac{1}{\alpha}} = \left\{{\rm E}_{C_+,C_-}[\gamma_j^\alpha]\right\}^{\frac{1}{\alpha}}.\end{aligned}$$ This proves the superposition $S_{N'}$ converges in density to $S(\alpha,\beta^*,\gamma^*,0)$. Figure \[fig:concept\] illustrates the concept of this proof. \ [p[6.2em]{}|c|p[2em]{} c p[2em]{}|c|]{} & $X_{11}$ & & $\cdots$ & & $X_{1N}$\ [superposition]{} & $\vdots$ && $\ddots$ && $\vdots$\ & $X_{M1}$ && $\cdots$ && $X_{MN}$\ \ ------------------------ -------------------------------- -- ---------- -- --------------------------------      [$M\to \infty$]{} [$\downarrow$]{} [$\downarrow$]{} \[4pt\] $S(\alpha,\beta_1,\gamma_1,0)$ $\cdots$ $S(\alpha,\beta_N,\gamma_N,0)$ ------------------------ -------------------------------- -- ---------- -- -------------------------------- \ $$\begin{aligned} \underbrace{ S(\alpha,\beta_1,\gamma_1,0),\ \cdots \ ,S(\alpha,\beta_N,\gamma_N,0)}_{\text{superposition}} \\ \overset{N\to \infty}{\longrightarrow} S(\alpha,\beta^*,\gamma^*,0) \end{aligned}$$ As above, the superposition $S_{N'}$ of non-identical stochastic processes converges in density to a unique stable distribution. Since the limiting distribution of $S_{N'}$ is the same as that of $S_N$, $S_N$ also converges to $S(x,\alpha,\beta^*,\gamma^*,0)$. When $\alpha=1$, this statement does not hold because of dependence between $M$ and $N$ in $A_{M_j}$, but we find that the limit distribution of the superposition $S_N$ generally converges in density to $S(x;\alpha,\beta^*,\gamma^*,0)$ as can be seen in the following numerical examples. [*Numerical confirmation—.*]{}As below, we confirm the claim of SGCLT by some numerical experiments. To verify the main claim numerically, we use two kinds of test: two-samples Kolmogorov-Smirnov (KS) test [@stephens1974edf] and two-samples Anderson-Darling (AD) test [@anderson1952asymptotic] with 5% significance level. We generate two data by different methods, and see the $P\mathalpha{-}values$ of both of tests. Then, unless the null hypothesis is rejected, we judge the two data follow the same distribution. For the first data, we generate non-identical stochastic processes satisfying Conditions 1-2, and prepare the superposition obtained in the same way as . For the second data, we generate the random numbers that follow the stable distribution, where the first data will converge to the stable distribution according to . Note that we compare the superposition with not a cumulative distribution function but random numbers obtained from another numerical method described below since a cumulative distribution function of a stable distribution cannot be expressed explicitly except for a few cases. For the first data, let us consider the chaotic dynamical system $x_{n+1}=g(x_n)$, where $g(x)$ is defined [@umeno1998superposition] as follows for $0<\alpha <2$: $$\begin{aligned} g(x)=\left\{ \begin{array}{lll} \frac{1}{\delta_1^2|x|}\left(\frac{|\delta_1 x|^{2\alpha}-1}{2}\right)^{1/\alpha}& \text{for} & \ \ x>\frac{1}{\delta_1}\\ -\frac{1}{\delta_1\delta_2|x|}\left(\frac{1-|\delta_1 x|^{2\alpha}}{2}\right)^{1/\alpha}& \text{for} & \ \ 0<x<\frac{1}{\delta_1}\\ \frac{1}{\delta_1\delta_2|x|}\left(\frac{1-|\delta_2 x|^{2\alpha}}{2}\right)^{1/\alpha}& \text{for} & \ \ -\frac{1}{\delta_2}<x<0\\ -\frac{1}{\delta_2^2|x|}\left(\frac{|\delta_2 x|^{2\alpha}-1}{2}\right)^{1/\alpha}& \text{for} & \ \ x<-\frac{1}{\delta_2}. \end{array} \right.\end{aligned}$$ This mapping has a mixing property and an ergodic invariant density for almost all initial points $x_0$. One of the authors (KU) obtained the following explicit asymmetric power-law distribution as an invariant density [@umeno1998superposition]: $$\begin{aligned} \rho_{\alpha,\delta_1,\delta_2}(x) =\left\{ \begin{aligned} & \frac{\alpha\delta_1^\alpha x^{\alpha-1}}{\pi(1+\delta_1^{2\alpha}x^{2\alpha})} &\text{if} &\ \ x\ge 0\\%$0<\alpha<2$\\ & \frac{\alpha\delta_2^\alpha |x|^{\alpha-1}}{\pi(1+\delta_2^{2\alpha}|x|^{2\alpha})} &\text{if} &\ \ x<0.%,$0<\alpha<2$ \end{aligned}\right.\end{aligned}$$ This asymmetric distribution behaves as follows for $x \to \pm\infty$: $$\begin{aligned} \rho_{\alpha,\delta_1,\delta_2}(x) \simeq\left\{ \begin{aligned} & \frac{\alpha}{\pi\delta_1^\alpha}x^{-(\alpha+1)} & \text{for} & \ \ x \to +\infty\\ & \frac{\alpha}{\pi\delta_2^\alpha}|x|^{-(\alpha+1)} & \text{for} & \ \ x \to -\infty. \end{aligned}\right.\end{aligned}$$ This is exactly the same expression with the condition of GCLT for random variables in $X$. Then, putting the variables $\delta_1$ and $\delta_2$ be distributed, we can obtain various [*different*]{} distributions with the same power-laws. We regard the parameters $\delta_{1i}$ and $\delta_{2i}$ as random samples obtained from $\Delta_1$ and $\Delta_2$, where $\Delta_1$ and $\Delta_2$ obey ${\rm P}_{\delta_1}(\delta)$ and ${\rm P}_{\delta_2}(\delta)$, respectively. These are defined for $\delta>0$ with finite mean. Then the parameters $c_{+i}$ and $c_{-i}$ are given as $c_{+i}=\frac{\alpha}{\pi\delta_{1i}^\alpha}$ and $c_{-i}=\frac{\alpha}{\pi\delta_{2i}^\alpha}$, and $\mathbb{E}[C_+]<\infty$, $\mathbb{E}[C_-]<\infty$ are also satisfied since $\delta_{1i}, \delta_{2i}$ are not 0 and samples from some random variables $\Delta_1$ and $\Delta_2$ with finite mean. As above, we can get some stochastic processes satisfying the Conditions 1-2. For the second data, the random numbers generated with the following procedure follow a stable distribution [@chambers1976method]. Let $\Theta$ and $\Omega$ be independent random numbers: $\Theta$ uniformly distributed in $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$, $\Omega$ exponentially distributed with mean $1$. In addition, let $R$ be as follows: $$\begin{aligned} R=\left\{ \begin{matrix} \frac{\sin\left(\alpha(\theta_0+\Theta)\right)}{(\cos\left(\alpha\theta_0\right) \cos\Theta)^{1/\alpha}}\left[\frac{\cos((\alpha-1)\Theta)}{\Omega}\right]^{(1-\alpha)/\alpha} & (\alpha\neq 1)\\ \frac{2}{\pi}\left[(\frac{\pi}{2}+\beta\Theta)\tan\Theta-\beta\log\left(\frac{\frac{\pi}{2}\Omega\cos\Theta}{\frac{\pi}{2}+\beta\Theta}\right)\right] & (\alpha=1), \end{matrix} \right.\end{aligned}$$ for $0< \alpha \le 2$ where $\theta_0=\arctan(\beta\tan(\pi\alpha/2))$. Then it follows that $R \sim S(x;\alpha,\beta,1,0)$. We get arbitrary stable distributions by the use of the property about the scale parameter and the location. $P\mathalpha{-}value$ (KS test) $P\mathalpha{-}value$ (AD test) ------------- ------------------ ------------------ ------- -------- --------------------------------- --------------------------------- \*[$0.5$]{} $1$(const) $1$(const) 10000 50000 0.122 0.074 ${\rm U}(1,2)$ ${\rm U}(1,2)$ 1000 100000 0.561 0.413 ${\rm U}(0.5,1)$ ${\rm U}(1,2)$ 1000 100000 0.865 0.546 \*[$1$]{} $1$ $1$ 1000 100000 0.226 0.308 ${\rm U}(1,2)$ ${\rm U}(1,2)$ 1000 100000 0.741 0.497 ${\rm U}(0.5,1)$ ${\rm U}(1,2)$ 1000 100000 0.659 0.301 \*[$1.5$]{} $1$ $1$ 1000 100000 0.916 0.529 ${\rm U}(1,1.2)$ ${\rm U}(1,1.2)$ 10000 20000 0.768 0.548 ${\rm U}(0.5,1)$ ${\rm U}(1.5,2)$ 10000 30000 0.108 0.099 : $P-values$ of two tests[]{data-label="tab:p-value"} $\alpha$ random variables N L KS test AD test ------------- --- --- ------------------------------- ------ ------- --------- --------- \*[$0.5$]{} 3 1 $ X_i - i/N$ 2000 10000 0.136 0.110 3 1 $ X_i - \ \text{Crand}(0,1)$ 1000 10000 0.289 0.190 \*[$1$]{} 3 1 $ X_i - i/N $ 1000 10000 0.305 0.081 3 1 $ X_i - \ \text{Crand}(0,1)$ 2000 10000 0.145 0.093 $1.5$ 3 1 $ X_i - \ \text{Crand}(0,1)$ 1000 10000 0.371 0.286 ![Comparison of two probability densities: the superposition ($N=10^3$, $L=10^5$ for $\alpha\mathalpha{=}1,\Delta_1\mathalpha{\sim} {\rm U}(0.5,1),\Delta_2\mathalpha{\sim} {\rm U}(1,2)$) and a stable distribution ($L\mathalpha=10^5$ for $\alpha\mathalpha{=}1,\beta^*\mathalpha{=}1/3,\gamma^*\mathalpha{=}1$)[]{data-label="fig:example"}](Rplot23alpha1.eps){width="80mm"} With two data obtained accordingly, we see whether the superposition $S_N=(\sum_{i=1}^NX_i-A_N)/N^{1/\alpha}$ numerically converges in density to a stable distribution $S(x;\alpha,\beta^*,\gamma^*,0)$ or not. Table \[tab:p-value\] and \[tab:p-value2\] show $P\mathalpha{-}values$ of the KS test and the AD test for each $\alpha,\Delta_1,\Delta_2$. The constant $L$ is the length of the sequence and $N$ is the number of sequences used for the superposition. The meaning of ${\rm U}(a,b)$ is the uniform distribution in $(a,b)$. Figure \[fig:example\] illustrates an example of correspondence when $\alpha=1$. “Crand$(0,1)$” is the random numbers follow the standard Cauchy distribution. This case shows that the integral average of the probability distribution function with the Cauchy distribution is not uniquely determined. ![Image of the convergence process: The left figure shows some samples of random variables $X_i - \text{Crand}(0,1)$, where $\alpha=1, \delta_1=3, \delta_2=1$. The integration of them does not have an explicit expression because of the indefinite mean of the Cauchy distribution. However the sum (the right figure) converges to the $S(1,-0.5,2/3,0)$.[]{data-label="fig:convergence"}](convergence.eps){width="8cm"} As can be seen from Table \[tab:p-value\] and \[tab:p-value2\], we cannot reject the null hypothesis in any case for $\alpha$. In other words, the distribution of superposition $S_N$ and the stable distribution $ S(x; \alpha, \beta^*, \gamma^*, 0)$ are close enough in density according to our SGCLT. In Figure \[fig:convergence\], we can see that the superposition of non-identical distributed random variables converges. [*Conclusions—.*]{}We have further generalized the GCLT for the sums of independent [*non-identical*]{} stochastic processes with the same power-law index $\alpha$. Our main claim of SGCLT can have more general applications since the various type of different power-laws exist in nature. Thus, our SGCLT can support the argument on the ubiquitous nature of stable laws such that the logarithmic return of the multiple stock price fluctuations follow a stable distribution with $1<\alpha<2$ by regarding them as the sums of non-identical random variables with power-laws. Take the data of the stock market as an example. Then, for the case that the distribution of the logarithmic return of each stock price fluctuation have the almost same power-law exponents and different scale parameters $(c_+,c_-)$, we get some trends or indicators according to this SGCLT. The authors thank Dr. Shin-itiro Goto (Kyoto University) for stimulating discussions. [30]{} B. Mandelbrot, Journal of Business, **36**, 394 (1963) R. N. Mantegna, H. E. Stanley, Phys. Rev. Lett. **73**, 2946 (1994) R. N. Mantegna, H. E. Stanley, Nature, **376**, 46 (1995) P. Gopikrishnana, M. Meyer, L. A. N. Amaral, H.E. Stanley, Eur. Phys. J. B, **3**, 139 (1998) X. Gabaix, P. Gopikrishnan, V. Plerou, H. E. Stanley, The Quarterly Journal of Economics, **121**, 2, 461 (2006) M. Denys, T. Gubiec, R. Kutner, M. Jagielski, H. E. Stanley, Phys. Rev. E, **94**, 042305 (2016) M. Tanaka, IEICE Technical Report, **116**, 27 (2016) (In Japanese) A. Dr$\breve{\text{a}}$gulescu, V. M. Yakovenko, Physica A: Statistical Mechanics and its Applications, **299**, 213 (2001) P. Bak, K. Christensen, L. Danon, T. Scanlon, Phys. Rev. Lett. **88**, 178501 (2002) D. C. Roberts, D. L. Turcotte, Fractals, **6**, 351 (1998) B. V. Gnedenko and A. N. Kolmogorov, [*Limit Distributions for Sums of Independent Random Variables*]{} (Addison-Wesley, Reading, MA, 1954). R. Weron, International Journal of Modern Physics C, **12**, 2, 209 (2001) J. Nolan, [*Stable distributions: models for heavy-tailed data*]{}, (Birkhauser Boston, 2003) M. A. Stephens, Journal of the American Statistical Association, **69**, 730 (1974) T. W. Anderson, D. A. Darling, The Annals of Mathematical Statistics, 193 (1952) K. Umeno, Phys. Rev. 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--- abstract: | In [@bowen], Bowen showed that for an expansive system $(X,T)$ with specification and a potential $\phi$ with the Bowen property, the equilibrium state is unique and fully supported. We generalize that result by showing that the same conclusion holds for non-uniform versions of Bowen’s hypotheses in which constant parameters are replaced by any increasing unbounded functions $f(n)$ and $g(n)$ with sublogarithmic growth (in $n$). We prove results for two weakenings of specification; the first is non-uniform specification, based on a definition of Marcus in ([@marcusmonat]), and the second is a significantly weaker property which we call non-uniform transitivity. We prove uniqueness of the equilibrium state in the former case under the assumption that $\liminf_{n \rightarrow \infty} (f(n) + g(n))/\ln n = 0$, and in the latter case when $\lim_{n \rightarrow \infty} (f(n) + g(n))/\ln n = 0$. In the former case, we also prove that the unique equilibrium state has the K-property. It is known that when $f(n)/\ln n$ or $g(n)/\ln n$ is bounded from below, equilibrium states may not be unique, and so this work shows that logarithmic growth is in fact the optimal transition point below which uniqueness is guaranteed. Finally, we present some examples for which our results yield the first known proof of uniqueness of equilibrium state. address: | Ronnie Pavlov\ Department of Mathematics\ University of Denver\ 2280 S. Vine St.\ Denver, CO 80208 author: - Ronnie Pavlov bibliography: - 'gapspec.bib' title: 'On non-uniform specification and uniqueness of the equilibrium state in expansive systems' --- [^1] Introduction {#intro} ============ A central question in the theory of topological pressure is knowing when a dynamical system $(X,T)$ and potential $\phi: X \rightarrow \mathbb{R}$ admit a unique equilibrium state. Often, such proofs use as hypotheses a mixing/shadowing property for the system $(X,T)$ and a regularity condition on the potential $\phi$. One of the first and most important results of this type was proved by Bowen in [@bowen], using the hypotheses of expansiveness and specification on $(X,T)$ and the so-called Bowen property on $\phi$. Informally, $(X,T)$ is expansive if there exists a fixed distance $\delta$ so that any unequal points of $X$ will be separated by distance at least $\delta$ under some iterate of $T$. Specification is the ability, given arbitrarily many orbit segments, to find a periodic point of $X$ whose orbit “shadows” (meaning it stays within some small distance of) those orbit segments, with gaps dependent only on the desired shadowing distance. The Bowen property is simply boundedness (w.r.t. $n$) of the differences of the partial sums $S_n \phi(x) = \phi(x) + \phi(Tx) + \ldots + \phi(T^{n-1}x)$ over pairs $(x,y)$ whose first $n$ iterates under $T$ stay within some predetermined distance. (See Section \[defs\] for formal definitions.) Bowen’s theorem can then be stated as follows. [([@bowen])]{}\[bowenthm\] If $(X,T)$ is an expansive system with specification and $\phi$ is a Bowen potential, then $(X,T)$ has a unique equilibrium state for $\phi$, which is fully supported. The assumptions of specification for $(X,T)$ and the Bowen property for $\phi$ each have associated constant bounds independent of a parameter $n$; for specification there is the bound $f(n)$ on the gap size required between shadowing orbit segments of length $n$, and for the Bowen property there is the bound $g(n)$ on the associated variation of the $n$th partial sum. The main results of this work show that the same conclusions hold even for unbounded $f(n)$ and $g(n)$, as long as they grow sublogarithmically with $n$. We have results using two different versions of specification; the first is called non-uniform specification, and the second, much weaker, property is called non-uniform transitivity. \[mainthm\] If $(X,T)$ is an expansive dynamical system (with expansivity constant $\delta$) with non-uniform specification with gap bounds $f(n)$ (at scale $\delta$), $\phi$ is a potential with partial sum variation bounds $g(n)$ (at scale $\delta$), and $\liminf_{n \rightarrow \infty} \frac{f(n) + g(n)}{\ln n} = 0$, then $X$ has a unique equilibrium state for $\phi$, which is fully supported. \[mainthm2\] If $(X,T)$ is an expansive dynamical system (with expansivity constant $\delta$) with non-uniform transitivity with gap bounds $f(n)$ (at scale $\delta$), $\phi$ is a potential with partial sum variation bounds $g(n)$ (at scale $\delta$), and if $\lim_{n \rightarrow \infty} \frac{f(n) + g(n)}{\ln n} = 0$, then $X$ has a unique equilibrium state for $\phi$, which is fully supported. Though our hypotheses for these results are stated for scale $\delta$ equal to the expansivity constant, standard arguments (here given as Lemmas \[gapscaleindep\] and \[sumscaleindep\]) show that they are equivalent at any scale less than $\delta$. For non-invertible surjective $(X,T)$, there is a canonical way to create an invertible system $(X', T')$ called the natural extension. It’s well-known that the natural extension has the same simplex of invariant measures as that of the original system, and so Theorems \[mainthm\] and \[mainthm2\] can also be applied to any $(X,T)$ whose natural extension satisfies their hypotheses. In particular, we note that whenever $(X,T)$ is positively expansive (see Theorem 2.2.32(3) of [@aokibook]), its natural extension is expansive. We also prove results about preservation of these properties under expansive factors and products, which are unavoidably a bit technical, and so we postpone formal statements to Section \[scaleindep\]. In particular, the preservation under products that we prove allows us to use an argument of Ledrappier ([@ledrappier]) to prove strong properties of the equilibrium state from Theorem \[mainthm\]. \[Kcor\] For any $(X,T)$ and $\phi$ satisfying the hypotheses of Theorem \[mainthm\] and associated unique equilibrium state $\mu$, $(X,T,\mu)$ is a K-system. Since K-systems have positive entropy, this also answers a question of Climenhaga from [@vaughntowers] about so-called hyperbolic potentials. Following [@IRRL], a potential $\phi$ is said to be hyperbolic for $(X,T)$ if every equilibrium state has positive entropy. Climenhaga’s question was the following: Is there an axiomatic condition on a subshift $(X,T)$, weaker than specification (perhaps some form of non-uniform specification), guaranteeing that every Hölder potential on $(X,T)$ is hyperbolic? Is there such a condition that is preserved under passing to (subshift) factors? Since K-systems have positive entropy and Hölder potentials on subshifts are Bowen (see, for instance, [@CT1]), Corollary \[Kcor\] gives the following positive answer. \[hypcor\] The class of subshifts $(X,T)$ with non-uniform specification with gap bounds $f(n)$ satisfying $\liminf_{n \rightarrow \infty} f(n)/\ln n = 0$ is closed under (subshift) factors, and for any such subshift, every Hölder potential is hyperbolic. We also collect results from the literature which demonstrate that when one of $f,g$ is $0$ and the other quantity grows logarithmically, uniqueness of the equilibrium state is still not guaranteed; this shows that our hypotheses cannot be weakened by too much. Full shifts correspond to the case $f = 0$, and for those there is the following example, based on a classical example of Hofbauer from [@hofbauer]. \[negpotthm\] For every $\epsilon > 0$, there exists a potential $\phi$ on the full shift $(X,T)$ on $\{0,1\}$ with partial sum variation bounds $g(n) < (1 + \epsilon) \ln n$ where $(X,T)$ has multiple equilibrium states for $\phi$, whose supports are disjoint. The example in [@CL], the so-called Double Hofbauer model, was actually for one-sided full shifts, rather than the two-sided ones treated here. Briefly, they define the potential $\phi$ in terms of the largest nonnegative integer $n$ where $x(0) = x(1) \ldots = x(n)$. It is not hard to adapt this to a two-sided version, which satisfies Theorem \[negpotthm\], by instead choosing maximal $n$ for which $x(-n) = \ldots = x(n)$. In fact, this is essentially the idea behind our later Example \[potex\]. The case $g = 0$ corresponds to equilibrium states for constant $\phi$, i.e. measures of maximal entropy. There, we have the following result, proved independently in [@kwietniaketal] and [@pavlovspec]. \[oldnegthm\] For any positive increasing $f$ with $\liminf_{n \rightarrow \infty} \frac{f(n)}{\ln n} > 0$, there exists a subshift $(X,T)$ with non-uniform specification with gap bounds $f(n)$ and multiple ergodic measures of maximal entropy whose supports are disjoint. Theorems \[mainthm\] and \[oldnegthm\] completely answer the question of when non-uniform specification forces uniqueness of the measure of maximal entropy for expansive systems; this is the case if and only if $\liminf_{n \rightarrow \infty} \frac{f(n)}{\ln n} = 0$. The reader may notice that there is a gap in the results we’ve presented for full shifts; Theorem \[negpotthm\] shows that non-uniqueness can happen for $g(n)$ with $\lim_{n \rightarrow \infty} \frac{g(n)}{\ln n}$ arbitrarily close to $1$, and Theorems \[mainthm\] and \[mainthm2\] guarantee uniqueness only when $\frac{g(n)}{\ln n}$ approaches $0$ in general or along a subsequence. In fact, a careful reading of our proofs shows that these theorems hold as long as the limit or liminf of $\frac{g(n)}{\ln n}$ is smaller than $\frac{1}{6}$. We chose not to use this as our hypothesis both due to an aesthetic preference for $f(n)$ and $g(n)$ to be on equal footing, and because we wanted hypotheses invariant under products in order to use Ledrappier’s arguments to prove the K-property for the unique equilibrium state. However, it seems that the actual “transition point” for $g(n)$ is likely of the form $C \log n$ for $\frac{1}{6} \leq C \leq 1$, and it would be an interesting technical problem to prove this and find the $C$ in question; we will not, however, treat that question in this work though. The techniques used to prove Theorems \[mainthm\] and \[mainthm2\] are somewhat similar to the proofs of previous weaker results from [@pavlovspec], which applied only to measures of maximal entropy on subshifts and showed only that two such measures $\mu$, $\nu$ could not have disjoint supports. Roughly speaking, the proof in [@pavlovspec] involved combining “large” collections of words based on $\mu$ and $\nu$ to create more words in $\mathcal{L}(X)$ than there should be by definition of $h(X,T)$. The assumption of disjoint supports of $\mu$ and $\nu$ implied that the words from the two collections could not have overlap above a certain length, which ensured that all words created were distinct. For the results in this work, several changes must be made. First of all, obviously ‘words’ must be replaced with ‘orbit segments shadowed by a very small distance’ for general expansive systems, and ‘number of words’ must be replaced by ‘partition function for an $(n,\delta)$-separated set’ for general potentials. These changes require some technical results about changes of scale for expansive systems (see Section \[scaleindep\]), but the ideas are all essentially present in previous work of Bowen and others. The main advance in this work is dealing with $\mu \neq \nu$ whose supports may not be disjoint. The best we can do then is to assume ergodicity of $\mu, \nu$, which implies their mutual singularity, and therefore the existence of disjoint compact sets $C, D$ with $\mu(C), \nu(D)$ arbitrarily small (and positive distance $d(C,D)$). The new idea here is the use of the maximal ergodic theorem, which allows us to define “large” collections of orbit segments based on $\mu$ and $\nu$ where all initial segments of $\nu$-orbit segments have a large proportion of visits to $C$, and all terminal segments of $\mu$-orbit segments have a large proportion of visits to $D$. This means that initial segments of $\nu$-orbit segments and terminal segments of $\mu$-orbit segments are separated by $d(C,D)$ at some point, mimicking the lack of overlap from the proof in [@pavlovspec]. This is enough to create an $(n, \delta)$-separated collection by combining segments from the two collections whose partition function is larger than the pressure $P(X, T, \phi)$ should allow, achieving the desired contradiction. In various works (including [@CT1], [@CT2], and [@CT3]), Climenhaga and Thompson have defined different weakenings of the specification property, which allowed them to both generalize Bowen’s results in a different direction, even treating some non-expansive systems and continuous flows. Without going into full detail here, their definitions involve decomposing all orbit segments of points in the system into prefixes, cores, and suffixes, where the sets of possible prefixes/suffixes are “small” in some sense, and where for any $N$, the collection of segments whose prefix and suffix are shorter than $N$ has specification (in the sense that one can always find a point which shadows arbitrarily many such segments, of any lengths, with constant gaps). Existence of such a Climenhaga-Thompson decomposition is less restrictive than non-uniform specification in that specification properties must hold only for a subset of orbit segments, but is more restrictive in that the property required for that subset (weak specification) is significantly stronger. To our knowledge, neither of non-uniform specification or a Climenhaga-Thompson decomposition implies the other. Finally, we summarize the structure of the paper: Section \[defs\] contains relevant definitions and background on topological dynamics, ergodic theory, and thermodynamic formalism. Section \[scaleindep\] contains some results about preservation of various hypotheses under changes of scale, expansive factors, and products. Section \[proofs\] contains the proofs of Theorems \[mainthm\] and \[mainthm2\], including various auxiliary results. Finally, Section \[examples\] contains some examples for which our results imply uniqueness of equilibrium state and for which we believe this to be previously not known. acknowledgments {#acknowledgments .unnumbered} =============== The author would like to thank Jerome Buzzi and Sylvain Crovisier for pointing out that the proof of Theorem \[mainthm\] could be easily adapted to use the weaker hypothesis of non-uniform transitivity (yielding Theorem \[mainthm2\]), and would also like to thank Fran[ç]{}ois Ledrappier for discussions about the use of the techniques from [@ledrappier] for proving the K-property for unique equilibrium states. Definitions {#defs} =========== A **dynamical system** is given by a pair $(X,T)$ where $X$ is a compact metric space and $T: X \rightarrow X$ is a homeomorphism. A dynamical system $(X, T)$ is **expansive** if there exists $\delta > 0$ (called an **expansivity constant**) so that for all unequal $x, y \in X$, there exists $n \in \mathbb{Z}$ for which $d(T^n x, T^n y) > \delta$. A particular class of expansive dynamical systems are given by subshifts, to which the next few definitions refer. Given a finite set $A$ called the **alphabet**, a subshift $(X,T)$ is given by $X \subset A^{\mathbb{Z}}$ which is closed (in the product topology) and invariant under the left shift map $T$ defined by $(Tx)(i) = x(i+1)$ for $i \in \mathbb{Z}$. Every subshift is expansive; simply choose any $\delta$ so that $d(x,y) \leq \delta \Longrightarrow x(0) = y(0)$. Then, any $x \neq y \in X$ must have $x(n) \neq y(n)$ for some $n$, and then $(T^n x)(0) \neq (T^n y)(0)$, so $d(T^n x, T^n y) > \delta$. The **language** of a subshift $(X,T)$, denoted by $\mathcal{L}(X)$, is the set of finite strings of letters from $A$ (called **words**) which appear in some $x \in X$. We now return to definitions for more general expansive systems. Given an expansive dynamical system $(X,T)$ and any $n \in \mathbb{N}$ and $\epsilon > 0$, a set $S$ is called **$(n, \epsilon)$-separated** if for all unequal $x, y \in S$, there exists $0 \leq k < n$ so that $d(T^k x, T^k y) > \epsilon$. Given a continuous function $\phi: X \rightarrow \mathbb{R}$ (called a **potential**), the **partial sums** of $\phi$ are the functions $S_n \phi: X \rightarrow \mathbb{R}$ defined by $S_n \phi(x) = \sum_{i=0}^{n-1} \phi(T^i x)$. For a dynamical system $(X,T)$ and potential $\phi$, the $n$th **partition function** of $\phi$ at scale $\eta$ are the functions $$Z(X, T, \phi, n, \eta) := \max_{S \textrm{ is } (n,\eta)-\textrm{separated}} \sum_{x \in S} e^{S_n \phi (x)}.$$ The **topological pressure at scale $\eta$** of $(X,T,\phi)$ is $$P(X, T, \phi, \eta) := \lim_{n \rightarrow \infty} \frac{\ln Z(X, T, \phi, n, \eta)}{n}.$$ The **topological pressure** of $(X,T)$ for a potential $\phi$ is $$P(X, T, \phi) := \lim_{\eta \rightarrow 0} P(X, T, \phi, \eta).$$ \[presscaleindep\] If $(X,T)$ is expansive with expansivity constant $\delta$, then for any $\eta \leq \delta$, $P(X, T, \phi) = P(X, T, \phi, \eta)$. We also need some definitions from measure-theoretic dynamics. All measures considered in this paper will be $T$-invariant Borel probability measures on $X$ for $(X,T)$ an expansive dynamical system, and we denote the space of such measures by $\mathcal{M}(X,T)$. A measure $\mu$ on $A^{\mathbb{Z}}$ is [**ergodic**]{} if any measurable set $C$ which is invariant, i.e. $\mu(C \bigtriangleup TC) = 0$, has measure $0$ or $1$. Not all $T$-invariant measures are ergodic, but a well-known result called the ergodic decomposition shows that any non-ergodic measure can be written as a “weighted average” (formally, an integral) of ergodic measures. Also, whenever ergodic measures $\mu$ and $\nu$ are unequal, in fact they must be mutually singular (written $\mu \perp \nu$), i.e. there must exist a set $R$ with $\mu(R) = \nu(R^c) = 0$. (See Chapter 6 of [@walters] for proofs and more information.) When a measure $\mu$ is ergodic and $f \in L^1(\mu)$, the ergodic averages $\displaystyle \frac{1}{N} \sum_{i=0}^{N-1} f(T^i x)$ converge $\mu$-a.e. to the “correct” value $\int f \ d\mu$; this is essentially the content of Birkhoff’s ergodic theorem. We will need the following related result, which deals with the supremum of such averages rather than their limit. \[maxerg\] For $f \in L^1(\mu)$, define $M^+f := \sup_{N \in \mathbb{N}} \frac{1}{N} \sum_{i=0}^{N-1} f(T^i x)$. Then for any $\lambda \in \mathbb{R}$, $$\lambda \mu(\{M^+f(x) > \lambda\}) \leq \int_{M^+f(x) > \lambda} f \ d\mu.$$ The following corollary is immediate. \[maxergcor\] For nonnegative $f \in L^1(\mu)$ and $Mf$ as in Theorem \[maxerg\], and any $\lambda \in \mathbb{R}$, $$\mu(\{M^+f(x) \leq \lambda\}) \geq 1 - \frac{\|f\|_1}{\lambda}.$$ We note that by considering $T^{-1}$ instead, both of these results also hold when $M^+f$ is replaced by $M^-f := \sup_{N \in \mathbb{N}} \frac{1}{N} \sum_{i=0}^{N-1} f(T^{-i} x)$. We also need concepts from measure-theoretic entropy/pressure; for more information/proofs, see [@walters]. \[ent1\] For any $\mu \in \mathcal{M}(X,T)$, and finite measurable partition $\mathcal{P}$ of $X$, the **information of $\mathcal{P}$ with respect to $(X,T,\mu)$** is $$H(X, T, \mu, \mathcal{P}) := \sum_{A \in \mathcal{P}} -\mu(A) \ln \mu(A),$$ where terms with $\mu(A) = 0$ are omitted from the sum. \[ent2\] For any $\mu \in \mathcal{M}(X,T)$ and finite measurable partition $\mathcal{P}$ of $X$, the **entropy of $\mathcal{P}$ with respect to $(X,T,\mu)$** is $$h(X, T, \mu, \mathcal{P}) := \lim_{n \rightarrow \infty} \frac{H\left(X, T, \mu, \bigvee_{i=0}^{n-1} T^i \mathcal{P}\right)}{n}.$$ Note that by subadditivity, it is always true that $H\left(X, T, \mu, \bigvee_{i=0}^{n-1} T^i \mathcal{P}\right) \geq nh(X, T, \mu, \mathcal{P})$. \[ent3\] For any $\mu \in \mathcal{M}(X,T)$, the **entropy of $(X,T,\mu)$** is $$h(X, T, \mu) := \sup_{\mathcal{P}} h(X, T, \mu, \mathcal{P}).$$ We say that $\mu \in \mathcal{M}(X,T)$ has the **K-property** if for every partition $\mathcal{P}$ consisting of nonempty sets, $h(X, T, \mu, \mathcal{P}) > 0$. Note that any $\mu \in \mathcal{M}(X,T)$ with the K-property trivially has $h(X, T, \mu) > 0$. A partition $\mathcal{P}$ is a **generating partition** for $(X,T,\mu)$ if $\bigvee_{i \in \mathbb{Z}} T^i \mathcal{P}$ separates $\mu$-a.e. points of $X$. If $\mathcal{P}$ is a generating partition, then $h(X, T, \mu) = h(X, T, \mu, \mathcal{P})$. By expansivity, any partition $\mathcal{P}$ of sets whose diameters are all less than $\delta$ is a generating partition for all $\mu$, and so we have the following fact: If $(X,T)$ is expansive with expansivity constant $\delta$ and $\mathcal{P}$ consists of sets whose diameters are all less than $\delta$, then $h(X, T, \mu) = h(X, T, \mu, \mathcal{P})$ for any measure $\mu \in \mathcal{M}(X,T)$. The relationship between topological pressure and measure-theoretic entropy is given by the following Variational Principle: For any dynamical system $(X,T)$ and continuous $\phi$, $$P(X, T, \phi) = \sup_{\mu} h(X, T, \mu) + \int \phi \ d\mu.$$ For any $(X,T)$, an **equilibrium state** for $(X,T)$ and $\phi$ is a measure $\mu$ on $X$ for which $P(X, T, \phi) = h(X, T, \mu) + \int \phi \ d\mu$. If $(X,T)$ is expansive, then the entropy map $\mu \mapsto h(X, T, \mu)$ is upper semi-continuous. As a corollary, if $(X,T)$ is expansive and $\phi$ is continuous, then it has an equilibrium state; the upper semi-continuous function $\mu \mapsto h(X, T, \mu) + \int \phi \ d\mu$ must achieve its supremum $P(X, T, \phi)$ on the compact space $\mathcal{M}(X,T)$ (endowed with the weak-$*$ topology). In fact, the ergodic decomposition, along with the fact that the entropy map is affine ([@walters], Theorem 8.1), implies that the extreme points of the simplex of equilibrium states are precisely the ergodic equilibrium states. In particular, any $(X,T,\phi)$ with multiple equilibrium states also has multiple ergodic equilibrium states. A potential $\phi$ on $(X,T)$ is called **hyperbolic** if every equilibrium state $\mu$ has $h(X, T, \mu) > 0$. Though this is not the original definition from [@IRRL], it was shown to be an equivalent one in their Proposition 3.1. Our remaining definitions pertain to the hypotheses used in Theorems \[mainthm\] and \[mainthm2\]. The first relates to the potential $\phi$. Given a dynamical system $(X,T)$ and a potential $\phi$, the **partial sum variations of $\phi$ at scale $\eta$** are given by $$V(X, T, \phi, n, \eta) = \max_{\{(x,y) \ : \ \forall 0 \leq i < n, \ d(T^i x, T^i y) < \eta\}} |S_n \phi(x) - S_n \phi(y)|.$$ We say that $\phi$ has **partial sum variation bounds $g(n)$ at scale $\eta$** if $g(n) \geq V(X, T, \phi, i, \eta)$ whenever $i \leq n$. Our remaining definitions are for specification properties on $(X,T)$. We first need a general notion of shadowing. \[shadow\] Given a dynamical system $(X,T)$, $\epsilon > 0$, points $z, x_1, \ldots, x_k \in X$, and integers $n_1$, $\ldots$, $n_k$, $m_1$, $\ldots$, $m_{k-1}$, we say that **$z$ $\eta$-shadows $(x_i)$ for $(n_i)$ iterates with gaps $(m_i)$** if for every $0 < i < k$ and $0 \leq m < n_i$, $$d(T^{m + \sum_{j=1}^{i-1} (n_j + m_j)} z, T^m x_i) < \eta.$$ \[spec\] A dynamical system $(X,T)$ has **specification** if for any $\eta > 0$, there exists a constant $C(\eta)$ so that for any $k$, any points $x_1$, $\ldots$, $x_k \in X$, and any integers $n_1$, $\ldots$, $n_{k-1}$, $n_k$, $m_1$, $\ldots$, $m_{k}$ satisfying $m_i \geq C(\eta)$ when $0 < i < k$, there exists a point $z \in X$ which $\eta$-shadows $(x_i)$ for $(n_i)$ iterates with gaps $(m_i)$ and for which $T^{\sum_{j=1}^{k} (n_j + m_j)} z = z$. A related property in the literature is **weak specification**, which is identical to the definition above except that no periodicity of $z$ is assumed. For expansive $(X,T)$, this distinction is irrelevant; weak specification in fact implies specification (see [@kwietniaketal2], Lemma 9). We now move to the mixing properties which we will consider in this work, both of which can be thought of as non-uniform generalizations of specification with no assumption of periodicity. \[nonunifdef\] For an increasing function $f: \mathbb{N} \rightarrow \mathbb{N}$, a dynamical system $(X,T)$ has **non-uniform specification with gap bounds $f(n)$ at scale $\eta$** if for any $k$, any points $x_1$, $x_2$, $\ldots$, $x_k \in X$, and any integers $n_1$, $\ldots$, $n_{k-1}$, $n_k$, $m_1$, $\ldots$, $m_{k-1}$ satisfying $m_i \geq \max(f(n_i), f(n_{i+1}))$, there exists a point $z \in X$ which $\eta$-shadows $(x_i)$ for $(n_i)$ iterates with gaps $(m_i)$. This property is almost the same as the main property used by Marcus in [@marcusmonat] (which was not there given a name). There are two differences: the first is that Marcus required $\frac{f(n)}{n} \rightarrow 0$ as part of his definition, and the second is that in Marcus’s definition $m_i$ was only assumed greater than or equal to $f(n_i)$. Essentially, non-uniform specification only guarantees the ability to shadow when gaps are large enough in comparison to the lengths of orbit segments being shadowed before and after the gap, and Marcus’s unnamed property requires only that gaps be large compared to the orbit segment before the gap. We also consider the following significantly weaker property of non-uniform transitivity, which is weaker than non-uniform specification in two important ways. The first is that it only guarantees the ability to shadow two orbit segments, rather than arbitrarily many, and the second is that it guarantees the existence of only a single gap length which allows for shadowing, rather than guaranteeing that all gaps above a certain threshold suffice. For an increasing function $f: \mathbb{N} \rightarrow \mathbb{N}$, a dynamical system $(X,T)$ has **non-uniform transitivity with gap bounds $f(n)$ at scale $\eta$** if for any $n$ and any points $x,y \in X$, there exists $i \leq f(n)$ and $z \in X$ which $\eta$-shadows $(x,y)$ for $(n,n)$ iterates with gap $i$. We note that for $(X,T)$ with this property and any $n$ and $j,k \leq n$, one can also find $z$ which $\eta$-shadows $(x,y)$ for $(j,k)$ iterates with some gap $i \leq f(n)$. This is because one can instead start with the pair $(T^{-(n-j)} x, y)$, and for any $z$ which $\eta$-shadows $(T^{-(n-j)} x, y)$ for $(n,n)$ iterates with gap $i$, it is immediate that $T^{n-j} z$ $\eta$-shadows $(x,y)$ for $(j,k)$ iterates with gap $i$. Preservation under factors, products, and changes of scale {#scaleindep} ========================================================== In this section, we summarize some simple results illustrating preservation of various properties/quantities under expansive factors, products, and changes of scale. The following results show that the hypotheses used for Theorems \[mainthm\] and \[mainthm2\] do not depend on the scale used. \[gapscaleindep\] If $(X,T)$ is expansive (with expansivity constant $\delta$) and has non-uniform specification (transitivity) at scale $\delta$ with gap bounds $f(n)$, then for every $\eta < \delta$ there exists a constant $C = C(\eta)$ so that $(X,T)$ has non-uniform specification (transitivity) at scale $\eta$ with gap bounds $f(n + C) + C$. We present only the proof for non-uniform specification, as the one for non-uniform transitivity is extremely similar. Choose $(X,T)$ as in the theorem, and any $\eta < \delta$. By expansivity, there exists $N$ so that if $d(T^i x, T^i y) < \delta$ for $-N \leq i \leq N$, then $d(x,y) < \eta$. Now, choose any $k,n \in \mathbb{N}$, any $x_1, \ldots, x_k \in X$, and any $n_1, \ldots, n_{k-1} \geq f(n + 2N) + 2N$. Use non-uniform specification to choose $y \in X$ which $\delta$-shadows $(T^{-N} x_1, \ldots, T^{-N} x_k)$ for $(n + 2N, \ldots, n + 2N)$ iterates, with gaps $(n_1 - 2N, \ldots, n_{k-1} - 2N)$. Then by definition of $N$, $T^N y$ $\eta$-shadows $(x_1, \ldots, x_k)$ for $(n, \ldots, n)$ iterates, with gaps $(n_1, \ldots, n_{k-1})$, proving the desired non-uniform specification at scale $\eta$. \[sumscaleindep\] If $(X, T)$ is expansive (with expansivity constant $\delta$), $\eta < \delta$, and $\phi$ is a potential with partial sum variation bounds $g(n)$ at scale $\eta$, then there exists a constant $D = D(\eta)$ so that for every $n$, $\phi$ has partial sum variation bounds $g(n) + D$ at scale $\eta$. Choose $(X,T)$ and $\phi$ as in the theorem, and any $\eta < \delta$. By expansivity, there exists $N$ so that if $d(T^i x, T^i y) < \delta$ for $-N \leq i \leq N$, then $d(x,y) < \eta$. For the second inequality, choose any $n > 2N$ and any $x,y$ for which $d(T^i x, T^i y) < \delta$ for $0 \leq i < n$. Then by definition of $N$, $d(T^i (T^N) x, T^i (T^N) y) < \eta$ for $0 \leq i < n - 2N$, and so $$\begin{gathered} |S_n \phi(x) - S_n \phi(y)| \leq 2N (\sup \phi - \inf \phi) + |S_{n - 2N} \phi(T^N x) - S_{n - 2N} \phi(T^N y)| \\ \leq 2N (\sup \phi - \inf \phi) + g(n-2N) \leq g(n) + 2N(\sup \phi - \inf \phi).\end{gathered}$$ Taking the supremum over such pairs $(x,y)$ completes the proof. \[hypfactor\] If $f: (X,T) \rightarrow (Y,S)$ is a factor map, then for all $\epsilon$ there exists $\delta$ such that if $(X,T)$ has non-uniform specification (transitivity) with gap bounds $f(n)$ at scale $\delta$, then $(Y,S)$ has non-uniform specification (transitivity) with gap bounds $f(n)$ at scale $\epsilon$. We give a proof only for non-uniform specification, as the proof for non-uniform transitivity is trivially similar. Given $\epsilon$, use uniform continuity of $f$ to choose $\delta$ so that $d(x,x') < \delta \Longrightarrow d(f(x), f(x')) < \epsilon$. Now, given any $y_1, \ldots, y_k \in Y$ and $n_1, \ldots, n_k$, $m_1$, $\ldots$, $m_{k-1}$, choose arbitrary $x_i \in f^{-1}(y_i)$ and use non-uniform specification of $(X,T)$ at scale $\delta$ to find $z \in X$ which $\delta$-shadows $(x_i)$ for $(n_i)$ iterates with gaps $(m_i)$. It is immediate that $f(z) \in Y$ $\epsilon$-shadows $(y_i)$ for $(n_i)$ iterates with gaps $(m_i)$, completing the proof. The following corollary follows immediately by using Lemma \[gapscaleindep\]. \[factorpres\] If $f: (X,T) \rightarrow (Y,S)$ is a factor map, $(X,T)$ and $(Y,S)$ are expansive (with expansivity constants $\delta$ and $\eta$ respectively), and $(X,T)$ has non-uniform specification (transitivity) with gap bounds $f(n)$ at scale $\delta$ where $\liminf_{n \rightarrow \infty} f(n)/\ln n = 0$ ($\lim_{n \rightarrow \infty} f(n)/\ln n = 0$), then there exists $\overline{f}(n)$ with $\liminf_{n \rightarrow \infty} \overline{f}(n)/\ln n = 0$ ($\lim_{n \rightarrow \infty} \overline{f}(n)/\ln n = 0$) for which $(Y,S)$ has non-uniform specification (transitivity) with gap bounds $\overline{f}(n)$ at scale $\eta$. The class of $(X,T)$ satisfying either hypothesis of this corollary is then closed under expansive factors, and by Theorems \[mainthm\] and \[mainthm2\], any such system has a unique equilibrium state for any potential $\phi$ with partial sum variation bounds $g(n)$ satisfying $\lim_{n \rightarrow \infty} g(n)/\ln n = 0$. We now move to products, with the goal of proving Corollary \[Kcor\]. All products of metric spaces will be endowed with the $d_{\infty}$ metric defined by $d_{\infty}((x_1, y_1), (x_2, y_2)) = \max(d_1(x_1, y_1), d_2(x_2, y_2))$. The proofs of the following results are left to the reader. \[productpres1\] If $(X_1,T_1)$, $(X_2,T_2)$ are expansive dynamical systems with the same expansivity constant $\delta$ and which have non-uniform specification with gap bounds $f_1(n)$ and $f_2(n)$ (at scale $\delta$) respectively, then $(X_1 \times X_2, T_1 \times T_2)$ is an expansive dynamical system with expansivity constant $\delta$ which has non-uniform specification with gap bounds $f(n) = \max(f_1(n), f_2(n))$ at scale $\delta$. If $(X_1, T_1)$ and $(X_2, T_2)$ are expansive dynamical systems with the same expansivity constant $\delta$ and $\phi_1$ and $\phi_2$ are potentials with partial sum variation bounds $g_1(n)$ and $g_2(n)$ (at scale $\delta$) respectively, then the potential $\phi$ on $(X_1 \times X_2, T_1 \times T_2)$ defined by $\phi(x_1, x_2) := \phi_1(x_1) + \phi_2(x_2)$ has partial sum variation bounds $g(n) = g_1(n) + g_2(n)$ at scale $\delta$. \[productpres2\] If $(X_1, T_1)$ and $(X_2, T_2)$ are expansive dynamical systems with non-uniform gap specification with gap bounds $f_1(n)$ and $f_2(n)$ respectively, and if $\phi_1$ and $\phi_2$ are potentials with partial sum variation bounds $g_1(n)$ and $g_2(n)$ respectively (all at scales equal to the relevant expansivity constants), and if there exists a sequence $n_k$ so that $\lim_{k \rightarrow \infty} \frac{f_1(n_k)}{\ln n_k} = \lim_{k \rightarrow \infty} \frac{f_2(n_k)}{\ln n_k} = \lim_{k \rightarrow \infty} \frac{g_1(n_k)}{\ln n_k} = \lim_{k \rightarrow \infty} \frac{g_2(n_k)}{\ln n_k} = 0$, and the potential $\phi$ on $(X_1 \times X_2, T_1 \times T_2)$ is defined by $\phi(x_1, x_2) := \phi_1(x_1) + \phi_2(x_2)$, then $(X_1 \times X_2, T_1 \times T_2, \phi)$ satisfies the hypotheses of Theorem \[mainthm\]. The reason no assumption need be made on the equality of expansivity constants in Corollary \[productpres2\] is that one can always render them equal (say to $1$) by normalizing the metrics. We can now apply the following result of Ledrappier. \[ktrick\] If $(X,T)$ is an expansive dynamical system, $\phi$ is a potential, and $(X \times X, T \times T)$ has a unique equilibrium state for the potential $\phi^{(2)}$ defined by $\phi^{(2)}(x,y) = \phi(x) + \phi(y)$, then $(X,T)$ has a unique equilibrium state $\mu$ for $\phi$, and $(X,T,\mu)$ is a K-system. Corollary \[Kcor\] is now implied by Corollary \[productpres2\] and Theorem \[ktrick\]. Corollary \[hypcor\] follows as well: the closure under expansive factors comes from Corollary \[factorpres\], Hölder potentials are Bowen (i.e. have bounded partial sum variation bounds) for subshifts, and positive entropy of the unique equilibrium state comes from Corollary \[Kcor\] since K-systems have positive entropy. Finally, we need some technical results about behavior of separated sets/partition functions under changes of scale. The proof of the following lemma is motivated by arguments from [@bowen]. \[sepscaleindep\] If $(X,T)$ is expansive (with expansivity constant $\delta$) and $\eta < \delta$, then there exists a constant $M = M(\eta)$ so that for every $n$, every $(n,\eta)$-separated set can be written as a disjoint union of $M$ sets which are each $(n,\delta)$-separated. Choose $(X,T)$ and $\eta$ as in the theorem. By expansiveness, there exists $N$ so that if $d(T^i x, T^i y) < \delta$ for $-N \leq i \leq N$, then $d(x,y) < \eta$. Since $T$ is a homeomorphism, there exists $\alpha > 0$ so that if $d(x,y) < \alpha$, then $d(T^i x, T^i y) < \delta$ for $-N \leq i \leq N$. Take any partition $\mathcal{P} = \{A_i\}_{i=1}^k$ of $X$ by sets of diameter less than $\alpha$. Now, choose any $n$ and $(n,\eta)$-separated set $S$. For each $1 \leq i,j \leq k$, define $S_{i,j} = S \cap A_i \cap T^{-n} A_j$. We claim that each $S_{i,j}$ is $(n,\delta)$-separated, which will complete the proof for $M = k^2$. To see this, fix any $i,j$ and any $x \neq y \in S_{i,j}$. Since $x,y \in S$ and $S$ is $(n,\eta)$-separated, there exists $0 \leq m < n$ so that $d(T^m x, T^m y) \geq \eta$. Since $x,y \in A_i$, $d(x,y) < \alpha$, and so $d(T^i x, T^i y) < \eta$ for $0 \leq i \leq N$. Similarly, since $x,y \in T^{-n} A_j$, $d(T^n x, T^n y) < \alpha$, and so $d(T^i x, T^i y) < \eta$ for $n - N \leq i \leq n$. Therefore, $N < m < n - N$. However, by definition of $N$, this means that there exists $i \in [m - N, m + N] \subseteq [0, n]$ so that $d(T^i x, T^i y) > \delta$, completing the proof. The following fact actually appears in [@bowen] (as Lemma 1), but as it is a simple corollary of Lemma \[sepscaleindep\], we give a proof here as well. \[partscaleindep\] If $(X,T)$ is expansive (with expansivity constant $\delta$) and $\eta < \delta$, then for any potential $\phi$, $$Z(X, T, \phi, n, \eta) \leq M(\eta) Z(X, T, \phi, n, \delta),$$ where $M(\eta)$ is as in Lemma \[sepscaleindep\]. Consider an $(n,\eta)$-separated set $U$ for which $\sum_{x \in S} e^{S_n \phi(x)} = Z(X,T, \phi,n,\eta)$. Then by Lemma \[sepscaleindep\], we can write $U = \bigcup_{i = 1}^{M(\eta)} U_i$, where each $U_i$ is $(n, \delta)$-separated. Then $$Z(X,T, \phi,n,\eta) = \sum_{x \in U} e^{S_n \phi(x)} = \sum_{i = 1}^{M(\eta)} \left(\sum_{x \in U_i} e^{S_n \phi(x)} \right) \leq M(\eta) Z(X, T, \phi, n, \delta).$$ Proofs of Theorems \[mainthm\] and \[mainthm2\] {#proofs} =============================================== The first tool that we need is quite basic; it is the existence of a sequence with helpful bounds on $f$ and $g$ under the hypotheses of either Theorem \[mainthm\] or \[mainthm2\]. \[goodseq\] If $(X,T)$ and $\phi$ satisfy the hypotheses of either Theorem \[mainthm\] or \[mainthm2\], then there exists a sequence $\{n_k\}$ so that for all $\eta < \delta$, there exist increasing $\overline{f}$ and $\overline{g}$ where $(X,T)$ has non-uniform transitivity with gap bounds $\overline{f}(n)$ at scale $\eta$, $\phi$ has partial sum variation bounds $\overline{g}(n)$ at scale $\eta$, and $\overline{f}(n_k)/\ln n_k, \overline{g}(n_k)/\ln n_k \rightarrow 0$. We first note that by Lemmas \[gapscaleindep\] and \[sumscaleindep\], there are positive constants $C,D$ so that we can take $\overline{f}(n) = C + f(n + C)$ and $\overline{g}(n) = D + g(n) \leq D + g(n + C)$. Under the hypotheses of Theorem \[mainthm\], there exists a sequence $\{m_k\}$ so that $\frac{f(m_k) + g(m_k)}{\ln m_k} \rightarrow 0$. Then if we take $n_k = m_k - C$, $\frac{\overline{f}(n_k) + \overline{g}(n_k)}{\ln n_k} \rightarrow 0$, so $\{n_k\}$ satisfies the conclusion of the lemma. Similarly, under the hypotheses of Theorem \[mainthm2\], $\frac{f(k) + g(k)}{\ln k} \rightarrow 0$, so $\frac{\overline{f}(k) + \overline{g}(k)}{\ln k} \rightarrow 0$ also, so taking $n_k = k$ satisfies the conclusion of the lemma. A sequence $\{n_k\}$ satisfying the conclusion of Lemma \[goodseq\] is called an **anchor sequence** for $(X,T)$ and $\phi$. The next tools that we’ll need are some upper bounds on the partition function for expansive systems with non-uniform specification, which generalize the well-known upper bound of of $e^{nP(X,T,\phi)}$ times a constant when specification is assumed (e.g. Lemma 3 of [@bowen]). \[specpartbdthm\] If $(X,T)$ is an expansive dynamical system (with expansivity constant $\delta$) and non-uniform specification with gap bounds $f(n)$ at scale $\delta/3$, and $\phi$ is a potential with partial sum variation bounds $g(n)$ at scale $\delta/3$, then for all $n$, $$Z(X, T, \phi, n,\delta) \leq e^{(n + f(n)) P(X,T, \phi) - f(n) \inf \phi + g(n)}.$$ Suppose that $(X,T)$ and $\phi$ are as in the theorem, denote $m = \inf \phi$, and fix any $n$. Then, choose any $k \in \mathbb{N}$ and an $(n,\delta)$-separated set $S$ so that $\sum_{x \in S} e^{S_n \phi(x)} = Z(X,T, \phi,n,\delta)$. Then, for any $x_1, \ldots, x_k \in S$, we can use non-uniform specification to choose a point $y = y(x_1, \ldots, x_k)$ which $\delta/3$-shadows $(x_1, \ldots, x_k)$ for $(n,\ldots,n)$ iterates, with gaps $(f(n), \ldots, f(n))$. Then $$S_{k(n+f(n))}\phi(y) \geq kmf(n) + \sum_{j=1}^{k} S_n \phi(T^{(j-1)(n + f(n))} y) \geq kmf(n) - kg(n) + \sum_{j=1}^{k} S_n \phi(x_j).$$ We note that the set $Y = \{y(x_1, \ldots, x_k) \ : \ x_i \in S\}$ is $(k(n+f(n)),\delta/3)$-separated by definition, and so $$\begin{gathered} Z(X, T,\phi, k(n+f(n)), \delta/3) \geq \sum_{y \in Y} e^{S_{k(n+f(n))}\phi(y)} \\ \geq e^{kmf(n) - kg(n)} \sum_{x_1, \ldots, x_k \in S} \prod_{i=1}^k e^{S_n \phi(x_i)} = e^{kmf(n) - kg(n)} Z(X, T,\phi, n, \delta)^k.\end{gathered}$$ Taking logarithms, dividing by $k(n+f(n))$, and letting $k \rightarrow \infty$ yields $$P(X,T, \phi) = P(X,T,\phi,\delta/3) \geq \frac{\ln Z(X,T, \phi, n, \delta) + mf(n) - g(n)}{n + f(n)}.$$ (The first equality comes from Lemma \[presscaleindep\].) Now, solving for $Z(X,T,\phi,n,\delta)$ completes the proof. \[speccor\] If $(X,T)$ satisfies the hypotheses of Theorem \[mainthm\] and $\{n_k\}$ is an anchor sequence, then for every $\epsilon > 0$, there exists $K$ so that for all $k > K$ and $1 \leq i \leq n_k$, $$Z(X,T, \phi, i, \delta) \leq e^{iP(X,T, \phi)} n_k^{\epsilon}.$$ By definition of anchor sequence, we can take $f(n)$ and $g(n)$ to be gap bounds and partial sum variation bounds at scale $\delta/3$, and can choose $K$ so that for $k > K$, $f(n_k) + g(n_k) < \frac{\epsilon}{P(X,T, \phi) + |\inf \phi| + 1} \ln n_k$. Then, for any such $k$ and $1 \leq i \leq n_k$, Theorem \[specpartbdthm\] (along with monotonicity of $f,g$) implies $$\begin{gathered} Z(X,T, \phi, i, \delta) \leq e^{(i + f(i)) P(X,T,\phi) - f(i) \inf \phi + g(i)} \leq \\ e^{iP(X,T, \phi)} e^{(P(X,T, \phi) + |\inf \phi| + 1)(f(i) + g(i))} \leq e^{iP(X,T, \phi)} n_k^{\epsilon}.\end{gathered}$$ We now prove a somewhat similar bound under the assumption of non-uniform transitivity, which requires information about $f,g$ for all large $n$ rather than a single value. \[transpartbdthm\] If $(X,T)$ is an expansive dynamical system (with expansivity constant $\delta$) and non-uniform transitivity with gap bounds $f(n)$ at scale $\delta/3$, $\phi$ is a potential with partial sum variation bounds $g(n)$, and $C > 0, M \geq 3$ satisfy $f(n) + g(n) \leq \min(C\ln n,n)$ for all $n \geq M$, then for all $n \geq M$, $$Z(X,T, \phi, n, \delta) \leq D e^{nP(X,T, \phi)} n^{CE},$$ where $D,E$ are constants depending only on $X$ and $\phi$. Suppose that $X$, $T$, $\phi$, $C$, and $M$ are as in the theorem, and fix any $n \geq M$. We use $m$ to denote $\inf \phi$. For every $j$, choose $U_j$ a $(j, \delta)$-separated set for which $\sum_{x \in U_j} e^{S_j \phi(x)} = Z(X, T, \phi, j, \delta)$. We will use non-uniform transitivity to give lower bounds on $Z(X, T, \phi, n_k, \delta)$ for a recursively defined sequence $\{n_k\}$. Define $n_0 = n$, and for $k \geq 0$ define $n_{k+1} = 2n_k + f(n_k)$. Note that all $n_k \geq M$, and so $n_{k+1} \leq 3n_k$ for all $k$, meaning that $n_k \leq 3^k n$ for all $k$. For a better bound, we see that by induction, for every $k$, $$\begin{gathered} \label{nkbd} n_k = 2^k n + 2^{k-1} f(n_1) + 2^{k-2} f(n_2) + \ldots + f(n_{k-1}) \\ \leq 2^k n + 2^k C [2^{-1} \ln (3n) + 2^{-2} \ln (3^2 n) + \ldots + 2^{-(k-1)} \ln (3^k n)] \\ = 2^k (n + C\ln (9n)).\end{gathered}$$ Then, for every $k \geq 0$, and for any $x, y \in U_{n_k}$, use non-uniform transitivity to create a point $z(x,y)$ which $\delta/3$-shadows $(x,y)$ for $(n_k,n_k)$ iterates, with a gap $i$ of length less than or equal to $f(n_k)$. Then, $$\begin{gathered} S_{n_{k+1}} \phi (z(x,y)) \geq S_{n_k} \phi(z(x,y)) + S_{n_k} \phi(T^{i + n_k} z(x,y)) - |m|f(n_k) \\ \geq S_{n_k} \phi(x) + S_{n_k} \phi(y) - 2g(n_k) - |m|f(n_k).\end{gathered}$$ This implies that $$\begin{gathered} \sum_{z(x,y)} e^{S_{n_{k+1}} \phi(z(x,y))} \geq \sum_{x,y} e^{S_{n_k} \phi(x) + S_{n_k} \phi(y) - 2g(n_k) - |m|f(n_k)} \\ = e^{-2g(n_k) - |m|f(n_k)} \left(\sum_{x \in U_{n_k}} e^{S_{n_k} \phi(x)} \right)^2 = e^{-2g(n_k) - |m|f(n_k)} Z(X, T, \phi, n_k, \delta)^2.\end{gathered}$$ Then there is a single gap $i_k \leq f(n_k)$ such that if we define $Y_k$ to be the set of $(x,y)$ for which $z(x,y)$ used a gap of length $i$, then $$\begin{gathered} \label{transbd1} \sum_{(x,y) \in Y_k} e^{S_{n_{k+1}} \phi(z(x,y))} \geq (f(n_k))^{-1} e^{-2g(n_k) - |m|f(n_k)} Z(X, T, \phi, n_k, \delta)^2 \\ \geq e^{-2g(n_k) - (|m|+1)f(n_k)} Z(X, T, \phi, n_k, \delta)^2.\end{gathered}$$ We now claim that $\{z(x,y) \ : \ (x,y) \in Y_k\}$ is $(n_{k+1}, \delta/3)$-separated. To see this, choose any unequal $(x,y), (x', y') \in Y_{n_{k+1}}$, and write $z = z(x,y)$ and $z' = z(x', y')$. Either $x \neq x' \in U_{n_k}$ or $y \neq y' \in U_{n_k}$. In the former case, since $U_{n_k}$ is $(n,\delta)$-separated, there exists $0 \leq j < n_k$ so that $d(T^j x, T^j x') > \delta$, and by definition of $z$ and $z'$, $d(T^j z, T^j z') > \delta/3$. In the latter case, there exists $0 \leq j < n_k$ so that $d(T^j y, T^j y') > \delta$, and again by definition of $z$ and $z'$, $d(T^{n_k + i_k + j} z, T^{n_k + i_k + j} z) > \delta/3$. So, $\{z(x,y) \ : \ (x,y) \in Y_k\}$ is $(n_{k+1}, \delta/3)$-separated as claimed. Therefore, Lemma \[partscaleindep\] and (\[transbd1\]) imply $$\begin{gathered} Z(X, T, \phi, n_{k+1}, \delta) \geq M(\delta/3)^{-1} Z(X, T, \phi, n_{k+1}, \delta/3) \geq \sum_{(x,y) \in Y_k} e^{S_{n_{k+1}} \phi(z(x,y))}\\ \geq M(\delta/3)^{-1} e^{-2g(n_k) - (|m|+1)f(n_k)} Z(X, T, \phi, n_k, \delta)^2.\end{gathered}$$ Now, by induction, $\ln Z(X, T, \phi, n_k, \delta)$ is greater than or equal to $$\begin{gathered} \label{transbd2} - k \ln M(\delta/3) + 2^k \ln Z(X, T, \phi, n, \delta) - \sum_{i = 0}^{k-1} 2^{k-i-1} (2g(n_i) - (m+1)f(n_i))\\ \geq - k \ln M(\delta/3) + 2^k \ln Z(X, T, \phi, n, \delta) - 2^k \sum_{i = 0}^{k-1} 2^{-i-1} (2 + |m+1|) C \ln (3^i n)\\ = -k \ln M(\delta/3) + 2^k(\ln Z(X, T, \phi, n, \delta) - C (2 + |m+1|) \ln(9n)). \end{gathered}$$ Therefore, by (\[nkbd\]) and (\[transbd2\]), $$\frac{\ln Z(X, T, \phi, n_k, \delta)}{n_k} \geq \frac{-k \ln M(\delta/3) + 2^k(\ln Z(X, T, \phi, n, \delta) - C (2 + |m+1|) \ln(9n))}{2^k(n + C\ln(9n))}.$$ Letting $n_k \rightarrow \infty$ yields $$P(X,T, \phi) \geq \frac{\ln Z(X, T, \phi, n, \delta) - C (2 + |m+1|) \ln(9n)}{n + C\ln(9n)},$$ and we can rewrite as $$Z(X, T, \phi, n, \delta) \leq D e^{Pn} n^{CE},$$ (here $D = 9^{C(P(X,T, \phi) + 2 + |m-1|)}$ and $E = P(X,T, \phi) + 2 + |m-1|$), completing the proof. \[transcor\] If $(X,T)$ and $\phi$ satisfy the hypotheses of Theorem \[mainthm2\] and $\{n_k\}$ is an anchor sequence, then for every $\epsilon > 0$, there exists $K$ so that for any $k > K$ and $1 \leq i \leq n_k$, $$Z(X, T, \phi, i, \delta) \leq e^{iP(X,T, \phi)} n_k^{\epsilon}.$$ Under the hypotheses of Theorem \[mainthm2\], we may choose $M \geq 3$ so that for $n \geq M$, $f(n) + g(n) < \min((\epsilon/2E) \ln n,n)$, where $E$ is as in Theorem \[mainthm2\]. Then, for all $n \geq M$, $$Z(X, T, \phi, i, \delta) \leq D e^{iP(X,T, \phi)} i^{\epsilon/2}.$$ This can clearly be improved to hold for all $i$ by changing $D$, i.e. there exists $D'$ so that for all $i$, $$Z(X, T, \phi, i, \delta) \leq D' e^{iP(X,T, \phi)} i^{\epsilon/2}.$$ We then just choose $K$ so that $n_k^{\epsilon/2} > D'$. Then, for any $k \geq K$ and $1 \leq i \leq n_k$, $$Z(X, T, \phi, i, \delta) \leq D' e^{iP(X,T, \phi)} i^{\epsilon/2} < e^{iP(X,T, \phi)} n_k^{\epsilon},$$ completing the proof. We will also need the following general lower bound on the sum of $e^{S_n \phi (x)}$ over separated sets within a set of positive measure for an equilibrium state. \[measbd\] If $(X,T)$ is an expansive dynamical system (with expansivity constant $\delta$), $\phi$ is a potential with partial sum variations $g(n)$ at scale $\delta/3$, and $A \subset X$ has $\mu(A) > 0$ for some equilibrium state $\mu$ of $X$ for $\phi$, then there exists an $(n,\delta/3)$-separated subset $U$ of $X$ with $$\sum_{x \in U} e^{S_n \phi(x)} \geq \left(e^{nP(X,T,\phi)}\right)^{1/\mu(A)} \left(Z(X,T, \phi,n,\delta/3)\right)^{(\mu(A)-1)/\mu(A)} M^{-1} e^{-g(n) - \frac{\ln 2}{\mu(A)}},$$ where $M = M(\delta/3)$ as defined in Lemma \[sepscaleindep\]. Consider such $X$, $T$, $\phi$, $\mu$, $A$, and $n$. Choose a maximal $(n, \delta/3)$-separated subset $U$ of $X$. As in [@bowen] or [@CT2], we can create a partition $\mathcal{P} = \{A_x\}_{x \in U}$ where for each $x \in U$ and $y \in A_x$, $y$ $\delta/3$-shadows $x$ for $n$ iterates, i.e. $d(T^i x, T^i y) < \delta/3$ for $0 \leq i < n$. Then, if two points are in the same element of $\bigvee_{m \in \mathbb{Z}} T^{mn} \mathcal{P}$, by expansivity they are equal, i.e. $\mathcal{P}$ is a generating partition for $(X,T^n,\mu)$. This means that $$\begin{gathered} \label{eqn1} nP(X, T, \phi) = n\left(h(X, T, \mu) + \int \phi \ d\mu\right) = h(X, T^n, \mu) + \int S_n \phi \ d\mu \\ = h(X, T^n, \mu, \mathcal{P}) + \int S_n \phi \ d\mu \leq H(X,T^n,\mu,\mathcal{P}) + \int S_n \phi \ d\mu \\ \leq \sum_{x \in U} \mu(A_x) \left[-\ln \mu(A_x) + \sup_{y \in A_x} S_n \phi(y)\right] \leq g(n) + \sum_{x \in U} \mu(A_x) \left[- \ln \mu(A_x) + S_n \phi(x)\right].\end{gathered}$$ We write $U' = \{x \in U \ : \ A_x \cap A \neq \varnothing\}$, and then can break up the final sum: $$\begin{gathered} \label{eqn2} \sum_{x \in U} \mu(A_x) \left[-\ln \mu(A_x) + S_n \phi(x)\right] = \sum_{x \in U'} \mu(A_x) \left[-\ln \mu(A_x) + S_n \phi(x)\right] \\ + \sum_{x \in U \setminus U'} \mu(A_x) \left[-\ln \mu(A_x) + S_n \phi(x)\right].\end{gathered}$$ For fixed positive reals $a_1, \ldots, a_n$ and positive $p_1, \ldots, p_N$ with fixed sum $S$, $\sum p_i(a_i - \ln p_i)$ has maximum value $S (-\ln S + \ln\sum e^{a_i})$. (See Lemma 9.9 in [@walters].) Therefore, if we write $A' = \bigcup_{x \in U'} A_x$, then $$\begin{gathered} \label{eqn3} \sum_{x \in U'} \mu(A_x) \left[-\ln \mu(A_x) + S_n \phi(x)\right] \leq \mu(A') \left(-\ln \mu(A') + \ln \sum_{x \in U'} e^{S_n \phi(x)}\right).\end{gathered}$$ Similarly, $$\begin{gathered} \label{eqn4} \sum_{x \in U \setminus U'} \mu(A_x) \left[-\ln \mu(A_x) + S_n \phi(x)\right] \\ \leq (1 - \mu(A')) \left(-\ln (1 - \mu(A')) + \ln \sum_{x \in U \setminus U'} e^{S_n \phi(x)}\right)\\ \leq (1 - \mu(A')) \left(\ln (1 - \mu(A')) + \ln Z(X, T, n, \phi, \delta/3)\right).\end{gathered}$$ Combining (\[eqn1\])-(\[eqn4\]) yields $$\begin{gathered} nP(X, T,\phi) \leq \mu(A') \left(-\ln \mu(A') + \ln \sum_{x \in U'} e^{S_n \phi(x)}\right) + \\ (1 - \mu(A')) \left(\ln (1 - \mu(A')) + \ln Z(X, T, \phi, n, \delta/3)\right)\\ \leq \ln 2 + \mu(A') \ln \sum_{x \in U'} e^{S_n \phi(x)} + (1 - \mu(A')) \ln Z(X, T, \phi,n, \delta/3).\end{gathered}$$ Finally, we solve for $\ln \sum_{x \in U'} e^{S_n \phi(x)}$: $$\begin{gathered} \label{bigeqn} \ln \sum_{x \in U'} e^{S_n \phi(x)} \geq \frac{nP(X, T,\phi)}{\mu(A')} - \frac{1 - \mu(A')}{\mu(A')} \ln Z(X, T, \phi, n, \delta/3) - \frac{\ln 2}{\mu(A')}\\ \geq \frac{nP(X,T, \phi)}{\mu(A)} - \frac{1 - \mu(A)}{\mu(A)} \ln Z(X, T, \phi, n, \delta/3) - \frac{\ln 2}{\mu(A)}.\end{gathered}$$ (Recall here that $\mu(A') \geq \mu(A)$.) Now, by Lemma \[sepscaleindep\], we can write $U'$ as the union of $M = M(\delta/3)$ sets which are each $(n, \delta)$-separated. Then (\[bigeqn\]) implies that there must exist one, call it $U''$, so that $$\label{bigeqn3} \sum_{x \in U''} e^{S_n \phi(x)} \geq M^{-1} \left(e^{nP(X,T, \phi)}\right)^{1/\mu(A)} \left(Z(X,T,\phi,n,\delta/3)\right)^{(\mu(A) - 1)/\mu(A)} e^{-\frac{\ln 2}{\mu(A)}}.$$ Recall that for every $x \in U''$, $A_x \cap A \neq \varnothing$. Therefore, we can define a new set $U'''$ which contains a single point from each $A_x \cap A$ for $x \in U''$. Recall that $U''$ was $(n, \delta)$-separated, and that every point of $A_x$ $\delta/3$-shadows $x$ for $n$ iterates; therefore, $U'''$ is $(n, \delta/3)$-separated. Then by (\[bigeqn3\]), $$\begin{gathered} \sum_{x \in U'''} e^{S_n \phi(x)} \geq \sum_{x \in U''} e^{S_n \phi(x) - g(n)}\\ \geq M^{-1} \left(e^{nP(X,T, \phi)}\right)^{1/\mu(A)} \left(Z(X,T,\phi,n,\delta/3)\right)^{(\mu(A) - 1)/\mu(A)} e^{-g(n) - \frac{\ln 2}{\mu(A)}},\end{gathered}$$ completing the proof. The following is an immediate corollary of Theorem \[measbd\] and Corollaries \[partscaleindep\], \[speccor\], and \[transcor\]. \[maincor\] If $(X,T)$ and $\phi$ satisfy the hypotheses of either Theorem \[mainthm\] or \[mainthm2\] and $\{n_k\}$ is an anchor sequence, then for every $\epsilon > 0$, there exists $K$ so that for any $A \subset X$ with $\mu(A) > 1/2$ for some equilibrium state $\mu$ of $X$ for $\phi$, any $k \geq K$, and any $1 \leq i \leq n_k$, there is an $(i,\delta/3)$-separated subset $T$ of $A$ with $$\sum_{x \in T} e^{S_i \phi(x)} \geq n^{-\epsilon} e^{iP(X,T, \phi)}.$$ We may now prove our main results. Choose $X$ and $\phi$ satisfying the hypotheses of either Theorem \[mainthm\] or \[mainthm2\], and a corresponding anchor sequence $\{n_k\}$. Define $m = \inf \phi$. Suppose for a contradiction that $X$ has more than one equilibrium state for $\phi$. Then, as noted in the introduction, $X$ has ergodic equilibrium states $\mu \neq \nu$ for $\phi$, and $\mu \perp \nu$. Then, there exists measurable $R \subset X$ with $\mu(R) = \nu(R^c) = 1$. Since $\mu, \nu$ are Borel measures, there exist open sets $U \supseteq R$ and $U' \supseteq R^c$ so that $\mu(U), \nu(U') < \frac{1}{5}$. We write $\eta = \min(\delta/9, d(U^c, U'^c)/3) > 0$. Define $W \subset X$ to be the set of $x \in X$ for which $$M^+(\chi_{U}) = \sup_N \frac{1}{N} \sum_{n=0}^{N-1} \chi_U(T^n x) \leq 2\mu(U) < \frac{2}{5}.$$ In other words, for every point of $W$ and any $i$, fewer than $\frac{2}{5}$ of its first $i$ iterates under $T$ are in $U$. By Corollary \[maxergcor\] (to the Maximal Ergodic Theorem), $\displaystyle \mu(W) \geq 1 - \frac{\int \chi_U \ d\mu}{2\mu(U)} = \frac{1}{2}$. Similarly, we define $V \subset X$ to be the set of $x \in X$ for which $$M^- (\chi_{T^m U'}) = \sup_N \frac{1}{N} \sum_{n=0}^{N-1} \chi_{T^m U'}(T^{-n} x) \leq 2\nu(U') < \frac{2}{5}.$$ For every point of $V$ and any $i$, fewer than $\frac{2}{5}$ of its first $i$ iterates under $T^{-1}$ are in $U'$. By Corollary \[maxergcor\] (applied to $T^{-1}$), $\displaystyle \mu(V) \geq 1 - \frac{\int \chi_{T^m U'} \ d\nu}{2\nu(U')} = \frac{1}{2}$. By Corollary \[maincor\], there exists $K$ so that for $k \geq K$ and all $j \leq n_k$, we can define $(j, \delta/3)$-separated sets $V_j \subset V$ and $W_j \subset W$ for which $$\label{VWbd} \sum_{x \in V_j} e^{S_j\phi(x)}, \sum_{x \in W_j} e^{S_j\phi(x)} \geq e^{jP(X,T, \phi)} n_k^{-1/5}.$$ Whether $(X,T)$ and $\phi$ satisfied the hypotheses of Theorem \[mainthm\] or \[mainthm2\], by Lemmas \[gapscaleindep\] and \[sumscaleindep\] we may assume that $(X,T)$ satisfies non-uniform transitivity at scale $\eta$ with gap bounds $f(n)$ and that $\phi$ has partial sum variation bounds $g(n)$ at scale $\eta$. The final step is to use non-uniform transitivity of $X$ to create an $(n_k,\eta)$-separated set by shadowing orbit segments from various $V_i$ and $W_j$. For large $k$, the sum of $e^{S_{n_k} \phi(x)}$ over this set will be large enough to contradict Corollary \[speccor\] or Corollary \[transcor\]. By definition of anchor sequence, we can increase $K$ so that for any $k \geq K$, $f(n_k), g(n_k) < \frac{1}{5(P(X,T, \phi) + |m| + 2)} \ln n_k$. Choose any $k \geq K$ and define $n := n_k$ for ease of notation. Then, for any integer $j$ in $[1, \frac{n - f(n)}{2f(n)}]$, any $v \in V_{2jf(n)}$, and any $w \in W_{n - f(n) - 2jf(n)}$, we use non-uniform transitivity to create a point $x(j,v,w)$ which $\eta$-shadows $(v,w)$ for $(2j f(n), n - f(n) - 2j f(n))$ iterates with gap $i \leq f(n)$. We first note that for all $j,v,w$, $$\begin{gathered} S_n \phi(x(j,v,w)) \geq \\ S_{2jf(n)} \phi(v) + S_{n - (2j+1)f(n)} \phi(w) - |m| f(n) - g(2jf(n)) - g(n - (2j+1)f(n)) \\ \geq S_{2jf(n)} \phi(v) + S_{n - (2j+1)f(n)} \phi(w) - |m| f(n) - 2g(n)\\ \geq S_{2jf(n)} \phi(v) + S_{n - 2jf(n) - f(n)} \phi(w) - 1/5 \ln n.\end{gathered}$$ This implies that for any $j$, $$\begin{gathered} \sum_{v,w} e^{S_n \phi (x(j,v,w))} \geq n^{-1/5} \left(\sum_v e^{S_{2jf(n)} \phi(v)}\right) \left(\sum_w e^{S_{n - 2jf(n) - f(n)} \phi(w)}\right)\\ \stackrel{(\ref{VWbd})}{\geq} e^{P(X,T,\phi)(n - f(n))} n^{-3/5} \geq e^{nP(X,T,\phi)} n^{-4/5}.\end{gathered}$$ Then there exists a set $T(j)$ of pairs $(v,w)$ so that all use the same gap $i_j \leq f(n)$, and $$\label{partbd2.5} \sum_{v,w \in T(j)} e^{S_n \phi(x(j,v,w))} \geq (f(n))^{-1} e^{nP(X,T,\phi)} n^{-4/5}.$$ We claim that the set $Z = \bigcup_{j=1}^{\lfloor (n-f(n))/2f(n) \rfloor} \{x(j,v,w) \ : \ (v,w) \in T(j)\}$ is $(n,\eta)$-separated. To see this, choose any triples $(j,v,w) \neq (j',v',w')$ with $(v,w) \in T(j)$ and $(v',w') \in T(j')$. We break into the cases $j = j'$ and $j \neq j'$, and for brevity write $x = x(j,v,w)$ and $x' = x(j',v',w')$. If $j = j'$, then either $v \neq v'$ or $w \neq w'$, and since $V_i$ and $W_i$ are $(i, \delta/3)$-separated for all $i$, either $d(T^k v, T^k v') > \delta/3$ for some $0 \leq k < 2j f(n)$ or $d(T^k w, T^k w') > \delta/3$ for some $0 \leq k < n - f(n) - 2j f(n)$. In the first case, since $x$ $\eta$-shadows $v$ for its first $2jf(n)$ iterates and $x'$ $\eta$-shadows $v'$ for its first $2j f(n)$ iterates (recall that $j = j'$), $d(T^k x, T^k x') > \delta/3 - 2\eta \geq \eta$. The second case is trivially similar, using the $\eta$-shadowing of $w$ and $w'$ and the fact that since $(v,w), (v', w') \in T(j)$, the gaps used for $x$ and $x'$ are equal. Now suppose that $j \neq j'$, and without loss of generality assume $j < j'$. Recall that $i_j$ denotes the gap used in the construction of $x$. Then by definition of $x$, $T^{2j f(n) + i_j} x$ $\eta$-shadows $w$ for $n - 2j f(n) - f(n) \geq (2j' - 2j) f(n) - i_j + 1$ iterates. Since $w \in W_{n - f(n) - 2jf(n)} \subset W$, fewer than $\frac{2}{5} ((2j' - 2j) f(n) - i_j + 1)$ of the first $(2j' - 2j) f(n) - i_j + 1$ iterates of $w$ are in $U$. Therefore, more than a proportion of $\frac{3}{5}$ of the points $\{T^m x \ : \ 2j f(n) + i_j \leq m \leq 2j'f(n)\}$ are within distance $\eta$ of $U^c$. Similarly, by definition, $x'$ $\eta$-shadows $v'$ for $2j' f(n) \geq (2j' - 2j) f(n) - i_j + 1$ iterates. Since $v' \in V_{2j'f(n)}$, $T^{2j'f(n)} v' \in V$, and so the proportion of the first $(2j' - 2j) f(n) - i_j + 1$ iterates under $T^{-1}$ of $T^{2j'f(n)} v'$ which are in $U'$ is less than $\frac{2}{5}$. Therefore, more than a proportion of $\frac{3}{5}$ of the points $\{T^m x' \ : \ 2j f(n) + i_j \leq m \leq 2j'f(n)\}$ are within distance $\eta$ of $U'^c$. So, there exists $k \in [2j f(n) + i_j, 2j'f(n)]$ so that $d(T^k x, U^c), d(T^k x', U'^c) < \eta$. By definition of $\eta$, $d(U^c, U'^c) \geq 3\eta$. Therefore, $d(T^k x, T^k x') > \eta$. In either case, we’ve shown that there exists $0 \leq k < n$ for which $d(T^k x, T^k x') > \eta$, and so $Z$ is $(n, \eta)$-separated as claimed. Then, $$\begin{gathered} \sum_{x \in Z} e^{S_n \phi(x)} = \sum_{j = 1}^{\lfloor (n-f(n))/2f(n) \rfloor} \sum_{(v,w) \in T(j)} e^{S_n \phi(x(j,v,w))}\\ \stackrel{(\ref{partbd2.5})}{\geq} \frac{n-f(n)}{2f(n)^2} e^{nP(X,T,\phi)} n^{-4/5}.\end{gathered}$$ As before, we can use Lemma \[sepscaleindep\] to pass to $T' \subseteq T$ which is $(n, \delta)$-separated and for which $$\sum_{x \in T'} e^{S_n \phi(x)} \geq \frac{n - f(n)}{2f(n)^2} (M(\eta))^{-1} e^{n P(X,T, \phi)} n^{-4/5},$$ implying that $$Z(X,T,\phi,n,\delta) \geq \frac{n - f(n)}{2f(n)^2} (M(\eta))^{-1} e^{n P(X,T, \phi)} n^{-4/5}.$$ However, this will contradict Corollary \[speccor\] or \[transcor\] for large $k$. Therefore, our original assumption of multiple equilibrium states on $X$ was false, and $X$ has a unique equilibrium state, which we denote by $\mu$.\ It remains to show that $\mu$ is fully supported, and so for a contradiction assume that there is a nonempty open set $U \subset X$ with $\mu(U) = 0$. Then the set $Y$ of points whose orbits under $T$ never visit $U$ has $\mu(Y) = 1$, and $Y$ contains some open ball $B_{\rho}(y)$. Define $\eta = \min(\delta/9, \rho/3)$. By the definition of anchor sequence, there exists $K$ so that for $k \geq K$, $f(n_k), g(n_k) < \frac{1}{3(3P(X, T, \phi) + 3|m| + 2)} \ln n_k$. By Theorem \[measbd\], for every $n$ there exists an $(n, \delta/3)$-separated set $Y_n \subseteq Y$ with $$\label{Ybd} \sum_{x \in Y_n} e^{S_n \phi(x)} \geq C e^{nP(X,T, \phi) - g(n)},$$ where $C = (2M(\delta/3))^{-1}$. Now, we will again use non-uniform transitivity to obtain a contradiction to one of Corollary \[speccor\] or Corollary \[transcor\]. Choose $k \geq K$ and denote $n := n_k$. Then, for any integer $j$ in $[1, \frac{n - 2f(n)}{2f(n)}]$, any $v \in U_{2jf(n)}$, and any $w \in U_{n - (2j+2)f(n) - 1}$, we use non-uniform transitivity to choose $z(j,v,w) \in X$ which $\eta$-shadows $(v,y,w)$ for $(2jf(n),1,n - (2j+2)f(n) - 1)$ iterates, with gaps $(i,i')$ both less than or equal to $f(n)$. We first note that for all $j,v,w$, $$\begin{gathered} S_n \phi(z(j,v,w)) \geq \\ S_{2jf(n)} \phi(v) + S_{n - (2j+2)f(n) - 1} \phi(w) - |m|(2f(n)+1) - g(2jf(n)) - g(n - (2j+2)f(n) - 1) \\ \geq S_{2jf(n)} \phi(v) + S_{n - (2j+2)f(n) - 1} \phi(w) - |m|(2f(n) + 1) - 2g(n)\\ \geq S_{2jf(n)} \phi(v) + S_{n - (2j+2) f(n) - 1} \phi(w) - 1/3 \ln n.\end{gathered}$$ Then, for any $j$, $$\begin{gathered} \sum_{v,w} e^{S_n \phi(z(j,v,w))} \geq n^{-1/3} \left( \sum_{v} e^{S_{2jf(n)} \phi(v)} \right) \left( \sum_{w} e^{S_{n - (2j+2)f(n) - 1} \phi(w)} \right)\\ \stackrel{(\ref{Ybd})}{\geq} C^2 e^{(n - 2f(n) - 1)P(X,T,\phi) - g(2jf(n)) - g(n - (2j + 2) f(n) - 1)} n^{-1/3}\\ \geq C^2 e^{(n - 2f(n) - 1)P(X,T,\phi) - 2g(n)} n^{-1/3} \geq C^2 e^{nP(X,T,\phi)} n^{-2/3}. \end{gathered}$$ Then there exists a set $U(j)$ of pairs $(v,w)$ so that all use the same gaps $i_j, i'_j \leq f(n)$, and $$\label{partbd3.5} \sum_{v,w \in U(j)} e^{S_n \phi(z(j,v,w))} \geq (f(n))^{-2} C^2 e^{nP(X,T,\phi)} n^{-2/3}.$$ We claim that the set $Z = \bigcup_{j=1}^{\lfloor (n-f(n))/2f(n) \rfloor} \{x(j,v,w) \ : \ (v,w) \in U(j)\}$ is $(n,\eta)$-separated. To see this, choose any triples $(j,v,w) \neq (j',v',w')$ with $(v,w) \in U(j)$ and $(v',w') \in U(j')$. We break into the cases $j = j'$ and $j \neq j'$, and for brevity write $z = z(j,v,w)$ and $z' = z(j',v',w')$. If $j = j'$, then the proof that there exists $0 \leq k < n$ for which $d(T^k z, T^k z') > \eta$ is the same as was done above in the proof of uniqueness of $\mu$ (again, recall that $z$ and $z'$ both use the same gap $j$.) If $j \neq j'$, then without loss of generality we assume $j < j'$, and recall that $i_j$ denotes the first gap used for $z$. Then by definition of $z$, $d(T^{2jf(n) + i_j + 1} z, y) < \eta$. Similarly, since $2jf(n) + i_j + 1 < 2j'f(n)$, $d(T^{2jf(n) + i_j + 1} z', T^{2j f(n) + i_j + 1} v) < \eta$ by definition of $z'$. Recall that $v \in Y$, and so $T^{2j f(n) + i_j + 1} v \notin U \Longrightarrow d(T^{2j f(n) + i_j + 1} v, y) > \rho \geq 3\eta$. Then, $d(T^{2jf(n) + i_j + 1} z', y) > 2\eta$, and so $d(T^{2jf(n) + i_j + 1} z, T^{2jf(n) + i_j + 1} z') > \eta$, completing the proof that $Z$ is $(n, \eta)$-separated. Then, $$\begin{gathered} \sum_{x \in Z} e^{S_n \phi(x)} = \sum_{j = 1}^{\lfloor (n-f(n))/2f(n) \rfloor} \sum_{(v,w) \in U(j)} e^{S_n \phi(z(j,v,w))}\\ \stackrel{(\ref{partbd3.5})}{\geq} \frac{n - f(n)}{2f(n)^3} C^2 e^{nP(X,T,\phi)} n^{-2/3}. \end{gathered}$$ Again we use Lemma \[sepscaleindep\] to pass to $Z' \subseteq Z$ which is $(n, \delta)$-separated and for which $$\sum_{x \in Z'} e^{S_n \phi(x)} \geq \frac{n - f(n)}{2f(n)^3} C^2 M(\eta)^{-1} e^{nP(X,T,\phi)} n^{-2/3},$$ implying that $$Z(X,T,\phi,n,\delta) \geq \frac{n - f(n)}{2f(n)^3} C^2 M(\eta)^{-1} e^{nP(X,T,\phi)} n^{-2/3}.$$ However, this will contradict Corollary \[speccor\] or \[transcor\] for large enough $k$. Therefore, our assumption was incorrect and $\mu$ is fully supported. Examples ======== Here we present some examples of $(X,T)$ and $\phi$ satisfying our hypotheses for which we believe our results to give the first proof of uniqueness of equilibrium state. We begin with $(X,T)$ with weakened specification properties. The following class of subshifts is defined in [@stanley], which as usual are endowed with $T$ the left shift map. Given any alphabet $A = \{0,1,\ldots,k\}$ and increasing subadditive $h: \mathbb{N} \rightarrow \mathbb{N}$, the **bounded density shift** associated to $k$ and $h$, denoted $X_{k,h}$, is the set of all $x \in A^{\mathbb{Z}}$ so that for all $i \in \mathbb{Z}$ and $n \in \mathbb{N}$, $x(i) + \ldots + x(i + n - 1) \leq h(n)$. By subadditivity, for any bounded density shift, $h(n)/n$ approaches some constant $\alpha$ (called the gradient), and $h(n) \geq n\alpha$ for all $n$. It was shown in [@stanley] that $X_{k,h}$ has specification if and only if $\alpha > 0$ and $h(n) - n\alpha$ is bounded. \[bdspec\] If $h(n) = n \alpha + e(n)$, where $e(n)$ is increasing, then $(X_{k,h},T)$ has non-uniform specification with gap bounds $f(n) := 2e(n)/\alpha$. We first recall that since $(X_{k,h}, T)$ is a subshift, it is expansive for a constant $\delta$ where $d(x,y) \leq \delta \Longrightarrow x(0) = y(0)$. This means that $\delta$-shadowing any $x$ for $n$ iterates is the same as agreeing with $x$ for $n$ letters. This means that the claimed non-uniform specification is implied by the following: for any $w_1, \ldots, w_k \in \mathcal{L}(X_{k,h})$, with lengths $n_1, \ldots, n_k$, and for any $m_1$, $\ldots$, $m_{k-1}$ with $m_i \geq \max(2e(n_i)/\alpha, 2e(n_{i+1})/\alpha) \newline \geq (e(n_i) + e(n_{i+1}))/\alpha$, the word $w = w_1 0^{m_1} w_2 \ldots 0^{m_{k-1}} w_k$ is in $\mathcal{L}(X_{k,h})$. Consider any such $(w_i)$, $(n_i)$, and $(m_i)$. It suffices to show that for every subword $v$ of $w$, the sum of the letters of $v$ is less than or equal to $h(|v|)$. Since $h$ is nondecreasing, it clearly suffices to consider only the case where $v$ neither begins nor ends with a subword of some $0^{m_i}$. We can then write $v$ as $$v = s 0^{m_i} w_{i+1} \ldots w_j 0^{m_j} p,$$ where $1 < i \leq j < k-1$, $s$ is a suffix of $m_{i-1}$ (say of length $a$), and $p$ is a prefix of $m_{j+1}$ (say of length $b$). Since each $w_i$ was in $\mathcal{L}(X_{k,h})$, the sum of the letters of any $w_i$ is at most $h(n_i)$. The sum of the letters of $v$ is then less than or equal to $h(a) + h(n_i) + \ldots + h(n_j) + h(b)$. Also, $$\begin{gathered} h(|v|) = h(a + m_i + n_i + \ldots + n_j + m_j + b) \geq \alpha(a + m_i + n_i + \ldots + n_j + m_j + b) \geq\\ \alpha (a + n_i + \ldots + n_j + b) + e(a) + e(n_i) + \ldots + e(n_j) + e(b) \geq h(a) + h(n_i) + \ldots + h(n_j) + h(b).\end{gathered}$$ Therefore, the sum of the letters of $v$ is less than or equal to $h(|v|)$. Since $v$ was arbitrary, $w$ is in $\mathcal{L}(X_{k,h})$, completing the proof. We do not believe that uniqueness of measure of maximal entropy is known for any bounded density shift without specification. Theorem \[mainthm\], however, yields the following corollary (by taking $\phi = 0$). Any bounded density shift $(X_{k,h},T)$ with $h(n) = n\alpha + e(n)$ for $e(n)$ nondecreasing with $\liminf_{n \rightarrow \infty} e(n)/\ln n = 0$ has a unique measure of maximal entropy (which is fully supported and has the K-property). We also present a class of subshifts with non-uniform transitivity (but not non-uniform specification) to which our results apply. Interestingly, these subshifts cannot have periodic points, and yet our results imply uniqueness of the equilibrium state. \[transex\] Fix any Sturmian subshift $S$ (see Chapter 6 of [@fogg] for an introduction to Sturmian subshifts) and sequence of integers $\{n_k\}$ where $n_k \geq 2n_{k-1} + 2k$ for every $k$. Define the associated subshift $X_{S, \{n_k\}}$ as the set of all $x \in \{0,1\}^{\mathbb{Z}}$ so that for every $i \in \mathbb{Z}$ and $k \in \mathbb{N}$, the word $x(i) \ldots x(i + n_k + 2k - 1)$ contains a $k$-letter word in the language of $S$. Note that by definition, for any $x \in X_{S, \{n_k\}}$, the orbit closure of $x$ contains a point of $S$. Since Sturmian shifts contain no periodic points, this means that $X_{S, \{n_k\}}$ contains no periodic points. We also show that $X_{S, \{n_k\}}$ cannot have non-uniform specification. Suppose for a contradiction that $X_{S, \{n_k\}}$ has non-uniform specification with gap bounds $f(n)$. Choose any $x \in X_{S, \{n_k\}}$ with $x(0) = 0$. Then by taking limits of points which $\delta$-shadow $(x,\ldots,x)$ for $(1,\ldots,1)$ iterates with gaps $(f(1),\ldots,f(1))$, we see that $X_{S, \{n_k\}}$ must contain a sequence $y$ of the form $\ldots 0 w_{-1} 0 w_0 0 w_1 0 \ldots$, i.e. $y(m(1+f(1))) = 0$ for all $m \in \mathbb{Z}$. Then the orbit closure of $y$ contains some $s \in S$, which also has the property that $s(m(1+f(1))) = 0$ for all $m \in \mathbb{Z}$. However, this is impossible; Sturmian shifts have a unique invariant measure with respect to which all powers are ergodic, and so the existence of $s$ would contradict the ergodic theorem. If $\lim_{n \rightarrow \infty} \frac{\ln n_k}{k} = \infty$, then $(X_{S, \{n_k\}}, T)$ has non-uniform transitivity with gap bounds $f(n)$ satisfying $\lim_{n \rightarrow \infty} \frac{f(n)}{\ln n} = 0$. Consider any such $S$, $\{n_k\}$, and associated subshift $X = X_{S, \{n_k\}}$. We claim that $(X, T)$ has non-uniform transitivity with gap bounds $f(n)$ where $f(n) = 2k$ for the minimal $k$ where $n \leq n_k$. This implies the desired result, since clearly $\lim_{n \rightarrow \infty} \frac{f(n)}{\ln n} = 0$ if $\lim_{n \rightarrow \infty} \frac{\ln n_k}{k} = \infty$. As above, $\delta$-shadowing orbit segments is just the same as containing words from the language, so it suffices to show that for any $v,w \in \mathcal{L}(X)$ with $|v| = |w| \leq n_k$, there exists $u$ with $|u| = 2k$ so that $vuw \in \mathcal{L}(X)$. Since $v$ and $w$ can be extended on the left and right to words of length $n_k$ in $\mathcal{L}(X)$, it suffices to treat only the case $|v| = |w| = n_k$; choose any such $v,w$. Of the prefixes of $v$, choose the one with maximal length which is in $\mathcal{L}(S)$, and denote it by $p_v$. Similarly define a prefix $p_w$ of $w$, and suffixes $s_v$ and $s_w$ of $v$ and $w$ respectively. Since $p_v, s_w \in \mathcal{L}(S)$, there exist a left-infinite sequence $x$ and a right-infinite sequence $y$ for which $x p_v, s_w y \in \mathcal{L}(S)$. Similarly, since $s_v, p_w \in \mathcal{L}(S)$, there exist $s, t$ with length $k$ so that $s_v s, t p_w \in \mathcal{L}(S)$. We claim that $x v s t w y \in X$, which will imply that $v st w \in \mathcal{L}(X)$, completing our proof by taking $u = st$. For this proof, we need to show that for every $j$ and every $(n_j + 2j)$-letter subword $z$ of $xvstwy$, $z$ contains a word in $\mathcal{L}(S)$. We break into cases, and first treat the case where $j > k$. Then $z$ has length $n_j + 2j > 2n_{j-1} + 4j \geq 2n_k + 2k + 2j$. Then $z$ must contain a $j$-letter subword of either $x$ or $y$, which by definition is in $\mathcal{L}(S)$. Suppose instead that $j \leq k$. If $z$ contains letters from both $s$ and $t$, then it contains a $j$-letter subword of one of them, which is in $\mathcal{L}(S)$. The remaining case is where $z$ is a subword of either $xvs$ or $twy$; without loss of generality, we assume the latter. If $j = k$, then since $|z| = 2n_k + 2k > n_k + 2k$, $z$ contains either a $j$-letter subword of $t$ or $y$, which is in $\mathcal{L}(S)$. So, we from now on assume $j < k$. This means that we can write $z = qr$, where either $q$ is a suffix of $t$ and $r$ is a prefix of $w$ or $q$ is a prefix of $w$ and $r$ is a prefix of $y$; without loss of generality, we assume the former. If $|q| + |p_w| \geq j$, then the $j$-letter prefix of $z = qr$ is a subword of $t p_w \in \mathcal{L}(S)$, so it would be in $\mathcal{L}(S)$ as well. The only remaining case is $|q| + |p_w| < j$. We claim that here, $r$ contains a $j$-letter word in $\mathcal{L}(S)$. To see this, recall that $w$ was in $\mathcal{L}(X)$, and so there exists $q'$ with $|q'| = |q|$ so that $q' w \in \mathcal{L}(X)$. In particular, this means that $q'r$, which is a subword of $q' w$ with length $n_j + 2j$, contains a $j$-letter word in $\mathcal{L}(S)$. If this word was not entirely contained in $r$, then it would contain a prefix of $w$ of length $|p_w| + 1$, which would be in $\mathcal{L}(S)$, contradicting maximality of $p_w$ in its definition. So, we know that $r$ contains a $j$-letter word in $\mathcal{L}(S)$, implying that $z = qr$ does as well. This shows that $vuw \in \mathcal{L}(X)$ (for $u = st$), completing the proof. Finally, we present a simple condition on a potential $\phi$ which guarantees slowly growing partial sum variation bounds, in the spirit of a proposition from [@bowen]. The proof is essentially identical to the one given in [@bowen], and so we omit it here. \[gbounds\] For $(X,T)$, a potential $\phi$, $n \in \mathbb{N}$, and $\eta > 0$, define $\textrm{Var}(X,T,\phi,n,\eta)$ to be the maximum of $|\phi(x) - \phi(y)|$ over pairs $(x,y)$ where $d(T^i x, T^i y) < \eta$ for all $|i| \leq n$. Then $\phi$ has partial sum variation bounds $g(n)$ at scale $\eta$ defined by $$g(n) = 2\sum_{i=0}^{\lfloor n/2 \rfloor} \textrm{Var}(X,T,\phi,n,\eta).$$ The following is an immediate corollary of Theorem \[gbounds\] and Lemma \[sumscaleindep\]. \[slowgrowth\] If $(X,T)$ is expansive, $\phi$ is a potential, $\eta > 0$, and $\phi$ is a potential with $\lim_{n \rightarrow \infty} n \textrm{Var}(X,T,\phi,n,\eta) = 0$, then $\phi$ has partial sum variation bounds $g(n)$ satisfying $\lim_{n \rightarrow \infty} g(n)/\ln n = 0$. It is simple to construct potentials for which $\textrm{Var}(X,T,\phi,n,\eta)$ grows as slowly as desired; the following is one example. \[potex\] For any increasing $h: \mathbb{N} \rightarrow \mathbb{R}^+$ and the full shift $(X,T)$ on symbols $0$ and $1$, define a potential $\phi_h$ by $\phi_h(x) = \frac{1}{h(k)}$, where $k$ is the maximal integer where $x(-k) = \ldots = x(k)$. (If $x$ consists entirely of $0$s or $1$s, then $\phi(x) = 0$.) If $\lim_{n \rightarrow \infty} h(n)/n = \infty$, then $\phi_h$ satisfies the hypotheses of Corollary \[slowgrowth\]. If $\frac{1}{h(n)}$ is not summable, then $\phi_h$ is not Bowen. We claim that $\textrm{Var}(X,T,\phi_h,n,\delta) = \frac{1}{h(n)}$. To see this, first note that a pair $(x,y)$ satisfies $d(T^i x, T^i y) < \delta$ for $|i| \leq n$ iff $x(-n) \ldots x(n) = y(-n) \ldots y(n)$. If $\phi_h(x) = \frac{1}{h(i)}$ for some $i < n$, then $x(-i) = \ldots = x(i)$ and either $x(-(i+1))$ or $x(i+1)$ is not equal to $x(0)$. The same is then true of $y$, so $\phi_h(y) = \phi_h(x)$. Therefore, if $\phi_h(x) \neq \phi_h(y)$, then both are less than or equal to $\frac{1}{h(n)}$, so $|\phi_h(x) - \phi_h(y)| \leq \frac{1}{h(n)}$, implying $\textrm{Var}(X,T,\phi_h,n,\delta) \leq \frac{1}{h(n)}$. Finally, $x = 0^{\mathbb{Z}}$ and $y$ defined by $y(i) = 0$ iff $|i| \leq n$ have $|\phi_h(x) - \phi_h(y)| = \frac{1}{h(n)}$, so $\textrm{Var}(X,T,\phi_h,n,\delta) = \frac{1}{h(n)}$. Since $\lim_{n \rightarrow \infty} h(n)/n = \infty$, $\lim_{n \rightarrow \infty} n \textrm{Var}(X,T,\phi_h,n,\delta) = 0$, and so $\phi_h$ satisfies the hypotheses of Corollary \[slowgrowth\]. Similarly to above, take $x = 0^{\mathbb{Z}}$ and $y$ defined by $y(i) = 0$ iff $i \geq 0$. Then $d(T^i x, T^i y) < \delta$ for $0 \leq i < n$, and $\left| S_n \phi_h(x) - S_n \phi_h(y) \right| = \sum_{i = 1}^n \frac{1}{h(i)}$. It is then clear that if $\frac{1}{h(n)}$ is not summable, then $\phi_h$ does not have the Bowen property, completing the proof. Theorems \[mainthm\] and \[mainthm2\] then provide uniqueness of equilibrium state, its full support, and sometimes the $K$-property, for many of these examples, including measures of maximal entropy for various bounded density shifts and the shifts $X_{S, \{n_k\}}$ of Example \[transex\]. However, the easiest new application is probably the uniqueness of equilibrium state for any expansive $(X,T)$ with weak specification and any $\phi$ satisfying Corollary \[slowgrowth\]. This even includes examples on manifolds. For instance, if $X = [0,1)$ and $T: x \mapsto 2x \pmod 1$, then though $(X,T)$ is non-invertible, its natural extension is invertible and has weak specification (for $f = 0$). A simple example of $\phi$ on $(X,T)$ which is not Bowen but has unique equilibrium state by Corollary \[slowgrowth\] is $\phi(x) = \frac{1}{1 + \log(1/x) \log \log(1/x)}$. [^1]: The author gratefully acknowledges the support of NSF grant DMS-1500685.
--- abstract: 'Most neural machine translation (NMT) models operate on source and target sentences, treating them as sequences of words and neglecting their syntactic structure. Recent studies have shown that embedding the syntax information of a source sentence in recurrent neural networks can improve their translation accuracy, especially for low-resource language pairs. However, state-of-the-art NMT models are based on self-attention networks (e.g., Transformer), in which it is still not clear how to best embed syntactic information. In this work, we explore different approaches to make such models syntactically aware. Moreover, we propose a novel method to incorporate syntactic information in the self-attention mechanism of the Transformer encoder by introducing attention heads that can attend to the dependency parent of each token. The proposed model is simple yet effective, requiring no additional parameter and improving the translation quality of the Transformer model especially for long sentences and low-resource scenarios. We show the efficacy of the proposed approach on NC11 English$\leftrightarrow$German, WMT16 and WMT17 English$\rightarrow$German, WMT18 English$\rightarrow$Turkish, and WAT English$\rightarrow$Japanese translation tasks.' author: - Emanuele Bugliarello - | Naoaki Okazaki\ \ Department of Computer Science, School of Computing, Tokyo Institute of Technology\ \ [emanuele.bugliarello@nlp.c.titech.ac.jp, okazaki@c.titech.ac.jp]{} bibliography: - 'references.bib' title: 'Improving Neural Machine Translation with Parent-Scaled Self-Attention' --- Introduction ============ Neural machine translation (NMT) models [@sutskever2014sequence; @bahdanau2014neural] have recently become the dominant paradigm to machine translation, obtaining outstanding empirical results by solving the translation task with simple end-to-end architectures that do not require tedious feature engineering typical of previous approaches. Most NMT models are only trained on corpora consisting of pairs of parallel sentences, disregarding any prior linguistic knowledge with the assumption that it can automatically be learned by an attention mechanism [@luong2015effective]. However, despite the impressive results achieved by such attention-based NMT models, [@shi2016does ]{} found that these models still fail to capture deep structural details. Being able to incorporate grammatical knowledge in a language is a promising approach to improve NMT models as statistical machine translation (SMT) models [@brown1990statistical; @koehn2003statistical] have shown relevant gains in translation accuracy in the past [@yamada2001syntax; @liu2006tree; @chiang2007hierarchical]. Over the past few years, there have been a number of studies showing that syntactic information has the potential to also improve NMT models [@luong2015multi; @sennrich2016linguistic; @P17-1064; @P17-2012; @AAAI1816060]. However, the majority of recent syntax-aware NMT models [@tran2018inducing; @bastings2019modeling] are based on recurrent neural networks (RNNs) [@elman1990finding], a class of attentional encoder-decoder models [@bahdanau2014neural] that sequentially process an input sentence, one token at a time. [@vaswani2017attention ]{} recently introduced the Transformer model, an encoder-decoder architecture solely based on self-attention [@cheng2016long; @lin2017structured]. Transformer models can achieve state-of-the-art results on various translation tasks with much faster training time. The key factor leading to the Transformer’s superior performance is its self-attention mechanism that allows to efficiently ($\mathcal{O}(1)$) access to any tokens in a sequence by directly attending to each pair of tokens. Nevertheless, recent studies have shown that self-attention networks (SANs) benefit from modeling local contexts, reducing the dispersion of the attention distribution by restricting it to neighboring representations [@shaw-etal-2018-self; @yang-etal-2018-modeling; @Yang2019ContextAwareSN]. Moreover, while SANs can focus on the entire sequence, recent work [@tran2018importance; @tang2018self] suggests that they might not succeed, especially in low-resource scenarios, at capturing the inherent syntactic structure of languages as well as recurrent models. This feature could prove useful for the models in order to reduce ambiguity and preserve agreement when translating. In response, we propose to enhance the self-attention mechanism by incorporating source-side syntactic information to further improve the performance of SANs in machine translation without compromising their flexibility. Specifically, we introduce *parent-scaled self-attention* (Pascal): a novel, parameter-free local attention mechanism that lets the model focus on the dependency parent of each token when encoding the source sentence. Our method is simple yet effective, resulting in better translation quality with no additional parameter to be learned or computational overhead. To demonstrate the effectiveness and generality of our approach, we run extensive experiments on the popular large-scale WMT16 and WMT17 English$\rightarrow$German (En-De), and WAT English$\rightarrow$Japanese (En-Ja) translation tasks, as well as on News Commentary v11 English$\leftrightarrow$German (En-De, De-En) and WMT18 English$\rightarrow$Turkish (En-Tr) as low-resource scenarios. Our results show that the proposed approach consistently exhibits significant improvements in translation quality, especially for long sentences, not only over previous NMT approaches using dependency information and the Transformer baseline, but also against other strong syntax-aware variants of this model. To the best of our knowledge, this is the first work that investigates and exploits the core properties of the Transformer architecture to incorporate source-side syntax to further improve its translation quality. Related Work ============ #### Notations. In this paper, vectors are column vectors represented by bold lowercase letters (e.g., $\mathbf{p}$, $\mathbf{q}$). Matrices and tensors are denoted by bold uppercase letters (e.g., $\mathbf{P}$, $\mathbf{Q}$). For a given matrix $\mathbf{A}$, $\mathbf{a}_i$ denotes its $i$-th row as a column vector and $a_{ij}$ is its entry in row $i$ and column $j$. Neural Machine Translation -------------------------- Neural Machine Translation (NMT) systems typically learn a probabilistic mapping $\mathbb{P}\left(\mathbf{y}\vert\mathbf{x}\right)$ from a source language sentence $\mathbf{x} = \left[x_1, x_2, \dots, x_{T_x}\right]$ to a target sentence $\mathbf{y} = \left[y_1, y_2, \dots, y_{T_y}\right]$, where $x_i$ and $y_t$ denote the $i$-th and $t$-th tokens in $\mathbf{x}$ and $\mathbf{y}$, respectively. Most NMT systems are based on an encoder-decoder framework. In this setting, the encoder reads the source sentence and generates $T_x$ internal representations (context vectors) $\mathbf{h}_i$ that are then used by the decoder to compute the conditional probability of each target token given its preceding tokens $\mathbf{y}_{<t}$ and the source sentence, $\mathbb{P}\left(y_t\vert\mathbf{y}_{<t}, \mathbf{x}\right) \propto \exp\left(y_t; \mathbf{r}_t, \mathbf{c}_t\right)$. Here, $\mathbf{r}_t$ denotes the decoder hidden state at time $t$ and $\mathbf{c}_t$ is the contextual information used in generating $y_t$ from the encoder hidden states. Nowadays, context vectors $\mathbf{c}_t$ are computed as a weighted average of $\mathbf{h}_i$, with weights given by an attention mechanism that assigns an alignment score to the pair of input at position $i$ and output at position $t$. Let $\mathbf{\Theta}$ be the parameters of the neural network and $D$ the source-target sentence pairs in the training data. The learning objective of the model is: $$\mathbf{\Theta}^* = \underset{\mathbf{\Theta}}{\operatorname*{argmax}} \sum_{\left(\mathbf{x}, \mathbf{y}\right)\in D}\sum_{t=1}^{T_y} \log \mathbb{P}\left(y_t\vert \mathbf{y}_{<t}, \mathbf{x}; \mathbf{\Theta}\right).$$ Syntax-Aware Neural Machine Translation --------------------------------------- Several approaches have been investigated in the literature in order to incorporate dependency syntax in NMT models. For instance, [@eriguchi2016tree ]{} integrate dependency trees into NMT models by a tree-based encoder that follows the phrase structure of a sentence. To alleviate the low efficiency of processing trees, [@P17-1064 ]{} linearize constituency parse trees into sequences of symbols mixed with words and syntactic tags. [@D17-1012 ]{} instead propose to combine together head information and sequential words as inputs to the source encoder. [@D17-1209 ]{} exploit graph-convolutional networks to produce syntax-aware representations of words. The majority of these studies focused on recurrent networks. [@wu2018dep2dep ]{} were also the first to evaluate an approach to embed syntactic information in NMT with a Transformer model. In this work, the authors first pair the source sentence encoder with two additional encoders that embed source dependency trees acquired by pre-order and post-order traversals. Then, the resulting context vectors are combined via a feed-forward layer and passed to two decoders, one modeling the target word sequence and the other modeling the parsing action sequence for the target dependency tree. The full model is trained to maximize the joint probability of target translations and their parsing trees. Concurrently to our work, two other studies have proposed methodologies to introduce syntactic knowledge into Transformer-based models. First, [@zhang-etal-2019-syntax ]{} integrate source-side syntax into NMT by concatenating the intermediate representations of a dependency parser to ordinary word embeddings. The authors rely on the hidden representations of the encoder of a dependency parser model in order to alleviate the error propagation problem from $1$-best tree outputs by syntax parsers. This approach, however, does not allow to learn sub-word units at the source side, requiring a larger vocabulary to minimize the number of out-of-vocabulary (OOV) words. In contrast, we explicitly account for sub-word units in our approach and also propose a regularization technique to make our attention mechanism robust to noisy parses. Second, [@currey-heafield-2019-incorporating ]{} propose two simple data augmentation techniques to incorporate source-side syntax. Their first method is a multi-task approach to parse and translate source sentences by prepending and appending special tags to each sentence and train the model to either translate the source sentence or output its linearized constituency parse. In their second approach, they train a Transformer model to translate both unparsed and parsed source sentences into unparsed target sentences. While these studies enhance the performance of the Transformer model, they treat it as a black box. Conversely, we explicitly enhance its self-attention mechanism (a core component of this architecture) to include syntactic information and further improve its translation accuracy. Model ===== In order to design a neural network that is efficient to train and that exploits syntactic information while producing high-quality translations, we base our model on the Transformer [@vaswani2017attention] architecture and upgrade its encoder with *parent-scaled self-attention* (Pascal) heads at layer $l_s$. Pascal is a novel, guided attention mechanism that enforces contextualization from the syntactic dependencies of each source token. In practice, we replace standard self-attention heads with Pascal ones in the first layer. In fact, the inputs to the first layer are word embeddings, which lack contextual information. By conditioning the representation of a given word on its parent, we can readily augment it and propagate it to upper layers. Our multi-head Pascal sub-layer has the same number $H$ of attention heads as other layers. #### Transformer encoder. The Transformer architecture follows the standard encoder-decoder paradigm, making use of positional embeddings to overcome the sequential nature of recurrent networks, which precludes them from parallelization among samples. The Transformer encoder consists of a stack of $N$ identical layers, each having two sub-layers coupled with layer normalization and residual connections. The first sub-layer computes attention weights for each token in a sequence via a multi-head attention mechanism. Specifically, $H$ heads are used to compute distributions $\mathbf{M}^h$ for each token in a sequence, which are then concatenated together to let the model attend to different representations: $$\mathbf{M}^h = {\rm softmax}\left(\frac{\mathbf{Q}^h\cdot \mathbf{K}^h}{\sqrt{d}}\right)\mathbf{V}^h,$$ $${\rm MultiHead} = {\rm concat}\left(\mathbf{M}^1, \dots, \mathbf{M}^H\right)\cdot\mathbf{W}_O,$$ where $\mathbf{Q}^h=\mathbf{X}\cdot\mathbf{W}_Q^h$, $\mathbf{K}^h=\mathbf{X}\cdot\mathbf{W}_K^h$, $\mathbf{V}^h=\mathbf{X}\cdot\mathbf{W}_V^h$ and $\mathbf{W}_Q^h$, $\mathbf{W}_K^h$, $\mathbf{W}_V^h$, $\mathbf{W}_O$ are parameter matrices, $\mathbf{X}$ is the representation of the input sequence from the previous layer, and $d$ is a constant that depends on the size of the model. The second sub-layer is a two-layer feed-forward network with a ReLU activation function between them. The Transformer decoder has a similar architecture; we refer the reader to [@vaswani2017attention ]{} for more details. Dependency position ------------------- Similarly to previous work, instead of just providing sequences of tokens, we supply the encoder with dependency relations given by an external parser. Our approach explicitly exploits sub-word units, which enable open-vocabulary translation: after generating sub-word units, we compute the middle position of each word in terms of number of tokens. For instance, if a word in position $4$ is split into three tokens, e.g., in positions $6$, $7$ and $8$, its middle position is $7$. We finally map each sub-word unit of a given word to the middle position of its parent. For the root word, we define its parent to be itself, resulting in a parse that is a directed graph. The input to our encoder is then a sequence of $T$ tokens and the absolute positions of their parents. Parent-Scaled Self-Attention {#sec:sasa} ---------------------------- Figure \[fig:sasa\] shows our parent-scaled self-attention sub-layer. Here, for a sequence of length $T$, the inputs to each head are a matrix $\mathbf{X}\in\mathbb{R}^{T\times d_{model}}$ of token embeddings and a vector $\textbf{p}\in\mathbb{R}^T$ whose $t$-th entry $p_t$ is the middle position of the $t$-th token’s dependency parent. Following [@vaswani2017attention ]{}, in each attention head $h$, we compute three vectors (called query, key and value) for each token, resulting in the three matrices $\mathbf{K}^h\in\mathbb{R}^{T\times d}$, $\mathbf{Q}^h\in\mathbb{R}^{T\times d}$, and $\mathbf{V}^h\in\mathbb{R}^{T\times d}$ for the whole sequence, where $d = d_{model}/H$. We then compute dot products between each query and all the keys, giving scores of how much focus to place on other parts of the input when encoding a token at a given position. The scores are divided by $\sqrt{d}$ to alleviate the vanishing gradient problem arising if dot products are large: $$\mathbf{S}^h = \mathbf{Q}^h\cdot{\mathbf{K}^h} / \sqrt{d}.$$ The main contribution of our work consists of weighing the scores of the token at position $t$, $\textbf{s}_t$, by the distance of each token from the position of $t$’s dependency parent: $$n^h_{tj} = s^h_{tj} ~ d^p_{tj}, ~~\text{ for } j=1,...,T,$$ where $\textbf{n}^h_{t}$ is the $t$-th row of the matrix $\mathbf{N}^h\in\mathbb{R}^{T\times T}$ representing scores normalized by the proximity to $t$’s parent. $d^p_{tj} = {\rm dist}(p_t, j)$ is the $\left(t, j\right)^{th}$ entry of the matrix $\mathbf{D}^p\in\mathbb{R}^{T\times T}$ containing, for each row $\mathbf{d}_t$, the distances of every token $j$ from the middle position of token $t$’s dependency parent $p_t$. In this paper, we compute this distance as the probability density of a normal distribution centered at $p_t$ and with variance $\sigma^2$, $\mathcal{N}\left(p_t,\sigma^{2}\right)$: $${\rm dist}(p_t, j) = f_{\mathcal{N}}\left(j\middle\vert p_t, \sigma^2\right) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{\left(j - p_t\right)^2}{2\sigma^2}}.$$ Finally, we apply a softmax function to yield a distribution of weights for each token over all the tokens in the sentence, and multiply the resulting matrix with the value matrix $\textbf{V}^h$, obtaining the final representations $\mathbf{M}^h$ for Pascal head $h$. One of the major strengths of our proposal is being parameter-free: no additional parameter is required to train a Pascal sub-layer as $\textbf{D}^p$ can be computed by a distance function that depends only on the vector of tokens’ parents positions and evaluated using fast matrix operations on GPUs. #### Parent ignoring. Due to the lack of parallel corpora with gold-standard parses, we rely on noisy annotations from an external parser. However, the performance of syntactic parsers drops abruptly when evaluated on out-of-domain data [@dredze2007frustratingly]. To prevent our model from overfitting to noisy dependencies, we introduce a regularization technique for our Pascal sub-layer: *parent ignoring*. In a similar vein as dropout [@JMLR:v15:srivastava14a], we disregard information during the training phase. Here, we ignore the position of the parent of a given token by randomly setting each row of $\mathbf{D}^p$ to $\mathbf{1}\in\mathbb{R}^T$ with some probability $q$. #### Gaussian weighing function. The choice of weighing each score by a Gaussian probability density is motivated by two of its properties: bell-shaped curve and non-zero values. First, it allows to concentrate most of the probability density at the mean of the distribution, which we set to the middle position of the sub-word units of the dependency parent of each token. In our experiments, we find that most words in the vocabularies are not split into sub-words, hence allowing Pascal to mostly focus on the actual parent. In addition, non-negligible weights are placed on the neighbors of the parent token, allowing the attention mechanism to also attend to them. This could be useful, for instance, to learn idiomatic expressions such as prepositional verbs in English. Second, we exploit the support of Gaussian-like distributions: while most of the weight is placed in a small window of tokens around the mean of the distribution, all the values in the sequence are actually multiplied by non-zero factors. This allows a token $j$ farther away from the dependency parent of token $t$, $p_t$, to still play a role in the representation of $t$ if its attention score $s_{tj}^h$ is high. #### Our attention mechanism can be seen as an extension of the local attention introduced by [@luong2015effective ]{}, with the alignment now guided by syntactic information. [@yang-etal-2018-modeling ]{} proposed a similar method that learns a Gaussian bias that is added to, instead of multiplied by, the original attention distribution. As we will see in the next section, our model significantly outperforms them. ----------------- ----------- ----------- ------- ------- NC11     En-De 238,843 233,483 2,169 2,999 WMT18 En-Tr 207,373 3,000 3,007 WMT16 En-De 4,500,962 4,281,379 2,169 2,999 WMT17 En-De 5,852,458 2,999 3,004 WAT       En-Ja 3,008,500 1,790 1,812 ----------------- ----------- ----------- ------- ------- : Number of sentences in our evaluation datasets.[]{data-label="tab:sizes"} Experiments =========== Experimental Setup ------------------ We evaluate the efficacy of the proposed approach on standard, large-scale benchmarks as well as on low-resource scenarios, where Transformer models were shown to induce poorer syntactic relationships than on high-resource ones. #### Pre-processing. Unless otherwise specified, we follow the same pre-processing steps as [@vaswani2017attention ]{}. We use Stanford CoreNLP [@manning-EtAl:2014:P14-5] to generate syntactic information in our experiments, and jointly learn byte-pair encodings (BPE) [@P16-1162] for source and target languages in each parallel corpus. #### Baselines. We report previous results in syntax-aware NMT for completeness and implement the following four Transformer-based approaches as stronger baselines: - **Transformer:** We train a Transformer model as a strong, standard baseline for our experiments using the hyperparameters in the latest Google’s Tensor2Tensor version ($3$). - $\textbf{+S\&H:}$ Following [@sennrich2016linguistic ]{}, we introduce syntactic information in the form of dependency labels in the embedding matrix of the Transformer encoder. More specifically, each token is associated with its dependency label which is first embedded into a vector representation of size $10$ and then used to replace the last $10$ embedding dimensions of the token embedding, ensuring a final size that matches the original one. - **+SISA:** Syntactically-Informed Self-Attention [@D18-1548]. In one attention head $h$, $\mathbf{Q}^h$ and $\mathbf{K}^h$ are computed through a feed-forward layer and the key-query dot product to obtain attention weights is replaced by a bi-affine operator $\mathbf{U}$. These attention weights are further supervised to attend to each token’s parent by interpreting each row $t$ as the distribution over possible parents for token $t$. Here, we extend the authors’ approach to BPE by defining the parent of a given token as its first sub-word unit (root of a word). The model is trained to maximize the joint probability of translations and parent positions. - $\textbf{+C\&H:}$ The multi-task approach from [@currey-heafield-2019-incorporating ]{} that uses a standard Transformer model to learn to both parse and translate source sentences. Each source sentence is first duplicated and associated its linearized parse as target sequence. To distinguish between the two tasks, a special tag indicating the desired task is prepended and appended to each source sentence. Finally, parsing and translation training data is shuffled together. #### Datasets. Following [@D17-1209 ]{}, we train on News Commentary v11 (NC11) dataset[^1] with English$\rightarrow$German (En-De) and German$\rightarrow$English (De-En) tasks so as to simulate low-resource cases and to evaluate the performance of our models for different source languages. We also train on the full WMT16 dataset for En-De, using *newstest2015* and *newstest2016* as validation and test sets, respectively, in each of these experiments. Moreover, we notice that these datasets contain sentences in different languages and use `langdetect`[^2] to remove sentences whose main language does not match with source and target ones. The sizes of the final datasets are listed in Table \[tab:sizes\], and we train our model on the filtered versions. We also train our models on WMT18[^3] English$\rightarrow$Turkish (En-Tr) as a standard low-resource scenario. Models are evaluated on *newstest2016* and tested on *newstest2017*. Previous studies on syntax-aware NMT have commonly been conducted on the WMT16 En-De and WAT English$\rightarrow$Japanese (En-Ja) tasks, while concurrent approaches are evaluated on the WMT17[^4] En-De task. In order to provide a generic and comprehensive evaluation of our proposed approach on large-scale data, we also train our models on the latter tasks. We follow the WAT18 pre-processing steps[^5] for experiments on En-Ja but use Cabocha[^6] to tokenize target sentences. On WMT17, we use *newstest2016* and *newstest2017* as validation and test sets. For each translation task, we jointly learn byte-pair encodings using $16,000$ merge operations on low-resource experiments, and $32,000$ merge operations on large-scale ones. #### Training details. **$lr$** **$(\beta_1, \beta_2)$** **$h_{Pascal}$** **$q$** ----------------- ---------- -------------------------- ------------------ --------- NC11     En-De 0.0007 (0.9, 0.997) 2 0.4 NC11     De-En 0.0007 (0.9, 0.997) 8 0.0 WMT18 En-Tr 0.0007 (0.9, 0.98) 7 0.3 WMT16 En-De 0.0007 (0.9, 0.98) 5 0.0 WMT17 En-De 0.0007 (0.9, 0.997) 7 0.3 WAT       En-Ja 0.0007 (0.9, 0.997) 7 0.0 : Hyperparameters for the reported models. $lr$ denotes the maximum learning rate, $(\beta_1, \beta_2)$ are Adam’s exponential decay rates, $h_{Pascal}$ is the number of parent-scaled self-attention heads, and $q$ is the parent ignoring probability.[]{data-label="tab:hypers"} We implement our models in PyTorch on top of the Fairseq toolkit[^7], which provides a re-implementation of the Transformer. All experiments are based on the base Transformer architecture and optimized following the learning schedule of [@vaswani2017attention ]{} with $8,000$ warm-up steps. Similarly, we use label smoothing $\epsilon_{ls} = 0.1$ during training and employ beam search with a beam size of $4$ and length penalty $\alpha = 0.6$ at inference time. We use a batch size of $32K$ tokens and run experiments on a cluster of $4$ machines, each having $4$ Nvidia P100 GPUs. For each model, we run a small grid search over the hyperparameters and select the ones giving the highest BLEU scores on the validation sets. Table \[tab:hypers\] lists the hyperparameters of the proposed model, including the number of Pascal heads. In our datasets, we observe that most words are not split after BPE and that the vast majority are within a few tokens. For instance, $92\%$ of the English words in our WMT16 training data is not split after BPE and $99.99\%$ of them are at most split into $7$ sub-word units. Hence, a window size of $3$ would be most suitable for Pascal to attend to dependency parents. This can be achieved by setting a variance of $1$ for the Gaussian weighing function, which gives non-negligible weights to tokens at most $3$ positions away from the mean. We use the <span style="font-variant:small-caps;">sacreBLEU</span>[^8] tool to compute case-sensitive BLEU [@Papineni:2002:BMA:1073083.1073135] scores. Statistical significance ($p < 0.01$) against the Transformer baseline via bootstrap re-sampling [@koehn-2004-statistical] is marked with $^\Uparrow$. When evaluating En-Ja translations, we follow the procedure at WAT by computing BLEU scores after tokenizing target sentences using KyTea[^9]. In addition, we also report RIBES [@isozaki2010ribes] scores on this translation task. Following [@vaswani2017attention ]{}, we train Transformer-based models for $100,000$ steps on large-scale data. On small-scale data, we train for $20,000$ steps and use a dropout probability $P_{drop} = 0.3$ as they let the Transformer baseline achieve higher performance on this size of data. For instance, in WMT18 En-Tr, our baseline reaches $+3.5$ BLEU points compared to the one in [@currey-heafield-2019-incorporating ]{}. Main Results ------------ ---------------------- --------------------- --------------------- --------------------- [@D17-1209 ]{} 16.1 Transformer 25.0 26.6 13.1 Transformer $+ S\&H$ 25.5 26.8 13.0 Transformer $+ SISA$ 25.5 27.2 13.6 Transformer $+ C\&H$ 24.8 26.7 **14.0** Proposed approach **25.9**$^\Uparrow$ **27.4**$^\Uparrow$ **14.0**$^\Uparrow$ ---------------------- --------------------- --------------------- --------------------- : Evaluation results on small-scale data.[]{data-label="tab:small_res"} [1]{} \[tab:full\] **WMT16** **WMT17** -------------------------------------------------- --------------------- ------------------- [@sennrich2016linguistic ]{} 28.4 [@D17-1209 ]{} 23.9 [@tran2018inducing ]{} 30.3 24.3 SE+SD-Transformer [@wu2018dep2dep] **26.2** Mixed Enc. [@currey-heafield-2019-incorporating] 26.0 Transformer 33.0 25.5 Transformer $+ S\&H$ 31.9 25.1 Transformer $+ SISA$ 33.6 25.9 Transformer $+ C\&H$ 32.4 24.6 Proposed approach **33.9**$^\Uparrow$ *26.1*$^\Uparrow$ : Evaluation results on large-scale data.[]{data-label="tab:results"} [1]{} \[tab:wat\] **BLEU** **RIBES** ---------------------------- --------------------- ---------------------- [@eriguchi2016tree ]{} 34.9 81.58 LGP-NMT+ [@D17-1012] 39.4 82.83 SE+SD-NMT [@wu2018dep2dep] 36.4 81.83 Transformer 43.1 83.46 Transformer $+ S\&H$ 42.8 83.88 Transformer $+ SISA$ 43.2 83.51 Transformer $+ C\&H$ 43.1 84.87 Proposed approach **44.0**$^\Uparrow$ **85.21**$^\Uparrow$ : Evaluation results on large-scale data.[]{data-label="tab:results"} #### Low-resource experiments. Table \[tab:small\_res\] presents the BLEU scores on the test data of the small-scale datasets introduced above. Clearly, the Transformer model vastly outperforms a previous syntax-aware RNN-based approach, proving it to be a strong baseline in our experiments. The proposed approach outperforms all the baselines, with consistent gains over the base Transformer regardless of the source language. Specifically, it leads to $+0.9$ BLEU points on NC11 En-De and WMT18 En-Tr, and $+0.8$ on NC11 De-En. The table also shows that the simple dependency-aware approach of [@sennrich2016linguistic ]{} does not lead to notable advantages when applied to the embeddings of the Transformer model. On the other hand, the SISA mechanism leads to modest but consistent improvements across all tasks, confirming that it can also be used to improve NMT. Finally, we observe that the multi-task approach of [@currey-heafield-2019-incorporating ]{} can benefit of better parameterization of the Transformer, leading to $+3.4$ BLEU points on the En-Tr task compared to the original results. While its performance matches our model on this task, it only attains comparable performance with the base Transformer on the NC11 tasks. **NC11 En-De** **NC11 De-En** **WMT18 En-Tr** **WMT16 En-De** **WMT17 En-De** **WAT En-Ja** ------------------- ---------------- ---------------- ----------------- ----------------- ----------------- --------------- Transformer 22.6 23.8 12.6 29.0 31.5 42.2 + data filtering 22.8 (+0.2) 24.0 (+0.2) 28.7 (-0.3) + Pascal 23.0 (+0.2) 24.6 (+0.6) 13.6 (+1.0) 29.2 (+0.5) 31.6 (+0.1) 43.5 (+1.3) + parent ignoring 23.2 (+0.2) 13.7 (+0.1) 32.1 (+0.6) [UTF8]{}[min]{} ---------- ------------------------------------------------------ **SRC** In a cooling experiment , **only** a tendency agreed **BASE** 冷却 実験 で は ，**わずか な** 傾向 が 一致 し た **OURS** 冷却 実験 で は 傾向 **のみ** 一致 し た ---------- ------------------------------------------------------ : Example of incorrect translation by the baseline.[]{data-label="tab:example"} [.5]{} **Layer** **En-De** **De-En** ----------- ----------- ----------- 1 **23.2** **24.6** 2 22.5 20.1 3 22.5 23.8 4 22.6 23.8 5 22.9 23.8 6 22.4 23.9 : []{data-label="tab:var"} [.5]{} **Variance** **En-De** **De-En** -------------- ----------- ----------- Parent-only 22.5 22.4 1 **23.2** **24.6** 4 22.7 24.3 9 22.8 24.3 16 22.7 24.4 25 22.8 24.1 : []{data-label="tab:var"} #### Large-scale experiments. Table \[tab:results\] lists the evaluation results on our large-scale datasets. As seen in small-scale experiments, our Transformer baseline outperforms previous RNN-based approaches. We now observe that adding syntactic information in the embeddings of the Transformer encoder leads to slightly lower BLEU scores than the baseline. This is also observed for our re-implementation of the multi-task approach of [@currey-heafield-2019-incorporating ]{}, which, however, again achieves a higher score than the one reported by the authors on WMT17. SISA, on the other hand, is consistent across small-scale and large-scale experiments but still giving modest improvements over the baseline. Finally, the proposed approach achieves the highest performance on the WMT16 and WAT tasks with a considerable $+0.9$ over the baseline’s BLEU scores. Moreover, our model also achieves a significant boost over the baseline ($+1.75$) in terms of RIBES, a metric with stronger correlation with human judgments than BLEU in En$\leftrightarrow$Ja translations. On WMT17, our slim model compares favorably to previous approaches, achieving the highest performance across all syntax-aware approaches that make use of source-side syntax. In fact, our method is only outperformed by the approach of [@wu2018dep2dep ]{}, which makes use of both source-side and target-side syntactic information. Not only does this limit the application of their method to low-resource target languages, but this model is also much more complex than ours, requiring three encoders and two decoders. Lastly, note that modest improvements in our WMT17 task should not be surprising as the data consists of $5.9M$ parallel sentences ($+1.4M$ compared to WMT16) and [@raganato-tiedemann-2018-analysis ]{} showed that the Transformer model can already learn better syntactic relationships on larger datasets. #### Discussion. These results show that previous approaches for syntax NMT might give lower gains in the Transformer model. We also find, on average, stronger and more consistent results across data sizes when embedding syntax in the attention mechanism and that more syntax-aware heads further improve the translation accuracy of the Transformer. Overall, our model achieves substantial gains given the grammatically rigorous structure of English and German. We expect performance gains to further increase on less rigorous sources and with better parses [@zhang-etal-2019-syntax]. Analysis -------- In this section, we further investigate the performance of the proposed approach, ground our design choices and show the performance of each component through an ablation study. #### Performance by sentence length. As shown in Figure \[fig:sasa2\], our model is particularly useful when translating long sentences, obtaining more than $+2$ BLEU points when translating long sentences in all low-resource experiments, and $+3.5$ BLEU points on the large En-Ja task. However, only a few sentences ($1\%$) in the evaluation datasets are long. #### Qualitative performance. Table \[tab:example\] presents an example where our model correctly translated the source sentence while the Transformer baseline made a syntactic error, misinterpreting the adverb “only" as an adjective of “tendency”. #### Pascal layer. When we introduced our model, we motivated our design choice of placing Pascal heads in the first layer in order to enrich the representations of words from their isolated embeddings by introducing contextualization from their parents. We ran an ablation study on the NC11 data in order to verify our hypothesis. As shown in Table \[tab:pascal\_layer\], the performance of our model on the validation sets is lower when placing Pascal heads in upper layers; a trend that we also observed with the SISA mechanism. These results corroborate the findings of [@raganato-tiedemann-2018-analysis ]{} who noticed that, in the first layer, more attention heads solely focus on the word to be translated itself rather than its context. We can then deduce that enforcing syntactic dependencies in the first layer effectively leads to better word representations, which further enhance the translation accuracy of the Transformer model. Investigating the performance of multiple syntax-aware layers is left as future work. #### Gaussian variance. Another design choice we made was the variance of the Gaussian weighing function. We set it to $1$ in our experiments motivated by the statistics of our datasets, where the vast majority of words is at most split into a few tokens after applying BPE. Table \[tab:var\] corroborates our choice, showing higher BLEU scores on the NC11 validation sets when the variance is equal to $1$. Here, “parent-only" is the case where weights are only placed on parents. #### Ablation. Table \[tab:ablation\] lists the contribution of each proposed technique (data filtering, Pascal and parent ignoring), in an incremental fashion, whenever they were used by a model. While removing sentences whose languages do not match the translation task can lead to better performance (NC11), the precision of the detection tool assumes a major role at large scale. In WMT16, `langdetect` removes more than $200K$ sentences and leads to performance losses. It would also drop $19K$ sentences on the clean WAT En-Ja dataset. The proposed Pascal mechanism is the component that most improves the performance of the models, achieving up to $+1.0$ and $+1.3$ BLEU on En-Tr and En-Ja, respectively. With the exception of NC11 En-De, we find parent ignoring useful on the noisier WMT18 En-Tr and WMT17 En-De datasets. In the former, low-resource case, the benefits of parent ignoring are minimal, but it proves fundamental on the large-scale WMT17 data, where it leads to significant improvements when paired with the Pascal mechanism. Finally, looking at the number of Pascal heads in Table \[tab:hypers\], we notice that most models rely on a large number of syntax-aware heads. [@raganato-tiedemann-2018-analysis ]{} found that only a few attention heads per layer encoded a significant amount of syntactic dependencies. Our study shows that the performance of the Transformer model can be improved by having more attention heads learn syntactic dependencies. Conclusion ========== This study provides a comprehensive investigation of approaches to induce syntactic knowledge into self-attention networks. We also introduce *Pascal*: a novel, parameter-free self-attention mechanism that enforces syntactic dependencies in the Transformer encoder with negligible computational overhead. Through extensive evaluations on a variety of translation tasks, we find that the proposed model leads to higher improvements in translation quality than other approaches. Moreover, we see that exploiting the core components of the Transformer model to embed linguistic knowledge leads to higher and more robust gains than treating it as a black box, and that multiple syntax-aware attention heads lead to superior performance. Our results show that the quality of the Transformer model can be improved by syntactic knowledge, motivating future research in this direction. Acknowledgments =============== The research results have been achieved by “Research and Development of Deep Learning Technology for Advanced Multilingual Speech Translation", the Commissioned Research of National Institute of Information and Communications Technology (NICT), Japan. [^1]: <http://www.statmt.org/wmt16/translation-task.html>. [^2]: https://pypi.org/project/langdetect. [^3]: <http://www.statmt.org/wmt18/translation-task.html>. [^4]: <http://www.statmt.org/wmt17/translation-task.html>. [^5]: <http://lotus.kuee.kyoto-u.ac.jp/WAT/WAT2018/baseline/dataPreparationJE.html>. [^6]: <https://taku910.github.io/cabocha/>. [^7]: <https://github.com/e-bug/pascal>. [^8]: <https://github.com/mjpost/sacreBLEU>.\ Signature: BLEU+c.mixed+\#.1+s.exp+tok.13a+v.1.2.12. [^9]: <http://www.phontron.com/kytea/>.
--- abstract: 'High-temperature expansions are presently the only viable approach to the numerical calculation of the higher susceptibilities for the spin and the scalar-field models on high-dimensional lattices. The critical amplitudes of these quantities enter into a sequence of universal amplitude-ratios which determine the critical equation of state. We have obtained a substantial extension through order 24, of the high-temperature expansions of the free energy (in presence of a magnetic field) for the Ising models with spin $s \geq 1/2$ and for the lattice scalar field theory with quartic self-interaction, on the simple-cubic and the body-centered-cubic lattices in four, five and six spatial dimensions. A numerical analysis of the higher susceptibilities obtained from these expansions, yields results consistent with the widely accepted ideas, based on the renormalization group and the constructive approach to Euclidean quantum field theory, concerning the no-interaction (“triviality”) property of the continuum (scaling) limit of spin-$s$ Ising and lattice scalar-field models at and above the upper critical dimensionality.' author: - 'P. Butera' - 'M. Pernici' title: | Triviality problem and the high-temperature expansions of\ the higher susceptibilities for the Ising and the scalar field\ models on four-, five- and six-dimensional lattices\ --- Introduction ============ The renormalization group theory (RG) theory[@wk; @zinn; @zinnb] predicts the value $d_c=4$ for the upper critical dimensionality of the $N$-component lattice scalar-field theory and of the short-range classical Heisenberg $N$-component spin systems with $O(N)$-symmetric interaction. When $d \geq 4$, the critical fluctuations become too weak to drive the leading critical exponents away from the “classical” values taken in the mean field (MF) approximation, and can only induce minor corrections to scaling. In particular, in 4D the simple MF asymptotic forms of the thermodynamical quantities at criticality should be corrected by logarithmic factors, whose precise structure is also predicted by the RG. In higher dimensions, the dominant singularities have purely MF forms and the fluctuations can only influence the critical amplitudes and the corrections to scaling. These RG predictions entail the “triviality”[@wk; @sok; @sokff; @sym; @calla] of the quantum $N$-component scalar-field theories in $d \geq d_c $, or, more precisely, the property that the continuum (scaling) limit of the lattice approximation of the theories (or of the spin models) describes fields whose connected fourth- and higher-order correlation-functions vanish and therefore are free or generalized-free. The main clues of this no-interaction property had been pointed out long ago[@land], but more stringent arguments were produced only by the modern developments of the RG theory[@wk; @zinn; @zinnb]. In the same years, a rigorous constructive approach[@sok; @sokff; @aize; @fro; @cara; @hara; @shlo] based on the representation[@sym] of the lattice scalar-field as a gas of polymers, made it possible to prove conclusively that the continuum Euclidean quantum field theory, built as the scaling limit of a lattice theory (with the simplest nearest-neighbor discretization of the Laplacian) in the symmetric phase, is “non-trivial”[@eck; @brydg] in $d\leq 3 $ and “trivial” in $d\geq 5 $ dimensions. The rigorous results that exist in 4D (and, in general, for $N>4$) are still incomplete, although they strongly suggest that nevertheless the triviality property still holds. Therefore some room is left not only to numerical studies, but also to a variety of efforts[@galla; @cons; @suslov] (and the related controversies[@balo]), aimed either to exploit possible gaps in the arguments, or to relax some of the hypotheses underlying the constructive approach, in order to make the definition of a “non-trivial” continuum theory feasible. For $d \geq 4$, the MonteCarlo (MC) simulation approach to the numerical verification of the RG predictions is not yet completely satisfactory. The detailed exploration of the near-critical behavior is hampered by the necessity of considering systems of very large sizes, and in particular, at $d=4$, by the difficulty of an accurate characterization of the slowly varying logarithmic deviations from MF behavior. For $d \geq 4$, also the finite-size-scaling theory and the confluent corrections to scaling have been debated[@dohm; @stauf; @luij; @akte; @merd; @jones]. Thus relatively few of the numerous available MC studies[@wein; @fox; @dru; @bern; @frick; @kim; @grass; @kenna; @cons; @luij; @bitt; @lundow] are likely to be extensive enough to yield a satisfactory overall description of these systems at criticality, in spite of the remarkable progress in the simulation algorithms with reduced critical slowdown. On the other hand, for these systems high-temperature (HT) expansions have been until now derived only for a small number of observables and are too short[@fishga; @gaunt; @bakerkinc], or perhaps barely adequate[@luscher; @vlad; @bcklein; @staufad; @hell] to extract reliable information in the critical region. We believe however, that the HT series methods might bring further insight into this context, provided that for a conveniently enlarged set of observables, the lengths of the expansions can be significantly extended. Recently, new stochastic algorithms[@prokof; @deng; @berche; @wolff] have shown promise of deriving extremely long, although approximate, HT expansions valid for finite-size lattice systems. The application[@wolff] of these methods also to the triviality problem is particularly interesting. The traditional graphical [@wortis; @fishga; @gaunt; @bakerkinc; @bcoenne; @bcg4; @bcisiesse; @butper; @nickelrehr; @bcfi4; @luscher] or iterative[@deneef; @arisue; @bcm; @bciter] methods of calculation, although severely limited by the fast increase of their combinatorial complexity with the order of expansion, remain necessary to derive the exact HT series coefficients, valid in the thermodynamical limit, which are needed for a reliable use of the known analytic extrapolation tools[@hunterbaker; @fishauy; @guttda], such as Padé approximants(PA) or differential approximants (DA). It is finally worth adding that, in the case of high-dimensional models, these exact HT expansions still seem to be the only practicable method to compute the higher-order field-derivatives of the free energy at zero field, usually called “higher susceptibilities”. In this paper, we focus on the HT series approach to provide further numerical evidence supporting the RG predictions in the case of the $N=1$ lattice scalar-field models and of the Ising spin-$s$ systems. For this purpose, we have computed and analyzed exact HT expansions of the higher susceptibilities, through order 24, to study their critical behavior and an important class of universal combinations of critical amplitudes (UCCAs), whose properties might also be of interest. The paper is organized as follows. In Section II we define the spin-$s$ Ising and the lattice scalar-field systems for which we have substantially extended the HT expansions of the specific free-energy in the presence of a uniform magnetic field. Then we make due reference to the few HT data already in the literature. In Section III, we introduce the higher susceptibilities and indicate how their expected critical behavior varies with the lattice dimensionality. In Section IV, we review the definition of the dimensionless $2n$-points renormalized coupling constants in terms of the higher susceptibilities and indicate their role in the discussion of the RG predictions. Then we introduce several classes of UCCAs related to the latter quantities. The following Section V is devoted to a detailed numerical analysis of our HT expansions including discussions of numerical estimates of the critical temperatures, exponents and several UCCAs for the models under scrutiny. The final Section contains our conclusions. Ising-type models. Definitions and notation ============================================ In what follows, we shall be concerned only with spin-$s$ Ising or one-component lattice scalar-fields, so that, unless explicitly needed, it will be convenient to drop the dependence of the physical quantities on the number $N$ of components of the spin or of the field. In a bounded region $\Lambda \subset {\cal Z}^d$ of the $d$-dimensional lattice $ {\cal Z}^d$, the spin-$s$ Ising model interacting with an external uniform magnetic field $H$ is described by the Hamiltonian[@wortis; @nickelrehr; @bcisiesse; @butper] $${\cal H}_{\Lambda}\{s\}= -\frac{J} {s^2} \sum_{<ij>} s_is_j-\frac{mH}{s}\sum_i s_i \label{isingesse}$$ where $s_i=-s,-s+1,...,s$ is the spin variable at the lattice site $\vec i$, $m$ is the magnetic moment of a spin, $J$ is the exchange coupling. Within the region $\Lambda$, the first sum extends over all distinct nearest-neighbor pairs of sites, the second sum over all lattice sites. Clearly, the conventional Ising model is obtained simply by setting $s=1/2$. The self-interacting one-component scalar-field lattice model in a magnetic field is described by the Hamiltonian[@bakerkinc; @luscher; @bcfi4; @butper] $${\cal H}_{\Lambda}\{\phi\}= -\sum_{<ij>} \phi_i \phi_j+\sum_i (V(\phi_i) + H\phi_i). \label{pdifi}$$ Here $-\infty <\phi_i< +\infty$ is a continuous variable associated to the site $\vec i$ and $V(\phi_i)$ is an even polynomial in the variable $\phi_i$. In this study, for brevity we have only discussed the particular model in which $ V(\phi_i)= \phi^2_i + g(\phi^2_i-1)^2$, but considering interactions of a more general form requires only simple changes in the computation. The Gibbs specific free energy ${\cal F}(K,h)$ is defined as usual by $${\cal F}(K,h)= -\lim_{|\Lambda| \rightarrow \infty} \frac {1} {|\Lambda| k_BT} {\rm ln} Z_{\Lambda} (K,h) =-\lim_{|\Lambda| \rightarrow \infty} \frac {1} {|\Lambda| k_BT} {\rm ln} \sum_{conf} {\rm exp}[-{\cal H}_{\Lambda}/k_BT]$$ Here $|\Lambda|$ is the volume of the region $\Lambda$, $K=J/k_BT$ (or $K=1/k_BT$ in the case of the scalar-field models), called inverse temperature for short, is the HT expansion variable, with $k_B$ the Boltzmann constant and $T$ the temperature, while $h=mH/k_BT$ denotes the reduced magnetic field. We have studied the models described by eqs.(\[isingesse\]) and (\[pdifi\]) on the hyper-simple-cubic (hsc) and the hyper-body-centered (hbcc) lattices. Following Ref.\[\], for $d \geq 4$, the hbcc lattices are defined as those in which the first neighbors $q \hat j$ of the site $\hat i$ are such that $ \hat i- \hat j=(\pm 1,\pm 1,...,\pm 1)$. This choice has the technical advantage, decisive for the computations on high dimensional lattices, that the “lattice free-embedding numbers”, that enter into the contribution of each graph to the HT expansion, factorize so that they can be expressed as powers of those referring to the 1D lattice. As a consequence of this drastic simplification, the computing time of the expansions is independent of the lattice dimensionality, whereas, in the case of the hsc lattices, it grows exponentially with the dimensionality. We should also notice that, for $d>2$, the coordination number $q=2^d$ of the hbcc lattice is much larger than the coordination number $q=2d$ of the hsc lattice and therefore the hbcc lattice expansions share the advantage of being notably smoother and faster converging than the hsc ones. The expansions presented here are based on a calculation of the HT and low-field expansion of the free energy of various models described by the Hamiltonians eqs.(\[isingesse\]) and (\[pdifi\]), in presence of an external uniform magnetic field, that we have extended through the order 24. In the case of the conventional Ising model, i.e. the model with spin $s=1/2$, such an expansion, through order 17, was already in the literature[@katsura; @mckenzie] in the case of the four-dimensional hsc lattice (h4sc). Our expansion agrees only up to order 16 with these data and, as a consequence, with the series coefficients of the ordinary susceptibility $\chi_2(K)$ and of the fourth field-derivative of the free energy $\chi_4(K)$, obtained from them and analyzed in Ref.\[\], as well as in some successive studies. For the conventional Ising model, in addition to the expansion in the case of the h4sc lattice, we have also computed the analogous expansion in the case of the hbcc lattice in 4D (h4bcc). For both the h4sc and the h4bcc lattices, we have moreover computed HT and low-field expansions in the case of the Ising models with spin $s=1,3/2,...,3$ and in the case of the Euclidean one-component scalar-field lattice models with an even-polynomial self-interaction. We have finally repeated the series derivation for the same set of models in 5D and 6D, but restricting ourselves to the five-dimensional hbcc (h5bcc) and the six-dimensional hbcc (h6bcc) lattices, for the reasons of computational simplification indicated above. All these expansions do not exist in the literature. Altogether, we have examined these Ising-type models in 28 cases distinct by spatial dimensionality, value of the spin and structure of the lattice or of the interaction. In a given dimension, all these models are expected to belong to the same critical universality class and therefore to be characterized by the same set of critical exponents and UCCAs. Finally, let us also mention that our HT expansions for the Ising models in a magnetic field can be readily transformed into low-temperature (LT) and high-field expansions, from which the spontaneous magnetization and the LT higher susceptibilities can be derived. In our calculation of the HT expansions, we have employed the linked-cluster graphical method of Ref.\[\]. We have used an algorithm of graph generation and series calculation already described in Ref.\[\]. The details of the computer implementation of this procedure, its validation, and its performance are discussed in the same paper, that was devoted to a study of the higher susceptibilities and the scaling equation of state for the 3D Ising universality class. Our extensions of the HT and low field expansions are summarized in Table \[tab1\]. The set of series coefficients cannot fit into this paper because of its large size and will be tabulated elsewhere. Available series expansions in zero field ----------------------------------------- It is appropriate to list here the few HT expansions of the higher susceptibilities for high-dimensional models at zero field that can already be found in the literature. All of them are restricted to the conventional spin-1/2 Ising model on the hsc lattices in zero field. The ordinary susceptibility $\chi_2(K)$ was derived[@fishga] through order 11 in dimensions $d=2,..,6$. More recently, these calculations were extended[@bakerkinc] to include, through the the same order, also $\chi_4(K)$ and the second moment of the correlation function $\mu_2(K)$, in $d=2,3,4$ dimensions and carried[@luscher] up to order 14. The expansion of the susceptibility $\chi_2(K)$ has been recently pushed[@hell] to order 19 on the h4sc and h5sc lattices. An expansion of $\chi_2(K)$ valid for any dimension $d$ was computed[@gofman] through order 15. For $\chi_2(K)$, $\chi_4(K)$ and the sixth field-derivative of the free energy $\chi_6(K)$, strong coupling expansions through order 11, i.e. expansions in powers of the second-moment correlation length $\xi^2(K)=\mu_2(K)/2d\chi_2(K)$, instead of $K$, valid for any $d$, have also been obtained[@benderboett]. Of course, the usual HT expansions in powers of $K$ can be recovered simply by reverting the appropriate expansion of $\xi^2(K)$. [|c|c|c|]{} & Existing data[@mckenzie] & This work\ ${\rm h4sc}$ Ising $s=1/2$& 17 & 24\ ${\rm h4sc}$ Ising $s>1/2$& 0 & 24\ ${\rm h4sc}$ scalar field & 0 & 24\ ${\rm h4bcc}$ Ising $s \geq 1/2$& 0 & 24\ ${\rm h4bcc}$ scalar field & 0 & 24\ ${\rm h5bcc}$ Ising $s \geq 1/2$& 0 & 24\ ${\rm h5bcc}$ scalar field & 0 & 24\ ${\rm h6bcc}$ Ising $s \geq 1/2$& 0 & 24\ ${\rm h6bcc}$ scalar field & 0 & 24\ \[tab1\] The HT expansions of the higher susceptibilities ================================================ The assumption of asymptotic scaling[@widom; @dombhunter; @pata; @kada; @fish67] for the singular part ${\cal F}_s (\tau,h)$ of the reduced specific free energy, valid for dimension $d \neq d_c$, when both $h$ and $\tau$ approach zero, is usually expressed in the form $${\cal F}_s (\tau,h) \approx |\tau|^{2-\alpha} Y_{\pm}(h/|\tau|^{\beta \delta}). \label{freescaling}$$ where $\tau=(1-K/K_c)$ is the reduced inverse temperature. The functions $Y_{\pm}(w)$ are defined for $0\le w \le \infty$ and the $+$ and $-$ subscripts indicate that different functional forms are expected to occur for $\tau<0$ and $\tau>0$. The exponent $\alpha$ specifies the divergence of the specific heat, $\beta$ describes the small $\tau$ asymptotic form of the spontaneous specific magnetization $M$ on the phase boundary $(h \rightarrow 0^+, \tau<0)$ $$M \approx B(-\tau)^{\beta} \label{ampmagsp}$$ and $B$ denotes the critical amplitude of $M$. The exponent $\delta$ characterizes the small $h$ asymptotic behavior of the magnetization on the critical isotherm $(h \ne 0,\tau=0)$, $$|M| \approx B_c |h|^{1/\delta} \label{amphiso}$$ and $B_c$ is the corresponding critical amplitude. For $d \geq d_c$, the MF values expected for the exponents $\alpha$, $\beta$ and $\delta$ are $\alpha=\alpha_{MF}=0$, $\beta=\beta_{MF}=1/2$ and $\delta=\delta_{MF}=3$, while for the susceptibility exponent we have $\gamma=\gamma_{MF}=1$ and for the correlation-length exponent $\nu=\nu_{MF}=1/2$. The usual scaling laws (but, of course, not the hyperscaling laws) follow from eq.(\[freescaling\]). From our calculation of the magnetic-field-dependent free energy, we have gained extensions of the existing HT expansions in zero field and, in addition, made available a large body of data not yet existing in the literature, in particular for the $n$-spin connected correlation functions at zero wave number and zero field (the “higher susceptibilities”), defined by the successive field-derivatives of the specific free energy $$\chi_{n}(K)=(\partial^n {\cal F}(K,h)/\partial h^n)_{h=0} =\sum_{s_2,s_3,...,s_{n}}<s_1 s_2...s_{n}>_c. \label{ncorr}$$ in the Ising model case, or by the analogous formula in the scalar field case. For odd values of $n$, the quantities $\chi_{n}(K) $ vanish in the symmetric HT phase, while they are nonvanishing for all $n$ in the broken-symmetry LT phase. For even values of $n$ in the symmetric phase, the RG theory predicts that, for $d>4$, we have $$\chi_{n}(\tau) \approx C^{+}_{n}|\tau|^{-\gamma_{n}}(1 + b^{+}_{n} |\tau|^{\theta} + \ldots). \label{2ncorras5}$$ as $\tau \rightarrow 0^+$ along the critical isochore ($h=0, \tau>0$). In eq.(\[2ncorras5\]), $b^{+}_n$ and $\theta$ denote, respectively, the amplitude and the exponent of the leading confluent correction to the asymptotic behavior. The explicit expressions obtained in the case of the spherical model[@joyce; @figut; @gutt] suggest that in 5D one should expect $\theta=1/2$, whereas, in 6D, $\theta=1$, with possible multiplicative logarithmic correction terms. At the marginal dimension $d_c$, the homogeneity property described by eq.(\[freescaling\]) is not strictly true, because of the expected logarithmic corrections. In this case, for even values of $n$, in the symmetric phase, the RG theory predicts for the higher susceptibilities the following asymptotic behavior $$\chi_{n}(\tau) \approx C^{+}_{n}\tau^{-\gamma_n} {\vert {\rm ln} (\tau)\vert^{G_n(N)}} \Big[1 + O\Big( {\rm ln}(\vert {\rm ln}(\tau)\vert)/{\rm ln}(\tau)\Big) \Big] \label{2ncorras4}$$ in the $\tau \rightarrow 0^+$ limit. In both eqs. (\[2ncorras5\]) and (\[2ncorras4\]), one has $\gamma_{n}=\gamma_{MF} +(n-2)\Delta_{MF}$, with the gap exponent $ \Delta_{MF}=\beta_{MF}\delta_{MF}=3/2$. The general expression for $G_n(N) $ is $$G_n(N)= (\frac{3} {2} n-2)\frac{N+2} {N+8} -n/2 +1 \label{gn}$$ so that in the $N=1$ case, $G_n(1)=G=1/3$, independently of $n$. Clearly, the usual hyperscaling relation $2\Delta=d\nu+\gamma$, which is valid for $d<d_c$, fails by a power when $d>4$, while it is only logarithmically violated in $d=4$. The simplest consequence of the usual scaling hypothesis eq.(\[freescaling\]), which will be tested using our HT expansions, is that the critical exponents of the successive derivatives of ${\cal F} (\tau,h)$ with respect to $h$ at zero field, are evenly spaced by the gap exponent $\Delta_{MF}$. Also in 4D, this property can be simply and accurately checked by a HT analysis of the higher susceptibilities. Renormalized couplings and related quantities ============================================= It is useful here to recall the definitions of the universal quantities $g^+_{2n} $, called zero-momentum $n-$spin dimensionless renormalized coupling constants (RCC’s) in the symmetric phase. They enter into the approximate representations of the scaling equation of state and moreover play a key role in the discussion of the triviality properties of the $d \geq 4$ systems. They are defined as the critical limit when $K \rightarrow K_c^-$ of the expressions $$g_4(K)=-\frac{{\it v} } {\xi^d(K) } \frac{\chi_4(K)} { \chi_2^2(K)} \label{g4}$$ $$g_6(K)=\frac{{\it v} ^2} {\xi^{2d}(K)}\Big [ -\frac{\chi_6(K)} {\chi_2^3(K)} + 10 \Big (\frac{\chi_4(K)} { \chi_2^2(K)}\Big)^2 \Big] \label{g6}$$ $$g_8(K)=\frac{{\it v} ^3} {\xi^{3d}(K)}\Big [ -\frac{\chi_8(K)} {\chi_2^4(K)} +56 \frac{\chi_6(K) \chi_4(K)} {\chi_2^5(K)} -280 \Big (\frac{\chi_4(K)} {\chi_2^2(K)}\Big)^3 \Big] \label{g8}$$ $$\begin{aligned} g_{10}(K)=\frac{{\it v} ^4} {\xi^{4d}(K)} \Big [-\frac{\chi_{10}(K)} {\chi_2^5(K)}+ 120\frac{\chi_8(K) \chi_4(K)} {\chi_2^6(K)} +126 \frac{\chi_6^2(K)} { \chi_2^6(K)}\\ \nonumber -4620 \frac{\chi_6(K) \chi_4^2(K)} { \chi_2^7(K)} +15400 \Big (\frac{\chi_4(K)} {\chi_2^2(K)}\Big)^4 \Big] \label{g10}\end{aligned}$$ $$\begin{aligned} g_{12}(K)=\frac{{\it v} ^5} {\xi^{5d}(K)} \Big [-\frac{\chi_{12}(K)} {\chi_2^6(K)}+ 220\frac{\chi_{10}(K) \chi_4(K)} {\chi_2^7(K)} +792 \frac{\chi_8(K)\chi_6(K)} { \chi_2^7(K)}\cr -17160 \frac{\chi_8(K)\chi_4^2(K)} { \chi_2^8(K)} -36036 \frac{\chi_6^2(K) \chi_4(K)} { \chi_2^8(K)} +560560 \frac{\chi_6(K)\chi_4^3(K)} {\chi_2^9(K)}\cr -1401400\Big (\frac{\chi_4(K)} {\chi_2^2(K)}\Big)^5 \Big] \label{g121}\end{aligned}$$ and so on. The constant ${\it v} $ is a lattice-dependent geometrical factor called the volume per lattice site[@bcg2n]. A longer list of the RCC’s appears in Ref.\[\], where the equation of state is discussed only for the 3D case. For technical reasons, we have not yet extended the HT expansions of $\mu_2(K)$ and, correspondingly, of the second-moment correlation length $\xi(K)$, so that in this paper we shall study only the ratios of RCC’s, for $n>2$, $$r_{2n}(K)=\frac {g_{2n}(K)}{g_{4}(K)^{n-1}} \label{r2n}$$ which share the computational advantage of being independent of $\xi(K)$. The critical limits of these ratios are universal quantities that will be denoted by $r^+_{2n}$. At the upper critical dimension $d_c$, the quantities $ g_{2n}(K)$ are expected to vanish like powers of $1/{\rm ln}(\tau)$, when $\tau \rightarrow 0^+$. Therefore the continuum limit theory is consistent only for vanishing renormalized coupling, i.e. it is trivial. We can check numerically that, in the same limit, the lowest ratios $r_{2n}(K)$ remain finite in 4D. For $d \geq 5$, both the $ g_{2n}(K)$ and the $r_{2n}(K)$ vanish in the critical limit like powers of $\tau$, so that the mentioned property of triviality is also true for $d>d_c$. Briefly recalling more detailed discussions[@zinnb; @guidazinn; @butper], we can also observe that, for $d>4$, in the small magnetization region, where the reduced magnetic field $ h(M,\tau) $ has a convergent expansion in odd powers of the magnetization $M$, the critical equation of state can be written in terms of an appropriate variable $z \propto M\tau^{-\beta}$ as $$h(M,\tau) = \bar h|\tau|^{\beta\delta}F(z) \label{eqstat}$$ where $\bar h$ is a constant and $F(z)$ is normalized by the equation $F'(0)=1$. In general, the small $z$ expansion of $F(z)$ can be written as $$F(z)=z+\frac{1}{3!}z^3+\frac{r^+_6} {5!} z^5+\frac{r^+_8} {7!} z^7+... \label{fz}$$ In the MF approximation, all $r^+_{2n}$ vanish, and $F(z)$ reduces to $F_{MF}(z)=z+\frac{1}{3!}z^3$. At the upper critical dimension, the following form of the critical universal equation of state for an $N$-component system is obtained[@zinn; @zinnb] from the RG $$H \propto \left [ M \tau |{\rm ln} M|^{(N+2)/(N+8)} + \frac{1} {(N+8)} \frac{M^3} {|{\rm ln}M |} \right ](1+\frac{const.} {|{\rm ln}M |}) \label{es4d}$$ valid for small $M$ and $H$. From eq.(\[es4d\]) the general formula eq.(\[2ncorras4\]) and the expression (\[gn\]) for $G_n(N)$ can be deduced. In terms of the higher susceptibilities, the simple sequence of quantities was defined[@watson] long ago $${\cal I}_{2r+4}(K)=\frac{ \chi^r_2(K) \chi_{2r+4}(K)}{\chi^{r+1}_4(K)} \label{watso}$$ with $r \geq 1 $. The finite and universal critical values $${\cal I}^+_{2r+4}=\frac{(C_2^+)^r C_{2r+4}^+} {(C_4^+)^{r+1}} \label{watsoncr}$$ of the functions ${\cal I}_{2r+4}(K) $ in the limit $K \rightarrow K_c^-$, include some of the UCCAs first described in the literature. Together with the sequence ${\cal I}^+_{2r+4}$ of UCCAs, the sequences ${\cal A}^+_{2r+4}$ and ${\cal B}^+_{2r+8}$, obtained as the critical limits of the functions $${\cal A}_{2r+4}(K)=\frac{\chi_{2r}(K) \chi_{2r+4}(K) } {(\chi_{2r+2}(K) )^2} \label{watsonar}$$ $${\cal B}_{2r+8}(K)=\frac{\chi_{2r}(K) \chi_{2r+8}(K) } {(\chi_{2r+4}(K) )^2} \label{watsonbr}$$ with $r \geq 1 $, were also defined in Ref.\[\]. In 4D the general formula eq.(\[gn\]) for $G_n(N)$ implies that the powers of the logarithms that enter into the leading critical singularities eq.(\[2ncorras4\]) cancel in the quantities ${\cal I}_{2r+4}(K)$, ${\cal A}_{2r+4}(K)$, and ${\cal B}_{2r+8}(K)$ at the critical limit. Conversely, eq.(\[gn\]) can also be obtained recursively from the knowledge of only $G_2(N)$ and $G_4(N)$ by requesting that such a cancellation occurs. The ratios $r_{2n}(K)$ can be simply expressed in terms of the functions ${\cal I}_{2r+4}(K)$. For example $$r_6(K)=\frac {g_{6}(K)}{g_{4}(K)^{2}} = -{\cal I}_6(K)+10 \label{r6}$$ $$r_8(K)=\frac {g_{8}(K)}{g_{4}(K)^{3}}= {\cal I}_8(K)-56 {\cal I}_6(K)+280 \label{r8}$$ and so on. Taking the $K \rightarrow K_c^-$ limit in the eqs.(\[r6\]), (\[r8\]), etc. and observing that the quantities $r_{2n}(K)$ vanish as $K \rightarrow K_c^-$ in the MF approximation[@zinn], the corresponding critical values $\hat{\cal I}^+_{2r+4}$ of the quantities in eq.(\[watso\]) can be simply evaluated, obtaining $\hat {\cal I}^+_6=10$, $\hat {\cal I}^+_8=280 $, $\hat {\cal I}^+_{10}=15400 $, $\hat {\cal I}^+_{12}=1401400 $, etc. It is also not difficult[@joyce] to compute the MF values of the first few terms of the sequences ${\cal A}^+_{2r+4}$ and ${\cal B}^+_{2r+8}$. For example: $\hat {\cal A}^+_{8}=14/5$, $\hat {\cal A}^+_{10}=55/28$ $\hat {\cal B}^+_{10}=154$, and $\hat {\cal B}^+_{12}=143/8$. In the next Section, we shall study numerically the first few terms of the sequences ${\cal I}^+_{2r+4} $, ${\cal A}^+_{2r+4}$ and ${\cal B}^+_{2r+8}$ and observe that they share similar properties. Results of the series analysis ============================== We address the reader to Refs.\[\], for a more detailed description of the numerical approximation techniques necessary to estimate the critical parameters in the models under study, i.e. the locations of the critical points, their exponents of divergence and the critical amplitudes for the various susceptibilities. We shall employ the DA method, a generalization[@hunterbaker; @fishauy; @guttda] of the elementary PA method to resum the HT expansions nearby the border of their convergence disks. This technique consists in estimating the values of the finite quantities or the parameters of the singularities of the expansions of the singular quantities from the solution, called [*differential approximant*]{}, of an initial value problem for an ordinary linear inhomogeneous differential equation of the first- or of a higher-order. The equation has polynomial coefficients defined in such a way that the series expansion of its solution equals, up to an appropriate order, the series under study. In addition to this technique, we shall also use a smooth and faster converging modification[@zinnmra; @guttda] of the traditional method of extrapolation of the coefficient-ratio sequence, sometimes called modified-ratio approximant (MRA) method, to determine the location and the exponents of critical points. Using PA or DA approximants, one can achieve, in some cases, evaluations of the parameters which are unbiased, i.e. obtained without using independent estimates of the critical temperature in the construction of the approximants. In some cases however, accurate estimates of the critical inverse temperatures are necessary to bias the determination of the critical exponents and amplitudes. As a general comment on the uncertainties of the estimates obtained by these methods, we have to observe that, inevitably, they are rather subjective. Therefore, we should be very cautious, compare the estimates obtained from the different approximation methods and also check how effectively our tools perform when applied to artificial model functions having the expected singularity structure. In our DA calculations, the uncertainties are taken as a small multiple of the spread among the estimates obtained from an appropriate sample of the highest-order approximants i.e. those using most or all available expansion coefficients. Similarly, in the case of the MRAs, the error bars will be defined as a small multiple of the uncertainty of an appropriate extrapolation[@bcisiesse] of the highest-order approximants. [|c|c|c|c|c|c|c|]{} lattice &$s=1/2$ &$s=1 $ &$s=3/2$ & $s=2$&$s=5/2$ &$s=3$\ ${\rm h4sc}$ &0.149693(3)&0.215597(3)&0.255641(3)&0.282568(3)&0.301919(3) &0.316497(3)\ ${\rm h4bcc}$&0.0690114(8)&0.101165(2)&0.120592(2)&0.133605(2)&0.142930(2) &0.149942(2)\ ${\rm h5bcc}$&0.0326478(3) &0.0484554(3)&0.0579714(3)&0.0643290(3)&0.0688769(3) &0.0722915(3)\ ${\rm h6bcc}$&0.0159390(2) &0.0237914(2) &0.0285102(2) &0.0316592(2) &0.0339099(2) &0.0355989(2)\ \[tabkc\] [|c|c|c|c|c|c|c|]{} lattice &$g=0.5 $ &$g=0.6$ & $g=0.9$&$g=1.1$ &$g=1.3$&$g=1.5$\ ${\rm h4sc}$& 0.283025(3)&0.280704(3) &0.270597(3) &0.262806(3)&0.254915(3) &0.247233(3)\ ${\rm h4bcc}$&0.136451(2)&0.134940(2)&0.129177(2) &0.124991(2) &0.120852(2) &0.116885(2)\ ${\rm h5bcc}$&0.0665777(2)&0.0657078(2)&0.0625961(2)&0.0604114(2) &0.0582806(2) &0.0562586(2)\ ${\rm h6bcc}$&0.0329566(1)&0.0324976(1)&0.0308921(1)&0.0297796(1) &0.0287008(1) &0.0276811(1)\ \[tabkcfi4\] The critical inverse temperatures of the models ----------------------------------------------- If in 4D, as predicted by the RG theory, logarithmic factors modify the structure of the leading critical singularities and also appear in the corrections to scaling, as described by eq.(\[2ncorras4\]), we should expect that the numerical procedures mentioned above might suffer from a slower convergence than in the case of pure power-law scaling. For the determination of the critical temperatures, different approximation methods such as PAs, DAs and extrapolated MRAs have been used to study the expansions of the ordinary susceptibility $\chi_2(K)$, the quantity which generally shows the fastest convergence. Independently of the lattice type and dimensionality, our best estimates of the critical inverse temperatures for the systems under study are obtained extrapolating to large order $r$ of expansion, a few (from four to seven) highest order terms of the MRA sequences of estimates $(K_c)_r$ of the critical inverse temperatures. To perform the extrapolation, we rely on the validity[@bcisiesse] of the simple asymptotic form $$(K_c)_r=K_c- \frac{\Gamma(\gamma)} {\Gamma(\gamma-\theta)} \frac{\theta^2(1-\theta)b_2}{r^{1+\theta}}+o(\frac{1}{r^{1+\theta}}). \label{kras}$$ In general, $\theta$ and $b_2$ indicate respectively the exponent and the amplitude of the leading confluent correction to scaling of $\chi_2(K)$. In the 4D case, in which the asymptotic critical behavior of $\chi_2(K)$ is described by eq.(\[2ncorras4\]), we can take $\theta=0$. Therefore the second term on the right-hand side of eq.(\[kras\]) vanishes and it must be replaced by a higher-order term depending on the exponent of the next-to-leading correction to scaling in eq.(\[2ncorras4\]). A similar argument applies in the 6D case in which we expect $\theta=1$. In the 5D case, in which we expect $\theta=1/2$, the coefficient of $1/r^{1+\theta}$ in eq.(\[kras\]) also appears to be negligible, so that the situation is similar to that of the 4D and the 6D cases. Since we do not know the values of the exponents of the next-to-leading correction to scaling, the simplest procedure of extrapolation might consist in assuming an asymptotic form $(K_c)_r=K_c+ w/r^{1+\epsilon}$ and in determining $K_c$, $w$, and $\epsilon$ by a best fit to our data. We obtain the values $\epsilon= 0.6(2)$ in 4D, $\epsilon= 1.1(2)$ in 5D and $\epsilon= 1.5(2)$ in 6D. These estimates are compatible with our previous remarks, indicating that the asymptotic behavior of eq.(\[kras\]) is determined by the next-to-leading rather than the leading correction to scaling. At the same time, as suggested by M.E. Fisher[@fpc], the expectations[@joyce; @figut; @gutt] concerning the exponent of the leading corrections to scaling, whose amplitudes are probably not negligible in spite of the fact that they are not seen by the MRAs, can be essentially confirmed studying by DAs the critical behavior of quantities like ${\cal I}_{2r+4}(K)$, ${\cal A}_{2r+4}(K)$, ${\cal B}_{2r+8}(K)$ etc. and of their derivatives. As above remarked, in these quantities the dominant critical singularities cancel, while the leading corrections to scaling should survive and could be detected by DAs. In particular, a study of the derivatives of ${\cal I}_{6}(K)$ and ${\cal I}_{8}(K)$, for the spin-$s$ Ising models, leads to the values $\theta=0.25(10)$ in 4D, $\theta=0.45(10)$ in 5D and $\theta=0.95(10)$, in very reasonable agreement with the predictions[@joyce; @figut; @gutt]. Our final results for the critical inverse temperatures of some spin-$s$ Ising and scalar-field models are collected in the Tables \[tabkc\] and \[tabkcfi4\]. In the 4D case, we have attached particularly generous error bars to our estimates. In $d>4$ dimensions, no logarithmic factors are expected to modify the leading MF behavior of the physical quantities, so that our approximation tools are likely to yield estimates of a higher accuracy, which moreover appear to improve with increasing lattice dimensionality, both because of the decreasing size of the corrections to scaling and of the increasing lattice coordination number. All these results are confirmed also by the analyses employing DAs. Only in the case of the Ising model with spin $s=1/2$ on the h4sc lattice, we can compare our estimates with those obtained in other studies by extrapolation of shorter HT series. In Ref.\[\] the estimate $K_c=0.149696(4)$ was obtained from a series of order 17, while in Ref.\[\] the result $K_c=0.149691(3)$ was derived from a series of order 19. As far as the most recent large-scale MC simulations are concerned, the estimate $K_c=0.149697(2)$ was obtained in Ref.\[\], the value $K_c=0.149697(2) $ in Ref.\[\], while the value $K_c=0.1496947(5) $ is reported in Ref.\[\]. Our result in Table \[tabkc\] is fully consistent with the older estimates. No comparison is possible either for higher values of the spin on the h4sc lattice, or for any value of the spin on the h4bcc lattice, since no studies are available for these systems. In the case of the higher dimensional lattices our analysis includes only the h5bcc and h6bcc lattices, which have not been studied elsewhere until now. The logarithmic corrections in 4D ---------------------------------- Also in the computation of the critical exponents, it is convenient to distinguish the 4D case from the higher dimensional ones. In 4D, when computing the exponent $\gamma$ of $\chi_2(K)$ by PAs or DAs, we obtain estimates very near to, but slightly larger than unity. These estimates should then be regarded as the values of “effective exponents” which reflect the presence of a small correction to the leading classical behavior (and of subleading corrections). If we assume that the leading correction to MF behavior has the multiplicative logarithmic structure predicted by the RG, we can resort to a variety of procedures proposed[@guttda; @gutt; @adler; @vlad] in the literature to isolate the logarithmic factor from the main power behavior and to measure its exponent. These prescriptions generally amount to cancel out the main power-singularity in favor of the weak logarithmic one and therefore they need to be biased with an estimate of the inverse critical temperature, to which, in turn, the values obtained for the exponent of the logarithm are very sensitive. For example, in the case of the ordinary susceptibility $\chi_2(K)$, one might study the auxiliary function $l(K;\tilde K_c)$ defined by $$l(K;\tilde K_c)=-(\tilde K_c-K){\rm ln} (\tilde K_c-K)\frac{d}{dK}{\rm ln}[(\tilde K_c-K)\chi_2(K)] %=l(\tilde K_c) +O({\rm ln|ln}\tau|)) \label{glog}$$ where $\tilde K_c$ is some accurate approximation of the true $K_c$. By eq.(\[2ncorras4\]), $l(K; K_c) = G +O({\rm ln|ln}\tau|)$, i.e. it yields the value of the exponent $G$ when $K \rightarrow K_c^-$ and $\tilde K_c= K_c$. Since $\tilde K_c$ enters as a parameter into the definition of this biased indicator, we should consider how the estimate $G(\tilde K_c)$ of the exponent depends on the choice of $\tilde K_c$ in a small vicinity of our best estimate of the critical inverse temperature reported in Tabs.\[tabkc\] or \[tabkcfi4\]. As a typical example, we show in Fig.\[figesplog\] the plots of $G(\tilde K_c)$ vs $\tilde K_c$ (normalized to our MRA estimate of $ K_c$), computed by PAs of various orders, in the case of the Ising model with spin $s=1$ on the h4bcc lattice. It is reasonable to expect that the value of $G(\tilde K_c)$ should depend slowly on $\tilde K_c$ near the exact value of the critical inverse temperature so that its best value might perhaps correspond to a stationary point. We observe that, for most PAs of $G(\tilde K_c)$, such a point does indeed exist and also that the curves obtained by various PAs touch nearby this point, which is generally not much different from our best estimate of $K_c$ as reported in our Tables. In the literature[@guttda; @gutt; @adler; @vlad], the value of $G(\tilde K_c)$ at the point where the various curves touch, is generally taken as the most accurate estimate of the exponent $G$. However, this choice may be questioned, since the result appears to be insensitive to the order of approximation. As shown in Fig.\[figesplog\], if we take the value of $G(\tilde K_c)$ at the stationary point as the best approximation, the estimates are also close to the expected value $G=1/3$. Unfortunately, also the choice of the stationary value as the best approximation is open to doubt, since in this case the successive approximations seem to worsen as the order of the series increases. We must moreover mention that, in the h4bcc Ising system, the values of $G$ computed in this way, range between $\approx 0.4$ and $\approx 0.3$, as the spin varies from $s=1/2$ to $s=3$. Finally, it is also unclear how to estimate the uncertainties involved in these procedures and thus how to interpret the spread of exponent estimates, which might be related to a strong spin-dependence of the slowly decaying corrections appearing in eq.(\[glog\]). Other prescriptions to study the exponent of the logarithmic corrections, do not lead to better results. The critical exponents of the higher susceptibilities ------------------------------------------------------ For each model under study, we have computed the exponent $\gamma$ of the susceptibility by second- or third-order DAs biased with our estimate of the inverse critical temperature, namely by resorting to the standard prescription of imposing that the approximants are singular at the values of $K_c$ reported in our tables \[tabkc\] and \[tabkcfi4\] and then computing the exponents. For $d>4$, it is rigorously proved[@aize; @fro] that the systems must exhibit a MF critical behavior (with non trivial subleading corrections). Let us discuss first how our numerical tools perform in the 5D and 6D cases. We shall then argue that the differences between the features of this computation and those of the 4D case can be simply ascribed to the expected presence of a multiplicative logarithmic correction to the dominant MF power behavior. In Fig.\[gamma456\], we have plotted our estimates of the exponent of the ordinary susceptibility vs the spin in the case of various spin-$s$ Ising models for $d=4,5,6$. For $d>d_c$, our estimates reproduce to a very good accuracy the expected value $\gamma_{MF}=1$, so that the small deviations from this value can be safely viewed as only the residual effects of the confluent corrections to scaling. These deviations also show the expected decreasing size as the dimensionality of the system increases. Moreover the critical universality, i.e. the independence of the exponents on the interaction structure, is well verified. On the contrary, at the upper critical dimension our calculations yield “effective” exponents larger than unity by $\approx 3 \%$. We can interpret this result as an indication that the leading critical singularity of the susceptibility is slightly stronger than a pure MF singularity so that it might indeed contain the logarithmic factor predicted by the RG, which is detected by the DAs as a power-like factor with a very small exponent. This is confirmed by observing that, if the expected logarithmic singularity is canceled by dividing out from the susceptibility the $ {\rm ln}(1-K/K_c)^{1/3}$ correction factor, the resulting estimate of the exponent $\gamma$ generally gets within $\approx 0.5 \%$ of the MF value. Thus the deviations are reduced to a smaller size and become compatible with the effects of the corrections to scaling. Very accurate estimates can be obtained also for the differences $D_n$ between the exponents of $\chi_{2n}(K)$ and $\chi_{2n-2}(K)$ $$D_n= \gamma_{2n}- \gamma_{2n-2}= 2 \Delta_{MF}=3 \label{dn}$$ They can be computed from the ratios $\chi_{2n}(K)/\chi_{2n-2}(K)$ by second- or third-order DAs biased with the critical inverse temperature. In the 4D case, we should not expect any effects from the logarithmic factors appearing in the leading singular behavior eq.(\[2ncorras4\]) of the higher susceptibilities, since these factors cancel in the above indicated ratios. Instead of the results of the biased prescription, we prefer to show here the estimates from a computation by the unbiased “critical point renormalization” method[@hunterbaker]. This procedure consists in determining the difference $D_n$ of eq.(\[dn\]) from the exponent of the singularity in $x=1$ of the series $\sum a_r x^r$ with coefficients $a_r=c_r^{2n}/c_r^{2n-2}$, where $c_r^s$ is the $r$-th coefficient of the expansion of $\chi_{s}(K)$. The biased DA calculation of the $D_n$, mentioned above, gives quite comparable results, so that it is not necessary to report the corresponding figures. The quantities $D_n$ with $n=2,3,...11$, obtained by the unbiased method in the case of the the scalar-field model on the h5bcc and h6bcc lattices, for several values of the coupling $g$, are plotted vs $n$ in Fig.\[diffga\_fi4h56bcc\]. The same computations for the spin-$s$ Ising models with various values of the spin on the h5bcc and h6bcc lattices yield completely similar results and therefore we do not report the corresponding figure. Our estimates of $D_n$ agree, generally within $0.1 \%$, with the expected value $2\Delta_{MF}=3$. Thus the small size of these deviations from the MF value suggests that they can safely be related with the confluent corrections to scaling. The critical universality is also well verified. On the other hand, our results in the 4D case reported in Fig.\[diffga\_fi4h4bcc\] in the case of the scalar-field model on the h4bcc lattice, those reported in Fig.\[diffga\_Isi\_h4bcc\] for the Ising model on the h4bcc lattice and those of Fig.\[diffga\_Isi\_h4sc\] for the same system in the case of the h4sc lattice, show relative deviations from the value of $2\Delta_{MF}$, five times larger than those in $d>4$ dimensions (i.e. of the order of $0.5 \%$), but still sufficiently small to reflect only the residual influence of the expected subleading logarithmic corrections to the critical behavior. Universal combinations of critical amplitudes ---------------------------------------------- For $d>4$, using second- or third-order DAs, the first few terms of the sequence of the UCCAs ${\cal I}^+_{2r+4}$, ${\cal A}^+_{2r+4}$ and ${\cal B}^+_{2r+8}$ can be evaluated to a good accuracy, by extrapolating to $K=K_c^-$ the estimates of the functions ${\cal I}_{2r+4}(K)$, ${\cal A}_{2r+4}(K)$ and ${\cal B}_{2r+8}(K)$. For convenience, we have introduced the ratios of these quantities to their MF values, and denoted them by ${\cal Q}_{2r+4}$, ${\cal R}_{2r+4}$ and ${\cal S}_{2r+8}$, respectively. In Fig.\[IsuImfh56bcc\], we have reported our estimates for the ratios ${\cal Q}_{6}$, ${\cal Q}_{8}$, ${\cal Q}_{10}$ and ${\cal Q}_{12}$ vs the value $s$ of the spin for Ising models on the h5bcc and h6bcc lattices. In complete agreement with the proven MF nature of the critical behavior, these ratios generally equal unity, within the accuracy expected from our approximations that, in this case, allows not only for the influence of the confluent corrections to scaling, but also for the uncertainties in the estimates of the critical temperatures needed to bias the calculations. Correspondingly, the critical limits of the ratios $r_{2n}(K)$ vanish and the equation of state takes the MF form. Quite similar results are shown in Fig.\[nuoveuccah56bcc\] for the other normalized UCCAs ${\cal R}^+_{8}$, ${\cal R}^+_{10}$, ${\cal S}^+_{10}$ and ${\cal S}^+_{12}$ which are plotted vs the spin $s$ for Ising models with various values of the spin in the case of the h5bcc and h6bcc lattices. Also in the 4D case, as shown in Fig.\[I6I8suImf\_fi4\_h4sc\_h4bcc\] for the first few UCCAs defined by eq.(\[watsoncr\]) in the case of the scalar-field model, and in Fig.\[nuoveuccah4bcc\] for a few UCCAs defined by eqs.(\[watsonar\]) and (\[watsonbr\]) in the case of the spin-$s$ Ising model, the various quantities probably have been evaluated with reasonable accuracy, because the logarithmic factors, expected to appear in the leading critical behavior of the higher susceptibilities, cancel in the ratios defining the UCCAs. As a consequence, the uncertainties in the critical temperatures and the influence of the corrections to scaling should still be considered as the main sources of error. However, we observe that generally the first few ratios ${\cal Q}_{2r+4}$, ${\cal R}_{2r+4}$ and ${\cal S}_{2r+8}$ are slightly, but definitely smaller than unity. We can imagine two possible explanations of this result: either the deviations from unity have to be related only to (unlikely) residual effects of the logarithms in the leading and subleading behavior of the higher susceptibilities, or the UCCAs are accurately estimated and they really do not take their MF values. Whatever the case, it is clear that also these data on the UCCAs confirm that, consistently with the RG predictions, the critical behavior in 4D is not MF-like. Conclusions =========== By analyzing our HT expansions of the zero-field higher-susceptibilities, extended through order 24, in the case of the $N=1$ lattice scalar-field models and of the spin-$s$ Ising systems, we have provided further numerical evidence consistent with the critical behavior predicted by the RG in this class of models. We have estimated the critical exponents of the ordinary and the higher susceptibilities and the values of a class of universal combinations of their critical amplitudes, which determine the form of the critical equation of state and are presently inaccessible by other computational methods. In 4D, the results of our analysis suggest that, within a good approximation, the critical exponents and this class of UCCAs, show small, but definitely nonvanishing deviations from their values in the MF approximation. For the UCCAs, this fact had been already predicted long ago also within the RG formalism, by showing[@brezin] that, at the upper critical dimension, at least one of the quantities in the above mentioned class does not take the MF value. More generally, in 4D the deviations from the MF critical behavior are compatible with the small effects associated to the logarithmic corrections predicted by the RG. Our direct numerical checks concerning in particular the exponents of the logarithmic corrections to the dominant power behavior of the higher susceptibilities have only a rather limited accuracy, due to the modest sensitivity of the DAs to the logarithmic singularities, either in the leading behaviors and in the confluent corrections. Quite on the contrary, the same kind of analysis performed on five- and six-dimensional lattices, shows no numerical evidence of deviations from the leading classical behavior by an extent larger than the expected numerical uncertainties: both the exponents and the UCCAs appear to take the MF values within a high approximation, so that the RG predictions concerning the triviality property are rather convincingly confirmed. Acknowledgements ================ We thank Ian Campbell, Michael E. Fisher, Ralph Kenna and Ulli Wolff for their patience in reading and commenting a preliminary draft of this paper. 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The dashed horizontal line represents the predicted value of the exponent of the logarithmic correction to the leading power behavior of the susceptibility.](FIG456a_1.eps){width="3.37"} ![ \[gamma456\] The exponent $\gamma$ of the susceptibility vs the value $s$ of the spin, in the case of the spin-$s$ Ising models on the h4sc, h4bcc, h5bcc and h6bcc lattices. For each value of the spin, the different estimates and their uncertainties are made more visible by slightly shifting their abscissas to avoid superpositions of the symbols. The exponents are denoted by open squares in the case of the h4sc lattice, or by open circles in the case of the h4bcc lattice. If the logarithmic correction to the leading critical singularity expected in 4D is canceled out from the susceptibility expansion, we are led to the estimates represented by crossed open squares in the case of the h4sc lattice and by crossed open circles in the case of the h4bcc lattice. The estimates on the h5bcc and the h6bcc lattices are represented by open triangles and open rhombs respectively. The dashed horizontal line represents the expected value $\gamma_{MF}=1$ of the exponent, and the continuous lines indicate a $1\%$ relative deviation from that value.](FIG456a_2.eps){width="3.37"} ![ \[diffga\_fi4h56bcc\] The unbiased estimates of the exponent differences $D_n=\gamma_{2n}-\gamma_{2n-2}$ for the susceptibilities $\chi_{2n}$ and $\chi_{2n-2}$, for $n=2,3,...11$ plotted vs $n$, in the case of the scalar-field model on the h5bcc and h6bcc lattices. In the case of the h5bcc lattice, for each value of $n$ we have reported a cluster of the four estimates of $D_n$ corresponding to the values $g=0.9$ (open squares), $g=1.1$ (open triangles), $g=1.3$ (open circles), $g=1.5$ (open rhombs) of the self-coupling of the field. The same quantities for the h6bcc lattice are represented by the corresponding black symbols. Although they correspond to the same value of $n$, the symbols within each cluster are slightly shifted apart to avoid cluttering and keep the spread of each estimate visible. The dashed horizontal line represents the expected value $2\Delta_{MF}=3$ of twice the gap exponent. ](FIG456a_3.eps){width="3.37"} ![ \[diffga\_fi4h4bcc\] The unbiased estimates of the exponent differences $D_n=\gamma_{2n}-\gamma_{2n-2}$ of the susceptibilities $\chi_{2n}$ and $\chi_{2n-2}$, for $n=2,3,...11$ plotted vs $n$, in the case of the scalar-field model on the h4bcc lattice. For each value of $n$ we have reported a cluster of the four estimates of $D_n$ corresponding to the values $g=0.9$ (squares), $g=1.1$ (triangles), $g=1.3$ (circles), $g=1.5$ (rhombs) of the self-coupling of the field. Although they correspond to the same value of $n$ the symbols within each cluster are slightly shifted apart to avoid cluttering and keep the spread of each estimate visible. The dashed horizontal line represents the expected value $2\Delta_{MF}=3$ of twice the gap exponent. The continuous horizontal lines indicate a relative deviation of $0.5 \%$ from the expected value.](FIG456a_4.eps){width="3.37"} ![ \[diffga\_Isi\_h4bcc\] The unbiased estimates of the exponent differences $D_n$ plotted vs $n$ in the case of the Ising model on the h4bcc lattice for the following values of the spin: $s=1/2$ (asterisks) $s=1$ (open squares), $s=3/2$ (open rhombs), $s=2$ (open circles), $s=5/2$ (open triangles), $s=3$ (open stars). The horizontal dashed line and the continuous lines have the same meaning as in Fig.\[diffga\_fi4h4bcc\]. ](FIG456a_5.eps){width="3.37"} ![ \[diffga\_Isi\_h4sc\] Same as Fig.\[diffga\_Isi\_h4bcc\] but for the Ising model on the h4sc lattice. The estimates of the exponent differences $D_n$ are plotted vs $n$ for spin $s=1/2$ (asterisks) $s=1$ (open squares), $s=3/2$ (open rhombs), $s=2$ (open circles), $s=5/2$ (open triangles), $s=3$ (open stars). The horizontal dashed line and the continuous lines have the same meaning as in Fig.\[diffga\_fi4h4bcc\] ](FIG456a_6.eps){width="3.37"} ![ \[IsuImfh56bcc\] The ratios ${\cal Q}_{2r+4}={\cal I}^+_{2r+4}/\hat {\cal I}^+_{2r+4}$ with $r=1,2,3,4$ vs the value $s$ of the spin for the Ising model on the h5bcc and h6bcc lattices. For each value of $s$ the various symbols are slightly shifted apart to avoid superpositions and to keep the spread of each estimate visible. In the case of the h5bcc lattice we have represented ${\cal Q}_{6}$ by open squares, ${\cal Q}_{8}$ by open triangles, ${\cal Q}_{10}$ by open circles, ${\cal Q}_{12}$ by open rhombs. The same ratios for the h6bcc lattice are represented by the corresponding black symbols. The horizontal dashed line represents the expected value of the ratios. ](FIG456a_7.eps){width="3.37"} ![ \[nuoveuccah56bcc\] The ratios $ {\cal R}^+_{8}$ (open squares), $ {\cal R}^+_{10}$ (open circles), $ {\cal S}^+_{10}$ (open rhombs),and $ {\cal S}^+_{12}$ (open triangles) vs the spin $s$ in the case of the Ising model with various values of the spin on the h5bcc lattice. The corresponding black symbols represent the estimates of the same quantities on the h6bcc lattice. For each value of $s$ the various symbols are slightly shifted apart to avoid superpositions and to keep the spread of each estimate visible. The horizontal dashed line represents the expected value of the ratios. ](FIG456a_8.eps){width="3.37"} ![ \[I6I8suImf\_fi4\_h4sc\_h4bcc\] The ratios ${\cal Q}_6={\cal I}^+_6/\hat {\cal I}^+_6$ (black squares) and ${\cal Q}_8={\cal I}^+_8/\hat {\cal I}^+_8$ (open squares) for the scalar field model on the h4sc lattice vs the coupling constant $g$ of the field. The same quantities ${\cal Q}_6$ (black circles), and ${\cal Q}_8$ (open circles) vs the coupling constant $g$ for the scalar-field model on the h4bcc lattice. Like in the preceding figure, symbols associated to the same value of $g$ are slightly shifted in order to avoid superpositions of the error bars. ](FIG456a_9.eps){width="3.37"} ![ \[nuoveuccah4bcc\] The ratios $ {\cal R}^+_{8}$ (open squares), $ {\cal R}^+_{10}$ (open circles), $ {\cal S}^+_{10}$ (open rhombs) and $ {\cal S}^+_{12}$ (open triangles) plotted vs the spin $s$ in the case of the Ising model on the h4bcc lattice.Like in the preceding figure, symbols associated to the same value of the spin are slightly shifted in order to avoid superpositions of the error bars. ](FIG456a_10.eps){width="3.37"}
--- abstract: 'The nonzero widths of heavy particles become significant when they appear in the final state of any decay occurring just around its kinematical threshold. To take into account such effects, a procedure, called the [*convolution method*]{}, was proposed by Altarelli, Conti and Lubicz. We expand their study which included only threshold effects for $t\to b W Z $ in the standard model. We discuss finite width effects in the three body decays $t\to cWW,cZZ$ and $ A^0(h^0)\to t b W$ in the type III version of a two Higgs doublet model. In particular, we find a substantial enhancement in the decay $t\to cZZ$, which brings its branching ratio to ${\rm BR}(t\to cZZ)\sim 10^{-3}$, and in the decay $A^0\to tbW$, which, unlike the $h^0$ case, becomes competitive with the $A^0$ two-body decay modes.' author: - 'Shaouly Bar-Shalom' - Gad Eilam - Mariana Frank - Ismail Turan title: Width Effects on Near Threshold Decays of the Top Quark and of Neutral Higgs Bosons --- -2.5cm Introduction ============ The finite width of a particle is directly related to its instability. When its width is small with respect to its physical mass, finite width effects (FWE) are usually neglected except for decays in which a resonance can emerge when the particle appears as an intermediate state, or in decays that are kinematically allowed only very close to threshold and the particle is involved in either the initial or the final state. The former case is usually handled with the Breit-Wigner prescription, while the latter case, i.e., taking into account the FWE in processes occurring just around their kinematical threshold, needs special attention. In this respect, there are two different methods proposed in the literature [@Altarelli:2000nt; @Mahlon:1994us; @Muta:1986is; @Calderon:2001qq]. They were referred to by Altarelli, Conti and Lubicz [@Altarelli:2000nt] as the [*decay-chain method*]{} (DCM) and the [*convolution method*]{} (CM).[^1] In the first approach (i.e., the DCM), the dominant decay modes of the unstable final state particles are taken into account as subsequent decays to obtain the “total” decay rate and then the branching ratio for the “signal” (i.e., with the unstable particle in the finite state) is calculated by taking the ratio of the “total” decay rate to the multiplication of rates of the subsequent decay modes [@Altarelli:2000nt; @Mahlon:1994us]. This method requires kinematical cuts in order to maintain the direct connection between the “signal” and the total number of events. That is, since the observed final state (with its subsequent decay products) could be produced through other channels, kinematical cuts are required to minimize this undesired background. Therefore, this method leads to physical quantities which depend on kinematical cuts and so it inherits some degree of experimental difficulties. Alternatively, in the CM the instability of a final state particle is described instead by a Breit-Wigner-like density function whose central value and half-width are governed by the width and the physical invariant mass of the particle. In this way, the unstable particle produced can be seen effectively as a real physical particle, having an invariant mass which is controlled by its density function. Although this method does not require any kinematical cut, it doubles the number of phase space integrals, making it computationally more challenging. In this paper we employ the CM to study FWE in the three-body flavor changing rare top decays $t\to cWW$ and $t\to cZZ$, by including the widths of $W$ and $Z$ bosons. These decay modes and other two and three-body rare flavor changing top decays [@mele], can provide a unique testing ground for the standard model (SM) Glashow-Iliopoulos-Maiani (GIM) mechanism and may give hints about - beyond the SM - flavor changing physics such as may occur in some variations of Two-Higgs Doublet Models (2HDM’s). FWE in these decay modes will be studied within the SM (in the case of $t\to cWW$) and in the context of the type III Two Higgs Doublet Model (in both $t\to cWW$ and $t\to cZZ$), which admits flavor changing neutral currents (FCNC) at the tree-level. The three-body top decays $t\to cWW,bWZ,cZZ$ have been considered before, without including FWE, in the SM [@Jenkins:1996zd; @Decker:1992wz; @Diaz-Cruz:1999ab], in 2HDM’s [@Diaz-Cruz:1999ab; @Bar-Shalom:1997tm; @Bar-Shalom:1997sj; @Li:zv; @DiazCruz:1999mq], in a generic formalism including scalar, vector or fermion exchanges [@9707229] and in topcolor-assisted Technicolor model [@0103081]. In addition, the top decays $t\to bWh^0$ and $t\to bWA^0$ have been analyzed in the context of a general 2HDM [@Iltan:2002am]. Among the above decay modes, a simple threshold analysis shows that $t\to cZZ$ and $t\to bWZ$ are potentially the most sensitive to FWE. In particular, according to the recent CDF analysis based on the Tevatron RUN II data, the top mass is ($1\sigma$) [@recenttop1]: $m_t= 173.5^{+2.7}_{-2.6}~(stat)\pm 4.0~(syst)$[^2] In fact, these later top mass measurements imply that for the stable Z-bosons case (i.e., without including FWE) the decay $t \to cZZ$ [*cannot*]{} occur if the top mass lies within its recent CDF and D0 $1\sigma$ limits. We, therefore, expect FWE to be substantial in this decay. Indeed, we find that FWE (due to the rather large ${\mathcal O} (GeV)$ Z-width) can give ${\rm BR}(t\to cZZ) \sim 10^{-5} - 10^{-3}$ (as opposed to null in the stable case), within some range of the allowed parameter space of the type III 2HDM. Moreover, even for the decay $t\to cWW$, for which the central value of the top-quark mass (i.e., $m_t=173.5$) is about 10 GeV away from the kinematical threshold, we find that FWE from the unstable W-bosons can cause a several orders of magnitudes enhancement in the type III 2HDM with a light neutral Higgs of mass $m_{h^0} \lsim 2m_W$, thus elevating the branching ratio from ${\rm BR}(t\to cWW) \sim 10^{-9} - 10^{-8}$ to ${\rm BR}(t\to cWW) \sim 10^{-4} - 10^{-3}$ in this case. Clearly, such large branching ratios would be accessible to the LHC and may even be detected at the Tevatron. A similar large enhancement due to FWE was found for the decay mode $t\to bWZ$ in both the CM [@Altarelli:2000nt] and the DCM [@Altarelli:2000nt; @Mahlon:1994us]. In particular, [@Altarelli:2000nt; @Mahlon:1994us] have found that, in the SM, the FWE increase this decay width by orders of magnitude (with respect to the stable final state gauge bosons), giving ${\rm BR}(t\to bWZ)\simeq 2\times 10^{-6}$ for $m_{t}\sim 176$ GeV. To demonstrate the potential importance of FWE in neutral Higgs decays, we also examine the three-body neutral Higgs decays $h^0 \to tbW$ and $A^0\to tbW$, within the type III 2HDM, assuming that either $h^0$ (the lighter CP-even neutral Higgs) or $A^0$ (the CP-odd neutral Higgs) have masses around $m_t+m_b+m_W$ (i.e., close to the threshold). It is well known that, for a SM-like Higgs, the two-body decay modes to the heaviest fermions and to the gauge bosons are dominant, since its couplings to these particles are proportional to their masses. Three-body sub-threshold decays (e.g., to $W^{*}W$ or $Z^{*}Z$ pairs) can also have sizable BR’s despite the suppression factors involved [@Rizzo:1980gz]. In the context of the minimal supersymmetric extension of the SM (MSSM) sub-threshold three-body decays of especially heavy Higgs bosons might also have a large branching ratio [@Djouadi:1995gv]. In this paper we show that, including the top quark and the W boson width in the framework of the CM, the three-body Higgs decays $h^0 \to tbW$ and $A^0\to tbW$ can be enhanced by about 3 orders of magnitudes in the type III 2HDM if they occur just around their kinematical threshold. For the case of $A^0\to tbW$, such an enhancement can push its BR to the level of tens of percents and may, therefore, become critical for experimental searches of $A^0$. The paper is organized as follows: In Section II we describe the [*convolution method*]{}. In Section III we give a brief overview of the type III 2HDM. In section IV we examine the FWE in the top decays $t\to cWW,cZZ$ and in section V we study the FWE in the three-body Higgs decays $h^{0}\to tbW$ and $A^{0}\to tbW$. In Section VI we summarize our results. The Convolution Method ====================== Particles with large width imply a large uncertainty in its mass from the mass uncertainty relation [@Matthews:1958sc]. The CM can be used to include such large width effects in decays involving unstable particles in the final state. Consider for example the main top decay $t \to bW$. Since the $W$ is unstable, we can define: $\displaystyle \Gamma(t \to bW) \equiv \Gamma=\sum_{i,j}\Gamma^0\left(t\to bf_i\bar{f}_j\right)$, where the sum runs over all the $W$ decay modes. Furthermore, $\Gamma$ can be decomposed into two parts corresponding to the transverse ($\Gamma_T$) and longitudinal ($\Gamma_L$) components of the intermediate $W$-boson (see e.g., [@Calderon:2001qq]): $$\begin{aligned} \label{transverse} \Gamma&=&\Gamma_T +\Gamma_L ~,\end{aligned}$$ where $$\begin{aligned} \displaystyle \Gamma_T&=&\frac{1}{\pi}\sum_{ij}\int_{(m_i+m_j)^2}^{(m_t-m_b)^2}dp^2 \frac{\sqrt{p^2}\,\,\,\Gamma^0\Bigl(t\to bW(p^2)\Bigr)\Gamma^0\Bigl(W(p^2)\to f_i\bar{f_j}\Bigr)}{\Bigl(p^2-m_W^2\Bigr)^2+\Bigl({\mathop{\mathrm{Im}}}\Pi_T(p^2)\Bigr)^2} ~,\end{aligned}$$ and $\Gamma_L \propto f(m_i,m_j)$, with $f\to 0$ as $m_i,m_j\to 0$. Also, $m_W$ is the mass of the $W$ boson and ${\mathop{\mathrm{Im}}}\Pi_T(p^2)$ and ${\mathop{\mathrm{Im}}}\Pi_L(p^2)$ (appearing in $\Gamma_L$) are the absorptive parts of the transverse and longitudinal vacuum polarization tensor (see e.g., [@Calderon:2001qq; @Atwood:nk]). Using the Cutkotsky rule in the limit of massless fermion $m_i,m_j \to 0$ ($f_i,~f_j$, are the fermions exchanged in the W self energy diagram), one obtains\ ${\mathop{\mathrm{Im}}}\Pi_L(p^2)\to 0$ and: $$\begin{aligned} Im\Pi_T(p^2)=\sqrt{p^2}\sum_{i,j}\Gamma^0\left(W(p^2)\to f_i\bar{f}_j\right) = \frac{p^2}{m_W}\Gamma_W^0 ~,\end{aligned}$$ where $\Gamma_W^0$ is the usual on-shell decay width of $W$ and $\sqrt{p^2}\ge m_i+m_j$. Thus, in this limit $\Gamma$ reduces to: $$\begin{aligned} \displaystyle \label{invariant} \Gamma=\Gamma_T&=&\int_{0}^{\bigl(m_t-m_b\bigr)^2}dp^2\, \rho\left(p^2,m_W,\Gamma_W^0\right)\Gamma^0\left(t\to bW(p^2)\right) ~,\end{aligned}$$ where $\rho\left(p^2,m_W,\Gamma_W^0\right)$ is the “invariant mass distribution function”, given by: $$\begin{aligned} \rho\left(p^2,m_W,\Gamma_W^0\right)&=&\frac{1}{\pi}\frac{\frac{p^2}{m_W}\Gamma_W^0}{\Bigl(p^2-m_W^2\Bigr)^2+ \left(\frac{p^2}{m_W}\Gamma_W^0\right)^2} \label{rho}~.\end{aligned}$$ Eqs. (\[invariant\]) and (\[rho\]) describe the factorization of the production and the decay modes of the $W$ boson (in the limit of massless fermions). The case of a stable $W$ boson (i.e., $\Gamma_W^0\to 0$) makes $\rho\to\delta(p^2-m_W^2)$ which sets $\Gamma=\Gamma^0(t\to bW)$, where $\Gamma^0$ is the width for an on-shell $W$ without FWE. The above prescription can be generalized to the case of a generic three-body decay of the form $a\to bV_1V_2$, where $V_1$ and $V_2$ are vector bosons: $$\begin{aligned} \label{twoinvariant} \!\!\!\!\!\Gamma(a\to bV_1V_2)=\int_{0}^{\bigl(m_a-m_b\bigr)^2}\!\!\!\!\!\!dp_1^2\int_{0}^{\bigl(m_a-m_b-\sqrt{p_1^2}\bigr)^2} \!\!dp_2^2&&\rho_1\left(p_1^2,m_{V_1},\Gamma_{V_1}^0\right) \rho_2\left(p_2^2,m_{V_2},\Gamma_{V_2}^0\right)\nonumber\\ &&\times\Gamma^0\left(a\to bV_1(p_1^2)V_2(p_2^2)\right) \label{v1v2}.\end{aligned}$$ Furthermore, for consistency of the CM one needs also the following modifications: 1. The sum over polarization vectors of a gauge-boson with an invariant mass $p^2$ should be taken as: $$\begin{aligned} \sum_{\lambda}\epsilon_{\lambda}^{\mu}(p)\epsilon_{\lambda}^{\nu *}(p)=-g^{\mu\nu}+\frac{p^{\mu}p^{\nu}}{p^2}~.\end{aligned}$$ 2. In calculating the “zeroth” width of the top-quark(Higgs) into the off-shell vector boson(s) \[i.e., $\Gamma^0\left(a\to bV_1(p_1^2)V_2(p_2^2)\right)$ in Eq. (\[v1v2\]) or $\Gamma^0\left({\cal H}\to a\,b(p_1)V(p_2^2)\right)$ for ${\cal H}=h^0$ in Eq. (\[tV\])\], the tree-level propagator of the massive vector bosons should be modified as (in the unitary gauge): $$\begin{aligned} \frac{-i}{p^2-m_{V}^2+im_{V}\Gamma_{V}^0}\left[g^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{m_{V}^2-im_{V}\Gamma_{V}^0}\right]\label{Vectorprop} ~.\end{aligned}$$ This substitution is required since in the CM the invariant mass $p^2$ is allowed to vanish, as can be seen from the integration limits in Eqs. (\[invariant\]) and (\[v1v2\]). 3. In order to restore gauge invariance in the $R_{\xi}$-gauge, the Feynman rules in which masses of such resonant intermediate particles appear should be modified to be functions of the corresponding invariant masses. Such a modification is, however, not necessary in the unitary gauge that we have used in all our calculations [@LopezCastro:1991nt]. The two Higgs doublet model of type III ======================================= One of the simplest extensions of the SM is obtained by enlarging the scalar sector with an additional $SU(2)_L$ doublet. In the most general case such a 2HDM gives rise to tree-level FCNC which are mediated by the physical Higgs bosons [@luke]. To avoid such potentially dangerous FCNC, one usually imposes an ad-hoc discrete symmetry [@Glashow:1976nt] that leads to the type I or type II 2HDM (see for example [@HHG] and [@ourreview]). An alternative way for suppressing FCNC in a general 2HDM (i.e., without imposing discrete symmetries) was suggested by Cheng and Sher in [@Cheng:1987rs]. In the Cheng and Sher Ansatz the arbitrary flavor changing couplings of the scalars to fermions are assumed to be proportional to the square root of masses of the fermions participating in the Higgs Yukawa vertex (see below).[^3] Within the most general 2HDM one can always choose a basis where only one of the doublets acquires a vacuum expectation value (VEV): $\langle \Phi_1\rangle=\left(0\,\,\,\,v/\sqrt{2}\right)^T\,\, \rm{and}\,\, \langle\Phi_2\rangle=0$. A general 2HDM in this basis is often referred to as the type III 2HDM (or Model III) [@Atwood:1996vj; @Sher:1991km; @Atwood:1995ej]. With this choice of basis, $\Phi_1$ corresponds to the usual SM doublet and all the new flavor changing couplings are attributed to $\Phi_2$. Note also that in this basis $\tan\beta=v_1/v_2$ has no physical meaning.[^4] As in any 2HDM, the physical Higgs sector of Model III consists of 3 neutral Higgs bosons (2 CP-even ones, $h^0$ and $H^0$, and one CP-odd state $A^0$) and a charged scalar with its conjugate $H^{\pm}$. The neutral bosons are given, in terms of the original SU(2) doublets, as: $$\begin{aligned} h^0&=&\sqrt{2}\Bigl[-\Bigl({\mathop{\mathrm{Re}}}\phi_1^0-v\Bigr)\,\sin\alpha+{\mathop{\mathrm{Re}}}\phi_2^0\, \cos\alpha\Bigr]\,,\nonumber\\ H^0&=&\sqrt{2}\Bigl[\Bigl({\mathop{\mathrm{Re}}}\phi_1^0-v\Bigr)\,\cos\alpha+{\mathop{\mathrm{Re}}}\phi_2^0\, \sin\alpha\Bigr]\,,\nonumber\\ A^0&=&-\sqrt{2}{\mathop{\mathrm{Im}}}\phi_2^0 ~.\end{aligned}$$ The flavor changing part of the Yukawa Lagrangian in Model III is given by [@luke; @Atwood:1996vj]: $$\begin{aligned} {\mathcal L}_{Y,FC}=\xi_{ij}^U \bar{Q}_{iL}\tilde{\phi}_2U_{jR}+\xi_{ij}^D \bar{Q}_{iL}\phi_2D_{jR}+{\rm H.c.} ~,\end{aligned}$$ where $\tilde{\phi}_2=i\tau_2\phi_2$, $Q$ stands for the quark $SU(2)_L$ doublets, $U(D)$ for up-type (down-type) quark $SU(2)_L$ singlets and $\xi^U,~\xi^D$ are $3\times3$ non-diagonal matrices (in family space) that parametrize the strength of the FCNC vertices in the neutral Higgs sector. Adopting the Cheng and Sher Ansatz we set:[^5] $$\begin{aligned} \xi_{ij}^{U,D}=\lambda_{ij}\frac{\sqrt{m_i m_j}}{v}\,,\,\,\,v=\left(\sqrt{2}G_F\right)^{-1/2} \label{xiud} ~,\end{aligned}$$ where for simplicity we assume the $\lambda_{ij}$’s to be real[^6] and symmetric (i.e., $\lambda_{ij}^*=\lambda_{ji}$) constants. For the Higgs-top-charm coupling we will take that $\lambda_{tc}=\lambda_{ct} \equiv \lambda \sim {\mathcal O}(1)$, which is compatible with all existing data, see [@Bar-Shalom:1997sj; @Atwood:1996vj] for details. Thus, for the top decays of our interest in this paper, the relevant terms in the Yukawa Lagrangian are [@Bar-Shalom:1997sj]: $$\begin{aligned} {\mathcal L}_{{\mathcal H}tc}&=&-\lambda \frac{\sqrt{m_c m_t}}{\sqrt{2}v}f_{{\mathcal H}} {\mathcal H} \bar{c}t\,,\nonumber\\ {\mathcal L}_{{\mathcal H}VV}&=&- g m_W G_V S_{{\mathcal H}}g_{\mu\nu}V^{\mu}V^{\nu}\,,\end{aligned}$$ where ${\mathcal H}=h^0$ or $H^0$, $V=W$ or $Z$ and $$\begin{aligned} f_{h^0;H^0}&=&\cos\alpha;\,\,\sin\alpha\,,\nonumber\\ S_{h^0;H^0}&=&\sin\alpha;\,\,-\cos\alpha\,,\nonumber\\ G_{W;Z}&=&1;\,\,\frac{m_Z^2}{m_W^2} ~.\end{aligned}$$ We will further need the ${\mathcal H}q_iq_i$ (with ${\mathcal H}=h^0,A^0$) and $H^{\pm}tb$ couplings [@Bar-Shalom:1997sj; @Atwood:1996vj]: $$\begin{aligned} {\mathcal L}_{{\mathcal H}q_iq_i}&=&-\frac{m_{q_i}}{v}\bar{q}_i\left[h^0\left(-\sin\alpha +\frac{\lambda_{ii}}{\sqrt{2}}\right)+iA^0\frac{\lambda_{ii}}{\sqrt{2}}\gamma_5\right]q_i\,,\nonumber\\ {\mathcal L}_{H^-t\bar{b}}&=&-\frac{1}{2v}V_{tb}^*H^-\bar{b}\Biggl[\biggl(\lambda_{bb}m_b-\lambda_{tt}m_t\biggr)-\biggr(\lambda_{bb}m_b+\lambda_{tt}m_t\biggr)\gamma_5\Biggr]t \label{yukawa}~.\end{aligned}$$ Finite width effects in the $t\to cWW$ and $t\to cZZ$ decays ============================================================ In this section we will use the CM to evaluate the FWE in the top decays $t\to cWW$ and $t\to cZZ$. Kinematically, the naive threshold (i.e., not including FWE) for the decay $t\to cZZ$ is about 4 GeV away (i.e., larger) from the recent CDF $1\sigma$ limit (from Tevatron RUN II) on the top mass, $m_t(1\sigma) \leq 180.2$ GeV [@recenttop1]. Also, as will be shown below, even for $t\to cWW$ the available phase space can be (depending on the top mass) small enough for the FWE to become significant. We will consider the decay $t\to cWW$ at the tree-level in both the SM and Model III, while $t\to cZZ$ will be analysed only within Model III, since in the SM this decay is doubly suppressed by both one-loop factors and non-diagonal Cabibbo-Kobayashi-Maskawa (CKM) elements and is, therefore, unobservably small. In the SM, the tree-level decay $t\to cWW$ proceeds via $t\to d^* W^+ \to c W^-W^+$ ($d=d,s$ or $b$ quarks), with a BR of the order of ${\mathcal O}(10^{-14}-10^{-13})$ (depending on the top-quark mass) if FWE are not taken into account [@Jenkins:1996zd]. The dominant SM diagram is $t\to b^*W^+ \to c W^-W^+$, since $V_{tb} \times V_{cb}$ is the largest out of the three possible products of CKM elements that enter this decay. In Model III there are two additional tree-level diagrams: $t\to ch^{0*}\to cW^+W^-$ and $t\to cH^{0*}\to cW^+W^-$ [@Bar-Shalom:1997tm; @Bar-Shalom:1997sj]. In this case, we will use the Breit-Wigner prescription for the propagators of ${\mathcal H}=h^0$ or $H^0$, i.e., $(q^2-m_{\mathcal H}^2+im_{\mathcal H}\Gamma_{\mathcal H})^{-1}$, where $\Gamma_{\mathcal H}$ is the total ${\mathcal H}$ width calculated from the dominant ${\mathcal H}$ decay modes: ${\mathcal H}\to b\bar{b},t\bar{t},t\bar{c},ZZ,WW,WW^*,ZZ^*$.[^7]$^,$[^8] Using the CM, the partial decay width for $t\to cWW$ in any given model $M$ can be written as \[see Eq. (\[twoinvariant\])\]: $$\begin{aligned} \displaystyle \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\Gamma^M_{\rm conv}(t\rightarrow cWW)=&&\!\! \frac{1}{512 \pi^3 m_{t}^3}\!\!\int_{0}^{\bigl(m_{t}-m_{c}\bigr)^2}\!\!dp_{W^+}^2 \left[\frac{p_{W^{+}}^2 \Gamma_{W}^0}{m_{W}\pi \left( \Bigl(p_{W^{+}}^2-m_{W}^2\Bigr)^2+ \Bigl(\frac{p_{W^{+}}^2 \Gamma_{W}^0}{m_{W}}\Bigr)^2\right)}\right]\nonumber\\ &&\!\!\!\times\int_{0}^{\bigl(m_{t}-m_{c}-\sqrt{p_{W^+}^2}\bigr)^2}\!\!dp_{W^-}^2 \left[\frac{p_{W^{-}}^2 \Gamma_{W}^0}{m_{W}\pi \left( \Bigl(p_{W^{-}}^2-m_{W}^2\Bigr)^2+ \Bigl(\frac{p_{W^{-}}^2 \Gamma_{W}^0}{m_{W}}\Bigr)^2 \right)}\right]\nonumber\\ &&\!\!\!\times\int_{\bigl(m_{c}+\sqrt{p_{W^-}^2}\bigr)^2}^{\bigl(m_{t}- \sqrt{p_{W^+}^2}\bigr)^2}\!\!dx_{1}\int_{x_{2,min}}^{x_{2,max}}\!dx_{2}\,\, \left|{\mathcal M}^M_{\rm conv}(x_{1},x_{2},p_{W^+}^2,p_{W^-}^2)\right|^2\!, \label{SMgamma} \end{aligned}$$ where the superscript $M$ stands for the model used for the calculation of the convoluted amplitude ${\mathcal M}^M_{\rm conv}$, and $$\begin{aligned} x_{2,min}&=&(E_{2}+E_{3})^{2}-\bigg(\sqrt{E_{2}^{2}-p_{W^{-}}^{2}}+\sqrt{E_{3}^{2}-p_{W^{+}}^{2}}\bigg)^{2} ~,\nonumber\\ x_{2,max}&=&(E_{2}+E_{3})^{2}-\bigg(\sqrt{E_{2}^{2}-p_{W^{-}}^{2}}-\sqrt{E_{3}^{2}-p_{W^{+}}^{2}}\bigg)^{2}~,\nonumber\\ E_{2}&=&\frac{x_{1}-m_{c}^2+p_{W^{-}}^2}{2\sqrt{x_1}};\;\;E_{3}=\frac{-x_{1}-p_{W^{+}}^2+m_{t}^2}{2\sqrt{x_1}}~.\end{aligned}$$ For the BR calculation, we approximate the total width of the top quark by its dominant decay $t\to bW$ which is computed at tree-level with the corresponding value of the top quark mass. In Fig. \[figtcWW1\] we plot the ${\rm BR}(t \to c W^+W^-)$ as a function of the top quark mass in the SM, with and without FWE. The case of stable $W$’s in the final state (i.e. without FWE) is obtained by taking the limit $\rho(p_W^2,m_W^2,\Gamma_W^0) \to \delta(p_W^2-m_W^2)$ \[see Eq. (\[rho\])\] which sets $p_{W^{\pm}}^2=m_W^2$ in the integrand of Eq. (\[SMgamma\]). The decay $t \to c W^+W^-$ in the SM with stable $W$’s was calculated in [@Jenkins:1996zd] and our result for this case agrees with hers. From Fig. \[figtcWW1\] we see that for the CDF central value of the top mass, $m_t =173.5$ GeV, FWE can enhance the ${\rm BR}(t \to c W^+W^-)$ by about an order of magnitude, reaching $\sim 2 \cdot 10^{-13}$. For the lower $1\sigma$ CDF limit $m_t \sim 167$ GeV, the enhancement due to FWE is of about two orders of magnitudes. Unfortunately, even with such large FWE in the decay $t \to c W^+W^-$, the BR in the SM is still too small to be measured - even at the LHC. 6.0in -2.3in 5.5in 5.5in -2.2in In Fig. \[figtcWW2\] we show the ${\rm BR}(t \to c W^+W^-)$ in Model III with $\lambda_{tc}=1$, $m_{H^0}=1$ TeV and $\alpha=\frac{\pi}{4}$[^9] (note that the SM tree-level contribution to $t \to c WW$, although included, is negligible in this case), as a function of $m_t$ with and without FWE, for several values of the light Higgs mass $m_{h^0}=130,~150,~170,~ {\rm and}~ 190$ GeV, and as a function of $m_{h^0}$ with FWE, for the lower, upper and central CDF values of the top-quark mass $m_t=166.9~,173.5$, and 180.2 GeV. As was found in [@Bar-Shalom:1997tm; @Bar-Shalom:1997sj], in Model III without FWE, the ${\rm BR}(t \to c W^+W^-)$ can at most reach the level of ${\rm few} \times 10^{-5}$ if $m_t$ lies within its $1\sigma$ CDF limits and [*only if*]{} $m_{h^0} \sim m_t$. On the other hand, when FWE are “turned on”, a huge enhancement to the width arises within a large range of the Higgs mass. In particular, for $ 100 ~{\rm GeV} \lsim m_{h^0} \lsim 165$ GeV, we find ${\rm BR}(t \to c W^+W^-) \gsim 10^{-4}$, if $ 167 ~{\rm GeV} \lsim m_t \lsim 180$ GeV, in Model III when FWE are included. Note that, for the lower $1\sigma$ limit $m_t \sim 167$ GeV, i.e., close to the threshold for producing $cWW$, the FWE causes an up to six orders of magnitudes enhancement to the ${\rm BR}(t \to c W^+W^-)$ if, e.g., $m_h \sim 130$ GeV. For the decay $t \to c ZZ$ in Model III we use the analytical results of $t\to cWW$ with the replacements $m_W\to m_Z/\cos\theta_W$ in the ${\mathcal H} VV$ vertex, $p_W^-\to p_{Z_1},\,p_W^+\to p_{Z_2}$ in Eq. (\[SMgamma\]) and with an additional overall factor of 1/2 to take into account the symmetry factor for identical particles in the final state (i.e., $Z$ bosons). Fig. \[figtcZZ1\] shows the scaled branching ratio ${\rm BR}(t\to cZZ)/\lambda^2$ ($\lambda \equiv \lambda_{tc}$) in Model III with $m_{H^0}=1$ TeV and $\alpha=\pi/4$ (see also footnote 9), as a function of $m_t$ with and without FWE, for $m_{h^0}=130,~150,~170,~{\rm and}~190$ GeV, and as a function of $m_{h^0}$ with FWE, for $m_t=166.9~,173.5$, and 180.2 GeV. Note that the decay $t\to cZZ$ is fundamentally different from $t\to cWW$, since, unlike $t\to cWW$, this decay channel [*cannot*]{} occur for stable Z-bosons if $m_t$ lies within its $1\sigma$ limits. Thus, the inclusion of FWE in $t\to cZZ$ is crucial in this case. In particular, from Fig. \[figtcZZ1\] we see that a remarkably large ${\rm BR}(t\to cZZ) \sim 10^{-5} - 10^{-3}$ is expected in Model III, if $m_{h^0}$ lies within $ 90 ~{\rm GeV} \lsim m_{h^0} \lsim 170$ GeV. Such a large BR will be accessible to the LHC and may even be detected at the Tevatron. Finally we note that, following [@Altarelli:2000nt] (who took $m_b=m_B$ for their calculation of $t\to bWZ$), we take $m_c=m_D=1.87$ GeV. 5.5in 5.5in -2.2in Finite width effects in the $A^0\to(\bar{t}b+t\bar{b})W$ and $h^0\to(\bar{t}b+t\bar{b})W$ decays ================================================================================================ In this section we will examine FWE in three-body decays of neutral Higgs bosons in Model III. We will focus on the decay channels $A^{0} \to \bar{t}bW^{+}$ and $h^0\to \bar{t}bW^{+}$ which can have both theoretical and experimental advantages for Higgs searches and for investigating Higgs properties in the Higgs mass range $200 ~{\rm GeV} \lsim m_{h^0},~m_{A^0} \lsim 300$ GeV. (200,90)(10,-5) (0,50)\[c\][$(a)$]{} (0,0)(50,0)[2]{} (50,50)(50,0) (50,0)(100,0) (100,0)(150,0)[3]{}[6]{} (50,50)(50,0) (100,0)(100,50) (55,30)(55,50) (25,10)\[c\][${A^{0}(h^{0})}$]{} (67,40)\[c\][$p_{t}$]{} (40,45)\[c\][$\bar{t}$]{} (75,10)\[c\][$t$]{} (105,45)\[c\][$b$]{} (75,10)\[c\][$t$]{} (145,13)\[c\][${W_{\mu}^+}$]{} (50,0)[2]{} (100,0)[2]{} (200,90)(130,0) (150,50)\[c\][$(b)$]{} (150,0)(200,0)[2]{} (200,0)(250,0)[2]{} (250,0)(300,0) (250,50)(250,0) (200,50)(200,0)[3]{}[6]{} (245,30)(245,50) (175,10)\[c\][${A^0(h^0)}$]{} (228,10)\[c\][${H^-}$]{} (275,10)\[c\][$b$]{} (260,45)\[r\][$\bar{t}$]{} (228,40)\[l\][$p_{t}\!\!$]{} (175,45)\[l\][${W^+}$]{} (200,0)[2]{} (250,0)[2]{} (200,90)(10,-5) (0,50)\[c\][$(c)$]{} (0,0)(50,0)[2]{} (50,0)(50,50) (100,0)(50,0) (100,0)(150,0)[3]{}[6]{} (100,50)(100,0) (95,30)(95,50) (25,10)\[c\][${A^{0}(h^{0})}$]{} (44,45)\[c\][$b$]{} (108,45)\[c\][$\bar{t}$]{} (78,10)\[r\][$b$]{} (78,40)\[l\][$p_{t}$]{} (145,13)\[c\][${W_{\mu}^+}$]{} (50,0)[2]{} (100,0)[2]{} (200,90)(130,-5) (150,50)\[c\][$(d)$]{} (150,0)(200,0)[2]{} (200,0)(250,0)[3]{}[6]{} (250,0)(300,0) (250,50)(250,0) (200,0)(200,50)[3]{}[6]{} (245,30)(245,50) (175,10)\[c\][${h^0}$]{} (228,10)\[c\][${W^-}$]{} (275,10)\[c\][$b$]{} (260,45)\[r\][$\bar{t}$]{} (228,40)\[l\][$p_{t}$]{} (170,45)\[l\][${W^+}$]{} (200,0)[2]{} (250,0)[2]{} The tree level diagrams contributing to these two decays in Model III are given in Fig. \[Feynm\] (note that, for the $A^0$ decay, the diagram with an intermediate $W$-boson is missing, i.e., diagram (d), due to the absence of a tree-level $A^0WW$ coupling). A fomula analogous to Eq. (\[v1v2\]) can be given for Higgs decays as $$\begin{aligned} \!\!\!\!\!\Gamma({\cal H}\to b\, \bar{a}\, V)=\int_{0}^{\bigl(m_{\cal H}-m_b\bigr)^2}\!\!\!\!\!\!dp_1^2\int_{0}^{\bigl(m_{\cal H}-m_b-\sqrt{p_1^2}\bigr)^2} \!\!dp_2^2&&\rho_1\left(p_1^2,m_t,\Gamma_{a}^0\right) \rho_2\left(p_2^2,m_V,\Gamma_V^0\right)\nonumber\\ &&\times\Gamma^0\left({\cal H}\to b\, \bar{a}(p_1^2)\, V(p_2^2))\right) \label{tV},\end{aligned}$$ where ${\cal H}=h^0$ or $A^0$ and $a(b)$ is the top(bottom) quark. Using the interaction terms in Section 3, we calculate the matrix element for each decay, where: 5.8in -2.3in 5.8in -2.3in - The propagator of the intermediate $W$ is taken from Eq. (\[Vectorprop\]). - In the calculation of $\Gamma^0$ in Eq. (\[tV\]), the usual sum over the spins of the outgoing top-quark is modified to $\sum{u(p_t)\bar{u}(p_t)}=\ptslash+\sqrt{p_{t}^2}$ since, using the prescription of the CM, the final state top-quark is allowed to be off-shell. - Throughout the following we assume that the Higgs mass spectrum respects $m_{h^0} < m_{A^0} \ll m_H^+,~m_H^0$, setting $m_H^+=m_H^0=1$ TeV. Thus, the contribution from the charged Higgs exchange, i.e., diagram (b) in Fig. \[Feynm\], becomes negligible. - The total width of $A^0$ is estimated from the decays $A^0\to \tau\bar{\tau},\,b\bar{b},\,h^0Z,h^0Z^*,\,(t\bar{b}+\bar{t}b)W$, and the total width of $h^0$ is estimated from the decays $h^0\to \tau\bar{\tau},\,b\bar{b},\,W^+W^-,ZZ$. - We set all the relevant flavor diagonal $\lambda$’s of the Higgs Yukawa couplings in Eq. (\[yukawa\]) to unity, i.e., $\lambda_{qq}=1$. With the above assumptions, the remaining relevant input parameters (in Model III) for evaluating the branching ratios under consideration are $m_{A^0}$, $m_{h^0}$ and the Higgs mixing angle $\alpha$. In Fig. \[BRA0\] we depict the branching ratio of $A^0\to(\bar{t}b+t\bar{b})W$ as a function of $m_{A^0}$, for two values of the light Higgs mass $m_{h^0}=170$ and 230 GeV and for $m_H^+=1$ TeV, $m_t=173.5$ GeV and $\alpha=\pi/4$. We see that near threshold, i.e., $m_{A^0} \sim 260$ GeV, there is an enhancement of several orders of magnitude due to FWE, wherein the the branching ratio can reach ${\rm BR}(A^0\to(\bar{t}b+t\bar{b})W) \sim 10^{-2}$. Away from threshold, the decay $A^0\to(\bar{t}b+t\bar{b})W$ is sensitive to the lightest neutral Higgs mass, $m_{h^0}$. In this case, the inclusion of FWE can increase the branching ratio by almost an order of magnitude, giving e.g. ${\rm BR}(A^0\to(\bar{t}b+t\bar{b})W) \sim {\rm few} \times 10^{-1}$ for $m_{A^0} \sim 300$ GeV and $m_{h^0}=230$ GeV. Thus, FWE in the three-body decay $A^0\to(\bar{t}b+t\bar{b})W$ can become very significant – bringing its BR to the level of tens of percents and making it competitive with the $A^0$ two-body decays and, therefore, a viable experimental signature for studies of the properties of the Higgs sector. Finally, let us consider the decay $h^0\to(\bar{t}b+t\bar{b})W$. In Fig. \[BRh0\] we plot its branching ratio as a function of $m_{h^0}$ for the same input parameters (of Model III) as in Fig. \[BRA0\]. In this case, in spite of the large enhancement near threshold due to FWE, the ${\rm BR}(h^0\to(\bar{t}b+t\bar{b})W)$ remains rather small, i.e., at most of ${\mathcal O}(10^{-5})$, mainly due to the much larger $h^0$ total width caused by its tree-level decays to a pair of gauge-bosons $h^0 \to WW,ZZ$. Summary ======= We have studied and emphasized the importance of FWE (finite width effects) in decays occurring just around their kinematical thresholds. For the inclusion of FWE we have adapted the so called CM (convolution method). In the CM, the unstable particle with 4-momentum $p$ is treated as a real physical particle with an invariant mass $\sqrt{p^2}$ and effectively weighted by a Breit-Wigner-like density function, which, becomes a Dirac-delta function in the limit that the particle’s total width approaches zero. We first examined the FWE within the SM in the rare and flavor-changing tree-level top decay $t \to c W^+W^-$ and then extended our analysis to FWE in the tree-level top decays $t \to c W^+W^-$, $t \to c ZZ$ and Higgs decays $A^0,~h^0 \to t \bar b W$ in a general two Higgs doublets model, the so called Model III, which gives rise to tree-level FCNC in the Higgs-fermion sector. In all these case we find that FWE can become substantial – enhancing the branching ratios for the above decays by several orders of magnitudes near threshold. Unfortunately, in the SM case, the top decay $t \to c W^+W^-$ remains too small to be of any value in the upcoming high energy colliders, i.e., ${\rm BR^{SM}}(t \to c W^+W^-) \sim 10^{-13}-10^{-12}$, in spite of the large enhancement due to FWE. On the other hand, in Model III, the large enhancement due to FWE in all these three-body top and Higgs decays can make a difference with respect to experimental studies in the upcoming hadron colliders. In particular, the branching ratios for the top-decays $t \to c W^+W^-$ and $t\to c ZZ$ can reach the level of $10^{-4}-10^{-3}$ near threshold – many orders of magnitudes larger than the corresponding branching ratio for the stable W and Z-bosons case (i.e., without FWE). For the $t\to c ZZ$ decay, the inclusion of FWE is essential since such a large branching ratio arises even though the naive threshold for this decay is a few GeV away from the most recent $1 \sigma$ upper limit on the top mass, $m_t(1\sigma) \sim 180$ GeV. In the Higgs decays, FWE are more noticeable in the pseudo-scalar Higgs decay $A^0 \to(\bar{t}b+t\bar{b})W$, elevating its branching ratio to the level of tens of percents, thus making this three-body decay channel dominant and competitive with its two-body decays and, therefore, extremely important for experimental studies. Thus, our study shows that FWE is essential for a proper treatment of otherwise neglected finite widths of particles which emerge at the final state of decays or scattering processes occurring just around the threshold. I.T. would like to thank the HEP group members at Technion for their support and kind hospitality during his stay there. The work of G.E. was supported in part by the United States Department of Energy under Grant Contract No. DE-FG02-95ER40896 and by the Israel Science Foundation. The work of M.F. is supported in part by NSERC under grant number 0105354. I.T. also thanks Marc Sher for useful conversations. [99]{} G. Altarelli, L. Conti and V. Lubicz, Phys. Lett. B [**502**]{}, 125 (2001). G. Mahlon and S. J. Parke, Phys. Lett. B [**347**]{}, 394 (1995). T. Muta, R. Najima and S. Wakaizumi, Mod. Phys. Lett. A [**1**]{}, 203 (1986). G. Calderon and G. Lopez Castro, arXiv: hep-ph/0108088. V. I. Kuksa, arXiv: hep-ph/0404281; arXiv: hep-ph/0508164. 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[^2]: Note that later D0 results from Tevatron RUN II, $m_t=170.6\pm 4.2~(stat)\pm 6.0~(syst)$ (see [@recenttop1]), are based on less accumulated data and has larger statistical and systematic uncertainties. [^3]: The Cheng and Sher Ansatz ensures the suppression of FCNC within the first two generations of quarks, as required by the experimental constraints on FCNC in meson transitions, see [@Atwood:1996vj]. [^4]: “Switching on” $\tan\beta$ by allowing $\langle\Phi_2\rangle \neq 0$ will not change any physical result. [^5]: Note that there is a factor of 1/2 difference between our definition for $\xi_{ij}^{U,D}$ in (Eq. \[xiud\]) and the one used in [@Bar-Shalom:1997sj]. This difference may be absorbed by redefining the arbitrary parameters $\lambda_{ij}$. [^6]: In this work we are not interested in CP-violating effects that may be driven by a possible phase contained in the $\lambda_{ij}$’s. [^7]: Note that in Model III the decay ${\mathcal H}\to t\bar{c}$ becomes important for $\lambda_{tc} \sim {\mathcal O}(1)$. [^8]: Depending on the ${\mathcal H}$ mass, only the kinematically allowed decays will be included in $\Gamma_{\mathcal H}$. [^9]: The dependence of ${\rm BR}(t \to c W^+W^-)$ and ${\rm BR}(t \to c ZZ)$ on the Higgs mixing angle $\alpha$ in Model III can be found in [@Bar-Shalom:1997tm; @Bar-Shalom:1997sj]. The maximum of these branching ratios with respect to $\alpha$ takes place at $\alpha=\pi/8$ (due to the dependency of the Higgs width on $\alpha$) and not at $\alpha=\pi/4$ which is used through out our analysis.
--- abstract: 'In previous work we derived the topological terms in the M-theory action in terms of certain characters that we defined. In this paper, we propose the extention of these characters to include the dual fields. The unified treatment of the M-theory four-form field strength and its dual leads to several observations. In particular we elaborate on the possibility of a twisted cohomology theory with a twist given by degrees higher than three.' --- hep-th/0509046 [**Duality symmetry and the form fields of M-theory**]{} [Hisham Sati]{} [^1] \ Introduction ============ Interesting global information is encoded in the Maxwell-like rank four field $G_4$ of M-theory, which is written locally as $G_4=dC_3$ where $C_3$ is the so-called C-field. So one concrete aim in this direction is to understand the nature of this C-field. Another is to understand Hodge duality that relates $G_4$ to its dual $*G_4$ in eleven dimensions. There is an analogous question in type II string theory where the fields are grouped into a total field strength containing the fields descending from $G_4$, by dimensional reduction, as well as their (ten-dimensional) Hodge duals. This package leads to the description in terms of twisted K-theory [@MW] [@DMW] [@MS]. We would like then to ask whether, in analogy to the type II case, we can unify both field strengths in eleven dimensions, namely the fields $G_4$ and $*G_4$. So we seek a generalized cohomology theory in which the eleven-dimensional fields are unified in the same way that the Ramond-Ramond fields (in the presence of Neveu-Schwarz fields) are unified into (twisted) K-theory. Earlier work [@KS1] [@KS2] [@KS3] with I. Kriz viewed elliptic cohomology as the right setting for type II string theory. The corresponding picture in M-theory leads to the question of whether the theory $\mathcal M$ proposed in [@S1] is new or whether it happens to be one of the known generalized cohomology theories. In [@S2] we proposed a unified quantization condition on $G_4$ and its dual by viewing the pair as components of the same total field strength. So the point we look at in the present paper is the possibility that this total field strength ‘lives’ in some generalized cohomology theory. One might argue that the problem can be looked at from the complementary picture of branes. In the same way that one has to talk about branes up to creation of other branes in type II string theory [@MM], here we ask whether one can talk about M-branes up to creation of other M-branes. While the picture is not precisely analogous, one can say that the existence of the $M5$-brane automatically requires the existence of the $M2$-brane, via the Hanany-Witten mechanism or via the dielectric effect. [^2] The supermultiplet $(g_{\mu \nu},\psi_{\mu}, C_3)$ of eleven-dimensional supergravity [@CJS] is composed of the metric, the gravitino and the C-field. Thus, in its standard formulation, the theory is manifestly duality-nonsymmetric. One can then ask about the role of the dual fields in the theory. One can get a free supersymmetric theory based on the dual 6-index field $C_6$, but the corresponding interacting theory is not consistent [@NTV]. There is also a duality-symmetric formulation of eleven-dimensional supergravity [@Ban]. [^3] However, such a formulation does not seem to accommodate nontrivial topology or fields that are nontrivial in cohomology. There is also the duality-symmetric formulation of the nongravitational fields in [@dual], again assuming $G_4=dC_3$, i.e. the field $G_4$ is trivial in cohomology, $[G_4]=0$. We need a degree four ‘Bott generator’ and either a degree seven or a degree eight gnerator for the dual. Using the rank seven field $*G_4$ as the dual field, we find the equations of motion (henceforth EOM) and the Bianchi identity as components of a unified expression of the total field strength, using a twisted differential, with the twist now given by the degree four field $G_4$ instead of $H_3$, in the usual case of type II string theory. Adding the one-loop term $I_8$ to the EOM serves a priori as an obstruction to having such a twisted cohomology. However, by absorbing $I_8$ in the definition of the dual field strength one still gets a twisting. One can ask about the relevance of the $E_8$ gauge theory. We know that the degree four field $G_4$ is intimately related to $E_8$, at least topologically [@W1]. What we are advocating is that there two ways of looking at the problem, one via $E_8$, and another via some generalized cohomology theory. But then adding the dual fields, one seems to break that connection, and in this case it seems possible to only look for a generalized cohomology interpretation, as the homotopy type of $E_8$ does not allow for a direct interpretation of the dual field(s). So we argue for two points of view regarding the fields. The first is the bundle picture in which only the lower-rank fields ‘electric’ fields are described, e.g. $G_4$ in M-theory via $E_8$, $F_2$ in type IIA via the M-theory $S^1$-bundle. The second is the generalized cohomology picture where the field strengths and their duals are grouped into one total field strength that lives in the corresponding generalized cohomology theory, e.g. twisted K-theory for type II. Thus taking the second point of view, the aim of this paper is to argue for a generalized cohomology theory for the case of the M-theory field strength $G_4$ and its dual. Such a unification was already started in [@S2] where the class of $G_4$ and the dual class $\Theta$ (realizing the RHS of the EOM) were given a unified expression that reflected their quantization laws. The existence of the corresponding generalized cohomology theory was proposed in [@S1] and further properties were given in [@S2]. The total field strength ======================== First note that, unlike the RR fields which have mod 2 periodicity, the fields of M-theory do not enjoy such a periodicity. This is obvious because one of the fields has even rank and the other has odd rank. Besides there are only two of them. One can ask first whether there is a Bott element of dimension three ($=$ the difference of the two ranks) that can take the role which the usual Bott element played in type II. The answer is negative and there is no such element in the class of theories descending directly from $MU$. So one can then ask whether there is another way to form a total M-theory field strength with a uniform degree. One is then forced to use more than one element to do the job. Again there is no element of odd degree, so in order to be able to say something useful, one seeks a modification of the point of view in which even degree fields are included. But what exactly should we do? Two things come to mind. First we can try to lift to the bounding twelve dimensional theory defined on $Z^{12}$ with $\partial Z^{12}=Y^{11}$. Here, one possibility is then to look at the four/eight combination $G^{(12)}=G_4 + *_{12}G_4$ in twelve dimensions. Then the arguments that hold for $G_4 + d*_{11}G_4$ in eleven dimensions hold for $G^{(12)}$ as well. [^4] Second, we can work with an eight-form in eleven dimensions, that we view as the dual field instead of the seven form. On the other hand, if we insist on working with odd forms, then this seems to suggest some deformation of cohomology rings which involves odd generators. We are looking for a generator of degree four that makes a degree zero form when multiplied with $G_4$. Since ${\rm dim}v_n=2p^n-2$, there is only one generator of degree four, which the first generator at $p=3$. What theory is a good candidate theory to include this generator? It is possible that this is either of the first Morava K-theories at $p=3$, i.e. either ${\widetilde K}(1)$ or $K(1)$ with coefficient rings ${\widetilde K}(1)_*={{\mathbb Z}}[v_1,v_1^{-1}]$, and ${K}(1)_*={{\mathbb Z}}/3[v_1,v_1^{-1}]$, respectively. We can then form the desired class [^5] $ (v_{1,p=3})^{-1}G_4.$ As in the case for $G_4$ we are looking for a generator whose degree is the same as the degree of the field, and which is inverted so that its inverse can be used to write down a uniform degree zero field. So here we need a degree eight generator. Now we would like to find an expression of total degree zero for the total M-theory field strength. The desired generator is the square of $v_{1,p=3}$, which has total dimension $4+4=8$. So with this possibility, we can write the following expression for the uniform total field strength [^6] $ G=(v_{1,p=3})^{-1}G_4 + (v_{1,p=3})^{-2}G_8. $ With this, we are using the same generator for the whole expression, which is the case analogous to the type II situation, One possibility that that we are then dealing with the $p=3$ first (integral) Morava K-theory. One can ask whether the problem can be looked at without specializing to a particular prime. The theory of Topological Modular Forms, $tmf$, has an interesting feature that it is not localized at a given prime, i.e. is not local and unifies all primes – see [@KS3] for a discussion on the relevance of TMF from a different but related point of view. This is attractive, and seems to be what a theory like M-theory should be doing. Besides, this might make sense since the vector bundles (or their ‘higher-degree’ analogs) are real, and TMF is a real theory – it is to elliptic cohomology $E$ as $KO$-theory is to $K$-theory. One can ask whether there are degree four and degree eight generators in $tmf$, which can be used for the total field strength. Indeed there are such generators, which were used in [@KS3]. As far as dynamics goes, it does not make much sense to talk about $*G_4$ or $d*G_4$ alone, because their dynamics involve $G_4$ (cf. the EOM of $G_4$). So in order to include the dual picture, one can at best look for a duality-symmetric formulation of the character, i.e. as opposed to a dual description. If we use the eight-form $d*G_4$ as the ‘dual’ form, then the corresponding exponential is $ e^{G_4 +d*G_4}. \label{8}$ We ask the question whether from this we can get the EOM and the Bianchi identity. By looking at the degrees of the forms, we see that while we can get the Bianchi identity by looking at the degree five component, i.e. $ \left[d\left( e^{G_4 +d*G_4} \right) \right]_{(5)}, $ we cannot get the EOM, simply because the degrees of forms would not match. [^7] One can then ask whether the exponential (\[8\]) can be looked at in some other way that would give the EOM and Bianchi. While the EOM can be obtained by some ‘flatness condition’ on the character, i.e. $ \left[e^{G_4 + d*G_4} \right]_{(8)}=0, \label{flat}$ the Bianchi identity does not follow. One instead gets a flatness condition on $G_4$ as well if one were to look at the degree four component of the expression (\[flat\]). Even though one can say we got both the EOM and the Bianchi identity, we actually did not do that by using the same expression, and this is obviously not satisfactory. This seems to indicate that while the quantization conditions on the forms [@S2] favors the four/eight combination, the dynamics favors instead the four/seven combinations of field strengths. Let us now look at the effect of including the generators– let us call them $v$ and ${\widetilde v}$ – in (\[8\]). Doing so results in the expression $ \left[\left( e^{v^{-1}G_4 + {\widetilde v}^{-1}d*G_4} \right) \right]_{(8)}=\frac{1}{2}v^{-2} G_4 \wedge G_4 + {\widetilde v}^{-1} d*G_4. \label{vv} $ So requiring that we get the EOM via factoring out the generators leads to the obvious condition that [^8] $ v^2={\widetilde v}. $ Naturally, we would like to see whether such a condition can occur in the generalized cohomology theories that we consider in this paper. We check the dimensions of the generators. Since in general that dimension at ‘level’ $n$ and prime $p$ is ${\rm dim}~v_n=2(p^n-1)$, we then need to satisfy the equality $ \left[2(p^n-1)\right]^2=2(p^m-1), \label{sq}$ where $m>n$. Even though such an expression is not expected to have many solutions in general, it is still more general than we want. It might be desirable to require that the total expression on the RHS of (\[vv\]) have degree zero. It turns out that this is not possible within the current context, and the next best thing is to require the first generator $v$ to have degree four. [^9] This then implies, via $2(p^n-1)=4$, that $p=3$ and $n=1$. Of course the equality is then satisfied and the dimension of ${\widetilde v}$ is $16$ with $m=2$ and the same prime $p=3$. Let us go back and look at what the above implies for the relationship between the dimensions of the generators and the dimensions of the field strengths. In the above we asked whether the expanded exponential expression has total degree zero. But then going back to the exponent, we see that it does not have total degree zero, because we have the generator ${\widetilde v}$, which we found to have dimension sixteen, multiplying $d*G_4$ which has rank eight as a form or a class. However, it is still true that the $G_4$ part has degree zero. What we learn from this is that what matters is for the degrees of the factors to match after expanding the exponential and not as they stand in the exponent. As mentioned earlier, generators of degree four and eight can be obtained from tmf (cf. [@KS3]). A twisted (generalized) cohomology? {#deg7} =================================== In this section, we would like to use the degree seven field as the dual field to $G_4$ and thus take the total field strength to be $G=G_4 + *_{11}G_4$. We would like to use such an expression (and slight variations on it –see below) as it is duality-symmetric [^10] in the electric-magnetic or membrane-fivebrane sense. Then it is interesting that one can write the Bianchi identity and the EOM of $G_4$, respectively, as the degree five and the degree eight component of the expression $ \left( d+ \frac{1}{2} G_4\wedge +\frac{1}{2}*G_4\wedge \right) G=0. \label{twist}$ The degree eleven component, i.e. the cross-terms between $G_4$ and $*G_4$, vanish because of the relative minus sign, [^11] and the $*G_4\wedge *G_4$ term vanishes because it involves the same form of odd degree. There are several interesting aspects to equation (\[twist\]). First, one can ask whether this has the form of some twisted structure in analogy to that associated with the RR fields in type II string theory, where one has for the total field strength $F$, $ dF=H_3 \wedge F.$ Written as $ d_{H_3}F=(d-H_3 \wedge)F=0,$ this leads to interpreting $d-H_3$ as the differential in twisted (de Rham) cohomology $H^*(X,H;{{\mathbb R}})$, even for type IIA and odd for type IIB [@MS]. One can easily check that $(d_{H_3})^2$ is indeed zero [@MS], which follows from the fact that the twisting field $H_3$ is closed and that the wedge product of two twisting fields $H_3\wedge H_3$ vanishes just because it is the wedge product of the same differential form of odd degree. Going back to (\[twist\]), we ask whether an analogous structure appears. Of course we have obvious differences from the type II case: what is to be interpreted as a ‘twisting field’, $\frac{1}{2}G$, is now part of the total field that is being twisted, namely $G$. [^12] The other difference is that the twist now involves an even rank field, which while it is closed in analogy to $H_3$, the wedge of two copies of which does not vanish since it is even-dimensional. If we interpret the combination $d+ \frac{1}{2}G_4+\frac{1}{2}*G_4$ as a new differential $d_{G}$ and hope that it forms a cohomology, then the nilpotency does not seem to be immediately obvious. However, it turns out that the situation is in fact encouraging. To see this, let us simply calculate the action of its square on the total field strength, $ d_{G}^2~G=\left( d+ \frac{1}{2}G_4\wedge +\frac{1}{2}*G_4 \wedge \right)^2 G, $ but this is zero for the same reasons that equation (\[twist\]) holds, namely by use of the EOM and the Bianchi identity, and by the fact that the rest of the terms have high degrees. Thus, $ d_{G}^2~G=0.$ This is on-shell and is valid when the differential acts on the field strength. In the case of type II string theory, $d_H=d+H_3$ was an actual differential, i.e. $d_H^2$ was zero without necessarily acting on the RR field $F$. Does this happen in our case of M-theory? Let us study the question one step at a time. To start, calculating $d_G^2$ gives the sum $ \frac{1}{2}G_4\wedge d + \frac{1}{2}G_4 \wedge d, $ i.e. $G_4\wedge d$. Obviously this is not zero, and so we need to modify the differential in order to have any hope at nilpotency. The problem can be traced back to the fact that $G_4$ has an even degree and so moving the differetnial over it does not pick a minus sign that would then cancel the other factor. Explicitly, the square gives the cross terms $d(G_4\wedge) + G_4\wedge d$, which when expanded gives $dG_4\wedge + G_4 \wedge d + G_4\wedge d$. The first term disappears because of the Bianchi identity but the second [*adds*]{} to the third (instead of [*subtracting*]{} had $G_4$ been of odd degree). Thus the problem does not arise for $*G_4$. Note that at this stage we can see that the somewhat artificial factor of half inside the differential does not seem to matter. We will see that this is indeed the case later. In order to get the two terms above to subtract instead of add, we need some form of grading. For that purpose, let us use the duality-symmetric total field strength introduced in [@dual], $ \mathcal{G}=vG_4+{\widetilde v} *G_4, $ and check whether this $\mathcal{G}$ can be used as a twist to form the desired differetial. As the problem above was due to the sign in the Leibnitz rule, let us consider the corresponding rule for $\mathcal{G}$. Due to the nature of $v$ and ${\widetilde{v}}$ [@dual], this is $ d(\mathcal{G}\wedge)=d\mathcal{G} \wedge ~ -\mathcal{G} \wedge d. $ Then using this Leibnitz rule to expand the expression $ (d\pm \mathcal{G})^2=d^2 \pm d (\mathcal{G}\wedge) \pm \mathcal{G}\wedge d + \mathcal{G}\wedge \mathcal{G} $ gives $ (d\pm \mathcal{G})^2=\pm d\mathcal{G} + \mathcal{G}\wedge \mathcal{G}. $ Now which sign to pick is determined simply by the vanishing of the right hand side. This happens for the minus sign [^13] because then the right hand side would be $ d\mathcal{G} - \mathcal{G} \wedge \mathcal{G}, $ which is zero as it is just the negative of the unified equation giving the EOM and the Bianchi identity derived in [@dual]. Then, $d-\mathcal{G}$ is indeed a differential, which we will denote by $d_{\mathcal{G}}$. At this point we can try to look for slight variations of this differential. - [*Scaling*]{}: From the expression $ (d+n\mathcal{G})^2=-nd\mathcal{G} + n^2 \mathcal{G}\wedge{G} $ we see that the constant $n$ can only be equal to one in order for the unified equation of motion to be satisfied. - [*Duality*]{}: We can derive the Leibnitz rule for the dual field $*\mathcal{G}$, $ d(*\mathcal{G}\wedge)=d(*\mathcal{G})\wedge + *\mathcal{G}\wedge d, $ which we use to show that $ (d\pm n* \mathcal{G})^2=\pm n d*\mathcal{G}\wedge~ \pm n *\mathcal{G}\wedge d ~\pm n *\mathcal{G}\wedge d~ + n^2 *\mathcal{G} \wedge *\mathcal{G}. \label{star} $ It is obvious then that $(d \pm n* \mathcal{G})$ is not a differetial since the terms $\pm n *\mathcal{G}\wedge d$ in (\[star\]) add, giving a result that cannot be zero without acting in a particular way on other forms. So does this mean we have twisted cohomology? This suggests that one gets such a structure if one uses the rank seven field $*_{11}G_4$ as the dual field of the M-theory rank four field $G_4$. At the level of differential forms, the differential $d_{{G}}$ is then interpreted as a map $ d_{{G}}: \Omega^m \oplus \Omega^{m-3} \longrightarrow \Omega^{m+1} \oplus \Omega^{m-2}, $ our case being $m=7$ of course. Such differentials (with one twist) were encountered in [@BHM]. One can also form a differential of uniform degree by introducing a formal parameter $t$ of degree $-3$ and write [^14] $d_{G_4}=d+ t G_4 + t^2 *G_4$. The interpretation of $t$ as a periodicity generator is desirable but is not very transparent again because it is of odd degree. This shift from even to odd degrees can be obtained by suspension or by looping (see below for relevance). Furthermore, we would like to interpret the above result at the level of twisted cohomology as the target via a generalized Chern character of some [*twisted generalized cohomology theory*]{} ${\mathcal M}(\bullet,G)$ or ${}^{\tau}{\mathcal M}$, where $\tau$ is $[G]$, the class of $G$, i.e. $ ch_{G}: {\mathcal M}(\bullet,G) \longrightarrow H^{4k}(\bullet,G). $ Note that elliptic cohomology theory can be thought of, at least heuristically, as the K-theory on the loop space, i.e. the elliptic cohomology of a space $X$ is the K-theory of $LX$. The twists of K-theory are given by its automorphism. This includes $H^3(X;{{\mathbb Z}})$. Applying this to the loop space gives the automorphism of elliptic cohomology, by which one can twist. [^15] By transgression, $H^3(LX;{{\mathbb Z}})$ gives $H^4(X;{{\mathbb Z}})$. For the $d*G_4$ part, we expect the arguement to be analogous. The $H^8$-twist in M-theory would descend to $H^7$-twist in string theory. [^16] Including the one-loop term {#one} --------------------------- The EOM after including the one-loop term (first introduced in [@DLM]) is modified to $ d*G_4=-\frac{1}{2}G_4 \wedge G_4 + I_8, $ where $I_8=-\frac{p_2 -(p_1/2)^2}{48}$ is the purely gravitational term, a polynomial in the Pontrjagin classes of the tangent bundle of the eleven dimensional spacetime $Y^{11}$. We can still group together $G_4$ and its dual in the presence of $I_8$. For the degree four/eight combination we simply add $I_8$ to $d*G_4$ and we are dealing with precisely the $\Theta$-class studied in [@DFM] and [@S2]. For the case of the degree four/seven combination, we can use the fact that $I_8=dX_7$ where $X_7$ is the transgression polynomial for $I_8$ in degree seven, and write the expressions using $*G_4 + X_7$. For example, $ \left(d + \frac{1}{2}G_4 \right)\left[G_4 +(*G_4+ X_7) \right], $ with the degree five and degree eight pieces giving respectively the Bianchi identity and the EOM upon using $dX_7=I_8$. Other formulae follow as well. We can see that when we add $I_8$ to the picture, it serves as an obstruction to having a twisted theory. However, if we absorb it in the defintion of the dual field as above, then we would still get a twist. [**Acknowledgements**]{} The author thanks Igor Kriz for helpful explanations on generalized cohomology theories, Edward Witten for very interesting discussions, Varghese Mathai for very useful comments, and Arthur Greenspoon for suggestions in improving the presentation. He also thanks the organizers of the Oberwolfach mini-workshop on Gerbes, Twisted K-theory and Conformal Field Theory, the Simons Workshop in Mathematics and Physics at Stony Brook, and the Theory Division at CERN for their hospitality during the final stages of this project. 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Freed, [*K-theory in quantum field theory*]{}, in Current developments in mathematics, 41, Int. Press, Somerville, MA, 2002, \[[arXiv:math-ph/0206031]{}\]. P. Bouwknegt, K. Hannabuss, and V. Mathai, [*T-duality for principal torus bundles and dimensionally reduced Gysin sequences*]{}, \[[arXiv:hep-th/0412268]{}\]. M. J. Duff, J. T. Liu, and R. Minasian, [*Eleven dimensional origin of string/string duality: a one loop test*]{}, Nucl. Phys. [**B452**]{} (1995) 261, \[[arXiv:hep-th/9506126]{}\]. [^1]: E-mail: [hisham.sati@adelaide.edu.au]{}\ Research supported by the Australian Research Council. [^2]: This was discussed briefly in [@ES]. [^3]: For ten-dimensional supergravity theories, this was dicussed in [@C1], [@C2] and [@faces]. [^4]: Throughout the paper, if the Hodge star operator has no explicit dimension label then it refers to the eleven-dimensional one. [^5]: In writing this expression and all the analogous ones, we are implicitly tensoring with ${{\mathbb R}}$ (or ${{\mathbb Q}}$). [^6]: This is meant to be analogous to the uniform degree zero expressions of the RR field strengths in [@F]. [^7]: unless the differential does not act on the exponential, which is not what is meant to happen. [^8]: It is interesting that if we interpret $v$ and ${\widetilde v}$ as the generators introduced in [@dual] and used in the next section, then the corresponding statement would be $\{v,v\}=-{\widetilde v}$, i.e. one of the relations of the gauge algebra for $G_4$ and $*G_4$. The minus sign would then make (\[vv\]) equal to $(d+ \mathcal{G})^2$, the obstruction to nilpotency. [^9]: In any case, even without requiring the $G_4$ term to have degree zero, one sees upon inspecting (\[sq\]), at least for relatively low $n$, $m$ and $p$ (which are the only relevant), that the result of the discussion does not change. [^10]: This is meant to be in the sense that the expression contains both $G_4$ and its dual $*G_4$, and that it is invariant under the exchange $G_4 \leftrightarrow *G_4$. It is not meant to be in the sense of exchanging $G$ and $*G$ as we will see explicitly later when the generators of the gauge algebra are included. [^11]: since $\alpha_k \wedge \beta_l=(-)^{kl}\beta_l \wedge \alpha_k$. [^12]: In order to make the equations and the statement symmetric, one might try to rescale and use both the total field strength and the twist as $\frac{1}{\sqrt{2}}G$. However, the equation of motion would then have an anomalous relative factor of $\sqrt{2}$. [^13]: One way this minus sign can be motivated is by saying it gives the differential $d- H$ in type IIA upon reduction (at least of the $G_4$-part of $\mathcal{G}$. [^14]: We are oversimplifying as we also have to include $v$ and $\widetilde{v}$. We hope to discuss this elsewhere. [^15]: We thank Constantin Teleman for explanations concerning this point. [^16]: We hope to discuss this in detail elsewhere.
--- abstract: 'The Willmore conjecture states that any immersion $F:\torus\to \mR^n$ of a 2-torus into euclidean space satisfies $\int_{\torus} H^2\geq 2\pi^2$. We prove it under the condition that the $L^p$-norm of the Gaussian curvature is sufficiently small.' author: - Bernd Ammann date: June 1999 title: The Willmore Conjecture for immersed tori with small curvature integral --- [**Keywords:**]{} Willmore integral, conformal metrics, two-dimensional torus [**Mathematics Classification:**]{} 53A05 (Primary), 53A30, 58G30 (Secondary) Introduction ============ Let $F:N\to \mR^n$ be an immersion of a closed surface $N$ into Euclidean space. The Willmore integral $\cW(F)$ is defined using the mean curvature $H$ of the immersion $F$ and the area form $\darea$ associated to the induced metric: $$\cW(F):=\int_N H^2\,\darea.$$ In the case $n= 3$ we easily get a lower estimate. If $\ka_1$ and $\ka_2$ are the principal curvatures we have $H^2=(\ka_1+\ka_2)^2/4\geq \max\{0,\ka_1\ka_2\}$. But $\ka_1\ka_2$ is just the Gaussian curvature $K$ of $(N, F^*\geukl)$ which is equal to the determinant of the Gauss map $N\to S^2$. Thus we obtain $$\cW(F)\geq \int_{N^+}K \, \darea \geq \int_{S^2} 1\,\darea=4\pi,$$ with $N^+=\{x\in N\,|\,K(x)\geq 0\}$ which is mapped onto $S^2$ via the Gauss map. The value $4\pi$ is attained if $F:S^2\to \mR^3$ is the standard embedding. And vice versa if $\cW(F)=4\pi$ we know that $N$ is $S^2$ whose image $F(S^2)$ is a round sphere. This well-known result has been improved and generalized by Li and Yau [@li.yau:82 Fact 3]. For arbitrary dimension $n\geq 3$ we assume that $F^{-1}(p)$ contains $k$ points for some $p\in \mR^n$. Then Li and Yau showed the estimate $$\cW(F)\geq 4\pi k.$$ If $N$ has positive genus, then the value $4\pi$ will never be attained. In this paper we will study the following conjecture attributed to Willmore [@willmore:65]. For any immersion $F:\torus \to \mR^n$ of the 2-dimensional torus into $\mR^n$, $n\geq 3$, the inequality $$\cW(F)\geq 2\pi^2$$ holds. Leon Simon proved in [@simon:93] that for any fixed dimension $n\geq 3$ the infimum $$\inf\{ \cW(F)\,|\, F:\torus\to \mR^n\}$$ is actually attained and he concludes that there is an estimate $W(F)\geq 4\pi+\ep_n$ with $\ep_n>0$ for any $n\geq 3$ without giving an explicit value for $\ep_n$. But the Willmore conjecture remains open until today. Nevertheless in many special cases the Willmore conjecture has been confirmed. If $n=3$ and if the image $F(\torus)$ has a rotational symmetry, the Willmore conjecture has been proven by Langer and Singer in [@langer.singer:84]. Shiohama and Takagi [@shiohama.takagi:70] and independently Willmore [@willmore:71] showed that the Willmore conjecture is true if $F(\torus)$ is the boundary of an $\ep$-neighborhood of a closed curve in $\mR^3$ with $\ep$ sufficiently small. It should be mentioned here that there is a natural generalization of the Willmore functional to immersions $F$ of a closed surface $N$ into a Riemannian manifold $(M,h)$ by defining $$\cW(F):=\int_N (H^2 + K_M)\,\darea,$$ where $K_M(p)$ is the sectional curvature of $(M,h)$ evaluated at the plane $dF(T_pN)$. This functional is invariant under conformal changes of $h$ [@thomsen:23; @weiner:78] . Hence, the Willmore conjecture for immersions $\torus\to\mR^n$ is equivalent to the Willmore conjecture for immersions $F:\torus\to S^n\subset\mR^{n+1}$. The Willmore conjecture for immersions of the latter kind has been proven by Ros [@ros:p97] under the additional condition that $n=3$ and that $F(\torus)$ is invariant under the antipodal map of $S^3$. Other partial solutions to the Willmore conjecture use spectral geometry. If $F:(N,g)\to S^n$ or equivalently $F:(N,g)\to \mR^n$ is a conformal immersion of a closed surface $N$ and if $\la_1$ is the first positive eigenvalue of the Laplace operator on $(N,g)$, then Li and Yau [@li.yau:82 Theorem 1] proved $$W(F)\geq {1\over 2}\la_1\area(N,g).$$ Every Riemannian 2-torus is conformally equivalent to a flat one, say $(\mR^2/\Ga_{xy},\geukl)$ where the lattice $\Ga_{xy}$ is generated by $(1,0)$ and $(x,y)$, $0\leq x \leq 1/2$, $x^2+y^2\geq 1$, $y>0$ and where $\geukl$ is the Euclidean standard metric on $\mR^2$. The first positve eigenvalue of the Laplace operator of $(\mR^2/\Ga_{xy},\geukl)$ is $4\pi^2/y^2$ and the area is $y$. Thus Li and Yau get the corollary $$\label{liyauhst} W(F)\geq {2\pi^2\over y}$$ which proves the Willmore conjecture for $y\leq 1$, for a subset of the moduli space that has positive measure (see figure \[modulpic\]). The set of conformal equivalence classes for which we know the Willmore conjecture has been enlarged by Montiel and Ros [@montiel.ros:85]. They proved that $y\leq 1$ could be replaced by the weaker condition $$\left(x-{1\over 2}\right)^2+ (y-1)^2\leq {1\over 4}.$$ =7.2cm (-.2,-.2)(2.2,2.2) (0,0)(-.2,-.2)(1.3,2.0) (1.42,0)[$x$]{} (0,2.16)[$y$]{} (.8,.4)[(0,-.15)(.3,.15)]{} (1.2,.4)[ ]{} (.8,1.1)[(0,-.15)(.3,.15)]{} (1.2,1.1)[ ]{} (.8,1.8)[(0,-.15)(.3,.15)]{} (1.2,1.8)[ ]{} Replacing the Laplace operator by the Dirac operator there is a similar result. If $F:(N,g)\to S^n$ or $F:(N,g)\to \mR^n$ is an isometric immersion, then Bär [@baer:98] established the estimate $$W(F)\geq \mu_1\area(\torus,g)$$ where $\mu_1$ is the first eigenvalue of a twisted Dirac operator on $\torus$. Unfortunately this statement is no longer true if we replace “isometric immersion” by “conformal immersion”, so in order to apply this estimate we have to get lower bounds for eigenvalues of Dirac operators on non-flat 2-tori. Such bounds have been found by the author in [@ammanndiss] or alternatively [@ammann:p99b]. With these estimates we proved the Willmore conjecture in an open subset of the spin-conformal moduli space provided that the $L^p$-norm of the Gaussian curvature of $N$ is sufficiently small. This open subset is disjoint from the subset in which Montiel and Ros proved the Willmore conjecture. A special situation is given if the immersion $F:\torus\to \mR^n$ is flat, the induced metric on $F(\torus)$ has vanishing Gaussian curvature. Examples of such tori are the Clifford torus $${1\over\sqrt{2}}S^1\times {1\over\sqrt{2}}\subset S^3$$ or the Hopf tori [@pinkall:85b]. For flat immersions the Willmore conjecture is true [@chen_bangyen:81]. Topping [@topping:p98] gave a proof of the Willmore conjecture for non-flat immersions $F:\torus\to S^3\subset\mR^4$ if the following condition is satisfied $$\int_{\torus}|K_{\gbel}|\,\darea_\gbel \leq {2\over \pi} \left(W(F)-\area(\torus,\gbel)\right),$$ where $\gbel$ is the induced metric. Note that the left hand side is the $L^1$-norm of the Gaussian curvature and the right hand side can be rewritten as $${2\over \pi}\,\int_{\torus} \left(H_{\torus\to S^3}\right)^2\darea_\gbel,$$ where $H_{\torus\to S^3}$ is the relative mean curvature of $\torus$ lying in $S^3$. The main theorems of this article now prove the Willmore conjecture under a similar condition on the Gaussian curvature, although our methods are completely different. Here we assume that the $L^p$-norm, $p>1$ of the Gaussian curvature is bounded by functions that only depend on intrinsic invariants of $(\torus,\gbel)$. So our assumptions are — in contrast to Topping’s results — purely intrinsic in the following sense: we construct many non-flat Riemannian metrics $g$ on $\torus$ such that any isometric immersion $F:(\torus,g)\to(\mR^n,\geukl)$ satisfies $W(F)\geq 2\pi^2$. For any real number $p>1$ and any conformal equivalence class $c$ on $\torus$ there exists $\tau(c,p)>0$ such that the following holds: If $F:\torus \to \mR^n$ is an immersion with induced metric $\gbel:=F^*\geukl$ and conformal equivalence class $[\gbel]$ satisfying $$\left\|K_\gbel\right\|_{L^p(\torus,\gbel)}\cdot \left(\area(\torus,\gbel)\right)^{1-{1\over p}}<\tau([\gbel],p),$$ then the Willmore conjecture $$W(F)\geq 2\pi^2$$ holds. There exists a function $\rho:\mo]0,\infty\mc[\times \mo]0,\infty\mc[\times \mo]1,\infty\mc[\to \mo]0,\infty\mc[$ with the following property: If $F:\torus \to \mR^n$ is an immersion whose induced metric $\gbel:=F^*\geukl$ satisfies $$\left\|K_\gbel\right\|_{L^p(\torus,\gbel)} < \rho\Bigl(\area(\torus,\gbel),\sys(\torus,\gbel),p\Bigr)$$ for some $p>1$, then the Willmore conjecture $$W(F)\geq 2\pi^2$$ holds. These results generalize most of the statements about the Willmore conjecture in [@ammanndiss]. The methods used in the proof are strongly related. The estimate of section \[streckabschsection\] can also be used to get spectral estimates on $2$-dimensional tori. This application will be presented in another article [@ammann:p99b] which is in preparation. I want to thank Christian Bär, Ernst Kuwert and Reiner Schätzle for many interesting discussions about the subject. Lower bounds for the Willmore functional ======================================== In this section we will prove the main theorems of the article using Theorem \[streckabsch\] which will be shown in section \[streckabschsection\]. At first we will define some geometric quantities of Riemannian 2-tori $(\torus,g)$. Let $K_g$ be the Gaussian curvature, $K_g^+:=\max\{K_g,0\}$ and $K_g^-:=\min\{K_g,0\}$. For $p\in [1,\infty[$ we set $$\begin{aligned} \cK_p^{\phantom{\pm}}(g)&:=& \left\|K_g^{\phantom{\pm}}\right\|_{L^p(\torus,g)}\cdot \left(\area(\torus,g)\right)^{1-{1\over p}}\\ \cK_p^\pm(g)&:=&\left\|K_g^\pm\right\|_{L^p(\torus,g)}\cdot \left(\area(\torus,g)\right)^{1-{1\over p}}.\end{aligned}$$ The Hölder inequality yields $\cK_p(g)\leq \cK_{p'}(g)$ and $\cK_p^\pm(g)\leq \cK_{p'}^\pm(g)$ for $p\leq p'$. Note that these quantities are invariant under rescaling of the metric. The *1-systole* or simply the *systole* $\sys(\torus,g)$ is the length of the shortest non-contracible loop in $\torus$. We define the geometric quantity $$\cV(\torus,\gbel):={\area(\torus,\gbel)\over \sys(\torus,\gbel)^2}$$ which is also invariant under rescaling of the metric. Furthermore we set $$\cM:=\left\{(x,y)\in\mR^2\,|\,0\leq x\leq 1/2,\;x^2+y^2\geq 1,\;y\geq 0\right\}.$$ As in the introduction we define for $(x,y)\in\cM$ the lattice $\Ga_{xy}$ in $\mR^2$ to be generated by $(1,0)$ and $(x,y)$. Unless otherwise stated $\mR^2/\Ga_{xy}$ always carries the Riemannian metric induced by the Euclidean metric of $\mR^2$. Every Riemannian metric on $\torus$ is conformally equivalent to exactly one torus of the form $\mR^2/\Ga_{xy}$ with $(x,y)\in\cM$. Hence $\cM$ can be identified with the moduli space of conformal structures on $\torus$. With these notations we have $$y=\cV(\mR^2/\Ga_{xy}).$$ The *oscillation* of a continuous function $u:\torus\to\mR$ is defined to be $\osc u:=\max u -\min u$ where the maximum and minimum is to be taken over $\torus$. Now, we will cite a lemma. \[konformkruem0lemma\] Let $g_1$ and $g_2=e^{2u}g_1$ be two conformal Riemannian metrics on a surface . Then their Gaussian curvatures are related via $$K_{g_2} - e^{-2u} K_{g_1} = \De_{g_2} u = e^{-2u} \De_{g_1} u.$$ We always use the convention $\De=-*d*d$, $\De$ has nonnegative eigenvalues on compact sets with Dirichlet boundary conditions. The formula of the lemma also yields a simple proof of the uniformisation theorem for $\torus$ stating that any Riemannian metric on $\torus$ is conformally equivalent to a flat one – a fact that has already been used several times in this article. \[loewnerprep\] Suppose that $\torus$ carries a flat metric $\gflach$ and another metric $\gbel$ conformal to $\gflach$. Then $$\cV(\torus,\gbel)\geq\cV(\torus,\gflach).$$ The proof of this lemma follows a proof of Loewner’s theorem, see [@gromov:81  4.1]. We write $\gbel=e^{2u}\gflach$. Obviously, $\area(\torus,\gbel)=\int_\torus e^{2u}\,\darea_\gflach$ and $\area(\torus,\gflach)=\int_\torus \darea_\gflach$. Let $c:S^1\to \torus$ be a non contractible loop of minimal length $l_0:=\sys(\torus,\gflach)$ with respect to $\gflach$. Then for $a\in\torus$ the translated loop $c_a(\,\cdot\,):=c(\,\cdot\,)+a$ has the same length with respect to $\gflach$. Let $l(a)$ be the length of $c_a$ with respect to $\gbel$. Then $$\begin{aligned} \int_\torus l(a)\,\darea_\gflach & = & \int_\torus \darea_\gflach \int _{S^1}dt\,\left|\dot{c}_a(t)\right|_\gbel\\ & = & \int_\torus \darea_\gflach \int _{S^1}dt\,e^{u\circ c_a(t)}\,\left|\dot{c}_a(t)\right|_\gflach\\ & = & l_0 \,\int_\torus e^u\,\darea_\gflach\\ & \leq & \l_0\,\area(\torus,\gflach)^{1/2}\,\area(\torus,\gbel)^{1/2}\end{aligned}$$ So there is a point $a\in\torus$ with Using the fact that $\cV(\torus,\gflach)\geq \sqrt{3}/2$ for any flat metric $\gflach$ we get the For any Riemannian 2-torus $(\torus,\gbel)$ we have $$\cV(\torus,\gbel)\geq {\sqrt{3}\over 2}.$$ Equality is attained only for the equilateral flat torus. \[loewnercor2\] Any isometric immersion $F:(\torus,\gbel)\to(\mR^n,\geukl)$ satisfies $$W(F)\geq {2\pi^2\over \cV(\torus,\gbel)}= {2\pi^2\,\sys(\torus,\gbel)^2\over \area(\torus,\gbel)}.$$ In particular, the Willmore conjecture is satisfied for any isometrically immersed torus with $\sys(\torus,\gbel)^2\geq \area(\torus,\gbel)$. We write $\gbel=e^{2u}\gflach$ with $\gflach$ flat. Let $(\torus,\gflach)$ be isometric to $\mR^2/\Ga_{xy}$ with $x,y\in\cM$. Then $$y=\cV(\torus,\gflach)\leq \cV(\torus,\gbel).$$ Now the corollary follows from inequality  of the introduction. The following lemma is a converse to Lemma \[loewnerprep\]. \[loewnerumk\] Suppose that $\torus$ carries a flat metric $\gflach$ and another metric $\gbel=e^{2u}\gflach$. Then $$\cV(\torus,\gbel)\leq e^{2\osc u}\,\cV(\torus,\gflach).$$ The proof is straightforward. We will prove our main theorems in a slightly stronger version than stated in the introduction. There exists a function $\tau:\mo[\sqrt{3}/2,\infty\mc[\times \mo]1,\infty\mc[\to\mo]0,\infty\mc[$ with the following property:If $F:\torus \to \mR^n$ is an immersion such that the induced metric $\gbel:=F^*\geukl$ satisfies $$\cK_p(\gbel)<\tau(y,p)\mbox{\ \ \ and\ \ \ $(T^2,\gbel)$ is conformally equivalent to }\mR^2/\Ga_{xy}$$ for some $p>1$, then the Willmore conjecture $$W(F)\geq 2\pi^2$$ holds. Let $F:(\torus ,\gflach) \to (\mR^n,\geukl)$ be a conformal immersion. Suppose that ($\torus,\gflach)$ is isometric to $(\mR^2/\Ga_{x2},\geukl)$, $0\leq x\leq 1/2$. From the explicit construction of $\tau$ in the proof we see that the Willmore conjecture is satisfied if $$\left\| K\right\|_{L^2(\torus ,\gbel)} \,{\area\Big(F(\torus )\Big)}^{1/2} \leq 0.1987553.$$ The $\tau$ constructed in the proof is continuous on $[\sqrt{3}/2,1\mc[\times\mo]1,\infty\mc[$ and on $\mo]1,\infty\mc[\times\mo]1,\infty\mc[$, but $$\lim_{y\searrow 1} \tau(y,p)=0\neq \tau(1,p)$$ for any $p\in \mo]1,\infty\mc[$. Hence $\tau$ is not continuous at $(y,p)$ with $y=1$. Nevertheless, if we view $\tau$ as a function on $\cM\times \mo]1,\infty\mc[$ we can combine Main Theorem I with the result of Montiel and Ros mentioned in the introduction [@montiel.ros:85] to get a similar function $\witi\tau:\cM\times \mo]1,\infty\mc[\to \mR^+$ that is continuous on $\left(\cM\ohne\{[\mR^2/\Ga_{01}]\}\right)\times \mo]1,\infty\mc[$ and such that Main Theorem I holds with $\tau$ replaced by $\witi\tau$. There exists a function $\si:\mo[\sqrt{3}/2,\infty\mc[\times \mo]1,\infty\mc[\to \mo]0,\infty\mc[$ with the following property:If $F:\torus \to \mR^n$ is an immersion such that the induced metric $\gbel:=F^*\geukl$ satisfies $$\cK_p(\gbel)<\si\left(\cV(\torus,\gbel),p\right)$$ for some $p>1$, then the Willmore conjecture $$W(F)\geq 2\pi^2$$ holds. In analogy to the previous remark, we cannot chose $\si$ to be continuous at $\cV=1$, $p$ arbitrary. But $\si$ can be chosen to be continuous on $\{(\cV,p)\,|\,\cV\neq 1\}$. A central role in the proof is played by \[beinahflachabsch\] Let $F:\torus\to \mR^n$ be an immersion. Let $\gflach$ denote the standard metric on $\mR^2$ and suppose that $\torus$ with the induced metric $F^*\geukl$ is isometric to $(\mR^2/\Ga_{xy},e^{2u}\gflach)$, $(x,y)\in \cM$, where $u$ is a smooth function. Then $$W(F)\geq e^{-2 \osc u}\,\pi^2 \Bigl(y+{1\over y}\Bigr)$$ This lemma will be shown at the end of this section. We now define $$\begin{aligned} \cQ(\cK,p,\cV)& := \exp& \Biggl[\;\Big|\log \left(1-{\cK\over 4\pi}\right)\Big|+ {\cK\over 4\pi -\cK}\,q\log (2q)\\ && \phantom{\Biggl[\;}+ {q\cK\over 2\pi} + {\cK\cV\over 4}\;\Biggr]\end{aligned}$$ with $q:=p/(p-1)$. From the previous lemma and from Theorem \[streckabsch\] we get a corollary. Under the conditions of the previous lemma we have $$\begin{aligned} W(F)& \geq & \cQ\left(\cK_p(g),p,\cV(\torus,g)\right)^{-1}\,\pi^2 \Bigl(y+{1\over y}\Bigr)\\ W(F)& \geq & \cQ\left(\cK_p(g),p,y\right)^{-1}\,\pi^2 \Bigl(y+{1\over y}\Bigr).\end{aligned}$$ if $\cK_p(g)<4\pi$. Let $(\torus,\gbel)$ be conformally equivalent to $\mR^2/\Ga_{xy}$, $(x,y)\in\cM$. We distinguish between two cases. In the case $y\leq 1$ we can use the result of Li-Yau [@li.yau:82] or Montiel-Ros [@montiel.ros:85] to see that $\tau(y,p)$ can be chosen as any arbitrary positive real number. On the other hand, for $y>1$ we get $$y+{1\over y}>2.$$ The function $\cQ(\cK,p,\cV)$ is continuous and for $\cK\to 0$ with $p$ and $\cV$ fixed $\cQ$ converges to $1$. Therefore there is a real number $\tau(y,p)>0$ such that $$\cQ\left(\cK,p,y\right)^{-1} \left(y+{1\over y}\right)>2$$ whenever $\cK\leq \tau(y,p)$. Obviously $\tau$ can be chosen as a continuous function on $\mo]1,\infty\mc[\times \mo]1,\infty\mc[$. As in the proof of Main Theorem I we will distinguish between two cases for the construction of $\si(\cV,p)$. In the first case, $\cV\leq 1$, according to Corollary \[loewnercor2\] the Willmore conjecture is satisfied for any immersion $F:\torus\to\mR^n$ with $\cV(\torus,F^*\geukl)=\cV$, hence Main Theorem II is true if $\si(\cV,p)$ is any positive number. In the remaining case, $\cV>1,$ we can choose $\si_1(\cV,p)>0$ such that $$\cQ(\cK,p,\cV)\leq \sqrt{\cV}$$ whenever $\cK\in [0,\si_1(\cV,p)]$. So let $\gbel=e^{2u}\gflach$ be an arbitrary metric on $\torus$ with $\gflach$ flat and suppose $\cK_p(\gbel)\leq \si_1(\cV(\torus,\gbel),p)$, then according to Theorem \[streckabsch\] $$e^{2\osc u}\leq \cQ\Bigl(\cK_p(\gbel),p,\cV(\torus,\gbel)\Bigr)\leq \sqrt{\cV(\torus,\gbel)}.$$ Using Lemma \[loewnerumk\] we get $$\cV(\torus,g_0)\geq \sqrt{\cV(\torus,\gbel)}>1.$$ Let $(\torus,\gbel)$ be conformally equivalent to $\mR^2/\Ga_{xy}$ with $(x,y)\in\cM$. Then the above inequality and Lemma \[loewnerprep\] yield $\sqrt{\cV(\torus,\gbel)}\leq y\leq \cV(\torus,\gbel)$. Now set $$\si(\cV,p):=\min\biggl\{\si_1(\cV,p),\min\Bigl\{\tau(v,p)\,\Bigm|\,\sqrt{\cV}\leq v\leq \cV\Bigr\}\biggr\}>0$$ with the function $\tau$ constructed in the proof of Main Theorem I. Because of the construction of $\si$ we get $\si\left(\cV(\torus,\gbel),p\right)\leq \tau(y,p)$. Therefore Main Theorem II follows from Main Theorem I. The continuity property is clear from the construction of $\si$. The lemma is a generalization of [@li.yau:82 Prop. 2, page 287]. We will adapt the proof of this proposition to our situation by introducing conformal factors into the formulas. Let $\gbel:=F^* \geukl$ be the metric on $\torus $ induced by the Euclidean metric on $\mR^n$. We decompose $F$ into its coordinate functions $F=(F_1,\dots,F_n)$, $F_i:\torus \to \mR$. We get $$\begin{aligned} 4 W(F) & = & 4 \int_{\torus } \left|H\right|^2 \,\darea_{\gbel}\\ & = & \sum_{i=1}^n \int_{\torus }\left(\De_{\gbel}F_i\right)^2\,\darea_{\gbel}\\ & = & \sum_{i=1}^n \int_{\torus } e^{-2u}\left(\De_{\gflach}F_i\right)^2\,\darea_{\gflach}\\ & \geq & \sum_{i=1}^n \int_{\torus } e^{-2\umax}\left(\De_{\gflach}F_i\right)^2\,\darea_{\gflach}.\end{aligned}$$ After a translation we can assume $$\int_{\torus } F_i\,\darea_{\gflach}= 0 \qquad \mbox{for any } i=1,\dots,n.$$ Now we make a Fourier decomposition for the functions $F_i$: $$\begin{aligned} F_i (w)& = & \sum_{p,q\in\mZ\atop (p,q)\neq (0,0)} A_{ipq}\, \sqrt{2\over y} \,\cos \left( 2\pi \left\lan \left(q, {p-qx\over y}\right),w\right\ran\right)\\ & & + \sum_{p,q\atop (p,q)\neq (0,0)} B_{ipq} \,\sqrt{2\over y}\, \sin \left( 2\pi \left\lan \left(q, {p-qx\over y}\right),w\right\ran\right).\end{aligned}$$ This means for our estimate of $W(F)$ $$\begin{aligned} 4 \,W(F) & \geq & e^{-2\umax}\,16 \pi^4 \left[\sum_{i,p,q\atop (p,q)\neq (0,0)} A_{ipq}^2\left( q^2+ \left({p-qx\over y}\right)^2\right)^2\right.\nonumber\\ && + \left. \sum_{i,p,q\atop (p,q)\neq (0,0)} B_{ipq}^2\left( q^2+ \left({p-qx\over y}\right)^2\right)^2\right]\nonumber\\ & \geq & e^{-2\umax} 16 \pi^4 \left[\sum_{i,p,q\atop (p,q)\neq (0,0)} \left(A_{ipq}^2+ B_{ipq}^2\right) \left(q^2 + {1\over y^2}\left({p-qx\over y}\right)^2\right)\right].\label{zwifor}\end{aligned}$$ For the last estimate we used that the inequality $$\left[q^2+ \left({p-qx\over y}\right)^2\right]^2 \geq q^2 + \left[2q^2 + \left({p-qx\over y}\right)^2\right] \left({p-qx\over y}\right)^2 \geq q^2 + {1\over y^2} \left({p-qx\over y}\right)^2$$ holds for any $(p,q)\in\mZ\times\mZ$ with $(p,q)\neq (0,0)$ and $x,y\in\mR$ with $2y^2\geq 1$. Now we transform the right hand side of inequality . The projections of the vectors $(1,0)$, $(0,1)$ of $\mR^2$ to the quotient $\torus =\mR^2/\lan (1,0),(x,y)\ran$ define two vector fields denoted $e_1$ and $e_2$. These vector fields form an orthonormal frame for the metric $\gflach$. The conformal factor $e^{2u}$ can be calculated as $$e^{2u} = \left|\pa_{e_1} F\right|_{\geukl}^2 = \sum_i \left( \pa_{e_1} F_i\right)^2.$$ Therefore we get $$\area(\torus ,\gbel)=\int e^{2u}\, \darea_{\gflach} = \sum_i \int_{\torus } \left(\pa_{e_1}F_i\right)^2\,\darea_{\gflach} = 4 \pi^2 \sum_{i,p,q\atop (p,q)\neq (0,0)} \left(A_{ipq}^2+ B_{ipq}^2\right) \,q^2.$$ If we replace $e_1$ by $e_2$ we get completely analogously $$\area(\torus ,\gbel)= 4 \pi^2 \sum_{i,p,q\atop (p,q)\neq (0,0)} \left(A_{ipq}^2+ B_{ipq}^2\right) \,\left({p-qx\over y}\right)^2.$$ On the other hand $$\area(\torus ,\gbel)=\int e^{2u}\,\darea_{\gflach} \geq e^{2 \umin}\area(\torus,\gflach)= e^{2 \umin}\cdot y.$$ All these inequalities together imply $$W(F) \geq e^{-2\umax}\, \pi^2\,\left(1+{1\over y^2}\right) \area(\torus ,\gbel)\geq e^{-2 \uosc}\, \pi^2 \,\left(y+{1\over y}\right).$$ Controling the conformal scaling function {#streckabschsection} ========================================= Consider the 2-dimensional torus with a Riemannian metric $g$. As in the previous section we write this metric in the form $$\gbel= e^{2u} \gflach,$$ where $\gflach$ is a flat metric. The function $u$ is uniquely determined by $g$ up to an additive constant. The aim of this section is to prove estimates for the oscillation $\osc u = \max u - \min u$. We will give such an estimate for the case $\cK_1(\gbel)=\int_\torus |K_\gbel|\,\darea_\gbel<4\pi$. In this situation we can choose $p>1$ such that $\cK_p(\gbel)<4\pi$. In order to motivate the condition $\cK_p(\gbel)<4\pi$ we construct some examples in section \[exsec\]. If $\cK_p(\gbel)$ is sufficiently big, then there is a bubbling phenomenon as described for example in [@chen_xiuxiong:98] for the special case $p=2$. \[streckabsch\] We assume for some $p\in \;\mo]1,\infty\mc[$ $$\cK_p:=\cK_p(\torus ,\gbel)<4\pi.$$ Then the oscillation of $u$ is bounded as follows where we use the definition $$\cS(\cK,p,\cV):={1\over 2}\bigg|\log \left(1-{\cK\over 4\pi}\right)\bigg|+ {\cK\over 8\pi -2\cK}\,q\log(2q) + {q\cK\over 4\pi} + {\cK\cV\over 8}$$ with $q:=p/(p-1)$. \[schrankedef\] $$\cQ(\cK,p,\cV) = \exp \Bigl(2\cS(\cK,p,\cV)\Bigr).$$ The bounds we get are related to a result of Brezis and Merle [@brezis.merle:91] proving the existence of a bound for $\|u\|_{L^q(\mR^2,\gflach)}$ for functions $u:\Om\to\mR$ defined on a bounded domain $\Om\subset \mR^n$ satisfying the Kazdan-Warner equation $$\De_\gflach u = K(x)e^{2u}$$ with Dirichlet boundary conditions. Their bound depends on $\|K_\gbel\|_{L^p(\mR^2,\gflach)}$ and $\Om$. The main difference to our results is that Brezis and Merle do not treat functions $u:\torus\to\mR$ and that their norms are taken with respect to the flat metric $\gflach$ whereas we bound $\osc u$ in terms of the $L^p$-norm with respect to the metric $\gbel=e^{2u}\gflach$. Now we turn to the proof of Theorem \[streckabsch\]. The theorem will follow from some propositions and corollaries that will take the rest of this section. For the proof we split the torus $\torus$ into three subsets two of which have the property that their fundamental group is mapped trivially to $\pi_1(\torus)$. On these two subsets $\osc u$ can be estimated using Corollary \[uupperboundcor\] and Proposition \[ulowerbound\]. The third part contains generators of $\pi_1(\torus)$ and will be dealt in Proposition \[umidbound\]. \[uupperbound\] Let $G$ be a bounded open subset of $\mR^2$ with the standard metric $\gflach$. For a smooth function $u:\ol{G}\to \mR$ with $u\geq 0$, $\res{u}{\partial G}\equiv 0$ we set $\gbel=e^{2u}\gflach$. We define $\mu_0:=\area(G,\gbel)$ and for the Gaussian curvature $K_\gbel$ with respect to $\gbel$ let $$k(A):=\sup \left\{\int_{\widehat G, \gbel} K_\gbel\; \Biggm|\;\widehat G \mbox{\rm \ open subset of $G$,\ } \area(\widehat G,\gbel)=A\right\}.$$ If there are $\ka\in\;]0,2\pi[$, $C>0$, $r\in\;]0,1]$ and $\mu_1\in\;]0,\mu_0]$, such that $$k(A)\leq \left\{\matrix{C\cdot A^r \leq \ka & \mbox{for}& 0\leq A \leq \mu_1\hfill\cr \ka \hfill &\mbox{for} & \mu_1 \leq A \leq \mu_0, \hfill }\right.$$ then $$\max u \leq {1\over 2}\left|\log \left(1-{\ka\over 2\pi}\right)\right|+ {\ka\over 4\pi -2\ka}\log \left({\mu_0\over \sqrr\mu_1}\right).$$ Before proving this proposition, we will prove a corollary. \[uupperboundcor\] Suppose the open subset $G$ of $\torus$ has the property that any loop $\ga:S^1\to \overline{G}$ is contractible in $\torus$. Let $\gflach$ be a flat metric on $\torus$ and $\gbel=e^{2u}\gflach$ another metric on $\torus$ with a smooth function $u:\torus\to \mR$ satisfying $\res{u}{\partial G}\equiv 0$ and $\res{u}{G}\geq 0$. Suppose for some $p\in\;\mo]1,\infty\mc[$ we have $$\cK_p=\cK_p(\torus ,\gbel)<4\pi.$$ Then we get the estimate $$\max_{x\in\ol G} u(x) \leq {1\over 2}\,\biggl|\log \left(1-{\cK_p\over 4\pi}\right)\biggr|\,+\, {\cK_p\over 8\pi -2\cK_p}\,q\log (2q),$$ with $q:=p/(p-1)$. Because of the Gauss-Bonnet theorem we have $$\left|\int_{\widehat G,\gbel}K_\gbel \right|= \left|\int_{\torus \ohne\widehat G,\gbel}K_\gbel \right|$$ for any open subset $\widehat G\subset G$. Therefore $$\left|\int_{\widehat G,\gbel}K_\gbel \right|\leq {1\over 2} \int_{\torus ,\gbel}\left|K_\gbel \right| \leq {1\over 2}\,\cK_p.$$ On the other hand, if we have $\area(\widehat G,\gbel)<2^{-q}\area(\torus,\gbel)$, then the estimate $$\left|\int_{\widehat G,\gbel}K_\gbel \right|\leq \left\|K_g\right\|_{L^p(\widehat G,\gbel)} \left(\area(\widehat G,\gbel)\right)^{1/q}\leq \cK_p\left({\area(\widehat G,\gbel)\over \area(\torus ,\gbel)}\right)^{1/q}$$ is better. Since any loop $\ga:S^1\to \overline{G}$ is contractible in $\torus$, we can lift $G$ to $\mR^2$. We can apply Proposition \[uupperbound\] with $\ka:=(1/2)\,\cK_p$, $r:=1/q$, $C:=\cK_p\,\area(\torus ,\gbel)^{-1/q}$, $\mu_0:=\area(G,\gbel)\leq \area(\torus ,\gbel)$ and $\mu_1:=\min\{\mu_0,2^{-q}\area(\torus ,\gbel)\}$. For $v\in \mR$ we define $G(v):=\{x\in G\,|\,u(x)>v\}$. The area of $G(v)$ with respect to $\gbel$ (or $\gflach$ resp.) will be denoted $A(v)$ (or $\Aflach(v)$ resp.). For the length of the boundary $\pa G(v)$ we write $l(v)$ (or $\lflach(v)$ resp.). The functions $A(v)$, $\Aflach(v)$, $l(v)$ and $\lflach(v)$ are differentiable at every regular value $v$ of the function $u$. Therefore we get for regular values $v$: $$\begin{aligned} e^{2v}\Aflach(v) & \leq & A(v)\; \leq\; e^{2\umax} \Aflach(v) \label{avunglei}\\ e^{2v}{d\over dv} \Aflach(v) & = & {d\over dv}A(v)\label{davglei}\\ e^v\lflach(v) & = & l(v) \label{lvglei}\\ \int_{G(v),\gbel}K_\gbel & \leq & k(A(v)). \label{kint1}\end{aligned}$$ On the other hand, according to Lemma \[konformkruem0lemma\] we have $$\int_{G(v),\gbel}K_\gbel = \int_{G(v),\gbel}\De_{\gbel} u = - \int_{\pa G(v)} *\, du = \int_{\pa G(v), \gflach} |du|_\gflach.\label{kint2}$$ The last equation follows since $u$ is equal to $v$ on $\pa G(v)$ and greater than $v$ on $G(v)$ and therefore $*\,du$ is negatively oriented on $\pa G(v)$. We also see that $\int_{G(v),\gbel}K_\gbel$ is positive for every regular value $v\in u(G)$ of the function $u$. Using and we calculate $$\begin{aligned} -{d\over dv} \Aflach(v) & = & \int_{\pa G(v),\gflach} {1\over |du|_\gflach} \;\geq\; {{\lflach(v)}^2\over \int_{\pa G(v), \gflach} |du|_\gflach} \nonumber \\ & \geq & {{\lflach(v)}^2\over k(A(v))}\label{davabsch}\end{aligned}$$ for any regular value $v\in u(G)$ of $u$. Now we apply the isoperimetric inequality $$\label{isoper} {\lflach(v)}^2\geq 4 \pi \, \Aflach(v)$$ and we get $$-{d\over dv} \Aflach(v)\geq {4 \pi\over k(A(v))}\Aflach(v). \label{vorfu}$$ We set $\mu_2:=\sqrr\mu_1$ and $u_2:=\inf\Bigl\{\al\in[0,\max u\mc[\;\Big|\;A(\al)\leq \mu_2\Bigr\}$. Let us distinguish between the two cases $v\geq u_2$ and $v<u_2$. We start with $v\geq u_2$. In this case the inequalities , , and the bound of $k(A)$ provide the estimate $$-{d\over dv}{A(v)}\geq {4\pi\over C\cdot {A(v)}^r} \,e^{2(v-\umax)} A(v),$$ and therefore $$-{1\over r}{d\over dv}\left({A(v)}^r\right)\geq {4\pi \over C}\,e^{2(v-\umax)}.$$ We use the following lemma. Note that any monotonically increasing function is differentiable almost everywhere. \[strealemma\] Let $f,g:[a,b]\to \mR$ be functions with $f$ monotonically increasing, $g$ continuously differentiable and $f(a)=g(a)$. If $f'\geq g'$ almost everywhere, then $f(b)\geq g(b)$. This lemma ensures that we can integrate the previous inequality from $u_2$ to $\umax$. We use $A(\umax)=0$ and get $${1\over r}{A(u_2)}^r \geq {2\pi \over C}\, e^{-2\umax}\left(e^{2\umax} -e^{2u_2}\right) = {2\pi \over C}\left(1 -e^{2(u_2-\umax)}\right).$$ Furthermore $A(u_2)=\lim\limits_{\al\searrow u_2} A(\al)\leq \mu_2=\sqrr\mu_1$ and $C{\mu_1}^r\leq \ka$ yield $$1-e^{2(u_2-\umax)} \leq {C\over 2\pi r}{A(u_2)}^r \leq {\ka\over 2\pi}$$ and therefore $$(\umax) - u_2 \leq {1\over 2}\,\biggl|\log \left(1-{\ka\over 2\pi}\right)\biggr|.\label{umaxschr}$$ Now we treat the case $v<u_2$. From the estimate  we know for small $\ep>0$ $$\Aflach(u_2-\ep) \geq e^{-2\umax}A(u_2-\ep)\geq e^{-2\umax}\mu_2\geq e^{-2u_2}\mu_3,$$ with $$\mu_3:=\left(1-{\ka\over 2\pi}\right)\mu_2.$$ Using we obtain $$\Aflach(v)\geq \int_v^{u_2-\ep} e^{-2\bar v}\left(-{d\over d\bar v}{A(\bar v)}\right)d\bar v \, + \,e^{-2u_2} \mu_3.$$ Here we used the fact that any monotonically decreasing function $h:[a,b]\to \mR$ satisfies $h(a)-h(b)\geq \int_a^b -h'(t)\, dt$. In particular we can integrate over the singularities. Inequality  then provides $$\begin{aligned} -{d\over dv}{A(v)} & \geq & {4\pi\over \ka}\, e^{2v} \left(\int_v^{u_2-\ep} e^{-2\bar v}\left(-{d\over d\bar v}{A(\bar v)}\right)d\bar v \, + \,e^{-2u_2} \mu_3\right).\end{aligned}$$ Let $f$ be the solution of the integral equality corresponding to this integral inequality, $$\begin{aligned} f(v)& = & {4\pi \over \ka}\, e^{2v} \left(\int_v^{u_2-\ep} e^{-2\bar v}f(\bar v)\, d\bar v \, + \,e^{-2u_2} \mu_3\right).\end{aligned}$$ Via differentiation we get the differential equation $${d\over dv} f (v) = 2 f(v) - {4\pi \over \ka}f(v)$$ with initial value $f(u_2-\ep)=(4\pi/\ka)e^{-2\ep}\mu_3$. So the solution is $$f(v)= {4\pi\over \ka}\,e^{-2\ep}\,\mu_3 \,e^{\left({4\pi\over \ka}-2\right)\left(u_2-v-\ep\right)}.$$ Here we use another elementary lemma. \[strealemma2\] Let $f_1$ and $f_2$ be $L^1$-functions on $[a,b]$ and $g_1,g_2:[a,b]\to \Rplus$ continuous functions. Let $C\in \Rplus$. We assume that for any $t\in [a,b]$ we have $$\begin{aligned} f_1(t) & \geq & g_1(t)\left(C+\int_a^t g_2(s) f_1(s) \,ds\right)\\ f_2(t) & = & g_1(t)\left(C+\int_a^t g_2(s) f_2(s) \,ds\right).\\ \end{aligned}$$ Then we get $f_1(t)\geq f_2(t)$ for any $t\in[a,b]$. Thus we obtain $$\begin{aligned} -{d\over dv}{A(v)} & \geq & f(v) = {4\pi\over \ka}\,\mu_3\,e^{-2\ep}\,e^{\left({4\pi\over \ka}-2\right)\left(u_2-v-\ep\right)}\\\end{aligned}$$ and the limit $\ep\to 0$ yields $$\begin{aligned} -{d\over dv}{A(v)} & \geq & f(v) = {4\pi\over \ka}\,\mu_3\,e^{\left({4\pi\over \ka}-2\right)\left(u_2-v\right)}\\\end{aligned}$$ Now integration from $0$ to $u_2$ using Lemma \[strealemma\] yields $$\mu_0-\mu_2 \geq A(0)-A(u_2) \geq \mu_3{\left(1- {\ka \over 2\pi}\right)}^{-1}\left(e^{\left({4\pi\over \ka}-2\right)u_2}-1\right).$$ Hence $$\begin{aligned} u_2& \leq & {\ka\over 4\pi - 2\ka}\log \left(1+ \left(1- {\ka \over 2\pi}\right) {(\mu_0-\mu_2)\over\mu_3}\right)\\ & = & {\ka\over 4\pi - 2\ka}\log \left({\mu_0\over\sqrr\mu_1}\right).\end{aligned}$$ This together with yields the estimate of the proposition. \[ulowerbound\] Let $u:\ol{G}\to \mR$, $u\leq 0$ be a smooth function on the bounded open subset $G\subset \mR^2$ satisfying the boundary condition $\res{u}{\partial G}\equiv 0$. Let $\gflach$ be the restriction of the standard metric on $\mR^2$. We set $\gbel:=e^{2u}\gflach$. For $p\in\;\mo]1,\infty\mc[$ we define $q:=p/(p-1)$. Then $$\min u \geq - {q\,\cKm_p(G,\gbel)\over 4\pi}.$$ This time we define $\ka:=\cKm_p(G,\gbel)$ and $G(v):=\{x\in G\,|\,u(x)<v\}$. As in the proof of Proposition \[uupperbound\] let $A(v)$ and $l(v)$ or $\Aflach(v)$ and $\lflach(v)$ resp.  be the area of $G(v)$ and the length of $\pa G(v)$ with respect to $\gbel$ or $\gflach$ resp. Again we have , and whereas we have to modify and : $$\begin{aligned} e^{2v}\Aflach(v) & \geq & A(v)\label{avunglei2}\\ \int_{G(v),\gbel}K_\gbel & = & - \int_{\pa G(v), \gflach} |du|_\gflach.\nonumber \end{aligned}$$ Furthermore we get $$\label{knabsch} k(v):=-\int_{G(v),\gbel}K_\gbel \leq \left\|K^-_\gbel\right\|_{L^p(G(v)),\gbel}\area(G(v),\gbel)^{1/q} \leq \ka \left({A(v)\over A(0)}\right)^{1/q}.$$ Inequality  holds with a different sign: $${d\over dv}\Aflach(v)\geq {{\lflach(v)}^2 \over k(v) }.\label{vdavm}$$ Together with and this yields $${d\over dv} \Aflach(v) \geq {4\pi\over \ka}\,\Aflach(v)\left(A(0)\over A(v)\right)^{1/q}.$$ Finally with and we obtain $${d\over dv} A(v) \geq {4\pi\over \ka} A(v) \left(A(0)\over A(v)\right)^{1/q}.$$ We use again Lemma \[strealemma\] in order to integrate this inequality from $\umin$ to $0$, and we get $$q\left(A(0)^{1/q}-A(\umin)^{1/q}\right)\geq {4\pi\over \ka} A(0)^{1/q}\,|\umin|.$$ Since $A(\umin)=0$ this implies \[boundest\] Let $G$ be an open set in $(\torus ,\gbel)$ with smooth boundary. We suppose that there are closed curves $c_1:[0,1]\to G$ and $c_2:[0,1]\to \torus \ohne \ol{G}$, whose corresponding homology classes in $H_1(\torus ,\mZ)$ are not zero. Then $$\length{\pa G}{\gbel}\geq 2 \,\sys(\torus ,\gbel).$$ If we regard $\overline{G}$ as a $2$-cycle, then clearly $\pa G$ is homologous to zero. Now decompose $\pa G$ into its components. Each component is diffeomorphic to $S^1$. We will show that there is at least one component non-homologous to zero. Together with $[\pa G]=0$ this implies that are at least two such components and therefore we get the statement of the lemma. So let us suppose that all components of $\pa G$ are homologous to zero. Let $\pi:\mR^2\to \torus $ be the universal covering. Then $\pi^{-1}(\pa G)$ is diffeomorphic to a disjoint union of countably many $S^1$. We write $$\pi^{-1}(\pa G)=\dot{\bigcup_{i\in \mN}} Y_i$$ with $Y_i\cong S^1$. We choose lifts $\ti c_i:\mR\to \mR^2$ of $c_i$, $\pi\left(\ti c_i(t+z)\right)=c_i(t)$ for all $t\in [0,1]$, $z\in \mZ$ and $i=1,2$. Then we take a path $\ti\ga:[0,1]\to \mR^2$ joining $\ti c_1(0)$ to $\ti c_2(0)$. We define $I$ to be the set of all $i\in \mN$ such that $Y_i$ meets the trace of $\ti\ga$. The set $I$ is finite. Using the Theorem of Jordan and Schoenfliess about simply closed curves in $\mR^2$ we can inductively construct a compact set $K\subset \mR^2$ with boundary $\bigcup_{i\in I}Y_i$. Either $\ti c_1(0)$ or $\ti c_2(0)$ is in the interior of $K$. But if $\ti c_i(0)$ is in the interior of $K$, then the whole trace $\ti c_i(\mR)$ is contained in $K$. Furthermore, $\ti c_i(\mR)=\pi^{-1}\left(c_i([0,1])\right)$ is closed and therefore compact. This implies that $c_i$ is homologous to zero in contradiction to our assumption. \[umidbound\] Suppose that $\torus $ carries a Riemannian metric $\gbel = e^{2u}\gflach$ with $\gflach$ flat. Let $c_i:S^1\to \torus $, $i=1,2$ be closed paths non-homologous to zero. We set $$v_1:=\max_{t\in S^1} u\circ c_1(t) \quad \mbox{and} \quad v_2:=\min_{t\in S^1} u\circ c_2(t).$$ Then The statement is void for $v_2\leq v_1$, therefore we can assume $v_2>v_1$. We set $\cK_1:=\cK_1(\torus ,\gbel)$. Let $v\in\;]v_1,v_2[$ be a regular value of $u$. As in the proof of the previous proposition we set $G(v):=\{x\in\torus \,|\,u(x)<v\}$, let $A(v)$ be the area $G(v)$ with respect to $\gbel$, $l(v)$ the length of $\pa G(v)$ with respect to $\gbel$, and $\Aflach(v)$ and $\lflach(v)$ the area and length with respect to $\gflach$. In analogy to we get $$\int_{G(v),\gbel}K_\gbel = - \int_{\torus \ohne G(v),\gbel}K_\gbel = -\int_{\pa G(v)} *\, du$$ and therefore $$\int_{\pa G(v), \gflach} |du|_\gflach=\int_{\pa G(v), \gbel} |du|_\gbel=\int_{\pa G(v)} *\, du \leq {1\over 2}\int_{\torus ,\gbel}\left|K_\gbel\right|\leq {\cK_1\over 2}.$$ We obtain $${d\over dv} \Aflach(v) = \int_{\pa G(v),\gflach} {1\over |du|_\gflach} \geq {{\lflach(v)}^2\over \int_{\pa G(v), \gflach} |du|_\gflach}\geq 2\,{{\lflach(v)}^2\over \cK_1}.$$ Using Lemma \[boundest\] we know that the right hand side of this inequality is greater than or equal to $8\,{\sys(\torus,\gflach)}^2/\cK_1$. Integration yields $$\area(\torus ,\gflach)\geq \Aflach(v_2)-\Aflach(v_1)\geq 8 \,{{\sys(\torus,\gflach)}^2\over \cK_1}\,(v_2-v_1),$$ and therefore the statement of (a). Similarly, we show statement (b): $${d\over dv} A(v) = \int_{\pa G(v),\gbel} {1\over |du|_\gbel} \geq {{l(v)}^2\over \int_{\pa G(v), \gbel} |du|_\gbel}\geq 8 \,{{\sys(\torus,\gbel)}^2\over \cK_1},$$ An alternative way to prove (b) is to use (a) together with Lemma \[loewnerprep\]. Again we set $G(v):=\{x\in\torus \,|\,u(x)>v\}$. Let $v_2$ be the supremum of all $v\in \mR$ with the property that there is a closed path $c_2(v):S^1\to G(v)$ that is non-homologous to zero in $\torus$. Similarly, we define $\widehat G(v):=\{x\in\torus \,|\,u(x)<v\}$, and let $v_1$ be the infimum of all $v\in \mR$ for which there is a closed path $c_1(v):S^1\to\widehat G(v)$ that is non-homologous to zero in $\torus$. For any $\ep>0$ we have $\max \bigl(u\circ c_1(v_1+\ep)\bigr) <v_1+\ep$ and $\min \bigl(u\circ c_2(v_2-\ep)\bigr)> v_2-\ep$. We apply Proposition \[umidbound\], and the limes $\ep\to 0$ yields $$\begin{aligned} v_2-v_1 & \leq &{\cK_1(\torus ,\gbel)\, \cV(\torus,\gflach)\over 8}\nonumber\\ &\leq & {\cK_p(\torus ,\gbel)\, \cV(\torus,\gflach)\over 8}. \label{streasummand1}\end{aligned}$$ Now we apply Corollary \[uupperboundcor\] for $G:=G(v)$ and $v:=v_2+\ep$ where we replace $u$ by $u-v$. We get in the limit $\ep\to 0$ $$(\umax)-v_2\leq {1\over 2}\biggl|\log \left(1-{\cK_p\over 4\pi}\right)\biggr|+ {\cK_p\over 8\pi -2\cK_p}\,q\log (2q), \label{streasummand2}$$ with $q:=p/(p-1)$. Similarly, Proposition \[ulowerbound\] yields $$v_1-(\umin)\leq {q\,\cK_p(G,\gbel)\over 4\pi}.\label{streasummand3}$$ Adding inequalities , and we obtain statement (a). The proof of statement (b) is completely analogous. An estimate on the disk ======================= From the results of the previous section is not difficult to derive another theorem. \[disktheo\] Let $\gbel$ be a Riemannian metric on a domain $G$ whose closure is diffeomorphic to the $2$-dimensional disk. We write $\gbel$ as $\gbel=e^{2u}\gflach$ with $\gflach$ flat and $\res{u}{\partial G}\equiv 0$. For $p\in\;\mo]1,\infty\mc[$ we assume $\cKpo_p:=\cKpo_p(G,\gbel)<2\pi$. Then we get the estimate $$\max u \leq {1\over 2}\,\Biggl|\log \left(1-{\cKpo_p\over 2\pi}\right)\Biggr|\,+\, {\cKpo_p\over 4\pi -2\cKpo_p}\,q\log q,$$ with $q:=p/(p-1)$. W.l.o.g. we can assume $u\geq 0$. For any open subset $\widehat{G}\subset G$ we have $$\begin{aligned} \int_{\widehat G,\gbel}K_\gbel & \leq & \left\|K^+_\gbel\right\|_{L^1(\widehat G,\gbel)} \nonumber\\ & \leq & \left\|K^+_\gbel\right\|_{L^p(\widehat G,\gbel)}\area(\widehat G,\gbel)^{1/q} \leq \cKpo_p \left({\area(\widehat G,\gbel)\over \area(G,\gbel)}\right)^{1/q}.\end{aligned}$$ We can apply Proposition \[uupperbound\] with $\ka=\cKpo_p$, $r=1/q$, $C=\cKpo_p\,\area(G,\gbel)^{-1/q}$ and $\mu_0=\mu_1=\area(G,\gbel)$ and we directly get the theorem. Cylindrical and conical examples {#exsec} ================================ In the previous section we gave a bound on $\osc u$ in terms of the $L^p$-norm of the Gaussian curvature $K$, $p>1$, the area and the systole. Now we will give some examples showing that $\osc u$ is **not** bounded by a function of the $L^1$-norm of $K$, the area and the systole. In contrast to this, note that the diameter of $\torus$ is bounded by a function depending on $\int_{\torus} |K_{\gbel}|$, $\area(\torus,\gbel)$ and and $\sys(\torus,\gbel)$, provided that $\int_{\torus} |K_{\gbel}|<4\pi$ [@ammanndiss Korollar 3.6.8]. (-4,-1.7)(11,5.4) (0,0) (-3.4,-.9)(1.6,-.9)(3.4,.9)(-1.6,.9) (1.2,0.4) (-1.2,0)(1.2,.4) (-1.2,0)(-.9,0)(-.6,.1)(-.6,.3) (1.2,0)(.9,0)(.6,.1)(.6,.3) (0,.3)(.6,.2) (-.6,.3)(.6,4.3) (-.6,.3)(-.6,4.3) (.6,.3)(.6,4.3) (0,4.3)(.6,.2) (-.6,4.3)(.6,4.5) (.6,.3)(1.3,.3) (.6,4.3)(1.3,4.3) (1,.3)(1,4.3) (1,2.3) (.6,.3)(.6,-1.9) (-.6,.3)(-.6,-1.9) (-.6,-1.6)(.6,-1.6) (0,-1.6) (0,4.3)[.6]{}[0]{}[180]{} (2,3.5)[$\be=0$]{} (6.5,0) (-3.4,-.9)(1.6,-.9)(3.4,.9)(-1.6,.9) (0,.8)(1.1,1.0) (1.2,0.4) (-1.2,0)(1.2,.4) (0,.3)(.6,.2) (-.6,.3)(.6,.5) (-.6,.3)(.6,.3)(0,1.5) (0,1.1)(.2,.066) (-.2,1.1)(.2,1.3) (-.2,1.1)(.8,3.1) (0,1.5)(0,3.3) (.6,.3)(.6,-1.9) (-.6,.3)(-.6,-1.9) (-.6,-1.6)(.6,-1.6) (0,-1.6) (.15,2.4)[$\be$]{} (0,1.5)[1.5]{}[63.45]{}[90]{} (.6,.3)(1.3,.3) (.2,1.1)(1.3,1.1) (1,.3)(1,1.1) (1,.7) In order to discuss the properties of our examples we will use a lemma that can easily be proven by using Lemma \[konformkruem0lemma\] and the Gauss-Bonnet theorem. Assume that a disk $D$ carries a rotationally symmetric Riemannian metric $\gbel$ and that in a neighborhood of the boundary $\gbel$ is isometric to a flat ring of the form $(B_R(0)\ohne B_r(0)\subset \mR^2,\geukl)$. Then there is a rotationally symmetric smooth function $u:B_R(0)\to\mR$ vanishing in a neighborhood of the boundary such that $(D,\gbel)$ is isometric to $(B_R(0),e^{2u}\,\geukl)$. The function $u$ is uniquely determined by these properties. The idea behind the construction of the metrics in this section is to start with a flat torus $(\torus,\gflach)$, to cut out a flat round disk and to replace it by a disk $D'$ with a rotationally symmetric metric $g'$. Because of the preceeding lemma the metric $\gbel$ obtained by this replacement can be written as $e^{2u}\,\gflach$ where $u$ is a smooth function supported on the disk. Viewed as a function on the disk, $u$ is rotationally symmetric. Therefore $\osc u$ can be easily estimated using polar coordinates. The disks $(D',g')$ we glue in are described by figure \[cynconpic\]. For the *cylindrical metric* we construct $(D',g')$ as follows: we take a cylinder of height $H$ and radius $R$, glue it together with a half sphere of radius $R$ at one end and a suitable socket on the other end. After smoothing we get $(D',g')$. The resulting metric on $\torus$, the *cylindrical metric*, will be denoted $g_{R,H,0}$. Similarly, for the *conical metric* $g_{R,H,\be}$ (Figures \[cynconpic\] and \[conicalpic\]) we take a truncated cone of height $H$, opening angle $\be>0$ and the two boundary components are circles of radius $R$ and $\rho=R-H\sin \be$. The end of the truncated cone corresponding to $\rho$ is closed smoothly by a topological disk and the other end is put on a socket. (-2.5,-0.4)(2.5,3.1) (0,0)(-2.3,-0.3)(2.3,2.5) (-.333,4pt)(-.333,-4pt) (-4pt,.5)(4pt,.5) (2.5,0)[$x$]{} (0,2.7)[$z$]{} (-0.333,-0.2)[$-\rho$]{} (-1,-0.2)[$-R$]{} (1,-0.2)[$R$]{} (-2,-0.2)[$-2R$]{} (2,-0.2)[$2R$]{} (-1.1,1.833)(.6,1.833) (-1.1,0.5)(-0.9,0.5) (.9,0.5)(1.1,0.5) (-1,1.833)(-1,.5) (-1,1.1) (-.95,.8)[$\be$]{} (-1,.5)[.5]{}[63.45]{}[90]{} Now we write $g'=e^{2u}\gflach$ and express $u$ in geodesic polar coordinates centered at the center of $D'$. On the cylinder or cone resp. $u$ is harmonic and therefore has the form $$u(r,\ph)=a + b \log r.$$ The Gauss-Bonnet theorem yields $b=\sin\be -1$. An elementary calculation shows $$\osc u\geq \left({1\over \sin \be}-1\right)\log\left({R\over \rho}\right)$$ for the conical metric and $$\osc u\geq H/R$$ for the cylindrical metric. Using Gauss-Bonnet we also get $$\int_{\torus}|K_{g_{R,H,\be}}|=4\pi(1-\sin\be).$$ If $R$ is sufficiently small, the systole does not depend on $H$, $R$ and $\beta$. Now for fixed $\be\geq 0$ choose sequences $H_i$ and $R_i$ such that $\area(\torus,g_{R_i,H_i,\be})$ is constant and such that $H_i/R_i\to \infty$ or $\rho_i/R_i\to 0$ resp. So we have constructed families of metrics $g_{R_i,H_i,\be}$ on $\torus$ with fixed $\int|K|$, fixed area and fixed systole but $\osc u_i\to \infty$. [Amm98]{} B. Ammann, *Spectral estimates on 2-tori*, Preprint in preparation. B. Ammann, *[S]{}pin-[S]{}trukturen und das [S]{}pektrum des [D]{}irac-[O]{}perators*, Ph.D. thesis, University of [F]{}reiburg, [G]{}ermany, 1998, Shaker-Verlag Aachen 1998. C. B[ä]{}r, *Extrinsic bounds for eigenvalues of the [D]{}irac operator*, Ann. Global Anal. Geom. **16** (1998), no. 6, 573–596. A. L. Besse, *Einstein manifolds*, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, no. 10, Springer-Verlag, 1987. H. Brezis and F. Merle, *Uniform estimates and blow-up behavior for solutions of $-{\De} u= {V}(x)\,e^u$ in two dimensions*, Comm. Partial Diff. Equat. **16** (1991), no. 8 & 9, 1223–1253. Bang-Yen Chen, *On the total curvature of immersed manifolds. [V]{}. ${C}$-surfaces in [E]{}uclidean $m$-space*, Bull. Inst. Math. Acad. Sinica **9** (1981), no. 4, 509–516. Xiuxiong Chen, *Weak limits of [R]{}iemannian metrics in surfaces with integral curvature bound*, Calc. Var. Partial Differential Equations **6** (1998), no. 3, 189–226. M. Gromov, *Structures m[é]{}triques pour les vari[é]{}t[é]{}s [R]{}iemanniennes*, CEDIC, Paris, 1981. J. Langer and D. Singer, *Curves in the hyperbolic plane and mean curvature of tori in 3-space*, Bull. London Math. Soc. **16** (1984), 531–534. P. Li and S.T. Yau, *A new conformal invariant and its applications to the [W]{}illmore [C]{}onjecture and the first eigenvalue of compact surfaces*, Invent. Math. **69** (1982), 269–291. S. Montiel and A. Ros, *Minimal immersions of surfaces by the first eigenfunctions and conformal area*, Invent. Math. **83** (1985), no. 1, 153–166. U. Pinkall, *Hopf tori in ${S}\sp 3$*, Invent. Math. **81** (1985), no. 2, 379–386. A. Ros, *The [W]{}illmore conjecture in the real projective space*, Preprint, 1997. L. Simon, *Existence of surfaces minimizing the [W]{}illmore functional*, Comm. Anal. Geom. **1** (1993), no. No. 2, 281–326. K. Shiohama and R. Takagi, *A characterization of a standard torus in $E^3$*, J. Diff. Geom. **4** (1970), 477–485. G. Thomsen, *[Ü]{}ber konforme [G]{}eometrie, [I]{}: [G]{}rundlagen der konformen [F]{}l[ä]{}chentheorie*, Abh. Math. Sem. Hamburg **3** (1923), 31–56. P. Topping, *Towards the [W]{}illmore conjecture*, Preprint, 1998. J. L. Weiner, *On a problem of [C]{}hen, [W]{}illmore, et al.*, Ind. Math. J. **27** (1978), no. 1, 19–35. T. J. Willmore, *Note on embedded surfaces*, An. Şti. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. (N.S.) **11B** (1965), 493–496. T. J. Willmore, *Mean curvature of [R]{}iemannian immersions*, J. London Math. Soc. (2) **3** (1971), 307–310. Bernd Ammann\ Mathematisches Institut\ Universität Freiburg\ Eckerstr. 1\ 79104 Freiburg\ Germany\ [ammann@mathematik.uni-freiburg.de]{}
--- abstract: 'The subgroup $K=GL_p \times GL_q$ of $GL_{p+q}$ acts on the (complex) flag variety $GL_{p+q}/B$ with finitely many orbits. We introduce a family of polynomials specializing to representatives for cohomology classes of the orbit closures in the Borel model. We define and study $K$-orbit determinantal ideals to support the geometric naturality of these representatives. Using a modification of these ideals, we describe an analogy between two local singularity measures: the $H$-polynomials and the Kazhdan-Lusztig-Vogan polynomials.' address: 'Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA' author: - 'Benjamin J. Wyser' - Alexander Yong date: 'March 4, 2014' title: 'Polynomials for $GL_p\times GL_q$ orbit closures in the flag variety' --- Introduction ============ Polynomial representatives in ordinary cohomology ------------------------------------------------- Consider the Levi subgroup $K=GL_p \times GL_q$ of $GL_n$ ($n=p+q$). (Throughout, we consider only complex general linear groups.) By a general result of T. Matsuki [@Matsuki Theorem 3], the flag variety $GL_n/B$ decomposes as a disjoint union of finitely many $K$-orbits: $$GL_n/B=\coprod_{\gamma} {\mathcal O}_{\gamma}.$$ The orbits ${\mathcal O}_{\gamma}$ are parameterized by [**$(p,q)$-clans**]{} $\gamma$, as described first by T. Matsuki-T. Oshima [@Matsuki-Oshima Theorem 4.1], and later elaborated upon by A. Yamamoto [@Yamamoto Theorem 2.2.8]. These clans are partial matchings of vertices $\{1,2,\ldots,n\}$, where unpaired vertices are assigned $+$ or $-$; the difference in the number of $+$’s and $-$’s must be $p-q$. Let ${\tt Clans}_{p,q}$ denote the set of all such clans. Three clans from ${\tt Clans}_{6,4}$ are shown below: $$\begin{picture}(280,10) \put(0,0){\epsfig{file=matchings.eps, height=.8cm}} \put(8,0){$+$} \put(17,0){$-$} \put(26,0){$+$} \put(43,0){$+$} \put(64,0){$-$} \put(83,0){$+$,} \put(104,0){$+$} \put(120,0){$-$} \put(130,0){$+$} \put(139,0){$-$} \put(159,0){$+$} \put(177,0){$+$} \put(196,0){,} \put(210,0){$-\!+\!+\!+\!-\!+\!-\!-\!+\!+$} \end{picture}$$ Let $Y_{\gamma}$ be the Zariski closure of ${\mathcal O}_{\gamma}$. This is the union of ${\mathcal O}_{\beta}$ for $\beta\prec \gamma$, where (by definition) $\prec$ is the [**closure order**]{} on clans. It is an irreducible variety. By the formula of [@Yamamoto Proposition 2.3.8], its dimension is ${p\choose 2}+{q\choose 2}+\ell(\gamma)$ where $$\label{eqn:Yam} \ell(\gamma)=\sum_{\mbox{\tiny{vertices $i<j$ are matched}}} j-i-\#\{\mbox{matchings of $s<t$ where $s<i<t<j$}\}.$$ $Y_{\gamma}$ admits a class in singular cohomology (with ${{{\mathbb}Z}}$ coefficients): $$[Y_{\gamma}]\in H^{\star}(GL_n/B)\cong {\mathbb Z}[x_1,\ldots,x_n]/I^{S_n},$$ where $I^{S_n}$ is the ideal generated by symmetric polynomials without constant term. The above isomorphism, due to A. Borel [@Borel] (cf. [@Fulton Section 10.2]), is suggestive of the following problem: > Describe a choice of polynomial representatives $\{\Upsilon_{\gamma}\}$ for the cosets associated to $\{[Y_{\gamma}]\}$ under Borel’s isomorphism. One solution begins by assigning polynomials to the ${n\choose p}$-many [**closed orbits**]{}. These orbits are indexed by [**matchless clans**]{} $\tau$, i.e., those consisting of $p$ many $+$’s and $q$ many $-$’s (the third displayed clan above is an example). We will typically use $\tau$ to denote a matchless clan, and $\gamma$ to indicate an arbitrary clan. The [**divided difference operator**]{} $\partial_i:{\mathbb Z}[x_1,\ldots,x_n]\to {\mathbb Z}[x_1,\ldots,x_n]$ is $$\partial_i f=\frac{f-f^{s_i}}{x_i-x_{i+1}}.$$ Representatives for all other orbits can be obtained by recursion using the $\partial_i$’s along a choice of path in *weak order* (defined in Section \[sec:defs\]). This approach was used by the first author in [@Wyser-13a]. We consider a different choice of polynomial representatives for the closed orbits than that found in *loc. cit*. From our perspective, this alternative choice of representatives is preferable for the following reasons: - It is provably “self-consistent”, by which we mean that each $\Upsilon_{\gamma}$ is a well-defined polynomial. Specifically, $\Upsilon_{\gamma}$ depends neither on the choice of closed $K$-orbit ${\mathcal{O}}_{\tau}$ at which we start the recurrence, nor on the aforementioned choice of path in weak order. - Each $\Upsilon_{\gamma}$ has nonnegative integer coefficients, and in many cases the geometric reason for this is transparent. - Our choice extends simply to $T$-equivariant cohomology and ($T$-equivariant) $K$-theory, where $T$ is the torus of diagonal matrices in $GL_n$. ([@Wyser-13a] covers the case of $T$-equivariant cohomology, but neither ordinary nor $T$-equivariant $K$-theory are discussed.) We mostly suppress discussion of these refinements until Section 2. To formulate our answer, we associate to a matchless $(p,q)$-clan $\tau$ a partition, which we will denote $\lambda(\tau)$. We will also associate a sequence of nonnegative integers denoted by ${\vec f}(\tau)$; this sequence is called a “flagging” in the context that we will use it below. The partition $\lambda(\tau)$ is formed as follows. Start from the upper-right corner of a $p \times q$ rectangle, and trace a lattice path to the lower-left corner, by moving down at step $i$ if the $i$th character of $\tau$ is a $+$, and left if it is a $-$. Then $\lambda(\tau)$ is the partition whose Young diagram is the portion of the $p \times q$ rectangle northwest of this path. Clearly, the assignment of $\lambda(\tau)$ to $\tau$ defines a bijection between matchless $(p,q)$-clans (or, equivalently, $p$-element subsets of $\{1,\hdots,n\}$) and partitions whose Young diagrams fit within a $p \times q$ rectangle. Now, ${\vec f}(\tau) = (f_1,\hdots,f_p)$ for $\lambda(\tau)$ is defined by $f_i = \text{index of $i$th $+$ of $\tau$}$. Next, let $\widehat\tau$ denote the $(q,p)$-clan obtained from $\tau$ by flipping all signs. Then we can also form the partition $\lambda(\widehat\tau)$ and the flagging ${\vec f}(\widehat\tau)$, as described above. Note that this partition has $q$ parts, and its flagging is a $q$-tuple. As an example, if $\tau=++--+-++$ then $\lambda(\tau)=(3,3,1,0,0)$ and ${\vec f}(\tau)=(1,2,5,7,8)$, while $\lambda(\widehat\tau)=(3,3,2)$ and ${\vec f}(\widehat\tau)=(3,4,6)$. The relevant pictures are as follows: (220,75) (0,0) (15,0)[(0,1)[75]{}]{} (30,0)[(0,1)[75]{}]{} (0,15)[(1,0)[45]{}]{} (0,30)[(1,0)[45]{}]{} (0,45)[(1,0)[45]{}]{} (0,60)[(1,0)[45]{}]{} (45,76)[(0,-1)[30]{}]{} (46,45)[(-1,0)[30]{}]{} (16,46)[(0,-1)[16]{}]{} (17,30)[(-1,0)[17]{}]{} (0,31)[(0,-1)[32]{}]{} (47,66)[$+$]{} (47,51)[$+$]{} (33,38)[$-$]{} (18,38)[$-$]{} (17,31)[$+$]{} (3,23)[$-$]{} (2,16)[$+$]{} (2,4)[$+$]{} (62,65)[$1$]{} (62,50)[$2$]{} (62,35)[$5$]{} (62,20)[$7$]{} (62,5)[$8$]{} (120,15) (135,15)[(0,1)[45]{}]{} (150,15)[(0,1)[45]{}]{} (165,15)[(0,1)[45]{}]{} (180,15)[(0,1)[45]{}]{} (120,30)[(1,0)[75]{}]{} (120,45)[(1,0)[75]{}]{} (196,60)[(-1,0)[30]{}]{} (166,61)[(0,-1)[31]{}]{} (166,31)[(-1,0)[16]{}]{} (151,31)[(0,-1)[16]{}]{} (151,16)[(-1,0)[31]{}]{} (185,52)[$-$]{} (170,52)[$-$]{} (167,46)[$+$]{} (167,33)[$+$]{} (154,22)[$-$]{} (152,16)[$+$]{} (138,7)[$-$]{} (123,7)[$-$]{} (200,50)[$3$]{} (200,35)[$4$]{} (200,20)[$6$]{} Now, given any partition $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq \lambda_m\geq 0)$ and a sequence of nonnegative integers $\vec f=(f_1,\ldots,f_m)$ (a flagging), one defines the [**flagged Schur polynomial**]{} to be $$s_{\lambda,\vec f}(X)=\sum_T {\bf x}^T,$$ where the sum is over all semistandard tableaux $T$ of shape $\lambda$ whose entries in row $i$ are weakly bounded above by $f_i$; see [@Manivel Section 2.6] for a textbook treatment of flagged Schur polynomials. So considering the partition $\lambda(\tau)=(3,3,1,0,0)$ and the flagging ${\vec f}(\tau) = (1,2,5,7,8)$ coming from the clan $\tau$ in our example, $$s_{(3,3,1,0,0),(1,2,5,7,8)}(x_1,x_2,x_3,x_4,x_5)=x_1^3 x_2^3 x_3+x_1^3 x_2^3 x_4 + x_1^3 x_2^3 x_5.$$ The three monomials correspond to the tableaux $${\vtop{\let\\\cr \baselineskip -16000pt \lineskiplimit 16000pt \lineskip 0pt \ialign{&{\def\thearg{##}\def\nothing{}\ifx\thearg\nothing \vrule width0pt height\cellsz depth0pt\else \hbox to 0pt{\usebox{\cell} \hss}\fi\vbox to \cellsz{ \vss \hbox to \cellsz{\hss$##$\hss} \vss}}\cr1 & 1& 1\\ 2& 2&2 \\3\crcr}}}, \ \ \ {\vtop{\let\\\cr \baselineskip -16000pt \lineskiplimit 16000pt \lineskip 0pt \ialign{&{\def\thearg{##}\def\nothing{}\ifx\thearg\nothing \vrule width0pt height\cellsz depth0pt\else \hbox to 0pt{\usebox{\cell} \hss}\fi\vbox to \cellsz{ \vss \hbox to \cellsz{\hss$##$\hss} \vss}}\cr1 & 1& 1\\ 2& 2&2 \\4\crcr}}} \ \ \ \mbox{and \ \ \ } {\vtop{\let\\\cr \baselineskip -16000pt \lineskiplimit 16000pt \lineskip 0pt \ialign{&{\def\thearg{##}\def\nothing{}\ifx\thearg\nothing \vrule width0pt height\cellsz depth0pt\else \hbox to 0pt{\usebox{\cell} \hss}\fi\vbox to \cellsz{ \vss \hbox to \cellsz{\hss$##$\hss} \vss}}\cr1 & 1& 1\\ 2& 2&2 \\5\crcr}}}.$$ Since $\lambda(\tau)$ and ${\vec f}(\tau)$ are determined by $\tau$, one can use the abbreviation ${\mathfrak s}_{\tau}(X) = s_{\lambda(\tau),{\vec f}(\tau)}(X)$, and similarly define ${\mathfrak s}_{\widehat\tau}(X)$. For matchless $\tau$, define: $$\label{eqn:defformatchless} \Upsilon_{\tau}:={\mathfrak s}_{\tau}(X) \cdot {\mathfrak s}_{\widehat\tau}(X).$$ Given a clan $\gamma$ which is not matchless, by [@Richardson-Springer Theorem 4.6] there exists a matchless clan $\tau$ and a sequence $s_1,\hdots,s_l$ of simple transpositions such that $\gamma = s_1 \cdot s_2 \cdot \hdots s_l \cdot \tau$. (This notation is explained in Section \[sec:defs\].) In general, neither $\tau$ nor the permutation $w=s_1 \hdots s_l$ is uniquely determined by $\gamma$. Our wish is to define $$\Upsilon_{\gamma}=\partial_1 \hdots \partial_l \Upsilon_{\tau}.$$ However, in light of the preceding sentence, it is not at all clear that this is a valid “definition”. The main purpose of this paper is to present the following (and its refinements): \[thm:intro\] Each $\Upsilon_{\gamma}$ is well-defined and represents $[Y_{\gamma}]$ under Borel’s isomorphism. We now make a few easy observations about the $\Upsilon_{\gamma}$’s. The flagged Schur polynomials from (\[eqn:defformatchless\]) are Schubert polynomials (see Proposition \[prop:KMY\]). It is a standard fact that any product of Schubert classes expands as a nonnegative linear combination of Schubert classes, and moreover the Schubert polynomials represent the Schubert classes under the Borel isomorphism; see, e.g., Chapter 10 (and specifically Section 10.4) of [@Fulton]. It follows that $\Upsilon_{\gamma}$ is a nonnegative linear combination of Schubert polynomials. Since Schubert polynomials have nonnegative integer coefficients, $$\Upsilon_{\gamma}\in {\mathbb Z}_{\geq 0}[x_1,\ldots,x_n] \mbox{ \ \ for all $\gamma\in {\tt Clan}_{p,q}$.}$$ We have emphasized the monomial expansion of $\Upsilon_{\gamma}$ since this positivity should have a geometric explanation (see Section 3). Finally, by our definition of $\Upsilon_{\tau}$ for $\tau$ matchless, it is easy to see that $\Upsilon_{\tau}$ has degree $\binom{n}{2} - \binom{p}{2} - \binom{q}{2} = pq$. This reflects the fact that any closed $K$-orbit is isomorphic to the flag variety for the group $K$, and hence has dimension equal to $\binom{p}{2} + \binom{q}{2}$. Combining this with the aforementioned dimension formula of A. Yamamoto (cf. (\[eqn:Yam\])), and with the fact that application of $\partial_i$ lowers the degree of any polynomial by $1$, it follows that the degree of $\Upsilon_{\gamma}$ for arbitrary $\gamma$ is $pq - l(\gamma)$, the codimension of ${\mathcal{O}}_{\gamma}$ in the flag variety. Further results and comparisons to the literature ------------------------------------------------- For a reductive algebraic group $G$ over ${\mathbb C}$, let $B$ be a Borel subgroup and $K\subset G$ be a [**spherical**]{} subgroup, i.e., one which acts by left translations on $G/B$ with finitely many orbits. The most widely analyzed case is when $K=B$, where the orbit closures are Schubert varieties. In this setting, the polynomial representatives problem was studied for Schubert varieties (in general type) by I. Bernstein-I. Gelfand-S. Gelfand [@BGG]. In type $A$, this led to the development of *Schubert polynomials* by A. Lascoux-M.-P. Schützenberger [@Lascoux.Schutzenberger]. Both papers begin with a choice of polynomial representative for the class of a point, with the remainder recursively obtained using $\partial_i$’s. However, the salient feature of Schubert polynomials is the nonnegativity of their coefficients. Since their discovery, many nice combinatorial properties of Schubert polynomials have been found, including combinatorial formulas for their expansion; see, e.g., the textbook [@Manivel]. We will use properties of Schubert polynomials to establish our main results. A spherical subgroup $K$ is [**symmetric**]{} if $K=G^{\theta}$ is the fixed point subgroup for a holomorphic involution $\theta$ of $G$. The symmetric pairs $(G,K)$ have a classification. For generalities, the reader may consult, e.g., [@Matsuki; @Springer; @Matsuki-Oshima; @Richardson-Springer]. The case of $(GL_{p+q},GL_p \times GL_q)$ corresponds to the involution $$\theta(A)=I_{p,q}A I_{p,q}$$ where $I_{p,q}$ is the diagonal $\pm 1$ matrix with $p$ many $1$’s followed by $q$ many $-1$’s. For more details about this case, see, e.g., [@Yamamoto; @McGovern; @McGovern-Trapa; @Wyser-13a]. The first author gave equivariant cohomology representatives for the closed orbits of cases of symmetric pairs $(G,K)$ with $G$ classical in [@Wyser-13a; @Wyser-13b]. For the case of $(GL_{p+q}, GL_p\times GL_q)$, small examples suggest that those representatives may also produce a self-consistent system, although we do not know a proof of this. At any rate, their ordinary cohomology specializations do not have nonnegative integer coefficients in general. To our best knowledge, this paper provides the first self-consistency proof of its kind for any symmetric pair $(G,K)$. In the case of Schubert varieties, the divided difference recurrence has only one initial condition (the class of a point). Further, minimal paths in the weak Bruhat order of $S_n$ correspond to reduced words of the same permutation. Since divided differences satisfy the braid relations, self-consistency is automatic for Schubert polynomials. As we have observed, neither of these two helpful properties hold for the symmetric pair we consider here. For some other symmetric pairs (also defined over the complex numbers), such as $(GL_{2n},Sp_{2n})$ or $(GL_n,O_n)$, the property of having only one initial condition — that is, a unique closed $K$-orbit — *does* hold. However, even in such cases, minimal chains in weak order can again correspond to reduced words of *different* permutations, so self-consistency is not a given in these cases either. The two aforementioned additional cases are considered in a sequel [@WyYo2]. There is further support for the choice of $\Upsilon_{\gamma}$. We use a geometric perspective originally applied by A. Knutson-E. Miller [@Knutson.Miller:annals] to justify Schubert polynomials. For a variety $X\subset GL_n/B$, consider the preimage $\pi^{-1}(X)\subset GL_n$ under the natural projection, and $\overline{\pi^{-1}(X)}\subset {\rm Mat}_{n\times n}$. Because $\pi^{-1}(X)$ is a union of left cosets of $B$, $\overline{\pi^{-1}(X)}$ is stable under right multiplication by $B$. Identifying $$[\overline{\pi^{-1}(X)}]_B\in H^\star_B({\rm Mat}_{n\times n}) \mbox{\ with $[\overline{\pi^{-1}(X)}]_T \in H^\star_T({\rm Mat}_{n\times n})\cong {\mathbb Z}[x_1,\ldots, x_n]$}$$ (see [@Knutson.Miller:annals Section 1.2]) uniquely picks out a polynomial representative for $[X]\in H^{\star}(GL_n/B)$. In the case $X=X_w:={\overline{B_{-}wB/B}}$ of Schubert varieties, to actually compute $[\overline{\pi^{-1}(X_w)}]_T$ they obtain, by Gröbner degeneration, the multidegree of Fulton’s *Schubert determinantal ideal* $I_w$, whose generators scheme-theoretically cut out $\overline{\pi^{-1}(X_w)}$. Their conclusion is $$[\overline{\pi^{-1}(X_w)}]_T={\mathfrak S}_w(x_1,\ldots,x_n),$$ the Schubert polynomial for $X_w$ [@Knutson.Miller:annals Theorem A]. To study the case $X=Y_{\gamma}$, we define the *$K$-orbit determinantal ideal* $I_{\gamma}$, generated by minors of the generic $n\times n$ matrix and certain auxiliary matrices. When $\gamma$ is [**non-crossing**]{}, i.e., no two arcs overlap (see the second of the displayed clans on page 2 for a non-example), these generators form a Gröbner basis with squarefree lead terms. The prime decomposition of the Gröbner limit is indexed by monomials of $\Upsilon_{\gamma}$. That is, $I_{\gamma}$ scheme-theoretically cuts out $\overline{\pi^{-1}(Y_\gamma)}$, and we show $$[\overline{\pi^{-1}(Y_\gamma)}]_T=\Upsilon_{\gamma}(x_1,\ldots,x_n), \text{\ \ for non-crossing $\gamma$}.$$ See Theorem \[claim:main\], whose proof uses [@Knutson.Miller:annals; @KMY; @Wyser12; @Wyser-13a]. This provides a geometric rationale for our choice of representatives, at least for the non-crossing case. Furthermore, we conjecture that the above equality holds for all $\Upsilon_{\gamma}$, whether $\gamma$ has crossings or not (cf. Section \[sec:conjectures\]). The non-crossing condition is special because then $Y_{\gamma}$ is a *Richardson variety* [@Wyser12], or the intersection of a Schubert variety with an opposite Schubert variety. Such varieties are so named because they were first studied by R. W. Richardson in [@Richardson]. Properties of Richardson varieties can be transparently deduced from the two Schubert varieties involved [@KWY]. These facts were our starting point for this project. In [@Brion], M. Brion proves (in a general setting, which applies in particular to the case at hand) a formula for $[Y_{\gamma}]$ as a sum of Schubert *classes*. In our example, this sum turns out to be multiplicity-free, meaning that all Schubert classes occurring in the sum occur with coefficient $1$. Thus taking Brion’s formula and replacing each Schubert *class* with its corresponding Schubert *polynomial* gives a cohomological representative of the type we are seeking. Indeed, our arguments will make it apparent that the representative so obtained is in fact equal to $\Upsilon_{\gamma}$. However, while Brion’s formula applies in both (ordinary) cohomology and $K$-theory, it does *not* apply $T$-equivariantly in either theory. Thus our representatives in the $T$-equivariant setting are truly “new“, in the sense that they cannot be easily be deduced from Brion’s formula. Finally, we consider a modification of the $K$-orbit determinantal ideal which we conjecture provides local equations of $Y_{\gamma}$, cf. Conjecture \[conj:D\]. Having such equations allows us to study the singularities of the orbit closures inside $G/B$. The *Kazhdan-Lusztig-Vogan polynomials* are one local measure of these singularities. We describe a conjectural analogy with another singularity measure, the *$H$-polynomials* of $Y_{\gamma}$, defined in Section \[sec:h-poly\]. This analogy parallels that between *Kazhdan-Lusztig polynomials* and $H$-polynomials of Schubert varieties described by L. Li and the second author in [@Li.Yong Section 2]. Organization ------------ In Section 2, we introduce a family of polynomials in two sets of variables, with a deformation parameter. This family is defined using Schubert polynomials and divided difference operators. With this, we state our choice of polynomial representatives for equivariant cohomology and equivariant $K$-theory. We establish our main theorems (Theorems \[thm:intro\], \[thm:equivver\] and \[thm:K\]) that they define a self-consistent system. In Section 3, we define the $K$-orbit determinantal ideal and establish our Gröbner basis theorem in the non-crossing case as well as formulate the more general conjectures. In Section 4, we use a modification of these ideals in our exploration of the singularities of $Y_{\gamma}$. More polynomial families and cohomology theories ================================================ Definition of $\Upsilon^{(\beta)}_{\gamma}$ {#sec:defs} ------------------------------------------- For non-crossing $\gamma$, define $u(\gamma)\in S_n$ by assigning - $-$’s and left endpoints of arcs the labels $1,2,\ldots,q-1,q$ from left to right, and - $+$’s and right endpoints of arcs the labels $q+1,q+2,\ldots,n$ from left to right. Define $v(\gamma)\in S_n$ by assigning - $+$’s and left endpoints of arcs the labels $1,2,\ldots,p-1,p$ from left to right, and - $-$’s and right endpoints of arcs the labels $p+1,p+2,\ldots,n$ from left to right. For the second clan $\gamma\in {\tt Clan}_{6,4}$ shown on page 2, $u(\gamma)=512637849 \ 10$ and $v(\gamma)=127389456 \ 10$. \[exa:nonmatching\] We are especially interested in matchless clans, which we typically denote by $\tau$. If $\tau=++--+-++$ (as in Section 1) then $u(\tau)=45126378\in S_8$ (in one-line notation) and $v(\tau)=12673845$. The discussion that follows freely uses facts about Schubert varieties, flag varieties and Schubert polynomials. Material on Schubert varieties and flag varieties may be found in Chapters 9 and 10 of [@Fulton]. Material about Schubert polynomials appears in Chapter 10.4 of *loc. cit* as well as Chapter 2 of [@Manivel]. Let $X=\{x_1,x_2,\ldots,x_n\}$ and $Y=\{y_1,y_2,\ldots,y_n\}$ be independent and commuting indeterminates. The $\beta$-[**double Schubert polynomial**]{} ${\mathfrak S}^{(\beta)}_w(X;Y)$ is defined by setting $${\mathfrak S}^{(\beta)}_{w_0}(X;Y)=\prod_{i=1}^{n-1}\prod_{j=1}^{n-i}(x_i-y_j+\beta x_i y_j)$$ where $w_0$ is the long element of $S_n$. Define $\partial^{(\beta)}_i$ by $$\partial^{(\beta)}_i(f)=\partial_i((1-\beta x_{i+1})f).$$ Now, if $i$ is any position such that $w(i)<w(i+1)$ then $${\mathfrak S}^{(\beta)}_w(X;Y)=\partial^{(\beta)}_i{\mathfrak S}^{(\beta)}_{ws_i}(X;Y)$$ where $s_i$ is the simple reflection transposing $i$ and $i+1$. Recall that $${\mathfrak S}_{w}(X;Y)={\mathfrak S}_{w}^{(0)}(X,Y)$$ is the [**double Schubert polynomial**]{} and $${\mathfrak S}_w(X)={\mathfrak S}_w^{(0)}(X;0)$$ is the [**single Schubert polynomial**]{}. Also, $${\mathfrak G}_w(X;Y)={\mathfrak S}^{(1)}_w\left(x_i\mapsto 1-x_i;y_j\mapsto \frac{1-y_j}{y_j}\right)$$ is the [**double Grothendieck polynomial**]{} ${\mathfrak G}_w(X;Y)$ and finally, $${\mathfrak G}_w(X)={\mathfrak G}_w(X;y_j\mapsto 1)$$ is the [**single Grothendieck polynomial**]{}. The use of a deformation parameter $\beta$ in Schubert polynomial theory is found in [@Fomin.Kirillov:groth]. Below we remind the reader in what sense the above substitutions give representatives of the Schubert classes. When $\tau$ is matchless, define $$\label{eqn:matchlessdef} \Upsilon^{(\beta)}_\tau(X,Y)={\mathfrak S}^{(\beta)}_{u(\tau)}(X;y_n,y_{n-1},\ldots,y_1)\cdot {\mathfrak S}^{(\beta)}_{v(\tau)}(X;Y).$$ For clans $\gamma$ which are not matchless, $\Upsilon_{\gamma}$ will be defined using divided difference operators according to the weak order on $K$-orbits, which we now define. Geometrically, we say that an orbit closure $Y_{\gamma}$ covers another orbit closure $Y_{\gamma'}$, and write $\gamma = s_i \cdot \gamma'$, if $$Y_{\gamma} = \pi_i^{-1}(\pi_i(Y_{\gamma'})),$$ where $\pi_i: G/B \rightarrow G/P_{s_i}$ is the natural projection. Here, $P_{s_i}$ is the standard minimal parabolic subgroup $B \cup Bs_iB$ of $G$. Note that this definition makes sense not only in our current example, but in any situation where we are dealing with varieties $Y$ which are closures of orbits of a spherical subgroup acting on $G/B$. Indeed, this is the appropriate definition of weak order in all such settings. In our example, the weak order has the following combinatorial description [@Matsuki; @Yamamoto]. The [**weak Bruhat order**]{} on ${\tt Clans}_{p,q}$ is the transitive closure of the covering relation $s_i \cdot \gamma \succ \gamma=(c_1,\ldots,c_n)$ if either: - $s_i \cdot \gamma=(\ldots,c_{i+1},c_i,\ldots)$ and - $c_i$ is a sign and $c_{i+1}$ is the end of an arc matching with a vertex to its right; - $c_i$ is the end of an arc matching with a vertex to its left and $c_{i+1}$ is a sign; or - $c_i$ and $c_{i+1}$ are endpoints of different arcs, and the mate of $c_i$ is left of the mate of $c_{i+1}$ - $s_i \cdot \gamma$ is obtained from $\gamma$ by replacing $c_i=\pm$ and $c_{i+1}=\mp$ by an arc. If $\gamma$ is not matchless, it follows from [@Richardson-Springer Theorem 4.6] that there is a matchless clan $\tau$ and a sequence of the form $$\gamma = s_1 \cdot s_2 \cdots s_l \cdot \tau.$$ (Here, $l = l(\gamma)$ in the notation of Section 1.) In this event, let $$\Upsilon^{(\beta)}_\gamma(X;Y) = \partial_1^{(\beta)} \hdots \partial_l^{(\beta)} \Upsilon^{(\beta)}_{\tau}(X;Y).$$ Just as representatives of Schubert classes are specializations of ${\mathfrak S}^{(\beta)}_w(X;Y)$, we will see that the same specializations of $\Upsilon^{(\beta)}(X;Y)$ give representatives of the classes of $Y_{\gamma}$’s: $$\begin{aligned} \nonumber \Upsilon_\gamma(X;Y) & := & \Upsilon^{(0)}_{\gamma}(X;Y)\\ \nonumber \Upsilon_{\gamma}(X) & := & \Upsilon_{\gamma}(X;0)\\ \nonumber \Upsilon^K_\gamma(X;Y) & := & \Upsilon^{(1)}_{\gamma}\left(x_i\mapsto 1-x_i;y_j\mapsto \frac{1-y_j}{y_j}\right)\\ \nonumber \Upsilon^K_{\gamma}(X) & := & \Upsilon^K_{\gamma}(X;y_j\mapsto 1)\nonumber\end{aligned}$$ Some combinatorial properties of $\Upsilon^{(\beta)}_{\gamma}$ -------------------------------------------------------------- We assume familarity with standard permutation combinatorics such as the Rothe diagram, essential set, code of a permutation and pattern avoidance; see, e.g., [@Manivel Sections 2.1-2.2]. A permutation is [**vexillary**]{} if it is $2143$-avoiding. \[lemma:isvex\] If $\gamma$ is non-crossing, then $u(\gamma)$ and $v(\gamma)$ are vexillary permutations. In addition $u(\gamma)$ and $v(\gamma)$ are inverse to Grassmannian permutations with descents at $q$ and $p$ respectively. Consider $u:=u(\gamma)$ and suppose $i_1<i_2<i_3<i_4$ where $u(i_1),u(i_2),u(i_3),u(i_4)$ are in the relative order $2143$. Then since $1,2,\ldots,q$ and $q+1,q+2,\ldots,p+q$ form rising sequences in $u$, $\gamma(i_1),\gamma(i_3)\in \{q+1,q+2,\ldots,p+q\}$ and $\gamma(i_2),\gamma(i_4)\in \{1,2,\ldots,q\}$. Hence $\gamma(i_1)>\gamma(i_4)$, a contradiction. Thus $u$ is vexillary. It is straightforward to see that the essential set of $u$ (provided $u$ is not the identity) must all lie in column $q$. This is equivalent to the inverse Grassmannian claim. The arguments for $v(\gamma)$ are similar. Continuing Example \[exa:nonmatching\], where $\tau=++--+-++$, the diagrams of $u(\tau)$ and $v(\tau)$ are given below. (The $\bullet$’s of $D(\pi)$ are in positions $(i,\pi(i))$.) $$\begin{picture}(330,96) \put(0,44){$D(u(\tau))=$} \put(60,0){\makebox[0pt][l]{\framebox(96,96)}} \thicklines \put(60,72){\makebox[0pt][l]{\framebox(36,24)}} \put(72,96){\line(0,-1){24}} \put(84,96){\line(0,-1){24}} \put(60,84){\line(1,0){36}} \put(84,36){\makebox[0pt][l]{\framebox(12,12)}} \thinlines \put(102,90){\circle*{4}} \put(102,90){\line(1,0){54}} \put(102,90){\line(0,-1){90}} \put(114,78){\circle*{4}} \put(114,78){\line(1,0){42}} \put(114,78){\line(0,-1){78}} \put(66,66){\circle*{4}} \put(66,66){\line(1,0){90}} \put(66,66){\line(0,-1){66}} \put(78,54){\circle*{4}} \put(78,54){\line(1,0){78}} \put(78,54){\line(0,-1){54}} \put(126,42){\circle*{4}} \put(126,42){\line(1,0){30}} \put(126,42){\line(0,-1){42}} \put(90,30){\circle*{4}} \put(90,30){\line(1,0){66}} \put(90,30){\line(0,-1){30}} \put(138,18){\circle*{4}} \put(138,18){\line(1,0){18}} \put(138,18){\line(0,-1){18}} \put(150,6){\circle*{4}} \put(150,6){\line(1,0){6}} \put(150,6){\line(0,-1){6}} \put(170,44){$D(v(\tau))=$} \put(230,0){\makebox[0pt][l]{\framebox(96,96)}} \thicklines \put(254,48){\makebox[0pt][l]{\framebox(36,24)}} \put(266,72){\line(0,-1){24}} \put(278,72){\line(0,-1){24}} \put(254,60){\line(1,0){36}} \put(266,24){\makebox[0pt][l]{\framebox(24,12)}} \put(278,24){\line(0,1){12}} \thinlines \put(236,90){\circle*{4}} \put(236,90){\line(1,0){90}} \put(236,90){\line(0,-1){90}} \put(248,78){\circle*{4}} \put(248,78){\line(1,0){78}} \put(248,78){\line(0,-1){78}} \put(296,66){\circle*{4}} \put(296,66){\line(1,0){30}} \put(296,66){\line(0,-1){66}} \put(308,54){\circle*{4}} \put(308,54){\line(1,0){18}} \put(308,54){\line(0,-1){54}} \put(260,42){\circle*{4}} \put(260,42){\line(1,0){66}} \put(260,42){\line(0,-1){42}} \put(320,30){\circle*{4}} \put(320,30){\line(1,0){6}} \put(320,30){\line(0,-1){30}} \put(272,18){\circle*{4}} \put(272,18){\line(1,0){54}} \put(272,18){\line(0,-1){18}} \put(284,6){\circle*{4}} \put(284,6){\line(1,0){42}} \put(284,6){\line(0,-1){6}} \end{picture}$$ The essential set boxes of $u(\tau)$ all lie in column $q=3$ while the essential set boxes of $v(\tau)$ lie in column $p=5$, in agreement with Lemma \[lemma:isvex\]. We now define [**pipe diagrams**]{} associated to $u(\gamma)$ for non-crossing $\gamma$. (The nomenclature alludes to the “pipe dreams” terminology of [@Knutson.Miller:annals].) To start, replace each box of $D(u(\gamma))$ by a $+$. The result is one of the pipe diagrams. All other pipe diagrams are obtained from this first one by iterating the use of the local operation $$\label{eqn:transformsconj} \begin{matrix} \cdot & \cdot\\ \cdot & + \end{matrix} \ \mapsto \ \begin{matrix} + & \cdot\\ \cdot & \cdot \end{matrix}$$ with the additional restriction that no $+$’s appear in columns $q+1,q+2,\ldots,n$. The collection of all such pipe diagrams is denoted ${\tt Pipe}(u(\gamma))$. We define ${\tt Pipe}(v(\gamma))$ in the same way but using $D(v(\gamma))$ and requiring that there are no $+$’s in columns $p+1,p+2,\ldots,n$. In addition, given any configuration ${\mathcal P}$ of $+$’s in the $n\times n$ grid define $${\tt wt}^{(\beta)}({\mathcal P})=\prod_{\tiny{\mbox{$+$ in position $(i,j)$}}} x_i-y_j+\beta x_i y_j.$$ We now explain why the initial conditions (\[eqn:matchlessdef\]) defining $\Upsilon_{\tau}(X)$ agree with the ones from Section 1. Actually, we have an extension. For $\gamma$ non-crossing, let $\tau^-$ be the matchless clan obtained by replacing each left end of an arc by $-$ and any right end of an arc by $+$. Also, let $\tau^+$ be the matchless clan obtained by replacing each left end of an arc by $+$ and each right end of an arc by $-$. Define $\lambda(\gamma)$ to be $\lambda(\tau^-)$, and $\lambda(\widehat\gamma)$ to be $\lambda(\widehat\tau^+)$, in the notation of the introduction. Define also flaggings ${\vec f}(\gamma)$ and ${\vec f}(\widehat \gamma)$ to be ${\vec f}(\tau^-)$ and ${\vec f}(\widehat\tau^+)$, respectively. The following result is straightforward from the results of [@KMY Section 5] (see specifically Theorem 5.8) and the definitions of $u(\gamma)$, $v(\gamma)$, $\lambda(\gamma)$, and $\lambda(\widehat\gamma)$: \[prop:KMY\] For non-crossing $\gamma$ we have $${\mathfrak S}^{(\beta)}_{u(\gamma)}(X;Y) = \sum_{{\mathcal P}\in {\tt Pipe}(u(\gamma))} {\tt wt}^{(\beta)}({\mathcal P}) \mbox{ \ and \ } {\mathfrak S}^{(\beta)}_{v(\gamma)}(X;Y) = \sum_{{\mathcal P}\in {\tt Pipe}(v(\gamma))} {\tt wt}^{(\beta)}({\mathcal P}).$$ There is a (weight preserving) bijection between ${\tt Pipe}(u(\gamma))$ and semistandard set-valued Young tableaux of shape $\lambda(\gamma)$ with flagging ${\vec f}(\gamma)$. The same holds for ${\tt Pipe}(v(\gamma))$ and semistandard set-valued Young tableaux of shape $\lambda(\widehat\gamma)$ with flagging ${\vec f}(\widehat\gamma)$. In particular, $${\mathfrak S}_{u(\gamma)}(X)=s_{\lambda(\gamma),{\vec f}(\gamma)}(X) \mbox{ \ \ and \ \ } {\mathfrak S}_{v(\gamma)}(X)=s_{\lambda(\widehat\gamma),{\vec f}(\widehat\gamma)}(X).$$ \[prop:Snonly\] Suppose $\gamma$ is non-crossing and $${\mathfrak S}^{(\beta)}_{u(\gamma)}(X;y_n,y_{n-1},\ldots,y_1){\mathfrak S}^{(\beta)}_{v(\gamma)}(X;Y)=\sum_{\kappa\in {\mathbb Z}_{\geq 0}^{\infty}} c^{(\beta)}_{\kappa}(Y){\bf x}^{\kappa},$$ where ${\bf x}^{\kappa}=x_1^{\kappa_1}x_2^{\kappa_2}\cdots$ and $c^{(\beta)}_{\kappa}(Y)\in {\mathbb Z}[\beta][Y]$. Then $c^{(\beta)}_{\kappa}(Y)=0$ unless $\kappa\leq (n-1,n-2,\ldots,3,2,1,0,0,0,\ldots )$ (component-wise comparison). Let us first show: \[claim:staircase\] If ${\bf x}^{\kappa}$ appears in ${\mathfrak S}_{u(\gamma)}(X){\mathfrak S}_{v(\gamma)}(X)$ then $\kappa\subseteq (n-1,n-2,\ldots,2,1,0)\in {\mathbb Z}_{\geq 0}^n$. Suppose ${\bf x}^\kappa=\cdots x_i^m \cdots$. Let $\omega$ be the width of the first non-empty row of $D(u(\gamma))$ that occurs in some row $s\geq i$ of $n\times n$. Let $\omega'$ be the width of the first nonempty row of $D(v(\gamma))$ that occurs in some row $t\geq i$ of $n\times n$. It is easy to see from the definitions that $$m\leq \omega+\omega'.$$ We may assume without loss that $s$ and $t$ exist and also $t\geq s$ (the alternate cases are proved similarly). Let $A$ be the number of $-$’s or left ends of an arc occuring in the leftmost $s$ positions of $\gamma$. Let $B$ be the number of $+$’s or left ends of an arc occuring in the leftmost $t$ positions of $\gamma$. Now $$\omega=q-A \mbox{ \ \ \ and \ \ \ } \omega'=p-B.$$ Since $$\omega+\omega'=p+q-A-B$$ it suffices to show $A+B\geq i$. Now, because in any left initial segment of $\gamma$, the number of right ends of an arc is at most the number of left ends of an arc, we have: $$\begin{aligned} \nonumber A+B & \geq & A+\#\{\mbox{$+$ or left end of an arc in first $s$ positions of $\gamma$}\}\\ \nonumber &\geq & A+\#\{\mbox{$+$ or right end of an arc in first $s$ positions of $\gamma$}\}\\ \nonumber & = & s \ \ \geq i,\nonumber\end{aligned}$$ as desired. Suppose the proposition is not true and there are set-valued tableaux $T$ and $U$ that contribute to ${\mathfrak S}^{(\beta)}_{u(\gamma)}(X;y_n,y_{n-1},\ldots,y_1)$ and ${\mathfrak S}^{(\beta)}_{v(\gamma)}(X;Y)$ respectively (under the bijection of Proposition \[prop:KMY\]) such that the number of $i$’s in $T$ and $U$ combined strictly exceeds $n-i$, for some $i$. Now let $T'$ be the ordinary tableau that picks each of those $i$’s as the representative of its box and picks any entry from the remaining boxes. Since $T$ is semistandard, $T'$ is semistandard as well and contributes to ${\mathfrak S}_{u(\gamma)}(X)$. Similarly, define $U'$, contributing to ${\mathfrak S}_{v(\gamma)}(X)$. Then in ${\mathfrak S}_{u(\gamma)}(X){\mathfrak S}_{v(\gamma)}(X)$ the monomial ${\bf x}^{T'}{\bf x}^{U'}$ appears, contradicting Claim \[claim:staircase\]. It is well known (see, e.g., [@Manivel Proposition 2.5.4]) that the single Schubert polynomials $\{{\mathfrak S}_w(X): w\in S_{n}\}$ form a ${\mathbb Z}$-linear basis of the vector space $\Gamma(X)$ of polynomials in $X$ using only monomials ${\bf x}^{\kappa}$ where $\kappa\leq (n-1,n-2,\ldots,3,2,1)$. Now, ${\mathfrak G}_w(X)$ has the same lead term as ${\mathfrak S}_w(X)$ under the reverse lexicographic order, namely ${\bf x}^{{\tt code}(w)}$. In addition, it is known (from [@Fomin.Kirillov:groth]) that ${\mathfrak G}_{w}(X)\in \Gamma(X)$. Thus $\{{\mathfrak G}_w(X):w\in S_n\}$ also forms a basis of $\Gamma(X)$. Similarly, $\{{\mathfrak S}^{(\beta)}_w(X;Y): w\in S_n\}$ is a ${\mathbb Z}[\beta][Y]$-module basis of ${\mathbb Z}[\beta][Y]\otimes_{\mathbb Z} \Gamma(X)$. This is since ${\mathfrak S}^{(\beta)}_w(X;Y)$ also has leading term of ${\bf x}^{{\tt code}(w)}$ and if any term $c^{(\beta)}_{\kappa}(Y){\bf x}^{\kappa}$ is any monomial then $\kappa\leq (n-1,n-2,\ldots,2,1,0)$. Therefore, by Proposition \[prop:Snonly\], when $\gamma$ is matchless $$\Upsilon^{(\beta)}_{\gamma}(X;Y)=\sum_{w\in S_n} c^{(\beta)}_{\gamma,w}(Y){\mathfrak S}^{(\beta)}_{w}(X;Y).$$ Since $\partial^{(\beta)}_i$ sends $\beta$-Schubert polynomials to $\beta$-Schubert polynomials (or zero), such an expression where the summation is over $S_n$ holds for all clans. Given a clan $\gamma$ let $-\gamma$ be the clan where the $+$’s of $\gamma$ are replaced by $-$’s and the $-$’s are replaced by $+$ (the arcs remain as is). We record the following property: Let $\gamma\in {\tt Clan}_{p,q}$. Then $$\Upsilon_{-\gamma}^{(\beta)}(X;Y)=\Upsilon_{\gamma}^{(\beta)}(X;y_n,y_{n-1},\ldots,y_2,y_1).$$ Let $\tau$ be a matchless clan such that $$\Upsilon_{\gamma}^{(\beta)}(X;Y)=\partial_{i_m}^{(\beta)}\cdots \partial_{i_1}^{(\beta)} \Upsilon_{\tau}^{(\beta)}(X;Y),$$ for some chain in weak Bruhat order from $\tau$ to $\gamma$ defined by $i_1,\ldots,i_m$. Now we are done since the same sequence defines a chain from $-\tau$ to $-\gamma$ and because the proposition is clear from the definitions for matchless $\tau$. In the ordinary cohomology, there is a further sense in which the choice of $\Upsilon_{\gamma}$ is simple. Consider the degree lexicographic term order on polynomials in ${\mathbb Q}[x_1,\ldots,x_n]$. The *Gröbner normal form* is a distinguished representative of any coset modulo $I^{S_n}$. The Schubert polynomials ${\mathfrak S}_w$ for $w\in S_n$ are the normal forms for their cosets; this is a fact due to [@FGP Section 12.1]. Thus any linear combination of these Schubert polynomials is the normal form for its coset modulo $I^{S_n}$. Concluding: \[cor:normalform\] $\Upsilon_{\gamma}(X)$ is the Gröbner normal form representative for the class of $[Y_{\gamma}]$ under the degree lexicographic term order. In other words, it is the unique representative that is a linear combination of $\{{\mathfrak S}_w:w\in S_n\}$. Representatives in the Borel models ----------------------------------- We first explain our proof for equivariant cohomology (the argument in equivariant $K$-theory is completely analogous). Let $T\subset GL_p\times GL_q$ be the torus of invertible diagonal matrices. Since each $Y_{\gamma}$ is $T$-stable, it admits a class $[Y_{\gamma}]_T\in H^\star_T(GL_n/B)$, a module over $H^\star_T(pt)\cong {\mathbb Z}[y_1,\ldots,y_n]$. The Borel-type model is $$\label{eqn:equivBorel} H^\star_T(GL_n/B)\cong {\mathbb Q}[X;Y]/J,$$ where $J$ is the ideal generated by $e_i(X)-e_i(Y)$ and $e_i(X)$ is the elementary symmetric function in $X$, etc. \[thm:equivver\] $\Upsilon_{\gamma}(X;Y)$ is well-defined and represents the coset of $[Y_{\gamma}]_T$ under (\[eqn:equivBorel\]). (The forgetful map from $H^\star_T(GL_n/B)\twoheadrightarrow H^{\star}(GL_n/B)$ in this context amounts to setting each $y_i=0$ and sends $[Y_{\gamma}]_T$ to $[Y_{\gamma}]$. Thus Theorem \[thm:intro\] follows from Theorem \[thm:equivver\] since the forgetful maps and the Borel isomorphisms commute.) The following is essentially standard. We include a proof for sake of completeness. \[prop:tworeps\] Suppose $f_1(X;Y)$ and $f_2(X;Y)$ are representatives of $[Y_{\gamma}]_T$ such that $$f_1(X;Y)=\sum_{w\in S_n} a_w(Y){\mathfrak S}_w(X;Y) \mbox{ \ \ and \ \ } f_2(X;Y)=\sum_{w\in S_n} b_w(Y){\mathfrak S}_w(X;Y).$$ Then $f_1(X;Y)=f_2(X;Y)$. We need that $a_w(Y)=b_w(Y)$ for all $w \in S_n$. Since $f_1$ and $f_2$ are equivariant cohomology class representatives of $[Y_\gamma]$ any substitution of $X$ by a permutation $Y_{\sigma}=(y_{\sigma(1)},\ldots,y_{\sigma(n)})$ gives $f_1(Y_{\sigma};Y)=f_2(Y_{\sigma};Y)$ (this is where we need that $\sigma\in S_n$). This follows from the localization theorem for equivariant cohomology, combined with the fact that restriction to the $T$-fixed point $\sigma$ is given by $$[Y_{\gamma}]_T|_{\sigma}= f_1(Y_{\sigma}; Y)=f_2(Y_{\sigma};Y).$$ These are standard facts, but the reader seeking a reference may consult [@Wyser-13b Section 1.2] for an expository treatment. Also, $$\label{eqn:vanishing} {\mathfrak S}_w(Y_\sigma;Y)=0 \mbox{\ if $\sigma\not\geq w$ in strong Bruhat order.}$$ Now, pick any linear extension $$\pi^{(1)}=id,\pi^{(2)},\ldots,\pi^{(n!)}=w_0$$ of Bruhat order. Hence $$a_{\pi^{(1)}}(Y){\mathfrak S}_{\pi^{(1)}}(Y_{\pi^{(1)}};Y)=f_1(Y_{\pi^{(1)}};Y)=f_2(Y_{\pi^{(1)}};Y)= b_{\pi^{(1)}}(Y){\mathfrak S}_{\pi^{(1)}}(Y_{\pi^{(1)}};Y).$$ Since ${\mathfrak S}_{w}(Y_{w};Y)\neq 0$, dividing we conclude $a_{\pi^{(1)}}(Y)=b_{\pi^{(1)}}(Y)$. Now set $$f'_1(X;Y)=f_1(X;Y)-a_{\pi^{(1)}}(Y){\mathfrak S}_{\pi^{(1)}}(X;Y),$$ and $$f'_2(X;Y)=f_2(X;Y)-a_{\pi^{(1)}}(Y){\mathfrak S}_{\pi^{(1)}}(X;Y).$$ Thus $$a_{\pi^{(2)}}(Y){\mathfrak S}_{\pi^{(2)}}(Y_{\pi^{(2)}};Y)=f'_1(Y_{\pi^{(2)}};Y)=f'_2(Y_{\pi^{(2)}};Y)= b_{\pi^{(2)}}(Y){\mathfrak S}_{\pi^{(2)}}(Y_{\pi^{(2)}};Y),$$ and so $a_{\pi^{(2)}}(Y)=b_{\pi^{(2)}}(Y)$. Repeating, set $$f''_1(X;Y)=f'_1(X;Y)-a_{\pi^{(2)}}(Y){\mathfrak S}_{\pi^{(2)}}(X;Y),$$ and $$f''_2(X;Y)=f'_2(X;Y)-a_{\pi^{(2)}}(Y){\mathfrak S}_{\pi^{(2)}}(X;Y).$$ In this manner, we conclude all $n!$ desired equalities. We will establish the assumption of the following claim at the end of this section, and in a different way, in the next section. Assuming $\Upsilon_{\tau}(X;Y)$ represents $[Y_{\tau}]_T$ when $\tau$ is matchless, then $\{\Upsilon_{\gamma}(X;Y)\}$ is self-consistent. Pick a (non-matchless) clan $\\gamma$ and suppose there are two matchless clans $\tau_1$ and $\tau_2$ (possibly with $\tau_1=\tau_2$) such that $$[Y_{\gamma}]_T=\partial_{i_m}\cdots\partial_{i_1}[Y_{\tau_1}]_T \mbox{\ \ and \ \ } [Y_{\gamma}]_T=\partial_{j_m}\cdots\partial_{j_1}[Y_{\tau_2}]_T;$$ where we have mildly abused $\partial_i$ to mean the geometrically defined (equivariant) push-pull operator on classes. We need to establish the *polynomial* equality: $$\partial_{i_m}\cdots\partial_{i_1}\Upsilon_{\tau_1}(X;Y)= \partial_{j_m}\cdots\partial_{j_1}\Upsilon_{\tau_2}(X;Y).$$ Since we know $\Upsilon_{\tau_1}(X;Y)$ and $\Upsilon_{\tau_2}(X;Y)$ expand into double Schubert polynomials (from $S_n$), the claim follows from Proposition \[prop:tworeps\]. Following [@Knutson.Miller:annals Section 2.3], the $K$-cohomology ring $K^{\circ}(GL_n/B)$ has the presentation $$K^{\circ}(GL_n/B)\cong {\mathbb Z}[x_1,\ldots,x_n]/K$$ where $K$ is the ideal generated by $e_d(x_1,\ldots,x_n)-{n\choose d}$ for $d\leq n$; here $e_d(x_1,\ldots,x_n)$ is the elementary symmetric function of degree $d$. Next, following [@Fulton.Lascoux] if we let $K^\circ_T(GL_n/B)$ denote the $T$-equivariant $K$-theory ring of $GL_n/B$ then $$K^\circ_T(GL_n/B)=K^\circ_T(pt)[x_1,\ldots,x_n]/J\cong {\mathbb Z}[y_1^{\pm 1},\ldots y_n^{\pm 1}][x_1,\ldots,x_n]/J,$$ where $J$ is as in (\[eqn:equivBorel\]). In these senses, one can speak of a (Laurent) polynomial “representing” the class of a structure sheaf of a ($T$-stable) variety in $GL_n/B$. \[thm:K\] The families $\{\Upsilon^K_{\gamma}(X)\}$ and $\{\Upsilon^K_{\gamma}(X;Y)\}$ are well-defined. Moreover, $\Upsilon^K_{\gamma}(X)$ represents $[{\mathcal O}_{Y_{\gamma}}]\in K^{\circ}(GL_n/B)$ and $\Upsilon^K_{\gamma}(X;Y)$ represents $[{\mathcal O}_{Y_{\gamma}}]_T \in K^{\circ}_T(GL_n/B)$. The proof is exactly the same as in equivariant cohomology, except one must use equivariant $K$-theory localization. This requires the now standard fact that, in equivariant $K$-theory, $[{\mathcal O}_{Y_{\gamma}}]_T|_{\sigma} = f(Y_{\sigma}; Y)$ when $f(X;Y)$ is a representative of $[{\mathcal O}_{Y_{\gamma}}]_T$ in the Borel model. We are unaware of a specific reference for it in the literature, so we remark here that the argument of [@Wyser-13b Proposition 1.3] can be modified to apply to $K$-theory simply by replacing the first Chern classes of the tautological line bundles by (the classes of) the bundles themselves. A recent reference for equivariant localization in $K$-theory is [@Harada]. The analogues of the vanishing conditions on Schubert classes (\[eqn:vanishing\]) also hold. One also needs the following, which should also be straightforward to experts, but for which we are also not aware of a proof in the literature: The isobaric divided difference operator $\pi_i=\partial^{(1)}_i$ takes a representative of the class of $Y_{\gamma}$ to one for $Y_{s_i\gamma}$ in (equivariant) $K$-theory of $GL_n/B$. Let $Y = Y_{\gamma}$, and $Y' = Y_{s_i \gamma}$. First, recall that all orbit closures for this case are multiplicity-free, meaning that their cycle classes in the Chow ring can be expressed in the Schubert basis with all coefficients $0$ or $1$. This is noted in [@Brion] and further elaborated upon in [@Wyser12]. Thus by [@Brion Theorem 6], $Y$ has rational singularities. Let $\pi: G/B \rightarrow G/P_{{\alpha}_i}$ be the natural projection where $P_{\alpha_i}$ is the minimal parabolic associated to $\alpha_i$. Since $Y' = \pi^{-1}(\pi(Y))$ is a ${\mathbb P}^1$-bundle over $\pi(Y)$, and since $Y'$ has rational singularities (being another multiplicity-free $K$-orbit closure), $\pi(Y)$ has rational singularities as well. Now we note that the proof of [@KK Lemma 4.12] or [@Fulton.Lascoux Theorem 3], given there for (equivariant) $K$-classes of Schubert varieties, applies to the case at hand. *Conclusion of proof of Theorems \[thm:intro\], \[thm:equivver\] and \[thm:K\]:* It remains to show that the proposed representatives are indeed representatives for the closed orbits. This follows from three facts. First, by [@Wyser12], when $\gamma$ is non-crossing, $$Y_{\gamma}=X_{v(\gamma)}^{w_0u(\gamma)}:={\overline{B_{-}v(\gamma)B/B}}\cap {\overline{Bw_0 u(\gamma)B/B}}.$$ Second, in the case of equivariant $K$-theory, the representative of the Schubert variety $X_{w}$ is ${\mathfrak G}_w(X;Y)$; this is proved in [@Fulton.Lascoux Theorem 3]. It also follows from *loc. cit.* that ${\mathfrak G}_{w_0w}(X;y_n,y_{n-1},\ldots,y_1)$ represents the opposite Schubert variety $X^w={\overline{BwB/B}}$. Similarly, it is known that ${\mathfrak S}_w(X)$, ${\mathfrak S}_w(X;Y)$ and ${\mathfrak G}_w(X)$ represent the Schubert classes in the corresponding cohomology theories, and ${\mathfrak S}_{w_0w}(X)$, ${\mathfrak S}_{w_0w}(X;y_n,\ldots,y_1)$ and ${\mathfrak G}_{w_0w}(X)$ represent the opposite Schubert classes. Finally, $[X_u^v]=[X_u][X^v]$ (interpreted in any of the cohomology theories we are using). The argument of the introduction that $\Upsilon_{\gamma}(X)\in {\mathbb Z}_{\geq 0}[x_1,\ldots,x_n]$ extends to prove appropriate notions of “positivity” for each of the given representatives associated to the other three cohomology theories. This is since in each case there is an available notion of positivity of Schubert calculus. See [@AGM] and the references therein. Consider $X_{2143}^{3412}$. It is true that ${\mathfrak S}_{2143}(x_1,x_2,x_3,x_4)^2=x_1^4+\ldots$ represents the class of the Richardson variety. However, this polynomial is not the normal form representative of its coset because it involves $x_1^4$ (cf. Proposition \[cor:normalform\] and see also [@Lenart.Sottile]). This emphasizes the role of Proposition \[prop:Snonly\] in our proofs. Our arguments show that if any collection of varieties in $GL_n/B$ have their classes related by (isobaric) divided difference operators then any choice of polynomial representatives for their minimal elements that expand into Schubert polynomials from $S_n$ gives a self-consistent family of representatives. In particular this can also be applied to the cases where $(G,K)=(GL_{2n}, Sp_{2n})$ and $(G,K)=(GL_n,O_n)$; cf. [@Wyser-13b] and [@WyYo2]. The $K$-orbit determinantal ideal ================================= Geometric naturality of $\Upsilon_{\gamma}^{(\beta)}$ ----------------------------------------------------- The [**$K$-orbit determinantal ideal**]{} $I_\gamma$ is defined as follows. Fix $\gamma\in {\tt Clan}_{p,q}$. For $i=1,\hdots,n$, let: - $\gamma(i;+)$ = the total number of $+$’s and matchings in the first $i$ vertices, and - $\gamma(i;-)$ = the total number of $-$’s and matchings in the first $i$ vertices. For $1 \leq i < j \leq n$, define - $\gamma(i;j) = \#\{k \in [1,i] \mid \mbox{$k$ and $l$ are matched} \text{ and } l > j\}$. Let $R_{+}(\gamma)$ be the vector with $i$-th entry equal to $i+1-\gamma(i;+)$ and $R_{-}(\gamma)$ be the vector with $i$-th entry $i+1-\gamma(i;-)$. Also, let $W(\gamma)$ be the $n\times n$ matrix whose $(i,j)$-th entry is $j+\gamma(i;j)+1$ if $i<j$ and is zero otherwise. Identify ${\rm Fun}({\rm Mat}_{n\times n})$ with ${\mathbb C}[z_{i,j}]$ where $z_{i,j}$ is the coordinate function of matrix coordinate $(i,j)$. Let $M_n$ be the generic $n\times n$ matrix with entry $z_{i,j}$. Now define $I_\gamma$ to have the following generators: 1. For each $i = 1,\hdots,n$, the minors of size $R_{+}(\gamma)_i$ of the lower-left $q \times i$ submatrix of $M_n$. 2. For each $i = 1,\hdots,n$, the minors of size $R_{-}(\gamma)_i$ of the upper-left $p \times i$ submatrix of $M_n$. 3. For each $1 \leq i < j \leq n$, the minors of size $W(\gamma)_{ij}$ of the following $n \times (i+j)$ matrix $P_{i,j}$: The upper-left $p \times i$ block coincides with the upper-left $p \times i$ block of $M_n$, the lower-left $q \times i$ block is zero, and the last $j$ columns coincide with the first $j$ columns of $M_n$. Let $\gamma= \begin{picture}(30,10) \put(0,0){\epsfig{file=matchingsSection4.eps, height=.4cm}} \put(11,1){$+-$} \end{picture}$. Then $$\begin{array}{ccc} \gamma_+ & = & (0,1,2,2)\\ R_{+}(\gamma) & = & (2,{\underline 2},{\underline 2},3) \end{array} , \ \begin{array}{ccc} \gamma_{-} & = & (0,1,1,2)\\ R_{-}(\gamma) & = & (2,{\underline 2},3,3) \end{array}, \ W(\gamma)=\begin{tabular}{c|cccc} $i$/$j$ & $1$ & $2$ & $3$ & $4$ \\ \hline $1$ & \ & \underline{$3$} & \underline{$4$} & $5$\\ $2$ & \ & \ & {\underline{$4$}} & $5$\\ $3$ & \ & \ & \ & $5$ \\ $4$ & \ & \ & \ & \ \\ \end{tabular}$$ Not all rank conditions give rise to non-trivial minors; we have underlined those that do. Specifically $R_+$ demands that the $2\times 2$ minors of the southwest $2\times 3$ submatrix of $$M_4=\left(\begin{matrix} z_{11} & z_{12} & z_{13} & z_{14}\\ z_{21} & z_{22} & z_{23} & z_{24}\\ z_{31} & z_{32} & z_{33} & z_{34}\\ z_{41} & z_{42} & z_{43} & z_{44} \end{matrix}\right)$$ be among the generators. $R_{-}$ contributes the $2\times 2$ northwest minor of this matrix. Here, $$P_{1,2}=\left(\begin{matrix} z_{11} & z_{11} & z_{12} \\ z_{21} & z_{21} & z_{22} \\ 0 & z_{31} & z_{32} \\ 0 & z_{41} & z_{42} \end{matrix}\right), P_{1,3}=\left(\begin{matrix} z_{11} & z_{11} & z_{12} & z_{13}\\ z_{21} & z_{21} & z_{22} & z_{23}\\ 0 & z_{31} & z_{32} & z_{33}\\ 0 & z_{41} & z_{42} & z_{43} \end{matrix}\right), P_{2,3}=\left(\begin{matrix} z_{11} & z_{12} & z_{11} & z_{12} & z_{13}\\ z_{21} & z_{22} & z_{21} & z_{22} & z_{23}\\ 0 & 0 & z_{31} & z_{32} & z_{33}\\ 0 & 0 & z_{41} & z_{42} & z_{43} \end{matrix}\right).$$ The conditions from $W(\gamma)$ say that we add the $3\times 3$ minors of $P_{1,2}$ and the $4\times 4$ minors of $P_{1,3}$ and of $P_{2,3}$. Actually, the rank conditions from $R_{+}$ and $R_{-}$ already imply the minors from $W(\gamma)$. This is true for all non-crossing $\gamma$, as explained below. Our reference for combinatorial commutative algebra, specifically the notion of multigrading, multidegree and $K$-polynomial is [@Miller.Sturmfels Chapter 8] as well as the connection to equivariant cohomology. For brevity, we refer the reader to that textbook for basic definitions and notions. Let $\prec_{p,q}$ be the lexicographic term order on monomials in $\{z_{i,j}\}$ that orders the variables by reading the bottom $q$ rows, from left to right and from bottom to top, followed by the top $p$ rows from left to right and from top to bottom. The $T\times T$ action on $M_n$ restricts to an action on $M_{\gamma}=\overline{\pi^{-1}(Y_{\gamma})}$. The associated grading associated to multidegrees assigns the variable $z_{ij}$ the weight $x_j-y_i$. For $K$-polynomials, the grading assigns $z_{ij}$ the weight $1-\frac{x_j}{y_i}$. The following result explains the geometric naturality of our choices for representatives of the closed orbits. It also applies more generally to orbit closures indexed by non-crossing clans. \[claim:main\] Suppose $\gamma$ is non-crossing. Then $M_{\gamma}$ is scheme-theoretically cut out by $I_{\gamma}$. Also: - The defining equations of $I_{\gamma}$ form a Gröbner basis with squarefree lead terms, with respect to the term order $\prec_{p,q}$. - The Gröbner limit ${\rm init}_{\prec_{p,q}}(I_{\gamma})$ has a prime decomposition whose components are naturally indexed by pairs of semistandard tableaux $(T,U)$ where - $T$ is a flagged tableaux of shape $\lambda(\gamma)$ with flagging ${\vec f}(\lambda(\gamma))$; and - $U$ is a flagged tableaux of shape $\mu(\gamma)$ with flagging ${\vec f}(\mu(\gamma))$. - ${\rm multidegree}_{{\mathbb Z}^{2n}}({\mathbb C}[Z]/I_{\gamma}) = \Upsilon_{\gamma}(X;Y)$ and ${\mathcal K}_{{\mathbb Z}^{2n}}({\mathbb C}[Z]/I_{\gamma}) = \Upsilon^K_{\gamma}(X;Y)$. Continuing our previous example, one checks that $$\begin{aligned} \nonumber {\rm init}_{\prec_{2,2}}(I_{\begin{picture}(30,10) \put(0,0){\epsfig{file=matchingsSection4.eps, height=.4cm}} \put(11,1){$+-$} \end{picture}})& =& \langle z_{42}z_{33}, z_{41}z_{33}, z_{41}z_{32}, z_{11}z_{22}\rangle\\ \nonumber \ & = & \langle z_{11}, z_{41},z_{42}\rangle \cap \langle z_{11},z_{41},z_{33}\rangle \cap \langle z_{11},z_{32}, z_{33}\rangle \cap \langle z_{22},z_{41},z_{42}\rangle \cap \\ \nonumber \ & \ & \langle z_{22},z_{41},z_{33}\rangle \cap \langle z_{22},z_{32},z_{33}\rangle. \nonumber\end{aligned}$$ Now consider the pipe diagrams associated to each prime component of ${\rm init}_{\prec_{2,2}}(I_{\begin{picture}(30,10) \put(0,0){\epsfig{file=matchingsSection4.eps, height=.4cm}} \put(11,1){$+-$} \end{picture}})$: we define them to be obtained by placing a “$+$” in position $(i,j)$ if $z_{ij}$ appears in the component. These are respectively: $$\left[\ \begin{matrix} + & . & . & . \\ . & . & . & . \\ \hline . & . & . & . \\ + & + & . & . \\ \end{matrix}\ \right], \left[\ \begin{matrix} + & . & . & . \\ . & . & . & . \\ \hline . & . & + & . \\ + & . & . & . \\ \end{matrix}\ \right], \left[\ \begin{matrix} + & . & . & . \\ . & . & . & . \\ \hline . & + & + & . \\ . & . & . & . \\ \end{matrix}\ \right], \left[\ \begin{matrix} . & . & . & . \\ . & + & . & . \\ \hline . & . & . & . \\ + & + & . & . \\ \end{matrix}\ \right],$$ $$\left[\ \begin{matrix} . & . & . & . \\ . & + & . & . \\ \hline . & . & + & . \\ + & . & . & . \\ \end{matrix}\ \right], \left[\ \begin{matrix} . & . & . & . \\ . & + & . & . \\ \hline . & + & + & . \\ . & . & . & . \\ \end{matrix}\ \right].$$ To compute the ${\mathbb Z}^{2n}$ multidegree one uses additive grading that assigns $z_{i,j}$ the weight $x_j-y_i$. Then $$\begin{aligned} \nonumber {\rm multidegree}_{{\mathbb Z}^{2n}}({\mathbb C}[Z]/I_{\gamma}) & = & (x_1-y_4)(x_2-y_4)\cdot (x_1-y_1) + (x_1-y_4)(x_3-y_3)\cdot (x_1-y_1) \\ \nonumber & & + (x_3-y_2)(x_3-y_3)\cdot (x_1-y_1) + (x_1-y_4)(x_1-y_3)\cdot (x_2-y_2) \\ \nonumber & & + (x_1-y_4)(x_3-y_3)\cdot (x_2-y_2) + (x_2-y_3)(x_3-y_3)\cdot (x_2-y_2) \nonumber\end{aligned}$$ In each term, we use “$\cdot$” to separate the factors coming from $+$’s below and above the horizontal line of the corresponding pipe diagram. Factoring gives $$\begin{aligned} \nonumber & [(x_1-y_4)(x_2-y_4)+(x_1-y_4)(x_2-y_3)+(x_2-y_3)(x_3-y_3)]\cdot [(x_1-y_1)+(x_2-y_2)]\\ \nonumber = & s_{(1,1),(2,3)}(x_1,x_2,x_3,x_4;y_4,y_3,y_2,y_1)s_{(1,0),(2,4)}(x_1,x_2,x_3,x_4;y_1,y_2,y_3,y_4)\nonumber\end{aligned}$$ in agreement with the theorem. One can also similarly verify the $K$-polynomial claim by computing the $K$-polynomial of the simplicial complex associated to ${\rm init}_{\prec_{p,q}}I_{\gamma}$. *Proof of Theorem \[claim:main\]:* We recall [@Wyser-13a Theorem 2.5]: Let $$E_p={\rm span}\{\vec e_1,\vec e_2,\ldots,\vec e_p\} \mbox{\ and \ } E^q={\rm span}\{\vec e_{p+1}, \vec e_{p+2},\ldots, \vec e_n\},$$ where $\vec e_i$ is the $i$-th standard basis vector of ${{{\mathbb}C}}^n$ and $\rho: {{{\mathbb}C}}^n \rightarrow E_p$ is the natural projection map. \[thm:wyserbruhat\] $Y_{\gamma}$ is the set of flags $F_{\bullet}$ such that the following three conditions hold: 1. $\dim(F_i \cap E_p) \geq \gamma(i;+)$ for all $i$; 2. $\dim(F_i \cap E^q) \geq \gamma(i;-)$ for all $i$; 3. $\dim(\rho(F_i) + F_j) \leq j + \gamma(i;j)$ for all $i<j$. Recall that $\pi:GL_n\to GL_n/B$ is the natural map. Consider the following diagram: $$\begin{CD} \pi^{-1}(Y_{\gamma}) @. \ \ \subset \ \ @. GL_n @. \subset {\rm Mat}_n\supset \overline{\pi^{-1}(Y_\gamma)}:=M_{\gamma}\subseteq V(I_\gamma)\\ @. @. @VV{\pi}V\\ Y_{\gamma} @.\ \subset \ \ @. GL_n/B @. \end{CD}$$ \[lemma:vanish\] Suppose $g\in GL_n$. Then $g\in \pi^{-1}(Y_{\gamma})$ if and only if $g$ vanishes on all generators (i), (ii) and (iii) of $I_{\gamma}$. We make the usual identification of $gB\in GL_n/B$ with the flag $$F_{\bullet}:\langle {\vec 0}\rangle\subset F_1\subset F_2\subset \ldots \subset F_{n-1}\subset {\mathbb C}^n,$$ where $F_i$ is spanned by the leftmost $i$ columns of $g$. Fix $F_{\bullet}\in Y_{\gamma}$. By Theorem \[thm:wyserbruhat\], the conditions (1), (2) and (3) of that theorem hold. We examine their implications on $g$: \(1) and (2): Consider the map $\phi:F_i\to E^q$ obtained by the projection of ${\vec v}\in F_i\subset {\mathbb C}^n$ onto ${\mathbb C}^n/E_p\cong E^q$. Since $\ker \phi = F_i\cap E_p$, by the rank-nullity theorem, (1) is equivalent to $${\rm rank} \ \phi = \dim F_i- \dim \ker \phi \leq i-\gamma(i,+).$$ Equivalently, the $g$ associated to $F_{\bullet}$ vanishes on the minors (i). Similarly, $F_{\bullet}$ satisfies (2) if and only if $g$ vanishes on the minors (ii). (3): $\rho(F_i) + F_j$ is isomorphic to the column space of the $n \times (i+j)$ matrix whose first $i$ columns coincide with the first $i$ columns of $g$, but with the lower-left $q \times i$ submatrix zeroed out, and whose next $j$ columns coincide with the first $j$ columns of $g$ (unaltered). Thus $g$ vanishes on the generators (iii) if and only if $F_{\bullet}$ satisfies (3). Since $I_{\gamma}$ vanishes on $\pi^{-1}(Y_{\gamma})$ we must have $M_{\gamma}:=\overline{\pi^{-1}(Y_{\gamma})}\subseteq V(I_{\gamma})$. We would know $M_{\gamma}=V(I_{\gamma})$ (as sets) if the latter is shown to be irreducible. Let ${\widetilde I}_{\gamma}$ be generated by the generators (i) and (ii). We will need to recall the following well-known and easy fact about Gröbner bases, stated in the specific form we need: \[lemma:easygrobner\] Let $A$ and $B$ be disjoint collections of commuting variables. Suppose $f_1,\ldots, f_n$ is a Gröbner basis of ${\Bbbk}[A]$ with respect to a pure lexicographic term order $\prec_A$, and that $g_1,\ldots,g_m$ is a Gröbner basis of ${\Bbbk}[B]$ with respect to a pure lexicographic term order $\prec_B$. Let $\prec_{A;B}$ be the pure lexicographic term order on ${\Bbbk}[A,B]$ extending $\prec_A$ and $\prec_B$ that favors $A$ over $B$. Then $G=\{f_1,\ldots,f_n, g_1,\ldots g_m\}$ is a Gröbner basis with respect to $\prec_{A,B}$. Indeed, if $S(f_i,g_j)$ is the $S$-polynomial then $$S(f_i,g_j):={\tt LT}(g_j)f_i - {\tt LT}(f_i)g_j=-(g_j-{\tt LT}(g_j))f_i+(f_i-{\tt LT}(f_i))g_j.$$ Thus, using the multivariate division algorithm, dividing $S(f_i,g_j)$ by $G$ (listed in the order $f_i,g_j,\ldots$) gives remainder $0$. Now apply Buchberger’s criterion [@Eisenbud Section 15.4]. \[claim:grobfortilde\] ${\widetilde I}_{\gamma}$ is a prime ideal that scheme-theoretically cuts out $\overline{\pi^{-1}(X_{u(\gamma)}^{v(\gamma)})}$. The generators form a Gröbner basis (with squarefree lead terms) with respect to $\prec_{p,q}$. By definition, ${\widetilde I}_\gamma$ is the ideal sum of a Schubert determinantal ideal associated to $v(\gamma)$ living in the first $p$ rows with a Schubert determinantal ideal associated to $u(\gamma)$ living in the bottom $q$ rows. The generators for each of these is individually Gröbner (with squarefree lead terms) for the term order given [@KMY Theorem 3.8]. Now the Gröbner assertion holds by Lemma \[lemma:easygrobner\]. Since $\pi^{-1}(X_{u(\gamma)}^{v(\gamma)})$ clearly vanishes on ${\widetilde I}_{\gamma}$ we have $\overline{\pi^{-1}(X_{u(\gamma)}^{v(\gamma)})}\subseteq V({\widetilde I}_{\gamma})$ (and both zero sets are of the same dimension). Since its generators are squarefree and Gröbner, by semicontinuity, ${\widetilde I}_{\gamma}$ is a radical ideal. On the other hand, $V({\widetilde I}_{\gamma})$ is clearly irreducible since it is the Cartesian product of two (irreducible) matrix Schubert varieties. Hence by the Nullstellensatz, ${\widetilde I}_{\gamma}$ is prime and so $\overline{\pi^{-1}(X_{u(\gamma)}^{v(\gamma)})}= V({\widetilde I}_{\gamma})$ (scheme-theoretic equality). Now we have $$\overline{\pi^{-1}(X_{u(\gamma)}^{v(\gamma)})}= V({\widetilde I}_{\gamma})\supseteq V(I_{\gamma})\supseteq M_{\gamma}.$$ However, by [@Wyser12] we know $Y_{\gamma}=X_{u(\gamma)}^{v(\gamma)}$ so $M_{\gamma}=\overline{\pi^{-1}(X_{u(\gamma)}^{v(\gamma)})}$ and hence $V({\widetilde I}_{\gamma})=V(I_{\gamma})$. Furthermore, by the Nullstellensatz, $I_\gamma\subseteq {\widetilde I}_{\gamma}(=\sqrt {\widetilde I}_{\gamma})$. However, by definition $I_{\gamma}\supseteq {\widetilde I}_{\gamma}$ and hence $I_{\gamma}={\widetilde I}_{\gamma}$. Thus (I) now follows by Claim \[claim:grobfortilde\] since the additional generators (with squarefree lead terms) that are in $I_{\gamma}$ but not ${\widetilde I}_{\gamma}$ do not affect Gröbnerness of the latter’s generators, for general reasons. Since $M_{\gamma}\subseteq V(I_\gamma)$ and now $I_{\gamma}={\widetilde I}_{\gamma}$ is prime, we must have $M_{\gamma}=V(I_{\gamma})$ and the first sentence of the theorem holds. In view of the equality $I_{\gamma}={\widetilde I}_{\gamma}$, (II) is now easy from [@KMY Section 4] since the latter is the ideal sum of two vexillary Schubert determinantal ideals. (Specifically note that our grading of $z_{ij}$ is transpose to the convention used in *loc. cit.*) Given (II), (III) follows from Proposition \[prop:KMY\] and the conclusion of our proof of the main theorems of Section 2. Alternatively, by the same line of reasoning as [@Knutson.Miller:annals Corollary 2.3.1], ${\rm multidegree}_{{\mathbb Z}^{2n}}({\mathbb C}[Z]/I_{\gamma})$ represents $[Y_{\gamma}]_T$. However, *a priori* this representative is not the same as $\Upsilon_{\gamma}(X;Y)$. That these are in fact equal follows from (II), Proposition \[prop:Snonly\] and Proposition \[prop:tworeps\]. The authors of *loc. cit.* in fact explain how their argument works in ordinary $K$-theory; cf. [@Knutson.Miller:annals Remarks 2.3.3 and 2.3.4]. Conjectures {#sec:conjectures} ----------- Some of the assertions of Theorem \[claim:main\] seem to hold generally. \[conj:A\] The generators of $I_{\gamma}$ are a Gröbner basis with respect to some lexicographic ordering. In particular, $I_{\gamma}$ is a radical ideal. We emphasize that the term order needed generally depends on $\gamma$. Conjecture \[conj:A\] has been verified exhaustively for $p+q\leq 6$ as well as in enough cases for larger $p+q$ for us to be convinced. When $(p,q)=(1,2),(2,1)$, all $\gamma$ are non-crossing. When $(p,q)=(2,2), (3,2)$, $\prec_{p,q}$ succeeds in making the defining generators of $I_{\gamma}$ Gröbner. This term order also succeeds for $(p,q)=(2,3)$ if one add some generators obtained by column operations on the $P_{i,j}$ matrices. The first interesting examples seem to be at $(p,q)=(3,3)$ where $$\begin{picture}(280,15) \put(0,0){\epsfig{file=matchingsSection3.eps, height=.8cm}} \put(13,0){$-$} \put(23,0){$+$} \put(49,0){$,$} \put(74,0){$+$} \put(84,0){$-$} \put(110,0){$,$} \put(176,0){$,$} \put(242,0){$,$} \put(139,0){$+$} \put(157,0){$-$} \put(272,0){$+$} \put(300,0){$-$} \end{picture}$$ are the instances where the defining generators (or the modification alluded to above) are not Gröbner with respect to $\prec_{p,q}$. \[conj:B\] Theorem \[claim:main\](III) holds for all $\gamma$. Equivalently, Conjecture \[conj:B\] claims that ${\rm multidegree}_{{\mathbb Z}^{2n}}({\mathbb C}[Z]/I_{\gamma})$ and ${\mathcal K}_{{\mathbb Z}^{2n}}({\mathbb C}[Z]/I_{\gamma})$ satisfy the divided difference and isobaric divided difference recurrences. This has been verified by exhaustive computer checks through $p+q=7$. We are not yet confident to assert that $I_\gamma$ is prime, although further discussion may appear elsewhere. These algebraic problems are closely related to two combinatorial questions: \[question:C\] Give a manifestly nonnegative combinatorial rule for the expansion of $\Upsilon^{(\beta)}_{\gamma}(X;Y)$ into monomials in $x_i-y_j+\beta x_i y_j$. \[question:D\] Give a manifestly nonnegative combinatorial rule for the expansion of $\Upsilon^{(\beta)}_{\gamma}(X;Y)$ into ${\mathfrak S}^{(\beta)}_{\gamma}(X;Y)$. Brion’s formula [@Brion] states: $$[Y_{\gamma}]=\sum_{w\in S_n} c_{\gamma,w}[X_w]\in H^{\star}(GL_n/B),$$ for explicit, combinatorially defined coefficients $c_{\gamma,w}\in\{0,1\}$. In view of Proposition \[prop:Snonly\], this formula implies a solution to Question \[question:D\] when $\beta=0$ and each $y_i=0$, by using any monomial expansion formula (e.g., [@Fomin.Kirillov; @BB]) for ${\mathfrak S}_{w}(X)$. A result of A. Knutson [@Knutson:multfree Theorem 3] shows how to obtain the $K$-theoretic expansion of a multiplicity-free subvariety (such as $Y_{\gamma}$) given the cohomology expansion. This provides answers to Questions 1 and 2 in ordinary $K$-theory. However, we are not aware of any formula (in ordinary cohomology or $K$-theory) that is geometrically natural from the perspective of Gröbner degenerations of $I_{\gamma}$. Question \[question:D\] in the case of $\Upsilon_{\gamma}(X;Y)$ for matchless $\gamma$ is equivalent to certain (yet unsolved) Schubert calculus problems. Once the matchless case is solved, a formula for the general case can be obtained by applying the operators $\partial_i$. These expansions involve coefficients in ${\mathbb Z}_{\geq 0}[y_{2}-y_1,\ldots,y_n-y_{n-1}]$. Singularities of the orbit closures =================================== We use a modification of $I_{\gamma}$ to compute measures of the singularities of $p\in Y_{\gamma}$. Representative points --------------------- We pick representative points of each ${\mathcal O}_{\gamma}$ to work with. Call a permutation $\sigma$ [**$\gamma$-shuffled**]{} if it is an assignment of - $1,2,\ldots,p$ (in any order) to the vertices of $\gamma$ that have a “$+$” or are the left end of an arc; and - $p+1,p+2,\ldots,n$ (in any order) to the remaining positions. Now let $F_{\bullet}^{\gamma,\sigma}=\langle {\vec v}_1,\ldots, {\vec v}_n\rangle$ be the flag given by $${\vec v}_i = \begin{cases} e_{\sigma(i)} & \text{ if vertex $i$ is a sign or the right end of an arc} \\ e_{\sigma(i)} + e_{\sigma(j)} & \text{ if $i$ and $j$ form an arc and $i<j$}. \end{cases}$$ We recall the following easy facts for convenience: Let $\gamma\in {\tt Clan}_{p,q}$ be given. - $F_{\bullet}^{\gamma,\sigma} \in {\mathcal O}_{\gamma}$ for any $\gamma$-shuffled $\sigma$. - $v(\gamma)$ is $\gamma$-shuffled. - If $\gamma$ is matchless then the $T$-fixed points ${\mathcal O}_{\gamma}^T=\{F_{\bullet}^{\gamma,\sigma}|\mbox{$\sigma$ is $\gamma$-shuffled}\}$. - If $\gamma$ is not matchless then ${\mathcal O}_{\gamma}$ contains no $T$-fixed points. - Every point of $Y_{\gamma}$ is locally isomorphic to some $F_{\bullet}^{\beta,\sigma}$ for some $\beta\prec\gamma$ and $\beta$-shuffled $\sigma$. - Let ${\mathcal P}$ be any upper-semicontinuous property of points of $Y_{\gamma}$. Then $Y_{\gamma}$ globally has property ${\mathcal P}$ if and only if some $T$-fixed point $F_{\bullet}^{\tau,\sigma}$ has property ${\mathcal P}$ for every matchless $\tau \prec \gamma$. \(I) follows easily from a theorem of T. Matsuki-T. Oshima [@Matsuki-Oshima] and A. Yamamoto [@Yamamoto] that $\mathcal{O}_{\gamma}$ is precisely the set of flags $F_{\bullet}$ such that - ${\rm dim}(F_i\cap E_p)= \gamma(i;+)$; - ${\rm dim}(F_i\cap E^q)= \gamma(i;-)$; - ${\rm dim}(\pi(F_i)+F_j)= j+\gamma(i;j)$. \(II) is immediate from the definitions. For (III) clearly the “$\supset$” inclusion is obvious. On the other hand, the set of $\gamma$-shuffled permutations is clearly a left coset in $S_p\times S_q \backslash S_n$, and so has order $|S_p\times S_q| = p!q!$. This is precisely the number of $T$-fixed points contained in any closed $K$-orbit, as each is isomorphic to the flag variety for the group $K$. Thus the inclusion is an equality. For (IV), simply note that there are $\binom{n}{p}$ closed orbits, each containing $p!q!$ $T$-fixed points (as just noted), for a total of $\binom{n}{p} \cdot p!q! = n!$ $T$-fixed points contained in the closed orbits. This means that no non-closed orbit can contain a $T$-fixed point. (Alternatively, (IV) follows directly from [@Springer Corollary 6.6].) For (V), the elements of $GL_p\times GL_q$ provide the isomorphisms. Finally, for (VI), the matchless clans are the minimal elements of the closure order. Part (IV) contrasts with Schubert varieties where every point is locally isomorphic to a $T$-fixed point. However, in view of (VI) these points of $Y_{\gamma}$ are still of special interest. The patch ideal --------------- Given a permutation $\sigma$, let $M_{n,\sigma}$ be the specialization of the generic matrix $M_n$ obtained by setting $z_{ij}=1$ if $i=\sigma(j)$ and $z_{ij}=0$ if $j>\sigma^{-1}(i)$. For example, if $\sigma=1324$ then (now writing the permutation matrix for $\pi$ with a $1$ in position $(\pi(i),i)$): $$M_{4,1324}= \begin{pmatrix} 1 & 0 & 0 & 0 \\ z_{2,1} & z_{2,2} & 1 & 0 \\ z_{31} & 1 & 0 & 0 \\ z_{4,1} & z_{4,2} & z_{4,3} & 1 \end{pmatrix}.$$ For a clan $\beta$, let $v=v(\beta)$ and let $L_{\beta}$ be the lower triangular unipotent matrix defined by having $1$’s in positions $(v(j),v(i))$ whenever $i<j$ is matched in $\beta$. Now define $M_{n,\beta}=L_{\beta}M_{n,v(\beta)}$. So if for example $\beta= \begin{picture}(30,10) \put(0,0){\epsfig{file=matchingsSection4.eps, height=.4cm}} \put(11,1){$+-$} \end{picture}$ then $v(\beta)=1324$, $$L_{\beta}= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \mbox{ \ \ \ and \ \ \ $M_{n,\beta}= \begin{pmatrix} 1 & 0 & 0 & 0 \\ z_{2,1} & z_{2,2} & 1 & 0 \\ z_{31}+1 & 1 & 0 & 0 \\ z_{4,1} & z_{4,2} & z_{4,3} & 1 \end{pmatrix}$.}$$ Finally, define the [**patch ideal**]{} $I_{\gamma,\beta}$ of $Y_{\gamma}$ at $\beta$ to be generated by the same polynomials as the $K$-orbit determinantal ideal except that $M_n$ is replaced by $M_{n,\beta}$ in the definition. Suppose $\gamma=\begin{picture}(27,10) \put(0,0){\epsfig{file=matchingsSection4KLV.eps, height=.4cm}} \end{picture}$ and we continue with $\beta= \begin{picture}(30,10) \put(0,0){\epsfig{file=matchingsSection4.eps, height=.4cm}} \put(11,1){$+-$} \end{picture}$. Then the reader can check that $I_{\gamma,\beta}$ is generated by the determinant of $$\begin{pmatrix} 1 & 1 & 0 & 0 \\ z_{2,1} & z_{2,1} & z_{2,2} & 1 \\ 0 & z_{31} & 1 & 0 \\ 0 & z_{4,1} & z_{4,2} & z_{4,3} \end{pmatrix}.$$ The following is standard; see the discussion of [@Insko.Yong]: \[prop:patch\] ${\rm Spec}({\rm Fun}(M_{n,\beta})/I_{\gamma,\beta})$ is set-theoretically equal to a local neighbourhood of $Y_{\gamma}$ around the point $F_{\bullet}^{\beta,v(\beta)}$. (The point ${\bf 0}$ corresponds to $F_{\bullet}^{\beta,v(\beta)}$.) Let $g=L_{\beta}v(\beta)$. Then $gB_{-}B/B\cap Y_{\gamma}$ is an affine open neighbourhood of $Y_{\gamma}$ around $gB$. Coordinates for $gB_{-}B/B$ are given by $M_{n,\beta}$. In view of Theorem \[thm:wyserbruhat\], any matrix of $M_{n,\beta}$ representing a flag in $Y_{\gamma}$ must vanish on generators of $I_{\gamma,\beta}$ and conversely, by Lemma \[lemma:vanish\]. \[conj:D\] $I_{\gamma,\beta}$ is a radical ideal. Conjecture \[conj:D\] has been verified for all patch ideals $I_{\gamma,\beta}$ with $\gamma \geq \beta$ through $p+q=6$. Additionally, it has been verified exhaustively for patch ideals $I_{\gamma,\tau}$ with $\tau$ a *matchless* clan for $n=7$, as well as for the cases $(p,q) = (2,6)$ and $(3,5)$. Numerous other successful checks of $I_{\gamma,\tau}$ with $\tau$ matchless in the case $(p,q) = (4,4)$ have also been performed. $H$-polynomials and Kazhdan-Lusztig-Vogan polynomials {#sec:h-poly} ----------------------------------------------------- We propose an analogy between two families of polynomials, one of which are the Kazhdan-Lusztig-Vogan (KLV) polynomials. Standard references on KLV polynomials are [@Vogan; @Lusztig-Vogan]. In their most general form, these polynomials are indexed by pairs $(Q,\mathcal{L})$ and $(Q',\mathcal{L}')$, where $Q,Q'$ are $K$-orbits on $G/B$, and $\mathcal{L},\mathcal{L}'$ are $K$-equivariant local systems on $Q,Q'$, respectively. For the associated polynomials to be nonzero, the pairs $(Q,\mathcal{L})$ and $(Q',\mathcal{L}')$ must be related in *$\mathcal{G}$-Bruhat order*, defined in [@Vogan]. Since all $K$-equivariant local systems on all orbits are trivial in the example we are considering, for us the KLV polynomials will be indexed simply by pairs of orbits (or rather, by the corresponding pairs of clans) $\beta,\gamma$ such that ${\mathcal{O}}_{\beta} \subseteq \overline{{\mathcal{O}}_{\gamma}}$. Furthermore, the coefficient of $q^i$ in the polynomial $P_{\beta,\gamma}(q)$ measures the dimension of the $2i$-th intersection homology group of $\overline{{\mathcal{O}}_{\gamma}}$ in a neighborhood of a point of ${\mathcal{O}}_{\beta}$, as follows from [@Lusztig-Vogan Theorem 1.12]. This mirrors the relationship between Schubert varieties and ordinary Kazhdan-Lusztig polynomials. Consider the ${\mathbb Z}$-graded Hilbert series of ${\rm gr}_{{\mathfrak m}_p}{\mathcal O}_{p,Z}$, the associated graded ring of the local ring ${\mathcal O}_{p,Z}$ of a variety $Z$. This is denoted by ${\rm Hilb}({\rm gr}_{{\mathfrak m}_p}{\mathcal O}_{p,Z},q)$. The [**$H$-polynomial**]{} $H_{p,Z}(q)$ is defined by $${\rm Hilb}({\rm gr}_{{\mathfrak m}_p}{\mathcal O}_{p,Z},q)=\frac{H_{p,Z}(q)}{(1-q)^{\dim Z}},$$ and $H_{p,Z}(1)$ is the [**Hilbert-Samuel multiplicity**]{} ${\rm mult}_{p,Z}$. \[conj:Hpoly\] - ${\rm gr}_{{\mathfrak m}_p}{\mathcal O}_{p,Y_{\gamma}}$ is Cohen-Macaulay. - $H_{p,Y_{\gamma}}(q)\in {\mathbb Z}_{\geq 0}[q]$. - $H_{p,Y_{\gamma}}(q)\in {\mathbb Z}_{\geq 0}[q]$ is upper-semicontinuous. In fact (i) implies (ii), by standard facts from commutative algebra. However, (i) and (ii) seem to be logically independent of (iii). Properties (ii) and (iii) are true for the KLV polynomial $P_{\beta,\gamma}(q)$. Property (ii) follows from [@Lusztig-Vogan Theorem 1.12], while property (iii) holds due to recent work of W.M. McGovern [@McGovern-preprint]. Thus the above conjecture is our rationale for drawing an analogy between $H_{\beta,\gamma}(q)$ and $P_{\beta,\gamma}(q)$. (Here, $H_{\beta,\gamma}(q)$ is the $H$-polynomial $H_{p,Y_{\gamma}}(q)$ where $p$ is any point of ${\mathcal{O}}_{\beta} \subseteq Y_{\gamma}$.) An analogous analogy and conjecture was proposed in the Schubert variety setting in [@Li.Yong]. \[prop:PHineq\] If $\gamma$ is non-crossing then $H_{\beta,\gamma}(q) \in {\mathbb Z}_{\geq 0}[q]$ and $P_{\beta,\gamma}(q)\leq H_{\beta,\gamma}(q)$ (coefficient-wise inequality). When $\gamma$ is non-crossing $Y_{\gamma}=X_{v(\gamma)}^{u(\gamma)}$. The KLV polynomial is the $IH$-Poincaré polynomial at a point of $X_{v(\gamma)}^{u(\gamma)}$. By [@KWY], this is therefore the product of Kazhdan-Lusztig polynomials for $X_{v(\gamma)}$ and for $X^{u(\gamma)}$. The same is true for the $H$-polynomial. However, $v(\gamma)$ and $u(\gamma)$ are vexillary. It is a theorem of [@Li.Yong] that for the Schubert varieties involved, the $H$-polynomials have nonnegative coefficients and bound the Kazhdan-Lusztig polynomials. Nonnegativity and this bound on polynomials is preserved by multiplication. The inequality of Theorem \[prop:PHineq\] does not always hold when $\gamma$ is not non-crossing. For example, if $\gamma=\begin{picture}(27,10) \put(0,0){\epsfig{file=matchingsSection4KLV.eps, height=.4cm}} \put(8,1){\tiny $+$} \end{picture}$ then $P_{-+++-,\gamma}(q)=1+q^2$, as one can verify using ATLAS (<http://www.liegroups.org>). However, we have $H_{-+++-,\gamma}(q)=1+q$. A. Woo and the first author have found an explicit combinatorial rule for $P_{\beta,\gamma}(q)$ when $\gamma$ is non-crossing. The following also seems true: \[conj:reduced\] ${\rm Spec}({\rm gr}_{{\mathfrak m}_p}{\mathcal O}_{p,Y_{\gamma}})$ is reduced. Using the patch equations one can exhaustively check Conjectures \[conj:Hpoly\] and \[conj:reduced\] for all $(p,q)$ where $p+q\leq 7$. We have also done checks for some larger cases. Appendix {#appendix .unnumbered} ======== Below we give the polynomials $\Upsilon_{\gamma}(X;Y)$ for all $\gamma\in {\tt Clans}_{2,2}$. --------------------------------------------------------------------------------------------------------------------------------------------------------- $\gamma$ $\Upsilon_{\gamma}(X;Y)$ -------------------------------------------------------------- ------------------------------------------------------------------------------------------ $--++$ $(x_2 - y_2) (x_2 - y_1) (x_1 - y_2) (x_1 - y_1)$ $-+-+$ $(x_1 - y_2) (x_1 - y_1) (x_1 - y_4 - y_3 + x_2) (x_2 - y_1 + x_3 - y_2)$ $-++-$ $(x_1 - y_2) (x_1 - y_1) (-x_1 y_3 + y_4 y_3 + y_3^2 - x_2 y_3 + x_1 x_3 - y_4 x_3$   $- x_3 y_3 + x_2 x_3 + x_2 x_1 - y_4 x_2 - y_4 x_1 + y_4^2)$ $+--+$ $(x_1 - y_4) (x_1 - y_3) (-x_1 y_2 + y_1 y_2 + y_2^2 - x_2 y_2 + x_1 x_3 - x_3 y_1$   $- x_3 y_2 + x_2 x_3 + x_2 x_1 - x_2 y_1 - y_1 x_1 + y_1^2)$ $+-+-$ $(x_1 - y_4) (x_1 - y_3) (x_1 - y_1 - y_2 + x_2) (x_3 - y_3 - y_4 + x_2)$ $++--$ $(x_2 - y_4) (x_2 - y_3) (x_1 - y_4) (x_1 - y_3)$ $\begin{picture}(30,10) $(x_1 - y_2) (x_1 - y_1) (x_2 - y_1 + x_3 - y_2)$ \put(10,0){\epsfig{file=matchingsSection4.eps, height=.4cm}} \put(0,1){$-$} \put(21,1){$+$} \end{picture}$ $\begin{picture}(30,10) $(x_1 - y_2) (x_1 - y_1) (x_1 - y_4 - y_3 + x_2)$ \put(18,0){\epsfig{file=matchingsSection4.eps, height=.4cm}} \put(0,1){$-+$} \end{picture}$ $\begin{picture}(30,12) $(x_1 - y_4 - y_3 + x_2) (-x_1 y_2 + y_1 y_2 + y_2^2 - x_2 y_2 + x_1 x_3 - x_3 y_1$ \put(0,0){\epsfig{file=matchingsSection4.eps, height=.4cm}} \put(11,1){$-+$} \end{picture}$   $- x_3 y_2 + x_2 x_3 + x_2 x_1 - x_2 y_1 - y_1 x_1 + y_1^2)$ $\begin{picture}(30,10) $(x_1 - y_1 - y_2 + x_2) (-x_1 y_3 + y_4 y_3 + y_3^2 - x_2 y_3 + x_1 x_3 - y_4 x_3$ \put(0,0){\epsfig{file=matchingsSection4.eps, height=.4cm}} \put(11,1){$+-$} \end{picture}$   $-x_3 y_3 + x_2 x_3 + x_2 x_1 - y_4 x_2 - y_4 x_1 + y_4^2)$ $\begin{picture}(30,12) $(x_1 - y_4) (x_1 - y_3) (x_1 - y_1 - y_2 + x_2)$ \put(18,0){\epsfig{file=matchingsSection4.eps, height=.4cm}} \put(0,1){$+-$} \end{picture}$ $\begin{picture}(30,10) $(x_1 - y_4) (x_1 - y_3) (x_3 - y_3 - y_4 + x_2)$ \put(10,0){\epsfig{file=matchingsSection4.eps, height=.4cm}} \put(0,1){$+$} \put(21,1){$-$} \end{picture}$ $\begin{picture}(30,10) $(x_1 - y_2) (x_1 - y_1)$ \put(10,0){\epsfig{file=matchingstableA.eps, height=.4cm}} \put(0,1){$-$} \put(15,1){$+$} \end{picture}$ $\begin{picture}(30,10) $-x_1 y_2 + y_1 y_2 + y_2^2 - x_2 y_2 + x_1 x_3 - x_3 y_1 \put(1,0){\epsfig{file=matchingstableA.eps, height=.4cm}} - x_3 y_2 + x_2 x_3 + x_2 x_1 - x_2 y_1 - y_1 x_1 + y_1^2$ \put(6,0){$-$} \put(21,1){$+$} \end{picture}$ $\begin{picture}(30,10) $(x_1 - y_4 - y_3 + x_2) (x_1 - y_1 - y_2 + x_2)$ \put(1,0){\epsfig{file=matchings1122.eps, height=.4cm}} \end{picture}$ $\begin{picture}(30,10) $-x_1 y_3 + y_4 y_3 + y_3^2 - x_2 y_3 + x_1 x_3 - y_4 x_3 - x_3 y_3 + x_2 x_3 + x_2 x_1 \put(1,0){\epsfig{file=matchingstableA.eps, height=.4cm}} - y_4 x_2 - y_4 x_1 + y_4^2$ \put(6,0){$+$} \put(21,1){$-$} \end{picture}$ $\begin{picture}(30,10) $(x_1 - y_4) (x_1 - y_3)$ \put(10,0){\epsfig{file=matchingstableA.eps, height=.4cm}} \put(0,1){$+$} \put(15,1){$-$} \end{picture}$ $\begin{picture}(30,10) $x_1 - y_1 - y_2 + x_2$ \put(0,0){\epsfig{file=matchingstableB.eps, height=.4cm}} \put(7,1){$-+$} \end{picture}$ $\begin{picture}(30,10) $x_2 - y_1 - y_3 + 2 x_1 - y_4 - y_2 + x_3$ \put(5,0){\epsfig{file=matchings1212.eps, height=.4cm}} \end{picture}$ $\begin{picture}(30,10) $x_1 - y_4 - y_3 + x_2$ \put(0,0){\epsfig{file=matchingstableB.eps, height=.4cm}} \put(7,1){$+-$} \end{picture}$ $\begin{picture}(30,10) $1$ \put(0,0){\epsfig{file=matchingstableC.eps, height=.4cm}} \end{picture}$ --------------------------------------------------------------------------------------------------------------------------------------------------------- Acknowledgements {#acknowledgements .unnumbered} ================ We wish to thank Bill Graham, Allen Knutson, William McGovern, Oliver Pechenik, Hal Schenck, Peter Trapa, Hugh Thomas and Alexander Woo for helpful correspondence. 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--- abstract: 'A $C\sharp$ implementation of a generalized k-means variant called [*relational k-means*]{} is described here. Relational k-means is a generalization of the well-known k-means clustering method which works for non-Euclidean scenarios as well. The input is an arbitrary distance matrix, as opposed to the traditional k-means method, where the clustered objects need to be identified with vectors.' author: - | Balázs Szalkai\ [Eötvös Loránd University, Budapest, Hungary]{} bibliography: - 'generalized-kmeans.bib' title: 'An implementation of the relational k-means algorithm' --- Introduction ============ Suppose that we are given a set of data points $p_1, ... p_n \in \mathbb{R}^d$ and a number of clusters $N$. We would like assort these data points “optimally” into $N$ clusters. For each point $p_i$ let $z_i$ denote the center of gravity of the cluster $p_i$ is assigned to. We call the $z_i$ vectors [*centroids*]{}. The standard k-means method [@kmeans] attempts to produce such a clustering of the data points that the sum of squared centroid distances $\sum_{i = 1}^{n} ||p_i - z_i||^2$ is minimized. The main difficulty of this method is that it requires the data points to be the elements of a Euclidean space, since we need to average the data points somehow. A generalization of k-means called [*relational k-means*]{} has been proposed [@2013arXiv1303.6001S] to address this issue. This generalized k-means variant does not require the data points to be vectors. Instead, the data points $p_1, ... p_n$ can form an abstract set $S$ with a completely arbitrary distance function $f : S \times S \to [0, \infty)$. We only require that $f$ is symmetrical, and $f(p_i, p_i) = 0$ for all $p_i \in S$. Note that $f$ need not even satisfy the triangle inequality. Relational k-means ================== First, we provide a brief outline of the algorithm. Let $A \in \mathbb{R}^{n \times n}$ be the squared distance matrix. That is, $A_{ij} = f(p_i, p_j)^2$. The algorithm starts with some initial clustering and improves it by repeatedly performing an iteration step. The algorithm stops if the last iteration did not decrease the [*value*]{} of the clustering (defined below). Of course, if the iteration increased the value of the clustering, the algorithm reverts the last iteration. Now we describe the algorithm in detail. At any time during execution, let $S_1, ... S_N \subset S$ denote the clusters. For each data point $p_i$, let $\ell(p_i)$ denote the index of the cluster $p_i$ is assigned to. Let $e_i \in \mathbb{R}^n$ denote the $i$th standard basis vector, and, for $i \in \{1, ... n\}$ and $j \in \{1, ... N\}$ let $v_{ij} := \frac{1}{|S_j|}\sum_{k \in S_j} e_k - e_i$. Let us call the quantity $q_{ij} := -\frac{1}{2}v_{ij}^\top A v_{ij}$ the [*squared centroid distance*]{} corresponding to the point $p_i$ and the cluster $S_j$. In [@2013arXiv1303.6001S] it is observed that, if the distance function is derived from a Euclidean representation of the data points, then $q_{ij}$ equals to the squared distance of $p_i$ and the center of gravity of $S_j$. Thus $q_{ij}$ is indeed an extension of the classical notion of squared centroid distances. Define the [*value*]{} of a clustering as $\sum_{i = 1}^n q_{i\ell{i}}$. We say that a clustering is [*better*]{} than another one if and only if its value is less than the value of the other clustering. The relational k-means algorithm takes an initial clustering (e.g. a random one), and improves it through repeated applications of an iteration step which reclusters the data points. The iteration step simply reassigns each data point to the cluster which minimized the squared centroid distance in the previous clustering. If the value of clustering does not decrease through the reassignment, then the reassignment is undone and the algorithm stops. In the non-Euclidean case there might be scenarios when an iteration actually [*worsens*]{} the clustering. Should this peculiar behavior be undesirable, it can be avoided by “stretching” the distance matrix and thus making it Euclidean. Stretching means replacing $A$ with $A + \beta (J-I)$, where $J$ is the matrix whose entries are all 1’s, $I$ is the identity matrix, and $\beta \geq 0$ is the smallest real number for which $A + \beta (J-I)$ is a Euclidean squared distance matrix, i.e. it equals to the squared distance matrix of some $n$ vectors. It can be easily deducted that such a $\beta$ exists. This method is called $\beta$-spread transformation (see [@Hathaway1994429]). The algorithm is thus as follows: - Start with some clustering, e.g. a random one - Calculate the value of the current clustering and store it in $V_1$ - For each $p_i$ data point, calculate and store the squared centroid distances $q_{i1}, ... q_{iN}$ - Reassign each $p_i$ data point to the cluster that yielded the smallest squared centroid distance in the previous step - Calculate the value of the current clustering and store it in $V_2$ - If $V_2 \geq V_1$, then undo the previous reassignment and stop - Go to line number 2 Time complexity =============== The algorithm is clearly finite because it gradually decreases the value of the current clustering and the number of different clusterings is finite. Each iteration step can easily be implemented in $\mathcal{O}(n^3)$ time: for each data point, we need to calculate $N$ quadratic forms, which can be done in $n \sum_{j=1}^N |S_j|^2 \leq n^3$ time. This is unfortunately too slow for practical applications. However, this can be improved down to $\mathcal{O}(n^2)$. The squared centroid distances can be transformed as follows (using $A_{ii} = 0$): $$q_{ij} = -\frac{1}{2}\left(\frac{1}{|S_j|}\sum_{k \in S_j} e_k - e_i\right)^\top A \left(\frac{1}{|S_j|}\sum_{k \in S_j} e_k - e_i\right) = -\frac{1}{2|S_j|^2} \sum_{a, b \in S_j} A_{ab} + \frac{1}{|S_j|} \sum_{k \in S_j} A_{ik}.$$ Here the first summand is independent of $i$ and thus needs to be calculated for each cluster only once per iteration. On the other hand, the second summand can be calculated in $\mathcal{O}(|S_j|)$ time. To sum up, the amount of arithmetic operations per iteration is at most constant times $\sum_{j = 1}^N |S_j|^2 + n \sum_{j = 1}^N |S_j| \leq 2 n^2$. The current implementation makes several attempts to find a better clustering. In each attempt, the full relational k-means algorithm is run, starting from a new random clustering. Every attempt has the possibility to produce a clustering which is better than the best one among the previous attempts. If the sofar best clustering has not been improved in the last $K$ attempts (where $K$ is a parameter), then it is assumed that the clustering which is currently the best is not too far from the optimum, and the program execution stops. Attempts do not depend on each other’s result and do not modify shared resources (apart from a shared random generator). Our implementation uses `Parallel.For` for launching multiple attempts at once. The number of parallel threads can be customized via a command-line switch, and by default equals to the number of logical processors. This results in near 100% CPU utilization. Launching less than $C$ threads allows leaving some CPU time for other processes. Test results and conclusion =========================== We implemented the above algorithm in $C\sharp$ and run the program on a set of &gt;1000 proteins with a Levenshtein-like distance function. The value of $K$ (maximum allowed number of failed attempts, i.e. the “bad luck streak”) was 20, and the value of $N$ (number of clusters) was 10. The testing was done on an average dual core laptop computer and finished in 30..60 seconds. This proves that relational k-means can be implemented in a way efficient enough to be applied to real-world datasets. Since the program is almost fully parallelized, we could expect it to finish in &lt;8 seconds for the same dataset on a 16-core machine. Note that the total runtime is proportional to the number of attempts made, which is highly variable due to the random nature of the algorithm. A C++ implementation could further reduce execution time. According to our estimate, it could make the program cca. twice as fast. Attachments =========== Program source code in $C\sharp$ -------------------------------- Example input ------------- The first $n$ lines of the input contain the names of the objects which need to be clustered. Then a line containing two slashes follows. After that, a matrix containing the pairwise distances is listed in semicolon-separated CSV format. Fractional distances must be input with the dot character (`.`) as decimal separator.
--- abstract: 'CeMnNi$_4$ exhibits an unusually large spin polarization, but its origin has baffled researchers for more than a decade. We use bulk sensitive hard x-ray photoelectron spectroscopy (HAXPES) and density functional theory based on the Green’s function technique to demonstrate the importance of electron-electron correlations of both the Ni 3$d$ ($U_{Ni}$) and Mn 3$d$ ($U_{Mn}$) electrons in explaining the valence band of this multiply correlated material. We show that Mn-Ni anti-site disorder as well as $U_{Ni}$ play crucial role in enhancing its spin polarization: anti-site disorder broadens a Ni 3$d$ minority-spin peak close to the Fermi level ($E_F$), while an increase in $U_{Ni}$ shifts it towards $E_F$, both leading to a significant increase of minority-spin states at $E_F$. Furthermore, rare occurrence of a valence state transition between the bulk and the surface is demonstrated highlighting the importance of HAXPES in resolving the electronic structure of materials unhindered by surface effects.' author: - 'Pampa Sadhukhan$^{1,\dagger}$, Sunil Wilfred D$^{\prime}$Souza$^{1,{\dagger}*}$, Vipin Kumar Singh$^{1}$, Rajendra Singh Dhaka$^{1a}$, Andrei Gloskovskii$^{2}$, Sudesh Kumar Dhar$^{3}$, Pratap Raychaudhuri$^{3}$, Ashish Chainani$^{4}$, Aparna Chakrabarti$^{5,6}$, Sudipta Roy Barman$^{1}$' title: 'Influence of anti-site disorder and electron-electron correlations on the electronic structure of CeMnNi$_4$ ' --- In recent years, hard x-ray photoelectron spectroscopy (HAXPES) has turned out to be a reliable tool to study the electronic structure of correlated systems, thin films and buried interfaces of materials, thus providing new insights into their physical properties[@Fadley10; @Wocik16; @Grayandothers]. In this work, we present the first study of the electronic structure of CeMnNi$_4$, an interesting material with  large spin transport polarization of 66%[@Singh06], using HAXPES and density functional theory calculations based on the spin polarized relativistic Korringa-Kohn-Rostoker (SPRKKR) method[@Ebert]. CeMnNi$_4$ has a cubic MgCu$_4$Sn-type structure[@Dhiman07]; it is ferromagnetic with a magnetic moment of 4.95$\mu_B$ and Curie temperature of 140 K[@Singh06]. These encouraging properties of CeMnNi$_4$ started a flurry of activity aimed at understanding its electronic structure[@Mazin06Voloshina06; @Lahiri10; @Bahramy10]. However, no photoemission study of its electronic structure has been reported to date, and the theoretical studies so far have been unable to explain the different aspects of its electronic structure and its spin polarization in particular. The early density functional theory (DFT) calculations[@Mazin06Voloshina06] reported a spin polarization[@p0formula] ($P_0$) value of about 16-20$\%$; and the much larger experimental polarization was attributed to disorder or non-stoichiometry of the specimens. In fact, in a subsequent x-ray absorption fine structure (XAFS) study, about 6$\%$ Mn-Ni anti-site disorder was reported[@Lahiri10]. The authors also performed a DFT calculation using the pseudopotential method as implemented in the VASP code including an ordered anti-site defect configuration of nearest neighbour Ni and Mn that were site-exchanged. Thus, in this approach, the effect of randomly disordered anti-site defects is not taken into account.  Their results however showed a significant increase in $P_0$, which was not related to disorder, but rather to enhanced minority spin states of the site-exchanged Mn 3$d$ partial density of states (PDOS) due to hybridization with neighboring Ni atom[@Lahiri10]. On the other hand, another DFT calculation that considered electron-electron correlation of the Mn 3$d$ electrons ($U_{Mn}$) but no anti-site defect showed that $P_0$ increases with $U_{Mn}$[@Bahramy10]. In the absence of any photoemission study and its direct comparison with theory that addresses the influence of both anti-site disorder and correlation, their role in determining the electronic structure and spin polarization of CeMnNi$_4$ has remained an unresolved question until date. In this letter, we show that both anti-site disorder and electron-electron correlations for Ni 3$d$ ($U_{Ni}$) and Mn 3$d$ ($U_{Mn}$) electrons have a crucial influence on the bulk electronic structure of CeMnNi$_4$. In addition, since $U_{Ce}$ is typically taken to be about 7 eV in Ce intermetallics[@Imer87], CeMnNi$_4$ can be regarded as a multiply correlated system, further complicated by the presence of inherent disorder[@Lahiri10]. $U_{Ni}$ and $U_{Mn}$ are responsible for determining the energy positions of the peaks in the valence band (VB) and their optimum values ($U_{Mn}$= 4.5 eV, $U_{Ni}$= 6.5 eV) are obtained by the best agreement between theoretically calculated and the experimental HAXPES VB. A surprising result is that the large $P_0$ of CeMnNi$_4$ has two origins: the anti-site disorder ($x$) and $U_{Ni}$. The former broadens a minority spin Ni 3$d$ peak close to $E_F$, while the latter shifts it towards $E_F$. Thus, in both cases, the minority spin total DOS at $E_F$ ($n_\downarrow$($E_F$)) increases, while the majority spin total DOS ($n_\uparrow$($E_F$)) remains essentially unchanged, resulting in a clear enhancement of $P_0$. The total magnetic moment exhibits contrasting variation: a decrease with $x$ and an increase with $U_{Ni}$. Furthermore, rare occurrence of a valence state transition on the surface of a ternary material is demonstrated: a bulk mixed valent state transforms to a nearly trivalent Ce$^{3+}$ state due to the weakened hybridization on the surface.\   \ [***Experimental and computational methods:***]{} HAXPES measurements were performed at the P09 beamline in PETRA III synchrotron center, Germany on polycrystalline CeMnNi$_4$ ingot that was cleaved under ultra high vacuum at 2$\times$10$^{-8}$ mbar pressure to expose a fresh surface. The spectra were recorded by using Phoibos 225 analyzer  with 30 eV pass energy at 50 K[@Andrei]. Photons were incident on the sample at a grazing angle (10$^{\circ}$) and the photoelectrons were collected in the nearly normal emission geometry. The total instrumental resolution (including both source and analyzer contributions), obtained from the least square fitting of the Au Fermi edge in electrical contact with the specimen, is 0.26 eV.  CeMnNi$_{4}$ ingot was prepared by an arc melting method and characterized for its structure using x-ray diffraction, as discussed in Ref. . The bulk ground state properties of CeMnNi$_4$ have been calculated in $F$$\bar{4}$$3m$ symmetry using the experimental lattice parameter ($a$= 6.9706Å) as determined by neutron powder diffraction at 17 K[@Dhiman07]. Disordered Mn-Ni anti-site defects have been considered by setting the 16$e$ site occupations to 1-0.25$x$ for Ni$_{\rm Ni}$ and 0.25$x$ for Mn$_{\rm Ni}$, while the occupancies at the 4$c$ site were set to $1-x$ for Mn$_{\rm Mn}$ and $x$ for Ni$_{\rm Mn}$, where X$_{\rm Z}$ refers to a X atom at a Z atom site (X, Z= Ni, Mn). Here, $x$ quantifies the amount of anti-site disorder as the fraction of Mn atoms occupying the Ni sites. In this work, we have varied $x$ from 0 to 0.12. The structures are shown in Fig. S1 of SM. Self-consistent band structure calculations were carried out using fully relativistic SPRKKR method in the atomic sphere approximation[@Ebert]. The exchange and correlation effects were incorporated within the generalized gradient approximation framework.[@Perdew96] The electron-electron correlation has been taken into account as described in the LSDA+U scheme[@Ebert03]. The parameters of screened on-site Coulomb interaction $U$ for all the components ($U_{Ni}$, $U_{Mn}$ and $U_{Ce}$) have been varied up to 7 eV, with the exchange interaction $J$ fixed at 0.8 eV. The static double counting of LSDA+U approach has been corrected using the atomic limit scheme. The angular momentum expansion up to $l_{max}$= 4 has been used for each atom. The energy convergence criterion and coherent potential approximation tolerance has been set to 10$^{-5}$ Ry. Brillouin zone integrations were performed on a 36$\times$36$\times$36 mesh of $k$-points in the irreducible wedge of the Brillouin zone. We have employed Lloyd’s formula, which provides an accurate determination of the Fermi level and density of states[@Llyod]. For calculating the angle integrated VB spectrum, all the PDOS contributions from $\textit{s}$, $\textit{p}$, $\textit{d}$ and $\textit{f}$ states of Ce, Mn and Ni were multiplied with their corresponding photoemission cross-sections[@Yeh85] and then added.  This is multiplied by the Fermi function and convoluted with the instrumental resolution and an energy dependent lifetime broadening 0.01$\times$($E_B$-$E_F$)[@Barman95] to obtain the VB. ![(a) The valence band (VB) HAXPES spectra of CeMnNi$_4$ at 50 K using 8 keV photon energy (black filled circles) compared with the calculated VB spectra for $x$= 0 (no disorder) and $x$= 0.06 (6% Mn-Ni anti-site disorder). The VB’s calculated with different $U_{Mn}$ are shown in the -3 to -8 eV range, where $x$= 0, $U_{Ni}$= $U_{Ce}$= 0 eV. The spectra are staggered along the vertical axis, zero of the horizontal scale corresponds to the Fermi level ($E_F$). (b) Mn 3$d$ contribution to the calculated VB as a function of $U_{Mn}$.[]{data-label="mnu"}](1R1_Layout77_3_fig1_ver3.pdf){width="98mm"}   \ [***Valence band of CeMnNi$_4$:***]{} The VB spectrum recorded with 8 keV photon energy at 50 K shows a step ($S$) close to $E_F$ at -0.4 eV; peaks at -1.5 ($A$), -2.2 ($B$), -3.6 ($C$), -4.2 ($D$), -5.2 eV ($E$) and a weak shoulder at -6 eV ($F$) (Fig. \[mnu\](a)). In order to ascertain their origin and study the influence of disorder on the spectral shape, we have calculated the VB spectra without (red line with open circles, $x$= 0) and with 6% Mn-Ni anti-site disorder (blue dashed line, $x$= 0.06). 6% disorder is considered because a previous XAFS study[@Lahiri10] inferred a disorder of this magnitude on a specimen that was prepared by the same procedure as ours. As discussed above,  the VB has been calculated from the partial DOS (PDOS) in Fig. S2. We find that disorder results in a small but finite broadening of the VB, but it has no effect on the position of the peaks. Comparison of the calculated VB with HAXPES shows glaring differences: the peaks corresponding to $A$ and $B$ (black arrows) are positioned at higher and lower energies, respectively and thus their separation (1.6 eV) is significantly larger compared to experiment (0.7 eV). The peak at -5.4 eV (red arrow) is shifted $w.r.t.$ peak $E$ of the experimental VB, the peak at -3.3 eV (red tick) appears at a dip, while there is no peak in the theory corresponding to $F$ (see the blue dashed arrows). In Fig. S2, DOS calculated with disorder up to $x$= 0.12 ($i.e.$ 12% aniti-site disorder) show increased broadening, but the positions of all the peaks remain unchanged. Thus, it is obvious from the above discussion that disorder is unable to explain the VB. So, we examine the possible role of correlation starting with $U_{Mn}$. As $U_{Mn}$ is increased, interesting modifications in the -3 to -6 eV region is observed in Fig. \[mnu\](a), which are primarily related to the systematic changes in the Mn 3$d$ PDOS (Fig. \[mnu\](b) and PDOS in Fig. S3). At $U_{Mn}$= 0, the Mn 3$d$ states are delocalized over 0 to -5 eV with the most intense peak at -3.3 eV. Increase of $U_{Mn}$ narrows the Mn 3$d$ PDOS, the peak intensity increases and it shifts by a large amount to lower energies [*i.e.*]{} away from $E_F$ ($e.g.$ -5.2 eV for $U_{Mn}$= 4.5 eV). The best agreement with experiment in the -3 to -6 eV region is obtained for $U_{Mn}$= 4.5 eV (black line), where the peaks at -3.6, -4.2, -5.2 and -6 eV appear at the same positions as $C$, $D$, $E$, and $F$, respectively of the experimental VB, as shown by the blue dashed arrows in Fig. \[mnu\](a).   The Mn 3$d$ states contribute primarily to the peak $E$, however, its intensity is relatively less due to smaller photoemission cross-section of Mn 3$d$ with respect to Ni 3$d$ at 8 keV[@Yeh85]. ![The valence band HAXPES spectrum of Fig. 1 (black filled circles)   compared with calculated VB spectra as a function of $U_{Ni}$, with $U_{Mn}$= 4.5 eV, $U_{Ce}$= 7 eV and $x$= 0.[]{data-label="niu"}](2R1_Layout76_3_Fig2_ver3.pdf){width="98mm"} Although $U_{Mn}$= 4.5 eV provides a good agreement for peaks $C$-$F$, the positions of the peaks $A$ and $B$ are not well reproduced, and these remain unaltered with $U_{Mn}$ (Fig. S3). It is evident that $A$ and $B$ originate primarily from Ni 3$d$ states, and so we calculate the VB by introducing $U_{Ni}$, with $U_{Mn}$ fixed at 4.5 eV. We find that as $U_{Ni}$ increases, the peak at -2.6 eV shifts to higher energy [*i.e.*]{} towards $E_F$ (blue dashed line) and appears close to the position of peak $B$ for $U_{Ni}$= 6.5 eV (Fig. \[niu\], see Fig. S4 for PDOS). On the other hand, the peak at -1.1 eV initially shifts to higher energies and eventually shifts back to lower energy (green dashed line) towards peak $A$. The separation of these two peaks is lowest (0.8 eV) at $U_{Ni}$= 7 eV. However, for $U_{Ni}$= 7 eV, a new peak appears at -0.7 eV in disagreement with experiment. Thus, we conclude that the best agreement is observed for $U_{Ni}$= 6.5 eV, where the positions as well as the separation (0.9 eV) of the calculated peaks agree well with $A$ and $B$ (black dashed arrows in Fig. \[niu\]). Note that the peaks in the -3 to -6 eV region are hardly affected by $U_{Ni}$. It is to be noted that in Fig. \[niu\] we also consider a value of $U_{Ce}$ (= 7 eV) for the Ce 4$f$ electrons that is generally observed in Ce intermetallic compounds[@Imer87]. However, $U_{Ce}$ does not have any discernible effect on the occupied states and the VB, since the Ce 4$f$ peak appears mostly above $E_F$ at 0.9 eV for $U_{Ce}$= 0  (Fig. S3(c)) and shifts to higher energy (1.2 eV) for $U_{Ce}$= 7 eV  (Fig. S4(a)). Thus, due to the significant variation of Ni and Mn 3$d$ states with $U_{Ni}$ and $U_{Mn}$, respectively and taking $U_{Ce}$ from literature[@Imer87], we are able to determine the optimum values of $U$ for CeMnNi$_4$ to be: $U_{Mn}$= 4.5 eV, $U_{Ni}$= 6.5 eV and $U_{Ce}$= 7 eV (referred henceforth as $U(4.5,6.5,7)$). The partial contributions of the different PDOS to each of the peaks in the VB for $U(4.5,6.5,7)$ are shown in Fig. S5 of SM. ![Spin polarization ($P_0$), majority ($n_\uparrow$($E_F$)) and minority ($n_\downarrow$($E_F$)) spin total DOS at $E_F$ (a) as a function of disorder ($x$) with $U_{Ni}$= $U_{Mn}$= $U_{Ce}$= 0; and (c) as a function of $U_{Ni}$, where $U_{Mn}$= 4.5 eV, $U_{Ce}$= 7 eV and $x$= 0. Majority  and minority  spin total DOS around $E_F$ corresponding to (a) and (c) as a function of (b) $x$ and (d) $U_{Ni}$, respectively.[]{data-label="spol"}](3R1_Layout74_9_Fig3_SP_dos_1200k_ver8.pdf){width="93mm"}   \   \ [***Spin polarization and magnetic moments:***]{} We find that the Mn-Ni anti-site disorder has an unexpected positive effect on the spin polarization ($P_0$). As shown in Fig. \[spol\](a) and Table[**I**]{} of SM, $P_0$ exhibits a monotonic increase with $x$, reaching a value of 45% (50%) for $x$= 0.06 (0.12). This is an important result since in half metals and Heusler alloys, a low experimental value of $P_0$ is generally attributed to disorder[@hm]. In order to understand the reason for this unusual behavior, we show the spin polarized total DOS around $E_F$ in Fig. \[spol\](b). A peak in the minority spin DOS close to $E_F$ at -0.1 eV progressively broadens and also shifts by a small amount ($\approx$ 15 meV) towards $E_F$ resulting in increase of $n_\downarrow$($E_F$) with $x$. On the contrary, the structureless majority spin DOS and consequently $n_\uparrow$($E_F$) remain almost unchanged. Thus, this contrasting behavior of $n_\downarrow$($E_F$) and $n_\uparrow$($E_F$) brings about the increase of $P_0$ with $x$ (Fig. \[spol\](a)). Table[**I**]{} of SM defines and shows the partial contributions from Ni 3$d$ ($P_{0_{{\rm Ni}3d}}$), Mn 3$d$ ($P_{0_{{\rm Mn}3d}}$) and Ce 4$f$ ($P_{0_{{\rm Ce}4f}}$) PDOS to $P_0$ for different $x$, and we find that $P_0$ increases solely because of $P_{0_{{\rm Ni}3d}}$. This is also confirmed in Fig. S6 where the peak in the minority spin DOS is clearly dominated by Ni 3$d$ PDOS (black tick). Turning to the influence of $U$ on $P_0$ (Fig. \[spol\](c)), we find that it increases with $U_{Ni}$ from about 3.8% for $U$(4.5,0,7) to 45% for $U_{Ni}$= 6.5 eV $i.e$ for the optimum $U$(4.5,6.5,7). This is related to increase of $n_\downarrow$($E_F$) due to a significant shift of the minority spin total DOS peak towards $E_F$ from -0.2 to -0.05 eV (Fig. \[spol\](d)). Clearly the total DOS is dominated by Ni 3$d$, black ticks in Fig. S7 show how the minority spin Ni 3$d$ PDOS peak shifts with $U_{Ni}$. In contrast, the majority spin total DOS is structureless and $n_\uparrow$($E_F$) remains almost unchanged (Fig. \[spol\](c,d)). The partial contributions to $P_0$ for different $U_{Ni}$ clearly show that the increase in $P_0$ is entirely due to $P_{0_{{\rm Ni}3d}}$ (Table[**I**]{} of SM). Due to disorder, the Ni 3$d$ minority spin peak  will broaden and also possibly shift by small amount towards $E_F$ and thus significantly increase $n_\downarrow$($E_F$) because of its proximity to $E_F$ ($e.g.$ at -0.05 eV for $U$(4.5,6.5,7)). On the other hand, $n_\uparrow$($E_F$) would remain unchanged due to the nearly flat nature of the majority spin total DOS. Thus, disorder would further increase $P_0$, and assuming that its effect is independent of $U$, we estimate $P_0$ for $U$(4.5,6.5,7) to increase from 45% to $>$55% ($>$60%) for $x$= 0.06 (0.12). This is in good agreement with the experimental value of 66%, given the fact that the measurements were performed in the diffusive limit[@Singh06] and here we calculate the static spin polarization. ![(a) Spin polarization P$_{0}$ as a function of electron-electron correlation $U$ for Ni 3$d$ ($U_{Ni}$), Mn 3$d$ ($U_{Mn}$) and Ce 4$f$ electrons ($U_{Ce}$). $P_0$ is plotted as a function of $U$ shown as a triplet ($U_{Mn}$, $U_{Ni}$, $U_{Ce}$), where the fixed $U$ values in the triplet are indicated by numbers in eV. For example, (5, $U_{Ni}$,0) means $U_{Ni}$ varies from 0 to 7 eV with $U_{Mn}$ and $U_{Ce}$ fixed at 5 eV and 0 eV, respectively. The total (spin plus orbital) moment, the total spin only moment of CeMnNi$_4$ and the local spin magnetic moments of Mn and Ni are plotted (b) as a function of $\textit{U}$$_{Ni}$ with $\textit{U}$$_{Mn}$= 4.5 eV, $\textit{U}$$_{Ce}$= 7 eV and $x$= 0; and (c) as a function of disorder ($\textit{x}$).  In all cases, the Ce atom possess a small opposite moment of -0.2 $\mu_B$.[]{data-label="S8&9"}](4R1_Layout15_2_moment_vs_NiU_x_1200k_ver4.pdf){width="93mm"} We have also studied how $U_{Mn}$ and $U_{Ce}$ affects $P_0$ and find that both have detrimental effect: in Fig. \[S8&9\](a), $P_0$($U_{Mn}$,0,0) shows a decrease from 33.4% to 11.4% with $U_{Mn}$ varying from 0 to 7 eV. In comparison, the effect of $U_{Ce}$ is milder with $P_0$(0,0,$U_{Ce}$) decreasing from 33.4% to 28%.   If $U_{Mn}$ and $U_{Ce}$ are set to 0, $P_0$ increases to a large value of 66% for $U_{Ni}$= 7 eV $i.e.$ for $U(0,7,0)$ (black filled squares in Fig. \[S8&9\](a)). On the other hand, a comparison of $P_0$($U_{Ni}$) for (0,$U_{Ni}$,0), (5,$U_{Ni}$,0), (4.5,$U_{Ni}$,7) shows that the extent of increase of $P_0$ is clearly arrested when $U_{Mn}$ and $U_{Ce}$ are non-zero. These results refute an earlier counterintuitive report[@Bahramy10], which concluded that $U_{Mn}$ increases $P_0$, while neither $U_{Ni}$ nor $U_{Ce}$ have any influence on $P_0$ (see Supplementary discussion SD1). The calculated magnetic moments show that the total moment of CeMnNi$_4$ is quite large $e.g.$ 5.43 $\mu_B$ for $U(4.5,6.5,7)$, the main contribution coming from the Mn spin moment (4.31 $\mu_B$). Fig. \[S8&9\](b) shows that both the total moment as well as the Ni spin moment increase with $U_{Ni}$, $e.g$ for $U(4.5,0,7)$ the total moment (Ni spin moment) is 5.15 (0.19) $\mu_B$, whereas for $U(4.5,6.5,7)$ it is 5.43 (0.3) $\mu_B$. The increase in the Ni spin moment is because of the shift of the Ni 3$d$ minority spin  states towards $E_F$ (Fig. \[spol\](d)) resulting in a decrease of the integrated occupied minority spin PDOS, while the majority spin PDOS remains largely unchanged. It may be noted that the total moment of 5.43 $\mu_B$ for $U(4.5,6.5,7)$ is somewhat overestimated compared to the experimental value of 4.95 $\mu_B$ from magnetization measurement at 5 K[@Singh06]. Interestingly, we find that the total magnetic moment decreases with increasing disorder (Fig. \[S8&9\](c)). This can be ascribed to the difference of the Mn$_{\rm Ni}$ (Mn atom in Ni position) and Mn$_{\rm Mn}$ (Mn atom in Mn position) 3$d$ spin-polarized PDOS, the latter having considerably reduced exchange splitting (Fig. S2). This difference is related to the change in hybridization due to different nearest neighbor configurations (Fig. S1). The local moment of Mn$_{\rm Ni}$ is thus substantially smaller (2.8 $\mu_B$) compared to Mn$_{\rm Mn}$ (3.8 $\mu_B$). Although the local moments hardly vary, the proportion of Mn$_{\rm Ni}$ increases with $x$, resulting in a decrease of the total moment. Thus, it can be argued that the overestimation of the total moment by theory with $U$(4.5,6.5,7) mentioned above could be somewhat compensated by its decrease caused by anti-site disorder. An additional interesting outcome of our study  is the demonstration of a valence state transition $i.e.$ a change of the valency of Ce between the bulk and the surface. Valence state transition could significantly alter the surface electronic structure compared to the bulk. It was first reported in Sm metal[@Wertheim78]  and later in binary Ce intermetallic compounds[@Laubschat90].   From the analysis of the Ce 3$d$ core-level spectra using HAXPES and XPS and using a simplified version of the Anderson single-impurity model[@Gunnarsson83] proposed by Imer and Wuilloud (IW)[@Imer87], we show that the Ce 4$f$ occupancy in the ground state ($n_f$) turns out to be 0.8 in the bulk, indicating a mixed valent state with 20% Ce in $f^0$ (Ce$^{4+}$) while 80% in $f^1$ (Ce$^{3+}$) configuration, where $f^{0}$ and $f^{1}$ are the satellite peaks in the Ce 3$d$ spectrum related to 3$d^{9}$4$f^{0}$ and 3$d^{9}$4$f^{1}$ final states, respectively[@Hillebrecht82; @Fuggle83]. In contrast, from the surface sensitive Ce 3$d$ XPS spectrum, $n_f$ increases to 0.98 and thus the surface has predominantly 3$d^{9}$4$f^{1}$ (Ce$^{3+}$) ground state. Thus, in the bulk, the Ce 4$f$ electron transfers to the valence states comprising primarily of Ni 3$d$ states making CeMnNi$_4$ a mixed valent system with 4$f$ occupancy of $n_f$= 0.8. However, at the surface, the reduced hybridization between the Ce 4$f$ and unsaturated 3$d$ states results in a lowering of the Ce 4$f$ states further below $E_F$. This increases the occupancy of the Ce 4$f$ level ($n_f$= 0.98) and results in the valence state transition. The detailed discussion on the valence state transition and comparison with surface sensitive XPS is provided in the Supplementary discussion SD2. In conclusion, we settle the long standing debate about the electronic structure of CeMnNi$_4$. We establish the importance of both anti-site disorder and electron-electron correlation in explaining its intriguing properties. Our work fundamentally alters the general notion that anti-site disorder is detrimental for spin polarization. We hope it will motivate further experimental work on CeMnNi$_4$ and related materials, mainly because disorder could be controlled and $P_0$ further enhanced. We find that the total magnetic moment exhibits contrasting behaviour, it decreases with $x$, but increases with $U_{Ni}$.  A valence state transition that originates due to the weakened hybridization on the surface is demonstrated. Our study highlights the power of HAXPES in combination with density functional theory for clarifying the electronic structure and properties of multiply-correlated materials with inherent anti-site disorder. [**Acknowledgments:**]{} The experiments were carried out at PETRA III of Deutsches Elektronen-Synchrotron, a member of Helmholtz-Gemeinschaft Deutscher Forschungszentren. Financial support by the Department of Science and Technology, Government of India within the framework of India@DESY collaboration is gratefully acknowledged. We would like to thank W. Drube and C. 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[*Supplementary material to the paper entitled:*]{}\   \ [**Influence of anti-site disorder and electron-electron correlations on the electronic structure of CeMnNi$_4$** ]{}   \   \   \ Pampa Sadhukhan$^{1,\dagger}$, Sunil Wilfred D$^{\prime}$Souza$^{1,{\dagger}*}$, Vipin Kumar Singh$^{1}$, Rajendra Singh Dhaka$^{1a}$, Andrei Gloskovskii$^{2}$, Sudesh Kumar Dhar$^{3}$, Pratap Raichaudhuri$^{3}$, Ashish Chainani$^{4}$, Aparna Chakrabarti$^{5}$, Sudipta Roy Barman$^{1}$\ $^{\dagger}$Both the authors have contributed equally to this work.\   \ [ This Supplementary material contains seven figures (S1 to S7), two tables (TABLE-I and II) and two Supplementary discussions (SD1 and SD2, which include figures S8-S13 ).]{} ![The crystal structure of (a) ordered CeMnNi$_4$ with Ce and Mn are placed at 4$a$ (0, 0, 0) and 4$c$ (0.25, 0.25, 0.25) sites, respectively, while Ni is placed at 16$e$ (0.624, 0.624, 0.624) site. The corresponding multiplicities of 4$a$, 4$c$ and 16$e$ atomic sites are 1, 1 and 4, respectively and (b) Disordered CeMnNi$_4$ with 12% ($x$= 0.12) Mn-Ni anti-site disorder. The crystallographic axes are represented by the arrows ($a, b, c$). The entire cubic structure has been rotated by 45$^{\circ}$ in the clockwise direction to provide a better view. Note that the first nearest neighbor (nn) of Mn$_{\rm Mn}$ (Mn atom in Mn position) are three Ni atoms, while the second nn are four Ce atoms. On the other hand, for Mn$_{\rm Ni}$ (Mn atom in Ni position), the first nn are six Ni atoms, while the second nn are three Mn$_{\rm Mn}$ atoms.](S1R1_S2_structure_CeMnNi4_12_final.pdf){width="85mm"} \[S2\_structure\] ![ Spin polarized (a) total DOS and the PDOS for (b) Ni$_{\rm Ni}$ 3$d$, (c) Ni$_{\rm Mn}$ 3$d$, (d) Mn$_{\rm Ni}$ 3$d$, (e) Mn$_{\rm Mn}$ 3$d$ and (f) Ce 4$f$ as a function of Mn-Ni anti-site disorder quantified by $x$= 0, 0.06 and 0.12.](S2R1_S1_Layout6_2_FigS1_new_ver2.pdf){width="93mm"} \[S1\] ![ Spin polarized total DOS and Ni 3$d$, Mn 3$d$ and Ce 4$f$ PDOS as a function of $U_{Mn}$ with $U_{Ni}$= $U_{Ce}$= 0 where for (a) $U_{Mn}$= 0 eV, (b) $U_{Mn}$= 2 eV, (c) $U_{Mn}$= 4 eV, and (d) $U_{Mn}$= 6 eV.](S3R1_Layout75_4_up_dn_dos_MnU_ver4.pdf){width="93mm"} \[S3\] ![Spin polarized total DOS and Ni 3$d$, Mn 3$d$ and Ce 4$f$ PDOS as a function of $U_{Ni}$ with fixed $U_{Mn}$= 4.5 eV and $U_{Ce}$= 7 eV where for (a) $U_{Ni}$= 0 eV, (b) $U_{Ni}$= 2 eV, (c) $U_{Ni}$= 4 eV and (d) $U_{Ni}$= 6.5 eV.](S4R1_Layout75_5_up_dn_dos_NiU_ver4.pdf){width="93mm"} \[S4\] ![ The calculated valence band spectrum for $U$(4.5,6.5,7) with the partial contributions from Ni 3$d$, Mn $3d$, Ni 4$s$, Ce 5$d$ and Ce 4$f$ compared with experimental HAXPES VB spectrum. The spectra are staggered along the vertical axis, and the zero for each are shown on the right vertical axis by red horizontal lines. The peaks in the VB are marked by $S, A, B, C, D, E$ and $F$. Light blue dashed arrows show the corresponding partial contributions to these features. $A$ and $B$ originates primarily from Ni 3$d$ like states with small amount of contribution from Ce 5$d$ and Ni 4$s$ in $B$. Mn 3$d$ along with Ni 4$s$ like states show major contribution to $E$. $C$ and $D$ are related to Ni 4$s$, Ce 5$d$ and Ni 3$d$. Features $F$ and $S$ mainly originate from Ni 4$s$ and Ni 3$d$, respectively. The disagreement in the intensity between experiment and theory could be related to the inelastic background that is not considered in the latter.](S5R1_Layout9_3_best_calculated_VB_NiU_ver4.pdf){width="93mm"} \[S5\] ![Spin polarized total, Ni 3$d$, Mn 3$d$ and Ce 4$f$ PDOS in a small range around $E_F$, as a function of anti-site disorder (a) $x$= 0, (b) $x$= 0.06 and (c) $x$= 0.12.[]{data-label="S6"}](S6R1_Layout14_1_FigS6_0p4_range_DOS_x.pdf){width="93mm"} ![Spin polarized total, Ni 3$d$, Mn 3$d$ and Ce 4$f$ PDOS in a small range around $E_F$, as a function of $U_{Ni}$ with $U_{Mn}$= 4.5 eV and $U_{Ce}$ =7 eV for $U_{Ni}$= (a) 0 eV, (b) 2 eV, (c) 4 eV and (d) 6.5 eV. The black ticks show the position of Ni 3$d$ minority spin peak that shifts towards $E_F$ with $U_{Ni}$.[]{data-label="S7"}](S7R1_Layout20_1_FigS7_updn_DOS_UNi_0p4range.pdf){width="93mm"} ------ ------ ------ ------ ----- ----- ------ ------ ----- ----- 0 33.4 13.4 10.4 7 3 18.2 4.9 3.3 5.8 0.04 40 18.6 9.2 7.5 4 20.6 9.1 3.2 5.2 0.06 45 21.4 9.1 8.4 5 26.7 15 3 5.2 0.08 47 23.1 9.2 9.3 6 38.6 25.3 2.7 5.6 0.10 48.4 24.3 8.8 8.9 6.5 45 32.2 2.4 5.4 0.12 50.1 25.3 8.8 9 7 54 42 2 5.2 ------ ------ ------ ------ ----- ----- ------ ------ ----- ----- \[pol\] **Supplementary Discussions** ============================= SD1: Discussion on the spin polarization reported in Ref. 9 ----------------------------------------------------------- Bahramy $et~al.$[@Bahramy10] found the cubic phase of CeMnNi$_4$ to be stable when $U_{Mn}$ is turned on and reported that the Mn 3$d$ states shift to lower energies with $U_{Mn}$. We also find similar behavior of the Mn 3$d$ states. However, the authors also reported that the static spin polarization ($P_0$) increases substantially with $U_{Mn}$, and commented that $U_{Ni}$ or $U_{Ce}$ have no effect on $P_0$ or any other ground state properties. This is not in agreement with the spin polarization results we obtain here and so we discuss below the possible reason for this. $P_0$, being proportional to the difference of $n_\downarrow$($E_F$) and $n_\uparrow$($E_F$), is highly sensitive to any small change of the DOS at $E_F$. So, it is very unlikely that the states at $E_F$ that are dominated by Ni 3$d$ PDOS will not be influenced by $U_{Ni}$, whose value (6.5 eV) we find to be larger than $U_{Mn}$ (4.5 eV) from the comparison of the experimental HAXPES VB with DFT calculations performed by us using SPRKKR. Our results clearly show that $U_{Ni}$ has large influence on both the position of the VB peaks $A$ and $B$ (Fig. 2 of MS) as well as the DOS close to $E_F$ (Fig. 3(d)). This leads to large increase of $P_0$ (Fig. 3b, Fig. S7 and Table[**I**]{}). On the other hand, the spin polarized DOS near $E_F$ with $U_{Ni}$ was not shown in Ref.. Fig. 4(b,c) of that work shows the total and $s$, $d$, and $f$ PDOS for $U_{Mn}$=0 and 6 eV. Based on this figure the authors conclude that $P_0$ increases from 10% to 30% with $U_{Mn}$ increasing from 0 to 6 eV. We have analyzed this figure critically and find that the increase in $P_0$ in Ref.  is due to an unusual variation of Ce 4$f$ PDOS at $E_F$ for $U_{Mn}$= 6. In their calculation, Ce 4$f$ PDOS at $E_F$ has a value of 0.3 for the minority spin, whereas in the majority spin PDOS it is largely reduced to 0.02. Thus, increase in $P_0$ to 30% for $U_{Mn}$= 6 eV results primarily from the variation of the Ce 4$f$ spin polarized states. Thus, we find $P_{0_{{\rm Ce}4f}}$ (defined in Table[**I**]{} of SM) increases from 7.9% to 30%, a whopping 380% increase, while $P_0$ increases from 10 to 30%. Thus at $U_{Mn}$= 6 eV, the whole spin polarization is contributed by Ce 4$f$ states only. The above discussed effect of $U_{Mn}$ on $P_0$ and Ce 4$f$ states is [**unlikely**]{}, since the Mn atom is surrounded by the Ni atoms and Ce is only the second nearest neighbor at a large distance of 3.02Å. Rather, one would expect that the Ni 3$d$ states that are dominant at $E_F$ would contribute to $P_0$, but between Fig. 4(b) and Fig. 4(c) of Ref. , the 3$d$ states hardly change (for $U_{Mn}$=0 these are 0.3 and 0.5 states/eV for majority and minority spin, respectively and for $U_{Mn}$= 6 eV these are 0.2 and 0.4 states/eV for majority and minority spin, respectively). Thus, strangely, $P_{0_{3d}}$ $i.e.$ spin polarization due to the 3$d$ states remains essentially similar ($P_{0_{3d}}$= 13% for $U_{Mn}$= 0 and $P_{0_{3d}}$= 16% $U_{Mn}$= 6 eV). Thus, while $P_{0_{{\rm Ce}4f}}$ increases by 380%, $P_{0_{3d}}$ hardly changes. This seems to be an unphysical result. We find that the increase of $P_0$ for both anti-site disorder ($x$) as well as electron-electron correlation of the Ni 3$d$ electrons ($U_{Ni}$) is primarily due to the changes in the Ni 3$d$ PDOS. Table[**I**]{} of SM clearly shows how only $P_{0_{{\rm Ni}3d}}$ increases as $P_0$, while $P_{0_{{\rm Ce}4f}}$ and $P_{0_{{\rm Mn}3d}}$ remain almost unchanged and thus do not play any role in the enhancement of $P_0$. ![Spin polarized total DOS and total $s$, $p$, $d$ and $f$ PDOS of CeMnNi$_{4}$ with (a) $U_{Mn}$=0 eV (b) $U_{Mn}$= 6 eV for comparison with Fig. 4(b,c) of Ref. .[]{data-label="SPdoswithUMn"}](S8R1_Layout3_1_SP_pdos_contribution_UMn.pdf){width="93mm"} \[S8\] For comparison with Ref. , in Fig. S8, we show the total and PDOS with $U_{Mn}$=6 eV, with $U_{Ni}$= $U_{Ce}$=0. We find that $P_0$ decreases from 33.4% to 13.2% with $U_{Mn}$ varying from 0 to 6 eV (Fig. 4(a) in MS). It is evident that the total $f$ PDOS is less than the total $d$ PDOS over the entire range and most importantly remains similar between $U_{Mn}$=0 and $U_{Mn}$= 6 eV. Thus, $P_{0_{4f}}$ is almost unchanged (rather decreases slightly) from 7.3% to 5.3% from $U_{Mn}$= 0 to 6 eV. This is in stark disagreement with the 380% increase of $P_{0_{4f}}$ that can be concluded from Ref. [@Bahramy10]. SD2: Surface valence transition ------------------------------- The Ce 3$d$ core-level spectrum displays two sets of triplet peaks corresponding to the spin-orbit split components (Fig. S\[Ce3d\]). The most intense among the triplet peaks is the $f^1$ satellite associated with a poorly screened 3$d^{9}$4$f^{1}$ final state occurring at 902.8 eV and 884.4 eV binding energies. The two additional satellite peaks that occur at relatively higher and lower binding energies are referred to as $f^{0}$ and $f^{2}$, respectively. The well screened $f^{2}$ satellite has an extra screening electron with 3$d^{9}$4$f^{2}$ final state, while the $f^{0}$ satellite is related to 3$d^{9}$4$f^{0}$ final state[@Hillebrecht82; @Fuggle83]. Notable in Fig. S\[Ce3d\] is the large $f^0$ ![ Ce 3$d$ core-level spectra (black dots) recorded with (a) 8 keV (HAXPES) and (b) 1.48 keV (XPS) photon energies. The spectra have been fitted (red line) using a least square (LS) error minimization routine and the $f^{n}$ satellite components for Ce 3$d_{3/2}$ are shown.   The calculated Ce 3$d_{3/2}$ spectra using IW theory along with the $f^{n}$ satellites are shown at the bottom and the residuals of fitting (black line) are shown at the top of each panel. []{data-label="Ce3d"}](S9R1_4_Layout31_1_fig4_Ce3d_refit_ver4.pdf){width="93mm"} intensity in HAXPES, which decreases drastically in soft x-ray PES (XPS). In order to extract quantitative information, the Ce 3$d$ core-level spectra were fitted using a least square error minimization routine with each peak assigned a Doniach and $\breve{S}$unji$\acute{c}$ (DS) line shape[@Doniach70]. This was further convoluted with a Gaussian function of fixed width to represent the instrumental broadening.  Since Ni 2$p$ that appears close in binding energy to Ce 3$d$ might contribute to the intensity in the Ce 3$d$ region, the Ni 2$p$ main and satellite peaks were also included in the fitting scheme. The whole region including Ni 2$p$ along with the components is shown in Fig. S10. A total 10 DS line shapes were used, 6 for Ce 3$d$ comprising of the three $f^n$ components for each spin-orbit (s.o.) peaks and 4 for Ni 2$p$ representing the main peak and satellite for both the s.o. components. The parameters defining each DS line shape are the intensity, position, width ($\Gamma$) and asymmetry parameter ($\alpha$). A Touguaard background was also included in the fitting scheme, where the $B_1$ parameter was varied and the $C$ parameter was kept fixed at 1643 eV$^2$[@Tougaard89]. Thus, a total of 35 parameters defined the full spectral shape including Ce 3$d$ and Ni 2$p$. However, some reasonable constraints were needed, for example (i) the life time broadening of $f^0$ for Ce 3$d_{3/2}$ was constrained to be greater than or equal to $f^0$ for Ce 3$d_{5/2}$, (ii) $\alpha$ was kept equal for all Ce 3$d$ DS components, (iii) for XPS fitting, the satellites of Ni 2$p$ have same width as HAXPES. ![Ce 3$d$ core-level spectra (black circles, includes Ni 2$p$ region) with (a) HAXPES compared with (b) XPS. The spectra has been fitted (red line) using a least square (LS) error minimization routine, and the residuals (black line) are shown at the top of each panel. The different components such as Ce 3$d$ $f^{0}$, $f^{1}$, and $f^{2}$, Ni 2$p$ main peaks and satellites as well as a Tougaard background are shown.[]{data-label="S10"}](S10R1_Layout30_S8_Ce3d_refit_ver3.pdf){width="93mm"} \[11\] From the least square fitting, we find that the normalized intensity of $f^0$ ($I_n(f^0)$) is 0.15 for HAXPES, where $I_n(f^0)$= $I(f^0)$/$\displaystyle\sum_{n=0}^2 I(f^n) $) (Table[**II**]{} of SM). Such large intensity of $f^0$ having almost similar height as $f^1$ is unusual and has not been observed in other Ce based intermetallic compounds[@Yano08; @Sundermann16]. In contrast, $I_n(f^0)$ is an order of magnitude less (0.04) in XPS.  This could be related to the bulk sensitivity of HAXPES with mean free path ($\lambda$) of 91 Å  for Ce 3$d$ electrons while XPS is surface sensitive with $\lambda$= 13 Å[@tpp2m].  In order to understand the differences between the above discussed bulk and surface Ce 3$d$ spectra, we turn to a simplified version of the Anderson single-impurity model[@Gunnarsson83] proposed by Imer and Wuilloud (IW), where the extended valence states are considered as a band of infinitely narrow width[@Imer87]. The Ce 3$d$ spectrum is calculated as a function of the energy of the unhybridized $4f$ state relative to $E_F$ ($\epsilon_f$), Coulomb repulsion between 4$f$ electrons at the same site ($U_{ff}$), Coulomb attraction between 4$f$ electron and the final-state core hole ($U_{fc}$), and hybridization between the 4$f$ states and the conduction band ($\Delta$). ------ --------- ----------------------- ------- ------ ------ ----- ---- -------------- ------ 8 $f^{0}$ [**0.15$\pm$0.01**]{} 913.5 0 [**0.15**]{} 0 $f^{1}$ 0.6$\pm$0.1 902.8 10.7 -1.0 1.5 10 0.53 10.5 $f^{2}$ 0.25$\pm$0.1 897.8 15.7 0.32 16.1 1.48 $f^{0}$ [**0.04$\pm$0.01**]{} 914.3 0 [**0.04**]{} 0 $f^{1}$ 0.5$\pm$0.1 904.1 10.2 -2.5 1.1 8 0.62 10.1 $f^{2}$ 0.45$\pm$0.1 899.9 14.4 0.34 14.7 ------ --------- ----------------------- ------- ------ ------ ----- ---- -------------- ------ \[Ce3dtable\] The above mentioned parameters are varied such that the $f^n$ satellites of the calculated Ce 3$d_{3/2}$ spectrum have similar intensities ($I_n$) and energy separations between $f^0$ and $f^n$ ($\delta_{0n}$), as obtained from the fitting of the experimental spectra. For example, besides the large change in $I_n$($f^0$), the binding energies of the $f^n$ satellites are lower in HAXPES (this is not due to recoil effect[@Fadley10], see Fig. S11), resulting in different $\delta_{0n}$ as shown in Table[**II**]{}. ![Ni 2$p$ core level spectra of CeMnNi$_4$ using 8, 6 and 1.48 kV photon energies, normalized to same height at the Ni 2$p_{3/2}$ peak and staggered along the vertical axis for clarity of presentation. The recoil effect that has been observed in the HAXPES spectra of light materials[@Fadley10] is absent here since it comprises of heavier 3$d$ and rare earth elements. The recoil effect, if present causes a uniform shift of the peaks to higher binding energies that increases with the kinetic energy of the electrons, which in turn depends on the photon energy used. We confirm the absence of any recoil effect here from the Ni 2$p$ spectra taken with different photon energies where any shift of the peaks for different photon energies is absent.[]{data-label="S11"}](S11R1_Layout14_1_Ni2p_diff_hv.pdf){width="93mm"} ![The XPS valence band spectrum of CeMnNi$_4$ taken with 1.25 keV photon energy (black line with dots) compared with calculated VB along with the Ni 3$d$, Mn 3$d$ ($\times$5) and Ce 4$f$ partial contributions. []{data-label="S12"}](S12R1_Layout19_2_XPS_calculated_VB_U7U4p5U6p5.pdf){width="93mm"} ![Spin integrated Ni 3$d$ and Ce 4$f$ PDOS for CeMnNi$_4$ with optimum $U(4.5,6.5,7)$. The integration of the Ce 4$f$ PDOS (blue line) that shows the occupancy ($n_f$) on the right axis.[]{data-label="S13"}](S13R1_Layout18_1_Ce_4f_Ni3d_integration.pdf){width="90mm"}   In order to simulate the Ce 3$d$ HAXPES spectra using IW theory, we note that $I_n$($f^0$) increases sensitively with $\epsilon_f$, and so this parameter is varied keeping the others fixed at the values suggested for Ce compounds ($\Delta$= 1.5 eV, $U_{ff}$= 7 eV, $U_{fc}$= 10 eV)[@Imer87]. For $\epsilon_f$= -1 eV, we find $I_n$($f^0$)= 0.15 in excellent agreement with experiment; and the other quantities such as $\delta_{0n}$, $I_n$($f^1$) and $I_n$($f^2$) are also in good agreement (Table [**II**]{}). The calculated spectrum obtained with $\epsilon_f$= -1 eV, $\Delta$= 1.5 eV, $U_{ff}$= 7 eV, $U_{fc}$= 10 eV is shown at the bottom of Fig. S\[Ce3d\](a), where the $f^n$ satellites have been broadened by their respective widths obtained from the fitting and a background[@Tougaard89] has also been added. The $f$ occupancy in the ground state ($n_f$) turns out to be 0.8, indicating a mixed valent state with 20% Ce in $f^0$ (Ce$^{4+}$) while 80% in $f^1$ (Ce$^{3+}$) configuration. In order to simulate the Ce 3$d$ XPS spectrum, we decrease $\epsilon_f$ to -2.5 eV from the HAXPES value of -1 eV and obtain $I_n$($f^0$)= 0.04. But concomitantly, both $\delta_{01}$ (=12.1 eV) and $\delta_{02}$ (=18.9 eV) become larger than experimental values of 10.2 eV and 14.4 eV, respectively (Table[**II**]{}). In order to decrease $\delta_{0n}$, both $\Delta$ and $U_{fc}$ need to be decreased, and thus, we obtain a good agreement with experiment for $\epsilon_f$= -2.5 eV, $\Delta$= 1.1 eV, $U_{fc}$= 8 eV, and $U_{ff}$= 7 eV  (bottom of Fig. S\[Ce3d\](b)). Due to the decrease of $\epsilon_f$, $n_f$ increases to 0.98, and thus, in contrast to bulk, at the surface Ce has predominantly 3$d^{9}$4$f^{1}$ (Ce$^{3+}$) ground state. Thus, in the bulk, since $\epsilon_f$ (= -1 eV) is closer to $E_F$ and $\Delta$ is larger, the Ce 4$f$ electron transfers to the valence states comprising of primarily Ni 3$d$ states making CeMnNi$_4$ a mixed valent system with 4$f$ occupancy of $n_f$= 0.8. However, at the surface, the reduced hybridization between the Ce 4$f$ and unsaturated 3$d$ states results in a lowering of the Ce 4$f$ states further below $E_F$. This increases the occupancy of the Ce 4$f$ level ($n_f$= 0.98) and results in the surface valence transition. Decrease in $U_{fc}$ from about 10 eV to 8 eV at the surface is also a manifestation of this transition possibly caused by the more efficient screening of the core hole due to increased $n_f$. It might be noted that although the surface valence transition is clearly manifested in the Ce 3$d$ core-level spectra, it does not however result in appearance of any Ce 4$f$ peak in the XPS VB (Fig. S12),   which could be expected due to enhanced $n_f$ at the surface. Firstly, this happens because the occupied part of Ce 4$f$ PDOS from -3 eV to $E_F$ is largely diminished, broad and featureless (Fig. S13).  Its integration (blue line) up to $E_F$ gives $n_f$= 0.96 in the bulk from DFT, which is in reasonable agreement with $n_f$= 0.8 from IW method, considering the assumptions of the latter model calculation[@Imer87]. The increase of $n_f$ by 0.18 at the surface obtained from IW method would manifest itself through a small shift of the Ce 4$f$ PDOS by 0.25 eV (obtained from integration of PDOS that gives $n_f$= 1.14) to lower energy in the rigid band model (red dashed line in Fig. S13). Thus, the main peak of Ce 4$f$ still remains above $E_F$ at the surface. Secondly, occupied Ni 3$d$ PDOS as well as its photoemission cross-section[@Yeh85] are much larger than Ce 4$f$ (Fig. S12, S2-S4) resulting in complete domination of the VB by Ni 3$d$ states at low photon energies too. This is reconfirmed by the relative contributions of Ni 3$d$ and Ce 4$f$ to the calculated XPS VB in Fig. S12.
--- abstract: 'In this paper, throughput and energy efficiency of cognitive multiple-input multiple-output (MIMO) systems operating under quality-of-service (QoS) constraints, interference limitations, and imperfect channel sensing, are studied. It is assumed that transmission power and covariance of the input signal vectors are varied depending on the sensed activities of primary users (PUs) in the system. Interference constraints are applied on the transmission power levels of cognitive radios (CRs) to provide protection for the PUs whose activities are modeled as a Markov chain. Considering the reliability of the transmissions and channel sensing results, a state-transition model is provided. Throughput is determined by formulating the effective capacity. First derivative of the effective capacity is derived in the low-power regime and the minimum bit energy requirements in the presence of QoS limitations and imperfect sensing results are identified. Minimum energy per bit is shown to be achieved by beamforming in the maximal-eigenvalue eigenspace of certain matrices related to the channel matrix. In a special case, wideband slope is determined for more refined analysis of energy efficiency. Numerical results are provided for the throughput for various levels of buffer constraints and different number of transmit and receive antennas. The impact of interference constraints and benefits of multiple-antenna transmissions are determined. It is shown that increasing the number of antennas when the interference power constraint is stringent is generally beneficial. On the other hand, it is shown that under relatively loose interference constraints, increasing the number of antennas beyond a certain level does not lead to much increase in the throughput.' author: - | \ [^1] [^2] title: On the Throughput and Energy Efficiency of Cognitive MIMO Transmissions --- [1.8]{} cognitive radio, effective capacity, energy efficiency, minimum energy per bit, multiple-input multiple-output (MIMO), quality of service (QoS) constraints, throughput. [1.8]{} Introduction ============ Cognitive Radio (CR), which has emerged as a method to tackle the spectrum scarcity and variability in both time and space, calls for dynamic access strategies that adapt to the electromagnetic environment [@barbarossa]. Performance of cognitive radio systems has been studied extensively in recent years, and a detailed description of different CR models and an overview of recent approaches can be found in [@akyildiz], [@Goldsmith2] and [@Zhao]. For instance, three different paradigms, namely underlay, overlay and interweave operation of cognitive radio systems, were discussed in [@Goldsmith2]. In underlay CR networks, cognitive secondary users (SUs) can coexist with the primary users (PUs) and transmit concurrently as long as they adhere to strict limitations on the interference inflicted on the PUs. This model is also known as spectrum sharing. On the other hand, in interweave CR networks, SUs initially perform channel sensing and opportunistically access only the spectrum holes in which the primary users are inactive. These two methods of spectrum sharing and opportunistic spectrum access can also be combined for improved performance. For instance, Kang *et al.* in [@Zhang] analyzed a hybrid model in which SUs first sense the frequency bands and detect the PU activity. Subsequently, cognitive radio transmission is performed at two different power levels depending on the sensed PU activity. More specifically, if the PUs are sensed to be active, secondary transmission still occurs but with reduced power level in order to lower the interference within tolerable levels. In such modes of cognitive operation, sensing the activities of PUs is a critical issue that has been studied and analyzed extensively (see e.g., [@Zhao2], [@Zhao3]) since the inception of the CR concept. Another advancement in communications technology is multiple-antenna communications. It is well-known that employing multiple antennas at the receiver and transmitter ends of a communication system can improve the performance levels by providing significant gains in the throughput and/or reliability of transmissions. Therefore, there has been much interest in understanding and analyzing multiple-input multiple-output (MIMO) channels and numerous comprehensive studies have been conducted [@goldsmith], [@telatar]. In most studies, ergodic Shannon capacity formulations are considered as the performance metrics [@Lozano], [@Lozano2], [@Sandhu]. For instance, the authors in [@Lozano] and [@Lozano2] studied multiple-antenna ergodic channel capacity and provided analytical characterizations of the impact of certain factors such as antenna correlation, co-channel interference, Ricean factors, and polarization diversity. It should be noted that ergodic capacity generally does not take into account any delay, buffer, or queueing constraints at the transmitter. In [@Gursoy_IT], the throughput of MIMO systems in the presence of statistical queuing constraints was investigated. Effective capacity was employed as the metric to measure the performance under quality-of-service (QoS) constraints. Effective capacity characterizes the maximum constant arrival rate that can be supported by a system under statistical limitations on buffer violations [@Wu]. There have been several studies on effective capacity in various communication settings [@Tang], [@Tang2]. Recently, the authors in [@Mittel] considered the maximization of effective capacity in a single-user multi-antenna system with covariance knowledge, and the authors in [@Liu] studied the effective capacity of a class of multiple-antenna wireless systems subject to Rayleigh flat fading. Recently, cognitive MIMO radio models have also been considered since having multiple antennas can provide higher performance levels for the SUs and lead to better protection of PUs. Modeling a channel setting with a single licensed user and a single cognitive user, that is equivalent to an interference channel with degraded message sets, the authors in [@sridharan] focused on the fundamental performance limits of a cognitive MIMO radio network, and they showed that under certain conditions, the achievable region is optimal for a portion of the capacity region that includes the sum capacity. In [@YingJun], three scenarios, namely when the secondary transmitter (ST) has complete, partial, or no knowledge about the channels to the primary receivers (PRs), was considered, and maximization of the throughput of the SU, while keeping the interference temperature at the PRs below a certain threshold, was investigated. Furthermore, in [@feifei], the authors proposed a practical CR transmission strategy consisting of three major stages, namely, environment learning that applies blind algorithms to estimate the spaces that are orthogonal to the channels from the PR, channel training that uses training signals and employs the linear-minimum-mean-square-error (LMMSE)-based estimator to estimate the effective channel, and data transmission. Considering imperfect estimations in both learning and training stages, they derived a lower bound on the ergodic capacity that is achievable by the CR in the data-transmission stage. In another study [@RuiZhang], the authors proposed a practical cognitive beamforming scheme that does not require any prior knowledge of the CR-PR channels, but exploits the time-division-duplex operation mode of the PR link and the channel reciprocities between CR and PR terminals, utilizing an idea called effective interference channel, that is estimated at the CR terminal via periodically observing the PR transmissions. It was also shown in [@Samir] that the asymptotes of the achievable transmission rates of the opportunistic (secondary) link are obtained in the regime of large numbers of antennas. Another study of cognitive MIMO radios was conducted in [@Zhang-Liang]. The above-mentioned references have not addressed considerations related to energy efficiency and QoS provisioning in cognitive MIMO channels. In our prior work, we studied the impact of QoS requirements in single-antenna cognitive radio systems. In particular, we considered a CR model in which SUs transmit with two different transmission rates and power levels depending on the activities of PUs under QoS constraints. In [@paper1], the ST senses only one channel and then depending on the channel sensing results, it chooses its transmission policy, whereas in [@paper2] the ST senses more than one channel and chooses the best channel for transmission under interference power limits and QoS constraints. In [@paper3], effective capacity limits of a CR model is analyzed with imperfect channel side information (CSI) at the transmitter and the receiver. In this article, we focus on a cognitive MIMO system operating under QoS constraints. In particular, we investigate the achievable throughput levels and also study the performance in the low-power regime in order to address the energy efficiency. We analyze the impact of imperfect sensing results and interference limitations on the performance, and determine energy-efficient transmission strategies in the low-power regime. In the system model, we consider two different transmission policies depending on the activities of PUs and interference power threshold required to protect the PUs. Essentially, we have a hybrid, sensing-based spectrum sharing model of cognitive radio operation as described in [@Zhang]. We consider a general cognitive MIMO link where fading coefficients have arbitrary distributions and are possibly correlated across antennas. Moreover, we model the received interference signals from the primary transmitters correlated as well. We assume that the ST and secondary receiver (SR) have perfect side information regarding their own channels. The contributions of the paper can be summarized as follows: 1. We identify a joint state-transition model, considering the reliability of the transmissions and taking into account the channel sensing decisions and their correctness. 2. We provide a formulation of the throughput metric (effective capacity) in terms of transmission rates and state transition probabilities which depend on sensing reliability and primary user activity. 3. We obtain expressions for the first and second derivative of the effective capacity at $\textsc{snr}=0$, and determine the minimum energy per bit in the presence of QoS limitations and imperfect sensing results. The organization of the paper is as follows. We provide the cognitive MIMO radio model and describe the transmission power and interference constraints in Section \[channel model\]. In Section \[state transition\], we construct a state transition model for CR transmission and identify the throughput under QoS constraints, and show the relation between the effective capacity and ergodic capacity. Finding the first and second derivatives of effective capacity at $\textsc{snr}=0$, we analyze in Section \[effective capacity low power\] the energy efficiency in the low-power regime. In Section \[Numeric\], we provide numerical results. We conclude in Section \[Conclusion\]. Proofs are relegated to the Appendix. Channel Model, Power Constraints, and Input Covariance {#channel model} ====================================================== Channel Model ------------- As seen in Figure \[fig:resimMIMO\], we consider a setting in which a single ST communicates with a single SR in the presence of possibly multiple PUs. We consider a cognitive MIMO radio model and assume that the ST and SR are equipped with $M$ and $N$ antennas, respectively. In a flat fading channel, we can express the channel input-output relation as $$\label{input-output 1} \textbf{y}=\textbf{H}\mathbf{x}+\mathbf{n}+\mathbf{s}$$ if the PUs are active in the channel, and as $$\label{input-output 2} \mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{n}$$ if the PUs are absent. Above, $\mathbf{x}$ denotes the $M\times1-$dimensional transmitted signal vector of ST, and $\mathbf{y}$ denotes the $N\times1-$dimensional received signal vector at the SR. In ($\ref{input-output 1}$) and ($\ref{input-output 2}$), $\mathbf{n}$ is an $N\times1-$dimensional zero-mean Gaussian random vector with a covariance matrix $\mathbb{E}\{\mathbf{n}\mathbf{n}^{\dag}\}=\sigma_{n}^{2}\mathbf{I}$ where $\mathbf{I}$ is the identity matrix. In ($\ref{input-output 1}$), $\mathbf{s}$ is an $N\times1-$dimensional vector of the sum of active PUs’ faded signals arriving at the secondary receiver. Considering that the vector $\mathbf{s}$ can have correlated components, we express its covariance matrix as $\mathbb{E}\{\mathbf{s}\mathbf{s}^{\dag}\}=N\sigma_{s}^{2} \mathbf{K}_{s}$ where $\sigma_s^2$ is the variance of each component of $\mathbf{s}$ and ${\mathbf{tr}}(\mathbf{K}_{s})=1$. Finally, in ($\ref{input-output 1}$) and ($\ref{input-output 2}$), $\mathbf{H}$ denotes the $N\times M$ dimensional random channel matrix whose components are the fading coefficients between the corresponding antennas at the secondary transmitting and receiving ends. We consider a block-fading scenario and assume that the realization of the matrix $\mathbf{H}$ remains fixed over a block duration of $T$ seconds and changes independently from one block to another. Power and Interference Constraints ---------------------------------- We assume that the SUs initially perform channel sensing to detect the activities of PUs, and then depending on the channel sensing results, they choose the transmission strategy. More specifically, if the channel is sensed as busy, the transmitted signal vector is $\mathbf{x}_1$. Otherwise, the signal is $\mathbf{x}_2$. When the channel is sensed as busy, the average energy of the channel input is $$\begin{aligned} \label{power constraint 1} \mathbb{E}\{||\mathbf{x}_{1}||^{2}\}&=\frac{P_{1}}{B}.\end{aligned}$$ On the other hand, if the channel is detected to be idle, the average energy becomes $$\label{power constraint 2} \mathbb{E}\{||\mathbf{x}_{2}||^{2}\} = \frac{P_{2}}{B}.$$ In ($\ref{power constraint 1}$) and ($\ref{power constraint 2}$), $B$ is the bandwidth of the system. Note that under the assumption that $B$ complex input vectors are transmitted every second, the above energy levels imply that the transmission powers are $P_1$ and $P_2$, depending on the sensing results. We first note that $P_1$ and $P_2$ are upper bounded by $P_{\max}$, which represents the maximum transmission power capabilities of cognitive transmitters. In a cognitive radio setting, transmission power levels are generally further restricted in order to limit the interference inflicted on the PUs. As a first measure, we assume that $P_{1}=\mu P_{2}$ where $0 \leq \mu \leq1$. Hence, smaller transmission power is used when the channel is sensed as busy, and we basically have $$\begin{gathered} P_1 \le P_2 \le P_{\max}.\end{gathered}$$ Additionally, we consider a practical scenario in which errors such as miss-detections and false-alarms possibly occur in channel sensing. We denote the correct-detection and false-alarm probabilities by $P_d$ and $P_f$, respectively. We note the following two cases. When PUs are active and this activity is sensed correctly (which happens with probability $P_d$ or equivalently $P_d$ fraction of the time on the average), then SUs transmit with average power $P_1$. On the other hand, if the PU activity is missed in sensing (which occurs with probability $1-P_d$), SUs send the information with average power $P_2$. In both cases, PUs experience interference proportional to the product of the transmission power, average fading power, and path loss in the channel between the ST and PUs. In order to limit the average interference, we impose the following constraint $$\label{Power Threshold} P_{d}P_{1}+(1-P_{d})P_{2}\leq P_{int}$$ where $P_{int}$ can be seen as the average interference constraint normalized by the average fading power and path loss[^3]. We note that a similar formulation for the average interference constraint was considered in [@Zhang]. Noting the assumption that $P_{1}=\mu P_{2}$ for some $\mu \in [0,1]$, we can rewrite ($\ref{Power Threshold}$) as $$\label{Power Threshold_new} P_{d}\mu P_{2}+(1-P_{d})P_{2}\leq P_{int},$$ which implies that $ P_2 \le \frac{P_{int}}{P_d \mu + (1-P_d)}. $ Considering the maximum of the average power, we can write $$\begin{gathered} \label{eq:P2upperbound} P_2 \le \min\left\{P_{\max}, \frac{P_{int}}{P_d \mu + (1-P_d)}\right\}.\end{gathered}$$ Note that for given $\mu$ and detection probability $P_d$, if the interference constraints are relatively relaxed and we have $\frac{P_{int}}{P_d \mu + (1-P_d)} \ge P_{\max}$, then we can choose to operate at $P_2 = P_{\max}$ and $P_1 = \mu P_{\max}$. Otherwise, interference constraints will dictate the transmission power levels. From (\[Power Threshold\]), we can also, for given $P_2$, $P_{int}$ and $P_d$, obtain $$\begin{gathered} \label{Power Threshold_new_values} \mu\leq \min\left\{\max\left\{\frac{P_{int}-P_{2}(1-P_{d})}{P_{2}P_{d}}, 0 \right\}, 1 \right\}.\end{gathered}$$ From above, we see that if $P_2 (1-P_d) \ge P_{int}$, then $\mu = 0$ and hence no transmission is performed by the ST when the channel is sensed as busy. In order to illustrate some of the interactions between the parameters discussed above, we plot, in Fig. $\ref{fig:fig3}$, the ratio $\mu = \frac{P_1}{P_2}$ as a function of $P_{2}$, the power level adapted when the channel is sensed as idle, for different values of power interference constraints $P_{int}$. In all cases, we have $\mu = 1$ for small values of $P_{2}$, while $\mu$ diminishes to zero as $P_2$ increases due to the presence of interference constraints. Note also that we reach $\mu = 0$ at smaller values of $P_2$ under more stringent interference constraints. Input Covariance Matrix ----------------------- Finally, we note that in addition to having different levels of transmission power, directionality of the transmitted signal vectors might also be different depending on the channel sensing results. We define the normalized input covariance matrix of $\mathbf{x}_1$ as $$\label{covariance 1} \mathbf{K}_{x_{1}}=\frac{\mathbb{E}\{\mathbf{x}_{1} \mathbf{x}_{1}^{\dag}\}}{P_{1}/B}$$ if the channel is busy, and that of $\mathbf{x}_2$ as $$\label{covariance 2} \mathbf{K}_{x_{2}}=\frac{\mathbb{E}\{\mathbf{x}_{2} \mathbf{x}_{2}^{\dag}\}}{P_{2}/B}$$ if the channel is idle. Note that the traces of normalized covariance matrices are $$\label{bound 1} {\mathbf{tr}}(\mathbf{K}_{x_{1}}) = 1$$ and $$\label{bound 2} {\mathbf{tr}}(\mathbf{K}_{x_{2}}) = 1.$$ State Transition Model and Channel Throughput {#state transition} ============================================= State Transition Model ---------------------- Depending on channel sensing results and their correctness, we have four scenarios: 1. Channel is busy, and is detected as busy (correct detection), 2. Channel is busy, but is detected as idle (miss-detection), 3. Channel is idle, but is detected as busy (false alarm), 4. Channel is idle, and is detected as idle (correct detection). Using the notation $\mathbb{E}\{(\mathbf{s}+\mathbf{n})(\mathbf{s}+ \mathbf{n})^{\dag}\}=\mathbb{E}\{\mathbf{s}\mathbf{s}^{\dag}\} +\mathbb{E}\{\mathbf{n} \mathbf{n}^{\dag}\}=N\sigma_s^2{\mathbf{K}}_s + \sigma_n^2 {\mathbf{I}}= \sigma_{n}^{2}\mathbf{K}_{z}$ where ${\mathbf{tr}}(\mathbf{K}_{z})=\frac{N(\sigma_{s}^{2}+ \sigma_{n}^{2})}{\sigma_{n}^{2}}$, we can express the instantaneous channel capacities in the above four scenarios as follows: $$\begin{aligned} \label{capacity} &C_{1}=B\max_{\substack{\mathbf{K}_{x_{1}}\succeq0\\{\mathbf{tr}}(\mathbf{K}_{x_{1}}) = 1}}\log_{2}\det\left[ \mathbf{I}+\frac{\mu P_{2}}{B\sigma_{n}^{2}}\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\right] =B\max_{\substack{\mathbf{K}_{x_{1}}\succeq0\\{\mathbf{tr}}(\mathbf{K}_{x_{1}})=1}}\log_{2}\det\left[\mathbf{I}+\mu N\textsc{snr}\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\right],\nonumber\\ &C_{2}=B\max_{\substack{\mathbf{K}_{x_{2}}\succeq0\\{\mathbf{tr}}(\mathbf{K}_{x_{2}})=1}}\log_{2}\det\left[\mathbf{I}+\frac{P_{2} }{B\sigma_{n}^{2}}\mathbf{H}\mathbf{K}_{x_{2}}\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\right] =B\max_{\substack{\mathbf{K}_{x_{2}}\succeq0\\{\mathbf{tr}}(\mathbf{K}_{x_{2}})=1}}\log_{2}\det\left[\mathbf{I}+ N\textsc{snr}\mathbf{H}\mathbf{K}_{x_{2}}\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\right],\nonumber\\ &C_{3}=B\max_{\substack{\mathbf{K}_{x_{1}}\succeq0\\{\mathbf{tr}}(\mathbf{K}_{x_{1}})=1}}\log_{2}\det\left[\mathbf{I}+\frac{\mu P_{2}}{B\sigma_{n}^{2}}\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag}\right] =B\max_{\substack{\mathbf{K}_{x_{1}}\succeq0\\{\mathbf{tr}}(\mathbf{K}_{x_{1}})=1}}\log_{2}\det\left[\mathbf{I}+\mu N\textsc{snr}\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag}\right],\nonumber\\ &C_{4}=B\max_{\substack{\mathbf{K}_{x_{2}}\succeq0\\{\mathbf{tr}}(\mathbf{K}_{x_{2}})=1}}\log_{2}\det\left[\mathbf{I}+\frac{P_{2} }{B\sigma_{n}^{2}}\mathbf{H}\mathbf{K}_{x_{2}}\mathbf{H}^{\dag}\right] =B\max_{\substack{\mathbf{K}_{x_{2}}\succeq0\\{\mathbf{tr}}(\mathbf{K}_{x_{2}})=1}}\log_{2}\det\left[\mathbf{I}+ N\textsc{snr}\mathbf{H}\mathbf{K}_{x_{2}}\mathbf{H}^{\dag}\right].\end{aligned}$$ Above, we define $\textsc{snr}=\frac{\mathbb{E}\{||\mathbf{x}_{2}||^{2}\}}{\mathbb{E}\{||\mathbf{n}||^{2}\}}=\frac{P_{2}}{NB\sigma_{n}^{2}}$ as the signal-to-noise ratio when the channel is sensed as idle. If, on the other hand, the channel is sensed as busy, signal-to-noise ratio is $\mu \textsc{snr}$ since the transmission power is $P_1 = \mu P_2$. We also note that since $\mathbf{K}_{z}$ is a positive definite matrix and its eigenvalues are greater than or equal to 1, $\mathbf{K}_{z}^{-1}$ is a positive definite matrix with eigenvalues $\frac{\sigma_{n}^{2}}{N(\sigma_{n}^{2}+\sigma_{s}^{2})}\le\lambda_{i}\le 1$. The secondary transmitter is assumed to send the data at two different rates depending on the sensing results. If the channel is detected as busy as in scenarios 1 and 3, the transmission rate is $$\label{r1} r_{1}=B\max_{\substack{\mathbf{K}_{x_{1}}\succeq0\\{\mathbf{tr}}(\mathbf{K}_{x_{1}})=1}}\log_{2}\det\left[\mathbf{I}+\mu N\textsc{snr}\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\right],$$ and if the channel is detected as idle as in scenarios 2 and 4, the transmission rate is $$\label{r2} r_{2}=B\max_{\substack{\mathbf{K}_{x_{2}}\succeq0\\{\mathbf{tr}}(\mathbf{K}_{x_{2}})=1}}\log_{2}\det\left[\mathbf{I}+ N\textsc{snr}\mathbf{H}\mathbf{K}_{x_{2}}\mathbf{H}^{\dag}\right].$$ In scenarios 1 and 4, sensing decisions are correct and transmission rates match the channel capacities, i.e., we have $r_1 = C_1$ in scenario 1, and $r_2 = C_4$ in scenario 4. In these cases, we assume that reliable communication is achieved. On the other hand, sensing errors in scenarios 2 and 3 lead to mismatches. We first establish the following result. Note that $\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag}$ and ${\mathbf{K}}_z^{-1}$ are are Hermitian matrices, they can be written as [@Matrix; @Analysis Theorem 4.1.5] $$\begin{aligned} \label{eq:spectral} \mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag} = \mathbf{A} = {\mathbf{U}}_A {\mathbf{\Lambda}}_A {\mathbf{U}}_A^{\dagger}\quad \text{and} \quad {\mathbf{K}}_z^{-1} = {\mathbf{U}}_{K_z^{-1}} {\mathbf{\Lambda}}_{K_z^{-1}} {\mathbf{U}}_{K_z^{-1}}^{\dagger}\end{aligned}$$ where ${\mathbf{U}}_A$ and ${\mathbf{U}}_{K_z^{-1}}$ are unitary matrices and ${\mathbf{\Lambda}}_A$ and ${\mathbf{\Lambda}}_{K_z^{-1}}$ are real diagonal matrices, consisting of the eigenvalues of $\mathbf{A }= \mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag}$ and ${\mathbf{K}}_z^{-1}$, respectively. Now, we can write $$\begin{aligned} \det\left[\mathbf{I}+\mu N\textsc{snr}\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\right]&=\det\left[\mathbf{I}+\mu N\textsc{snr}\mathbf{A}\mathbf{K}_{z}^{-1}\right]\label{eq:det-inequality0}\\ &=\det\left[\left( \begin{array}{cc} \mathbf{U}_{A} & 0 \\ 0 & \mathbf{U}_{K_{z}^{-1}} \\ \end{array} \right)\left( \begin{array}{cc} \mu N\textsc{snr}\mathbf{\Lambda}_{A} & -\mathbf{I} \\ \mathbf{I} & \mathbf{\Lambda}_{K_{z}^{-1}} \\ \end{array} \right)\left( \begin{array}{cc} \mathbf{U}_{A}^{\dag} & 0 \\ 0 & \mathbf{U}_{K_{z}^{-1}}^{\dag} \\ \end{array} \right) \right]\nonumber\\ &=\det\left[\mathbf{U}_{A}\mathbf{U}_{K_{z}^{-1}}\right]\det\left[\mathbf{I}+\mu N\textsc{snr}\mathbf{\Lambda}_{A}\mathbf{\Lambda}_{K_{z}^{-1}}\right]\det\left[\mathbf{U}_{A}^{\dag}\mathbf{U}_{K_{z}^{-1}}^{\dag}\right]\\ &\leq\det\left[\mathbf{U}_{A}\mathbf{U}_{K_{z}^{-1}}\right]\det\left[\mathbf{I}+\mu N\textsc{snr}\mathbf{\Lambda}_{A}\right]\det\left[\mathbf{U}_{A}^{\dag}\mathbf{U}_{K_{z}^{-1}}^{\dag}\right] \label{eq:det-inequality}\\ &=\det\left[\mathbf{I}+\mu N\textsc{snr}\mathbf{A}\right]=\det\left[\mathbf{I}+\mu N\textsc{snr}\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag}\right]\label{eq:det-inequality1}.\end{aligned}$$ The inequality in (\[eq:det-inequality\]) follows from the following observation: $$\begin{aligned} \det\left[\mathbf{I}+\mu N\textsc{snr}\mathbf{\Lambda}_{A}\mathbf{\Lambda}_{K_{z}^{-1}}\right] &= \prod_i(1 + \mu N\textsc{snr} \lambda_{A,i} \lambda_{K_{z}^{-1},i}) \\ &\le \prod_i(1 + \mu N\textsc{snr} \lambda_{A,i}) \label{eq:det-inequality2} \\ &= \det\left[\mathbf{I}+\mu N\textsc{snr}\mathbf{\Lambda}_{A}\right].\end{aligned}$$ Above, $\lambda_A$ and $\lambda_{K_{z}^{-1}}$ denote the eigenvalues of $\mathbf{A}$ and ${\mathbf{K}}_z^{-1}$, respectively. The inequality in (\[eq:det-inequality2\]) follows from the fact that the eigenvalues of ${\mathbf{K}}_z^{-1}$ are smaller than 1, i.e., $\frac{\sigma_{n}^{2}}{N(\sigma_{n}^{2}+\sigma_{s}^{2})}\le\lambda_{K_z^{-1},i}\le 1$ as mentioned before, and the fact that $\lambda_{A,i} \ge 0$ which is due to the positive semi-definiteness of $\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag}$ [^4]. From the inequality established through (\[eq:det-inequality0\]) – (\[eq:det-inequality1\]), we see that, in scenario 3, the transmission rate is less than the capacity (i.e., $r_{1} \leq C_{3}$). Hence, although reliable transmission is achieved at the rate of $r_1$, channel is not fully utilized due to the false alarm in channel sensing. On the other hand, in a similar manner, it can be shown that in scenario 2, we have the transmission rate $r_2$ exceeding the channel capacity $C_{2}$ because sensing has not led to the successful detection of the active PUs, and the PUs’ interference on the SUs’ signals is not taken into account. In this case, we assume that reliable communication cannot be achieved. Hence, the transmission rate is effectively zero, and retransmission is required in scenario 2. In the other three scenarios, communication is performed reliably. These four scenarios or equivalently states are depicted in Figure $\ref{fig:fig1}$. Following the discussion above, we assume that the channel is ON in states 1,3, and 4, in which data is sent reliably, and is OFF in state 2. Next, we determine the state-transition probabilities. We use $p_{ij}$ to denote the transition probability from state $i$ to state $j$ as seen in Fig. $\ref{fig:fig1}$. Due to the block fading assumption, state transitions occur every $T$ seconds. We also assume that PU activity does not change within each frame. We consider a two-state Markov model to describe the transition of the PU activity between the frames. This Markov model is depicted in Figure \[Markov\]. Busy state indicates that the channel is occupied by the PUs, and idle state indicates that there is no PU present in the channel. Probability of transitioning from busy state to idle state is denoted by $\textit{a}$, and the probability of transitioning from idle state to busy state is denoted by $\textit{b}$. Let us first consider in detail the probability of staying in the topmost ON state in Fig. $\ref{fig:fig1}$. This probability, denoted by $p_{11}$, is given by $$\begin{aligned} p_{11}&= \Pr\left\{ \substack{\text{channel is busy and is detected busy} \\ \text{in the $l^{th}$ frame}} \,\, \Big | \,\, \substack{\text{channel is busy and is detected busy} \\ \text{in the $(l-1)^{th}$ frame}} \right\} \label{pro11_1} \\ &= \Pr\left\{ \substack{\text{channel is busy } \\ \text{in the $l^{th}$ frame}} \,\, \Big | \,\, \substack{\text{channel is busy } \\ \text{in the $(l-1)^{th}$ frame}} \right\} \times \Pr\left\{ \substack{\text{channel is detected busy} \\ \text{in the $l^{th}$ frame}} \,\, \Big | \,\, \substack{\text{channel is busy} \\ \text{in the $l^{th}$ frame}} \right\} \nonumber \\ &=(1-a)P_{d}\end{aligned}$$ where $P_{d}$ is the probability of detection in channel sensing. Channel being busy in the $l^{th}$ frame depends only on channel being busy in the $(l-1)^{th}$ frame and not on the other events in the condition. Moreover, since channel sensing is performed individually in each frame without any dependence on the channel sensing decision and PU activity in the previous frame, channel being detected as busy in the $l^{th}$ frame depends only on the event that the channel is actually busy in the $l^{th}$ frame. Similarly, the probabilities for transitioning from any state to state 1 (topmost ON state) can be expressed as $$\begin{aligned} \label{p11ler} p_{b1}=p_{11}&=p_{21}=(1-a)P_{d}\quad\textrm{and}\quad p_{i1}=p_{31}=p_{41}=bP_{d}.\end{aligned}$$ Note that we have common expressions for the transition probabilities in cases in which the originating state has a busy channel (i.e., states 1 and 2) and in cases in which the originating state has an idle channel (i.e., states 3 and 4). In a similar manner, the remaining transition probabilities are given by the following: For all $b \in \{1,2\}$ and $i \in \{3,4\}$, $$\begin{aligned} \label{prob2} \begin{array}{ll} p_{b2}=(1-a)(1-P_{d}),\quad\textrm{and}\quad p_{i2}=b(1-P_{d}),\\ p_{b3}=aP_{f},\quad\textrm{and}\quad p_{i3}=(1-b)P_{f},\\ p_{b4}=a(1-P_{f}),\quad\textrm{and}\quad p_{i4}=(1-b)(1-P_{f}). \end{array}\end{aligned}$$ Now, we can easily see that the $4\times4$ state transition matrix can be expressed as $$\begin{aligned} \label{R} R=\left( \begin{array}{cccc} p_{11} & . & . & p_{14} \\ p_{21} & . & . & p_{24} \\ p_{31} & . & . & p_{34} \\ p_{41} & . & . & p_{44} \\ \end{array} \right)=\left( \begin{array}{cccc} p_{b1} & . & . & p_{b4} \\ p_{b1} & . & . & p_{b4} \\ p_{i1} & . & . & p_{i4} \\ p_{i1} & . & . & p_{i4} \\ \end{array} \right).\end{aligned}$$ Effective Capacity ------------------ In [@Wu], Wu and Negi defined the effective capacity as the maximum constant arrival rate that a given service process can support in order to guarantee a statistical QoS requirement specified by the QoS exponent $\theta$. If we define $Q$ as the stationary queue length, then $\theta$ is defined as the decay rate of the tail distribution of the queue length $Q$: $$\label{decayrate} \lim_{q\rightarrow \infty}\frac{\log \Pr(Q\geq q)}{q}=-\theta.$$ Hence, we have the following approximation for the buffer violation probability for large $q_{max}$: $\Pr(Q\geq q_{max})\approx e^{-\theta q_{max}}$. Therefore, larger $\theta$ corresponds to more strict QoS constraints, while the smaller $\theta$ implies looser constraints. In certain settings, constraints on the queue length can be linked to limitations on the delay and hence delay-QoS constraints. It is shown in [@Liu] that $\Pr\{D\geq d_{max}\}\leq c\sqrt{\Pr\{Q\geq q_{max}\}}$ for constant arrival rates, where $D$ denotes the steady-state delay experienced in the buffer. In the above formulation, $c$ is a positive constant, $q_{max}=ad_{max}$ and $a$ is the source arrival rate. Therefore, effective capacity provides the maximum arrival rate when the system is subject to statistical queue length or delay constraints in the forms of $\Pr(Q \ge q_{\max}) \le e^{-\theta q_{max}}$ or $\Pr\{D \ge d_{\max}\} \le c \, e^{-\theta a \, d_{max}/2}$, respectively, for large thresholds $q_{\max}$ and $d_{\max}$. Since the average arrival rate is equal to the average departure rate when the queue is in steady-state [@ChangZajic], effective capacity can also be seen as the maximum throughput in the presence of such constraints. The effective capacity for a given QoS exponent $\theta$ is formulated as $$\label{exponent} -\lim_{t\rightarrow \infty}\frac{1}{\theta t}\log_{e}\mathbb{E}\{e^{-\theta S(t)}\}=-\frac{\Lambda(-\theta)}{\theta}$$ where $\Lambda(\theta) = \lim_{t\rightarrow \infty}\frac{1}{t}\log_{e}\mathbb{E}\{e^{\theta S(t)}\}$ is a function that depends on the logarithm of the moment generating function of $S(t)$, $S(t)=\sum_{k=1}^{t}r(k)$ is the time-accumulated service process, and $\{r(k),k=1,2,\dots\}$ is defined as the discrete-time, stationary and ergodic stochastic service process. Note that the service rate in each transmission block is $r(k) = Tr_{1}$ if the cognitive system is in Scenario 1 or 3 at time $k$. Similarly, the service rate is $r(k) = Tr_{2}$ in Scenario 4. In the OFF state in Scenario 2, the service rate is effectively zero. Considering the effective rates in each scenario and the probabilities of the scenarios, we have the following theorem. \[theo:effective capacity\] For the CR channel with the aforementioned state transition model , the normalized effective capacity in bits/s/Hz/dimension is given by $$\begin{aligned} \label{effective capacity} C_{E}(\textsc{snr},\theta)&=\max_{\substack{0 \le \mu \le 1\\0 \le P_2 \le \min\left\{P_{\max}, \frac{P_{int}}{P_d \mu + (1-P_d)}\right\}}}-\frac{1}{\theta TBN}\log_{e}\mathbb{E}\bigg\{\frac{1}{2}\left[\left(p_{b1}+p_{i3}\right)e^{-\theta Tr_{1}}+p_{i4}e^{-\theta Tr_{2}}+p_{b2}\right]\nonumber\\ &\frac{1}{2}\left\{\left[\left(p_{b1}-p_{i3}\right)e^{-\theta Tr_{1}}-p_{i4}e^{-\theta Tr_{2}}+p_{b2}\right]^{2}+4\left(p_{i1}e^{-\theta Tr_{1}}+p_{i2}\right)\left(p_{b3}e^{-\theta Tr_{1}}+p_{b4}e^{-\theta Tr_{2}}\right)\right\}^{1/2}\bigg\}\end{aligned}$$ where $T$ is the frame duration over which the fading stays constant, $r_1$ and $r_2$ are the transmission rates given in (\[r1\]) and (\[r2\]), and $\{p_{bk},p_{il}\}$ for $k,l\in{1,2,3,4}$ are the state transition probabilities given in (\[p11ler\]) and (\[prob2\]). *Proof:* See Appendix \[app:effective capacity\]. Note that above we have assumed that $\mathbf{H}$ is perfectly known at the transmitter, which, equipped with this knowledge, can choose the input covariance matrices to maximize the instantaneous channel capacities as seen in (\[r1\]) and (\[r2\]). If, on the other hand, only statistical information related to $\mathbf{H}$ are known at the transmitter, then the input covariance matrix can be chosen to maximize the effective capacity. In that case, the normalized effective capacity will be expressed as $$\begin{aligned} \label{capacity with H statistics} C_{E}&(\textsc{snr},\theta)=\max_{\substack{0 \le \mu \le 1\\0 \le P_2 \le \min\left\{P_{\max}, \frac{P_{int}}{P_d \mu + (1-P_d)}\right\}}} \max_{\substack{\mathbf{K}_{x_{1}},\mathbf{K}_{x_{2}}\succeq0\\{\mathbf{tr}}(\mathbf{K}_{x_{1}}) = {\mathbf{tr}}(\mathbf{K} _{x_{2}}) = 1}} -\frac{1}{\theta TBN}\log_{e}\mathbb{E}\bigg\{\frac{1}{2}\left[\left(p_{b1}+p_{i3}\right)\Theta_{r_{1}}+p_{i4} \Theta_ { r_ { 2 } } +p_ { b2 }\right]\nonumber\\ &+\frac{1}{2}\left\{\left[\left(p_{b1}-p_{i3}\right)\Theta_{r_{1}}-p_{i4}\Theta_{r_{2}}+p_{b2}\right] ^ {2}+4\left(p_{i1}\Theta_{r_{1}}+p_{i2}\right)\left(p_{b3}\Theta_{r_{1}}+p_{b4}\Theta_{r_{2}} \right)\right\}^{1/2}\bigg\}\textrm{bits/s/Hz/dimension}\end{aligned}$$ where $\Theta_{r_{1}}=e^{-\theta TB\log_{2}\det\left[\mathbf{I}+\mu N\textsc{snr}\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\right]}$ and $\Theta_{r_{2}}=e^{-\theta TB\log_{2}\det\left[\mathbf{I}+N\textsc{snr}\mathbf{H}\mathbf{K}_{x_{2}}\mathbf{H}^{\dag}\right]}$. Now, the input covariance matrices are selected to maximize the effective rate. For given $\mu$ and $P_2$, and for given input covariance matrices $\mathbf{K}_{x_{1}}$ and $\mathbf{K}_{x_{2}}$, we express the effective rate as $$\begin{aligned} \label{effective rate} R_{E}&(\textsc{snr},\theta)=-\frac{1}{\theta TBN}\log_{e}\mathbb{E}\bigg\{ \frac{1}{2}\left[\left(p_{b1}+p_{i3}\right)\Theta_{r_{1}}+p_{i4}\Theta_{r_{2}}+p_{b2 }\right]\nonumber\\ &+\frac{1}{2}\left\{\left[\left(p_{b1}-p_{i3}\right)\Theta_{r_{1}}-p_{i4}\Theta_{r_{2}}+p_{b2}\right] ^ {2}+4\left(p_{i1}\Theta_{r_{1}}+p_{i2}\right)\left(p_{b3}\Theta_{r_{1}}+p_{b4}\Theta_{r_{2}} \right)\right\}^{1/2}\bigg\}\textrm{bits/s/Hz/dimension}.\end{aligned}$$ Ergodic Capacity ---------------- As $\theta$ vanishes, the QoS constraints become loose and it can be easily verified that the effective capacity approaches the ergodic channel capacity, i.e., $$\begin{aligned} \label{capacity teta 0} \hspace{-.1cm}\lim_{\theta\rightarrow0}C_{E}(\textsc{snr},\theta)&=\frac{1}{N}\max_{\substack{0 \le \mu \le 1\\0 \le P_2 \le \min\left\{P_{\max}, \frac{P_{int}}{P_d \mu + (1-P_d)}\right\}}} \frac{bP_{d}+aP_{f}}{a+b}\mathbb{E}\left\{\max_{\substack{\mathbf{K}_{x_{1}}\succeq0\\{\mathbf{tr}}(\mathbf{K}_{x_{1}}) = 1}}\log_2\det\left[\mathbf{I}+\mu N\textsc{snr}\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\right]\right\}\nonumber\\ &\hspace{5.4cm}+\frac{a(1-P_{f})}{a+b}\mathbb{E}\left\{\max_{\substack{\mathbf{K}_{x_{2}}\succeq0\\\\tr(\mathbf{K} _{x_{2}}) = 1}}\log_2\det\left[\mathbf{I}+ N\textsc{snr}\mathbf{H}\mathbf{K}_{x_{2}}\mathbf{H}^{\dag}\right]\right\}.\end{aligned}$$ In order to gain further insight on the ergodic capacity expression, we note the following: $$\begin{aligned} \Pr\{\substack{\text{channel is}\\\text{detected busy}}\} &= \Pr\{\substack{\text{channel}\\\text{is busy}}\}\Pr\{\substack{\text{channel is }\\\text{detected busy}}\mid \substack{\text{channel}\\\text{is busy}}\} + \Pr\{\substack{\text{channel}\\\text{is idle}}\}\Pr\{\substack{\text{channel is }\\\text{detected busy}}\mid \substack{\text{channel}\\\text{is idle}}\} \\ &= \frac{b}{a+b} P_d + \frac{a}{a+b} P_f \\ &= \frac{bP_d + aP_f}{a+b} \label{eq:probofbusydet}\end{aligned}$$ where we used the fact that for the two-state Markov model of the PU activity depicted in Fig. \[Markov\], the probability of being in the busy state is $a/(a+b)$. Similarly, we have $$\begin{aligned} \Pr\{\substack{\text{channel is idle and}\\\text{is detected idle}}\} = \Pr\{\substack{\text{channel}\\\text{is idle}}\}\Pr\{\substack{\text{channel is }\\\text{detected idle}}\mid \substack{\text{channel}\\\text{is idle}}\} = \frac{a}{a+b} (1-P_f). \label{eq:probofidleandidledet}\end{aligned}$$ Recall that when the channel is detected busy, the transmitter sends the data at the rate $r_{1}$ given in (\[r1\]), and the transmission is successful because we are in either state 1 or 3 (of the state transition model in Fig. \[fig:fig1\]) which are both ON. If the channel is idle and is detected idle, then we are in state 4, which is also ON, and data is transmitted successfully at the rate $r_2$ given in (\[r2\]). On the other hand, when the channel is busy but is detected idle, the rate $r_2$ cannot be supported by the channel and reliable communication cannot be achieved. Consequently, in this scenario (which is state 2 in Fig. \[fig:fig1\]), the successful transmission rate is zero. From this discussion, we immediately realize that the ergodic capacity in (\[capacity teta 0\]) is proportional to the average of these transmission rates weighted by the probabilities of the corresponding scenarios. Energy Efficiency in the Low-Power Regime {#effective capacity low power} ========================================= In this section, we investigate the performance of cognitive MIMO transmissions in the low-power regime. For this analysis, we consider the following second-order low-[[<span style="font-variant:small-caps;">snr</span>]{}]{} expansion of the effective capacity: $$\label{effective capacity second order} C_{E}(\textsc{snr},\theta)=\dot{C}_{E}(0,\theta)\textsc{snr}+\ddot{C}_{E}(0,\theta)\frac{\textsc{snr}^{2}}{2}+ o(\textsc{snr}^{2})$$ where $\dot{C}_{E}(0,\theta)$ and $\ddot{C}_{E}(0,\theta)$ denote the first and second derivatives of the effective capacity with respect to $\textsc{snr}$ at $\textsc{snr}=0$. Note that the above expansion provides an accurate approximation of the effective capacity at low ${\textsc{snr}}$ levels. The benefits of a low-${\textsc{snr}}$ analysis are mainly twofold. First, operating at low power levels limits the interference inflicted on the PUs which is an important consideration in practice. Secondly, as will be seen below, energy efficiency improves as one lowers the transmission power. Hence, in this section, we consider a practically appealing and ambitious scenario in which cognitive users, in addition to their primary goal of efficiently utilizing the spectrum by filling in the spectrum holes, strive to operate energy efficiently while at the same time severely limiting the interference they cause on the PUs. For the energy efficiency analysis, we adopt the *energy per bit* given by $$\begin{gathered} \frac{E_b}{N_0} = \frac{\textsc{snr}}{C_{E}(\textsc{snr},\theta)},\end{gathered}$$ as the performance metric. It is shown in [@Verdu] that the bit energy requirements diminish as [[<span style="font-variant:small-caps;">snr</span>]{}]{} is lowered and the minimum energy per bit is achieved as ${\textsc{snr}}$ vanishes, i.e., $$\label{Bit Energy} \frac{E_{b}}{N_{0}}_{\min}=\lim_{\textsc{snr}\rightarrow0}\frac{\textsc{snr}}{C_{E}(\textsc{snr}, \theta)}=\frac{1}{\dot{C}_{E}(0,\theta)}.$$ Note that $\frac{E_{b}}{N_{0}}_{\min}$ is characterized only by the first derivative $\dot{C}_{E}(0,\theta)$. At $\frac{E_{b}}{N_{0}}_{\min}$, the slope $\mathcal{S}_{0}$ of the effective capacity versus $E_{b}/N_{0}$ (in dB) curve is defined as [@Verdu] $$\label{Slope} \mathcal{S}_{0}=\lim_{\frac{E_{b}}{N_{0}}\downarrow\frac{E_{b}}{N_{0}}_{\min}}\frac{C_{E}\left(\frac{E_{b}}{N_{0}}\right)}{10\log_{10}\frac{E_{b}}{N_{0}} -10\log_{10}\frac{E_{b}}{N_{0}}_{\min}}10\log_{10}2.$$ Considering the expression for the effective capacity, the wideband slope can be found from [@Verdu] $$\label{slope2} \mathcal{S}_{0}=\frac{2\left[\dot{C}_{E}(0,\theta)\right]^{2}}{-\ddot{C}_{E}(0,\theta)}\log_{e}2\quad \textrm{bits/s/Hz/(3 dB)/receive antenna}.$$ Hence, the wideband slope is obtained from both the first and second derivatives at ${\textsc{snr}}= 0$. The wideband slope $\mathcal{S}_0$ together with the minimum energy per bit $\frac{E_{b}}{N_{0}}_{\min}$ provide a linear approximation of the effective capacity as a function of the energy per bit in the low-${\textsc{snr}}$ regime and enable us to gain insight on the energy efficiency of cognitive transmissions. The next result identifies the first derivative of the effective capacity and the minimum bit energy. \[theo:first-deriv\] In the cognitive MIMO channel considered in this paper, the first derivative of the effective capacity with respect to ${\textsc{snr}}$ at ${\textsc{snr}}= 0$ is $$\label{derivative first effective capacity} \dot{C}_{E}(0,\theta)=\frac{1}{\log_{e}2}\bigg\{\frac{bP_{d}+aP_{f}}{a+b}\mathbb{E}\left[\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})\right] +\frac{a(1-P_{f})}{a+b}\mathbb{E}\left[\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{H})\right]\bigg\}.$$ Consequently, the minimum energy per bit is given by $$\label{min-bit-energy} \frac{E_{b}}{N_{0}}_{\min}=\frac{\log_{e}2}{\frac{bP_{d}+aP_{f}}{a+b}\mathbb{E}\left[\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})\right] +\frac{a(1-P_{f})}{a+b}\mathbb{E}\left[\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{H})\right]}.$$ *Proof:* See Appendix \[app:first-deriv\]. As detailed in the proof of Theorem \[theo:first-deriv\], the first derivative of the effective capacity at ${\textsc{snr}}= 0$ and hence the minimum energy per bit is achieved by transmitting in the maximal-eigenvalue eigenspaces of $\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H}$ and $\mathbf{H}^{\dag}\mathbf{H}$, when the channel is sensed as busy and idle, respectively. For instance, input covariance matrices in the cases of busy- and idle-sensed channels can be chosen, respectively, as $$\mathbf{K}_{x_{1}}=\mathbf{u}_{1}\mathbf{u}_{1}^{\dag} \quad \text{ and } \quad \mathbf{K}_{x_{2}}=\mathbf{u}_{2}\mathbf{u}_{2}^{\dag}$$ where $\mathbf{u}_{1}$ and $\mathbf{u}_{2}$ are the unit-norm eigenvectors associated with the maximum eigenvalues $\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})$ and $\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{H})$, respectively. Hence, beamforming in the eigenvector directions corresponding to the maximum eigenvalues of $\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H}$ and $\mathbf{H}^{\dag}\mathbf{H}$ is optimal in terms of energy efficiency. Note that when the channel is sensed as busy, the possible interference arising from the primary users’ transmissions is taken into account by incorporating $\mathbf{K}_{z}^{-1}$ into the transmission strategy. Note further that as shown in (\[eq:probofbusydet\]) and (\[eq:probofidleandidledet\]), $\frac{bP_{d}+aP_{f}}{a+b}$ is the probability of detecting the channel as busy, and $\frac{a(1-P_{f})}{a+b}$ is the probability that channel is idle and is detected as idle. The expressions in (\[derivative first effective capacity\]) and (\[min-bit-energy\]) do not depend on the QoS exponent $\theta$, indicating that the performance in the low power regime as ${\textsc{snr}}\to 0$ does not get affected by the presence of QoS requirements. Indeed, $\frac{E_{b}}{N_{0}}_{\min}$ in (\[min-bit-energy\]) is the minimum energy per bit attained when no QoS constraints are imposed. It is also interesting to note that the sensing performance has an impact on the energy efficiency. In particular, we can immediately notice that $\frac{E_{b}}{N_{0}}_{\min}$ decreases with increasing detection probability $P_d$. Similarly, $\frac{E_{b}}{N_{0}}_{\min}$ decreases as the false alarm probability $P_f$ decreases. This can be seen by noticing that decreasing $P_f$ leads to an increased weight on $\mathbb{E}\left[\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{H})\right]$ and a decreased weight on $\mathbb{E}\left[\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})\right] $, and noting that using Ostrowski’s Theorem [@Matrix; @Analysis Theorem 4.5.9 and Corollary 4.5.11] and its extension to non-square transforming matrices in [@Higham Theorems 3.2 and 3.4], we have $$\begin{aligned} \lambda_{\max}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H}) \le \lambda_{\max}(\mathbf{K}_{z}^{-1}) \lambda_{\max}(\mathbf{H}^{\dag}\mathbf{H}) \le \lambda_{\max}(\mathbf{H}^{\dag}\mathbf{H})\end{aligned}$$ where the last inequality follows from the property that $\lambda_{\max}(\mathbf{K}_{z}^{-1}) \le 1$. Since minimum energy per bit is a metric in the asymptotic regime in which ${\textsc{snr}}$ vanishes, we next consider the wideband slope in order to identify the performance at low but nonzero ${\textsc{snr}}$ levels. Wideband slope in (\[slope2\]) depends on the both the first and second derivatives of the effective capacity at ${\textsc{snr}}= 0$. In obtaining the second derivative, we essentially make use of the fact that the optimal input covariance matrices in the low ${\textsc{snr}}$ regime, which are required to achieve the minimum bit energy and hence the wideband slope, can be expressed as $$\mathbf{K}_{x_{1}}=\sum_{i=1}^{m_1}\kappa_{1i}\mathbf{u}_{1,i}\mathbf{u}_{1,i}^{\dag} \quad \text{ and } \quad \mathbf{K}_{x_{2}}=\sum_{i=1}^{m_2}\kappa_{2i}\mathbf{u}_{2,i}\mathbf{u}_{2,i}^{\dag}$$ where $\kappa_{1i},\kappa_{2i}\in[0,1]$ are the weights satisfying $\sum_{i=1}^{m_1}\kappa_{1i}=1$ and $\sum_{i=1}^{m_2}\kappa_{2i}=1$, and $m_1 \ge 1$ and $m_2 \ge 1$ are the multiplicities of $\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})$ and $\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{H})$, respectively. Moreover, $\{\mathbf{u}_{1,i}\}$ and $\{\mathbf{u}_{2,i}\}$ are the orthonormal eigenvectors that span the maximal-eigenvalue eigenspaces of $\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H}$ and $\mathbf{H}^{\dag}\mathbf{H}$, respectively. Despite this characterization, obtaining a general closed-form expression for the second-derivative seems intractable and we concentrate on the special case in which $a + b = 1$. Note that this case represents a scenario where there is no memory in the two-state Markov model for the PU activity. Hence, for instance, transitioning from busy state to busy state has the same probability as transitioning from idle state to busy state. \[theo:second-deriv\] In the special case in which the transition probabilities satisfy $a + b = 1$ in the two-state model for the PU activity, the second derivative of the effective capacity with respect to ${\textsc{snr}}$ at ${\textsc{snr}}= 0$ is $$\begin{aligned} \ddot{C}_{E}(0,\theta) &= \frac{\theta TBN}{\log_{e}^{2}2}\mathbb{E}^{2}\left[\ell_{1}\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})+\ell_{2} \lambda_{\max}(\mathbf{H}^{\dag}\mathbf{H})\right]\nonumber\\&-\frac{\theta TBN}{\log_{e}^{2}2}\mathbb{E}\left[\ell_{1}\lambda_{\max}^{2}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H}) +\ell_{2}\lambda_{\max}^{2}(\mathbf{H}^{\dag}\mathbf{H})\right]\nonumber\\ &-\frac{N}{\log_{e}2}\mathbb{E}\left[\frac{\ell_{1}\lambda_{\max}^{2}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})}{m_{1}} +\frac{\ell_{2}\lambda_{\max}^{2}(\mathbf{H}^{\dag}\mathbf{H})}{m_{2}}\right]\nonumber\end{aligned}$$ where $m_1$ and $m_2$ are the multiplicities of the eigenvalues $\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})$ and $\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{H})$, respectively, and we have defined $\ell_{1}=(bP_{d}+aP_{f})$ and $\ell_{2}=a(1-P_{f})$. The wideband slope is $$\begin{aligned} \label{widebandslope} \mathcal{S}_{0}=\frac{2\mathbb{E}^{2}\left[\ell_{1}\lambda_{\max,1}+\ell_{2}\lambda_{\max,2}\right]}{\theta TBN\left\{\mathbb{E}\left[\ell_{1}\lambda_{\max,1}^{2}+\ell_{2}\lambda_{\max,2}^{2}\right]-\mathbb{E}^{2} \left[\ell_{1}\lambda_{\max,1}+\ell_{2}\lambda_{\max,2}\right]\right\}+N\mathbb{E}\left[\frac{\ell_{1}\lambda_{\max,1}^{2}}{m_{1}} +\frac{\ell_{2}\lambda_{\max,2}^{2}}{m_{2}}\right]\log_{e}2}.\end{aligned}$$ where we used the notation $\lambda_{\max,1} = \lambda_{\max}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})$ and $\lambda_{\max,2} = \lambda_{\max}(\mathbf{H}^{\dag}\mathbf{H})$. *Proof:* See Appendix \[app:second-deriv\]. Unlike the minimum energy per bit, second derivative and the wideband slope depend on the QoS exponent $\theta$. In particular, we immediately notice that as $\theta$ increases (i.e., the QoS constraints become more stringent), wideband slope decreases, worsening the energy efficiency. Note that lower slopes imply that the same throughput is attained at an increased level of energy per bit. When we have equal power allocation, i.e., $\mathbf{K}_{x}=\frac{1}{M}\mathbf{I}$, and with the assumption that $\mathbf{s}$ with dimension $N\times1$ is a zero-mean Gaussian random vector with a covariance matrix $\mathbb{E}\{\mathbf{s}\mathbf{s}^{\dag}\}=\sigma_{s}^{2}\mathbf{I}$ where $\mathbf{I}$ is the identity matrix, we can obtain $$\label{obtained 1_1} \frac{E_{b}}{N_{0}}_{min}=\frac{\log_{e}2}{\left(\frac{\ell_{1}}{\sigma_{s}^{2}}+\ell_{2}\right) \mathbb{E}\left[{\mathbf{tr}}(\mathbf{H}^{\dag}\mathbf{H})\right]}$$ $$\begin{aligned} \label{obtained 2_1} \mathcal{S}_{0}=\frac{2\left(\frac{\ell_{1}}{\sigma_{s}^{2}}+\ell_{2}\right)^{2}\mathbb{E}^{2}\left[{\mathbf{tr}}(\mathbf{H}^{\dag}\mathbf{H})\right]}{\theta TBN\left\{\left[\frac{\ell_{1}}{\sigma_{s}^{4}}+\ell_{2}\right]\mathbb{E}\left[{\mathbf{tr}}^{2}(\mathbf{H}^{\dag}\mathbf{H})\right] -\left[\frac{\ell_{1}}{\sigma_{s}^{2}}+\ell_{2}\right]^{2}\mathbb{E}^{2}\left[{\mathbf{tr}}(\mathbf{H}^{\dag}\mathbf{H})\right]\right\} +N\left[\frac{\ell_{1}}{\sigma_{s}^{4}}+\ell_{2}\right]\mathbb{E}\left[{\mathbf{tr}}\left((\mathbf{H}^{\dag}\mathbf{H})^{2}\right)\right]\log_{e}2}.\end{aligned}$$ Now, assuming that $\mathbf{H}$ has independent zero-mean unit-variance complex Gaussian random entries, we have [@Lozano] $$\begin{aligned} \label{HHHH} \mathbb{E}\left[{\mathbf{tr}}(\mathbf{H}^{\dag}\mathbf{H})\right]=NM,\quad \mathbb{E}\left[{\mathbf{tr}}^{2}(\mathbf{H}^{\dag}\mathbf{H})\right]=NM(NM+1),\quad \mathbb{E}\left[{\mathbf{tr}}\left((\mathbf{H}^{\dag}\mathbf{H})^{2}\right)\right]=NM(N+M).\end{aligned}$$ Using these facts, we can write the following minimum bit energy and wideband slope expressions for the case of uniform power allocation: $$\label{New_Bit_Energy} \frac{E_{b}}{N_{0_{min}}}=\frac{\log_{e}2}{\left(\frac{\ell_{1}}{\sigma_{s}^{2}}+\ell_{2}\right)NM}$$ $$\label{New_Wideband_Slope} \mathcal{S}_{0}=\frac{2\left(\frac{\ell_{1}}{\sigma_{s}^{2}}+\ell_{2}\right)^{2}M^{2}}{\theta TB\left\{\left[\frac{\ell_{1}}{\sigma_{s}^{4}}+\ell_{2}\right]M(NM+1)- \left[\frac{\ell_{1}}{\sigma_{s}^{2}}+\ell_{2}\right]^{2}M^{2}\right\} +\left[\frac{\ell_{1}}{\sigma_{s}^{4}}+\ell_{2}\right]M(N+M)\log_{e}2}.$$ Numerical Results {#Numeric} ================= In our numerical results, we consider a Rayleigh fading channel model where the components of the channel matrix $\mathbf{H}$ are independent and identically distributed (i.i.d.) zero-mean, unit variance, circularly symmetric Gaussian random variables. Moreover, we assume that input covariance matrix is $\mathbf{K}_{x}=\frac{1}{M}\mathbf{I}$ and that the components of the received signal coming from PUs are i.i.d. and have a variance $\sigma_{s}^{2}$ so that $\mathbf{K}_{z}=\frac{\sigma_{s}^{2}+\sigma_{n}^{2}}{\sigma_{s}^{2}}\mathbf{I}$. Furthermore, as the objective function we consider the effective rate which is given as $$\begin{aligned} \label{effective rate numerical} R_{E}(\textsc{snr},\theta)=-\frac{1}{\theta TB}\log_{e}\mathbb{E}\bigg\{& \ell_{1}e^{-\theta TB\log_{2}\det\left[ \mathbf{I}+\frac{\mu N\sigma_{n}^{2}}{M(\sigma_{s}^{2} +\sigma_{n}^{2})}\textsc{snr}\mathbf{H}\mathbf{H}^{\dag}\right]} +\ell_{2}e^{-\theta TB\log_{2}\det\left[ \mathbf{I}+\frac{ N}{M}\textsc{snr}\mathbf{H}\mathbf{H}^{\dag} \right]}\nonumber\\&+b(1-P_{d})\bigg\}\textrm{bits/Hz/s}.\end{aligned}$$ With these assumptions, effective rate can be computed by using the expression for the moment generating function of instantaneous mutual information given by Wang and Giannakis in [@Giannakis Theorem 1]. After adopting this expression into our effective rate formulation ($\ref{effective rate numerical}$), we obtain $$\begin{aligned} \label{effective rate numerical extra} R_{E}(\textsc{snr},\theta)=-\frac{1}{\theta TB}\log_{e}\bigg\{ &[bP_{d}+aP_{f}]\frac{\det\left[\mathbf{G}\left(\theta,\frac{\mu\sigma_{n}^{2}\textsc{snr}}{\sigma_{s}^{2}+\sigma_{n}^{2}}\right)\right]}{\prod_{i=1}^{k}\Gamma(d+i)\Gamma(i)} +a(1-P_{f})\frac{\det\left[\mathbf{G}\left(\theta,\textsc{snr}\right)\right]}{\prod_{i=1}^{k}\Gamma(d+i)\Gamma(i)}\nonumber\\&+b(1-P_{d})\bigg\}\textrm{bits/Hz/s}\end{aligned}$$ where $k=\min(M,N)$, $d=\max(M,N)-\min(M,N)$, and $\Gamma(.)$ is the Gamma function. Here, $\mathbf{G}(\theta,\textsc{snr})$ is a $k\times k$ Hankel matrix whose $(m,n)^{th}$ component is $$\label{HAnkel} g_{m,n}=\int_{0}^{\infty}\left(1+\frac{N}{M}\textsc{snr}\textit{z}\right)^{-\theta TB\log_{2}e}\textit{z}^{m+n+d-2}e^{-\textit{z}}d\textit{z} \qquad \text{for }{m,n=1,2,...,k}.$$ In our numerical results, we assume $T=0.1$ s, $B=100$ Hz, $\sigma_{n}^{2}=\sigma_{s}^{2}=1$, $P_{d}=0.92$, $P_{f}=0.21$, and $P_{\max}=10$ dB. In Figure \[fig:fig4-5\], we plot the effective rate as a function of $P_{int}$ for different values of the QoS exponent, $\theta$. In this figure, number of transmit and number of receive antennas are both 3, i.e., $ M = N = 3$. When the interference power threshold is low, the optimal ratio of power level $P_{1}$ to the power level $P_{2}$ is very small, i.e., $\mu = P_1/P_2 \sim 0$. Therefore, there is almost no transmission when the channel is detected as busy. Note in this case that false alarms lead to almost no transmission even if the channel is not occupied by the PUs. In addition, from (\[eq:P2upperbound\]), we see that if the detection probability $P_d < 1$, then $P_2$, the transmission power when the channel is sensed as idle, scales with $P_{int}$ if $P_{int}$ is sufficiently small. Consequently, we see in Fig. \[fig:fig4-5\], the throughput diminishes to zero as $P_{int}$ gets smaller. On the other hand, as $P_{int}$ increases beyond a certain threshold, we observe that the effective rate becomes fixed due to OFF state (the state in which there is no data transmission and/or unreliable transmission), which becomes dominant in the effective rate expression, and the fact that even if $P_{int}$ is very high or there is no interference power threshold, average peak power, $P_{\max}$, limits the transmission powers. Another remark regarding the plots in Fig. \[fig:fig4-5\] is that, as expected, the higher the QoS exponent $\theta$ (or equivalently the more strict the QoS constraints), the smaller the effective rate is. In Fig. \[fig:fig7\], we plot the corresponding energy-per-bit requirements, $\frac{E_{b}}{N_{0}}$, as a function of SNR. Confirming our results, we observe that the minimum bit energy given in (\[New\_Bit\_Energy\]) is indeed approached as SNR is diminished, and since the minimum energy per bit is independent of $\theta$, all curves converge as SNR vanishes. In Figure \[fig:fig6\], we plot the effective rate for different numbers of transmit and receive antennas as a function of $P_{int}$. We set $\theta=0.1$. We observe that increasing the number of antennas beyond a certain level does not improve the transmission quality for higher values of $P_{int}$. On the other hand, for smaller values of $P_{int}$ in the range \[-30dB, 0dB\] (i.e., under relatively stringent interference constraints), with higher number of antennas, improvements in the throughput can be realized. In Fig. \[fig:fig8\], we plot the effective rate as a function of probability of detection, $P_d$, when $P_{int} = 0$ dB. In this figure, we observe the impact of channel sensing performance on the throughput of cognitive MIMO transmissions. The curves with thick lines are obtained when probability of false alarm is $P_{f}=0.21$. Curves with thin lines are obtained when $P_f = 0.1$. With the increasing $P_{d}$, the effective rate increases as a result of efficient power allocation when the channel is sensed as idle. The interference caused by the primary users is controlled by allocating less power when the channel is sensed as busy. However, since the optimal power ratio $\mu = P_{1} / P_{2}$ depends on the value of the detection probability, $P_{d}$, the power allocated to transmission when the channel is sensed as idle decreases with the increasing $P_{d}$ but does not go to zero, which is because of the non-zero probability of false alarm, $P_{f}$. Therefore, we also observe that with the decreasing probability of false alarm, the effective rate decreases due to less power allocated when the channel is sensed as busy. Furthermore, in Fig. \[fig:fig9\], we plot the effective rate as a function of power ratio $\mu$ for different power interference values, $P_{int}$. We observe that with decreasing $P_{int}$, the optimal $\mu$ is decreasing for the aforementioned $P_{d}$ and $P_{f}$ values. Note that the optimal $\mu$ is 1 when $P_{int}=P_{max}$. Finally, in Figure \[fig:fig10\], we plot the effective rate as a function of the QoS exponent, $\theta$. As expected with the increasing $\theta$ values, the effective rate is decreasing due to more strict buffer/delay constraints. We also note that smaller $P_{int} $ and hence more strict interference constraints lead to reduced throughput for smaller values of $\theta$. On the other hand, if $\theta$ is large, the impact of $P_{int}$ lessens and curves converge. Conclusion {#Conclusion} ========== In this paper, we have investigated the throughput and energy efficiency of cognitive MIMO wireless communication systems operating under queuing constraints, interference limitations, and imperfect channel sensing. We have considered effective capacity and rate as our throughput metrics and formulated them in terms of instantaneous transmission rates and state transition probabilities, which in turn depend on the primary user activities and sensing reliability. Through numerical results, we have investigated the impact of QoS and interference constraints and sensing performance, and the benefit of multiple antenna transmissions. For the energy efficiency analysis, we have studied the effective capacity in the low-power regime. We have obtained expressions for the first and second derivatives of the effective capacity. We have determined the minimum energy per bit required in the cognitive MIMO system. We have remarked that while the minimum energy per bit does not get affected by the presence of the QoS constraints, it decreases as the channel sensing reliability improves. We have seen that the second derivative and the wideband slope depend on the QoS exponent $\theta$. We have also shown that the minimum energy per bit and wideband slope are achieved by performing beamforming in the maximal-eigenvalue eigenspace of the matrices $\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H}$ and $\mathbf{H}^{\dag}\mathbf{H}$. Proof of Theorem \[theo:effective capacity\] {#app:effective capacity} -------------------------------------------- The proof follows along similar lines as in [@paper3] in which a single-antenna case with channel uncertainty is studied. In [@Performance Chap. 7, Example 7.2.7], it is shown for Markov modulated processes that $$\begin{gathered} \label{eq:theta-envelope} \frac{\Lambda(\theta)}{\theta} = \frac{1}{\theta} \log_e sp(\phi(\theta)R)\end{gathered}$$ where $sp(\phi(\theta)R)$ is the spectral radius (i.e., the maximum of the absolute values of the eigenvalues) of the matrix $\phi(\theta)R$, $R$ is the transition matrix of the underlying Markov process, and $\phi(\theta) = \text{diag}(\phi_1(\theta), \ldots, \phi_F(\theta))$ is a diagonal matrix whose components are the moment generating functions of the processes in $F$ states. The rates supported by the CR channel with the state transition model described above can be seen as a Markov modulated process and hence the setup considered in [@Performance] can be immediately applied to our setting. Note that the transmission rates are random in each state in the cognitive channel. Therefore, the corresponding moment generating functions are $\phi_{1}(\theta)=\phi_{3}(\theta)=\mathbb{E}\{e^{T\theta r_{1}}\}$, $\phi_{4}(\theta)=\mathbb{E}\{e^{T\theta r_{2}}\}$ and $\phi_{2}(\theta)=1$. Then, using , we can write $$\begin{aligned} \phi(\theta)R=\left[ \begin{array}{cccc} \phi_{1}(\theta)p_{b1} & . & . & \phi_{1}(\theta)p_{b4} \\ \phi_{2}(\theta)p_{b1} & & & \phi_{2}(\theta)p_{b4} \\ \phi_{3}(\theta)p_{i1} & & & \phi_{3}(\theta)p_{i4} \\ \phi_{4}(\theta)p_{i1} & . & . & \phi_{4}(\theta)p_{i4} \\ \end{array} \right] = \left[\begin{array}{cccc} \mathbb{E}\{e^{T\theta r_{1}}\}p_{b1} & . & . & \mathbb{E}\{e^{T\theta r_{1}}\}p_{b4}\\ p_{b1} & . & . & p_{b4} \\ \mathbb{E}\{e^{T\theta r_{1}}\}p_{i1} & & & \mathbb{E}\{e^{T\theta r_{1}}\}p_{i4} \\ \mathbb{E}\{e^{T\theta r_{2}}\}p_{i1} & & & \mathbb{E}\{e^{T\theta r_{2}}\}p_{i4} \\ \end{array} \right]\end{aligned}$$ Since $\phi(\theta)R$ is a matrix with rank 2, we can readily find that $$\begin{aligned} \label{eq:sp} &sp(\phi(\theta)R) = \text{trace}(\phi(\theta)R)\nonumber\\ &=\frac{1}{2}\left\{\phi_{1}(\theta)p_{b1}+\phi_{2}(\theta)p_{b2} +\phi_{3}(\theta)p_{i3}+\phi_{4}(\theta)p_{i4}\right\}\nonumber\\ &+\frac{1}{2}\left\{\left[\phi_{1}(\theta)p_{b1}+\phi_{2}(\theta)p_{b2} -\phi_{3}(\theta)p_{i3}-\phi_{4}(\theta)p_{i4}\right]^{2}+4\left(\phi_{1}(\theta)p_{i1}+\phi_{2} (\theta)p_{i2}\right)\left(\phi_{3}(\theta)p_{b3}+\phi_{4}(\theta)p_{b4}\right)\right\}^{1/2} \nonumber \\ &=\frac{1}{2}\left\{\left(p_{b1}+p_{i3}\right)\mathbb{E}\{e^{T\theta r_{1}}\}+p_{i4}\mathbb{E}\{e^{T\theta r_{2}}\}+p_{b2}\right\}\nonumber\\ &+\frac{1}{2}\left\{\left[\left(p_{b1}-p_{i3}\right)\mathbb{E}\{e^{T\theta r_{1}}\}-p_{i4}\mathbb{E}\{e^{T\theta r_{2}}\}+p_{b2}\right]^{2}+4\left(p_{i1}\mathbb{E}\{e^{T\theta r_{1}}\}+p_{i2}\right)\left(p_{b3}\mathbb{E}\{e^{T\theta r_{1}}\}+p_{b4}\mathbb{E}\{e^{T\theta r_{2}}\}\right)\right\}^{1/2}.\end{aligned}$$ Then, combining (\[eq:sp\]) with (\[eq:theta-envelope\]) and (\[exponent\]), we obtain the expression inside the maximization on the right-hand side of (\[effective capacity\]). $\square$ Proof of Theorem \[theo:first-deriv\] {#app:first-deriv} ------------------------------------- We define a new function $$\begin{aligned} \label{f} f(\textsc{snr},\theta)&=\frac{1}{2}\left[\left(p_{b1}+p_{i3}\right)e^{-\theta Tr_{1}}+p_{i4}e^{-\theta Tr_{2}}+p_{b2 }\right]\nonumber\\ &+\frac{1}{2}\underbrace{\left\{\left[\left(p_{b1}-p_{i3}\right)e^{-\theta Tr_{1}}-p_{i4}e^{-\theta Tr_{2}}+p_{b2}\right] ^ {2}+4\left(p_{i1}e^{-\theta Tr_{1}}+p_{i2}\right)\left(p_{b3}e^{-\theta Tr_{1}}+p_{b4}e^{-\theta Tr_{2}} \right)\right\}^{1/2}}_{\chi},\end{aligned}$$ and we can write the effective rate in (\[effective rate\]) as $$\label{effective rate_f} R_{E}(\textsc{snr},\theta)=D\log_{e}\mathbb{E}\left[f(\textsc{snr},\theta)\right]$$ where $D=-\frac{1}{\theta TBN}$. The derivative of the effective rate with respect to ${\textsc{snr}}$ will be $$\begin{aligned} \label{derivative first 1} &\dot{R}_{E}(\textsc{snr},\theta)=\frac{D}{\mathbb{E}\left[f(\textsc{snr},\theta)\right]}\mathbb{E}\left[\dot{f}(\textsc{snr},\theta)\right]\end{aligned}$$ where $$\label{f first derivative} \dot{f}(\textsc{snr},\theta)=-\theta T\alpha(\textsc{snr},\theta)\dot{r}_{1}e^{-\theta Tr_{1}}-\theta T\beta(\textsc{snr},\theta)\dot{r}_{2}e^{-\theta Tr_{2}}$$ and $\alpha(\textsc{snr},\theta)=\frac{1}{2}(p_{b1}+p_{i3})+\frac{(p_{b1}-p_{i3})\left[(p_{b1}-p_{i3})e^{-\theta Tr_{1}}- p_{i4}e^{-\theta Tr_{2}}+p_{b2}\right]}{2\chi}+\frac{p_{i1}\left(p_{b3}e^{-\theta Tr_{1}}+p_{b4}e^{-\theta Tr_{2}}\right) + p_{b3}\left(p_{i1}e^{-\theta Tr_{1}}+p_{i2}\right)}{\chi}$, $\beta(\textsc{snr},\theta)=\frac{1}{2}p_{i4}-\frac{p_{i4}\left[\left(p_{b1}-p_{i3}\right)e^{-\theta T_{r_{1}}}-p_{i4}e^{-\theta Tr_{2}}+p_{b2}\right]}{2\chi}+\frac{p_{b4}\left(p_{i1}e^{-\theta Tr_{1}}+p_{i2}\right)}{\chi}$, and $\chi$ is defined in (\[f\]). Note that we can write $r_{1}$ and $r_{2}$ as $$\label{r1u_new} r_{1}=\frac{B}{\log_{e}2}\sum_{i}\log_{e}\left[1+\mu N\textsc{snr}\lambda_{i}(\Phi_{1})\right]$$ and $$\label{r2u_new} r_{2}=\frac{B}{\log_{e}2}\sum_{i}\log_{e}\left[1+N\textsc{snr}\lambda_{i}(\Phi_{2})\right]$$ where $\Phi_{1}=\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}$ and $\Phi_{2}=\mathbf{H}\mathbf{K}_{x_{2}}\mathbf{H}^{\dag}$, and $\lambda_{i}$ is the eigenvalue of the matrices given in the parentheses. Now, we can write the derivatives of $r_{1}$ and $r_{2}$ with respect to ${\textsc{snr}}$ as $$\label{r1u_derivative} \dot{r}_{1}=\frac{B}{\log_{e}2}\sum_{i}\frac{\mu N\lambda_{i}(\Phi_{1})}{1+\mu N\textsc{snr}\lambda_{i}(\Phi_{1})}$$ and $$\label{r2u_derivative} \dot{r}_{2}=\frac{B}{\log_{e}2}\sum_{i}\frac{N\lambda_{i}(\Phi_{2})}{1+N\textsc{snr}\lambda_{i}(\Phi_{2})}.$$ Noting that the function $f(\textsc{snr},\theta)$ evaluated at $\textsc{snr}=0$ is 1, i.e., $f(0,\theta)=1$, and $\alpha(0,\theta)$ and $\beta(0,\theta)$ are constants denoted by $\bar{\alpha}$ and $\bar{\beta}$, respectively, we can easily see that the value of the first derivative of the effective rate at $\textsc{snr}=0$ is $$\label{capacity first derivative at snr 0} \dot{R}_{E}(0,\theta)=\frac{1}{\log_{e}2}\mathbb{E}\left[\bar{\alpha}\mu{\mathbf{tr}}\{\Phi_{1}\}+\bar{\beta}{\mathbf{tr}}\{\Phi_{2}\}\right].$$ Note that by definition, $\mathbf{K}_{x_{1}}$ and $\mathbf{K}_{x_{2}}$ are positive semi-definite Hermitian matrices. As Hermitian matrices, $\mathbf{K}_{x_{1}}$ and $\mathbf{K}_{x_{2}}$ can be written as follows $$\label{K_Unitary 1} \mathbf{K}_{x_{1}}=\mathbf{U}_{1}\Lambda_{1}\mathbf{U}_{1}^{\dag}=\sum_{i=1}^{M}\lambda_{1,i}\mathbf{u}_{1,i}\mathbf{u}_{1,i}^{\dag}$$ and $$\label{K_Unitary 2} \mathbf{K}_{x_{2}}=\mathbf{U}_{2}\Lambda_{2}\mathbf{U}_{2}^{\dag}=\sum_{i=1}^{M}\lambda_{2,i}\mathbf{u}_{2,i}\mathbf{u}_{2,i}^{\dag}$$ where $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$ are the unitary matrices, $\{\mathbf{u}_{1,i}\}$ and $\{\mathbf{u}_{2,i}\}$ are the column vectors of $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$, respectively. $\Lambda_{1}$ and $\Lambda_{2}$ are the real diagonal matrices with diagonal components $\{\lambda_{1,i}\}$ and $\{\lambda_{2,i}\}$, respectively. Since $\mathbf{K}_{x_{1}}$ and $\mathbf{K}_{x_{2}}$ are positive semi-definite, we have $\lambda_{1,i}\geq0$ and $\lambda_{2,i}\geq0$. Furthermore, since all the available energy should be used for transmission, we have ${\mathbf{tr}}(\mathbf{K}_{x_{1}})=\sum_{i=1}^{M}\lambda_{1,i}=1$ and ${\mathbf{tr}}(\mathbf{K}_{x_{2}})=\sum_{i=1}^{M}\lambda_{2,i}=1$. Now, we can write $$\begin{aligned} \dot{R}_{E}(0,\theta)&=\frac{1}{\log_{e}2}\mathbb{E}\left[\bar{\alpha}\mu{\mathbf{tr}}(\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}) +\bar{\beta}{\mathbf{tr}}(\mathbf{H}\mathbf{K}_{x_{2}}\mathbf{H}^{\dag})\right]\nonumber\\ &=\frac{1}{\log_{e}2}\mathbb{E}\left[\bar{\alpha}\mu{\mathbf{tr}}(\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag}\mathbf{U}_{z}\Lambda_{z}\mathbf{U}_{z}^{\dag}) +\bar{\beta}{\mathbf{tr}}(\mathbf{H}\mathbf{K}_{x_{2}}\mathbf{H}^{\dag})\right]\nonumber\\ &=\frac{1}{\log_{e}2}\mathbb{E}\left[\bar{\alpha}\mu{\mathbf{tr}}(\Lambda_{z}^{1/2}\mathbf{U}_{z}^{\dag}\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag}\mathbf{U}_{z}\Lambda_{z}^{1/2}) +\bar{\beta}{\mathbf{tr}}(\mathbf{H}\mathbf{K}_{x_{2}}\mathbf{H}^{\dag})\right]\nonumber\\ &=\frac{1}{\log_{e}2}\sum_{i=1}^{M}\bigg\{\lambda_{1,i}\bar{\alpha}\mu\mathbb{E}[{\mathbf{tr}}(\Lambda_{z}^{1/2}\mathbf{U}_{z}^{\dag}\mathbf{H}\mathbf{u}_{1,i}\mathbf{u}_{1,i}^{\dag}\mathbf{H}^{\dag}\mathbf{U}_{z}\Lambda_{z}^{1/2})] +\lambda_{2,i}\bar{\beta}\mathbb{E}[{\mathbf{tr}}(\mathbf{H}\mathbf{u}_{2,i}\mathbf{u}_{2,i}^{\dag}\mathbf{H}^{\dag})]\bigg\}\nonumber\\ &=\frac{1}{\log_{e}2}\sum_{i=1}^{M}\bigg\{\lambda_{1,i}\bar{\alpha}\mu\mathbb{E}[{\mathbf{tr}}(\mathbf{u}_{1,i}^{\dag}\mathbf{H}^{\dag}\mathbf{U}_{z}\Lambda_{z}^{1/2}\Lambda_{z}^{1/2}\mathbf{U}_{z}^{\dag}\mathbf{H}\mathbf{u}_{1,i})] +\lambda_{2,i}\bar{\beta}\mathbb{E}[{\mathbf{tr}}(\mathbf{u}_{2,i}^{\dag}\mathbf{H}^{\dag}\mathbf{H}\mathbf{u}_{2,i})]\bigg\}\nonumber\\ &=\frac{1}{\log_{e}2}\sum_{i=1}^{M}\bigg\{\lambda_{1,i}\bar{\alpha}\mu\mathbb{E}[{\mathbf{tr}}(\mathbf{u}_{1,i}^{\dag}\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H}\mathbf{u}_{1,i})] +\lambda_{2,i}\bar{\beta}\mathbb{E}[{\mathbf{tr}}(\mathbf{u}_{2,i}^{\dag}\mathbf{H}^{\dag}\mathbf{H}\mathbf{u}_{2,i})]\bigg\}\nonumber\\ &\leq\frac{1}{\log_{e}2}\bigg\{\bar{\alpha}\mu\mathbb{E}\left[\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})\right] +\bar{\beta}\mathbb{E}\left[\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{H})\right]\bigg\} \label{rate_upperbound_zero_derivative}\end{aligned}$$ where $\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})$ and $\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{H})$ denote the maximum eigenvalues of the matrices $\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H}$ and $\mathbf{H}^{\dag}\mathbf{H}$. The upper bound in ($\ref{rate_upperbound_zero_derivative}$) can be achieved by choosing the normalized input covariance matrices as $$\label{input covariance 1} \mathbf{K}_{x_{1}}=\mathbf{u}_{1}\mathbf{u}_{1}^{\dag}$$ and $$\label{input covariance 2} \mathbf{K}_{x_{2}}=\mathbf{u}_{2}\mathbf{u}_{2}^{\dag}$$ where $\mathbf{u}_{1}$ and $\mathbf{u}_{2}$ are the unit-norm eigenvectors that correspond to the maximum eigenvalues $\lambda_{max}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})$ and $\lambda_{max}(\mathbf{H}^{\dag}\mathbf{H})$. This lets us conclude that $$\label{derivative first effective capacity-app} \dot{C}_{E}(0,\theta)=\frac{1}{\log_{e}2}\bigg\{\bar{\alpha}\mu\mathbb{E}\left[\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})\right] +\bar{\beta}\mathbb{E}\left[\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{H})\right]\bigg\}.$$ Final expression in (\[derivative first effective capacity\]) is derived by noticing that $\bar{\alpha} = \frac{bP_{d}+aP_{f}}{a+b}$ and $\bar{\beta} = \frac{a(1-P_{f})}{a+b}$, which are obtained by making use of the transition probability expressions in (\[p11ler\]) and (\[prob2\]). Note also that we set $\mu = 1$ since $\dot{C}_{E}(0,\theta)$ is achieved in the low-power regime as ${\textsc{snr}}$ and hence $P_2$ approach zero, and constraint in (\[Power Threshold\_new\]) is eventually satisfied in this regime for any interference power constraint $P_{int} >0$ regardless of the value of $\mu$. Choosing $\mu =1$ maximizes the first derivative and leads to the smallest value of the minimum energy per bit. Proof of Theorem \[theo:second-deriv\] {#app:second-deriv} -------------------------------------- We first note that the upper bound in (\[rate\_upperbound\_zero\_derivative\]) and hence the first derivative of the effective capacity and the minimum energy per bit is achieved only if the cognitive radio transmits in the maximal-eigenvalue eigenspaces of the matrices $\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H}$ and $\mathbf{H}^{\dag}\mathbf{H}$. More specifically, input-covariance matrices should be selected as $$\label{covariance_l 1} \mathbf{K}_{x_{1}}=\sum_{i=1}^{m_{1}}\kappa_{1i}\mathbf{u}_{1,i}\mathbf{u}_{1,i}^{\dag}$$ and $$\label{covariance_l 2} \mathbf{K}_{x_{2}}=\sum_{i=1}^{m_{2}}\kappa_{2i}\mathbf{u}_{2,i}\mathbf{u}_{2,i}^{\dag}$$ for some $\kappa_{1i},\kappa_{2i}\in[0,1]$ satisfying $\sum_{i=1}^{m_1}\kappa_{1i}=1$ and $\sum_{i=1}^{m_2}\kappa_{2i}=1$. Above, $m_1$ and $m_2$ denote the multiplicities of the maximum eigenvalues $\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})$ and $\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{H})$, respectively, and $\{\mathbf{u}_{1,i}\}$ and $\{\mathbf{u}_{2,i}\}$ are the orthonormal eigenvectors that span the maximal-eigenvalue eigenspaces of $\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H}$ and $\mathbf{H}^{\dag}\mathbf{H}$, respectively. The above input covariance structure, which is needed to achieve the minimum energy per bit, is consequently required to achieve the second derivative of the effective capacity and hence the wideband slope. As for the second derivative, we differentiate $\dot{R}_{E}(\textsc{snr},\theta)$ in $(\ref{derivative first 1})$ with respect to $\textsc{snr}$ once more. In order to obtain a closed-form solution, we concentrate on the special case in which $a+b=1$. Now, we obtain $$\begin{aligned} \label{derivative second 1} \ddot{R}_{E}(\textsc{snr},\theta)&=\frac{D}{\mathbb{E}\left[f(\textsc{snr},\theta)\right]}\mathbb{E}\left[\ddot{f}(\textsc{snr},\theta)\right] -\frac{D}{\mathbb{E}^{2}\left[f(\textsc{snr},\theta)\right]}\mathbb{E}^{2}\left[\dot{f}(\textsc{snr},\theta)\right]\end{aligned}$$ where $$\begin{aligned} \label{f_first_derivative_simple} \dot{f}(\textsc{snr},\theta)=-\theta T(aP_{f}+bP_{d})\dot{r}_{1}e^{-\theta Tr_{1}} -\theta Ta(1-P_{f})\dot{r}_{2}e^{-\theta Tr_{2}}\end{aligned}$$ and $$\begin{aligned} \label{f second derivative} \ddot{f}(\textsc{snr},\theta)=&\theta^{2}T^{2}(bP_{d}+aP_{f})\dot{r}_{1}^{2}e^{-\theta Tr_{1}}+\theta^{2}T^{2}a(1-P_{f})\dot{r}_{2}^{2}e^{-\theta Tr_{2}} -\theta T(bP_{d}+aP_{f})\ddot{r}_{1}e^{-\theta Tr_{1}}\nonumber\\&-\theta Ta(1-P_{f})\ddot{r}_{2}e^{-\theta Tr_{2}}.\end{aligned}$$ Now, we can write the second derivatives of $r_{1}$ and $r_{2}$ as $$\label{r1u_second derivative} \ddot{r}_{1}=-\frac{B}{\log_{e}2}\sum_{i}\frac{\mu^{2}N^{2}\lambda_{i}^{2}(\Phi_{1})}{\left[1+\mu N\textsc{snr}\lambda_{i}(\Phi_{1})\right]^{2}}$$ and $$\label{r2u_second derivative} \ddot{r}_{2}=-\frac{B}{\log_{e}2}\sum_{i}\frac{N^{2}\lambda_{i}^{2}(\Phi_{2})}{\left[1+N\textsc{snr}\lambda_{i}(\Phi_{2})\right]^{2}}.$$ We can easily see that when $\textsc{snr}$ goes to 0, we can express the first and second derivatives of $f(\textsc{snr},\theta)$ $$\label{f_first_derivative} \dot{f}(0,\theta)=-\frac{(bP_{d}+aP_{f})\theta TBN\mu}{\log_{e}2}{\mathbf{tr}}\{\Phi_{1}\}-\frac{a(1-P_{f})\theta TBN}{\log_{e}2}{\mathbf{tr}}\{\Phi_{2}\}$$ and $$\begin{aligned} \label{f_second_derivative} \ddot{f}(0,\theta)&=\frac{\ell_{1}\theta TBN^{2}\mu^{2}}{\log_{e}2}{\mathbf{tr}}\{\Phi_{1}^{\dag}\Phi_{1}\} + \frac{\ell_{2}\theta TBN^{2}}{\log_{e}2}{\mathbf{tr}}\{\Phi_{2}^{\dag}\Phi_{2}\}\nonumber\\ &+ \frac{\ell_{1}\theta^{2}T^{2}B^{2}N^{2}\mu^{2}}{\log_{e}^{2}2}{\mathbf{tr}}^{2}\{\Phi_{1}\} + \frac{\ell_{2}\theta^{2}T^{2}B^{2}N^{2}}{\log_{e}^{2}2}{\mathbf{tr}}^{2}\{\Phi_{2}\},\end{aligned}$$ and $\ell_{1}=(bP_{d}+aP_{f})$ and $\ell_{2}=a(1-P_{f})$. We know $f(0,\theta)=1$. Then, we write $$\ddot{R}(0,\theta)=\frac{1}{\theta TBN}\left\{\mathbb{E}^{2}\left[\dot{f}(0,\theta)\right]-\mathbb{E}\left[\ddot{f}(0,\theta)\right]\right\}.$$ We can easily verify that $$\begin{aligned} \label{verification 1} \mathbb{E}\left\{{\mathbf{tr}}(\Phi_{1})\right\}&=\mathbb{E}\left\{{\mathbf{tr}}(\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1})\right\} =\mathbb{E}\left\{\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})\right\}\\ \mathbb{E}\left\{{\mathbf{tr}}(\Phi_{2})\right\}&=\mathbb{E}\left\{{\mathbf{tr}}(\mathbf{H}\mathbf{K}_{x_{2}}\mathbf{H}^{\dag})\right\} =\mathbb{E}\left\{\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{H})\right\}\end{aligned}$$ and $$\begin{aligned} \label{verification 2} \mathbb{E}\left\{{\mathbf{tr}}(\Phi_{1}^{\dag}\Phi_{1})\right\}&=\mathbb{E}\left\{{\mathbf{tr}}(\mathbf{K}_{z}^{-1}\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag} \mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1} )\right\}\nonumber\\&=\mathbb{E}\left\{{\mathbf{tr}}(\mathbf{K}_{z}^{-1}\mathbf{K}_{z}^{-1}\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag} \mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag})\right\}\\ &\geq\mathbb{E}\left\{{\mathbf{tr}}(\mathbf{K}_{z}^{-1}\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag} \mathbf{K}_{z}^{-1}\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag})\right\}\label{kitaptan}\\ &=\mathbb{E}\left\{\sum_{i,j}^{m_1}\kappa_{1i}\kappa_{1j}{\mathbf{tr}}(\mathbf{K}_{z}^{-1}\mathbf{H}\mathbf{u}_{i}\mathbf{u}_{i}^{\dag}\mathbf{H}^{\dag} \mathbf{K}_{z}^{-1}\mathbf{H}\mathbf{u}_{j}\mathbf{u}_{j}^{\dag}\mathbf{H}^{\dag})\right\}\label{follow1}\\ &=\mathbb{E}\left\{\sum_{i}^{m_1}\kappa_{1i}^{2}{\mathbf{tr}}(\mathbf{K}_{z}^{-1}\mathbf{H}\mathbf{u}_{i}\mathbf{u}_{i}^{\dag}\mathbf{H}^{\dag} \mathbf{K}_{z}^{-1}\mathbf{H}\mathbf{u}_{i}\mathbf{u}_{i}^{\dag}\mathbf{H}^{\dag})\right\}\label{follow2}\\ &=\mathbb{E}\left\{\sum_{i}^{m_1}\kappa_{1i}^{2}\lambda_{\max}(\mathbf{H}^{\dag} \mathbf{K}_{z}^{-1}\mathbf{H}){\mathbf{tr}}(\mathbf{K}_{z}^{-1}\mathbf{H}\mathbf{u}_{i}\mathbf{u}_{i}^{\dag}\mathbf{H}^{\dag})\right\}\\ &=\mathbb{E}\left\{\sum_{i}^{m_1}\kappa_{1i}^{2}\lambda_{\max}(\mathbf{H}^{\dag} \mathbf{K}_{z}^{-1}\mathbf{H}){\mathbf{tr}}(\mathbf{u}_{i}^{\dag}\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H}\mathbf{u}_{i})\right\}\\ &=\mathbb{E}\left\{\sum_{i}^{m_1}\kappa_{1i}^{2}\lambda_{\max}^{2}(\mathbf{H}^{\dag} \mathbf{K}_{z}^{-1}\mathbf{H})\right\}\\ &=\mathbb{E}\left\{\lambda_{\max}^{2}(\mathbf{H}^{\dag} \mathbf{K}_{z}^{-1}\mathbf{H})\sum_{i}^{m_1}\kappa_{1i}^{2}\right\}\\ &\geq\frac{1}{m_1}\mathbb{E}\left\{\lambda_{\max}^{2}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})\right\}\label{follow3}\end{aligned}$$ where ($\ref{kitaptan}$) comes from the fact that if $A,B\in M_{n}$ are Hermitian, ${\mathbf{tr}}(AB)^{2}\leq{\mathbf{tr}}(A^{2}B^{2})$ [@Matrix; @Analysis Chap. 4, Problem 4.1.11]. ($\ref{follow1}$) and ($\ref{follow2}$) follow from the fact that $\{\mathbf{u}_{1i}\}$ are the eigenvectors that correspond to $\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})$ and hence $\mathbf{u}_{1,i}^{\dag}\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H}\mathbf{u}_{1,j}=0$ if $i\neq j$, which comes from the orthonormality of $\{\mathbf{u}_{1,i}\}$. Finally, ($\ref{follow3}$) follows from the properties that $\kappa_{1i}\in[0,1]$ and $\sum_{i=1}^{m_1}\kappa_{1i}=1$, and the fact that $\sum_{i=1}^{m_1}\kappa_{1i}^{2}$ is minimized by choosing $\kappa_{1i}=\frac{1}{m_1}$, that leads us to the lower bound $\sum_{i=1}^{m_1}\kappa_{1i}^{2}\geq\frac{1}{m_1}$. Same procedure can be applied to $\mathbb{E}\left\{{\mathbf{tr}}(\Phi_{2}^{\dag}\Phi_{2})\right\}$, and we can easily see that $$\begin{aligned} \label{verification 2_1} \mathbb{E}\left\{{\mathbf{tr}}(\Phi_{2}^{\dag}\Phi_{2})\right\}=\mathbb{E}\left\{{\mathbf{tr}}(\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag} \mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag})\right\}&=\mathbb{E}\left\{\sum_{i,j}^{m_2}\kappa_{2,i}\kappa_{2,j} {\mathbf{tr}}(\mathbf{H}\mathbf{u}_{2,i}\mathbf{u}_{2,i}^{\dag}\mathbf{H}^{\dag}\mathbf{H}\mathbf{u}_{2,j}\mathbf{u}_{2,j}^{\dag} \mathbf{H}^{\dag})\right\}\\ &=\mathbb{E}\left\{\sum_{i}^{m_2}\kappa_{2,i}^{2} {\mathbf{tr}}(\mathbf{H}\mathbf{u}_{2,i}\mathbf{u}_{2,i}^{\dag}\mathbf{H}^{\dag}\mathbf{H}\mathbf{u}_{2,i}\mathbf{u}_{2,i}^{\dag} \mathbf{H}^{\dag})\right\}\\ &=\mathbb{E}\left\{\sum_{i}^{m_2}\kappa_{2,i}^{2}\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{H}) {\mathbf{tr}}(\mathbf{H}\mathbf{u}_{2,i}\mathbf{u}_{2,i}^{\dag}\mathbf{H}^{\dag})\right\}\\ &=\mathbb{E}\left\{\sum_{i}^{m_2}\kappa_{2,i}^{2}\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{H}) {\mathbf{tr}}(\mathbf{u}_{2,i}^{\dag}\mathbf{H}^{\dag}\mathbf{H}\mathbf{u}_{2,i})\right\}\\ &=\mathbb{E}\left\{\sum_{i}^{m_2}\kappa_{2,i}^{2}\lambda_{\max}^{2}(\mathbf{H}^{\dag}\mathbf{H})\right\}\\ &=\mathbb{E}\left\{\lambda_{\max}^{2}(\mathbf{H}^{\dag}\mathbf{H})\sum_{i}^{m_2}\kappa_{2,i}^{2}\right\}\\ &\geq\frac{1}{m_2}\mathbb{E}\left\{\lambda_{\max}^{2}(\mathbf{H}^{\dag}\mathbf{H})\right\}\end{aligned}$$ Now, we can write the second derivative of effective rate as $$\begin{aligned} \label{second derivative of effective rate} \ddot{R}_{E}(0,\theta)=&\frac{1}{\theta TBN}\bigg\{\mathbb{E}^{2}\left[\frac{\ell_{1}\theta TBN\mu}{\log_{e}2}{\mathbf{tr}}(\Phi_{1})+\frac{\ell_{2}\theta TBN}{\log_{e}2}{\mathbf{tr}}(\Phi_{2})\right]-\mathbb{E}\bigg[ \frac{\ell_{1}\theta^{2}T^{2}B^{2}N^{2}\mu^{2}}{\log_{e}^{2}2}{\mathbf{tr}}^{2}(\Phi_{1})\nonumber\\ &+ \frac{\ell_{2}\theta^{2}T^{2}B^{2}N^{2}}{\log_{e}^{2}2}{\mathbf{tr}}^{2}(\Phi_{2})\bigg] -\mathbb{E}\bigg[\frac{\ell_{1}\theta TBN^{2}\mu^{2}}{\log_{e}2}{\mathbf{tr}}(\Phi_{1}^{\dag}\Phi_{1})+ \frac{\ell_{2}\theta TBN^{2}}{\log_{e}2}{\mathbf{tr}}(\Phi_{2}^{\dag}\Phi_{2}) \bigg]\bigg\}\\ =&\frac{\theta TBN}{\log_{e}^{2}2}\mathbb{E}^{2}\left[\ell_{1}\mu{\mathbf{tr}}(\Phi_{1})+\ell_{2}{\mathbf{tr}}(\Phi_{2})\right]-\frac{\theta TBN}{\log_{e}^{2}2}\mathbb{E}\left[\ell_{1}\mu^{2}{\mathbf{tr}}^{2}(\Phi_{1})+\ell_{2}{\mathbf{tr}}^{2}(\Phi_{2})\right]\nonumber\\ &-\frac{N}{\log_{e}2}\mathbb{E}\left[\ell_{1}\mu^{2}{\mathbf{tr}}(\Phi_{1}^{\dag}\Phi_{1})+\ell_{2}{\mathbf{tr}}(\Phi_{2}^{\dag}\Phi_{2})\right]\\ \leq&\frac{\theta TBN}{\log_{e}^{2}2}\mathbb{E}^{2}\left[\ell_{1}\mu\lambda_{\max}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})+\ell_{2} \lambda_{\max}(\mathbf{H}^{\dag}\mathbf{H})\right]\nonumber\\&-\frac{\theta TBN}{\log_{e}^{2}2}\mathbb{E}\left[\ell_{1}\mu^{2}\lambda_{\max}^{2}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H}) +\ell_{2}\lambda_{\max}^{2}(\mathbf{H}^{\dag}\mathbf{H})\right]\nonumber\\ &-\frac{N}{\log_{e}2}\mathbb{E}\left[\frac{\ell_{1}\mu^{2}\lambda_{\max}^{2}(\mathbf{H}^{\dag}\mathbf{K}_{z}^{-1}\mathbf{H})}{m_1} +\frac{\ell_{2}\lambda_{\max}^{2}(\mathbf{H}^{\dag}\mathbf{H})}{m_2}\right]\nonumber\\=&\ddot{C}_{E}(0,\theta)\label{second derivative of effective rate and capacity}\end{aligned}$$ Finally, we again set $\mu =1$ following the same reasoning discussed at the end of Appendix \[app:first-deriv\]. 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[^3]: For instance, if average transmission power is limited by $P_{int}$ when the primary users are active, the average interference experienced at a given primary receiver will be limited by $P_{int} \frac{c}{d^{\varsigma}} {\mathbb{E}}\{z\}$ where $z$ is the magnitude square of the fading in the channel between the secondary transmitter and primary receiver, $d$ is the distance between them, $\varsigma$ is the path loss exponent, and $c$ is some constant related to the path loss model. [^4]: The positive semi-definiteness can be easily seen from the following simple argument. For any vector $\mathbf{a}$, we can write $\mathbf{a}^{{\dagger}}\mathbf{H}\mathbf{K}_{x_{1}}\mathbf{H}^{\dag} \mathbf{a} = \mathbf{b}^{{\dagger}}\mathbf{K}_{x_{1}} \mathbf{b} \ge 0 $, where we have defined $\mathbf{b} = \mathbf{H}^{\dag} \mathbf{a}$ and used the fact that ${\mathbf{K}}_{x_1}$ is positive semi-definite.
--- abstract: 'The study and understanding of human behaviour is relevant to computer science, artificial intelligence, neural computation, cognitive science, philosophy, psychology, and several other areas. Presupposing cognition as basis of behaviour, among the most prominent tools in the modelling of behaviour are computational-logic systems, connectionist models of cognition, and models of uncertainty. Recent studies in cognitive science, artificial intelligence, and psychology have produced a number of cognitive models of reasoning, learning, and language that are underpinned by computation. In addition, efforts in computer science research have led to the development of cognitive computational systems integrating machine learning and automated reasoning. Such systems have shown promise in a range of applications, including computational biology, fault diagnosis, training and assessment in simulators, and software verification. This joint survey reviews the personal ideas and views of several researchers on neural-symbolic learning and reasoning. The article is organised in three parts: Firstly, we frame the scope and goals of neural-symbolic computation and have a look at the theoretical foundations. We then proceed to describe the realisations of neural-symbolic computation, systems, and applications. Finally we present the challenges facing the area and avenues for further research.' author: - | Tarek R. Besold tarek-r.besold@city.ac.uk\ Department of Computer Science, City, University of London Artur d’Avila Garcez a.garcez@city.ac.uk\ Department of Computer Science, City, University London Sebastian Bader sebastian.bader@uni-rostock.de\ Department of Computer Science, University of Rostock Howard Bowman h.bowman@kent.ac.uk\ School of Computing, University of Kent Pedro Domingos pedrod@cs.washington.edu\ Department of Computer Science & Engineering, University of Washington Pascal Hitzler pascal.hitzler@wright.edu\ Department of Computer Science & Engineering, Wright State University Kai-Uwe Kühnberger kkuehnbe@uni-osnabrueck.de\ Institute of Cognitive Science, University of Osnabrück Luis C. Lamb luislamb@acm.org\ Instituto de Informatica, Universidade Federal do Rio Grande do Sul Daniel Lowd lowd@cs.uoregon.edu\ Department of Computer and Information Science, University of Oregon Priscila Machado Vieira Lima priscilamvl@gmail.com\ NCE, Universidade Federal do Rio de Janeiro Leo de Penning leo.depenning@illuminoo.com\ Illuminoo B.V. Gadi Pinkas pinkas@gmail.com\ Center for Academic Studies and Gonda Brain Research Center, Bar-Ilan University, Israel Hoifung Poon hoifung@microsoft.com\ Microsoft Research Gerson Zaverucha gerson@cos.ufrj.br\ COPPE, Universidade Federal do Rio de Janeiro bibliography: - 'neural-symbolic\_survey\_arxiv.bib' title: | Neural-Symbolic Learning and Reasoning:\ A Survey and Interpretation --- Overview ======== Prolegomena of Neural-Symbolic Computation {#prolegomena} ========================================== [NSCA]{} as Application Example for Neural-Symbolic Computing {#learning_and_reasoning} ============================================================= Neural-Symbolic Integration in and for Cognitive Science: Building Mental Models {#mental_models} ================================================================================ Putting the Machinery to Work: Binding and First-Order Inference in a Neural-Symbolic Framework {#logical_neurons} =============================================================================================== Connectionist First-Order Logic Learning Using the Core Method {#connectionist_first-order_logic} ============================================================== Markov Logic Networks Combining Probabilities and First-Order Logic {#markov_logic} =================================================================== Relating Neural-Symbolic Systems to Human-Level Artificial Intelligence {#hlai} ======================================================================= (Most) Recent Developments and Work in Progress from the Neural-Symbolic Neighbourhood {#recent_developments} ====================================================================================== Challenges and Future Directions {#future_directions} ================================ Concluding Remarks {#conclusion} ================== 0.2in
--- abstract: 'We measure the mass, decay width and production rate of orbitally excited B mesons in $1.25$ million hadronic Z decays registered by the L3 detector in 1994 and 1995. B meson candidates are inclusively reconstructed and combined with charged pions produced at the event primary vertex. An excess of events above the expected background is observed in the ${\mathrm{B}}\pi$ mass spectrum near $5.7~{\mathrm{\,Ge\hspace{-0.1em}V}}$. These events are interpreted as resulting from the decay ${\mathrm{B}^{**}}\rightarrow {\mathrm{B}^{(*)}}\pi$, where ${\mathrm{B}^{**}}$ denotes a mixture of $L=1$ B meson spin states. The masses and decay widths of the ${\mathrm{B}_2^{*}}$ ($j_q = 3/2$) and ${\mathrm{B}_1^{*}}$ ($j_q = 1/2$) resonances and the relative production rate for the combination of all spin states are extracted from a fit to the mass spectrum.' address: | Institute for Particle Physics, ETH Zurich\ CH-8093 Zurich, Switzerland author: - 'Vuko Brigljević[^1]' title: | Measurement of the Spectroscopy of Orbitally Excited B Mesons\ with the L3 detector --- Introduction ============ Detailed understanding of the resonant structure of orbitally excited B mesons provides important information regarding the underlying theory. A symmetry (Heavy Quark Symmetry) arises from the fact that the mass of the $b$ quark is large relative to $\Lambda_{\mathrm{QCD}}$. In this approximation, the spin of the heavy quark ($\vec{s}_Q$) is conserved independently of the total angular momentum ($\vec{j}_q = \vec{s}_q + \vec{l}$) of the light quark. Excitation energy levels are thus degenerate doublets in total spin and can be expressed in terms of the spin-parity of the meson $J^P$ and the total spin of the light quark $j_q$. Corrections to this symmetry are a series expansion in powers of $1/m_Q$[@Isgur], calculable in Heavy Quark Effective Theory (HQET). The $L=0$ mesons, for which $j_q = 1/2$, have two possible spin states: a pseudo-scalar $P$ ($J^P = 0^-$) and a vector $V$ ($J^P = 1^-$). If the spin of the heavy quark is conserved independently, the relative production rate of these states is $V/(V+P) = 0.75$.[^2] Recent measurements of this rate for the ${\mathrm{B}}$ system [@BstarL3; @BstarDELPHI; @BstarOPAL; @BstarALEPH] agree well with this ratio. In the case of orbitally excited $L=1$ mesons, two sets of degenerate doublets are expected: one corresponding to $j_q = 1/2$ and the other to $j_q = 3/2$. Their relative production rates follow from spin state counting ($2J+1$ states). Rules for the decay of these states to the $1S$ states are determined by spin-parity conservation [@Isgur; @Rosner]. For the dominant two-body decays, the $j_q = 1/2$ states can decay via an $L=0$ transition (S-wave) and their decay widths are expected to be broad in comparison to those of the $j_q = 3/2$ states which must decay via an $L=2$ transition (D-wave). Table \[tab:decays\] presents the nomenclature of the various spin states for $L=1$ ${\mathrm{B}}$ mesons containing either a $u$ or $d$ quark, with the predicted production rates and two-body decay modes. Several models, based on HQET and on the charmed $L=1$ meson data, have made predictions for the masses and widths of orbitally excited ${\mathrm{B}}$ mesons. Some of these models [@Gronau; @Eichten; @Falk] place the average mass of the $j_q = 3/2$ states above that of the $j_q = 1/2$ states, while others [@Isgur; @Ebert] predict the opposite (“spin-orbit inversion”). Name $j_q$ $J^P$ Production Decay mode Transition ---------------------- ------- ------- ------------ -------------------------------------------------------------------- ------------ ${\mathrm{B}_0^{*}}$ $1/2$ $0^+$ 1/12 ${\mathrm{B}_0^{*}}\rightarrow{\mathrm{B}}\pi$ S-wave ${\mathrm{B}_1^{*}}$ $1/2$ $1^+$ 3/12 ${\mathrm{B}_1^{*}}\rightarrow{\mathrm{B}^{*}}\pi$ S-wave ${\mathrm{B}_1}$ $3/2$ $1^+$ 3/12 ${\mathrm{B}_1}\rightarrow{\mathrm{B}^{*}}\pi$ D-wave ${\mathrm{B}_2^{*}}$ $3/2$ $2^+$ 5/12 ${\mathrm{B}_2^{*}}\rightarrow{\mathrm{B}^{*}}\pi,{\mathrm{B}}\pi$ D-wave : Spin states of the $L=1$ mesons with their predicted production rates and decay modes.[]{data-label="tab:decays"} Recent analyses at LEP combining a charged pion produced at the primary event vertex with an inclusively reconstructed ${\mathrm{B}}$ meson [@BstarALEPH; @BdstarOPAL; @BdstarDELPHI] have measured an average mass of $M_{{\mathrm{B}^{**}}} = 5700-5730{\mathrm{\,Me\hspace{-0.1em}V}}$, where ${\mathrm{B}^{**}}$ indicates a mixture of all $L=1$ spin states. An analysis [@BdstarALEPH] combining a primary charged pion with a fully reconstructed ${\mathrm{B}}$ meson, measures $M_{{\mathrm{B}_2^{*}}} = (5739{\phantom{0}}^{+8}_{-11}{(\mathrm{stat})}{\phantom{0}}^{+6}_{-4}{(\mathrm{syst})}){\mathrm{\,Me\hspace{-0.1em}V}}$ by performing a fit to the mass spectrum which fixes the mass differences, widths and relative rates of all spin states according to the predictions of Eichten, [*et al.*]{}.[@Eichten]. The analysis presented here [@L3Bdstar] is based on the combination of primary charged pions with inclusively reconstructed ${\mathrm{B}}$ mesons. Several new analysis techniques make it possible to improve on the resolution of the ${\mathrm{B}}\pi$ mass spectrum and to unfold this resolution from the signal components. As a result, measurements are obtained for masses and widths of D-wave ${\mathrm{B}_2^{*}}$ decays and of S-wave ${\mathrm{B}_1^{*}}$ decays. Event Selection =============== Selection of ${\mathrm{Z}}\rightarrow b\bar{b}$ decays ------------------------------------------------------ The analysis is performed on data collected by the L3 detector [@L3] in 1994 and 1995, corresponding to an integrated luminosity of $90{\mathrm{\,pb^{-1}}}$ with LEP operating at the Z mass. Hadronic Z decays are selected[@Hadrons] which have an event thrust direction satisfying $|\cos\theta| < 0.74$, where $\theta$ is the polar angle. The events are also required to contain an event primary vertex reconstructed in three dimensions, at least two calorimetric jets, each with energy greater than $10{\mathrm{\,Ge\hspace{-0.1em}V}}$, and to pass stringent detector quality criteria for the vertexing, tracking and calorimetry. A total of $1,248,350$ events are selected. A cut on a $Z \rightarrow b\bar{b}$ event discriminant based on track DCA significances [@bbtag] yields a $b$-enriched sample of $176,980$ events. To study the content of the selected data, a sample of 6 million hadronic Z decays have been generated with JETSET 7.4 [@Jetset], and passed through a GEANT based [@Geant] simulation of the L3 detector. From this sample, the $Z \rightarrow b\bar{b}$ event purity is determined to be $\pi_{b\bar{b}} = 0.828$. Selection of ${\mathrm{B}^{**}}\rightarrow {\mathrm{B}^{(*)}}\pi$ decays ------------------------------------------------------------------------ Secondary decay vertices and primary event vertices are reconstructed in three dimensions by an iterative procedure such that a track can be a constituent of no more than one of the vertices. A calorimetric jet is selected as a ${\mathrm{B}}$ candidate if it is one of the two most energetic jets in the event, if a secondary decay vertex has been reconstructed from tracks associated with that jet, and if the decay length of that vertex with respect to the event primary vertex is greater than $3\sigma$, where $\sigma$ is the estimated error of the measurement. The decay of a ${\mathrm{B}^{**}}$ to a ${\mathrm{B}^{(*)}}$ meson and a pion is carried out via a strong interaction and thus occurs at the primary event vertex. In addition, the predicted masses for the $L=1$ states correspond to relatively small $Q$ values, so that the decay pion ($\pi^{**}$) direction is forward with respect to the ${\mathrm{B}}$ meson direction. We take advantage of these decay kinematics by requiring that, for each ${\mathrm{B}}$ meson candidate, there is at least one track which is a constituent of the event primary vertex and which is located within 90 degrees of the jet axis. A total of $60,205$ track-jet pairs satisfy these criteria. To decrease background, typically due to charged fragmentation particles, only the track with the largest component of momentum in the direction of the jet is selected. This choice has been found [@BoscCDF; @BdstarALEPH] to improve the purity of the signal. The track is further required to have a transverse momentum with respect to the jet axis larger than $100{\mathrm{\,Me\hspace{-0.1em}V}}$, to reduce background due to charged pions from ${\mathrm{D}^{*}}\rightarrow {\mathrm{D}}\pi$ decays. These selection criteria are satisfied by $48,022$ ${\mathrm{B}}\pi$ pairs with a $b$ hadron purity of $\pi_{{\mathrm{B}}} = 0.942$. ### B meson direction reconstruction The direction of the ${\mathrm{B}}$ candidate is estimated by taking a weighted average in the $\theta$ (polar) and $\phi$ (azimuthal) coordinates of directions defined by the vertices and by particles with a high rapidity relative to the jet axis. A numerical error-propagation method [@Swain] makes it possible to obtain accurate estimates for the uncertainty of the angular coordinates measured from vertex pairs. These errors, as well as the error for the decay length measurement used in the secondary vertex selection, are calculated for each pair of vertices from the associated error matrices. Particles coming from the decay of $b$ hadrons produced in Z decays have a characteristically high rapidity relative to the original direction of the hadron when compared to that of particles coming from fragmentation. A cut on the particle rapidity distribution is thus a powerful tool for selecting the ${\mathrm{B}}$ meson decay constituents[@BstarDELPHI]. A second estimate for the direction of the ${\mathrm{B}}$ is obtained by summing the momenta of all charged and neutral particles (excluding the $\pi^{**}$ candidate) with rapidity $y > 1.6$ relative to the original jet axis. Estimates for the uncertainty of the coordinates obtained by this method are determined from simulated ${\mathrm{B}}$ meson decays as an average value for all events. The final ${\mathrm{B}}$ direction coordinates are taken as the error-weighted averages of these two sets of coordinates. The resolution for each coordinate is parametrized by a two-Gaussian fit to the difference between the reconstructed and generated values. For $\theta$, the two widths are $\sigma_1 = 18{\mathrm{\,mrad}}$ and $\sigma_2 = 34{\mathrm{\,mrad}}$ with $68\%$ of the ${\mathrm{B}}$ mesons in the first Gaussian. For $\phi$, the two widths are $\sigma_1 = 12{\mathrm{\,mrad}}$ and $\sigma_2 = 34{\mathrm{\,mrad}}$ with $62\%$ of the ${\mathrm{B}}$ mesons in the first Gaussian. ### B meson energy reconstruction The energy of the ${\mathrm{B}}$ meson candidate is estimated by taking advantage of the known center of mass energy at LEP to constrain the measured value. The energy of the ${\mathrm{B}}$ meson from this method [@BdstarOPAL] can be expressed as $$\label{eq:Benergy} E_{{\mathrm{B}}} = \frac{M^2_{{\mathrm{Z}}} + M^2_{{\mathrm{B}}} - M^2_{\mathrm{recoil}}}{2M_{{\mathrm{Z}}}} \quad,$$ where $M_{{\mathrm{Z}}}$ is the mass of the Z boson and $M_{\mathrm{recoil}}$ is the mass of all particles in the event other than the ${\mathrm{B}}$. To determine $M_{\mathrm{recoil}}$, the energy and momenta of all particles in the event with rapidity $y < 1.6$, including the $\pi^{**}$ candidate (regardless of its rapidity), are summed and $M^2_{\mathrm{recoil}} = E^2_{y<1.6} - p^2_{y<1.6}$. Fitting the difference between reconstructed and generated values for the B meson energy with an asymmetric Gaussian yields a maximum width of $2.8{\mathrm{\,Ge\hspace{-0.1em}V}}$. Analysis of the ${\mathbf{B}}{\mathbf{\pi}}$ Mass Spectrum ========================================================== The combined ${\mathrm{B}}\pi$ mass is defined as $$\label{eq:BpiMass} M_{{\mathrm{B}}\pi} = \sqrt{M^2_{{\mathrm{B}}} + m^2_{\pi} + 2 E_{{\mathrm{B}}} E_{\pi} - 2 p_{{\mathrm{B}}} p_{\pi} cos\alpha} \quad,$$ where $M_{{\mathrm{B}}}$ and $m_{\pi}$ are set to $5279{\mathrm{\,Me\hspace{-0.1em}V}}$ and $139.6{\mathrm{\,Me\hspace{-0.1em}V}}$, respectively, and $\alpha$ is the measured angle between the ${\mathrm{B}}$ meson and the $\pi^{**}$ candidate. The data mass spectrum is shown in Figure \[fig:Voigt\].a together with the expected Monte Carlo background. Background function ------------------- The background distribution is estimated from the Monte Carlo data sample, excluding ${\mathrm{B}^{**}}\rightarrow{\mathrm{B}^{(*)}}\pi$ decays, and fitted with a six-parameter threshold function given by $$\label{eq:BgdEqn} p_1 \times (x - p_2)^{p_3} \times e^{(p_4 \times (x - p_2) + p_5 \times (x - p_2)^2 + p_6 \times (x - p_2)^3)} \quad.$$ Parameters $p_2$ through $p_6$ are fixed to the shape of the simulated background, while the overall normalization factor $p_1$ is allowed to float freely in order to obtain a correct estimate of the contribution of the background to the statistical error of the signal. Signal function --------------- To examine the underlying structure of the signal, it is necessary to unfold effects due to detector resolution. The $\pi^{**}$ candidates are expected to have typical momenta of a few ${\mathrm{\,Ge\hspace{-0.1em}V}}$. In this range, the single track momentum resolution is no more than a few percent with an angular resolution better than $2{\mathrm{\,mrad}}$. The dominant sources of uncertainty for the mass measurement are thus the B meson angular and energy resolutions. Monte Carlo studies confirm that these two components are dominant and roughly equal in magnitude. This analysis thus concentrates on unfolding the effects of these components by parametrizing and removing their contribution to the mass resolution. ### Signal resolution and efficiency The dependence of the ${\mathrm{B}}\pi$ mass resolution and selection efficiency on $Q$ value is studied by generating signal events at several different values of ${\mathrm{B}^{**}}$ mass and Breit-Wigner width. The simulated events are passed through the same event reconstruction and selection as the data. The resulting ${\mathrm{B}}\pi$ mass distributions are each fitted with a Breit-Wigner function convoluted with a Gaussian resolution (Voigt function) and the detector resolution is extracted by fixing the Breit-Wigner width to the generated value. The Gaussian width is found to increase linearly from $20{\mathrm{\,Me\hspace{-0.1em}V}}$ to $60{\mathrm{\,Me\hspace{-0.1em}V}}$ in the ${\mathrm{B}^{**}}$ mass range $5.6-5.8{\mathrm{\,Ge\hspace{-0.1em}V}}$. This increase with $Q$ value is mainly due to the angular component of the uncertainty, which increases as a function of the opening angle $\alpha$. The resolution is parametrized as a linear function of the ${\mathrm{B}^{**}}$ mass from a fit to the extracted widths. Similarly, the selection efficiency is found to increase slightly with $Q$ value and the dependence is parametrized with a linear function. Agreement between data and Monte Carlo for the ${\mathrm{B}}$ meson energy and angular resolution is confirmed by analyzing ${\mathrm{B}^{*}}\rightarrow{\mathrm{B}}\gamma$ decays selected from the same sample of ${\mathrm{B}}$ mesons. The photon selection for this test is the same as that described in reference [@BstarL3]. A ${\mathrm{B}^{*}}$ meson decays electromagnetically and hence has a negligible decay width compared to the detector resolution. As in the case of the ${\mathrm{B}}\pi$ mass resolution, the ${\mathrm{B}}$ meson energy and angular resolution are the dominant components of the reconstructed ${\mathrm{B}}\gamma$ mass resolution. Fits to the $M_{{\mathrm{B}}\gamma}-M_{{\mathrm{B}}}$ spectra are performed with the combination of a Gaussian signal and the background function described above. For the Monte Carlo, the Gaussian mean value is found to be $M_{{\mathrm{B}}\gamma}-M_{{\mathrm{B}}} = (46.5 \pm 0.6{(\mathrm{stat})}){\mathrm{\,Me\hspace{-0.1em}V}}$ with a width of $\sigma = (11.1 \pm 0.7{(\mathrm{stat})}){\mathrm{\,Me\hspace{-0.1em}V}}$. The input generator mass difference is $46.0{\mathrm{\,Ge\hspace{-0.1em}V}}$. For the data, the Gaussian mean value is found to be $M_{{\mathrm{B}}\gamma}-M_{{\mathrm{B}}} = (45.1 \pm 0.6{(\mathrm{stat})}){\mathrm{\,Me\hspace{-0.1em}V}}$ with a width of $\sigma = (10.7 \pm 0.6{(\mathrm{stat})}){\mathrm{\,Me\hspace{-0.1em}V}}$. Good agreement between the widths of the data and Monte Carlo signals provides confidence that the ${\mathrm{B}}$ energy and angular resolution are well understood and simulated. ### Combined signal According to spin-parity rules, five mass resonances are expected, corresponding to five possible ${\mathrm{B}^{**}}$ decay modes: ${\mathrm{B}_2^{*}}\rightarrow{\mathrm{B}}\pi$, ${\mathrm{B}_2^{*}}\rightarrow{\mathrm{B}^{*}}\pi$, ${\mathrm{B}_1}\rightarrow{\mathrm{B}^{*}}\pi$, ${\mathrm{B}_1^{*}}\rightarrow{\mathrm{B}^{*}}\pi$ and ${\mathrm{B}_0^{*}}\rightarrow{\mathrm{B}}\pi$. No attempt is made to tag subsequent ${\mathrm{B}^{*}}\rightarrow{\mathrm{B}}\gamma$ decays, as the efficiency for selecting the soft photon is relatively low. As a result, the effective ${\mathrm{B}}\pi$ mass for a decay to a ${\mathrm{B}^{*}}$ meson is shifted down by the $46{\mathrm{\,Me\hspace{-0.1em}V}}$ ${\mathrm{B}^{*}}-{\mathrm{B}}$ mass difference. ------------ ------------ 3.3 truein 3.3 truein a) b) ------------ ------------ -.2 cm The five resonances are fitted with five Voigt functions, with the relative production fractions determined by spin counting rules. The Gaussian convolutions to the widths are determined by the resolution function. Additional physical constraints are applied to the mass differences and relative widths in order to obtain the most information possible from the data sample. Predictions for the mass differences $M_{{\mathrm{B}_2^{*}}} - M_{{\mathrm{B}_1}}$ and $M_{{\mathrm{B}_1^{*}}} - M_{{\mathrm{B}_0^{*}}}$ depend on several factors, including the $b$ and $c$ quark masses and, in some cases, input from experimental data of the D meson system. The values are predicted to be roughly equal and in the range $5-20{\mathrm{\,Me\hspace{-0.1em}V}}$ [@Gronau; @Eichten; @Falk; @Isgur; @Ebert]. We constrain both of the mass differences to $12{\mathrm{\,Me\hspace{-0.1em}V}}$. Predictions for the Breit-Wigner widths of the $j_q = 3/2$ are extrapolated from measurements in the D meson system [@Ddstar] and are expected to be roughly equal and about $20-25~{\mathrm{\,Me\hspace{-0.1em}V}}$. No precise predictions exist for the $j_q=1/2$ states as there are no corresponding measurements in the D system. In general, however, they are also expected to be roughly equal, although broader than those of the $j_q=3/2$ states. We constrain $\Gamma_{{\mathrm{B}_1}} = \Gamma_{{\mathrm{B}_2^{*}}}$ and $\Gamma_{{\mathrm{B}_0^{*}}} = \Gamma_{{\mathrm{B}_1^{*}}}$, but allow the widths of the ${\mathrm{B}_2^{*}}$ and ${\mathrm{B}_1^{*}}$ to float freely in the fit. Fit results ----------- Monte Carlo events for each of the expected ${\mathrm{B}^{**}}$ decays are generated and passed through the simulation and reconstruction programs and the ${\mathrm{B}}\pi$ event selection. The resulting mass spectra are combined with background and fitted with the signal and background functions under the constraints described above. Mass values and decay widths for the ${\mathrm{B}_2^{*}}$ and ${\mathrm{B}_1^{*}}$ resonances and the overall normalization are extracted from the fit and found to agree well with the generated values. All differences lie within the statistical error and have no systematic trend. The data ${\mathrm{B}}\pi$ mass spectrum is fitted with the combined signal and background functions, allowing the normalization parameters to float freely. The resulting fit, shown in Figure \[fig:Voigt\], has a $\chi^2$ of $39$ for $74$ degrees of freedom. A total of $2652$ events occupy the signal region corresponding to a relative ${\mathrm{B}^{**}_{u,d}}$ production rate of $\sigma({\mathrm{B}^{**}_{u,d}})/\sigma({\mathrm{B}_{u,d}}) = 0.39 \pm 0.05{(\mathrm{stat})}$. The mass and width of the ${\mathrm{B}_2^{*}}$ are found to be $M_{{\mathrm{B}_2^{*}}} = (5770 \pm 6{(\mathrm{stat})}){\mathrm{\,Me\hspace{-0.1em}V}}$ and $\Gamma_{{\mathrm{B}_2^{*}}} = (21 \pm 24{(\mathrm{stat})}){\mathrm{\,Me\hspace{-0.1em}V}}$ and the mass and width of the ${\mathrm{B}_1^{*}}$ are found to be $M_{{\mathrm{B}_1^{*}}} = (5675 \pm 12{(\mathrm{stat})}){\mathrm{\,Me\hspace{-0.1em}V}}$ and $\Gamma_{{\mathrm{B}_1^{*}}} = (75 \pm 28{(\mathrm{stat})}){\mathrm{\,Me\hspace{-0.1em}V}}$. Systematic uncertainty ---------------------- Sources of systematic uncertainty and their estimated contributions to the errors of the measured values are summarized in Table \[tab:syst\]. The $b$ hadron purity of the sample is varied from $91\%$ to $96\%$. The fraction of $b$ quarks hadronizing to ${\mathrm{B}_{u,d}}$ mesons is taken to be $79\%$ and is varied between $74\%$ and $83\%$ in accordance with the recommendations of the LEP ${\mathrm{B}}$ Oscillation Working Group [@LEPBoscWG]. These variations effect only the overall ${\mathrm{B}^{**}}$ production fraction. Systematic effects due to background modelling are studied by varying the shape parameters of the background function and by performing the fit with other background functions to study the effect on the measured values. Contributions to the error due to modelling of the signal are estimated for the mass and width constraints: the $M_{{\mathrm{B}_2^{*}}}-M_{{\mathrm{B}_1}}$ and $M_{{\mathrm{B}_1^{*}}}-M_{{\mathrm{B}_0^{*}}}$ mass differences are varied in the range $6-18{\mathrm{\,Me\hspace{-0.1em}V}}$ and the $\Gamma_{{\mathrm{B}_1}}/\Gamma_{{\mathrm{B}_2^{*}}}$ and $\Gamma_{{\mathrm{B}_0^{*}}}/\Gamma_{{\mathrm{B}_1^{*}}}$ ratios are varied between $0.8$ and $1$. Effects due to uncertainty in the resolution and efficiency functions are estimated by varying the slopes and offsets of the linear parametrizations. Three-body decays of the type ${\mathrm{B}_2^{*}}\rightarrow{\mathrm{B}}\pi\pi$ have been generated and passed through the simulation and reconstruction programs and the ${\mathrm{B}}\pi$ event selection. ${\mathrm{B}}\pi$ pairs, for which only one of the pions is tagged, are studied as a possible source of resonant background. The resulting reflection is found to contribute insignificantly to the background in regions of small $Q$ value. Similarly, generated ${\mathrm{B}^{**}_{s}}\rightarrow{\mathrm{B}}{\mathrm{K}}$ decays, where the ${\mathrm{K}}$ is mistaken for a $\pi$ are found to contribute only slightly to the low $Q$ value region and their effects are included in the background modelling uncertainty contribution. Conclusion ========== We measure for the first time the masses and decay widths of the ${\mathrm{B}_2^{*}}$ ($j_q=3/2$) and ${\mathrm{B}_1^{*}}$ ($j_q=1/2$) mesons. From a constrained fit to the ${\mathrm{B}}\pi$ mass spectrum, we find $$\begin{aligned} M_{{\mathrm{B}_2^{*}}} & = & (5770 \pm 6 {(\mathrm{stat})}\pm 4 {(\mathrm{syst})}){\mathrm{\,Me\hspace{-0.1em}V}}\\ \Gamma_{{\mathrm{B}_2^{*}}} & = & (23 \pm 26 {(\mathrm{stat})}\pm 15 {(\mathrm{syst})}){\mathrm{\,Me\hspace{-0.1em}V}}\\ M_{{\mathrm{B}_1^{*}}} & = & (5675 \pm 12 {(\mathrm{stat})}\pm 4 {(\mathrm{syst})}){\mathrm{\,Me\hspace{-0.1em}V}}\\ \Gamma_{{\mathrm{B}_1^{*}}} & = & (76 \pm 28 {(\mathrm{stat})}\pm 15 {(\mathrm{syst})}){\mathrm{\,Me\hspace{-0.1em}V}}\quad .\end{aligned}$$ The relative ${\mathrm{B}^{**}_{u,d}}$ production rate, including all $L=1$ spin states, is measured to be $$\begin{aligned} \frac{{\mathrm{Br}}(b\rightarrow{\mathrm{B}^{**}_{u,d}}\rightarrow{\mathrm{B}^{(*)}}\pi)} {{\mathrm{Br}}(b\rightarrow{\mathrm{B}_{u,d}})} = 0.39 \pm 0.05 {(\mathrm{stat})}\pm 0.06 {(\mathrm{syst})}\end{aligned}$$ where isospin symmetry is employed to account for decays to neutral pions. [**Sources**]{} [**$M_{{\mathrm{B}_2^{*}}}$**]{} [**$\Gamma_{{\mathrm{B}_2^{*}}}$**]{} [**$M_{{\mathrm{B}_1^{*}}}$**]{} [**$\Gamma_{{\mathrm{B}_1^{*}}}$**]{} [**$f^{**}$**]{} ------------------------------- ---------------------------------- --------------------------------------- ---------------------------------- --------------------------------------- ------------------ $b$ purity — — — — $\pm 0.02$ ${\mathrm{B}_{u,d}}$ fraction — — — — $\pm 0.03$ background $\pm 2$ $\pm 9$ $\pm 3$ $\pm 9$ $\pm 0.05$ M constraints $\pm 3$ $\pm 7$ $\pm 3$ $\pm 7$ $<0.01$ $\Gamma$ constraints $<1$ $\pm 2$ $<1$ $\pm 2$ $<0.01$ resolution $\pm 2$ $\pm 9$ $\pm 1$ $\pm 9$ $\pm 0.01$ efficiency $<1$ $\pm 1$ $<1$ $\pm 1$ $\pm 0.01$ [**Total**]{} $\pm 4$ $\pm 15$ $\pm 4$ $\pm 15$ $\pm 0.06$ : Sources of systematic uncertainty and their estimated contributions to the errors of the measured values.\[tab:syst\] Acknowledgments {#acknowledgments .unnumbered} =============== I wish to thank my colleagues Steven Goldfarb and Franz Muheim for their help in preparing this talk. 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--- abstract: 'The close correlation between cooling flows and emission-line nebulae in clusters of galaxies has been recognized for over a decade and a half, but the physical reason for this connection remains unclear. Here we present deep optical spectra of the nebula in Abell 2597, one of the nearest strong cooling-flow clusters. These spectra reveal the density, temperature, and metal abundances of the line-emitting gas. The abundances are roughly half-solar, and dust produces an extinction of at least a magnitude in $V$. The absence of \[O III\] $\lambda$4363 emission rules out shocks as a major ionizing mechanism, and the weakness of He II $\lambda$4686 rules out a hard ionizing source, such as an AGN or cooling intracluster gas. Hot stars are therefore the best candidate for producing the ionization. However, even the hottest O-stars cannot power a nebula as hot as the one we see. Some other non-ionizing source of heat appears to contribute a comparable amount of power. We show that the energy flux from a confining medium can become important when the ionization level of a nebula drops to the low levels seen in cooling-flow nebulae. We suggest that this kind of phenomenon, in which energy fluxes from the surrounding medium augment photoelectric heating, might be the common feature underlying the diverse group of objects classified as LINERS.' author: - 'G. Mark Voit and Megan Donahue' title: 'A Deep Look at the Emission-Line Nebula in Abell 2597' --- Introduction ============ A magnificent ionization nebula fills the central 20 kpc of Abell 2597, a relatively nearby cluster of galaxies at $z=0.0821$. The H$\alpha$+\[N II\] luminosity of this nebula, uncorrected for reddening, is $2.7 \times 10^{42} \, {\rm erg \, s}^{-1}$ (Heckman 1989). This intracluster emission-line display is not unique. Such nebulae are common among clusters whose central cooling times are shorter than a Hubble time but are absent in clusters with longer central cooling times (Hu, Cowie, & Wang 1985; Heckman et al. 1989). The strong correlation between central cooling time and optical nebulosity suggests that the cooling of the intracluster medium somehow excites the line emission. Yet, despite this tantalizing hint, the true source of the ionizing energy in these clusters has remained mysterious. The puzzle is not new. Over 65 years ago Hubble & Humason (1931) had already noted the unusual appearance of NGC 1275, the central galaxy of the Perseus cluster. It is quite blue for an elliptical galaxy, and its spectrum bristles with strong emission lines. Two decades later, Baade & Minkowski (1954) identified NGC 1275 with the radio source Perseus A, and Minkowski (1957) discovered that its extended emission-line system separated into two subsystems differing by 3000 km s$^{-1}$ in radial velocity. Minkowski suggested that the two velocity subsystems arose from a galaxy-galaxy collision. Soon thereafter, the Burbidges speculated that NGC 1275 might be an exploding galaxy (Burbidge, Burbidge, & Sandage 1963; Burbidge & Burbidge 1965), an interpretation given impetus by the the spectacular H$\alpha$ image of Lynds (1970) showing a complex of filaments extending up to 100 kpc from the galaxy’s center. X-ray astronomers came upon the mystery of emission lines in clusters from a different direction. Once X-ray telescopes began to resolve the hot intracluster medium (ICM) in the nearest clusters, it became apparent that the central cooling times in clusters were frequently shorter than a Hubble time. The brevity of these cooling times meant that, if there were no heat source to replenish the radiative losses, the ICM at the cores of such clusters must be cooling and settling towards the center in a “cooling flow” (see Fabian 1994 for a review). Significantly, many of the earliest known cooling-flow clusters also displayed emission-line systems, and advocates of the cooling-flow hypothesis suggested that condensing intracluster gas somehow emitted the optical lines as it cooled from several times $10^7$ K down through $10^4$ K (Fabian & Nulsen 1977; Mathews & Bregman 1978). Quantitative followup of cooling-flow clusters revealed that the connection between ICM cooling and line emission was complicated. The H$\alpha$ luminosity of a hot homogenous gas cooling through H$\alpha$-emitting temperatures is $\sim (3.8 \times 10^{39} \, {\rm erg \, s^{-1}}) \dot{M}_{100}$, where $\dot{M}_{100}$ is the mass cooling rate in units of 100 solar masses per year. The H$\alpha$ luminosities of intracluster nebulae, if they arose from simple cooling, would require $\dot{M}$ rates up to $10^4 \, M_\odot \, {\rm yr^{-1}}$, tens to hundreds of times higher than the X-ray derived cooling rates. Even the more modest cooling rates inferred from X-ray observations, typically $\dot{M}_X \sim 10-1000 \, M_\odot \, {\rm yr^{-1}}$, face a crisis which continues: we have not yet figured out where the cold gas goes. Normal star formation in cooling-flow clusters progresses at only 1-10% of $\dot{M}_X $ (e.g. McNamara & O’Connell 1989, 1992; O’Connell & McNamara 1989). Enigmatic soft X-ray absorption in many clusters seems to indicate a large mass of cold intracluster gas (White 1991; Allen 1993; Ferland, Fabian, & Johnstone 1994), but this presumably molecular gas has not yet been detected in any other waveband (McNamara, Bregman, & O’Connell 1990; O’Dea 1994; Braine 1995; Voit & Donahue 1995). While these difficulties with the cooling-flow hypothesis leave ample room for skepticism, intracluster nebulosity and a short central cooling time are clearly connected. All clusters known to have extended H$\alpha$ emission at their centers also have central cooling times shorter than a Hubble time (Hu, Cowie, & Wang 1985; Heckman 1989; Baum 1992; Donahue 1997). Moreover, the majority of clusters with short central cooling times contain such nebulae, and their line luminosities correlate with $\dot{M}_X$. In hopes of nailing down this crucial piece of the cooling-flow puzzle, astronomers have invented a variety of schemes to link line emission to the cooling ICM. The models proposed have included repressurizing shocks (Cowie, Fabian, & Nulsen 1980; David, Bregman, & Seab 1987), high-velocity shocks (Binette, Dopita, & Tuohy 1985), self-irradiated cooling condensations (Voit & Donahue 1990; Donahue & Voit 1991), and turbulent mixing layers (Begelman & Fabian 1990; Crawford & Fabian 1992). None have been entirely successful. Recent searches for \[Fe X\] 6374 Å  emission from cooling-flow clusters now seem to rule out any mechanism that relies on the ionizing photons from cooling of hot gas (Donahue & Stocke 1994; Yan & Cohen 1996). Photoionization by a central source also appears unlikely because the ionization level in cooling-flow nebulae remains constant while the pressure drops like $1/r$ (Johnstone & Fabian 1988; Heckman 1989). Johnstone, Fabian, & Nulsen (1987), motivated by the anticorrelation they found between the strength of the 4000 Å  break and the H$\beta$ luminosities of cooling flows, proposed that hot stars forming at the centers of cooling flows might photoionize cooling flow nebulae. Initially, this idea did not look promising because the emission-line spectra of cooling-flow nebulae look very different from H II region spectra. However, some recent work has strengthened the link between emission lines and star formation in cooling-flow clusters. Unpolarized excess blue continuum emission correlates with both the spatial distributions aand luminosities of cooling-flow nebulae (Allen 1995; Cardiel, Gorgas, & Aragon-Salamanca 1995, 1997; McNamara 1996a,b). In the meantime, we have obtained deep optical spectra of cooling flow clusters that also implicate hot stars as the main stimuli of intracluster H$\alpha$. Here we present our analysis of the emission lines from Abell 2597. Section 2 describes the observations, and § 3 discusses them. Because we can detect a large number of important optical lines, we can determine the density, temperature, and metallicity at the center of the emission-line nebula without resorting to model-dependent assumptions. Our analysis rules out both shocks and photoionization by a hard continuum, leaving hot stars as the most plausible option. However, hot stars have difficulty accounting for the elevated temperatures we measure. Section 4 discusses why an additional source of heating might be necessary and suggests that some sort of energy transfer from the confining hot gas might well supply it. Section 5 summarizes our results. Observations ============ We observed Abell 2597 on August 12, 1993, in nearly photometric conditions with the 5m Hale Telescope at the Palomar Observatory. Using the Double Spectrograph, we gathered both blue and red spectra simultaneously. In these observations, a dichroic split the spectrum at approximately 5500 Å, directing the blue light (3770-5500Å) onto a grating with 300 lines/mm, a dispersion of 2.15 Å/pixel, and an effective resolution of $\sim5$Å, and the red light (5550Å- 7980Å) onto a grating with 316 lines/mm, a wavelength scale of 3.06 Å/pixel, and an effective resolution of $\sim 7-8$Å. Our three 3 exposures of Abell 2597 totalled 1.5 hours, and in each exposure a 2 slit was placed on the central galaxy of Abell 2597 at a position angle of 30$^\circ$. The position angle remained within 30$^\circ$ of the parallactic angle throughout the observation. The data were processed in a standard way, using IRAF, in December 1993. We removed pixel-to-pixel variations with a normalized dome flat. Our calibration of the wavelength scale, using arc lamps of HC (blue) and NeAr (red) and some sky lines, yielded wavelength solutions with 0.5Å RMS (red) and 1.2Å RMS (blue). Several exposures of a star at different positions along the slit were used to remove curvature along the dispersion axis. The IRAF task [*fitcoords*]{} in the [*noao.twodspec.longslit*]{} package was used to correct the data and the task [*background*]{} was used to select the source-free regions along the slit and to subtract the sky contribution. The spectra were corrected for atmospheric extinction and were flux corrected with standard star exposures of Feige 110 and BD33+2642, taken directly before the observations of Abell 2597. Galactic extinction is predicted to be only 0.08 ($B$) magnitudes, so we did not deredden the spectrum. Any reddening we measure thus includes the effects of Galactic dust. We extracted a single spectrum, along 8 arcseconds of the slit centered on the nucleus, from each CCD (Figure \[all\_a2597\]). The signal-to-noise ratio of this spectrum exceeded 100 in the continuum and proved suitable for studying the faint emission lines of interest (Figures \[b\_a2597\] and \[r\_a2597\]). The emission line fluxes, measured using the [*splot*]{} task, are listed in Table 1. Errors in the fluxes of the red emission lines were assessed automatically by [*splot*]{}, which generates Monte Carlo simulations of the data and computes a 1$\sigma$ (68.3%) confidence range. The quoted errors in the red lines thus include uncertainties in the background subtraction and Poissonian noise. The complexity of the stellar continuum underlying the blue lines in our spectrum necessitated a more detailed line-measuring procedure. To measure the blue lines, we constructed a template spectrum of nearby dwarf elliptical galaxies without emission lines. These spectra were acquired during the same night with the same instrumental setup. The absorption features in the template spectrum match the absorption features in the A2597 spectrum quite well. We fitted the ratio of the A2597 spectrum to the template spectrum with a low-order polynomial, used this polynomial to scale the template spectrum, and subtracted the scaled template from the A2597 spectrum. Template subtraction proved to be very important. The A2597 emission line + stellar continuum spectrum showed a possible feature at the expected position of \[O III\] $\lambda$4363, but this feature vanished upon subtraction of the stellar continuum because it was an artifact of the stellar absorption lines (Figure \[fig4363\]). Measuring the He II 4686Å  recombination line also depends critically on template subtraction (Figure \[fig4686\]). In general, most of the error in the blue line measurements stems from uncertainties in the subtraction of the background continuum. To gauge this uncertainty, we measured the lines manually many times, evaluating how different methods of background removal changed the total line fluxes. Line flux estimates generally stayed within 5% from measurement to measurement. For the weakest lines, this variation sometimes approached 10%. Our line-flux uncertainties reflect this variation. Upper limits for the undetected lines were evaluated by measuring the residual RMS variation after subtraction of the stellar continuum template. For a FWHM=10Å, the $3\sigma$ upper limit is $1.1 \times 10^{-16} \, {\rm erg \, cm^{-2} \, s^{-1}}$. HeI 4471 was detected just barely above the 3$\sigma$ limit of $1.2-1.5 \times 10^{-16} \, {\rm erg \, cm^{-2} \, s^{-1}}$, and we will treat this flux as an upper limit. The lines all had a FWHM consistent with a velocity dispersion ($\sigma$) of 270 km/sec (FWHM$=2.355\sigma$). The best-fit redshift is $0.0821 \pm 0.0002$. No correction for the LSR was required because $V_{\rm LSR} = -13.5 \, {\rm km \, s^{-1}}$, smaller than our resolution. Emission-Line Analysis ====================== Most emission-line studies of cooling-flow nebulae have relied on accurate measurements of the strongest lines. The relative fluxes of features like H$\alpha$, H$\beta$, \[N II\] $\lambda$6584, and \[O III\] $\lambda$5007 are generally handy for broad classification of extragalactic nebulae, but by themselves they are not quite so useful for measuring physical quantities such as electron temperature and metallicity. With information on only a small set of lines, we are usually left having to fit underconstrained photoionization models to the data. This dataset on Abell 2597 enables us to do much more. The extensive set of line fluxes in Table 1 can be used to measure the reddening of the emission-line spectrum, the density and temperature of the nebular gas, and the nebula’s approximate metallicity, all without recourse to underconstrained photoionization models. We can also put strong limits on the shock-excited \[O III\] line at 4363 Å and the He II recombination line at 4686 Å, measurements that rule out shocks as a major ionizing mechanism and point towards hot stars as the primary ionizing agent in the intracluster nebula. Oxygen Lines & Shocks --------------------- One of our primary motivations in obtaining a deep blue spectrum of Abell 2597 was to measure the 4363 Å  emission line of \[O III\]. Shocks radiate this line much more efficiently than photoionized gas, making it an effective diagnostic for distinguishing between these ionization processes. Most shock models predict $$\ROIII \equiv \frac {F_{4363}} {F_{4959} + F_{5007}} \approx 0.05 - 0.07$$ for the ratio between the \[O III\] 4363 Å line and the sum of the 4959 Å and 5007 Å lines. Photoionization models predict smaller $R_{\rm O \, III}$ values because the \[O III\] emitting gas is cooler. At a temperature of 10,000 K, $R_{\rm O \, III} \approx 0.005$ in low-density gas. We find that at the center of Abell 2597, $R_{\rm O \, III} < 0.02 \, (3\sigma)$, strongly indicating that shocks are not the dominant ionization process there. Figure \[o3rat\] shows $R_{\rm O \, III}$ predictions for shocks of various velocities from the models of Shull & McKee (1979; SM79), Binette (1985; BDT85), Hartigan, Raymond, & Hartmann (1987; HRH87), and Dopita & Sutherland (DS95). Figure \[o3\_hb\_rat\] gives the corresponding ratios of \[O III\] to H$\beta$. As the shock velocity $v_s$ rises through $80 \, {\rm km \, s^{-1}}$, the shock begins moving fast enough to collisionally ionize O$^+$ to O$^{++}$, so the \[O III\] fluxes rise rapidly. In the cooling postshock gas, the mean temperature of the \[O III\] emitting gas does not vary much with shock velocity, and the $R_{\rm O \, III}$ ratio remains close to 0.05 up to shock velocities of a couple hundred km s$^{-1}$. At higher shock velocities most of the \[O III\] flux comes from photoionized gas either upstream or downstream from the shock. Ultraviolet radiation from the cooling gas behind the shock propagates in both directions, ionizing whatever it encounters. In the DS95 (s) and BDT95 models, which consider photoionization of only the downstream gas, $R_{\rm O \, III}$ almost drops below the observed limits, but the \[Fe X\] limits rule out shocks this fast (Yan & Cohen 1995). The DS95 (sp) model also includes the photoionizing effects of postshock radiation on the lower-density upstream gas. The photoionizing precursor creates a more highly ionized region that can radiate strongly in \[O III\]. Such a shock satisfies the observed limits on $R_{\rm O \, III}$ but produces an \[O III\]/H$\beta$ ratio far larger than observed. In a very narrow range of shock velocities near 80 km s$^{-1}$ shock models come close to satisfying the observed constraints on $R_{\rm O \, III}$ and \[O III\]/H$\beta$ but fail for other reasons. These shocks require extremely unlikely fine tuning and do not reproduce the other forbidden-line ratios. At 80 km s$^{-1}$, the SM79 models predict \[N II\] 6584/H$\beta \approx 0.27$ and \[S II\] 6717/H$\beta \approx 0.24$, whereas our data show \[N II\] 6584/H$\beta \approx 0.80$ and \[S II\] 6717/H$\beta \approx 0.54$. The disagreement of shock models with the observed \[O III\] lines leaves photoionization as the most likely mechanism for generating the emission lines in Abell 2597. Balmer Lines & Reddening ------------------------ Our spectrum of Abell 2597 contains hydrogen Balmer recombination lines ranging from H$\alpha$ to H$\zeta$. Assuming Case B recombination applies, we can use the Balmer series to measure the reddening of the visible spectrum. The Balmer-line ratios decline systematically from red to blue, relative to Case B expectations at a density of $10^2 \, {\rm cm}^{-2}$ and a temperature of 10,000 K, indicating that significant amounts of dust obscure our view of the nebula. The Galactic H I column towards Abell 2597 is a modest $2.5 \times 10^{20} \, {\rm cm}^{-2}$, so the large majority of the obscuring dust must reside in the cluster. Figure \[redden\_errs\] illustrates how the inferred amount of dust depends on the geometry of the dusty gas. To convert reddening to extinction, we assume a Galactic reddening law (Fitzpatrick 1986). If the dust forms a screen interposed between us and the nebula, the optical depth of the screen is $\approx 1.2$ at the wavelength of H$\beta$ ($A_V \sim 1$), corresponding to $N_{\rm H \, I} \approx 2 \times 10^{21} \, {\rm cm}^{-2}$ for a Galactic dust-to-gas ratio (Draine & Lee 1984). The observed reddening of the Balmer lines turns out to be quite consistent with the properties of Galactic dust. Alternatively, the grains could be intermixed with the line-emitting gas. This scenario requires a significantly larger dust column, one we cannot limit from above. The expected reddening in the large $N_{\rm H \, I}$ limit is essentially indistinguishable from that in the screen model. Density & Ionization Parameter ------------------------------ The \[S II\] line ratio straightforwardly gives the electron density, $n_e$, at the center of the nebula. Figure \[dens\_new\] shows that $n_e = 100 - 300 \, {\rm cm}^{-3}$, with a best value of $200 \, {\rm cm}^{-3}$. Such a density is typical of the inner regions of cooling-flow nebulae (e.g. Heckman 1989). We can combine this density measurement with the H$\alpha$ surface brightness to estimate the ionization level and column density of the nebula. The H$\alpha$ brightness at the center of the nebula is $4.3 \times 10^{-15} \, {\rm erg \, cm^{-2} \, s^{-1} \, arcsec^{-2}}$, implying an emission measure of $2100 \, {\rm cm^{-6} \, pc}$. At an electron density of $200 \, {\rm cm^{-3}}$, the inferred column density of ionized gas is then $N_{\rm H \, II} \approx 3 \times 10^{19} \, {\rm cm^{-2}}$. Because the 21 cm absorption line in Abell 2597 indicates a much higher column density of H I, the ionized layers are likely to be thin skins on the surfaces of thick neutral clouds (O’Dea, Gallimore, & Baum 1994). The ionization parameter $U$ of a photoionized nebula is defined to be the ionizing photon density divided by the number density of hydrogen nuclei. A thick photoionized slab of hydrogen gas illuminated from one side has an ionized column $\sim (10^{23} \, {\rm cm}^{-2}) U$, so the observed H II column implies $U \sim 10^{-4}$, if the nebulae are ionization-bounded and only a few distinct photoionized surfaces lie along a given line of sight through the nebula. This value of $U$ is similar to that indicated by the \[O III\]/ratio (Voit, Donahue, & Slavin 1994). Temperature ----------- Our spectra contain temperature sensitive line sets from both \[O II\] (3726 Å, 3729 Å, 7320 Å, 7330 Å) and \[S II\] (4068 Å, 4076 Å, 6717 Å, 6731 Å). The two ratios of interest are $$\ROII \equiv \frac {F_{7320}+F_{7330}} {F_{3726}+F_{3729}}$$ and $$\RSII \equiv \frac {F_{4068}+F_{4076}} {F_{6716}+F_{6731}} \; .$$ To determine $\ROII$, we need to remove the contribution of \[Ca II\] 7324 to the red \[O II\] line blend by subtracting 0.68 times the \[Ca II\] 7291 flux. Thus, $\ROII = 0.041 \pm 0.004$ and $\RSII = 0.042 \pm 0.004$. To measure accurate temperatures, we need to correct these line ratios for reddening. Luckily, reddening affects the temperatures derived from these line ratios in opposite ways. In the temperature-reddening plane, the true temperature should lie at the intersection of the loci defined by $\ROII$ and $\RSII$. Figure \[temps\_shade\] illustrates how the temperatures inferred from these ratios in Abell 2597 change as the dust optical depth at H$\beta$ ($\tau_{{\rm H}\beta}$) increases, given an electron density of 200 cm$^{-3}$ and a screen model for the obscuring dust. We also include the locus defined in this same plane by the H$\delta$/H$\alpha$ ratio. Within the observational errors, these three loci all intersect in a region where $9500 \, {\rm K} < T_e < 12,000 {\rm K}$. Note that these temperature limits do not depend significantly on the reddening model. The red \[S II\] lines are near H$\alpha$ and the blue \[S II\] lines are near H$\delta$, so the necessary Balmer-line correction gives the \[S II\] reddening correction directly. The exact correction for the \[O II\] lines should differ only slightly from the \[S II\] correction. Note also that the 9,500 K lower limit on the electron temperature does not depend on $\ROII$. Metallicity ----------- Because we know both the temperature and density of the ionized gas, the ratio of each forbidden-line flux to the hydrogen Balmer lines tells us the abundance of each line-emitting species. To derive elemental abundances from these lines, we would also need to know the ionization level in the plasma. Given the small ionization parameter (§ 3.2), the ionization levels are likely to be low, with most species predominantly singly ionized. Figure \[nii\_rat\] shows how the derived abundance of N$^+$ varies with electron temperature for the \[N II\] 6584/H$\alpha$ line ratio of 0.83 observed in Abell 2597. If the electron temperature is $\sim 9,500$ K, the N$^+$/H$^+$ ratio is about half of the solar N/H ratio. At the upper end of the allowed temperature range, near 12,000 K, the derived N$^+$/H$^+$ ratio drops to about 1/4 of the solar N/H. The story for S$^+$ is similar to that for N$^+$. Figure \[sii\_rat\] shows the derived range of S$^+$/H$^+$, which runs from half-solar at 9,500 K to 1/4 solar at 12,000 K. Sulphur is somewhat more likely to be doubly ionized than nitrogen, as the ionization potential for S$^+$ is 23.3 eV, compared to 29.6 eV for N$^+$. Nevertheless, the close correspondence between the relative abundances of N$^+$ and S$^+$ suggests that sulphur is mostly singly-ionized. Our determination of the oxygen abundance benefits from lines radiated by three different ionization states. Figure \[o\_rat\] shows the relative abundances of O$^+$ and O$^\circ$, with respect to H$^+$. As with N and S, their sum corresponds to half of the solar O/H ratio at 9,500 K and to 1/4 the solar value at 12,000 K. The relatively modest \[O III\] lines indicate that O$^{++}$ is a minority species; both O$^+$ and O$^\circ$ are more common. The consistency of these abundances with those for nitrogen and sulphur is comforting; however, some of the \[O I\] flux could be coming from X-ray heated neutral gas behind the ionized region, so the derived O$^\circ$/H$^+$ is really an upper limit for the ionized gas. Helium Recombination Lines -------------------------- The forbidden lines in cooling-flow nebulae are generally quite strong, relative to the Balmer recombination lines, indicating a large amount of heat input per photoionization. One possible way to supply the requisite heating is with a hard photoionizing continuum that remains strong into the soft X-ray band. Models that have used such an ionizing spectrum to reproduce some of the major line ratios include fast-shock models (Binette et al. 1985), self-irradiation models (Voit & Donahue 1990, Donahue & Voit 1991), and mixing-layer models (Crawford & Fabian 1992). The helium recombination lines provide a model-independent way to check the hardness of the incident continuum. Our spectrum of Abell 2597 includes the He II recombination line at 4686 Å and the He I recombination lines at 6678 Å, 5876 Å, and 4471 Å. Broad Na I absorption, possibly associated with the dusty H I gas, contaminates the 5876 Å line, reducing its usefulness. The story told by the 4471 Å  line is also unclear because the underlying continuum is so noisy. We barely detect this line at a level of $F_{4471}/\Hb \approx 0.02$, whereas we would expect $F_{4471}/\Hb \approx 0.05$ if all the helium were singly ionized. The 6678 Å  line, located in a cleaner part of the spectrum, is probably the most reliable. At the center of the cluster, $F_{6678}/F_{\Ha} = 0.01$, as expected in a nebula where singly-ionized helium predominates. In contrast, $F_{4686}/F_{\Hb} \approx 0.02$, implying very little He$^{++}$. The lack of He II recombination-line flux from Abell 2597 indicates that the photoionizing continuum cannot be especially hard. If the incident spectrum follows a $F_\nu \propto \nu^\alpha$ power law from 13.6 eV into the soft X-ray band, then $\alpha \approx -2.3$ for $U \sim 10^{-4}$. This kind of spectrum is too soft to supply the necessary heating per ionization. The He II/H I line ratio argues against AGN irradiation, fast shocks, and cooling gas as sources of the photoionizing continuum, leaving hot stars as the only plausible possibility.[^1] Energetics ========== The line-ratio analysis of the previous section eliminated many of the most frequently mentioned energy sources for cooling-flow nebulae. Hot stars were not ruled out by these model-independent diagnostics, but can we construct a hot-star photoionization model that actually works? In this section, we attempt to do so, but we find that standard hot-star photoionization fails to provide the requisite heating by a factor of about two. We speculate that some kind of energy transfer from the hot surrounding medium might supply the other half of the heating. Stellar Photoionization: Insufficient ------------------------------------- To investigate whether hot stars alone can produce the emission lines observed in Abell 2597, we constructed numerous models with the photoionization code CLOUDY, version 84 (Ferland 1993). The code includes a grid of Kurucz model atmospheres up to effective temperatures of 50,000 K, as well as blackbody spectra at any temperature. We tested stellar models with $T_{\rm eff} = 35,000 - 50,000$ K and blackbody models with $T_{\rm bb} = 30,000 - 100,000$ K incident upon a gas with 0.5 solar metallicity, Galactic depletions, and $n_e = 200 \, {\rm cm^{-3}}$. Figure \[tmodels\] compares the temperatures derived from the \[S II\] and \[O II\] lines in these models with the temperature ranges derived from the observed lines, assuming $\tau_{{\rm H}\beta} = 1$. Only in the very hottest blackbody models do the electron temperatures approach the observed values. Thus, the continuua of hot stars are not hard enough to produce temperatures exceeding 9,500 K in gas with half-solar metallicity. If stars are responsible for photoionizing the nebula in Abell 2597, some other energy source must be providing additional heating comparable in magnitude to the photoelectric heating. Figure \[heat50\_50\] shows a specific example of stellar insufficiency. In these models, a hot stellar continuum with $T_{\rm eff} = 50,000$ K irradiates a gas with half-solar metallicity. When the illuminating star is this hot, the O III/ratio constrains the ionization parameter to be $U \approx 10^{-4.0} - 10^{-4.5}$, in agreement with our estimate from the H$\alpha$ surface brightness (§ 3.3). Refractory elements are depleted into dust grains, as in our own interstellar medium, and photoelectric heating from dust contributes energy to the nebula. Both depletion of coolants and dust heating raise the expected equilibrium temperature. Even so, the equilibrium temperatures set by photoelectric heating barely rise above 8000 K. The extra heating required to boost $T_e$ above 9,500 K is comparable to the photoelectric heating itself. In equilibrium at these higher temperatures the sum of photoelectric heating and some other supplementary form of heating must balance line cooling. To achieve temperatures in the observed range, the total heating must be roughly twice the photoelectric heating. Even when the “extra heating” is several times the photoelectric input, Ly$\alpha$ cooling keeps the equilibrium temperatures below 12,000 K. Lowering the metallicity does not fix the mismatch between photoelectric heating and line cooling. In § 3.5 we showed that metallicities around 1/4 solar are possible if $T_e \approx 12,000$ K. Figure \[heat50\_25\] illustrates that stellar photoionization indeed generates higher equilibrium temperatures in lower metallicity gas, but the extra heating needed to boost $T_e$ all the way up to 12,000 K is still $2-4$ times the photoelectric heating. Models with half-solar metallicities actually require smaller proportions of additional heating. Extra Heating ------------- The failure of stellar photoionization to account fully for the heating of the nebula might be surprising, given the convincing evidence in favor of stellar photoionization as the primary ionizing mechanism. On the other hand, another ample source of heat energy, namely the pervasive hot intracluster gas, lies close at hand and presumably confines the ionized surfaces of the nebula. Several authors have pointed out that the energy fluxes necessary to power the most luminous cooling-flow nebulae are similar to the product of the pressure $P$ and the sound speed $v_{\rm th}$ in the surrounding medium (Heckman 1989; Donahue & Voit 1991; Crawford & Fabian 1992). Here we argue that this property of cooling-flow nebulae may not be entirely coincidental. Energy can conceivably flow from the hot ICM into the cooler nebula in numerous ways. A few specific examples are electron thermal conduction (e.g. Sparks 1992), acoustic wave heating (e.g. Pringle 1989), and MHD wave heating (e.g. Friaca 1997). All of these heat fluxes saturate at a rate $\sim Pv_{\rm th}$. Assume for the moment that thermal energy passes from the hot medium into the photoionized surfaces of the nebula at the saturated flux $$F_{\rm sat} = f_{\rm sat} P v_{\rm th} \; \; ,$$ where $f_{\rm sat}$ is a parameter of order unity. Meanwhile, the flux of photoelectric heat energy into the nebula is $$F_{\rm ph} \approx \frac {PcU(\bar{E}_{\rm ph} - 13.6 \, {\rm eV})} {2.3 k(10,000 \, {\rm K})}$$ where $\bar{E}_{\rm ph}$ is the mean energy per ionizing photon. The ratio of these energy fluxes is $$\frac {F_{\rm sat}} {F_{\rm ph}} = 1.5 \, f_{\rm sat} \left( \frac {U} {10^{-4}} \right)^{-1} \left( \frac {v_{\rm th}} {300 \, {\rm km s^{-1}}} \right) \left( \frac {\bar{E}_{\rm ph}} {13.6 \, {\rm eV}} - 1 \right)^{-1} \; \; .$$ Note that in normal H II regions, with $U \sim 10^{-2} - 10^{-3}$, the photoelectric heat flux overwhelms any mechanism operating at the rate $F_{\rm sat}$. Such processes become energetically significant only where $U$ is low, in nebulae with dilute radiation fields or high pressures. At the low ionization parameters observed in Abell 2597 and other cooling-flow nebulae, a saturated mechanical energy flux could easily be comparable to the photon energy flux. While these alternative mechanisms for supplying “extra heating” appear promising, and maybe even inevitable, on the broad level of the energy budget, further work is required to see if they will dissipate and distribute their energy throughout the nebula. For instance, saturated electron thermal conduction can transfer energy at a maximum flux of $$F_{e^-} \approx 5 P v_{\rm th}$$ (Cowie & McKee 1977), but the penetration depths of hot electrons into the nebula depend strongly on their velocities. An electron of energy $T_{\rm keV} (1 \, {\rm keV})$ passing into a fully ionized $10^4$ K plasma of density $n_e = 200 \, {\rm cm^{-3}}$ will stop after traversing a column density $$N_{\rm stop} \approx (2 \times 10^{17} \, {\rm cm^{-2}}) T_{\rm keV}^2$$ (Voit 1991). Comparing $N_{\rm stop}$ to the typical thickness of a low-ionization H II layer, $\sim (10^{19} \, {\rm cm^{-2}})(U/10^{-4})$, shows that the distribution of conductive heating in the nebula will depend sensitively on the exterior temperature. However, electrons at the expected energies of several keV are clearly energetic enough to penetrate most of the ionized layer, as long as the magnetic field geometry is favorable. A coupled photoionization/conduction code will be needed to solve for the actual temperature structure and line emission from such a cloud. LINERs in General ----------------- Classifiers of extragalactic nebulae place cooling-flow nebulae among the LINERs, a heterogeneous class of objects whose \[O II\] $\lambda$3727 lines are stronger than their \[O III\] $\lambda$5007 lines and whose \[O I\] $\lambda$6300 lines are greater than 1/3 of \[O III\] $\lambda$5007 (see Filippenko 1996 for a recent review). Generally, LINERs also have \[N II\] $\lambda$6584 / $>$ 0.6, unusually high for extragalactic H II regions (e.g. Ho 1996). These large forbidden-line fluxes, relative to and , imply high electron temperatures difficult to attain with normal O-star photoionization. While photoionization by an AGN-like nuclear X-ray source might explain the line ratios in compact LINERs, the line emission in a significant fraction of LINERs extends over a few kiloparsecs and appears to be powered by local processes (Filippenko 1996). Furthermore, the He II $\lambda$4686/ratios in LINERs are frequently lower than AGN-like models would predict (Netzer 1990). Quite possibly, supplementary heating of the kind proposed here for cooling-flow nebulae could account for the strong forbidden lines of LINERs in general, regardless of the ionizing source. Mechanical and conductive forms of heating probably operate constantly at some level in all kinds of photoionized nebulae. Usually, the photon fluxes incident on these nebulae overwhelm any other energy source. However, when the ambient pressure rises or the photon flux decreases, the ionization parameter drops, reducing the dominance of photoelectric heating. If mechanical or conductive processes transfer heat from a hot confining medium to photoionized clouds at the saturation rate ($\sim P v_{\rm th}$), then we would expect their contributions to become significant when $U \lesssim 10^{-3.5}$, the ionization range characteristic of LINERs. A variety of physical processes, differing from one astrophysical environment to another, could conspire to produce this characteristic combination of low ionization and supplementary heating in many different situations. The spectral signatures of these objects would all look LINER-like, even though the underlying phenomena differ. Their only shared property would be a value of $U$ low enough for other forms of heating to augment photoelectric heating. Summary ======= Our deep spectra of the cooling-flow nebula in Abell 2597 strongly constrain the possible ionizing mechanisms. The lack of \[O III\] $\lambda$4363 emission and the modest \[O III\] $\lambda$5007 ratio rule out shocks. The Balmer sequence indicates substantial reddening. After we correct for reddening, we find that the \[O II\] and \[S II\] temperatures agree, placing the nebula between 9,500 K and 12,000 K. Temperatures like this in low-ionization nebulae usually signify a hard photoionizing source that extends into the X-ray band, but the small He II $\lambda$4686 / ratio we observe shows that the role of photons $> 54.4$ eV must be very minor. Hot stars are the only plausible ionizing sources that remain. Our finding that hot stars are the most likely ionizing source agrees with observations of excess blue light and dilution of stellar absorption features in cooling-flow nebulae. However, the hottest O stars are still too cool to generate the high temperatures and strong forbidden lines we observe. In a pure photoionization model, the observed line ratios require a blackbody-like spectrum exceeding 100,000 K with a luminosity of a few times $10^{44} \, {\rm erg \, s^{-1}}$. The photoelectric heating provided by O stars with effective temperatures of 50,000 K supplies only about half the necessary heating. Conduction or some other mechanical form of heating might supplement photoelectric heating in cooling-flow nebulae and other LINERs. At the characteristic ionization parameter $U \sim 10^{-4}$, the maximum heat flux from the confining medium $(\sim P v_{\rm th})$ is similar to the photoionizing energy flux. Heat sources like this might always be present in photoionized nebulae, at levels too low to make a difference. However, in low-ionization nebulae their contributions can grow significant, boosting the forbidden-line output, especially if the temperature of the surrounding medium exceeds $10^7$ K, as in cooling flows or the hot galactic winds expected from starbursts and AGNs. M. D. thanks the Carnegie Observatories for the Carnegie Fellowship that enabled her to gather these data. The authors would also like to acknowldge Daniela Calzetti for donating her reddening-curve software, Ari Laor for an enlightening Christmas Eve message, and Michaela Voit for her continuing inspiration and enthusiasm. Allen, S. 1995, , 276, 947. Allen, S. W., Fabian, A. C., Johnstone, R. M., White, D. A., Daines, S. J., Edge, A. C., & Stewart, G. C. 1993, , 262, 901 Baade, W., & Minkowski, R. 1954, , 119, 215. Baum, S. A. 1992, in Clusters and Superclusters of Galaxies, ed. A. C. Fabian (Dordrecht: Kluwer), p. 171 Begelman, M. C., & Fabian, A. C. 1990, , 244, 26P Binette, L., Dopita, M. A., & Tuohy, I. 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A. 1991, , 252, 72 Yan, L., & Cohen, J. 1995, , 454, 54 [lccc]{}   & Observed & Line Flux & &  \ Line ID & Wavelength & 1$\sigma$ Error & FWHM\   & (Å) & (10$^{-15}$ erg s$^{-1}$ cm$^{-2}$) & (Å)\   &   &   &  \ OII 3727 & 4033.65 & 37. $\pm$ 1.0 & 13.1\ NeIII 3869 & 4186.3 & 1.4 $\pm$ 0.10 & 11.5\ H$\zeta$ 3888.1 & 4208.6 & 0.6 $\pm$ 0.10 & 11.5\ H$\epsilon$+NeIII 3966 & 4292.1 & 1.95 $\pm$ 0.20 & 13.5\ SII 4069 & 4405.0 & 1.5 $\pm$ 0.15 & 13.9\ H$\delta$ 4101 & 4437.8 & 1.7 $\pm$ 0.15 & 13.9\ H$\gamma$ 4340 & 4696.7 & 2.9 $\pm$ 0.3 & 11.3\ OIII 4363 &   & $< 0.13 \, (3\sigma)$ &\ HeII 4686 & 5073. & 0.26 $\pm$ 0.04 &  \ OIII 4969 & 5367.0 & 1.4 $\pm$ 0.2 & 14.0\ H$\beta$ 4861 & 5260.4 & 8.4 $\pm$ 0.2 & 14.0\ OIII 5007 & 5420.8 & 4.8 $\pm$ 0.3 & 14.0\ NI 5200 & 5628.5 & 2.7 $\pm$ 0.14 & 14.8\ OI 6300 & 6817.8 & 10.9 $\pm$ 0.1 & 14.7\ OI 6363 & 6886.9 & 2.37 $\pm$ 0.07 & 14.7\ NII 6548 & 7086. & 10.6 $\pm$ 0.09 & 15.5\ H$\alpha$ 6563 & 7102.35 & 35.4 $\pm$ 0.12 & 15.5\ NII 6584 & 7124.48 & 29.5 $\pm$ 0.12 & 15.5\ HeI 6678 & 7226. & 0.4 $\pm$ 0.1 & 16.2\ SII 6717 & 7268.2 & 20.0 $\pm$ 0.1 & 16.2\ SII 6731 & 7284.3 & 16.0 $\pm$ 0.11 & 16.2\ CaII 7290 & 7887.0 & 0.7 $\pm$ 0.1 & 20.0\ OII+CaII 7320 & 7924.5 & 2.0 $\pm$ 0.2 & 20.0\ [^1]: We note, however, that a significant minority of quasars do have He II 4686/ratios this small (e.g. Boroson & Green 1992). The ionizing continuua of these objects must be far weaker than the standard AGN continuum at the He II edge and cannot sustain the high nebular temperatures we observe.
--- abstract: 'We report high resolution angle-scanned photoemission and Fermi surface (FS) mapping experiments on the layered transition-metal dichalcogenide 1$T$-TaS$_{2}$ in the quasi commensurate (QC) and the commensurate (C) charge-density-wave (CDW) phase. Instead of a nesting induced partially removed FS in the CDW phase we find a pseudogap over large portions of the FS. This remnant FS exhibits the symmetry of the one-particle normal state FS even when passing from the QC-phase to the C-phase. Possibly, this Mott localization induced transition represents the underlying instability responsible for the pseudogapped FS.' address: | $^{(1)}$Institut de Physique, Université de Fribourg, CH-1700 Fribourg, Switzerland\ $^{(2)}$ Institut de Physique Appliquée, EPF, CH-1015 Lausanne, Switzerland\ [submitted to Phys. Rev. Lett.,]{.nodecor} author: - 'Th. Pillo$^{(1)}$, J. Hayoz$^{(1)}$, H. Berger$^{(2)}$, M. Grioni$^{(2)}$, L. Schlapbach$^{(1)}$, P. Aebi$^{(1)}$' title: 'Remnant Fermi surface in the presence of an underlying instability in layered 1$T$-TaS$_{2}$' --- epsf [2]{} The layered transition metal dichalcogenide 1$T$-TaS$_{2}$ is a model system being the first material where charge density waves (CDW) have been experimentally discovered by means of superlattice spots in X-ray diffraction (XRD) experiments[@wilson74]. It provides, however, one crucial difference to isostructural 1$T$-type materials, because it shows, besides the Peierls transition at approximately 550 K, a Mott localization induced transition at 180 K, where electrons stemming from the Ta $5d$-band manifold become more and more localized with decreasing temperature and, suddenly, yield a commensurate locked-in CDW[@fazekas80]. As a consequence, 1$T$-TaS$_{2}$ exhibits a rich phase diagram, where several phases exist as a function of temperature[@wilson75]. The relevant phases here are, first, the quasi-commensurate (QC) phase, stable at room temperature (RT) and known to exhibit hexagonal arrays of commensurate domains with $(\sqrt{13} \times \sqrt{13})$ symmetry[@stm]. Second, the commensurate (C) phase below 180 K where the CDW is completely locked in. The electronic structure of 1$T$-TaS$_{2}$ is considerably influenced by the CDW expressed by the decay of the one-particle Ta $5d$-band, which is split off into three dispersionless submanifolds already in the QC-phase[@smith85; @manzke88]. Angle-resolved photoelectron spectroscopy (ARPES) investigations have given experimental evidence[@manzke89; @dardel92a; @dardel92b] for a so-called $'$star-of-David$'$ model of Fazekas and Tosatti (FT)[@fazekas80]. Thirteen Ta atoms form two outerlying bonding shells with six Ta atoms each, displayed in ARPES spectra as the two low energetic manifolds. The shallow band containing the remaining thirteenth electron is susceptible to Mott localization and splits into a lower occupied (LHB) and upper unoccupied Hubbard subband. The LHB is manifest as a sharp, dispersionless peak near the Fermi level E$_{F}$ in near-normal emission ARPES spectra of the C phase[@manzke89; @dardel92a; @dardel92b]. Complementary tunneling spectroscopy data[@kim94] indicates a symmetric splitting of the LHB and the unoccupied upper band with respect to E$_{F}$. Temperature dependent near normal emission measurements showed, that the C-phase reveals a pseudogap with residual spectral weight at E$_{F}$ down to very low temperatures, explaining low temperature resistivity data in terms of a variable range hopping mechanism[@dardel92a; @dardel92b]. Moreover, the local Coulomb correlation energy, the Hubbard U$_{dd}$, depends on random disorder[@grioni98]. Very recently Fermi surface (FS) measurements using ARPES have gained particular interest with respect to the mechanism behind high-temperature superconductivity[@ronning98; @norman98]. In underdoped cuprates a remnant FS has been detected at temperatures around the transition temperature T$_{c}$[@norman98]. The important point seems to be that there is an underlying electronic instability which drives the pseudogap and the remnant FS behaviour[@ding97; @ding96]. One might ask whether to expect a comparable behaviour of the FS for other materials with underlying electronic phase transitions. As a possible candidate appears 1$T$-TaS$_{2}$, which has as an underlying instability the Mott localization derived transition at 180 K, where no new symmetry is dictacted[@fazekas80]. Above the transition temperature, in the QC phase an intact FS should be present except for new zone boundaries where a (Peierls) gap is opened[@gruener]. In the present Letter we show, using temperature dependent *scanned* ARPES and FS mapping (FSM) measurements on 1$T$-TaS$_{2}$ that the FS is (pseudo)gapped although only a partial removal of the FS due to nesting is expected. Actually, it is a remnant FS yielding the symmetry of the normal (metallic) state (NS) even below 180 K with residual spectral weight at E$_{F}$. Experiments have been performed in a modified VG ESCALAB Mk II spectrometer using monochromatized He-I$\alpha$ (21.2 eV) photons[@pillo98].The FSM measuring mode including sequential motorized sample rotation has been outlined previously[@fsm]. Note that this type of *scanned* ARPES measurement yields *direct* information about relevant points in **k**-space, in contrast to traditional ARPES work, where the datasets are intrinsically smaller and possibly omit information. ARPES and FSM experiments were performed with energy and angular resolution of $30$ meV and $\pm 0.5^{\circ}$, respectively. Pure 1$T$-TaS$_{2}$ samples were prepared by standard flux growing techniques[@dardel92a; @dardel92b] and cleaved *in situ* at pressures in the upper $10^{-11}$ mbar region. Surface quality and cleanness have been checked by low energy electron diffraction (LEED) and X-ray photoelectron spectroscopy, respectively. X-ray photoelectron diffraction[@osterwalder] allows us to determine the sample orientation *in situ* with an accuracy of better than 0.5$^\circ$. Figure 1 shows FSM data of 1$T$-TaS$_{2}$ taken with monochromatized He-I$\alpha$ radiation (21.2 eV) in the QC-phase (a) and in the C-phase (b). The (1x1), i.e., NS-surface Brillouin zones (SBZ) are given by the hexagons with the corresponding high symmetry points indicated. Additionally, in panel (b) the (expected) new SBZ due to the commensurate CDW is illustrated by the small hexagons. The raw data has been normalized[@norm] such that the overall polar intensity variation has been removed and symmetrized according to the space group D$_{3d}^3$[@wilson75] of the CdI$_{2}$-type materials. In all panels the measurements are given in a parallel-projected linear gray scale representation with maximum intensity corresponding to white. The center of the plots denotes normal emission ($\Gamma$, polar emission angle $\theta=0^\circ$) and the outer circle represents grazing emission ($\theta=90^\circ$). Elliptic branches, centered around M, as expected from band-structure calculations[@calc] are seen. A comparable shape has been found in a dichroism study of the valence band of 1$T$-TaS$_{2}$[@matsushita] When comparing the data at 295 K and 140 K, two features are surprising. First, at 295 K as well as at 140 K, the symmetry is threefold and not broken according to the $(\sqrt{13} \times \sqrt{13})$ reconstruction as indicated by the small hexagons in panel (b). This might be explained by a small Fourier component of the CDW potential[@claessen90], at least for 295 K. Nevertheless, this is surprising since 1$T$-TaS$_{2}$ is reported to have a very strong CDW amplitude[@wilson75; @stm]. Second, for 140 K, one would not expect to see any FS at all due to the reported rigid quasiparticle (QP) band shift of 180 meV in the C-phase[@manzke89]. Instead we observe small but finite spectal weight at E$_{F}$ all over the $''$FS$''$. In order to get more detailed information about the actual behaviour of the FS, ARPES spectra were measured along the FS contour. Spectra are given in Fig. 2 for 295 K and 140 K, respectively. The arrow on the right side indicates the **k**-space location of the spectra 1–41 as displayed in Fig. 3 (bottom) where the NS-FS is mimicked by ellipses. To obtain the exact FS contour locations, we measured the spectral weight at E$_{F}$ within the BZ wedge $\Gamma$KMK$\Gamma$ (see Fig. 3) with a very high point density including approximately 5400 angular positions. For 295 K one observes two broad QP peaks (Fig. 2), denoted A and B in the spectra at energy positions of about 220 meV and 700 meV, respectively. Both peaks show significant modulation along the FS contour but practically no dispersion. Most importantly, there is no clear Fermi edge visible. Instead we observe a leading edge shift of at least 30 meV as compared ot the Fermi edge of the polycrystalline Cu sampleholder. In the low temperature C-phase spectra (right panel) along the FS contour the situation is very similar expect that both peaks yield a rigid shift of about 120 meV and become narrower. Despite the energy shift the gap does not open completely and the ARPES spectra retain small but finite spectral weight at E$_{F}$. Note that the same **k**-space dispersion is observed as in (a), however with the maximal intensity reduced by approximately an order of magnitude. This means that the QC-C transition does not change the symmetry of the $''$FS$''$ anymore. We interpret this in terms of a pseudogap due to finite hybridization of the overlapping tails of the two Hubbard subbands. The pseudogap remains open all over the $''$FS$''$, indicating that there is no clear Fermi level crossing of a one-particle peak, even along the $\left[ \Gamma \mbox{M} \right]$ close to $\Gamma$, where the intensity is enhanced[@pillo99]. Instead one has a remnant Fermi surface (RFS) already at room temperature. Over all, this RFS has, as seen from Fig. 1(a) and (b) the symmetry retained from the elliptic one-particle NS-FS. For convenience, Fig. 3 shows a sketch of the situation in 1$T$-TaS$_{2}$. The upper part shows the E(**k**) of the one-particle NS Ta 5$d$ band with the Fermi level crossing in the normal phase at about 1/3 of the $\Gamma$M distance[@smith85; @manzke88; @grioni98; @calc; @pillo99]. The $''$shady$''$ areas indicate the practically dispersionless subbands due to the CDW potential splitting the Ta 5$d$ band[@fazekas80; @smith85]. The analoguous **k$_{\|}$**(E$_{F}$) is plotted in the lower part as the NS-FS. In addition we superimposed on the NS-FS the Peierls gap region as a gray part on the ellipse around the small half-axis. Since the loss of intensity of peak B appears exactly in this region (Fig. 2), it may be attributed to removal of spectral weight due to the Peierls gap[@pillo99]. From Fig. 2 it is now clear, that the FS is disrupted to a RFS yet retaining the overall NS symmetry even below 180 K. In order to corroborate the pseudogap along the RFS contour probed in Fig. 2, we measured azimuthal ARPES spectra along the two circular arcs shown in Fig. 3, i.e., along a$\rightarrow$b and c$\rightarrow$d. Results for RT are shown in Fig. 4 (a) and (b), respectively. Figure 4(a) shows the spectra for a polar angle of 32$^\circ$ (a$\rightarrow$b) thus crossing the Peierls gap region as indicated in Fig. 3. Figure 4(b) shows the spectra for a polar angle of 50$^\circ$ (c$\rightarrow$d), being outside the region influenced by the CDW. In both panels we display a gray scale representation and the spectra. The arrows indicate the Fermi level crossing of the NS band as obtained from plots of the occupation number n(**k**). In both panels a single dispersing band can be seen sitting on an incoherent background which is built from non-dispersing QP peaks, corresponding to the decay of the NS band into three submanifolds[@fazekas80]. At $\theta=32^\circ$ polar angle, the dispersing band reaches down to 0.95 eV binding energy in the $\left [\Gamma \mbox{M} \right ]$ azimuth, and for $\theta=50^\circ$ we have 0.5 eV as the maximum binding energy. The maximum binding energies fit well with the fact that the a$\rightarrow$b scan crosses at the full depth of the band near M (see Fig. 3) whereas the c$\rightarrow$d scan probes a more shallow part near the end of the ellipse. As a guide to the eye, the peak positions are shown, respectively, as white circles overplotted on the gray scale map and as small ticks in the spectra. Strikingly, one can see the backdispersing of the band when it reaches the region indicated by the arrows. However, there is *no* QP band crossing the Fermi level anywhere in the irreducible BZ wedge confirming that the original FS is completely pseudogapped and remains a RFS. These findings are remarkable insofar as one would expect a simple removal of FS portions due to nesting, either via large parallel areas (i.e., the gray shaded portion of the ellipse; bottom of Fig.÷3) of the NS-FS or via a very strong electron-phonon coupling constant. Other nested FS regions than those indicated by the gray zones in Fig. 3 are not expected because XRD[@wilson74] and LEED data[@pillo99b] clearly show the existence of a nesting vector corresponding to the superimposed ($\sqrt{13}$x$\sqrt{13}$) symmetry (indicated by the small hexagons in Fig. 1(b)). Originally the lattice distortion may be driven by nesting, but the electronic structure is more and more influenced as the localization of star-centered Ta electrons sets in. This remaining thirteenth electron in the star, the LHB, becomes more and more localized and gives rise to the one order of magnitude increase of the in-plane resistivity at the QC-C transition[@fazekas80]. The QC-phase exhibits commensurate domains with an incoherent superposition of domains. Hence, the overlap of the QP peaks is considerable but decreases with temperature[@claessen90]. At the QC-C transition there is probably a sudden increase of U$_{dd}$ which pops the two Hubbard subbands apart. Interestingly, the remaining spectral weight distribution keeps the same symmetry. From Fig. 4 we note that at RT we are left with *one* dispersing QP band whereas the other two bands are visible as shoulders in the QP peak, and already exhibit a dispersionless behaviour. Nonetheless we do not have a Fermi edge crossing. We may explain this when we consider the Mott transition at 180 K as an underlying transition. At RT the pseudogap might be interpreted as a kind of precursor. There is a striking similarity with underdoped cuprates[@norman98; @ding97; @ding96] where a crossover temperature scale T$^\star >$ T$_{c}$ has been introduced[@ding97]. Between T$^\star$ and T$_{c}$ a pseudogap opens leading to the complete opening of the superconducting gap at T$_{c}$. One is tempted to introduce such a T$^{\star}$ as well for 1$T$-TaS$_{2}$, i.e. at the onset of the localization where the CDW becomes quasicommensurate. These findings should also motivate a reconsideration of isoelectronic systems, where electron-phonon and electron-electron interactions do interfere. In summary, we have shown by means of temperature dependent scanned ARPES and FSM mesurements that at room temperature 1$T$-TaS$_{2}$ yields a remnant Fermi surface which is not only affected by a comparingly small influence of the CDW formation, i.e., nesting, but rather it is the underlying Mott localization induced transition which seems to tune FS properties. This remnant FS is retained even below the QC-C transition with the same symmetry than in the metallic normal state phase. 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--- author: - Nguyen Tien Quang and Nguyen Thu Thuy title: 'RING EXTENSION PROBLEM, SHUKLA COHOMOLOGY AND ANN-CATEGORY THEORY' --- Every ring extension of $A$ by $R$ induces a pair of group homomorphisms $\mathcal{L}^{*}:R\to End_{\mathbb{Z}}(A)/L(A);\mathcal{R}^{*}:R\to End_{\mathbb{Z}}(A)/R(A),$ preserving multiplication, satisfying some certain conditions. A such $4$-tuple $(R,A,\mathcal{L}^{*},\mathcal{R}^{*})$ is called a ring pre-extension. Each ring pre-extension induces a $R$-bimodule structure on bicenter $K_A$ of ring $A,$ and induces an obstruction $k,$ which is a $3$-cocycle of ${\mathbb{Z}}$-algebra $R,$ with coefficients in $R$-bimodule $K_A$ in the sense of Shukla. Each obstruction $k$ in this sense induces a structure of a regular Ann-category of type $(R,K_A).$ This result gives us the first application of Ann-category in extension problems of algebraic structures, as well as in cohomology theories. Introduction ============ Group extension problem has been presented with group of automorphisms $Aut(G)$ and quotient group $Aut(G)/In(G)$ by group of inner automorphisms. For ring extension problem, Mac Lane \[1\] has replaced the above groups by the ring of bimultiplications $M_A$ and the quotient ring $P_A$ of $M_A$ upon ring of inner bimultiplications. Besides, Maclane has replaced commutative, associative laws for addition in ring $R$ by commutative-associative law $(u+ v)+(r+s)=(u+r)+(v+s)$ and therefore proved that each obstruction of ring extension problem is an element of $3$-dimensional cohomology group in the sense of Maclane, and the number of solutions corresponds to $2$-dimensional cohomology group of ring under a bijection.\ The idea of solving group extension problem by groups $Aut(G)$ and $Aut(G)/In(G)$ can be applied to ring extension theory in a different way. In this way, we use separately associative, commutative laws to construct obstruction of ring extension problem and therefore give the solution to ring extension problem in the terms of Shukla cohomology \[5\] of ${\mathbb{Z}}$-algebras. This cohomology, in our opinion, is more convenient than cubical resolution in Maclane\[1\].\ ${\mathbb{Z}}$-split ring extension problem relates close to Hochschild cohomology and is regarded as an application of Ann-category theory. In this paper, we establish the relationship between ring extension problem in the general case and Ann-category theory. In this way, we may use Ann-category as a united terms to interpret cohomology of different algebraic systems. Cohomology of an associative algebra ==================================== Shukla cohomology (see \[5\]) of ring $R$ (regarded as a ${\mathbb{Z}}$-algebra) with coefficients in $R$-bimodule $M$ is the group $$H^*_S(R,M)=H^*(\sum_{n\geq 0} Hom_{{\mathbb{Z}}}(U^n,M)),$$ where $U$ is a graded differential ${\mathbb{Z}}$-algebra as well as a free resolusion of $R$ on ${\mathbb{Z}}.$ To facilitate the calculation, in \[2\], N.T.Quang has built nomalized complex $B(U) = \sum U{\otimes }(U/{\mathbb{Z}})^n{\otimes }U,$ in which $(U/{\mathbb{Z}})^0={\mathbb{Z}}, U/{\mathbb{Z}}=U/{I{\mathbb{Z}}},$ where $I:{\mathbb{Z}}\longrightarrow U$ is a canonical homomorphism. Each generator of $B(U)$ has the form $$x=u_0[u_1|...u_n]u_{n+1}, n\geq 0$$ This element is equal to $0$ if there exists one in $u_i (i = 1,...,n)$ belonging to $I{\mathbb{Z}}.$ Then, the normalized complex $B(U)$ is a graded differential bimodule on $DG$-algebra $U$ with grading is defined by $$deg(u_0[u_1|...|u_n]u_{n+1})=n+degu_0 + ... +degu_{n+1}$$ and with differential $\partial=\partial_r+\partial_s$ is defined by $$\partial_r(u_0[u_1|...|u_n]u_{n+1})=du_0[u_1|...|u_n]u_{n+1}$$ $$-\sum_{i=1}^n(-1)^{e_{i-1}}u_0[u_1|...|du_i|...|u_n]u_{n+1}+(-1)^{e_n}u_0[u_1|...|u_n](du_{n+1})$$ $$\partial_s(u_0[u_1|...| u_n]u_{n+1})=(-1)^{e_0}u_0u_1[u_2|...|u_n]u_{n+1}$$ $$+\sum_{i=1}^{n-1}(-1)^{e_i}u_0[u_1|...|u_iu_{i+1}|...|u_n]u_{n+1}+(-1)^{e_n}u_0[u_1|...|u_{n-1}]u_nu_{n+1}$$ where $e_0=0,\ e_i=i+degu_0+\cdots +degu_i.$ Classical problem: Singular ring extension ========================================== A *ring extension* is a ring epimorphism $\sigma:S \longrightarrow R$ which carries the identity of $S$ to the identity of $R$. Then, $A=Ker\sigma$ is a two-sided ideal in $S$ and therefore, we have the short exact sequence of rings and ring homomorphisms $$\begin{diagram}\node{E:\quad 0} \arrow{e,t}{} \node{A} \arrow{e,t}{\chi} \node{S}\arrow{e,t}{\sigma} \node{R} \arrow{e,t}{} \node{0\quad}\node{,\quad\sigma(1_S)=1_R.} \end{diagram}$$ Extension $E$ is called *singular* if $A$ is a ring with *null multiplication*, i.e., $A^2 = 0$. Then, $A$ becomes a $R$-bimodule with operators $$xa = u(x)a; ax = au(x), a\in A, x\in R.$$ where $u(x)$ is a representative of $R$ in $S$ in the sense $\sigma u(x) = x$ (note that we always choose $u(0)=0, u(1_R)=1_S$). Let $E$ be a given singular extension of $A$ by $R$ and $u(x)$ be one of its representatives. Then, addition and multiplication in $S$ induce two factor sets $f,g$ determined by $$\begin{split} \begin{aligned} u(x)+u(y)&= f(x,y)+u(x+y)\\ u(x).u(y)&= g(x,y)+u(xy) \end{aligned} \end{split}$$ where $f, g: R^2 \longrightarrow A$\ Since $u(0)=0,u(1_R)=1_S,$ $f$ and $g$ satisfy normalization condition in the sense $$\begin{split} \begin{aligned} f(x,0)&= f(0,y)=0\\ g(x,0)&= g(0,y)=g(1,y)=g(y,1)=0 \end{aligned} \end{split}$$ From commutative, associative laws for addition in $S,$ respectively, we have $$\begin{split} f(x,y)=f(y,x) \end{split}$$ $$\begin{split} f(y,z)-f(x+y,z)+f(x,y+z)-f(x,y)=0 \end{split}$$ From associative law for multiplication in $S,$ we have $$\begin{split} xg(y,z)-g(xy,z)+g(x,yz)-g(x,y)z=0 \end{split}$$ From left and right distributive laws for addition and multiplication, respectively, we have $$\begin{split} \begin{aligned} xf(y,z)-f(xy,xz)&=g(x,y)+g(x,z)-g(x,y+z)\\ (f(x,y))z-f(xz,yz)&=g(x,z)+g(y,z)-g(x+y,z) \end{aligned} \end{split}$$ The pair $(f, g)$ satisfies relations (2)-(6) is called *a factor set* of extension $E$. A factor set of a singular extension of $A$ by $R$ is a $2$-cocycle of ring $R$ with coefficients in $R$-module $A$ in the sense of Mac Lane-Shukla. This result is obtained from calculation of group of $2$-cocycles of ring $R$ with coefficients in $R$-bimodule $A.$ Calculation of group $H^2(R,A)$ based on definitions of Maclane as well as of Shukla coincide with each other. Differences in representation as well as complexity in calculation of these two cohomologies occur when the dimension is down to $3.$ If we choose factor set $u'(x)$ instead of $u(x)$ such that $u'(0)=0,u'(1)=1,$ we will have $\sigma(u'(x)-u(x))=0,$ so $u'(x)=u(x)+t(x),$ where $t(x)\in A.$ Then we have $$\begin{split} \begin{aligned} f'(x,y)&= f(x,y)-t(x+y)+t(x)+t(y)=f(x,y)+(\delta_1t)(x,y)\\ g'(x,y)&= g(x,y)+xt(y)-t(xy)+t(x)y=g(x,y)+(\delta_2t)(x,y) \end{aligned} \end{split}$$ This shows that each singular ring extension $E$ responds to a factor set $(f', g')$ which is equal to $(f, g)$ up to a $2$-coboundary and therefore each singular extension $E$ of $A$ by $R$ responds to an element of cohomology group $H^2(R,A)$ in the sense of Mac Lane - Shukla. Conversely, if we have $R$-bimodule $A$ together with functions $f,g$ satisfying relations (2)-(6), we can construct extension $E$ of $A$ by $R,$ where ring $S$ is determined by $$\begin{aligned} S&=&\{(a,x)\mid a\in A,x\in R\}\nonumber\end{aligned}$$ together with two operations $$\begin{split} \begin{aligned} (a,x)+(b,y)&=&(a+b+f(x,y),x+y)\\ (a,x)(b,y)&=&(ay+xb+g(x,y),xy) \end{aligned} \end{split}$$ Pre-extensions of rings ======================= We now consider the general case, i.e., ring extension $$\begin{diagram}\node{E:\quad 0} \arrow{e,t}{} \node{A} \arrow{e,t}{\chi} \node{S}\arrow{e,t}{\sigma} \node{R} \arrow{e,t}{} \node{0\quad}\node{,\quad\sigma(1_S)=1_R.} \end{diagram}$$ which is unnecessarily singular. Then $A$ is a two-sided ideal of ring $S$ and therefore $A$ is regarded as a $S$-bimodule and $\chi$ is a $S$-bimodule homomorphism. To determine necessary conditions for ring estension problem, we will expand the technology which is used for group extension problem. This way is a bit different from the one of Maclane in \[1\]. In this way, we establish directly the relationship between ring extension problem, Shukla cohomology and Ann-category theory. We consider the following two sets of mappings instead of the set of bimultiplications of ring $A$ which Maclane as well as Shukla have done $$\begin{aligned} L(A)&=&\{l_a: A\longrightarrow A,a\in A\mid l_a(b)=ab,\forall b\in A\}\nonumber\\ R(A)&=&\{r_a:A\longrightarrow A,a\in A\mid r_a(b)=ba,\forall b\in A\}\nonumber\end{aligned}$$ Clearly, each $l_a$ (resp. $r_a$) is a right (resp. left) $A$-module endomorphism of $A$ and $L(A), R(A)$ are two subrings of ring $End_{{\mathbb{Z}}}(A)$ of endomorphisms of group $A.$\ Consider ring homomorphism $$\begin{aligned} \mathcal{L}:\quad S&\rightarrow&End_{{\mathbb{Z}}}(A)\nonumber\\ s&\mapsto&\mathcal{L}_s,\mathcal{L}_s(a)=sa,\forall a\in A.\nonumber\end{aligned}$$ Clearly, $\mathcal{L}_s$ is a right $A$-module endomorphism of $A,$ and $\mathcal{L}$ induces group homomorphism $\mathcal{L}^{*}:R\to End_{{\mathbb{Z}}}(A)/L(A)$ such that following diagram $$\begin{diagram} \node{0} \arrow[1]{e,t}{} \node{A} \arrow[1]{e,t}{\chi} \arrow[1]{s,r}{l} \node{S} \arrow{e,t}{\sigma}\arrow{s,r}{\mathcal{L}} \node{R} \arrow{e,t}{}\arrow{s,r}{\mathcal{L}^{*}}\node{0}\\ \node{0} \arrow{e,t}{} \node{L(A)} \arrow{e,t}{i} \node{End_{{\mathbb{Z}}}(A)} \arrow{e,t}{p} \node{End_{{\mathbb{Z}}}(A)/L(A)} \arrow{e,t}{} \node{0} \end{diagram}$$ commute.\ Similarly, we have ring homomorphism $$\begin{aligned} \mathcal{R}:\quad S&\rightarrow&End_{{\mathbb{Z}}}(A)\nonumber\\ s&\mapsto&\mathcal{R}_s,\mathcal{R}_s(a)=as,\forall a\in A.\nonumber\end{aligned}$$ $\mathcal{R}_s$ is a left $A$-module endomorphism of $A,$ and $\mathcal{R}$ induces group homomorphism $\mathcal{R}^{*}:R\to End_{{\mathbb{Z}}}(A)/R(A)$ such that following diagram $$\begin{diagram} \node{0} \arrow[1]{e,t}{} \node{A} \arrow[1]{e,t}{\chi} \arrow[1]{s,r}{r} \node{S} \arrow{e,t}{\sigma}\arrow{s,r}{\mathcal{R}} \node{R} \arrow{e,t}{}\arrow{s,r}{\mathcal{R}^{*}}\node{0}\\ \node{0} \arrow{e,t}{} \node{R(A)} \arrow{e,t}{i} \node{End_{{\mathbb{Z}}}(A)} \arrow{e,t}{p} \node{End_{{\mathbb{Z}}}(A)/R(A)} \arrow{e,t}{} \node{0} \end{diagram}$$ commute.\ Moreover, $\mathcal{L}_s$ and $\mathcal{R}_s$ satisfy relations $$\begin{split} l_a\circ \mathcal{L}_s=l_{\mathcal{R}_s(a)}\qquad,\mathcal{L}_s\circ l_a=l_{\mathcal{L}_s(a)}\\ \mathcal{R}_s\circ r_a=r_{\mathcal{R}_s(a)}\qquad,r_a\circ \mathcal{R}_s=r_{\mathcal{L}_s(a)} \end{split}$$ $$\mathcal{L}_s\circ\mathcal{R}_{s'}=\mathcal{R}_{s'}\circ\mathcal{L}_s\nonumber$$ forall $a\in A$ and $s,s'\in S.$\ From relations (9), we can see that $L(A)$ and $R(A)$ are, respectively, two-sided ideals in $\mathcal{L}(S)$ and $\mathcal{R}(S)$. Therefore, $\mathcal{L}^{*}(R)$ and $\mathcal{R}^{*}(R)$ are rings, $\mathcal{L}^{*}$ and $\mathcal{R}^{*}$ are ring homomorphisms from $R$ to its images. Let $A$ be a ring (without identity) and $R$ be a ring with identity $1\ne 0.$ Assume that we have group homomorphisms $$\mathcal{L}^{*}:R\to End_{{\mathbb{Z}}}(A)/L(A);\mathcal{R}^{*}:R\to End_{{\mathbb{Z}}}(A)/R(A)$$ such that for each $x\in R,$ there exists a pair $\varphi_x\in \mathcal{L}^{*}(x),\psi_x\in\mathcal{R}^{*}(x)$ satisfying relations $$\begin{split} \phi_1=\psi_1=id\qquad\qquad\qquad\qquad\\ \begin{matrix} l_a\circ \varphi_x&=&l_{\psi_x(a)}\qquad,\qquad \varphi_x\circ l_a&=& l_{\varphi_x(a)}\\ \psi_x\circ r_a&=&r_{\psi_x(a)}\qquad,\qquad r_a\circ \psi_x&=& r_{\varphi_x(a)} \end{matrix}\\ \varphi_x\circ\psi_y=\psi_y\circ\varphi_x\qquad\qquad\qquad \end{split}$$ Moreover, $\mathcal{L}^{*}$ and $\mathcal{R}^{*}$ preserve multiplication. Then, we call $4$-tuple $(R,A,\mathcal{L}^{*},\mathcal{R}^{*})$ *a pre-extension* of $A$ by $R$ inducing $\mathcal{L}^{*},\mathcal{R}^{*}.$\ Ring extension problem is to find whether pre-extension $(R,A,\mathcal{L}^{*},\mathcal{R}^{*})$ has extension and, if so, how many extensions of $A$ by $R$ are. The obstruction of a ring pre-extension ======================================= We now present the concept of *obstruction* of pre-extension $(R,A,\mathcal{L}^{*},\mathcal{R}^{*}).$\ Since $\mathcal{L}^{*}(x).\mathcal{L}^{*}(y)=\mathcal{L}^{*}(xy),$ we have $$\varphi_x.\varphi_y=\varphi_{xy}+l_{g(x,y)}\qquad,\qquad g(x,y)\in A$$ Then, from associative law for multiplication in $End_{{\mathbb{Z}}}(A),$ we have $$l_{\varphi_x[g(y,z)]}+l_{g(xy,z)}=l_{g(xy,z)}+l_{\psi_z[g(x,y)]}$$ which yeilds $$\begin{split} \varphi_x[g(y,z)]-g(xy,z)+g(x,yz)-\psi_z[g(x,y)]=\alpha(x,y,z), \end{split}$$ in which $\alpha(x,y,z)\in K_A,$ where $K_A$ is a two-sided ideal of $A.$ $$K_A=\{c\in A\mid ca=0=ac \ \ \forall a\in A\}.$$ We call $K_A$ *bicenter* of $A.$ Associative, commutative laws for addition in $End_{{\mathbb{Z}}}(A),$ respectively, give us $$\begin{split} \begin{matrix} f(y,z)-f(x+y,z)+f(x,y+z)-f(x,y)&=&\xi(x,y,z)\\ f(x,y)-f(y,x)&=&\eta(x,y) \end{matrix} \end{split}$$ where $\xi:R^3\to K_A,\ \eta:R^2\to K_A.$\ Finally, from distributive law in $End_{{\mathbb{Z}}}(A),$ we have $$\begin{split} \varphi_x[f(y,z)]-f(xy,xz)+g(x,y+z)-g(x,y)-g(x,z)=\lambda(x,y,z)\\ \psi_z[f(x,y)]-f(xz,yz)+g(x+y,z)-g(x,z)-g(y,z)=\rho(x,y,z) \end{split}$$ where $\lambda,\rho:R^3\to K_A.$\ We call the family $(\xi,\eta,\alpha,\lambda,\rho)$ which is so determined *an obstruction* of pre-extension $(R,A,\mathcal{L}^{*},\mathcal{R}^{*}).$\ Clearly, $K_A$ is a $R$-bimodule with operations $xa=\varphi_x(a),ax=\psi_x(a),$ which are independent of the choice of $\varphi_x,\psi_x.$ We may note that if commutative-associative law $$(u+v)+(r+s)=(u+r)+(v+s)$$ for addition in $End_{{\mathbb{Z}}}(A)$ is used, we will have a function $\gamma:R^4\to K_A,$ given by $$\gamma(x,y,z,t)=f(x+y,z+t)-f(x,z)-f(y,t)-f(x+z,y+t)+f(x,y)+f(z,t)\quad(12')$$ Then, an obstruction of pre-extension $(R,A,\mathcal{L}^{*},\mathcal{R}^{*})$ is a family of 4 functions $(\alpha,\gamma,\lambda,\rho)$ satisfying relations (11), (12’) and (13). We can see that two concepts of above-mentioned obstruction are equivalent. Due to the theory of cohomology of Maclane, each obstruction $(\alpha,\gamma,\lambda,\rho)$ is a $3$-cocycle in $Z^3_M(R,K_A).$ One of the main results in this paper is showing that each obstruction regarded as a family of 5 functions $(\xi,\eta,\alpha,\lambda,\rho)$ is a $3$-cocycle in the sense of Shukla. First, we may prove two following lemmas for obstructions. If we fix $\varphi_x\in\mathcal{L}^{*}(x), \psi_x\in\mathcal{R}^{*}(x),$ and replace functions $g,f$ by functions $g',f'$ such that $$l_{g(x,y)}=l_{g'(x,y)}\ \ \textrm{and}\ \ l_{f(x,y)}=l_{f'(x,y)}$$ then $g'=g+\nu$ and $f'=f+\mu,$ where $\nu,\mu:R^2\to K_A.$ Then, $$\xi'=\xi+\partial_1\mu,\qquad\eta'=\eta+ant\mu,\qquad\alpha'=\alpha+\partial_2\nu$$ $$(\partial_1\mu)(x,y,z)=\mu(y,z)-\mu(x+y,z)+\mu(x,y+z)-\mu(x,y)$$ $$\begin{split} (\partial_2\nu)(x,y,z)=x\nu(y,z)-\nu(xy,z)+\nu(x,yz)-\nu(x,y)z \end{split}$$ $$\lambda'(x,y,z)=\lambda(x,y,z)+\nu(x,y+z)-\nu(x,y)-\nu(x,z)+x\mu(y,z)-\mu(xy,xz)$$ $$\rho'(x,y,z)=\rho(x,y,z)+\nu(x+y,z)-\nu(x,z)-\nu(y,z)+\mu(x,y)z-\mu(xz,yz)$$ Moreover, two functions $\nu,\mu$ can be chosen arbitrarily. If we replace functions $\varphi_x,\psi_x$ by functions $\varphi'_x,\psi'_x,$ we will be able to choose functions $g',f'$ such that family $(\xi,\eta,\alpha,\lambda,\rho)$ is unchanged. Two obstructions $(\xi,\eta,\alpha,\lambda,\rho)$ and $(\xi',\eta',\alpha',\lambda',\rho')$ of the same pre-extension $(R,A,\mathcal{L}^{*},\mathcal{R}^{*})$ are *cohomologous* if they satisfy relations (14) where $\nu,\mu$ are certain functions. Two above lemmas show that each pre-extension $(R,A,\mathcal{L}^{*},\mathcal{R}^{*})$ determines uniquely the cohomology class of any one of its obstructions. We now show the solution of ring extension problem by Shukla cohomology of ring (regarded as ${\mathbb{Z}}$-algebra) The cohomology class of obstruction $(\xi,\eta,\alpha,\lambda,\rho)$ of pre-extension $(R,A,\mathcal{L}^{*},\mathcal{R}^{*})$ is an element $\overline{k}\in H^3_{Sh}(R,K_A).$ Pre-extension $(R,A,\mathcal{L}^{*},\mathcal{R}^{*})$ has extension iff $\overline{k}=0.$ Then, there exists a bijection between the set $Ext(R,A)$ of equivalence classes of extension and the set $H^3_{Sh}(R,K_A).$ In \[2\], N.T.Quang has built acyclic and ${\mathbb{Z}}$-free complex as follows $$U= \sum_{i=0}^4 U_i,$$ where $U_i$ are elements of the exact sequence $$0\rightarrow U_4\stackrel{d_4}{\rightarrow}U_3\stackrel{d_3}{\rightarrow}U_2\stackrel{d_2}{\rightarrow}U_1\stackrel{d_1}{\rightarrow}U_0\stackrel{\epsilon}{\rightarrow}R\rightarrow 0$$ of free abelian groups\ $$\begin{aligned} U_0 &= {\mathbb{Z}}(R^0), R^0 = R\setminus\{ 0\}\\ U_1 &= {\mathbb{Z}}(R^0\times R^0)\\ U_2 &= {\mathbb{Z}}(R^0\times R^0\times R^0)\oplus U_1\\ U_3 &= {\mathbb{Z}}(R^0\times R^0\times R^0\times R^0)\oplus U_2\oplus U_0\\ U_4 &= Kerd_3 \end{aligned}$$\ Homomorphisms $d_i$ and $\epsilon$ are given by\ $$\begin{aligned}\epsilon[x]&=x\\ d_1[x,y]&=[y]-[x+y]+[x]\\ d_2[x,y,z]&=[y,z]-[x+y,z]+[x,y+z]-[x,y]\\ d_2[x,y]&=[x,y]-[y,x]\\ d_3[x,y,z,t]&=[y,z,t]-[x+y,z,t]+[x,y+z,t]-[x,y,z+t]+[x,y,z]\\ d_3[x,y,z]&= [x,y,z]-[x,z,y]+[z,x,y]-[y,z]+[x+y,z]-[x,z]\\ d_3[x,y]&=[x,y]+[y,x]\\ d_3[x]&=[x,x] \end{aligned}$$\ $d_4$ is a canonical embedding.\ After that, we determine an associative multiplication (which satisfies Leibniz formula for differential $d$) to make $U$ become a DG-algebra on ${\mathbb{Z}}.$ 1\) First, for $[x]\in U_0$ we set $$[x][x_1,\ldots,x_n]=[xx_1,\ldots,xx_n]$$ $$[x_1,\ldots,x_n][x]=[x_1x,\ldots,x_nx]$$ $$[x,y][z,t]=-[xz+yz]+[xz,yz,xt]+[xz+xt,yz,yt]-[xz,xt,yz]+[xt,yz]$$ 2\) In other cases, for $a_i,a_j$ which are, respectively, generators of $U_i,\ U_j$ we have $$(da_i)a_j+(-1)^ia_i(da_j)\in Ker(d_{i+j-1})$$ so they vanish under operation of differentials $d.$ Besides, the short exact sequence of abelian groups $$\begin{diagram} \node{0} \arrow{e,t}{} \node{Ker(d_{i+j})} \arrow{e,t}{} \node{U_{i+j}}\arrow{e,t}{d_{i+j}} \node{Ker(d_{i+j-1})} \arrow{e,t}{} \node{0} \end{diagram}$$ splits since $Ker(d_{i+j-1})$ is ${\mathbb{Z}}$-free. So there exists a group injection $s:Ker(d_{i+j-1})\to U_{i+j}$ such that $d_{i+j}\circ s=id.$ Then, we set $$a_ia_j=s[(da_i)a_j+(-1)^ia_i(da_j)].$$ With the above-constructed $DG$-algebra $U,$ a $3$-cocycle in $Z^3_{Sh}(R,K_A)$ is an obstruction $(\xi,\eta,\alpha,\lambda,\rho)$ of pre-extension $(R,A,\mathcal{L}^{*},\mathcal{R}^{*}).$ Moreover, two $3$-cocycles $(\xi,\eta,\alpha,\lambda,\rho)$ and $(\xi,\eta,\alpha,-\lambda,\rho)$ belong to the same cohomology class of $H^3_{Sh}(R,K_A)$ iff they are cohomologous obstructions. So we deduce that pre-extension $(R,A,\mathcal{L}^{*},\mathcal{R}^{*})$ determine uniquely element $\overline{k}\in H^3_{Sh}(R,K_A).$ If $(R,A,\mathcal{L}^{*},\mathcal{R}^{*})$ has extension, obviously, $\overline{k}=0.$ Conversely, if $\overline{k}=0,$ $k=(\xi,\eta,\alpha,\lambda,\rho)$ where $$\xi=\partial_1\mu,\qquad\eta=ant\mu,\qquad\alpha=\partial_2\nu$$ $$\lambda(x,y,z)=\nu(x,y+z)-\nu(x,y)-\nu(x,z)+x\mu(y,z)-\mu(xy,xz)$$ $$\rho(x,y,z)=\nu(x+y,z)-\nu(x,z)-\nu(y,z)+\mu(x,y)z-\mu(xz,yz).$$ Applying Lemma 4 by replacing $g,f$ by $g'=g-\nu,f'=f-\mu,$ we will have $$k'=(\xi',\eta',\alpha',\lambda',\rho')=0$$ Then, we may construct extension $S$ of $A$ by $R$ as in relations (8), in which $f,g$ are replaced by $f',g'.$ The rest of the theorem is proved by the above-mentioned events, without new technology. Ann-category and solution of ring extension problem =================================================== In this section, we show the relationship between Ann-category theory and ring extension problem. See \[2\], \[3\] for definitions and fundamental results of Ann-categories. Let $R$ be a ring with identity $1\neq 0$ and $A$ be a $R$-bimodule. An Ann-category ${\mathcal{I}}$ of type $(R,A)$ is a category whose objects are elements of $R$ and whose morphisms are endomorphisms. In the concrete, for $r,s\in R,$ we define $$Hom(r,s) = \emptyset \ \ \textrm{where} \ \ r\ne s$$ $$Hom(s,s) = Aut(s)=\{s\}\times A.$$ The composite of two morphisms is induced by addition in $A.$\ Two tensor products ${\otimes },{\oplus}$ on ${\mathcal{I}}$ are defined by $$\begin{aligned} r{\oplus}s&=&r+s \ \ \textrm{(addition in ring $R$)}\nonumber\\ (r,u){\oplus}(s,v)&=&(r+s,u+v)\nonumber\\ r{\otimes }s&=&rs \ \ \textrm{(product in ring $R$)}\nonumber\\ (r,u){\otimes }(s,v)&=&(rs,rv+us).\nonumber\end{aligned}$$ Constraints on ${\mathcal{I}}$ are defined to be a family $$(\xi,\eta,(0,id,id),\alpha,(1,id,id),\lambda,\rho)$$ where $\eta:R^2\to A,\ \alpha,\lambda,\rho:R^3\to A$ are functions so that the set of axioms of an Ann-category is satisfied. Ann-category of type $(R,A)$ is *regular* if its commutivity constraint satisfies $\eta(x,x)=0$ for all $x\in R.$ The following propositions give us the first relationship between ring extension problem and Ann-category theory. Each obstruction $(\xi,\eta,\alpha,\lambda,\rho)$ of pre-extension $(R,A,\mathcal{L}^{*},\mathcal{R}^{*})$ is a structure of regular Ann-category ${\mathcal{I}}$ of type $(R,K_A).$ From relations (10)-(13), we may verify directly that family $(\xi,\eta,\alpha,\lambda,\rho)$ satisfies following relations 1. $\xi(y,z,t)-\xi(x+y,z,t)+\xi(x,y+z,t)-\xi(x,y,z+t)+\xi(x,y,z)=0$ 2. $\xi(0,y,z)=\xi(x,0,z)=\xi(x,y,0)=0$ 3. $\xi(x,y,z)-\xi(x,z,y)+\xi(z,x,y)-\eta(x,z)+\eta(x+y,z)-\eta(y,z)=0$ 4. $\eta(x,y)+\eta(y,x)=0$ 5. $\eta(x,x) = 0$ 6. $x\eta(y,z)-\eta(xy,xz)=\lambda(x,y,z)-\lambda(x,z,y)$ 7. $\eta(x,y)z-\eta(xz,yz)=\rho(x,y,z)-\rho(y,x,z)$ 8. $x\xi(y,z,t)-\xi(xy,xz,xt)=\lambda(x,z,t)-\lambda(x,y+z,t)+\lambda(x,y,z+t)-\lambda(x,y,z)$ 9. $\xi(x,y,z)t-\xi(xt,yt,zt)=\rho(y,z,t)-\rho(x+y,z,t)+\rho(x,y+z,t)-\rho(x,y,t)$ 10. $\rho(x,y,z+t)-\rho(x,y,z)-\rho(x,y,t)+\lambda(x,z,t) +\lambda(y,z,t)-\lambda(x+y,z,t)\\=-\xi(xz+xt,yz,yt) +\xi(xz,xt,yz)-\eta(xt,yz)+\xi(xz+yz,xt,yt)-\xi(xz,yz,xt)$ 11. $\alpha(x,y,z+t)-\alpha(x,y,z)-\alpha(x,y,t)= x\lambda(y,z,t)+\lambda(x,yz,yt)-\lambda(xy,z,t)$ 12. $\alpha(x,y+z,t)-\alpha(x,y,t)-\alpha(x,z,t)\\= x\rho(y,z,t)-\rho(xy,xz,t)+\lambda(x,yt,zt)-\lambda(x,y,z)t$ 13. $\alpha(x+y,z,t)-\alpha(x,z,t)-\alpha(y,z,t)= -\rho(x,y,z)t-\rho(xz,yz,t)+\rho(x,y,zt)$ 14. $x\alpha(y,z,t)-\alpha(xy,z,t)+\alpha(x,yz,t) -\alpha(x,y,zt)+\alpha(x,y,z)t=0$ 15. $\alpha(1,y,z)=\alpha(x,1,z)=\alpha(x,y,1)=0$ 16. $\alpha(0,y,z)=\alpha(x,0,t)=\alpha(x,y,0)=0$ 17. $\lambda(1,y,z)=\lambda(0,y,,z)=\lambda(x,0,z)=\lambda(x,y,0)=0$ 18. $\rho(x,y,1)=\rho(0,y,z)=\rho(x,0,z)=\rho(x,y,0)=0$ where $x,y,z,t\in R.$ These relations (without the fifth relation) show that the family of constraints $(\xi,\eta,(0,id,id),\alpha,(1,id,id),\lambda,\rho)$ satisfies the set of axioms of an Ann-category. Then, it is said that family $(\xi,\eta,\alpha,\lambda,\rho)$ is a structure of Ann-category of type $(R,A).$ From the fifth relation, Ann-category $(R,A)$ is regular. $[4]$ Family $(\xi,\eta,\alpha,\lambda,\rho)$ is a structure of regular $Ann$-category of type $(R,A)$ iff $(\xi,\eta,\alpha,-\lambda,\rho)$ is a $3$-cocycle of ring $R$ with coefficients in $R$-bimodule $A$ in the sense of Shukla cohomology. The proof is deduced from the proof of Theorem 5 and Proposition 6 Let ${\mathcal{A}},{\mathcal{B}}$ be Ann-categories. An Ann-functor from ${\mathcal{A}}$ to ${\mathcal{B}}$ is a triple $(F,{\breve{F}},{\widetilde{F}})$ where $(F,{\breve{F}})$ is a symmetric monoidal ${\oplus}$-functor and $(F,{\widetilde{F}})$ is a monoidal ${\otimes }$-functor so that following diagrams [ $$\divide \dgARROWLENGTH by 2 \begin{diagram} \node{F(X(Y{\oplus}Z))}\arrow{e,t}{{\widetilde{F}}}\arrow{s,l}{F({\mathcal{L}})}\node{FX F(Y{\oplus}Z)}\arrow{e,t}{id {\otimes }{\breve{F}}}\node{FX(FY{\oplus}FZ)}\arrow{s,r}{{\mathcal{L}}'}\\ \node{F((XY){\oplus}(XZ))}\arrow{e,t}{{\breve{F}}}\node{F(X Y) {\oplus}F(X Z)}\arrow{e,t}{{\widetilde{F}}\oplus{\widetilde{F}}}\node{(FX FY){\oplus}(FX FZ)} \end{diagram}$$]{} [ $$\divide \dgARROWLENGTH by 2 \begin{diagram} \node{F((X{\oplus}Y){\otimes }Z)}\arrow{e,t}{{\widetilde{F}}}\arrow{s,l}{F({\mathcal{R}})} \node{F(X{\oplus}Y) FZ}\arrow{e,t}{{\breve{F}}{\otimes }id}\node{(FX{\oplus}FY)FZ}\arrow{s,r}{{\mathcal{R}}'}\\ \node{F((X Z){\oplus}(Y Z))}\arrow{e,t}{{\breve{F}}}\node{F(X Z) {\oplus}F(Y Z)}\arrow{e,t}{{\widetilde{F}}\oplus{\widetilde{F}}}\node{(FX FZ){\oplus}(FY FZ)} \end{diagram}$$]{} commute Two structures $f =(\xi,\eta,\alpha,\lambda,\rho)$ and $f'=(\xi,\eta,\alpha,\lambda,\rho)$ of Ann-category of type $(R,A)$ are cohomologous iff there is an Ann-functor $(F,{\breve{F}},{\widetilde{F}}): (R,A,f)\longrightarrow (R,A,f')$ where $F = id.$ From the definition of Ann-functor, we have following proposition Two structures $f$ and $f'$ of Ann-category of type $(R,A)$ are cohomologous iff they satisfy relations mentioned in Lemma 3. Proposition 7 and 8 give us the following corollary. Each cohomology class of structures of regular Ann-categories of type $(R,A)$ is an element in $H^3_{Sh}(R,K_A).$ Indeed, if Ann-category of type $(R,A)$ is regular, $f$ and $f'$ are $3$-cocycles of group $H^3_{Sh}(R,K_A).$ Since they satisfy Lemma 3, they are equal up to a $3$-coboundary, and therefore it completes the proof. [99]{} S. Mac Lane. *Extension and obstruction for rings.* Illinois J. of Math. Vol 2(1958), 316-345. N.T.Quang. *Ann-categories.* Doctoral Dissertation. Hanoi, Vietnamese, 1988. N.T.Quang. *Introduction to Ann-categories.* ArXiv: math. CT/0702588v2 21 Feb 2007. N.T.Quang. *Structure of Ann-catogories and Mac Lane - Shukla Cohomology* East - West J. of Mathematics: Vol. 5, No 1 (2003) pp. 51-66. U. Shukla *Cohomologie des algebras associatives*. These, Paris (1960). Math. Dept., Hanoi University of Education\ E-mail adresses: nguyenquang272002@gmail.com
--- abstract: 'We study phase structure of mass-deformed ABJM theory which is a three dimensional $\mathcal{N}=6$ superconformal theory deformed by mass parameters and has the gauge group $\text{U}(N)\times \text{U}(N)$ with Chern-Simons levels $(k,-k)$ which may have a gravity dual. We discuss that the mass deformed ABJM theory on $S^3$ breaks supersymmetry in a large-$N$ limit if the mass is larger than a critical value. To see some evidence for this conjecture, we compute the partition function exactly, and numerically by using the Monte Carlo Simulation for small $N$. We discover that the partition function has zeroes as a function of the mass deformation parameters if $N\ge k$, which supports the large-$N$ supersymmetry breaking. We also find a solution to the large-$N$ saddle point equations, where the free energy is consistent with the finite $N$ result.' --- [KIAS-P18012]{}\ [YITP-18-76]{} [**Supersymmetry Breaking in a Large $N$ Gauge Theory\ with Gravity Dual** ]{}\ [${}^{1}$]{} [Tomoki Nosaka,[^1]]{}[${}^{2}$]{} [Kazuma Shimizu[^2]]{}[${}^{3}$]{}\ [${}^{3}$]{}\ [${}^{1}$]{}: [*Department of Particle Physics and Astrophysics, Weizmann Institute of Science\ Rehovot 7610001, Israel* ]{}\ [${}^{2}$]{}: [*School of Physics, Korea Institute for Advanced Study\ 85 Hoegiro Dongdaemun-gu, Seoul 02455, Republic of Korea* ]{}\ [${}^{3}$]{}: [*Yukawa Institute for Theoretical Physics, Kyoto University\ Kitashirakawa-Oiwakecho, Sakyo, Kyoto 606-8502, Japan* ]{} Introduction ============ Spontaneous supersymmetry (SUSY) breaking in string/M-theory is one of the most important subjects and has been discussed intensively in the context of phenomenology and cosmology. The SUSY breaking in string/M-theory is the super-Higgs phenomenon in general since the theory has gauged SUSY rather than global one. In contrast, various situations in string/M-theory are expected to have holographic descriptions by SUSY quantum field theories (QFT), whose SUSY are global. Therefore it is interesting to discuss SUSY breaking in QFT with a gravity dual which is typically large-$N$ and strongly coupled. As far as we know, the only explicit examples of such problem are the models discussed in [@KS1; @KS2; @KS3; @EKSS] in which the SUSY is kinematical, i.e. the SUSY algebra does not includes the Hamiltonian.[^3] Main reason for the existence of the very few examples is that it is technically hard since we typically need non-perturbative analysis in this type of problem. In this paper we study the SUSY breaking problem in so-called massive ABJM theory [@HLLLP; @GRVV] which is a three dimensional $\mathcal{N}=6$ superconformal theory known as ABJM theory [@ABJM] deformed by two mass parameters and has the gauge group $\text{U}(N)\times \text{U}(N)$ with Chern-Simons levels $(k,-k)$. It is expected that the ABJM theory without the masses is the low-energy effective theory of $N$ coincident M2-branes and dual to the M-theory on $AdS_4 \times S^7 /\mathbb{Z}_k$. The mass deformation in this situation corresponds to the introduction of the background flux. Then the massive ABJM theory in the large-$N$ limit is holographically dual to M-theory on asymptotic $AdS_4$ geometry [@Lin:2004nb; @CKK]. We argue that the mass deformed ABJM theory on $S^3$ breaks supersymmetry in the large-$N$ limit with $k$ fixed if the mass is larger than a critical value. We can adress this because the partition function of the theory on $S^3$ can be exactly computed by the localization technique [@KWY; @J; @HHL]. Note that the localization technique can be applied to our theory with finite $N$ on $S^3$, like the Witten index on $S^1 \times M$ where $M$ is a compact manifold, even if it will break the supersymmetry spontaneously in the large-$N$ or large volume limit. The arguments are based on the existences of the zeroes of the partition function which will be related to the SUSY breaking and the phase transition at the critical mass which is expected from the large-$N$ saddle point solution found in [@NST2]. A summary of our arguments for the SUSY breaking of the theory will be explained in sec. \[susybreaking\]. In the previous work [@NST2] a part of the authors considered this theory with the two equal mass parameters, which enjoys the ${\cal N}=6$ supersymmetry. We studied the partition function in the M-theory limit ($N\rightarrow\infty$ with $k$ kept finite) by using the saddle point approximation, and found that the saddle point solution which gives the free energy $F\sim N^{3/2}$ disappears as we increase the mass deformation parameter to some critical value. Although this would suggest that a phase transition occurs at that point, the whole phase structure is still unclear. ![ Our proposal on phase structure of the massive ABJM theory on $S^3$ in the large-$N$ limit. The dashed line denotes expected zeroes of the sphere partition function: $\frac{\zeta_1 \zeta_2}{k^2}=\frac{1}{16}$. []{data-label="fig:proposal"}](result.eps){width="7.5cm"} The rest of this paper is organized as follows. After introducing the mass deformed ABJM theory with two mass parameters $\zeta_1$, $\zeta_2$ in the next section, in sec. \[susybreaking\] we argue that this theory with $\zeta_1=\zeta_2$ breaks the supersymmetry for $\zeta_1/k>1/4$. In the same section we also argue that the supersymmetry breaking does not occur if $\zeta_1=0$ or $\zeta_2=0$. In sec. \[zeta10\] we first study the latter cases with $\zeta_1=0$ in detail, and indeed find the large-$N$ free energy obeys $N^{3/2}$-law for any value of $\zeta_2$. Then, in sec. \[bothnonzero\] we consider the case with both $\zeta_1$ and $\zeta_2$ are non-zero. In this case the partition function $Z(k,N,\zeta_1,\zeta_2)$ for $N\ge k$ can have zeroes at some finite $\zeta_1,\zeta_2$, this was explicitly shown for $N=2$. We provide positive evidence for the existence of zeroes from the Monte Carlo simulation. We also argue the physical interpretation for the zeroes, and estimate how the partition function behaves in the large-$N$ limit. Our proposal on the phase structure in the large-$N$ limit is summarized in fig. \[fig:proposal\]. We expect that the partition function vanishes when $\frac{\zeta_1 \zeta_2}{k^2}=\frac{1}{16}$ and the theory is in the SUSY breaking phase for $\frac{\zeta_1 \zeta_2}{k^2}\geq \frac{1}{16}$. In sec. \[discuss\] we summarize our analysis and propose future directions. We summarize technical details in appendices. In app. \[Sdualsection\], starting from the localization formula we rewrite the partition function to a simpler matrix model , which we use in the subsequent sections. In particular, in the new expression the integration is absolutely convergent, hence we can evaluate the partition function numerically by applying the Monte Carlo method. In app. \[app:detail\_exact\] we display the exact computation of the partition function with $\zeta_1=0$ with $k$ and $N$ being small integers, and summarize the results in app. \[Z1exact\_results\]. As $N$ increases these results match with the saddle point approximation in sec. \[saddle\], which support the validity of the saddle point approximation. We also compare the exact partition function with the partition function of the linear quiver theory with single hypermultiplet obtained in [@NY1]. Review on Mass deformed ABJM theory =================================== In this section we review some basic facts on the mass-deformed ABJM theory on $S^3$. The field content of the ABJM theory consists of, in the 3d ${\cal N}=2$ SUSY notation, a $\text{U}(N)_k$ vector multiplet ${\cal V}=(A_\mu,\sigma,\chi,D)$, a $\text{U}(N)_{-k}$ vector multiplet ${\widetilde {\cal V}}=({\widetilde A}_\mu,{\widetilde\sigma},{\widetilde\chi},{\widetilde D})$, two chiral multiplets ${\cal Z}_\alpha=(A_\alpha,\phi_\alpha,F_\alpha)$ in $(\Box,\bar{\Box})$ representation under $\text{U}(N)_k\times \text{U}(N)_{-k}$ and two chiral multiplets ${\cal W}_{\dot{\alpha}}=(B_{\dot{\alpha}},\psi_{\dot{\alpha}},G_{\dot{\alpha}})$ in $(\bar{\Box},\Box)$ representation.[^4] Here the vector multiplets obey the Chen-Simons action with level $\pm k$, while the action for the chiral multiplets consists of the superpotential together with the following minimal coupling to the vector multiplets $$\begin{aligned} S_\text{kin} &=\int d^3x\sqrt{g}\operatorname{Tr}\Bigl[|D_\mu A_a|^2+|D_\mu W_{\dot{a}}|^2 +\frac{3}{4r_{S^3}^2}(|A_a|^2+|W_{\dot{a}}|^2)\nonumber \\ &\quad +\frac{1}{r_{S^3}}|\sigma A_a-A_a{\widetilde\sigma}|^2 +i(\bar{A}^aDA_a-A_a{\widetilde D}\bar{A}^a)\nonumber \\ &\quad +\frac{1}{r_{S^3}} |{\widetilde \sigma}B_{\dot{a}} -B_{\dot{a}}\sigma |^2 +i(\bar{B}^{\dot{a}}{\widetilde D}B_{\dot{a}}-B_{\dot{a}}D\bar{B}^{\dot{a}}) \Bigr]+(fermions) . \label{minimalcoupling}\end{aligned}$$ We can introduce a mass by turning on a background vector multiplet ${\cal V}^{\text{(bgd)}}=(A_\mu^{\text{(bgd)}},\sigma^{\text{(bgd)}}$, $\chi^{\text{(bgd)}},D^{\text{(bgd)}})$ of a global symmetry in the following supersymmetric configuration[^5] [@FP] $$\begin{aligned} A_\mu^{\text{(bgd)}}=0,\quad \sigma^{\text{(bgd)}}=\delta,\quad \chi^{\text{(bgd)}}=0,\quad D^{\text{(bgd)}}=-\delta. \label{background}\end{aligned}$$ where we have set the radius of $S^3$ to $r_{S^3}=1$. Here we turn on the background multiplets of the flavor symmetries $\text{U}(1)_1\times\text{U}(1)_2\times\text{U}(1)_3$ commuting with the ${\cal N}=2$ supersymmetry under which the chiral multiplets are charged[^6] as in table \[tab:charges\]. The background gauge fields also minimally couples to the chiral multiplets in the same way as , hence it modifies the action as $$\begin{aligned} S &\rightarrow S+\int \sqrt{g}\operatorname{Tr}\Biggl[ \Bigl(\frac{\delta^1}{2}+\frac{\delta^2}{2}+\frac{\delta^3}{2}\Bigr) (-i|A_1|^2 -2(\bar{A}^1\sigma A_1-\bar{A}^1A_1{\widetilde\sigma}) ) +\Bigl(\frac{\delta^1}{2}+\frac{\delta^2}{2}+\frac{\delta^3}{2}\Bigr)^2|A_1|^2\nonumber \\ &\quad+\Bigl(\frac{\delta^1}{2}-\frac{\delta^2}{2}-\frac{\delta^3}{2}\Bigr) (-i|A_2|^2 -2(\bar{A}^2\sigma A_2-\bar{A}^2A_2{\widetilde\sigma}) ) +\Bigl(\frac{\delta^1}{2}-\frac{\delta^2}{2}-\frac{\delta^3}{2}\Bigr)^2|A_2|^2\nonumber \\ &\quad+\Bigl(-\frac{\delta^1}{2}+\frac{\delta^2}{2}-\frac{\delta^3}{2}\Bigr) ( -i|B_{\dot{1}}|^2 -2({\bar B}^{\dot{1}}{\widetilde\sigma}B_{\dot{1}}-{\bar B}^{\dot{1}}B_{\dot{1}}\sigma) ) +\Bigl(-\frac{\delta^1}{2}+\frac{\delta^2}{2}-\frac{\delta^3}{2}\Bigr)^2|B_{\dot{1}}|^2\nonumber \\ &\quad+\Bigl(-\frac{\delta^1}{2}-\frac{\delta^2}{2}+\frac{\delta^3}{2}\Bigr) ( -i|B_{\dot{2}}|^2 -2({\bar B}^{\dot{2}}{\widetilde\sigma}B_{\dot{2}}-{\bar B}^{\dot{2}}B_{\dot{2}}\sigma) ) +\Bigl(-\frac{\delta^1}{2}-\frac{\delta^2}{2}+\frac{\delta^3}{2}\Bigr)^2|B_{\dot{2}}|^2 \Biggr]. \label{massterms}\end{aligned}$$ Here $\delta^{i}$ $(i=1,2,3)$ are the vacuum expectation values of the background vector multiplets ${\cal V}^{(\text{bgd},i)}$ for $\text{U}(1)_i$. In this paper we choose $\delta^i$ as[^7] $$\begin{aligned} \delta^1=\frac{2(\zeta_1+\zeta_2)}{k},\quad \delta^2=\frac{2(\zeta_1-\zeta_2)}{k},\quad \delta^3=0,\end{aligned}$$ so that $\zeta_1$, $\zeta_2$ are interpreted as the mass parameters for the chiral multiplets as $m_1=2\zeta_1/k$ for ${\cal Z}_1$, ${\cal W}_{\dot{2}}$ and $m_2=2\zeta_2/k$ for ${\cal Z}_2$, ${\cal W}_{\dot{1}}$. $\text{U}(1)_1$ $\text{U}(1)_2$ $\text{U}(1)_3$ ------------------------- ----------------- ----------------- ----------------- $\mathcal{Z}_1$ $\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{2}$ $\mathcal{Z}_2$ $\frac{1}{2}$ $-\frac{1}{2}$ $-\frac{1}{2}$ $\mathcal{W}_{\dot{1}}$ $-\frac{1}{2}$ $\frac{1}{2}$ $-\frac{1}{2}$ $\mathcal{W}_{\dot{2}}$ $-\frac{1}{2}$ $-\frac{1}{2}$ $\frac{1}{2}$ : Charges of the $\text{U}(1)_1\times\text{U}(1)_2\times\text{U}(1)_3$ flavor symmetry. []{data-label="tab:charges"} Applying the localization method [@KWY; @J; @HHL], the sphere partition function of the massive ABJM theory is given by the following matrix model [@JKPS] $$\begin{aligned} Z =\frac{1}{(N!)^2}\int\frac{d^N\lambda}{(2\pi)^N}\frac{d^N{\widetilde\lambda}}{(2\pi )^N} e^{\frac{ik}{4\pi}\sum_i(\lambda_i^2-{\widetilde\lambda}_i^2)} \frac{\prod_{i\neq j }^N 2\sinh\frac{\lambda_i-\lambda_j}{2} \cdot 2\sinh\frac{{\widetilde\lambda}_i-{\widetilde\lambda}_j}{2} } {\prod_{i,j=1}^N 2\cosh\frac{\lambda_i-{\widetilde\lambda}_j -4\pi \zeta_1 /k}{2} \cdot 2\cosh\frac{\lambda_i-{\widetilde\lambda}_j -4\pi \zeta_2 /k}{2} } . \label{Original}\end{aligned}$$ In the rest of sections, we practically analyze another equivalent representation for $Z$: $$\begin{aligned} Z(N,k,\zeta_1,\zeta_2)=\frac{1}{N!}\int\frac{d^Nx}{(2\pi k)^N}\prod_{i=1}^N\frac{e^{\frac{2i\zeta_1}{k}x_i}}{2\cosh\frac{x_i}{2}} \frac{ \prod_{i<j}^N(2\sinh\frac{x_i-x_j}{2k})^2 }{ \prod_{i,j=1}^N2\cosh\frac{x_i-x_j+4\pi\zeta_2}{2k} }, \label{Sdual}\end{aligned}$$ which we derive in app. \[Sdualsection\]. For $k=1$ and $\zeta_1=\zeta_2=0$, this latter expression coincides with the partition function of the ${\cal N}=8$ $\text{U}(N)$ Yang-Mills theory coupled with a fundamental chiral multiplet, which is dual to the ABJM theory under the $\text{SL}(2,\mathbb{Z})$ transformation in the type IIB brane setup. Because of this reason we simply refer to as the S-dual representation even for general $(k,\zeta_1,\zeta_2)$. Note that the integration in the S-dual representation is absolutely convergent in contrast to the representation where the convergence is achieved by the rapidly oscillating factors. Because of this fact, it is much easier to apply the Monte Carlo simulation of the partition function to than . With the help of the Monte Carlo simulation of we will observe a novel behavior of the partition function: the partition function vanishes at some finite values of $\zeta_1,\zeta_2$, which was not encountered in the undeformed case or the case of the R-charge deformation ($\zeta_1,\zeta_2\in i\mathbb{R}$). Evidence for SUSY breaking {#susybreaking} ========================== In this section, we discuss why we expect the SUSY breaking of the mass deformed ABJM theory on $S^3$ in the large-$N$ limit at some finite $(\zeta_1 ,\zeta_2 )$ and explain our criterion for the SUSY breaking which we will examine in the following sections. First, in the case of $\zeta_1=\zeta_2 =\zeta$, there is a large-$N$ saddle point solution for the original matrix model which exist only for $0 \leq \frac{\zeta}{k} < \frac14$ [@NST2]. This solution becomes the saddle point solution of the massless ABJM theory [@HKPT] in the $\zeta \rightarrow 0$ limit and gives the $N^{3/2}$-law of the free energy: - = ( 1+ ) N\^[3/2]{} . However, this saddle point solution becomes singular in the $\frac{\zeta}{k} \rightarrow \frac14$ limit. There is another large-$N$ solution for any value of $\zeta$. The free energy of this solution is proportional to $N^2$ and this solution may correspond to a confinement vacuum.[^8] Although it would be possible that there are other solutions,[^9] these results strongly indicate a phase transition at $\frac{\zeta}{k} \rightarrow \frac14$. We expect that this phase transition comes from SUSY breaking as follows. Let us take the mass very large, i.e. $\frac{\zeta}{k} \gg 1$, then, at least naively, the hypermultiplets become heavy and decouple from the vector multiplets. The remaining ${\cal N}=2$ SUSY pure Chern-Simons theory will spontaneously break SUSY as shown in [@Witten; @Ohta], and becomes the confinement phase in the large-$N$ limit. This expectation is consistent with the above large-$N$ solutions. However, for the mass deformed ABJM theory, the SUSY index was computed to be non-zero in [@KK; @CKK]. In this theory, there are infinitely many discrete classical vacua which are characterized by the fuzzy $S^3$ solutions given in [@Terashima; @GRVV], which represent M5-branes. Although the contribution to the index for the trivial vacuum, where all the scalar fields are zero, vanishes as in the pure SUSY Chern-Simons theory, other vacua give the non-zero contributions to the index if there are no coincident M5-branes. This result seems to contradict with the above argument of the SUSY breaking. However, this results is for the theory on $T^3$, not on $S^3$. For the ${\cal N}=2$ SUSY theory on $S^3$, there are mass terms for the hypermultiplets proportional to the curvature of $S^3$. The mass term will lift all of the vacua except the trivial vacuum at the origin classically.[^10] Thus, the result of [@KK; @CKK] on the SUSY index does not exclude the possibility that the mass deformed ABJM theory on $S^3$ has the SUSY breaking phase.[^11] Here note that we do not take the large volume limit. Criterion for SUSY breaking --------------------------- By now, we have not defined what the spontaneous SUSY breaking on $S^3$ is. Usually, the SUSY breaking means that there is no states with zero energy in the theory. For $S^3$, we can not define states with an appropriate Hamiltonian and time, thus it is difficult to use this definition. Instead of this definition for the SUSY breaking, the spontaneous breaking of a symmetry $\hat{Q}$ can be defined as ${}^\exists{\cal \hat{O}}$ s.t. $\langle 0|[\hat{Q},{\cal \hat{O}}]|0\rangle \neq 0$. In the path-integral formalism, this corresponds to $$\begin{aligned} Q\text{ is spontaneously broken}\ \mathop{\Longleftrightarrow}^{\text{def}}\ {}^\exists{\cal O}\text{ s.t. }\langle Q{\cal O}\rangle \neq 0, \label{susybreaking180212a}\end{aligned}$$ where the condensation is the order parameter. Note that this correspondence is valid for the theory with enough number of non-compact space directions, in which the notion of vacuum is meaningful, otherwise, $\langle Q{\cal O} \rangle$ corresponds to $\operatorname{Tr}[\hat{Q},{\cal \hat{O}}]$,[^12] not to $\langle 0|[\hat{Q},{\cal O}]|0\rangle$. Because $Q$ is a symmetry generator which behaves well, we expect that $\langle Q{\cal O}\rangle=0$ ($\operatorname{Tr}[Q,{\cal O}]=0$) is trivial identity due to the invariance of path integral measure (cyclic invariance of $\operatorname{Tr}$).[^13] For example, for SUSY quantum mechanics case, the invariance of the Witten index means $\operatorname{Tr}[(-1)^{\hat{F}} \hat{Q},{ \hat{Q}^\dagger}] \sim \operatorname{Tr}(-1)^{\hat{F}} \hat{H} = 0$. Thus, the definition of (\[susybreaking180212a\]) is meaningful for the theory with some space with enough number of non-compact directions. Since $S^3$ is compact, we need to take the large volume limit or large-$N$ limit which can effectively gives extra dimension. If this happens, there should exist a massless Goldstone fermion in the theory which makes the (SUSY) partition function $Z$ vanished. Instead of the large volume limit, we take a large-$N$ limit in which the SUSY breaking is meaningful. Thus, we need a criterion of the SUSY breaking in the large-$N$ limit from a finite $N$ result. For the theory in which we can define the Witten index, the vanishing of it, i.e. $Z=0$, is the necessary condition for the SUSY breaking for the finite volume. For the other theories also, we expect that the massless Goldstone fermion makes $Z=0$. Indeed, for a superconformal theory on $S^3$, the theory can break the SUSY if $Z=0$ because the radius of $S^3$ is not physical. Such theories were discussed in [@MN; @KWY3; @Suyama1; @Suyama2]. For our case, the theory is not conformal, but we take the large-$N$ limit. Thus, we regard $Z=0$ as a criterion of the SUSY breaking.[^14] [^15] It is worth to note that $Z=0$ does not necessarily mean SUSY breaking as in Witten index. However, for our case, interpreting $Z=0$ as SUSY breaking is the most natural possibility because the mass deformed ABJM theory on $S^3$ will be smoothly connected to the pure SUSY CS theory in the large mass limit whose SUSY is broken for $k\leq N$. In the following sections, we will give further supporting arguments for the above picture of the SUSY breaking phase for the mass deformed ABJM theory on $S^3$ using the S-dual representation of the matrix model. Here we will summarize these argument for the SUSY breaking shortly. The S-dual representation of the matrix model (for $k=1$) is obtained from the $\text{U}(N)$ Yang-Mills theory with an adjoint and fundamental matter fields where $\zeta_1$ and $\zeta_2$ corresponds to the FI term and the mass for the adjoint matter, respectively. Because of the FI term (and the mass term), this theory will break the SUSY at the origin of the Coulomb branch moduli space which will be favored by the mass terms induced from the curvature of $S^3$. This picture will be right for a generic large value of $\zeta_1$ and $\zeta_2$. However, for $\zeta_1=0$, the FI term vanishes and the SUSY will not break.[^16] For this case, as we will see later, we can construct a large-$N$ solution for any value of $\zeta_2$, thus there are no critical mass for this case. This is consistent with the above picture. In order to investigate further, we will compute the partition function $Z$ for finite $N$ exactly and numerically using the Monte Carlo method for various points of $(\zeta_1,\zeta_2 )$. We expect that some values of $N$ for which we computed $Z$ are not very large, but enough large for the large-$N$ expansion. Indeed, the computed values of $Z$ are consistent with the large-$N$ solutions for $\zeta_1=\zeta_2<k/4$ and $\zeta_1=0$. These actual computations of $Z$ for finite $N$ shows that as increasing $\zeta_i$, $Z$ is decreasing and oscillating, thus $Z=0$ for some values of $\zeta_i$. We expect that this zero corresponds to the SUSY breaking in the large-$N$ limit. Furthermore, if we increase $N$ with other parameters fixed, the smallest value of $\zeta_i$ which gives $Z=0$ decreasingly approaches to the critical point of the large-$N$ solution. Therefore, the extrapolation of this to the large-$N$ limit may be consistent with the SUSY breaking picture above. The case with one massless hypermultiplet ($\zeta_1=0$) {#zeta10} ======================================================= In this section we consider the case with $\zeta_1=0$. In this case we find a solution to the saddle point equation for the partition function in the S-dual representation . We can also compute the exact values of the partition function for finite $(N,k)$ by a slight generalization [@Nosaka; @OZ] of the technique used in the ABJM theory [@TW; @PY]. We will see a good agreement of the both results. Saddle point analysis in the large-$N$ limit {#saddle} -------------------------------------------- In this subsection we compute the partition function in the large-$N$ limit N, (k,\_2 ). In this limit, we can evaluate the partition function by the saddle point method. To perform the saddle point analysis, we first introduce the effective action $S_\text{eff}$ by $$\begin{aligned} Z=\frac{1}{N!} \int \frac{d^Nx}{(2\pi k)^N}\ e^{-S_\text{eff}(x)} \label{ZinS}\end{aligned}$$ where $$\begin{aligned} S_\text{eff}(x) &=& -\frac{2i\zeta_1}{k}\sum_{i=1}^N x_i +\sum_{i=1}^N\log \left( 2\cosh\frac{x_i}{2} \right) \nonumber\\ && -\sum_{i<j}^N\log\Bigl(2\sinh\frac{x_i-x_j}{2k}\Bigr)^2 +\sum_{i,j=1}^N\log \left( 2\cosh\frac{x_i-x_j+4\pi\zeta_2}{2k} \right). \label{S}\end{aligned}$$ We rearrange the eigenvalues $x_i$ such that $x_{i+1}\geq x_i$ by the permutation symmetry and regard $x_i$ as a function of $s=i/N -1/2$, which becomes the continuous variable in the large-$N$ limit: x\_i x( s ), s   [and]{}   0 . Then the summations over $i$ are replaced by the integral over $s$ $$\begin{aligned} \sum_{i} \rightarrow N\int_{-\frac{1}{2}}^{\frac{1}{2}}ds.\end{aligned}$$ We look for saddle point solutions by the approach taken in [@HKPT] which has been used to derive $\mathcal{O}(N^{\frac{3}{2}})$ behaviors of free energies, rather than the traditional approach often applied for matrix models in the planar limit.[^17] This is achieved by taking the following ansatz $$\begin{aligned} x(s) =\sqrt{N}z (s) , \label{eq:ansatz}\end{aligned}$$ with an ${\cal O}(1)$ real[^18] function $z(s)$, and perform large-$N$ expansion of $S_\text{eff}(x)$ to simplify the saddle point equation. It is easy to write down the leading part for the first and second terms in : $$\begin{aligned} -\frac{2i\zeta_1}{k}\sum_{i=1}^N x_i +\sum_{i=1}^N\log \left( 2\cosh\frac{x_i}{2} \right) =N^{\frac{3}{2}}\int_{-\frac{1}{2}}^{\frac{1}{2}} ds\Bigl(-\frac{2i\zeta_1z}{k}+\frac{|z|}{2}\Bigr)+{\cal O}(N).\end{aligned}$$ We can also expand the third and fourth terms in respectively by using the techniques of [@NST2] to see that the leading part of $S_\text{eff}$ in the large-$N$ limit is proportional to $N^{\frac{3}{2}}$. First we rewrite these terms as $$\begin{aligned} &-\sum_{i<j}^N\log\Bigl(2\sinh\frac{x_i-x_j}{2k}\Bigr)^2 =-\frac{N^2}{2}\int ds ds'\log\Bigl(2\sinh\frac{\sqrt{N}(z-z')}{2k}\Bigr)^2\nonumber \\ =& \frac{N^2}{2}\int dsds' \Biggl[\operatorname{sgn}(z-z')\frac{\sqrt{N}(z-z')}{k} + \log \left( 1-e^{-\sqrt{N}\mid\frac{z-z^{\prime}}{k}\mid}\right)^2 \Biggr], \label{sinh}\end{aligned}$$ and[^19] $$\begin{aligned} &\sum_{i,j}^N\log \left( 2\cosh\frac{x_i-x_j+4\pi\zeta_2}{2k} \right)\nonumber \\ =& N^2\int dsds' \Biggl[ \operatorname{sgn}(z-z')\frac{\sqrt{N}(z-z')}{2k} +\log \left( 1+e^{-\operatorname{sgn}(z-z^{\prime})\frac{\sqrt{N}(z-z^{\prime})+4\pi\zeta_{2}}{k}}\right) \Biggr] . \label{cosh}\end{aligned}$$ where $z'$ is the abbreviation for $z(s')$. Note that the $\mathcal{O}(N^{5/2})$ terms, which are the first terms in and , are canceled and only the second terms remain. Here we use the following formula in the large-$N$ limit:[^20] $$\begin{aligned} \int_{s_0}^{1/2} ds \ln (1 \pm e^{-2y(s)}) &\sim \frac{1}{\sqrt{N} \dot{v}(s_{0})} \int_{w(s_0 )}^\infty dt \ln (1 \pm e^{-2t}), \label{logcoshapprox1} \\ \int_{-1/2}^{s_0} ds \ln ( 1 \pm e^{2y(s)}) &\sim \frac{1}{\sqrt{N} \dot{v}(s_{0})} \int_{-w(s_0 )}^\infty dt \ln (1 \pm e^{-2t}), \label{logcoshapprox2}\end{aligned}$$ where $y(s)=\sqrt{N}v(s)+w(s)$ and $s_{0}$ is the zero of $v(s)$. These formulas are obtained by changing the integration variable and reflected with the fact that the contribution to the integral in l.h.s of and is coming from only $s\sim s_{0}$ region in the large-$N$ limit. Using these formulas the second terms in and can be evaluated as && 2kN\^\ &&= (1+ ). Putting the above computations together, we find the following large-$N$ expansion for the effective action $$\begin{aligned} S_\text{eff}=N^{3/2}\int ds F(z,\dot{z})+{\cal O}(N), \label{SinlargeN}\end{aligned}$$ with $$\begin{aligned} F(z,\dot{z})=\Bigl[-\frac{2i\zeta_1z}{k}+\frac{|z|}{2}+\frac{\pi^2k}{2}\Bigl(1+\frac{16\zeta_2^2}{k^2}\Bigr)\frac{1}{\dot{z}}\Bigr].\end{aligned}$$ The overall scaling $N^{3/2}$ in implies that the integration is dominated in the large-$N$ limit by the saddle point configuration satisfying the following equation of motion $$\begin{aligned} 0=\frac{\partial F}{\partial z}-\frac{d}{dt}\frac{\partial F}{\partial\dot{z}}=-\frac{2i\zeta_1}{k}+\frac{\operatorname{sgn}(z)}{2}-\frac{d}{ds}\Bigl[-\frac{\pi^2k}{2}\Bigl(1+\frac{16\zeta_2^2}{k^2}\Bigr)\frac{1}{\dot{z}^2}\Bigr] \label{saddleeq}\end{aligned}$$ together with the boundary condition $$\begin{aligned} 0=\frac{\partial F}{\partial \dot{z}}=-\frac{\pi^2k}{2}\bigg[\bigl(1+\frac{16\zeta_2^2}{k^2}\Bigr)\frac{1}{\dot{z}^2}\bigg]_{\text{boundary}} . \label{bc}\end{aligned}$$ First let us consider the case for $\zeta_1=0$. First of all the equation of motion has the following two local solutions depending on $\operatorname{sgn}(z)$ $$\begin{aligned} z^{(+)}(s)&=\sqrt{2\pi^2k\Bigl(1+\frac{16\zeta_2^2}{k^2}\Bigr)}(z_b-\sqrt{2(s_b-s)}),\quad (\operatorname{sgn}(z)=+1)\nonumber \\ z^{(-)}(s)&=-\sqrt{2\pi^2k\Bigl(1+\frac{16\zeta_2^2}{k^2}\Bigr)}(z_b-\sqrt{2(-s_b+s)}),\quad (\operatorname{sgn}(z)=-1) \label{solntosaddleeq}\end{aligned}$$ where $s_b$ and $z_b$ are the integration constants.[^21] The bulk solution would be obtained by connecting these solutions appropriately and determining the integration constants so that $z(s)$ satisfies the boundary condition $\dot{z}(s)=\pm \infty$ at every point where $z(s)$ or $\dot{z}(s)$ is discontinuous. Notice that both of $z^{(\pm)}(s)$ satisfies $\dot{z}^{(\pm)}(s)=\infty$ at only a single point $s=s_b$. Therefore, if we split the support $-1/2<s<1/2$ into segments by the points of discontinuity, $z(s)$ on each segment must be given as a smooth junction of $z^{(-)}(s)$ and $z^{(+)}(s)$. Since $z^{(+)}(s)$ cannot be followed by $z^{(-)}(s)$ due to the assumption that $z(s)$ is monotonically increasing, we conclude that the solution is given by a single junction of $z^{(-)}(s)$ with $s_b=-1/2$ ($-1/2<s<s_0$) and $z^{(+)}(s)$ with $s_b=1/2$ ($s_0<s<1/2$) with some $s_0$. The remaining constants $s_0,s_b$ are determined from $z^{(-)}(s_0)=z^{(+)}(s_0)$ and $\dot{z}^{(-)}(s_0)=\dot{z}^{(+)}(s_0)$ as $s_0=0$, $z_b=1$ (for both domain). In summary we obtain the following unique solution as the saddle point configuration: $$\begin{aligned} z(s)=\operatorname{sgn}(s)\sqrt{2\pi^2k\Bigl(1+\frac{16\zeta_2^2}{k^2}\Bigr)}(1-\sqrt{1-2|s|}). \label{zsfinal}\end{aligned}$$ In the language of the eigenvalue density, this solution corresponds to \(z) = = ( 1 - ) . Substituting this solution to , we find that the partition function in the large-$N$ limit is given as $$\begin{aligned} \left. -\log Z \right|_{\zeta_1 =0} \approx \frac{\pi\sqrt{2k}}{3}\sqrt{1+\frac{16\zeta_2^2}{k^2}}N^{\frac{3}{2}}. \label{Zinsaddle}\end{aligned}$$ For $\zeta_{1}\neq 0$ we could not solve the saddle point equation with the ansatz we used here because the solution can not satisfy the boundary condition due to the existence of the imaginary term $\frac{2i\zeta_{1}}{k}$ in . However, the partition function with $\zeta_{1}=0,\ \zeta_{2}\neq0$ and that with $\zeta_1\neq 0, \ \zeta_{2}=0$ is the same because the partition function is invariant under exchanging $\zeta_{1}$ and $\zeta_{2}$. This fact suggests that even when $\zeta_{2}=0,\zeta_{1} \neq 0$, there exists the solution of the saddle point equation in large-$N$ limit and the free energy can be evaluated by the saddle point approximation. Exact partition function for finite $(N, k)$ {#TWPY} -------------------------------------------- Next we compute the partition function for some finite $(N,k)$ by the technique used in [@Nosaka]. We start with the partition function written in the Fermi gas formalism $$\begin{aligned} Z(N,k,0,\zeta_2)=\frac{1}{N!}\int\frac{d^Nx}{(2\pi)^N}\det_{i,j}\langle x_i|{\widehat\rho} (\hat{q} ,\hat{p}) |x_j\rangle,\end{aligned}$$ where $[\hat{q}, \hat{p}] = i\hbar$ with $\hbar =2\pi k$ and[^22] $$\begin{aligned} {\widehat\rho}= \sqrt{\frac{1}{2\cosh\frac{\widehat q}{2}}} \frac{e^{\frac{2i\zeta_2}{k}{\widehat p}}}{2\cosh\frac{\widehat p}{2}} \sqrt{\frac{1}{2\cosh\frac{\widehat q}{2}}}.\end{aligned}$$ If we consider the generating function of the partition function or equivalently the grand partition function $\sum_{N=0}^\infty z^NZ(N)$, we can show that it is written as the following Fredholm determinant $$\begin{aligned} \sum_{N=0}^\infty z^NZ(N)=\operatorname{Det}(1+z{\widehat\rho})\equiv \exp\Bigl[\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}z^n \operatorname{Tr}{\widehat\rho}^n\Bigr]. \label{eq:grand}\end{aligned}$$ Comparing the coefficient of $z^N$ on the both sides, we find that the partition function $Z(N)$ is determined by $\operatorname{Tr}{\widehat\rho}^n$ with $n\le N$, as $$\begin{aligned} Z(1)=\operatorname{Tr}{\widehat\rho},\quad Z(2)=\frac{1}{2}(\operatorname{Tr}{\widehat\rho})^2-\frac{1}{2}\operatorname{Tr}{\widehat\rho}^2,\quad Z(3)=\frac{1}{6}(\operatorname{Tr}{\widehat\rho})^3-\frac{1}{2}\operatorname{Tr}{\widehat\rho}\operatorname{Tr}{\widehat\rho}^2+\frac{1}{3}\operatorname{Tr}{\widehat\rho}^3,\quad \cdots.\end{aligned}$$ We can compute $\operatorname{Tr}{\widehat\rho}^n$ by completely the same way as that in the case of R-charge deformation [@Nosaka]. First we notice that the matrix element $\langle x|{\widehat \rho}|y\rangle$ has the following structure $$\begin{aligned} \langle x|{\widehat\rho}|y\rangle =\frac{1}{2\cosh\frac{x}{2}}\frac{1}{2k\cosh\frac{x-y+4\pi \zeta_2}{2k}} =\frac{E(x)E(y)}{k(\alpha M(x)+\alpha^{-1}M(y))}, \label{TWstructure}\end{aligned}$$ with $$\begin{aligned} E(x)=\frac{e^{\frac{x}{2k}}}{\sqrt{2\cosh\frac{x}{2}}},\quad M(x)=e^{\frac{x}{k}},\quad \alpha=e^{\frac{2\pi\zeta_2}{k}}.\end{aligned}$$ For $\alpha =1$, this form is in the range of application of Tracy-Widom’s lemma [@Tracy:1995ax] which has been very powerful tool to systematically compute ${\rm Tr}\hat{\rho}^n$ in various M2-brane theories without masses [@Hatsuda:2012hm; @PY; @Hatsuda:2012dt; @Matsumoto:2013nya; @Honda:2014npa; @Hatsuda:2014vsa; @Moriyama:2014gxa; @Moriyama:2014nca; @Moriyama:2017gye]. We can easily extend it to general $\alpha$ as follows. The structure can be expressed as a quasi-commutation relation for ${\widehat\rho}$ $$\begin{aligned} \alpha {\widehat M}{\widehat\rho}+\alpha^{-1}{\widehat\rho}{\widehat M}={\widehat E}|0{\rangle\!\rangle}{\langle\!\langle}0|{\widehat E},\quad\quad ({\widehat E}=E({\widehat q}),\quad {\widehat M}=M({\widehat q})) ,\end{aligned}$$ where $|p {\rangle\!\rangle}$ is momentum eigenstate satisfying $$\begin{aligned} \langle x|x'\rangle =2\pi\delta(x-x'),\quad {\langle\!\langle}p|p'{\rangle\!\rangle}=2\pi\delta(p-p'),\quad \langle x|p{\rangle\!\rangle}=\frac{1}{\sqrt{k}}e^{\frac{ixp}{\hbar}},\quad {\langle\!\langle}p|x\rangle=\frac{1}{\sqrt{k}}e^{-\frac{ixp}{\hbar}}. \label{eq:braket}\end{aligned}$$ This relation can be generalized straightforwardly for ${\widehat\rho}^n$ as $$\begin{aligned} \alpha^n{\widehat M}{\widehat\rho}^n-(-1)^n\alpha^{-n}{\widehat\rho}{\widehat M} =\sum_{\ell=0}^{n-1}(-1)^\ell \alpha^{n-1-2\ell}{\widehat\rho}^\ell {\widehat E}|0{\rangle\!\rangle}{\langle\!\langle}0| {\widehat E} {\widehat\rho}^{n-1-\ell}.\end{aligned}$$ This implies that we can compute the matrix element of ${\widehat\rho}^n$ from two sets of functions $\phi_\ell(x)$ and $\psi_\ell(x)$ as $$\begin{aligned} \langle x|{\widehat \rho}^n|y\rangle=\frac{E(x)E(y)}{\alpha^nM(x)-(-1)^n\alpha^{-n}M(y)}\sum_{\ell=0}^{n-1}(-1)^\ell\phi_\ell(x)\psi_{n-1-\ell}(y),\end{aligned}$$ where $$\begin{aligned} \phi_\ell(x)=\alpha^{-\ell}\langle x|{\widehat E}^{-1}{\widehat\rho}^\ell {\widehat E}|0{\rangle\!\rangle},\quad \psi_\ell(x)=\alpha^\ell{\langle\!\langle}0|{\widehat E}^{-1}{\widehat\rho}^\ell {\widehat E}|x{\rangle\!\rangle}=\phi_\ell(x)|_{\alpha\rightarrow\alpha^{-1}} .\end{aligned}$$ We can show that the function $\phi_\ell(x)$ satisfies the following recursion relation $$\begin{aligned} \phi_{\ell+1}(x)&=\int\frac{dy}{2\pi}\frac{1}{E(x)}\alpha^{-1}\rho(x,y)E(y)\phi_\ell(y)\nonumber \\ &=\int \frac{dy}{2\pi k}\frac{1}{e^{\frac{y}{k}}+\alpha^2e^{\frac{x}{k}}}\frac{e^{\frac{y}{k}}}{e^{\frac{y}{2}}+e^{-\frac{y}{2}}}\phi_\ell(y), \label{recursive_int}\end{aligned}$$ as well as $\psi_\ell (x)$. In app. \[app:detail\_exact\], we explain how to practically solve the recursion relation for integer $k$ while their details are slightly different between odd $k$ and even $k$ cases. According to the algorithm, we have computed $Z(N,k,1,\zeta_2)$ by Mathematica for $(k=1 ,N\le 12)$, $(k=2,N\le 9 )$, $(k=3 ,N\le 5)$, $(k=4 ,N\le 5 )$ and $(k=6 ,N\le 4 )$. In app. \[Z1exact\_results\], we explicitly write down a part of the results and also compare them with the result of saddle point approximation . General deformation with $\zeta_1,\zeta_2\neq 0$ {#bothnonzero} ================================================ In this section we consider the case for $\zeta_1, \zeta_2\neq 0$. Note that this may affect the sign of the partition function because the integrand of for $\zeta_1\neq 0$ has the oscillation factor $e^{\frac{2i \zeta_1}{k}\sum_i x_i}$ in contrast to the $\zeta_1=0$ case, where the integrand was positive semi-definite. Therefore the partition function may be negative or zero depending on the parameters $(N,k,\zeta_1 ,\zeta_2 )$. For large $\zeta_1,\zeta_2$, we can easily see that this actually happens as follows. In this limit, the hypermultiplets become very massive and integrating them out leads us to the $\mathcal{N}=2$ SUSY $\text{U}(N)_k\times \text{U}(N)_{-k}$ pure Chern-Simons theory schematically.[^23] It is known that the sphere partition function of the pure Chern-Simons theory vanishes for $k<N$. In this section, we will see that the zeroes appear also for finite $(\zeta_1 ,\zeta_2 )$. In this case we could not find a solution to the saddle point equation. The technique for small integers $k,N$ in sec. \[TWPY\] is not applicable either. Nevertheless we can evaluate the partition function exactly for $N=1,2$, which suggest the partition function has zeroes as a function of $\zeta_1,\zeta_2$ only for $(N,k)=(2,1),(2,2)$. We argue a possible interpretation for this zeroes. We further conjecture the zeroes for general $k,N$, and provide positive evidence from the numerical computation of the partition function for $N\ge 3$. Exact expression for $N=1,2$ {#sec:exactN2} ---------------------------- In this subsection we review the exact results for $N=1,2$ obtained in [@RS].[^24] The relation between the partition function and ${\rm Tr}{\widehat\rho}^n $ is correct also for general $\zeta_1$ if we take ${\widehat\rho}$ as $$\begin{aligned} \langle x|{\widehat\rho}|y\rangle =\frac{e^{\frac{2i\zeta_1}{k}x}}{2\cosh\frac{x}{2}}\frac{1}{2k\cosh\frac{x-y+4\pi \zeta_2}{2k}}.\end{aligned}$$ For $N=1$, the partition function is simply given by $Z(1,k,\zeta_1,\zeta_2)=\operatorname{Tr}{\widehat\rho}$, which can be exactly computed as $$\begin{aligned} Z(1,k,\zeta_1,\zeta_2) =\int_{-\infty}^\infty\frac{dx}{2\pi} \frac{e^{\frac{2i\zeta_1x}{k}}}{2\cosh\frac{x}{2}}\frac{1}{2k\cosh\frac{2\pi\zeta_2}{k}} =\frac{1}{4k\cosh\frac{2\pi\zeta_1}{k}\cosh\frac{2\pi\zeta_2}{k}}. \label{eq:exactN1}\end{aligned}$$ For $N=2$, we need to compute $\operatorname{Tr}{\widehat\rho}^2$, which is given by the following two dimensional integration $$\begin{aligned} \operatorname{Tr}{\widehat\rho}^2 =\int_{-\infty}^\infty \frac{dx}{2\pi}\frac{dy}{2\pi} \frac{e^{\frac{2i\zeta_1(x+y)}{k}}}{16k^2\cosh\frac{x}{2}\cosh\frac{y}{2}\cosh\frac{x-y+4\pi\zeta_2}{2k}\cosh\frac{x-y-4\pi\zeta_2}{2k}}.\end{aligned}$$ After changing the integration variables to $x_\pm=x\pm y$, we can easily perform the $x_+$-integration, which leads to $$\begin{aligned} \operatorname{Tr}{\widehat\rho}^2 =\frac{1}{16\pi k^2\sin\frac{4\pi\zeta_1}{k}} \int_{-\infty}^\infty dx_- \frac{\sin\frac{2\zeta_1x_-}{k}} {\sin\frac{x_-}{2}\cosh\frac{x_-+4\pi\zeta_2}{2k}\cosh\frac{x_--4\pi\zeta_2}{2k}} .\end{aligned}$$ For $k\in \mathbb{Z}_+$, this integral can be evaluated by considering an integral with the same integrand along a rectangular whose corners are $x_- = (-\infty,\infty ,\infty +2\pi ik ,-\infty +2\pi ik)$ [@Okuyama; @RS], and we obtain \^2 = , where R\_k = (-1)\^ & [for odd]{} k (-1)\^[+1]{} & [for even]{} k . For example, the final results for $k=1,2,3,4$ are explicitly given by $$\begin{aligned} Z(2,1,\zeta_1,\zeta_2) &=\frac{\sin 8\pi\zeta_1\zeta_2}{8\sinh 4\pi\zeta_1\sinh 4\pi\zeta_2 \cosh 2\pi\zeta_1 \cosh 2\pi\zeta_2},\nonumber \\ Z(2,2,\zeta_1,\zeta_2) &=\frac{\sin^2 2\pi\zeta_1\zeta_2}{8\sinh^2 2\pi\zeta_1 \sinh^2 2\pi\zeta_2},\nonumber \\ Z(2,3,\zeta_1,\zeta_2) &=\frac{1}{24(\cosh\frac{4\pi\zeta_1}{3}+\cosh\frac{8\pi\zeta_1}{3})(\cosh\frac{4\pi\zeta_2}{3}+\cosh\frac{8\pi\zeta_2}{3})} \left( 2-\frac{\sin\frac{8\pi\zeta_1\zeta_2}{3}}{\sinh\frac{2\pi\zeta_1}{3}\sinh\frac{2\pi\zeta_2}{3}}\right),\nonumber \\ Z(2,4,\zeta_1,\zeta_2) &=\frac{1}{128\sinh^2\pi\zeta_1\sinh^2\pi\zeta_2} \left(1-\frac{1}{\cosh\pi\zeta_1}-\frac{1}{\cosh\pi\zeta_2}+\frac{\cos 2\pi\zeta_1\zeta_2}{\cosh\pi\zeta_1\cosh\pi\zeta_2}\right) . \label{ZN2exact}\end{aligned}$$ We easily see from these results that the partition function for $(N,k)=(2,1),(2,2)$ has zeroes at finite $(\zeta_1 /k ,\zeta_2 /k)$. $N\ge 3$ from Monte Carlo Simulation {#sec:MC} ------------------------------------ In this subsection we provide numerical evidence that the partition function has zeroes at finite $(\zeta_1 /k ,\zeta_2 /k)$ also for $N\geq 3$. For this purpose, we apply (Markov chain) Monte Carlo method to the partition function in the $S$-dual representation : Z(k,N,\_1 ,\_2 ) =  e\^[-S(k,N,\_1 ,\_2 ;x)]{} , where S(k,N,\_1 ,\_2 ;x) &=& -\_[i&lt;j]{}\^N +\_[i,j=1]{}\^N\ && +\_[i=1]{}\^N - . ### Algorithm First we explain our algorithm. There are two subtleties in applying the Monte Carlo method to our problem. The first subtlety, which will not be problematic as explained below, is that Monte Carlo simulation can directly calculate only “expectation values" or equivalently ratio of two functions rather than $Z$ itself. The second one is that the Boltzmann weight $e^{-S}$ is not positive semi-definite for $\zeta_1 \neq 0$ and hence cannot be regarded as probability. This problem appears in many contexts such as finite density QCD, real time systems and theories with CS terms. We take care of these subtleties as follows. Instead of $Z$ itself, we consider the ratio $$\begin{aligned} Z_{\rm MC}(N,k,\zeta_1 ,\zeta_2 ) =\frac{Z(N,k,\zeta_1,\zeta_2)}{Z(N,k,0,\zeta_2)} =\Biggl\langle \cos\left(\frac{2\zeta_1}{k}\sum_{i=1}^Nx_i \right) \Biggr\rangle_{\zeta_1 =0}, \label{ZcMC}\end{aligned}$$ where $\langle \mathcal{O}(x) \rangle_{\zeta_1 =0}$ denotes the expectation value of $\mathcal{O}(x)$ under the action $S(N,k,\zeta_1 =0 ,\zeta_2 )$. Then we approximate the ratio[^25] by Hybrid Monte Carlo simulation[^26] by taking samples generated with the probability $\sim \left. e^{-S} \right|_{\zeta_1 =0}$. Note that studying only the ratio is sufficient for our purpose since $Z(N,k,0,\zeta_2)$ is real positive and we are interested in the sign of the partition function.[^27] Since we are taking samples of the oscillating function, whose oscillation is controlled by $\zeta_1 /k$, we typically need more statistics for larger $\zeta_1 /k$ to obtain precise approximations. Note also that the $S$-dual representation has much milder oscillation than the original matrix model . This is why we are using the $S$-dual representation as in [@KEK]. ### Results ![ The ratio computed by Monte Carlo simulation is plotted against $\zeta_1$ for $(N,k,\zeta_2 )=(4,1,1)$. The right panel is the zoomup of the left panel around the negative peak of the partition function. \[fig:N4k1zeta2\_1\] ](N4k1zeta2_1.eps "fig:"){width="7.8cm"} ![ The ratio computed by Monte Carlo simulation is plotted against $\zeta_1$ for $(N,k,\zeta_2 )=(4,1,1)$. The right panel is the zoomup of the left panel around the negative peak of the partition function. \[fig:N4k1zeta2\_1\] ](N4k1zeta2_1_zoom.eps "fig:"){width="7.8cm"} Now we present numerical results for the ratio $Z_{\rm MC}$ , which has the same sign as the partition function $Z(N,k,\zeta_1 ,\zeta_2 )$ itself. Fig. \[fig:N4k1zeta2\_1\] plots $Z_{\rm MC}$ for $(N,k,\zeta_2 )=(4, 1,1)$ as a function of $\zeta_1$. The statistical errors are estimated by Jackknife method although they are practically almost invisible in the figures. The right panel of fig. \[fig:N4k1zeta2\_1\] is the zoomup of the left figure in the range $\zeta_1 \in [0.1,0.2]$. From the right figure, we easily see that the partition function takes negative values when $\zeta_1$ $=$ $0.110$, $0.115$, $\cdots ,0.135$ even if we take into account the errors. Therefore there must be a zero of the partition function for $\zeta_1 \leq 0.110$ and the plot indicates that the zero is located at $0.105 <\zeta_1 <0.110$. We have found similar results for other values of $(N,k,\zeta_2 )$ whose samples are shown in fig. \[fig:various\_cases\]. These figures indicate that the partition function has the zeroes at finite $\zeta_1 /k$ for various $(N,k,\zeta_2 )$. Note also that we sometimes encounter subtle cases. For example, in the case of $(N,k,\zeta_2 )=(4,2,1)$ shown in the right-bottom of fig. \[fig:various\_cases\], the minimum is consistent with both positive and negative $Z$ within the numerical errors.[^28] We expect that this type of behavior appear when the partition function is positive semidefinite but has zeroes as in the case of $(N,k)=(2,2)$ whose analytic result is given in the second line of . For this type of cases, any numerical simulation with nonzero errors cannot establish existence of zeroes since numerical values at the zeroes must be consistent with all the possible signs of $Z$ within errors. Therefore, for this type of cases, the best thing we can do by numerical simulation is to check existence of points consistent with $Z=0$. For all values of $(N\geq 2,k,\zeta_2 )$ which we have analyzed, we have checked that there exists at least one value of $\zeta_1$ consistent with $Z=0$ within errors. ![ The numerical plots of $Z_{\rm MC}$ as functions of $\zeta_1$ for various $(N,k,\zeta_2 )$ with their zoomups around the minima. \[fig:various\_cases\] ](N4k1zeta2_2.eps "fig:"){width="7.8cm"} ![ The numerical plots of $Z_{\rm MC}$ as functions of $\zeta_1$ for various $(N,k,\zeta_2 )$ with their zoomups around the minima. \[fig:various\_cases\] ](N4k1zeta2_5.eps "fig:"){width="7.8cm"}\ ![ The numerical plots of $Z_{\rm MC}$ as functions of $\zeta_1$ for various $(N,k,\zeta_2 )$ with their zoomups around the minima. \[fig:various\_cases\] ](N7k1zeta2_1.eps "fig:"){width="7.8cm"} ![ The numerical plots of $Z_{\rm MC}$ as functions of $\zeta_1$ for various $(N,k,\zeta_2 )$ with their zoomups around the minima. \[fig:various\_cases\] ](N9k1zeta2_2.eps "fig:"){width="7.8cm"}\ ![ The numerical plots of $Z_{\rm MC}$ as functions of $\zeta_1$ for various $(N,k,\zeta_2 )$ with their zoomups around the minima. \[fig:various\_cases\] ](N3k2zeta2_1.eps "fig:"){width="7.8cm"} ![ The numerical plots of $Z_{\rm MC}$ as functions of $\zeta_1$ for various $(N,k,\zeta_2 )$ with their zoomups around the minima. \[fig:various\_cases\] ](N4k2zeta2_1.eps "fig:"){width="7.8cm"} For the cases where we have established existence of first zeroes of $Z$, we write down bounds on the zeroes in tables \[tab:zeta1\], \[tab:zeta2\] and \[tab:zeta5\] for fixed $(k,\zeta_2 )$ (see tab. \[negativeextrema\] for $Z_{\rm MC}$ at the first negative peaks and their errors). We also estimate their precise locations by constructing[^29] interpolating functions of all the data points of $Z_{\rm MC}$ for fixed $(N,k,\zeta_2 )$ and finding zeroes of the interpolating functions. We will discuss implications of these values in sec. \[conjlargeN\]. $N$ Bounds on the zeroes Estimate of the zeroes ----- ----------------------- ------------------------------------------------ $2$ $\zeta_1=0.125$ $\zeta_1=0.125$ $4$ $0.105<\zeta_1<0.11$ $\zeta_1 =0.108084 \pm 0.000016 $ $5$ $0.105<\zeta_1<0.11$ $\zeta_1 =0.105249 \pm 0.000041$ $7$ $0.095<\zeta_1<0.1$ $\zeta_1 =0.0975822^{+0.0004201}_{-0.0003715}$ $9$ $0.085<\zeta_1<0.095$ $\zeta_1=0.0898839^{+0.0003752}_{-0.0004039}$ : Bounds on first zeroes of the partition function and estimate of their precise locations by interpolating functions for $(k,\zeta_2 )=(1,1)$. The value for $N=2$ is the exact value. []{data-label="tab:zeta1"} $N$ Bounds on the zeroes Estimate of the zeroes ----- ------------------------ ------------------------------------------------ $2$ $\zeta_1=0.0625$ $\zeta_1=0.0625$ $4$ $0.055<\zeta_1<0.0575$ $\zeta_1 = 0.0565766 \pm 0.0000060$ $5$ $0.0525<\zeta_1<0.055$ $\zeta_1 =0.0543974\pm 0.0000068 $ $7$ $0.05<\zeta_1<0.0525$ $\zeta_1 =0.0518753^{+0.0000324}_{-0.0000320}$ $9$ $0.0475<\zeta_1<0.05$ $\zeta_1 =0.0496673^{+0.0000700}_{-0.0000677}$ : Bounds and estimate of first zeroes of the partition function for $(k,\zeta_2 )=(1,2)$. []{data-label="tab:zeta2"} $N$ Bounds on the zeroes Estimate of the zeroes ----- ------------------------ ------------------------------------------------- $2$ $\zeta_1=0.025$ $\zeta_1=0.025$ $4$ $0.0023<\zeta_1<0.024$ $\zeta_1 =0.0238516 \pm 0.0000102$ $5$ $0.023<\zeta_1<0.024$ $\zeta_1 =0.023177 \pm 0.000007$ $7$ $0.022<\zeta_1<0.023$ $\zeta_1 =0.022638^{+0.000042}_{-0.000039}$ $9$ $0.021<\zeta_1<0.023$ $\zeta_1 =0.0218204^{+0.0000247}_{-0.0000241} $ : Bounds and estimate of first zeroes of $Z$ for $(k,\zeta_2 )=(1,5)$. []{data-label="tab:zeta5"} $N$ $\zeta_2$ $\zeta_1$ $Z_{\text{MC}}$ Errors ----- ----------- ----------- ----------------- ------------------------- $4$ $1$ $0.12$ $-0.00242055$ $7.70257\times 10^{-6}$ $2$ $0.0625$ $-0.00536328$ $0.0000115859$ $5$ $0.025$ $-0.00206855$ $0.000035207$ $5$ $1$ $0.115$ $-0.000807839$ $7.76448\times 10^{-6}$ $2$ $0.06$ $-0.00579732$ $0.0000125459$ $5$ $0.025$ $-0.00481262$ $0.000036708$ $7$ $1$ $0.105$ $-0.0000473376$ $7.52208\times 10^{-6}$ $2$ $0.055$ $-0.000501799$ $0.0000102382$ $5$ $0.035$ $-0.00411775$ $0.000018267$ $9$ $1$ $0.1$ $-0.0000167033$ $2.0722\times 10^{-6}$ $2$ $0.0525$ $-0.000239292$ $0.0000109368$ $5$ $0.035$ $-0.00176051$ $0.0000142852$ : $Z_{\rm MC}$ at the first negative peaks and their statistical errors for various $(N,k,\zeta_2 )$. []{data-label="negativeextrema"} Physical origins of the zeroes and Fermi gas formalism {#sec:origin} ------------------------------------------------------ In this subsection we discuss physical origins of the zeroes of the partition function. For this purpose, we apply Fermi gas formalism and identify which effects trigger the change of the sign of $Z$. Note that some techniques in the Fermi gas formalism are not available for $\zeta_1 \neq 0$ since the Hamiltonian is not hermitian. However, there is a technique which is still available. This is a formal $\hbar$-expansion of ${\rm Tr}{\widehat\rho}^n$ via Wigner transformation where ${\rm Tr}{\widehat \rho}^n$ is expressed as a phase space integral of a function whose explicit representation can be obtained by acting differential operators on $\rho (q,p)$. In this technique, it does not matter whether or not the Hamiltonian is hermitian since the problem is reduced to compute a perturbative series of the explicit two dimensional integral with respect to $\hbar$. Fortunately, this analysis has been already done in [@Nosaka] for imaginary $(\zeta_1 ,\zeta_2 )$ in the context of the R-charge deformation and hence we can obtain the $\hbar$-expansion simply by analytic continuation of the result in [@Nosaka] up to a subtlety discussed below.[^30] Once we find ${\rm Tr}{\widehat \rho}^n$ approximated in this way, one can compute the grand potential $J(\mu )$ by the following Mellin-Barnes expression J() =-\_[-i]{}\^[+i]{} (t) (-t) (t) e\^[t]{} (0&lt;&lt;1 ), where $\mathcal{Z}(t) = {\rm Tr}{\widehat \rho}^{t}$ and the canonical partition function can be obtained from $J(\mu )$ by Z(N) =d e\^[J() -N ]{} . The $\hbar$-expansion of $\mathcal{Z}(n)$ takes the form \(n) = \_[s=0]{}\^\^[2s-1]{} \_[2s]{}(n) +(e\^[-]{}) \[eq:small\_h\] where the second term denotes non-perturbative effects of the $\hbar$-expansion which we are ignoring. The work [@Nosaka] computed the first four coefficients $\mathcal{Z}_0$, $\mathcal{Z}_2$, $\mathcal{Z}_4$ and $\mathcal{Z}_6$ which are explicitly written down in app. A of [@Nosaka]. For example, the leading order coefficient $\mathcal{Z}_0$ is given by \_0 (n) = B B, where we are keeping $(\zeta_1 /k ,\zeta_2 /k)$ fixed and B(x,y) = . The large-$N$ behavior of $Z(N)$ can be easily derived by the large-$\mu$ behavior of $J(\mu )$ which has the following structure J() =J\^[pert]{}() + ( e\^[-]{} , e\^[-]{},e\^[-]{} ) +( e\^[-]{}) , \[eq:J\] where J\_[pert]{}() = \^3 +B(\_1 ,\_2 ,k) +A(\_1 ,\_2 ,k) . \[eq:large\_mu\] Several comments are in order. First, the $\hbar$-expansions for the coefficients $C$ and $B$ are terminated at leading and sub-leading orders respectively: && C= ,\ && B= -( + ) + . The coefficient $A$ receives all order corrections and it has been conjectured in [@Nosaka] that the exact answer for $A$ is given by A =, where [@KEK; @Hatsuda:2014vsa] $$\begin{aligned} A_\text{ABJM}(k) =\frac{2\zeta(3)}{\pi^2k}\Bigl(1-\frac{k^3}{16}\Bigr)+\frac{k^2}{\pi^2}\int_0^\infty dx\frac{x}{e^{kx}-1}\log(1-e^{-2x}) .\end{aligned}$$ If the approximation by $J_{\rm pert}(\mu )$ is reliable, then the canonical partition function is approximated by $$\begin{aligned} Z \simeq Z_{\rm pert} ,\quad Z_{\rm pert} =\int d\mu\ e^{J_{\rm pert}(\mu ) -\mu N } =e^A C^{-\frac{1}{3}}\operatorname{Ai}\left[ C^{-\frac{1}{3}}(N-B)\right] . \label{eq:Zpert}\end{aligned}$$ The large-$N$ limit of this formula exhibits the $N^{3/2}$-law:[^31] - = C\^[-1/2]{} N\^[3/2]{} +(N\^[1/2]{}) = N\^[3/2]{} +(N\^[1/2]{}) , \[eq:FlargeN\] which agrees with for $\zeta_1 =0$ and the result of [@NST2] for $\zeta_1 =\zeta_2 =\zeta$. The second term in is non-perturbative corrections of the large-$\mu$ expansion whose exponents can be explicitly derived by the $\hbar$-expansion . These corrections for the massless case have been identified with membrane instanton effects whose type IIA picture is D2-branes wrapping (warped) $\mathbb{RP}^3$ in $AdS_4 \times \mathbb{CP}^3$ [@Drukker:2011zy]. The third term in takes over the non-perturbative correction of the $\hbar$-expansion in whose exponents cannot be determined by the above arguments. It has been conjectured in [@Nosaka] that the exponent for imaginary $(\zeta_1 ,\zeta_2 )$ is given by $\mathcal{O}(e^{-\frac{4\mu}{k(1\pm 4i\zeta_1 /k)(1\pm 4i\zeta_2 /k)}} )$. These corrections for the massless case have been identified with worldsheet instanton effects coming from fundamental strings wrapping $\mathbb{CP}^1$ [@Cagnazzo:2009zh]. Let us estimate when we can trust the approximation by the perturbative part $J_{\rm pert}(\mu )$ in the large-$\mu$ expansion , or equivalently when the canonical partition function is well approximated by . We easily see that the second term in is exponentially suppressed for any $(\zeta_1 ,\zeta_2 )$ and therefore we can ignore the second term in the large-$N$ limit. Then let us focus on the third term which comes from non-perturbative effects of the $\hbar$-expansion. We have not estimated the exponent of the third term for real $(\zeta_1 ,\zeta_2 )$ precisely. However, the exponent for real $(\zeta_1 ,\zeta_2 )$ should be the same as the naive analytic continuation of the one for imaginary $(\zeta_1 ,\zeta_2 )$ in the domain where the partition function is holomorphic with respect to $(\zeta_1 ,\zeta_2 )$. Therefore, if we assume the above conjecture on the exponent for imaginary $(\zeta_1 ,\zeta_2 )$ in [@Nosaka], then we should have the following correction in for real $(\zeta_1 ,\zeta_2 )$: ( e\^[-]{} ) = ( e\^[- ]{} ) . \[eq:NPhbar\] Note that this correction is no longer exponentially suppressed for , and we cannot approximate the grand potential $J(\mu )$ by $J_{\rm pert}(\mu )$ in this region. This also implies that the holomorphy of the partition function with respect to $(\zeta_1 ,\zeta_2 )$ is broken at $\frac{16\zeta_1 \zeta_2}{k^2} = \frac{1}{4}$ because if we start with imaginary $(\zeta_1 ,\zeta_2 )$, then the naive analytic continuation to real $(\zeta_1 ,\zeta_2 )$ does not commute with the large-$N$ limit. Namely, if we take the large-$N$ limit first, then the free energy behaves as $\sim N^{3/2}$ and its continuation to real $(\zeta_1 ,\zeta_2 )$ is also described by the same formula for any $(\zeta_1 ,\zeta_2 )$ which is very likely different from the large-$N$ limit after the continuation in the domain $\frac{16\zeta_1 \zeta_2}{k^2}\geq \frac{1}{4}$. ![ The quantity $-\log{Z_{\rm MC}}$ with $Z_{\rm MC}(N,k,\zeta_1 ,\zeta_2 )$ $=$ $\frac{Z(N,k,\zeta_1,\zeta_2)}{Z(N,k,0,\zeta_2)}$ is plotted against $N^{3/2}$ for $\frac{\zeta_1 \zeta_2}{k^2} <\frac{1}{16}$. The symbols are the numerical results obtained by the Monte Carlo simulation. The red line denotes the result computed by the Airy function formula , namely $-\log{\frac{Z_{\rm pert}(N,k,\zeta_1,\zeta_2)}{Z_{\rm pert}(N,k,0,\zeta_2)} }$. \[fig:VS\_Airy\] ](Airy_k1zeta0o005and1.eps "fig:"){width="7.8cm"} ![ The quantity $-\log{Z_{\rm MC}}$ with $Z_{\rm MC}(N,k,\zeta_1 ,\zeta_2 )$ $=$ $\frac{Z(N,k,\zeta_1,\zeta_2)}{Z(N,k,0,\zeta_2)}$ is plotted against $N^{3/2}$ for $\frac{\zeta_1 \zeta_2}{k^2} <\frac{1}{16}$. The symbols are the numerical results obtained by the Monte Carlo simulation. The red line denotes the result computed by the Airy function formula , namely $-\log{\frac{Z_{\rm pert}(N,k,\zeta_1,\zeta_2)}{Z_{\rm pert}(N,k,0,\zeta_2)} }$. \[fig:VS\_Airy\] ](Airy_k1zeta0o05and1.eps "fig:"){width="7.8cm"}\ ![ The quantity $-\log{Z_{\rm MC}}$ with $Z_{\rm MC}(N,k,\zeta_1 ,\zeta_2 )$ $=$ $\frac{Z(N,k,\zeta_1,\zeta_2)}{Z(N,k,0,\zeta_2)}$ is plotted against $N^{3/2}$ for $\frac{\zeta_1 \zeta_2}{k^2} <\frac{1}{16}$. The symbols are the numerical results obtained by the Monte Carlo simulation. The red line denotes the result computed by the Airy function formula , namely $-\log{\frac{Z_{\rm pert}(N,k,\zeta_1,\zeta_2)}{Z_{\rm pert}(N,k,0,\zeta_2)} }$. \[fig:VS\_Airy\] ](Airy_k2zeta0o02and1.eps "fig:"){width="7.8cm"} ![ The quantity $-\log{Z_{\rm MC}}$ with $Z_{\rm MC}(N,k,\zeta_1 ,\zeta_2 )$ $=$ $\frac{Z(N,k,\zeta_1,\zeta_2)}{Z(N,k,0,\zeta_2)}$ is plotted against $N^{3/2}$ for $\frac{\zeta_1 \zeta_2}{k^2} <\frac{1}{16}$. The symbols are the numerical results obtained by the Monte Carlo simulation. The red line denotes the result computed by the Airy function formula , namely $-\log{\frac{Z_{\rm pert}(N,k,\zeta_1,\zeta_2)}{Z_{\rm pert}(N,k,0,\zeta_2)} }$. \[fig:VS\_Airy\] ](Airy_k3zeta0o045and2.eps "fig:"){width="7.8cm"} The above estimate is consistent with our numerical results obtained in sec. \[sec:MC\]. In fig. \[fig:VS\_Airy\] we compare the ratio $Z_{\rm MC}(N,k,\zeta_1 ,\zeta_2 )$ $=$ $\frac{Z(N,k,\zeta_1,\zeta_2)}{Z(N,k,0,\zeta_2)}$ obtained by the Monte Carlo simulation with the one computed by the approximation for some cases with $\frac{\zeta_1 \zeta_2}{k^2} <\frac{1}{16}$ where we expect to be good approximation. The plots show that our numerical results agree with the Airy function formula and exhibit the $N^{3/2}$-law. Although we explicitly present only the four cases, we have observed similar behaviors for various other values of $(k,\zeta_1 ,\zeta_2 )$ satisfying $\frac{\zeta_1 \zeta_2}{k^2} <\frac{1}{16}$. Figure \[fig:VS\_Airy2\] shows similar plots for $\frac{\zeta_1 \zeta_2}{k^2} \geq \frac{1}{16}$ where we expect that we cannot trust due to the correction . In contrast to fig. \[fig:VS\_Airy\], we easily see that the numerical results do not agree with and no longer exhibit the $N^{3/2}$-law. We have also found similar behaviors for various other values of $(k,\zeta_1 ,\zeta_2 )$ with $\frac{\zeta_1 \zeta_2}{k^2} \geq \frac{1}{16}$. Thus our numerical results support our expectation that the approximation by is valid for $\frac{\zeta_1 \zeta_2}{k^2} < \frac{1}{16}$. ![ Similar plots to fig. \[fig:VS\_Airy\] for $\frac{\zeta_1 \zeta_2}{k^2} \geq \frac{1}{16}$. \[fig:VS\_Airy2\] ](Airy_k1zeta0o2and1.eps "fig:"){width="7.8cm"} ![ Similar plots to fig. \[fig:VS\_Airy\] for $\frac{\zeta_1 \zeta_2}{k^2} \geq \frac{1}{16}$. \[fig:VS\_Airy2\] ](Airy_k2zeta0o6and1.eps "fig:"){width="7.8cm"} Conjecture on phase structure in the large-$N$ limit {#conjlargeN} ---------------------------------------------------- We discuss the phase structure of the mass deformed ABJM theory in the large-$N$ limit. Let us first recall the results obtained so far by the various analyzes: - In the case of $\zeta_1=\zeta_2=\zeta$, the partition function in the representation has been analyzed by the saddle point method in [@NST2] as reviewed in sec. \[susybreaking\]. The saddle point configuration in [@NST2] realizes the $\mathcal{O}(N^{3/2})$ free energy and becomes singular at $\zeta/k=1/4$. - In sec. \[saddle\], we have constructed the saddle point solution for $\zeta_1 =0$ in the $S$-dual representation, which gives the $\mathcal{O}(N^{3/2})$ free energy . This behavior is consistent with the exact results for finite $N$ obtained in sec. \[TWPY\]. Note also that we can obtain the result for $\zeta_1 \neq 0$, $\zeta_2 =0$ by the replacement $\zeta_2 \rightarrow \zeta_1$ in since the partition function is symmetric under $\zeta_1 \leftrightarrow \zeta_2$. - In sec. \[sec:exactN2\], we have written down the exact results for $N=1,2$ and arbitrary $(k,\zeta_1 ,\zeta_2)$ obtained in [@RS]. It has turned out that the partition function for $N=2$ has the zeroes at finite $(\zeta_1 ,\zeta_2)$ while the one for $N=1$ does not. - In sec. \[sec:MC\], we have performed the Monte Carlo simulation for higher $N$. We have observed that the partition function has the zeroes at finite $(\zeta_1 ,\zeta_2 )$ given $(N,k)$. The bounds and estimates on the zeroes given in tables \[tab:zeta1\], \[tab:zeta2\] and \[tab:zeta5\], imply that the first zeroes do not increase by $N$. It is natural to expect that the partition function becomes zero at some finite $(\zeta_1, \zeta_2 )$ also in the large-$N$ limit. - In sec. \[sec:origin\], we have argued when one can trust the approximation in terms of the perturbative grand potential in the Fermi gas formalism, which gives the $\mathcal{O}(N^{3/2})$ free energy in the large-$N$ limit. We have found that the approximation is reliable for $\frac{\zeta_1 \zeta_2}{k^2}<\frac{1}{16}$ while in the other regime $\frac{\zeta_1 \zeta_2}{k^2}\geq \frac{1}{16}$, the expected non-perturbative effects of the $\hbar$-expansion are no longer exponentially suppressed. Note that for $\zeta_1=\zeta_2=\zeta$, the approximation starts to be invalid at $\zeta /k =1/4$, which is the same as the condition that the saddle point in [@NST2] becomes singular. Based on the above results, we propose the following scenario (see fig. \[fig:proposal\] for schematic picture): - For small $(\zeta_1,\zeta_2 )$, the large-$N$ free energy behaves as $F\sim N^{3/2}$ whose explicit form is given by . We expect that this formula is valid for $\frac{\zeta_1 \zeta_2}{k^2}<\frac{1}{16}$, and becomes invalid for $\frac{\zeta_1 \zeta_2}{k^2}\geq \frac{1}{16}$. - The partition function vanishes at some finite values of $(\zeta_1,\zeta_2 )$. We expect that this occurs at the boundary of the validity of : $\frac{\zeta_1 \zeta_2}{k^2}= \frac{1}{16}$. We interpret this as the SUSY breaking at this point. ![ Fitting of the estimated zeroes for $(k,\zeta_2)=(1,2)$ given in table \[tab:zeta2\] by the function $a(k,\zeta_2 )+\frac{b(k,\zeta_2 )}{\sqrt{N}}+\frac{c(k,\zeta_2 )}{N}$. The three thick lines denote the fitting functions of the estimates from the interpolating functions of $Z_{\rm MC}$ and $Z_{\rm MC}$ plus/minus the errors. The black dashed line denotes our expectation on the zero in the large-$N$ limit: $\zeta_1 =\frac{k^2}{16 \zeta_2}$, which is $\frac{1}{32}=0.03125$ for $(k,\zeta_2)=(1,2)$. The result of the fitting is $a(k=1,\zeta_2 =2 ) = 0.0309507^{+0.0003777}_{-0.0003645}$. []{data-label="fig:fit"}](fit_zeta2.eps){width="8.2cm"} We already have strong evidence of the first point by the saddle point analysis for $\zeta_1 =\zeta_2$ in [@NST2] and Fermi gas analysis in sec. \[sec:origin\]. Now we provide further evidence for the second point. From tables \[tab:zeta1\], \[tab:zeta2\] and \[tab:zeta5\], we observe that the locations of the first zeroes decrease slowly as $N$ increases. Therefore it is plausible that the first zeroes in the large-$N$ limit are at some finite values of $(\zeta_1, \zeta_2 )$. It would be nontrivial whether or not the first zeroes in the large-$N$ limit coincide with our expectation $\frac{\zeta_1 \zeta_2}{k^2}=\frac{1}{16}$. We perform consistency checks of this by fitting analysis of our numerical data. Note that the fitting analysis in the current situation is subtle in the following two reasons. First we do not know asymptotic behaviors of the first zeroes for large-$N$. In other words, we do not know what appropriate fitting functions are a priori. Second, we do not have sufficient data of the first zeroes since it is only for the five values of $N$ ($N=2,3,5,7,9$). Nevertheless the fitting analysis provides quite nontrivial consistency checks as we will see soon. We have constructed fitting functions for the first zeroes $\zeta_1 (N, k,\zeta_2 )$ with fixed $(k,\zeta_2 )$ and varied $N$. As a conclusion, we have found that when we find a fitting function nicely interpolating numerical data, asymptotic value of the fitting function at $N\rightarrow\infty$ agrees with our expectation $\zeta_1 =\frac{k^2}{16\zeta_2}$. ![ Similar plots to fig. \[fig:fit\] for $(k,\zeta_2)=(1,1)$ and $(1,5)$. The intercepts of the fitting functions are $a(k=1,\zeta_2 =1 )$ $=$ $0.0620729^{+0.0014133}_{-0.0011213}$ and $a(k=1,\zeta_2 =5 )$ $=$ $0.0194694^{+0.0000579}_{-0.0000551}$. []{data-label="fig:fit2"}](fit_zeta1_1.eps "fig:"){width="7.0cm"} ![ Similar plots to fig. \[fig:fit\] for $(k,\zeta_2)=(1,1)$ and $(1,5)$. The intercepts of the fitting functions are $a(k=1,\zeta_2 =1 )$ $=$ $0.0620729^{+0.0014133}_{-0.0011213}$ and $a(k=1,\zeta_2 =5 )$ $=$ $0.0194694^{+0.0000579}_{-0.0000551}$. []{data-label="fig:fit2"}](fit_zeta5_1.eps "fig:"){width="7.0cm"} In fig. \[fig:fit\], we plot the estimated zeroes for $(k,\zeta_2 )=(1,2)$ given in tab. \[tab:zeta2\] against $1/\sqrt{N}$. We construct a fitting function for this data by the ansatz a(k,\_2 )++ , \[eq\_fit1\] where $a(k,\zeta_2 )$ corresponds to the zero in the large-$N$ limit with respect to $\zeta_1$. We easily see that the fitting function nicely interpolates the data points. Therefore it is natural to compare the asymptotic value of the fitting function at large-$N$ with our expectation. As a result, we have found a(k=1,\_2 =2 ) = 0.0309507\^[+0.0003777]{}\_[-0.0003645]{} , which includes our expectation on the zero at large-$N$: $\zeta_1$ $=$ $\frac{k^2}{16 \zeta_2}|_{(k,\zeta_2)=(1,2)}$ $=$ $0.03125$. This strongly supports our expectation on the large-$N$ phase structure. Fig. \[fig:fit2\] shows results by the same fitting function for the other values of $\zeta_2$. It is clear that the fitting functions for $(k,\zeta_2 )=(1,1)$ and $(1,5)$ do not nicely interpolate the data as much as for the case of $(k,\zeta_2 )=(1,2)$ although the fitting for $(k,\zeta_2 )=(1,5)$ is better than the one for $(k,\zeta_2 )=(1,1)$. From the fitting functions, we have found $a(k=1,\zeta_2 =1 )$ $=$ $0.0577778^{+0.0038557}_{-0.0033802}$ and $a(k=1,\zeta_2 =5 )$ $=$ $0.0154572^{+0.0001219}_{-0.000114526}$ which do not include our expectation $\zeta_1 =\frac{k^2}{16 \zeta_2}$ although the result for $(k,\zeta_2 )=(1,5)$ is not far from the expectation. We interpret that this does not mean invalidity of our expectation since the fitting ansatz does not exhibit very nice interpolations for $(k,\zeta_2 )=(1,1)$ and $(1,5)$, and we need another fitting functions or more data. In fig. \[fig:fit3\] we also present similar plots to fig. \[fig:fit2\] by the different fitting function $a(k,\zeta_2 )+\frac{b(k,\zeta_2 )}{\sqrt{N}}$ for $(k,\zeta_2 )=(1,1)$ and $(1,5)$. Again the fitting functions do not interpolate the data very nicely and have different intercepts from the ones obtained by the ansatz although the intercept for $(k,\zeta_2 )=(1,1)$ includes our expectation: $a(k=1,\zeta_2 =1 )$ $=$ $0.0577778^{+0.0038557}_{-0.0033802}$. To summarize we need to find more appropriate fitting functions or data points for larger $N$ in order to further check our expectation except for $(k,\zeta_2 )=(1,2)$. We leave this for future work. ![ Similar analysis to fig. \[fig:fit2\] by using the different fitting function $a(k,\zeta_2 )+\frac{b(k,\zeta_2 )}{\sqrt{N}}$. The values of the intercepts are $a(k=1,\zeta_2 =1 )$ $=$ $0.0577778^{+0.0038557}_{-0.0033802}$ and $a(k=1,\zeta_2 =5 )$ $=$ $0.0154572^{+0.0001219}_{-0.000114526}$. []{data-label="fig:fit3"}](fit_zeta1_2.eps "fig:"){width="7.0cm"} ![ Similar analysis to fig. \[fig:fit2\] by using the different fitting function $a(k,\zeta_2 )+\frac{b(k,\zeta_2 )}{\sqrt{N}}$. The values of the intercepts are $a(k=1,\zeta_2 =1 )$ $=$ $0.0577778^{+0.0038557}_{-0.0033802}$ and $a(k=1,\zeta_2 =5 )$ $=$ $0.0154572^{+0.0001219}_{-0.000114526}$. []{data-label="fig:fit3"}](fit_zeta5_2.eps "fig:"){width="7.0cm"} Moreover, the correlation between the supersymmetry breaking and the singularity in the saddle point approximation was argued for the pure Chern-Simons theory [@MoNi]. Hence it would be more than just a minimal scenario for our theory to relate the singularity in the saddle point approximation with the supersymmetry breaking. It would be interesting to test this conjecture by studying the partition function for larger $N$ in future. Discussion {#discuss} ========== In this paper we have studied the mass deformed ABJM theory on the three sphere. Based on the argument in sec. \[susybreaking\], we expect that this theory exhibits a spontaneous supersymmetry breaking in large-$N$ limit at $\zeta_1=\zeta_2=k/4$. To gain an evidence for this conjecture we have analyzed the partition function of the mass deformed ABJM theory for finite $k$ and $N$ by using the Monte Carlo simulation. As a result we have found that the partition function vanishes at some finite values of $\zeta_1,\zeta_2$. The numerical results also indicate that the zeroes exist for general $N$, and that the locus of the first zero stays finite as $N$ increases. These observations are consistent with the expectation in the end of sec. \[susybreaking\] from the large-$N$ supersymmetry breaking. Our result would shed new light to the phase structure of the mass deformed ABJM theory in the M-theory limit, which was unclear in the previous works [@NST; @NST2]. Precisely speaking, the correct physical interpretation for the zeroes of the partition function for finite $N$ is not clear, since a spontaneous symmetry breaking can happens only in the limit of large degree of freedom. To test our conjecture it is important to study the partition function in the large-$N$ limit. One possible method would be the saddle point approximation. In the previous work we found the saddle point solution only for the special case $\zeta_1=\zeta_2$. In this paper we found a solution for a new slice $\zeta_1=0$ by rewriting the matrix model into the S-dual representation. Another direction is to improve the algorithm of the numerical simulation. In this paper, we have treated the oscillation factor of in the quite naive way where we just regard the factor as the observable in the system with $\zeta_1 =0$. In this approach, we need much more statistics than simulations without oscillating factors so that the simulation at large-$N$ becomes harder. It is nice if one can find more appropriate algorithm such as complex Langevin method and Lefschetz thimble. Lastly, it would be interesting to compare the exact partition functions for $\zeta_1=0$ with those in [@NY1]. In that paper they computed the partition function of the $\text{U}(N)_k\times \text{U}(N+M)_{-k}$ linear quiver superconformal Chern-Simons theory. Naively, for $M=0$ this theory can be obtained by taking the decoupling limit $\zeta_2\rightarrow\infty$ in the mass deformed ABJM theory. Indeed we observe for $N<k/2$ that, if we take the limit $\zeta_2\rightarrow \infty$ in the exact expressions for the partition function $Z(N,k,0,\zeta_2)$ they precisely coincide with those in [@NY1] up to the contribution of decoupled hypermultiplet $e^{-2\pi N^2\zeta_2/k}$ (see app. \[comarewithNY\]). On the other hand we also observe that for some cases with $N\ge k/2$ the decay is slower than $e^{-2\pi N^2\zeta_2/k}$. Actually in these cases the corresponding linear quiver theory is a bad theory [@Nosaka:2018eip], hence it should not be the right decoupling limit of the mass deformed ABJM theory. Though it is still not clear, the correct description of the decoupling limit might be obtained by expanding the Coulomb branch moduli around a configuration which depends on $\zeta_2$ in a non-trivial way. Acknowledgement {#acknowledgement .unnumbered} --------------- We would like to thank Sungjay Lee, Sanefumi Moriyama and Shuichi Yokoyama for valuable discussions. T. N.  would appreciate Jin-beom Bae, Joonho Kim, Takaya Miyamoto and Dario Rosa for several pieces of advice on numerical simulation. The numerical analysis were performed on the ATOM server which is supported by Korea Institute for Advanced Study. The work of S. T.  was supported by JSPS KAKENHI Grant Number 17K05414. K. S.  is supported by JSPS fellowship. K. S.  is also supported by Grant-in-Aid for JSPS Fellow No. 18J11714. Partition function in S-dual representation {#Sdualsection} =========================================== In this appendix we derive the S-dual representation of the partition function, which we practically use in the main text. The computation is essentially the same as those in the Fermi gas formalism for the ABJM theory [@MP]. Let us start with the integral . Changing the integration variables as $\lambda\rightarrow \lambda /k+\pi (\zeta_1 +\zeta_2 )$ and ${\widetilde\lambda}\rightarrow{\widetilde\lambda}/k -\pi (\zeta_1 +\zeta_2 )$, we find $$\begin{aligned} Z=\frac{1}{(N!)^2}\int\frac{d^N\lambda}{(2\pi k)^N}\frac{d^N{\widetilde\lambda}}{(2\pi k)^N}e^{\frac{i}{4\pi k}\sum_i(\lambda_i^2-{\widetilde\lambda}_i^2)+\frac{i\zeta}{k}\sum_i(\lambda_i+{\widetilde\lambda}_i)} \frac{ \prod_{i<j}^N(2\sinh\frac{\lambda_i-\lambda_j}{2k})^2 \prod_{i<j}^N(2\sinh\frac{{\widetilde\lambda}_i-{\widetilde\lambda}_j}{2k})^2 }{ \prod_{i,j=1}^N \prod_\pm 2\cosh\frac{\lambda_i-{\widetilde\lambda}_j\pm \mu}{2k} }.\end{aligned}$$ where $$\begin{aligned} \zeta=\frac{\zeta_1+\zeta_2}{2},\quad \mu=2\pi(\zeta_1-\zeta_2).\end{aligned}$$ Next we rewrite the 1-loop determinant into pair of determinants of $N\times N$ matrices by using the Cauchy determinant formula $$\begin{aligned} \frac{ \prod_{i<j}^N2\sinh\frac{x_i-x_j}{2} \prod_{i<j}^N2\sinh\frac{y_i-y_j}{2} }{ \prod_{i,j=1}^N2\cosh\frac{x_i-y_j}{2} } =\det_{i,j}\frac{1}{2\cosh\frac{x_i-y_j}{2}}, \label{Cauchy}\end{aligned}$$ and then combine them by using the formula $$\begin{aligned} \frac{1}{N!}\int d^Nx \det_{i,j}f_i(x_j) \det_{i,j}g_i(x_j)=\det_{i,j} \int dxf_i(x)g_j(x). \label{MM}\end{aligned}$$ After these manipulations we obtain the following expression for the partition function $$\begin{aligned} Z=\frac{1}{N!}\int\frac{d^N\lambda}{(2\pi)^N}\det_{i,j}f(\lambda_i,\lambda_j),\end{aligned}$$ where $$\begin{aligned} f(x,y)=\int\frac{dz}{2\pi}e^{\frac{i}{4\pi k}x^2+\frac{i\zeta}{k}x}\frac{1}{2k\cosh\frac{x-z-\mu}{2k}}e^{-\frac{i}{4\pi k}z^2+\frac{i\zeta z}{k}}\frac{1}{2k\cosh\frac{z-y-\mu}{2k}}.\end{aligned}$$ Hence the partition function takes the form of the partition function of 1d $N$ particle non-interacting Fermi gas if we regard $f(\lambda',\lambda'')$ as the matrix element of a one-particle density matrix $\langle \lambda|{\widehat\rho}|\lambda'\rangle$ with position eigenstates $|\cdot \rangle$ $$\begin{aligned} Z=\frac{1}{N!}\int\frac{d^Nx}{(2\pi)^N}\det_{i,j}\langle x_i|{\widehat\rho}|x_j\rangle, \label{Fermigas}\end{aligned}$$ where[^32] $$\begin{aligned} {\widehat\rho}=e^{\frac{i}{4\pi k}{\widehat q}^2+\frac{i\zeta}{k}{\widehat q}}\frac{e^{\frac{i\mu}{2\pi k}{\widehat p}}}{2\cosh\frac{\widehat p}{2}}e^{-\frac{i}{4\pi k}{\widehat q}^2+\frac{i\zeta}{k}{\widehat q}}\frac{e^{\frac{i\mu}{2\pi k}{\widehat p}}}{2\cosh\frac{\widehat p}{2}}.\end{aligned}$$ Now it is obvious from that the partition function is invariant under any similarity transformation of the density matrix ${\widehat\rho}$. We can simplify ${\widehat\rho}$ by the similarity transformation ${\widehat\rho}\rightarrow e^{-\frac{i}{4\pi k}{\widehat p}^2-\frac{i\zeta}{k}{\widehat p}}{\widehat\rho}e^{\frac{i}{4\pi k}{\widehat p}^2+\frac{i\zeta}{k}{\widehat p}}$ as $$\begin{aligned} {\widehat\rho}=\frac{e^{\frac{2i\zeta_1}{k}{\widehat q}}}{2\cosh\frac{\widehat q}{2}} \frac{e^{\frac{2i\zeta_2}{k}{\widehat p}}}{2\cosh\frac{\widehat p}{2}}, \label{rhosimplified}\end{aligned}$$ whose matrix element is $$\begin{aligned} \langle x|{\widehat\rho}|y\rangle=\frac{e^{\frac{2i\zeta_1}{k}x}}{2\cosh\frac{x}{2}}\frac{1}{2k\cosh\frac{x-y+4\pi \zeta_2}{2k}}. \label{rhosimplifiedME}\end{aligned}$$ Applying the Cauchy determinant formula reversely to $\det_{i,j}\langle x_i|{\widehat\rho}|x_j\rangle$ in the Fermi gas formalism with this new ${\widehat\rho}$, we finally obtain the S-dual representation for the partition function . Technical details on exact computation of $Z(N,k,0,\zeta_2)$ {#app:detail_exact} ============================================================ In this appendix we explain details on how to solve the recursion relation for integer $k$. ### {#section .unnumbered} If $k\in 2\mathbb{N}$, we can introduce a new variable $u=e^{\frac{x}{k}}$ to rewrite the integration as $$\begin{aligned} \phi_{\ell+1}(u)=\frac{1}{2\pi}\int_0^\infty dv\frac{v^{\frac{k}{2}}}{(v+\alpha^2u)(v^k+1)}\phi_\ell(v). \label{int_rec_phitophi}\end{aligned}$$ If we assume that $\phi_\ell(u)$ can be expanded as the following finite series (inductively correct) $$\begin{aligned} \phi_\ell(u)=\sum_{j\ge 0}\phi_\ell^{(j)}(u) (\log u)^j,\quad\quad (\phi_\ell^{(j)}(u)\text{ are some rational functions of }u)\end{aligned}$$ we can compute the integration as [@PY] $$\begin{aligned} \phi_{\ell+1}(u)&=\frac{1}{2\pi}\sum_{j\ge 0}\Bigl[-\frac{(2\pi i)^j}{j+1}\int_\gamma\frac{v^{\frac{k}{2}}}{(v+\alpha^2u)(v^k+1)}\phi_\ell^{(j)}(v)B_{j+1}\Bigl(\frac{\log^{(+)}v}{2\pi i}\Bigr)\Bigr]\nonumber \\ &=\frac{1}{2\pi}\sum_{j\ge 0}\biggl[-\frac{(2\pi i)^{j+1}}{j+1}\sum_{w\in poles} \operatorname{Res}\Bigl[\frac{v^{\frac{k}{2}}}{(v+\alpha^2u)(v^k+1)}\phi_\ell^{(j)}(v)B_{j+1}\Bigl(\frac{\log^{(+)}v}{2\pi i}\Bigr),v\rightarrow w\Bigr]\biggr]. \label{TWPYRes}\end{aligned}$$ Here $\log^{(+)}$ is logarithm function with the branch cat located on $\mathbb{R}^+$ and the integration contour $\gamma$ is as depicted in figure \[TWPYpath\]. ![The integration contour $\gamma$ in (blue) and the deformed contour to use the Cauchy theorem (green). The cut of $\log^{(+)}$ is depicted by wavy red line.[]{data-label="TWPYpath"}](twpypath_v2.eps){width="6cm"} The poles to be collected in the step $\phi_\ell\rightarrow \phi_{\ell+1}$ are at most $$\begin{aligned} v&=-\alpha^2 u,\nonumber \\ v&=\alpha^{-2a}e^{\frac{\pi i(2b+1)}{k}},\quad (a=0,1,\cdots,\ell;\,\,b=0,1,\cdots,k-1),\end{aligned}$$ which can be seen from the same argument as in [@Nosaka]. After obtaining $\phi_\ell$ for $\ell=0,1,\cdots,n-1$, we can compute $\operatorname{Tr}{\widehat \rho}^n$ by $$\begin{aligned} \operatorname{Tr}{\widehat \rho}^n&=\frac{1}{2\pi(\alpha^n-(-1)^n\alpha^{-n})}\int_0^\infty du\frac{u^{\frac{k}{2}-1}}{u^k+1}\Psi_n(u)\quad\quad\Bigl(\Psi_n(u)=\sum_{\ell=0}^{n-1}(-1)^\ell\phi_\ell(u)\psi_{n-1-\ell}(u)\Bigr)\nonumber \\ &=\frac{1}{2\pi(\alpha^n-(-1)^n\alpha^{-n})}\sum_{j\ge 0}\biggl[-\frac{(2\pi i)^{j+1}}{j+1}\sum_{w\in poles}\operatorname{Res}\Bigl[\frac{u^{\frac{k}{2}-1}}{u^k+1}\Psi_n^{(j)}(u)B_{j+1}\Bigl(\frac{\log^{(+)}u}{2\pi i}\Bigr),u\rightarrow w\Bigr]\biggr],\end{aligned}$$ where $\Psi_n=\sum_{j\ge 0}\Psi_n^{(j)}(u)(\log u)^j$ and $poles$ are (at most) $$\begin{aligned} u=\alpha^{-2a} e^{\frac{\pi i(2b+1)}{k}}.\quad (a=-(n-1),-(n-2),\cdots,n-1;\,\, b=0,1,\cdots,k-1)\end{aligned}$$ ### {#section-1 .unnumbered} For odd $k$, we define $u=e^{\frac{x}{2k}}$ to obtain the following formulas $$\begin{aligned} \phi_{\ell+1}(u)&=\frac{1}{\pi}\sum_{j\ge 0}\biggl[-\frac{(2\pi i)^{j+1}}{j+1}\sum_{v\in poles}\operatorname{Res}\Bigl[\frac{1}{v^2+\alpha^2 u^2}\frac{v^{k+1}}{v^{2k}+1}\phi_\ell^{(j)}(v)B_{j+1}\Bigl(\frac{\log^{(+)}v}{2\pi i}\Bigr),v\rightarrow w\Bigr]\biggr],\end{aligned}$$ where $\phi_\ell^{(j)}(u)$ are the rational functions given by $\phi_\ell(u)=\sum_{j\ge 0}(\log u)^j\phi_\ell^{(j)}(u)$ and the poles to be collected are $$\begin{aligned} v&=\pm i\alpha u,\nonumber \\ v&=\alpha^{-2a} e^{\frac{\pi i(2b+1)}{2k}},\quad \Bigl(a=0,1,\cdots,\Bigl[\frac{\ell}{2}\Bigr];\,\,b=0,1,\cdots,2k-1\Bigr)\nonumber \\ v&=\alpha^{-(2a+1)}e^{\frac{\pi ib}{k}}.\quad \Bigl(a=0,1,\cdots,\Bigl[\frac{\ell-1}{2}\Bigr];\,\,b=0,1,\cdots,2k-1\Bigr)\end{aligned}$$ The traces of ${\widehat \rho}^n$ can be computed as $$\begin{aligned} \operatorname{Tr}{\widehat \rho}^n&=\frac{1}{\pi(\alpha^n-(-1)^n\alpha^{-n})}\sum_{j\ge 0}\biggl[-\frac{(2\pi i)^{j+1}}{j+1}\sum_{w\in poles}\operatorname{Res}\Bigl[\frac{u^{k-1}}{u^{2k}+1}\Psi_n^{(j)}(u)B_{j+1}\Bigl(\frac{\log^{(+)} v}{2\pi i}\Bigr),u\rightarrow w\Bigr]\biggr]\end{aligned}$$ where $\psi(u)=\sum_{\ell=0}^{n-1}(-1)^\ell \phi_\ell(u)\psi_{n-1-\ell}(u)=\sum_{j\ge 0}(\log u)^j\Psi_\ell^{(j)}(u)$ and the poles are $$\begin{aligned} u&=\alpha^{2a} e^{\frac{\pi i(2b+1)}{2k}},\quad \Bigl(a=-\Bigl[\frac{n-1}{2}\Bigr],-\Bigl[\frac{n-1}{2}\Bigr]+1,\cdots,\Bigl[\frac{n-1}{2}\Bigr];\,\,b=0,1,\cdots, 2k-1\Bigr),\nonumber \\ u&=\alpha^{\pm(2a+1)} e^{\frac{\pi ib}{k}}.\quad \Bigl(a=0,1,\cdots,\Bigl[\frac{n-2}{2}\Bigr];\,\,b=0,1,\cdots, 2k-1\Bigr)\nonumber \\\end{aligned}$$ Exact expressions for $Z(N,k,0,\zeta_2)$ {#Z1exact_results} ======================================== The technique introduced in sec. \[TWPY\] allows us to compute the partition function of the mass deformed ABJM theory $Z(N,k,0,\zeta_2)$ with $\zeta_1=0$ and for small integers $N,k$. We have computed $Z(N,k,0,\zeta_2)$ for $(k=1 ,N\le 12)$, $(k=2,N\le 9 )$, $(k=3 ,N\le 5)$, $(k=4 ,N\le 5 )$ and $(k=6 ,N\le 4 )$. Here we display the first few results ($\alpha=e^{2\pi\zeta_2/k}$). $$\begin{aligned} Z(1,1,0,\zeta_2)&= \frac{\alpha}{2(1+\alpha^2)},\quad Z(2,1,0,\zeta_2)= -\frac{\zeta_2\alpha^3}{(1+\alpha^2)(1-\alpha^4)},\nonumber \\ Z(3,1,0,\zeta_2)&= \frac{\alpha^5(1+12 \zeta_2 \alpha-\alpha^2)}{8(1+\alpha^2)(1-\alpha^4)(1+\alpha^6)},\quad \cdots \label{exactk1}\end{aligned}$$ $$\begin{aligned} Z(1,2,0,\zeta_2)&= \frac{\alpha}{4(1+\alpha^2)},\quad Z(2,2,0,\zeta_2)= \frac{\zeta_2^2\alpha^4} {(1-\alpha^4)^2},\nonumber \\ Z(3,2,0,\zeta_2)&= \frac{\alpha^7(-1+4 \zeta_2^2+(2+32 \zeta_2^2) \alpha^2+(-1+4 \zeta_2^2) \alpha^4)}{32(1+\alpha^4)(1-\alpha^4)^2(1+\alpha^6)},\quad \cdots \label{exactk2}\end{aligned}$$ $$\begin{aligned} Z(1,3,0,\zeta_2)&=\frac{\alpha}{6(1+\alpha^2)},\nonumber \\ Z(2,3,0,\zeta_2)&=\frac{\alpha^4(1+(1+4\zeta_2)\alpha+4\zeta_2\alpha^2+(-1+4\zeta_2)\alpha^7\alpha^3-\alpha^4)}{12(1+\alpha)(1+\alpha^2)(1-\alpha^3)(1+\alpha^6)},\nonumber \\ Z(3,3,0,\zeta_2)&=-\frac{\alpha^8}{18\sqrt{3}(1+\alpha^6)^3(1-\alpha^6)} (2+\sqrt{3}\zeta_2-3\sqrt{3}\alpha+(4-2\sqrt{3}\zeta_2)\alpha^2+3\sqrt{3}\zeta_2\alpha^4\nonumber \\ &\quad +(-4-2\sqrt{3}\zeta_2)\alpha^6+3\sqrt{3}\alpha^7+(-2+\sqrt{3}\zeta_2)\alpha^8) ,\quad \cdots \label{exactk3}\end{aligned}$$ $$\begin{aligned} Z(1,4,0,\zeta_2)&=\frac{\alpha}{8(1+\alpha^2)},\quad Z(2,4,0,\zeta_2)=\frac{\alpha^4(1+(-2-8\zeta_2^2)\alpha^2+\alpha^4)}{64(1+\alpha^4)(1-\alpha^4)^2},\nonumber \\ Z(3,4,0,\zeta_2)&= \frac{\alpha^9}{256(1 - \alpha^2)^2 (1 + \alpha^2)^2 (1 + \alpha^4)^2(1 + \alpha^6) (1 + \alpha^8)} (5 + 8 \zeta_2 + 4 \zeta_2^2 + (-7 + 8 \zeta_2) \alpha^2\nonumber \\ &\quad + (5 - 8 \zeta_2 + 4 \zeta_2^2) \alpha^4 + (-6 - 32 \zeta_2^2) \alpha^6 + (5 + 8 \zeta_2 + 4 \zeta_2^2) \alpha^8 + (-7 - 8 \zeta_2) \alpha^{10}\nonumber \\ &\quad + (5 - 8 \zeta_2 + 4 \zeta_2^2) \alpha^{12}),\quad \cdots \label{exactk4}\end{aligned}$$ $$\begin{aligned} Z(1,6,0,\zeta_2)&=\frac{\alpha}{12(1+\alpha^2)},\quad Z(2,6,0,\zeta_2)=\frac{\alpha^4(1-9\alpha^2+8(2+3\zeta_2^2)\alpha^4-9\alpha^6+\alpha^8)}{432(1-\alpha^4)(1-\alpha^{12})},\nonumber \\ Z(3,6,0,\zeta_2)&=\frac{\alpha^9}{5184 (1 + \alpha^2)^2 (-1 + \alpha^6)^2 (1 + \alpha^6) (1 + \alpha^{12})} (1 + (-54 - 32 \sqrt{3} \zeta_2 - 24 \zeta_2^2) \alpha^2\nonumber \\ &\quad + (-15 - 64 \sqrt{3} \zeta_2) \alpha^4 + (30 - 32 \sqrt{3} \zeta_2 + 96 \zeta_2^2) \alpha^6 + (76 + 192 \zeta_2^2) \alpha^8\nonumber \\ &\quad + (30 + 32 \sqrt{3} \zeta_2 + 96 \zeta_2^2) \alpha^{10} + (-15 + 64 \sqrt{3} \zeta_2) \alpha^{12} + (-54 + 32 \sqrt{3} \zeta_2 - 24 \zeta_2^2) \alpha^{14}\nonumber \\ &\quad + \alpha^{16}),\quad\cdots. \label{exactk6}\end{aligned}$$ Comparison with saddle point approximation ------------------------------------------ Let us compare the exact partition function - with the result of the saddle point approximation . In figure \[Zexact\_vs\_saddle\] we plot the difference between two results $$\begin{aligned} F_{\text{saddle}-\text{exact}}=\frac{\pi\sqrt{2k}}{3}\sqrt{1+\frac{16\zeta_2^2}{k^2}}N^{\frac{3}{2}}-(-\log Z(N,k,0,\zeta_2)) \label{Fsaddle-exact}\end{aligned}$$ for $k=1,2,3,4,6$ and $\zeta_2=1$. The plot indicates $F_{\text{saddle}-\text{exact}}\sim \sqrt{N}$ for large-$N$, hence the leading part of the two results ($\sim N^{3/2}$) agree with each other. ![ Plot of $F_{\text{saddle}-\text{exact}}$ for $k=1,2,3,4,6$, $\zeta_2=1$. []{data-label="Zexact_vs_saddle"}](180719_Fsaddleexact_vs_sqrtN_z21_k12346_Nmax12_9_5_7_6.eps){width="12cm"} We can make a more refined comparison between the exact results and large-$N$ expansion as follows. First we notice that the saddle point approximation agree with the following expression in the large-$N$ limit $$\begin{aligned} Z_\text{pert}=e^AC^{-\frac{1}{3}}\operatorname{Ai}[C^{-\frac{1}{3}}(N-B)] \label{Airy}\end{aligned}$$ where $$\begin{aligned} C&=\frac{2}{k\pi^2(1+\frac{16\zeta_2^2}{k^2})},\quad B=\frac{k}{24}-\frac{1}{6k}+\frac{1}{2k(1+\frac{16\zeta_2^2}{k^2})},\nonumber \\ A&=\frac{2A_\text{ABJM}(k) +A_\text{ABJM}(k+4i\zeta_2) +A_\text{ABJM}(k-4i\zeta_2) }{4} ,\end{aligned}$$ which is obtained from the partition function of the ABJM theory with R-charge deformation by ignoring the large-$N$ non-perturbative effects ($e^{-\sqrt{N}}$) and replacing the real deformation parameters $\xi,\eta$ formally as $\xi\rightarrow 0$, $\eta\rightarrow 4i\zeta_2/k$ (see eq(1.4) in [@Nosaka]). By comparing the numerical values of - and we find good agreement. As an example, in figure \[Zexact\_vs\_Airy\] we display the comparison of the free energy for $\zeta_2=1$. ![ Blue points: exact values of $-\log Z(N,k,0,\zeta_2)$ -; Red line: $-\log Z_\text{pert}(N)$ . []{data-label="Zexact_vs_Airy"}](180719_exact_vs_Airy_k1_Nmax12_z21.eps "fig:"){width="8cm"} ![ Blue points: exact values of $-\log Z(N,k,0,\zeta_2)$ -; Red line: $-\log Z_\text{pert}(N)$ . []{data-label="Zexact_vs_Airy"}](180719_exact_vs_Airy_k2_Nmax9_z21.eps "fig:"){width="8cm"}\ ![ Blue points: exact values of $-\log Z(N,k,0,\zeta_2)$ -; Red line: $-\log Z_\text{pert}(N)$ . []{data-label="Zexact_vs_Airy"}](180719_exact_vs_Airy_k3_Nmax5_z21.eps "fig:"){width="8cm"} ![ Blue points: exact values of $-\log Z(N,k,0,\zeta_2)$ -; Red line: $-\log Z_\text{pert}(N)$ . []{data-label="Zexact_vs_Airy"}](180719_exact_vs_Airy_k4_Nmax7_z21.eps "fig:"){width="8cm"}\ ![ Blue points: exact values of $-\log Z(N,k,0,\zeta_2)$ -; Red line: $-\log Z_\text{pert}(N)$ . []{data-label="Zexact_vs_Airy"}](180719_exact_vs_Airy_k6_Nmax6_z21.eps "fig:"){width="8cm"} Decoupling limit $\zeta_2\rightarrow\infty$ {#comarewithNY} ------------------------------------------- To compare the mass deformed ABJM theory in the decoupling limit with the $\text{U}(N)_k\times \text{U}(N)_{-k}$ linear quiver superconformal Chern-Simons theory (Gaiotto-Witten theory), it is reasonable to divide the partition function $Z(N,k,0,\zeta_2)$ by $e^{2\pi N^2\zeta_2/k}$, the naive contribution from the massive hypermultiplet. For $(N,k)=(1,1)$, $(1,2)$, $(1,3)$, $(2,3)$, $(1,4)$, $(2,4)$, $(1,6)$, $(2,6)$, $(3,6)$ the result of $\lim_{\zeta_2\rightarrow \infty} Z(N,k,0,\zeta_2)/e^{-2\pi N^2\zeta_2/k}$ is finite and coincide with the partition function of the Gaiotto-Witten theory with the same $(N,k)$ obtained in [@NY1]. For the other $(N,k)$ we have found the following asymptotic behavior $$\begin{aligned} Z(N,k,0,\zeta_2)\rightarrow f_{N,k}(\zeta_2)e^{-\frac{2\pi p_{N,k}\zeta_2}{k}},\end{aligned}$$ where $p_{N,k}$ are some integers and $f_{N,k}(\zeta_2)$ are some polynomials of $\zeta_2$, which are listed in the following tables. $$\begin{aligned} &\begin{tabular}{|c|c|c|} \multicolumn{3}{l}{$k=1$}\\ \hline $N$&$p_{N,k}$&$f_{N,k}(\zeta_2)$\\ \hline 1& 1&$\frac{1}{2}$\\ \hline 2& 3&$\zeta_2$\\ \hline 3& 5&$\frac{1}{8}$\\ \hline 4& 8&$-\frac{1}{32}+\frac{\zeta_2^2}{2}$\\ \hline 5&11&$-\frac{3}{64}+\frac{\zeta_2^2}{4}$\\ \hline 6&14&$\frac{1}{64}$\\ \hline 7&18&$-\frac{5\zeta_2}{192}+\frac{\zeta_2^3}{12}$\\ \hline 8&22&$\frac{9}{1024}-\frac{\zeta_2^2}{24}+\frac{\zeta_2^4}{12}$\\ \hline 9&26&$-\frac{11\zeta_2}{768}+\frac{\zeta_2^3}{48}$\\ \hline 10&30&$\frac{1}{1024}$\\ \hline 11&35&$\frac{3}{16384}-\frac{7\zeta_2^2}{1536}+\frac{\zeta_2^4}{192}$\\ \hline 12&40&$-\frac{45}{65536}+\frac{151\zeta_2^2}{36864}-\frac{19\zeta_2^4}{2304}+\frac{\zeta_2^6}{144}$\\ \hline \end{tabular} \quad \begin{tabular}{|c|c|c|} \multicolumn{3}{l}{$k=2$}\\ \hline $N$&$p_{N,k}$&$f_{N,k}(\zeta_2)$\\ \hline 1& 1&$\frac{1}{4}$\\ \hline 2& 4&$\frac{\zeta_2^2}{2}$\\ \hline 3& 7&$-\frac{1}{32}+\frac{\zeta_2^2}{8}$\\ \hline 4&10&$\frac{1}{256}$\\ \hline 5&15&$-\frac{\zeta_2^2}{192}+\frac{\zeta_2^4}{96}$\\ \hline 6&20&$-\frac{1}{1024}+\frac{17\zeta_2^2}{4608}-\frac{\zeta_2^4}{144}+\frac{\zeta_2^6}{144}$\\ \hline 7&25&$\frac{19\zeta_2^2}{9216}-\frac{7\zeta_2^4}{2304}+\frac{\zeta_2^6}{576}$\\ \hline 8&30&$\frac{3}{65536}-\frac{\zeta_2^2}{3072}+\frac{\zeta_2^4}{6144}$\\ \hline 9&35&$\frac{1}{262144}$\\ \hline \end{tabular}\nonumber \\ &\begin{tabular}{|c|c|c|} \multicolumn{3}{l}{$k=3$}\\ \hline $N$&$p_{N,k}$&$f_{N,k}(\zeta_2)$\\ \hline 1& 1&$\frac{1}{6}$\\ \hline 2& 4&$\frac{1}{12}$\\ \hline 3& 8&$-\frac{1}{9\sqrt{3}}+\frac{\zeta_2}{18}$\\ \hline 4& 12&$\frac{1}{432}$\\ \hline 5& 17&$\frac{1}{2592}$\\ \hline \end{tabular} \quad \begin{tabular}{|c|c|c|} \multicolumn{3}{l}{$k=4$}\\ \hline $N$&$p_{N,k}$&$f_{N,k}(\zeta_2)$\\ \hline 1& 1&$\frac{1}{8}$\\ \hline 2& 4&$\frac{1}{64}$\\ \hline 3& 9&$\frac{5}{256}-\frac{\zeta_2}{32}+\frac{\zeta_2^2}{64}$\\ \hline 4&14&$\frac{1}{1024}-\frac{\zeta_2}{256}+\frac{\zeta_2^2}{512}$\\ \hline 5&19&$\frac{1}{32768}$\\ \hline \end{tabular} \quad \begin{tabular}{|c|c|c|} \multicolumn{3}{l}{$k=6$}\\ \hline $N$&$p_{N,k}$&$f_{N,k}(\zeta_2)$\\ \hline 1& 1&$\frac{1}{12}$\\ \hline 2& 4&$\frac{1}{432}$\\ \hline 3& 9&$\frac{1}{5184}$\\ \hline 4&16&$\frac{1}{1296}-\frac{\zeta_2}{972\sqrt{3}}+\frac{\zeta_2^2}{7776}$\\ \hline \end{tabular}\end{aligned}$$ [99]{} T. Kuroki and F. Sugino, “Spontaneous supersymmetry breaking by large-N matrices,” Nucl. Phys. B [**796**]{} (2008) 471 doi:10.1016/j.nuclphysb.2007.12.003 \[arXiv:0710.3971 \[hep-th\]\]. 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For $\delta^2=0$ case, the background gauge fields couple uniformly to $A_\alpha$, $\bar{B}^{\dot{\alpha}}$ in and hence can be absorbed into the shift of $(\sigma,D,{\widetilde\sigma},{\widetilde D})\rightarrow(\sigma,D,{\widetilde\sigma},{\widetilde D})+(-\pi\zeta/k,\pi\zeta/k,\pi\zeta/k,-\pi\zeta/k)$ with $\zeta =(\zeta_1 +\zeta_2 )/2$. These field redefinitions generate the Fayet-Illiopoulos terms out of the Chern-Simons term instead. [^8]: This statement is not precise because the Chern-Simons interaction remains and theory may be in a gapped phase. Nevertheless we will call the confinement phase for such case also. Note that here we take $k/N \rightarrow 0$ limit, thus the Chern-Simons interaction will be ignored for the leading order in the large-$N$ limit and the Yang-Mills term always induced by the renormalization flow. We also note that the ${\cal N}=2$ SUSY pure Yang-Mills theory does not have SUSY vacua. [^9]: With some numerical methods, we can not find any solution other than the two solutions. [^10]: We expect that the energy of the possible metastable SUSY breaking vacuum is proportional to $\zeta$ and the free energy will be proportional to $\zeta r_{S^3}$. The extra contribution by the curvature induced mass term to the free energy for the fuzzy sphere solutions will also proportional to $\zeta r_{S^3}$ because the size of the fuzzy sphere grows as $\zeta$ grows. Of course, this is not valid except the weak coupling limit and the phase of the theory can be non-trivial. [^11]: Here, we assume that the theory is regarded as a deformation of the ABJM theory on $S^3$ for a small $\zeta/k$ case. For a enough large $\zeta/k$ case, we think that the curvature effect of $S^3$ is almost negligible, but still remains. This picture will lead the SUSY breaking scenario explained here. [^12]: For the SUSY, it corresponds to $\operatorname{Tr}(-1)^{\hat{F}} \{ \hat{Q},{\cal \hat{O}} \}= \operatorname{Tr}[ (-1)^{\hat{F}} \hat{Q},{\cal \hat{O}}]$. [^13]: In the case of $Q=$SUSY, $Q{\cal O}$ is such as $F$-term and $D$-term. Unfortunately we cannot compute $\langle F\rangle$ or $\langle D\rangle$ by using the supersymmetry localization. We can compute $\langle \int F\rangle$ and $\langle \int D\rangle$, but they are trivially zero. This is consistent with the fact that there is no SUSY breaking for the theory on $S^3$ with $N$ finite. [^14]: In the gravity dual, the SUSY is gauged and the theory is described by a supergravity. In the supergravity, there are massless fermions, however, there are no zero modes around the SUSY vacuum which is an asymptotic $AdS_4$ background. In a SUSY breaking vacuum, some fermions near the boundary have zero modes. [^15]: From the analogy to the case with bosonic zero mode, an appropriate analysis would be to add an explicit-susy-breaking deformation to kill the zero mode and see what happens in the limit of zero deformation. In this approach, however, we cannot use the result of the localization. [^16]: The mass deformed ABJM also will not break the SUSY for this case because a half of the hypermultiplets remain massless and do not decouple. [^17]: The traditional approach was taken in [@Grassi:2014vwa; @Hatsuda:2015lpa] for $\zeta_1 =0=\zeta_2$ identifying ’t Hooft coupling with $N/N_f$ where $N_f$ denotes an additional power put on the $\cosh$ (For our case, $N_f =1$). [^18]: In our actual analysis, we have looked for solutions with complex $z(s)$ under the ansatz but we have found only a real solution as a result. Because of this, we take $z(s)$ to be real for simplicity of explanations in the main text. Precisely speaking, we should first take the variation $\frac{\delta S_\text{eff}}{\delta z(s)}$ with $z(s)\in\mathbb{C}$ before assuming $z(s)\in\mathbb{R}$. This induces a new constraint $\frac{\delta S_\text{eff}}{\delta \text{Im}(z(s))}=0$ in addition to and ; nevertheless the final result remains the same. [^19]: Note that odd functions of $z-z'$ do not contribute. [^20]: The condition that this evaluation is valid is following [@NST2]: $$-\frac{1}{4} < {\rm Im} (w)- {\rm Re} (w) \frac{{\rm Im}(\dot{v}) }{{\rm Re} (\dot{v})} < \frac{1}{4}.$$ In this case this condition is satisfied. [^21]: We have excluded the other two solutions by the condition $\dot{z}(s)\geq 0$. [^22]: For a later convenience we have symmetrized the density matrix by another similarity transformation from . [^23]: More precisely, integrating out the matter fields induces level shifts of all the possible mixed CS terms which are among the gauge symmetry $\text{U}(N)\times \text{U}(N)$, flavor symmetry $\text{U}(1)_1 \times \text{U}(1)_2$ and $\text{U}(1)_\text{R}$ symmetry. In the case of the massive ABJM theory, most of the shifts are canceled and we have only contributions from the gauge-$\text{U}(1)_\text{R}$ and flavor $\text{U}(1)_\text{R}$ CS terms but these terms do not affect the zeroes of the partition function. Here the integration of the matter fields is assumed to be at the origin of the Coulomb moduli space. Thus, the decoupling of the matter fields in the large mass limit is possible. [^24]: The notation in [@RS] is related to ours by $$Z_{\rm ours}(N,k,\zeta_1 ,\zeta_2 ) =2^{-2N}Z_{\rm Russo-Silva}( N,k, m_1 =-4\pi\zeta_2/k ,\zeta_2 =-4\pi\zeta_1 /k ).$$ [^25]: This is so-called reweighting method. [^26]: The application to a similar system is explained in app. A of [@KEK]. [^27]: Of course we can also compute $Z(N,k,\zeta_1 ,\zeta_2 )$ itself by combining $Z_{\rm MC}(N,k,\zeta_1 ,\zeta_2 )$ with $Z(N,k,0 ,\zeta_2 )$ computed in another way. For example, we know the exact values of $Z(N,k,0 ,\zeta_2 )$ for various $(N,k,\zeta_2)$ obtained in sec.  and Monte Carlo simulation of $Z(N,k,0 ,\zeta_2 )$ is much easier than the $\zeta_1 \neq 0$ case if we use the algorithm in [@KEK]. [^28]: Similar behaviors have been observed for $(N,k,\zeta_2 )$ $=$ $(3,1,1)$, $(4,2,2)$, $(5,2,1)$, $(5,2,2)$. [^29]: This is done by the command “Interpolation" in Mathematica. The values without “$\pm$" are (first) zeroes of the interpolating functions for the average values of $Z_{\rm MC}$. The values including “$\pm$" denotes zeroes of interpolating functions for the average values plus/minus the errors. [^30]: This analysis was done in sec. 4 of [@Nosaka]. The result in our notation can be obtained by taking $p\rightarrow 1$, $q\rightarrow 1$, $\xi \rightarrow \frac{4i}{k}\zeta_1$ and $\eta \rightarrow \frac{4i}{k}\zeta_2$. [^31]: In the large-$N$ limit, the $\mu$-integral is dominated by $\mu =\sqrt{\frac{N-B}{C}}$. Therefore the non-perturbative effects in contribute to $Z$ like $\sim \mathcal{O}(e^{-\sqrt{kN}})$, $\mathcal{O}(e^{-\sqrt{N/k}})$. [^32]: See for the notation for the 1d quantum mechanics.
--- abstract: 'We use Kashiwara’s theory of crystal bases to study plactic monoids for $U_{q}(so_{2n+1})$ and $U_{q}(so_{2n})$. Simultaneously we describe a Schensted type correspondence in the crystal graphs of tensor powers of vector and spin representations and we derive a Jeu de Taquin for type $B$ from the Sheats sliding algorithm.' author: - | Cedric Lecouvey\ lecouvey@math.unicaen.fr date:   title: 'Schensted-type correspondences and plactic monoids for types $B_{n}$ and $D_{n}$' --- Introduction ============= The Schensted correspondence based on the bumping algorithm yields a bijection between words $w$ of length $l$ on the ordered alphabet $\mathcal{A}_{n}=\{1\prec2\prec\cdot\cdot\cdot\prec n\}$ and pairs $(P^{A}(w),Q^{A}(w))$ of tableaux of the same shape containing $l$ boxes where $P^{A}(w)$ is a  semi-standard Young tableau on $\mathcal{A}_{n}$ and $Q^{A}(w)$ is a standard tableau. By identifying the words $w$ having the same tableau $P^{A}(w)$, we obtain the plactic monoid $Pl(A_{n})$ whose defining relations were determined by Knuth: $$\begin{aligned} yzx & =yxz\text{ \ \ and \ \ }xzy=zxy\text{ if }x\prec y\prec z,\\ xyx & =xxy\text{ \ \ and \ \ }xyy=yxy\text{ if }x\prec y.\end{aligned}$$ The Robinson-Schensted correspondence has a natural interpretation in terms of Kashiwara’s theory of crystal bases [@DJM], [@Ka2], [@LLT]. Let $V_{n}^{A}$ denote the vector representation of $U_{q}(sl_{n})$. By considering each vertex of the crystal graph of $\underset{l\geq0}{\bigoplus (}V_{n}^{A})^{\otimes l}$ as a word on $\mathcal{A}_{n}$, we have for any words $w_{1}$ and $w_{2}$: - $P^{A}(w_{1})=P^{A}(w_{2})$ if and only if $w_{1}$ and $w_{2}$ occur at the same place in two isomorphic connected components of this graph. - $Q^{A}(w_{1})=Q^{A}(w_{2})$ if and only if $w_{1}$ and $w_{2}$ occur in the same connected component of this graph. Replacing $V_{n}^{A}$ by the vector representation $V_{n}^{C}$ of $sp_{2n}$ whose basis vectors are labelled by the letters of the totally ordered alphabet $$\mathcal{C}_{n}=\{1\prec\cdot\cdot\cdot\prec n-1\prec n\prec\overline{n}\prec\overline{n-1}\prec\cdot\cdot\cdot\prec\overline{1}\},$$ we have obtained in [@Lec] a Schensted type correspondence for type $C_{n}$. This correspondence is based on an insertion algorithm for the Kashiwara-Nakashima’s symplectic tableaux [@KN] analogous to the bumping algorithm. It may be regarded as a bijection between words $w$ of length $l$ on $\mathcal{C}_{n}$ and pairs $(P^{C}(w),Q^{C}(w))$ where $P^{C}(w)$ is a symplectic tableau and $Q^{C}(w)$ an oscillating tableau of type $C$ and length $l,$ that is, a sequence $(Q_{1},...,Q_{l})$ of Young diagrams such that two consecutive diagrams differ by exactly one box. Moreover by identifying the words of the free monoid $\mathcal{C}_{n}^{\ast}$ having the same symplectic tableau we also obtain a monoid $Pl(C_{n})$. This is the plactic monoid of type $C_{n}$ in the sense of [@Lit] and [@LLT]. The vector representations $V_{n}^{B}$ and $V_{n}^{D}$ of $U_{q}(so_{2n+1})$ and $U_{q}(so_{2n})$ have crystal graphs whose vertices may be respectively labelled by the letters of $$\mathcal{B}_{n}=\{1\prec\cdot\cdot\cdot\prec n-1\prec n\prec0\prec\overline {n}\prec\overline{n-1}\prec\cdot\cdot\cdot\prec\overline{1}\}$$ and $$\mathcal{D}_{n}=\{1\prec\cdot\cdot\cdot\prec n-1\prec\begin{array} [c]{l}n\\ \overline{n}\end{array} \prec\overline{n-1}\prec\cdot\cdot\cdot\prec\overline{1}\}.$$ Let $G_{n}^{B}$ and $G_{n}^{D}$ be the crystal graphs of $\underset{l\geq 0}{\text{ }\bigoplus}(V_{n}^{B})^{\otimes l}$ and $\underset{l\geq0}{\bigoplus}(V_{n}^{D})^{\otimes l}.$ Then it is possible to label the vertices of $G_{n}^{B}$ and $G_{n}^{D}$ by the words of the free monoids $\mathcal{B}_{n}^{\ast}$ and $\mathcal{D}_{n}^{\ast}$. However the situation is more complicated than in the case of types $A$ and $C$. Indeed there exist a fundamental representation of $U_{q}(so_{2n+1})$ and two fundamental representations of $U_{q}(so_{2n})$ that do not appear in the decompositions of $\bigoplus(V_{n}^{B})^{\otimes l}$ and $\underset{l\geq0}{\bigoplus}(V_{n}^{D})^{\otimes l}$ into their irreducible components. They are called the spin representations and denoted respectively by $V(\Lambda_{n}^{B}),$ $V(\Lambda_{n}^{D})$ and $V(\Lambda_{n-1}^{D})$. In [@KN], Kashiwara and Nakashima have described their crystal graphs by using a new combinatorical object that we will call a spin column. Write $SP_{n}$ for the set of spin columns of height $n$ and set $\frak{B}_{n}=\mathcal{B}_{n}\cup SP_{n},$ $\frak{D}_{n}=\mathcal{D}_{n}\cup SP_{n}$. Then each vertex of the crystal graphs $\frak{G}_{n}^{B}$ and $\frak{G}_{n}^{D}$ of $\underset{l\geq 0}{\bigoplus}\left( V_{n}^{B}\bigoplus V(\Lambda_{n}^{B})\right) ^{\otimes l}$ and$\underset{l\geq0}{\text{ }\bigoplus}\left( V_{n}^{D}\bigoplus V(\Lambda_{n}^{D})\bigoplus V(\Lambda_{n-1}^{D})\right) ^{\otimes l}$ may be respectively identified with a word on $\frak{B}_{n}$ or $\frak{D}_{n}.$ We can define two relations $\overset{B}{\sim}$ and $\overset{D}{\sim}$ by: $w_{1}\overset{B}{\sim}w_{2}$ if and only if $w_{1}$ and $w_{2}$ occur at the same place in two isomorphic connected components of $\frak{G}_{n}^{B},$ $w_{1}\overset{D}{\sim}w_{2}$ if and only if $w_{1}$ and $w_{2}$ occur at the same place in two isomorphic connected components of $\frak{G}_{n}^{D}.$ In this article, we prove that $Pl(B_{n})=\mathcal{B}_{n}^{\ast}/\overset {B}{\sim},$ $Pl(D_{n})=\mathcal{D}_{n}^{\ast}/\overset{D}{\sim}$, $\frak{Pl(}B_{n}\frak{)=B}_{n}^{\ast}\frak{/}\overset{B}{\sim}$ and $\frak{Pl}(D_{n})=\frak{D}_{n}^{\ast}/\overset{D}{\sim}$ are monoids and we undertake a detailed investigation of the corresponding insertion algorithms. We summarize in part 2 the background on Kashiwara’s theory of crystals used in the sequel. In part 3, we first recall Kashiwara-Nakashima’s notion of orthogonal tableau (analogous to Young tableaux for types $B$ and $D$) and we relate it to Littelmann’s notion of Young tableau for classical types. Then we derive a set of defining relations for $Pl(B_{n})$ and $Pl(D_{n})$ and we describe the corresponding column insertion algorithms. Using the combinatorial notion of oscillating tableaux (analogous to standard tableaux for types $B$ and $D$), these algorithms yield the desired Schensted type correspondences in $G_{n}^{B}$ and $G_{n}^{D}$. In part 4 we propose an orthogonal Jeu de Taquin for type $B$ based on Sheats’ sliding algorithm for type $C$ [@SH]. Finally in part 5, we bring into the picture the spin representations and extend the results of part 3 to the graphs $\frak{G}_{n}^{B}$, $\frak{G}_{n}^{D}$ and the monoids $\frak{Pl(}B_{n}\frak{)}$, $\frak{Pl(}D_{n}\frak{).}$ In the sequel, we often write $B$ and $D$ instead of $B_{n}$ and $D_{n}$ to simplify the notation. Moreover, we frequently define similar objects for types $B$ and $D$. When they are related to type $B$ (respectively $D$), we attach to them the label $^{B}$ (respectively the label $^{D}$). To avoid cumbersome repetitions, we sometimes omit the labels $^{B}$ and $^{D}$ when our statements are true for the two types. Conventions for crystal graphs ============================== Kashiwara’s operators --------------------- Let $\frak{g}$ be simple Lie algebra and $\alpha_{i},$ $i\in I$ its simple roots. Recall that the crystal graphs of the $U_{q}(\frak{g})$-modules are oriented colored graphs with colors $i\in I$. An arrow $a\overset {i}{\rightarrow}b$ means that $\widetilde{f}_{i}(a)=b$ and $\widetilde{e}_{i}(b)=a$ where $\widetilde{e}_{i}$ and $\widetilde{f}_{i}$ are the crystal graph operators (for a review of crystal bases and crystal graphs see [@Ka2]). Let $V,V^{\prime}\ $be two $U_{q}(\frak{g})$-modules and $B,B^{\prime}$ their crystal graphs. A vertex $v^{0}\in B$ satisfying $\widetilde{e}_{i}(v^{0})=0$ for any $i\in I$ is called a highest weight vertex. The decomposition of $V$ into its irreducible components is reflected into the decomposition of $B$ into its connected components. Each connected component of $B$ contains a unique vertex of highest weight. We write $B(v^{0})$ for the connected component containing the highest weight vertex $v^{0}$. The crystals graphs of two isomorphic irreducible components are isomorphic as oriented colored graphs. We will say that two vertices $b_{1}\ $and $b_{2}$ of $B$ occur at the same place in two isomorphic connected components $\Gamma_{1}$ and $\Gamma_{2}$ of $B$ if there exist $i_{1},...,i_{r}\in I$ such that $w_{1}=\widetilde{f}_{i_{i}}\cdot\cdot \cdot\widetilde{f}_{i_{r}}(w_{1}^{0})$ and $w_{2}=\widetilde{f}_{i_{i}}\cdot\cdot\cdot\widetilde{f}_{i_{r}}(w_{2}^{0})$, where $w_{1}^{0}$ and $w_{2}^{0}$ are respectively the highest weight vertices of $\Gamma_{1}$ and $\Gamma_{2}$. The action of $\widetilde{e}_{i}$ and $\widetilde{f}_{i}$ on $B\otimes B^{\prime}=\{b\otimes b^{\prime};$ $b\in B,b^{\prime}\in B^{\prime}\}$ is given by: $$\begin{aligned} \widetilde{f_{i}}(u\otimes v) & =\left\{ \begin{tabular} [c]{c}$\widetilde{f}_{i}(u)\otimes v$ if $\varphi_{i}(u)>\varepsilon_{i}(v)$\\ $u\otimes\widetilde{f}_{i}(v)$ if $\varphi_{i}(u)\leq\varepsilon_{i}(v)$\end{tabular} \right. \label{TENS1}\\ & \text{and}\nonumber\\ \widetilde{e_{i}}(u\otimes v) & =\left\{ \begin{tabular} [c]{c}$u\otimes\widetilde{e_{i}}(v)$ if $\varphi_{i}(u)<\varepsilon_{i}(v)$\\ $\widetilde{e_{i}}(u)\otimes v$ if $\varphi_{i}(u)\geq\varepsilon_{i}(v)$\end{tabular} \right. \label{TENS2}$$ where $\varepsilon_{i}(u)=\max\{k;\widetilde{e}_{i}^{k}(u)\neq0\}$ and $\varphi_{i}(u)=\max\{k;\widetilde{f}_{i}^{k}(u)\neq0\}$. Denote by $\Lambda_{i},$ $i\in I$ the fundamental weights of $\frak{g}$. The weight of the vertex $u$ is defined by $\mathrm{wt}(u)=\underset{I}{\sum}(\varphi _{i}(u)-\varepsilon_{i}(u))\Lambda_{i}$. Write $s_{i}=s_{\alpha_{i}}$ for $i\in I$. The Weyl group $W$ of $\frak{g}$ acts on $B$ by: $$\begin{aligned} s_{i}(u) & =(\widetilde{f_{i}})^{\varphi_{i}(u)-\varepsilon_{i}(u)}(u)\text{ if }\varphi_{i}(u)-\varepsilon_{i}(u)\geq0,\label{actionW}\\ s_{i}(u) & =(\widetilde{e_{i}})^{\varepsilon_{i}(u)-\varphi_{i}(u)}(u)\text{ if }\varphi_{i}(u)-\varepsilon_{i}(u)<0.\nonumber\end{aligned}$$ We have the equality $\mathrm{wt}(\sigma(u))=\sigma(\mathrm{wt}(u)$ for any $\sigma\in W$ and $u\in B.$ The following lemma is a straightforward consequence of (\[TENS1\]) and (\[TENS2\]). \[lem\_phi\_tens\]Let $u\otimes v$ $\in$ $B\otimes B^{\prime}$. Then: - $\mathrm{(i)}$ $\varphi_{i}(u\otimes v)=\left\{ \begin{tabular} [c]{l}$\_[i]{}(v)+\_[i]{}(u)-\_[i]{}(v)$ if $\_[i]{}(u)&gt;\_[i]{}(v)$\\ $\_[i]{}(v)$ otherwise. \end{tabular} \right. .$ - $\mathrm{(ii)}$ $\varepsilon_{i}(u\otimes v)=\left\{ \begin{tabular} [c]{l}$\_[i]{}(v)+\_[i]{}(u)-\_[i]{}(u)$ if $\_[i]{}(v)&gt;\_[i]{}(u)$\\ $\_[i]{}(u)$ otherwise. \end{tabular} \right. .$ - $\mathrm{(iii)}$ $u\otimes v$ is a highest weight vertex of $B\otimes B^{\prime}$ if and only if for any $i\in I$ $\widetilde{e}_{i}(u)=0$ (i.e. $u$ is of highest weight) and $\varepsilon_{i}(v)\leq\varphi_{i}(u).$ For any dominant weight $\lambda\in P_{+}$, write $B(\lambda)$ for the crystal graph of $V(\lambda),$ the irreducible module of highest weight $\lambda$ and denote by $u_{\lambda}$ its highest weight vertex. Kashiwara has introduced in [@Ka3] an embedding of $B(\lambda)$ into $B(m\lambda)$ for any positive integer $m$. He uses this embedding to obtain a simple bijection between Littelmann’s path crystal associated to $\lambda$ and $B(\lambda)$ [@Lit3]. \[th\_strech\](Kashiwara) There exists a unique injective map $$\begin{gathered} S_{m}:B(\lambda)\rightarrow B(m\lambda)\subset B(\lambda)^{\otimes m}\\ u_{\lambda}\mapsto u_{\lambda}^{\otimes m}$$ such that for any $b\in B(\lambda)$: $$\begin{aligned} \text{$\mathrm{(i)}$\ \ }S_{m}(\widetilde{e}_{i}(b)) & =\widetilde{e}_{i}^{m}(S_{m}(b)),\nonumber\\ \text{$\mathrm{(ii)}$ \ }S_{m}(\widetilde{f}_{i}(b)) & =\widetilde{f}_{i}^{m}(S_{m}(b)),\nonumber\\ \text{$\mathrm{(iii)}$ \ }\varphi_{i}(S_{m}(b)) & =m\varphi_{i}(b),\label{stretch}\\ \text{$\mathrm{(iv)}$ \ }\varepsilon_{i}(S_{m}(b)) & =m\varepsilon _{i}(b),\nonumber\\ \text{$\mathrm{(v)}$ \ }\mathrm{wt}(S_{m}(b)) & =m\mathrm{wt}(b).\nonumber\end{aligned}$$ \[cor\_strech\]Let $\lambda_{1},...,\lambda_{k}\in P_{+}.$ Then, the map: $$\begin{gathered} \text{{\Large S}}_{m}:B(\lambda_{1})\otimes\cdot\cdot\cdot\otimes B(\lambda_{k})\rightarrow B(m\lambda_{1})\otimes\cdot\cdot\cdot\otimes B(m\lambda_{k})\\ \text{\ \ \ \ \ \ \ \ \ \ \ }b_{1}\otimes\cdot\cdot\cdot\otimes b_{k}\mapsto S_{m}(b_{1})\otimes\cdot\cdot\cdot\otimes S_{m}(b_{k})\end{gathered}$$ is injective and satisfies the relations (\[stretch\]) with $b=b_{1}\otimes\cdot\cdot\cdot\otimes b_{k}.$ Moreover the image by [S]{}$_{m}$ of a highest weight vertex of $B(\lambda_{1})\otimes\cdot\cdot\cdot\otimes B(\lambda_{k})$ is a highest weight vertex of $B(m\lambda_{1})\otimes \cdot\cdot\cdot\otimes B(m\lambda_{k})$. By induction, we can suppose $k=2$. [S]{}$_{m}$ is injective because $S_{m}$ is injective. Let $u\otimes v\in B(\lambda_{1})\otimes B(\lambda _{2}).$ Suppose that $\varphi_{i}(u)\leq\varepsilon_{i}(v)$. We derive the following equalities from Formulas (\[TENS1\]) and (\[TENS2\]): $$\begin{gathered} \text{{\Large S}}_{m}\widetilde{f}_{i}(u\otimes v)=\text{{\Large S}}_{m}(u\otimes\widetilde{f}_{i}v)=S_{m}(u)\otimes S_{m}(\widetilde{f}_{i}v)=S_{m}(u)\otimes\widetilde{f}_{i}^{m}S_{m}(v)\\ \text{and }\widetilde{f}_{i}^{m}(\text{{\Large S}}_{m}(u\otimes v))=\widetilde {f}_{i}^{m}(S_{m}(u)\otimes S_{m}(v))=S_{m}(u)\otimes\widetilde{f}_{i}^{m}S_{m}(v).\end{gathered}$$ Indeed, $\varepsilon_{i}(S_{m}(v))=m\varepsilon_{i}(v)\geq m\varphi _{i}(u)=\varphi_{i}(S_{m}(u))$ and for $p=1,...,m$ $\varepsilon_{i}(\widetilde{f}_{i}^{p}S_{m}(v))>\varepsilon_{i}(S_{m}(v)).$ Hence [S]{}$_{m}\widetilde{f}_{i}(u\otimes v)=\widetilde{f}_{i}^{m}($[S]{}$_{m}(u\otimes v)).$ Now suppose $\varepsilon_{i}(v)<\varphi_{i}(u)$ i.e. $\varepsilon_{i}(u)\leq\varphi_{i}(v)+1$. We obtain: $$\begin{gathered} \text{{\Large S}}_{m}\widetilde{f}_{i}(u\otimes v)=\text{{\Large S}}_{m}(\widetilde{f}_{i}u\otimes v)=S_{m}(\widetilde{f}_{i}u)\otimes S_{m}(v)=\widetilde{f}_{i}^{m}S_{m}(u)\otimes S_{m}(v)\\ \text{and }\widetilde{f}_{i}^{m}(\text{{\Large S}}_{m}(u\otimes v))=\widetilde {f}_{i}^{m}(S_{m}(u)\otimes S_{m}(v))=\widetilde{f}_{i}^{m}S_{m}(u)\otimes S_{m}(v)\end{gathered}$$ because $\varepsilon_{i}(S_{m}(v))=m\varepsilon_{i}(v)\leq m\varphi _{i}(u)+m=\varphi_{i}(S_{m}u)+m$. Hence we have [S]{}$_{m}\widetilde {f}_{i}(u\otimes v)=\widetilde{f}_{i}^{m}($[S]{}$_{m}(u\otimes v)).$ Similarly we prove that [S]{}$_{m}\widetilde{e}_{i}(u\otimes v)=\widetilde{e}_{i}^{m}($[S]{}$_{m}(u\otimes v)).$ So [S]{}$_{m}$ satisfies the formulas $\mathrm{(i)}$and $\mathrm{(ii)}$. By Lemma \[lem\_phi\_tens\] $\mathrm{(i)}$and $\mathrm{(ii)}$ we obtain then that [S]{}$_{m}$ satisfies $\mathrm{(iii)}$, $\mathrm{(iv)}$ and $\mathrm{(v)}$. Suppose that $u\otimes v$ is a highest weight vertex of $B(\lambda_{1})\otimes B(\lambda_{2})$. By Lemma \[lem\_phi\_tens\] $\mathrm{(iii)}$, $u$ is the highest weight vertex of $B(\lambda_{1})$ and $\varepsilon_{i}(v)\leq \varphi_{i}(u)$ for $i\in I$. Then by definition of $S_{m},$ $S_{m}(u)$ is the highest weight vertex of $B(m\lambda_{1})$. Moreover for any $i\in I,$ $\varepsilon_{i}(S_{m}(v))=m\varepsilon_{i}(v)\leq m\varphi_{i}(u)=\varphi _{i}(S_{m}(u))$. So $S_{m}(u)\otimes S_{m}(v)=$[S]{}$_{m}(u\otimes v)$ is of highest weight in $B(m\lambda_{1})\otimes B(m\lambda_{2})$. By this corollary, the connected component of $B(\lambda_{1})\otimes\cdot\cdot\cdot\otimes B(\lambda_{k})$ of highest weight vertex $u^{0}=u_{1}\otimes\cdot\cdot\cdot\otimes u_{k}$, may be identified with the sub-graph of $B(m\lambda_{1})\otimes\cdot\cdot\cdot\otimes B(m\lambda_{k})$ generated by the vertex $S_{m}(u_{1})\otimes\cdot\cdot\cdot\otimes S_{m}(u_{k})$ and the operators $\widetilde{f}_{i}^{m}$ for $i\in I$. Tensor powers of the vector representations ------------------------------------------- We choose to label the Dynkin diagram of $so_{2n+1}$ by: $$\overset{1}{\circ}-\overset{2}{\circ}-\overset{3}{\circ}\cdot\cdot \cdot\overset{n-2}{\circ}-\overset{n-1}{\circ}\Longrightarrow\overset{n}{\circ}$$ and the Dynkin diagram of $so_{2n}$ by: $$\overset{1}{\circ}-\overset{2}{\circ}-\overset{3}{\circ}\cdot\cdot \cdot\overset{n-3}{\circ}- \begin{tabular} [c]{l}$\ \ \ \ \overset{n}{\circ}$\\ $\ \ \ /$\\ $\overset{n-2}{\circ}$\\ $\ \ \ \backslash$\\ \ \ \ $\underset{n-1}{\circ}$\end{tabular} .$$ Write $W_{n}^{B}$ and $W_{n}^{D}$ for the Weyl groups of $so_{2n+1}$ and $so_{2n}$. Denote by $V_{n}^{B}$ and $V_{n}^{D}$ the vector representations of $U_{q}(so_{2n+1})$ and $U_{q}(so_{2n}).$ Their crystal graphs are respectively: $$1\overset{1}{\rightarrow}2\cdot\cdot\cdot\rightarrow n-1\overset {n-1}{\rightarrow}n\overset{n}{\rightarrow}0\overset{n}{\rightarrow}\overline{n}\overset{n-1}{\rightarrow}\overline{n-1}\overset{n-2}{\rightarrow }\cdot\cdot\cdot\rightarrow\overline{2}\overset{1}{\rightarrow}\overline{1} \label{vect_B}$$ and $$1\overset{1}{\rightarrow}2\overset{2}{\rightarrow}\cdot\cdot\cdot\overset {n-3}{\rightarrow}n-2\overset{n-2}{\rightarrow} \begin{tabular} [c]{c}$\overline{n}$ \ \ \\ \ \ $\overset{n}{\nearrow}$ $\ \ \ \overset{n-1}{\text{ \ }\searrow}$ \ \ \ \\ $n-1\ \ \ \ \ \ \ \ \ \ \overline{n-1}$\\ \ $\underset{n-1}{\searrow}$ \ \ \ $\underset{n}{\nearrow}$ \ \ \ \\ $n$ \ \end{tabular} \overset{n-2}{\rightarrow}\overline{n-2}\overset{n-3}{\rightarrow}\cdot \cdot\cdot\overset{2}{\rightarrow}\overline{2}\overset{1}{\rightarrow }\overline{1}. \label{vect_D}$$ By induction, formulas (\[TENS1\]), (\[TENS2\]) allow to define a crystal graph for the representations $(V_{n}^{B})^{\otimes l}$ and $(V_{n}^{D})^{\otimes l}$ for any $l$. Each vertex $u_{1}\otimes u_{2}\otimes \cdot\cdot\cdot\otimes u_{l}$ of the crystal graph of $(V_{n}^{B})^{\otimes l}$ will be identified with the word $u_{1}u_{2}\cdot\cdot\cdot u_{l}$ on the totally ordered alphabet $$\mathcal{B}_{n}=\{1\prec\cdot\cdot\cdot\prec n-1\prec n\prec0\prec\overline {n}\prec\overline{n-1}\prec\cdot\cdot\cdot\prec\overline{1}\}.$$ Similarly each vertex $v_{1}\otimes v_{2}\otimes\cdot\cdot\cdot\otimes v_{l}$ of the crystal graph of $(V_{n}^{D})^{\otimes l}$ will be identified with the word $v_{1}v_{2}\cdot\cdot\cdot v_{l}$ on the partially ordered alphabet $$\mathcal{D}_{n}=\{1\prec\cdot\cdot\cdot\prec n-1\prec\begin{array} [c]{l}n\\ \overline{n}\end{array} \prec\overline{n-1}\prec\cdot\cdot\cdot\prec\overline{1}\}.$$ By convention we set $\overline{0}=0$ and for $k=1,\cdot\cdot\cdot,n,$ $\overline{\overline{k}}=k$. The letter $x$ is barred if $x\succeq\overline {n}$ unbarred if $x\preceq n$ and we set: $$\left| x\right| =\left\{ \begin{tabular} [c]{l}$x$ if $x$ is unbarred\\ $\overline{x}$ otherwise. \end{tabular} \right.$$ Write $\mathcal{B}_{n}^{\ast}$ and $\mathcal{D}_{n}^{\ast}$ for the free monoids on $\mathcal{B}_{n}$ and $\mathcal{D}_{n}$. If $w$ is a word of $\mathcal{B}_{n}^{\ast}$ or $\mathcal{D}_{n}^{\ast}$, we denote by $\mathrm{l}(w)$ its length and by $d(w)=(d_{1},...,d_{n})$ the $n$-tuple where $d_{i}$ is the number of letters $i$ in $w$ minus the number of letters $\overline{i} $. Let $G_{n}^{B}$ and $G_{n,l}^{B}$ be respectively the crystal graphs of $\underset{l}{\bigoplus}(V_{n}^{B})^{\otimes l}$ and $(V_{n}^{B})^{\otimes l}$. Then the vertices of $G_{n}^{B}$ are indexed by the words of $\mathcal{B}_{n}^{\ast}$ and those of $G_{n,l}^{B}$ by the words of $\mathcal{B}_{n}^{\ast}$ of length $l$. Similarly $G_{n}^{D}$ and $G_{n,l}^{D}$, the crystal graphs of $\underset{l}{\bigoplus}(V_{n}^{B})^{\otimes l}$ and $(V_{n}^{B})^{\otimes l}$ are indexed respectively by the words of $\mathcal{D}_{n}^{\ast}$ and by the words of $\mathcal{D}_{n}^{\ast}$ of length $l$. If $w$ is a vertex of $G_{n}$, write $B(w)$ for the connected component of $G_{n}$ containing $w$. Denote by $\Lambda_{1}^{B},...,\Lambda_{n}^{B}$ and $\Lambda_{1}^{D}...,\Lambda_{n}^{D}$ the fundamental weights of $U_{q}(so_{2n+1})$ and $U_{q}(so_{2n}).$ Let $P_{+}^{B}$ and $P_{+}^{D}$ be the sets of dominant weights of their weight lattices. We set $$\begin{aligned} \omega_{n}^{B} & =2\Lambda_{n}^{B},\\ \omega_{i}^{B} & =\Lambda_{i}^{B}\text{ for }i=1,...,n-1\end{aligned}$$ and $$\begin{aligned} \omega_{n}^{D} & =2\Lambda_{n}^{D},\\ \overline{\omega}_{n}^{D} & =2\Lambda_{n-1}^{D},\\ \omega_{n-1}^{D} & =\Lambda_{n}^{D}+\Lambda_{n-1}^{D},\\ \omega_{i}^{D} & =\Lambda_{i}^{D}\text{ for }i=1,...,n-2.\end{aligned}$$ Then the weight of a vertex $w$ of $G_{n}$ is given by: $$\mathrm{wt}(w)=d_{n}\omega_{n}+\overset{n-1}{\underset{i=1}{\sum}}(d_{i}-d_{i+1})\omega_{i}.$$ Thus we recover the well-known fact that there is no connected component of $G_{n}^{B}$ isomorphic to $B(\Lambda_{n}^{B})$ and no connected component of $G_{n}^{D}$ isomorphic to $B(\Lambda_{n}^{D})$ or $B(\Lambda_{n-1}^{D})$. Recall that in the cases of the types $A$ and $C,$ every crystal graph of an irreducible module may be embedded in the crystal graph of a tensor power of the vector representation. For $\lambda\in P_{+}^{B}$,$\ B^{B}(\lambda)$ may be embedded in a tensor power of the vector representation $V_{n}^{B}$ if and only if $\lambda$ lies in the weight sub-lattice $\Omega^{B}$ generated by the $\omega_{i}^{B}$’s. Similarly, for $\lambda\in P_{+}^{D},$ $B^{D}(\lambda)$ may be embedded in a tensor power of the vector representation $V_{n}^{D}$ if and only if $\lambda$ lies in the weight sub-lattice $\Omega^{D}$ generated by the $\omega_{i}^{D}$’s. Set $\Omega_{+}^{B}=P_{+}^{B}\cap\Omega^{B}$ and $\Omega_{+}^{D}=P_{+}^{D}\cap\Omega^{D}$. Now we introduce the coplactic relation. For $w_{1}$ and $w_{2}\in$ $\mathcal{B}_{n}^{\ast}$ (resp. $\mathcal{D}_{n}^{\ast}$), write $w_{1}\overset{B}{\longleftrightarrow}w_{2}\ $(resp. $w_{1}\overset {D}{\longleftrightarrow}w_{2}$) if and only if $w_{1}$ and $w_{2}$ belong to the same connected component of $G_{n}^{B}$ (resp. $G_{n}^{D}$). The proof of the following lemma is the same as in the symplectic case [@Lec]. \[lem\_coplactic\]If $w_{1}=u_{1}v_{1}$ and $w_{2}=u_{2}v_{2}$ with $l(u_{1})=l(u_{2})$ and $l(v_{1})=l(v_{2})$$$w_{1}\longleftrightarrow w_{2}\Longrightarrow\left\{ \begin{tabular} [c]{l}$u_{1}\longleftrightarrow u_{2}$\\ $v_{1}\longleftrightarrow v_{2}$\end{tabular} \right. .$$ Crystal graphs of the spin representations ------------------------------------------ The spin representations of $U_{q}(so_{2n+1})$ and $U_{q}(so_{2n})$ are $V(\Lambda_{n}^{B}),$ $V(\Lambda_{n}^{D})$ and $V(\Lambda_{n-1}^{D})$. Recall that $\dim V(\Lambda_{n}^{B})=2^{n}$ and $\dim V(\Lambda_{n}^{D})=\dim V(\Lambda_{n-1}^{D})=2^{n-1}$. Now we review the description of $B(\Lambda _{n}^{B}),$ $B(\Lambda_{n}^{D})$ and $B(\Lambda_{n-1}^{D})$ given by Kashiwara and Nakashima in [@KN]. It is based on the notion of spin column. To avoid confusion between these new columns and the classical columns of a tableau that we introduce in the next section, we follow Kashiwara-Nakashima’s convention and represent spin columns by column shape diagrams of width $1/2$. Such diagrams will be called K-N diagrams. A spin column $\frak{C}$ of height $n$ is a K-N diagram containing $n$ letters of $\mathcal{D}_{n}$ such that the word $x_{1}\cdot\cdot\cdot x_{n}$ obtained by reading $\frak{C}$ from top to bottom does not contain a pair $(z,\overline{z})$ and verifies $x_{1}\prec\cdot\cdot\cdot\prec x_{n}$. The set of spin columns of length $n$ will be denoted $SP_{n}$. $B(\Lambda_{n}^{B})=\{\frak{C;}$ $\frak{C}\in SP_{n}\}$ where Kashiwara’s operators act as follows: if $n\in\frak{C}$ then $\widetilde{f}_{n}\frak{C}$ is obtained by turning $n$ into $\overline{n}$, otherwise $\widetilde{f}_{n}\frak{C}=0$, if $\overline{n}\in\frak{C}$ then $\widetilde{e}_{n}\frak{C}$ is obtained by turning $\ \overline{n}$ into $n,$ otherwise $\widetilde{e}_{n}\frak{C}=0$, if $(i,\overline{i+1})\in\frak{C}$ then $\widetilde{f}_{i}\frak{C}$ is obtained by turning $(i,\overline{i+1})$ into $(i+1,\overline{i})$, otherwise $\widetilde{f}_{i}\frak{C}=0$, if $(i+1,\overline{i})\in\frak{C}$ then $\widetilde{e}_{i}\frak{C}$ is obtained by turning $(i+1,\overline{i})$ into $(i,\overline{i+1})$, otherwise $\widetilde{e}_{i}\frak{C}=0$. $B(\Lambda_{n}^{D})=\{\frak{C}\in SP_{n};$ the number of barred letters in $\frak{C}$ is even$\}$ and $B(\Lambda_{n-1}^{D})=\{\frak{C}\in SP_{n};$ the number of barred letters in $\frak{C}$ is odd$\}$ where Kashiwara’s operators act as follows: if $(n-1,n)\in\frak{C}$ then $\widetilde{f}_{n}\frak{C}$ is obtained by turning $(n-1,n)\ $into $(\overline{n},\overline{n-1})$, otherwise $\widetilde{f}_{n}\frak{C}=0$, if $(\overline{n},\overline{n-1})\in\frak{C}$ then $\widetilde{e}_{n}\frak{C}$ is obtained by turning $(\overline{n},\overline{n-1})$ into $(n-1,n)$, otherwise $\widetilde{e}_{n}\frak{C}=0$, for $i\neq n,$ $\widetilde{f}_{i}$ and $\widetilde{e}_{i}$ act like in $B(\Lambda_{n}^{B}).$ \[ptb\] [Figure1.eps]{} In the sequel we denote by $v_{\Lambda_{n}}^{B}$ the highest weight vertex of $B(\Lambda_{n}^{B})$, by $v_{\Lambda_{n}}^{D}$ and $v_{\Lambda_{n-1}}^{D}$ the highest weight vertices of $B(\Lambda_{n}^{D})$ and $B(\Lambda_{n-1}^{D}) $. Note that $v_{\Lambda_{n}}^{B}$ and $v_{\Lambda_{n}}^{D}$ correspond to the spin column containing the letters of $\{1,...,n\}$ and $v_{\Lambda_{n-1}}^{D}$ corresponds to the spin column containing the letters of $\{1,...,n-1,\overline{n}\}$. \[sec\_in\_vect\]Schensted correspondences in $G_{n}^{B}$ and $G_{n}^{D}$ ========================================================================= Orthogonal tableaux ------------------- Let $\lambda\in\Omega_{+}$. We are going to review the notion of standard orthogonal tableaux introduced by Kashiwara and Nakashima [@KN] to label the vertices of $B(\lambda)$. ### \[subsubsec\_colo\]Columns and admissible columns A column of type $B$ is a Young diagram $$C= \begin{tabular} [c]{|l|}\hline $x_{1}$\\\hline $\cdot$\\\hline $\cdot$\\\hline $x_{l}$\\\hline \end{tabular}$$ of column shape filled by letters of $\mathcal{B}_{n}$ such that $C$ increases from top to bottom and $0$ is the unique letter of $\mathcal{B}_{n} $ that may appear more than once. A column of type $D$ is a Young diagram $C$ of column shape filled by letters of $\mathcal{D}_{n}$ such that $x_{i+1}\nleqslant x_{i}$ for $i=1,...,l-1$. Note that the letters $n$ and $\overline{n}$ are the unique letters that may appear more than once in $C$ and if they do, these letters are different in two adjacent boxes. The height $h(C)$ of the column $C$ is the number of its letters. The word obtained by reading the letters of $C$ from top to bottom is called the reading of $C$ and denoted by $(C)$. We will say that the column $C$ contains a pair $(z,\overline{z})$ when a letter $0$ or the two letters $z\preceq n$ and $\overline{z}$ appear in $C$. (Kashiwara-Nakashima) Let $C$ be a column such that $(C)=x_{1}\cdot\cdot\cdot x_{h(C)}$. Then $C$ is admissible if $h(C)\leq n$ and for any pair $(z,\overline{z})$ of letters in $C$ satisfying $z=x_{p}$ and $\overline{z}=x_{q}$ with $z\preceq n$ we have $$\left| q-p\right| \geq h(C)-z+1. \label{cond_admissi}$$ (Note that $0\succ n$ on $\mathcal{B}_{n}$ and we may have $q-p<0$ for type $D$ and $z=n).$ For $n=4,$ $40\bar{4}\bar{2}$ and $3\overline{4}4\overline{3}$ are readings of admissible columns respectively of type $B$ and $D$. Let $C$ be a column of type $B$ or $D$ and $z\preceq n$ a letter of $C$. We denote by $N(z)$ the number of letters $x$ in $C$ such that $x\preceq z$ or $x\succeq\overline{z}$. Then Condition (\[cond\_admissi\]) is equivalent to $N(z)\leq z.$ Suppose that $C$ is non admissible and does not contain a pair $(z,\overline{z})$ with $z\preceq n$ and $N(z)>z.$ Then $h(C)>n$. Hence $C$ is of type $B$ and $0\in C.$ Indeed, if $0\notin C,$ $C$ contains a letter $z$ maximal such that $z\preceq n$ and $\overline{z}\in C$. It means that for any $x\in\{z+1,...,n\},$ there is at most one letter $y\in C$ with $\left| y\right| =x.$ We have a contradiction because in this case $N(z)>n-(n-z)$. We obtain the \[not\_N(z)\]A column $C$ is non admissible if and only if at least one of the following assertions is satisfied: $\mathrm{(i)}:C$ contains a letter $z\preceq n$ and $N(z)>z$ $\mathrm{(ii)}:C$ is of type $B,$ $0\in C$ and $h(C)>n.$ If we set $v_{\omega_{k}}^{B}=1\cdot\cdot\cdot k$ for $k=1,...,n,$ then $B(v_{\omega_{k}}^{B})$ is isomorphic to $B(\omega_{k}^{B})$. Similarly, if we set $v_{\omega_{k}}^{D}=1\cdot\cdot\cdot k$ for $k=1,...,n$ and $v_{\overline{\omega}_{n}}^{D}=1\cdot\cdot\cdot(n-1)\overline{n},$ then $B(v_{\omega_{k}}^{D})$ and $B(v_{\overline{\omega}_{n}}^{D})$ are respectively isomorphic to $B(\omega_{k}^{D})$ and $B(\overline{\omega}_{n}^{D})$. \[prop\_KNf\](Kashiwara-Nakashima) - The vertices of $B(v_{\omega_{k}}^{B})$ are the readings of the admissible columns of type $B$ and length $k$. - The vertices of $B(v_{\omega_{k}}^{D})$ with $k<n$ are the readings of the admissible columns of type $D$ and length $k$. - The vertices of $B(v_{\omega_{n}}^{D})$ are the readings of the admissible columns$\ C$ of type $D$ such that $\mathrm{w}(C)=x_{1}\cdot \cdot\cdot x_{n}$ and $x_{k}=n$ (resp. $x_{k}=\overline{n}$) implies $n-k$ is even (resp. odd). - The vertices of $B(v_{\overline{\omega}_{n}}^{D})$ are the readings of the admissible columns $C$ of type $D$ such that $\mathrm{w}(C)=x_{1}\cdot\cdot\cdot x_{n}$ and $x_{k}=\overline{n}$ (resp. $x_{k}=n$) implies $n-k$ is odd (resp. even). We can obtain another description of the admissible columns by computing, for each admissible column $C$, a pair of columns $(lC,rC)$ without pair $(z,\overline{z})$. This duplication was inspired by the description of the admissible columns of type $C$ in terms of De Concini columns used by Sheats in [@SH]. Let $C$ be a column of type $B$ and denote by $I_{C}=\{z_{1}=0,...,z_{r}=0\succ z_{r+1}\succ\cdot\cdot\cdot\succ z_{s}\}$ the set of letters $z\preceq0$ such that the pair $(z,\overline{z})$ occurs in $C$. We will say that $C$ can be split when there exists (see the example below) a set of $s$ unbarred letters $J_{C}=\{t_{1}\succ\cdot\cdot\cdot\succ t_{s}\}\subset \mathcal{B}_{n}$ such that: $\ \ \ t_{1}$ is the greatest letter of $\mathcal{B}_{n}$ satisfying: $t_{1}\prec z_{1},t_{1}\notin C$ and $\overline{t_{1}}\notin C,$    for $i=2,...,s$, $t_{i}$ is the greatest letter of $\mathcal{B}_{n}$ satisfying: $t_{i}\prec\min(t_{i-1,}z_{i}),$ $t_{i}\notin C$ and $\overline{t_{i}}\notin C.$ In this case we write: - $rC$ for the column obtained first by changing in $C$ $\overline{z}_{i}$ into $\overline{t}_{i}$ for each letter $z_{i}\in I,$ next by reordering if necessary. - $lC$ for the column obtained first by changing in $C$ $z_{i}$ into $t_{i}$ for each letter $z_{i}\in I,$ next by reordering if necessary. \[def\_C\_hat\]Let $C$ be a column of type $D$. Denote by $\widehat{C}$ the column of type $B$ obtained by turning in $C$ each factor $\overline{n}n$ into $00$. We will say that $C$ can be split when $\widehat{C}$ can be split. In this case we write $lC=l\widehat{C}$ and $rC=l\widehat{C}.$ \[exam\_splitting\]Suppose $n=9$ and consider the column $C$ of type $B$ such that $\mathrm{w}(C)=458900\bar{8}\bar{5}\bar{4}$. We have $I_{C}=\{0,0,8,5,4\}$ and $J_{C}=\{7,6,3,2,1\}$. Hence $$\mathrm{w}(lC)=123679\bar{8}\bar{5}\bar{4}\text{ and }\mathrm{w}(rC)=4589\bar{7}\bar{6}\bar{3}\bar{2}\bar{1}.$$ Suppose $n=8$ and consider the column $C^{\prime}$ of type $D$ such that $\mathrm{w}(C^{\prime})=56\bar{8}8\bar{8}\bar{6}\bar{5}\bar{2}$. Then $\mathrm{w}(\widehat{C^{\prime}})=5600\bar{8}\bar{6}\bar{5}\bar{2},$ $I_{\widehat{C^{\prime}}}=\{0,0,6,5\}$ and $J_{\widehat{C^{\prime}}}=\{7,4,3,1\}$. Hence $$\mathrm{w(}lC^{\prime})=1347\bar{8}\bar{6}\bar{5}\bar{2}\text{ and }\mathrm{w(}rC^{\prime})=56\bar{8}\bar{7}\bar{4}\bar{3}\bar{2}\bar{1}.$$ \[Lem\_dC\_impl\_Cadm\]Let $C$ be a column of type $B$ or $D$ which can be split . Then $C$ is admissible. Suppose $C$ of type $B$. We have $h(C)\leq n$ for $C$ can be split. If there exists a letter$\ z\prec0$ in $C$ such that the pair $(z,\overline{z})$ occurs in $C$ and $N(z)\geq z+1$, $C$ contains at least $z+1$ letters $x$ satisfying $\left| x\right| \preceq z$. So $rC$ contains at least $z+1$ letters $x^{\prime}$ satisfying $\left| x^{\prime}\right| \preceq z$. We obtain a contradiction because $rC$ does not contain a pair $(t,\overline{t})$. When $C$ is of type $D,$ by applying the lemma to $\widehat{C}$ we obtain that $\widehat{C}$ is admissible. So $C$ is admissible. The meaning of $lC$ and $rC$ is explained in the following proposition. \[prop\_imag\_S2\]Let $\omega\in\{\omega_{1}^{B},...,\omega_{n}^{B})$ or $\omega\in\{\omega_{1}^{D},...,\omega_{n-1}^{D},\omega_{n}^{D},\overline {\omega}_{n}^{D}\}$. The map $$S_{2}:B(v_{\omega})\rightarrow B(v_{\omega})\otimes B(v_{\omega})$$ defined in Theorem \[th\_strech\] satisfies for any admissible column $C\in B(v_{\omega})$: $$S_{2}(\mathrm{w(}C))=\mathrm{w(}rC)\otimes\mathrm{w(}lC).$$ Consider $\omega=\omega_{2}^{B}$ for $U_{q}(so_{5})$. The following graphs are respectively those of $B(\omega)$ and $S_{2}(B(\omega)).$$$\begin{tabular} [c]{l}$12\overset{2}{\rightarrow}10\overset{2}{\rightarrow}1\bar{2}\overset {1}{\rightarrow}2\bar{2}\overset{1}{\rightarrow}2\bar{1}$\\ \ \ \ \ \ \ \ \ $\downarrow1$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\downarrow2$\\ $\text{\ \ \ \ \ \ \ }20\overset{2}{\rightarrow}00\overset{2}{\rightarrow }0\bar{2}\overset{1}{\rightarrow}0\bar{1}\overset{2}{\rightarrow}\bar{2}\bar{1}$\end{tabular}$$$$\begin{tabular} [c]{l}$(12)\otimes(12)\overset{2^{2}}{\rightarrow}(1\bar{2})\otimes(12)\overset {2^{2}}{\rightarrow}(1\bar{2})\otimes(1\bar{2})\overset{1^{2}}{\rightarrow }(2\bar{1})\otimes(1\bar{2})\overset{1^{2}}{\rightarrow}(2\bar{1})\otimes(2\bar{1})$\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\downarrow1^{2}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\downarrow 2^{2}$\\ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2\bar{1})\otimes(12)\overset{2^{2}}{\rightarrow}(\bar{2}\bar{1})\otimes(12)\overset{2^{2}}{\rightarrow}(\bar {2}\bar{1})\otimes(1\bar{2})\overset{1^{2}}{\rightarrow}(\bar{2}\bar {1})\otimes(2\bar{1})\overset{2^{2}}{\rightarrow}(\bar{2}\bar{1})\otimes (\bar{2}\bar{1})$\end{tabular}$$ (of proposition \[prop\_imag\_S2\]) In this proof we identify each column with its reading to simplify the notations. When $C=v_{\omega}$ is the highest weight vertex of $B(v_{\omega})$, $r(v_{\omega})=l(v_{\omega})=v_{\omega}$ because $v_{\omega}$ does not contain a pair $(z,\overline{z})$. So $S_{2}(v_{\omega})=rC\otimes lC.$ Each vertex $C$ of $B(\omega)$ may be written $C=\widetilde{f}_{i_{1}}\cdot\cdot\cdot\widetilde{f}_{i_{r}}(v_{\omega})$. By induction on $r$, it suffices to prove that for any $\mathrm{w}(C)\in B(v_{\omega})$ such that $\widetilde{f}_{i}(C)\neq0$ we have $$S_{2}(C)=rC\otimes lC\Longrightarrow S_{2}(\widetilde{f}_{i}C)=r(\widetilde {f}_{i}C)\otimes l(\widetilde{f}_{i}C).$$ For any column $D$ we denote by $[D]_{i}$ the word obtained by erasing all the letters $x$ of $D$ such that $\widetilde{f}_{i}(x)=\widetilde{e}_{i}(x)=0$. It is clear that only the letters of $[D]_{i}$ may be changed in $D$ when we apply $\widetilde{f}_{i}$. Suppose $\omega\in\{\omega_{1}^{B},...,\omega_{n}^{B})$. Consider $C\in B(v_{\omega})$ such that $S_{2}(C)=rC\otimes lC$ and $\widetilde{f}_{i}(C)\neq0$. When $i\neq n$, the letters $x\notin\{\overline{i+1},\overline {i},i,i+1\}$ do not interfere in the computation of $\widetilde{f}_{i}$. It follows from the condition $\widetilde{f}_{i}(C)\neq0$ and an easy computation from (\[TENS1\]) and (\[TENS2\]) that we need only consider the following cases: $\mathrm{(i)}$$[C]_{i}=i$, $\mathrm{(ii)}$ $[C]_{i}=\overline{i+1}$, $\mathrm{(iii)}$ $[C]_{i}=(i+1)(\overline{i+1}),$ $\mathrm{(iv)}$ $[C]_{i}=(i)(\overline{i+1}),$ $\mathrm{(v)}$ $[C]_{i}=i(i+1)(\overline{i+1})$ and $\mathrm{(vi)}$ $[C]_{i}=i(\overline {i+1})\overline{i}$. In the case $\mathrm{(i)}$, if $i+1\notin J_{C},$ we have $[lC]_{i}=i$ and $[rC]_{i}=i$. Then $[\widetilde{f}_{i}(C)]_{i}=i+1$ and $J_{\widetilde{f}_{i}C}=J_{C}$ (hence $i\notin J_{\widetilde{f}_{i}C}$). So $[l(\widetilde{f}_{i}C)]_{i}=i+1$ and $[r(\widetilde{f}_{i}C)]_{i}=i+1.$ That means that $S_{2}(\widetilde{f}_{i}C)=\widetilde{f}_{i}^{2}(rC\otimes lC)=\widetilde{f}_{i}(rC)\otimes\widetilde{f}_{i}(lC)=r(\widetilde{f}_{i}C)\otimes l(\widetilde{f}_{i}C)$ by definition of the map $S_{2}$. If $i+1\in J_{C}$, we can write $[rC]_{i}=(i)(\overline{i+1})$ and $[lC]_{i}=(i)(i+1)$. Then $[\widetilde{f}_{i}C)]_{i}=i+1$ and $J_{\widetilde{f}_{i}C}=J_{C}-\{i+1\}+\{i\}$. So $[r(\widetilde{f}_{i}C)]=(i+1)(\overline{i})$ and $[l(\widetilde{f}_{i}C)]=(i)(i+1).$ Hence $S_{2}(\widetilde{f}_{i}C)=\widetilde{f}_{i}^{2}(rC\otimes lC)=\widetilde{f}_{i}^{2}(rC)\otimes lC=r(\widetilde{f}_{i}C)\otimes l(\widetilde{f}_{i}C)$. The proof is similar in the cases $\mathrm{(ii)}$ to $\mathrm{(vi)}$. When $i=n$, only the letters of $\{\overline{n},0,n\}$ interfere in the computation of $\widetilde{f}_{n}$. We obtain the proposition by considering the cases: $[C]_{n}=\underset{0\text{ p times}}{\underbrace{0\cdot\cdot\cdot0}},$ $[C]_{n}=n\underset{0\text{ p times}}{\underbrace{0\cdot\cdot\cdot0}}$ and $[C]_{n}=n$. Suppose $\omega\in\{\omega_{1}^{D},...,\omega_{n-1}^{D},\overline{\omega}_{n}^{D},\omega_{n}^{D}\}$. When $i<n-1$ the proof is the same than above. When $i\in\{n-1,n\},$ the proposition follows by considering successively the cases: $$\left\{ \begin{tabular} [c]{l}$\lbrack C]_{i}=n-1(\overline{n}n)^{r}$,\\ $\lbrack C]_{i}=n(\overline{n}n)^{r}\overline{n}$,\\ $\lbrack C]_{i}=(n-1)n(\overline{n}n)^{r}\overline{n}$,\\ $\lbrack C]_{i}=(\overline{n}n)^{r}\overline{n}$,\\ $\lbrack C]_{i}=(n-1)(\overline{n}n)^{r}\overline{n}$,\\ $\lbrack C]_{i}=(n-1)(\overline{n}n)^{r}\overline{n}(\overline{n-1}).$\end{tabular} \right. \text{ if }i=n-1\text{ and }\left\{ \begin{tabular} [c]{l}$\lbrack C]_{i}=n-1(n\overline{n})^{r}$,\\ $\lbrack C]_{i}=\overline{n}(n\overline{n})^{r}n$,\\ $\lbrack C]_{i}=(n-1)\overline{n}(n\overline{n})^{r}n$,\\ $\lbrack C]_{i}=(n\overline{n})^{r}n$,\\ $\lbrack C]_{i}=(n-1)(n\overline{n})^{r}n$,\\ $\lbrack C]_{i}=(n-1)(n\overline{n})^{r}n(\overline{n-1}).$\end{tabular} \right. \text{ if }i=n.$$ where $(\overline{n}n)^{r}$ (resp. $(n\overline{n})^{r}$) is the word containing the factor $\overline{n}n$ (resp. $n\overline{n}$) repeated $r$ times. Using Lemma \[Lem\_dC\_impl\_Cadm\] we derive immediately the A column $C$ of type $B$ or $D$ is admissible if and only if it can be split. \[exa\_splitC\]From Example \[exam\_splitting\], we obtain that $C$ is admissible for $n=9$ and $C^{\prime}$ is admissible for $n=8.$ With the notations of the previous proposition, denote by $W_{n}/W_{\omega}$ the set of cosets of the Weyl group $W_{n}$ with respect to the stabilizer $W_{\omega}$ of $\omega$ in $W_{n}$. Then we obtain a bijection $\tau$ between the orbit *O*$_{\omega}$* *of $v_{\omega}$ in $B(\omega)$ under the action of $W_{n}$ defined by (\[actionW\]) and $W_{n}/W_{\omega}$. Using Formulas (\[actionW\]) it is easy to prove that *O*$_{\omega}$ consists of the vertices of $B(v_{\omega})$ without pair $(z,\overline{z})$. Moreover if $C_{1},C_{2}$ are two columns such that $\mathrm{w(}C_{1})=x_{1}\cdot\cdot\cdot x_{p},\mathrm{w(}C_{2})=y_{1}\cdot\cdot\cdot y_{p}\in\textsl{O}_{\omega}$, we have $$C_{1}\preceq C_{2}\Longleftrightarrow\tau_{\mathrm{w(}C_{1})}\vartriangleleft _{\omega}\tau_{\mathrm{w(}C_{2})}$$ where $C_{1}\preceq C_{2}$ means that $x_{i}\preceq y_{i},i=1,...,p$ and $"\vartriangleleft_{\omega}"$ denotes the projection of the Bruhat order on $W_{n}/W_{\omega}.$ Then Proposition \[prop\_imag\_S2\] may be regarded as a version of Littelmann’s labelling of $B(v_{\omega})$ by pairs $(\tau _{\mathrm{w(}rC)},\tau_{\mathrm{w(}lC)})\in W_{n}/W_{\omega}\times W_{n}/W_{\omega}$ satisfying $\tau_{\mathrm{w(}lC)}\vartriangleleft_{\omega }\tau_{\mathrm{w(}rC)}$ [@Lit2]. ### Orthogonal tableaux Every $\lambda\in\Omega_{+}^{B}$ has a unique decomposition of the form $\lambda=\overset{n}{\underset{i=1}{\sum}}\lambda_{i}\omega_{i}^{B}$. Similarly, every $\lambda\in\Omega_{+}^{D}$ has a unique decomposition of the form $\mathrm{(\ast)}$ $\lambda=\overset{n}{\underset{i=1}{\sum}}\lambda_{i}\omega_{i}^{D}$ or $\mathrm{(\ast\ast)}$ $\lambda=\lambda_{n}\bar{\omega}_{n}^{D}+\overset{n-1}{\underset{i=1}{\sum}}\lambda_{i}\omega _{i}^{D} $ with $\lambda_{n}\neq0,$ where $(\lambda_{n},...,\lambda_{n})\in\mathbb{N}^{n}.$ We will say that $(\lambda_{1},...,\lambda_{n})$ is the positive decomposition of $\lambda\in\Omega_{+}$. Denote by $Y_{\lambda}$ the Young diagram having $\lambda_{i}$ columns of height $i$ for $i=1,...,n$. If $\lambda\in\Omega_{+}^{D},$ $Y_{\lambda}$ may not suffice to characterize the weight $\lambda$ because a column diagram of length $n$ may be associated to $\omega_{n}$ or to $\overline{\omega}_{n}$. In Subsection \[sub\_sec\_Cor\_in\_Gn\] we will need to attach to each dominant weight $\lambda\in\Omega_{+}$ a combinatorial object $Y(\lambda)$. Moreover it will be convenient to distinguish in $\mathrm{(\ast)}$ the cases where $\lambda _{n}=0$ or $\lambda_{n}\neq0$. This leads us to set: $$\begin{aligned} \mathrm{(i)} & :Y(\lambda)=Y_{\lambda}\text{ if }\lambda\in\Omega_{+}^{B},\nonumber\\ \mathrm{(ii)} & :Y(\lambda)=(Y_{\lambda},+)\text{ in case }\mathrm{(\ast )}\text{ with }\lambda_{n}\neq0,\nonumber\\ \mathrm{(iii)} & :Y(\lambda)=(Y_{\lambda},0)\text{ in case }\mathrm{(\ast )}\text{ with }\lambda_{n}=0,\label{Def_Y(lambda)}\\ \mathrm{(iv)} & :Y(\lambda)=(Y_{\lambda},-)\text{ in case }\mathrm{(\ast \ast).}\nonumber\end{aligned}$$ When $\lambda\in\Omega_{+}^{D},$ $Y(\lambda)$ may be regarded as the generalization of the notion of the shape of type $A$ associated to a dominant weight. Now write $$\begin{aligned} v_{\lambda}^{B} & =(v_{\omega_{1}^{B}})^{\otimes\lambda_{1}}\otimes \cdot\cdot\cdot\otimes(v_{\omega_{n}^{B}})^{\otimes\lambda_{n}}\text{ in case }\mathrm{(i),}\\ v_{\lambda}^{D} & =(v_{\omega_{1}^{D}})^{\otimes\lambda_{1}}\otimes \cdot\cdot\cdot\otimes(v_{\omega_{n}^{D}})^{\otimes\lambda_{n}}\text{ in case$\mathrm{(ii)}$,}\\ v_{\lambda}^{D} & =(v_{\omega_{1}^{D}})^{\otimes\lambda_{1}}\otimes \cdot\cdot\cdot\otimes(v_{\omega_{n-1}^{D}})^{\otimes\lambda_{n-1}}\text{ in case $\mathrm{(iii)}$ and}\\ v_{\lambda}^{D} & =(v_{\omega_{1}^{D}})^{\otimes\lambda_{1}}\otimes \cdot\cdot\cdot\otimes(v_{\overline{\omega}_{n}^{D}})^{\otimes\lambda_{n}}\text{ in case $\mathrm{(iv).}$}$$ Then $v_{\lambda}^{B}$ and $v_{\lambda}^{D}$ are highest weight vertices of $G_{n}^{B}$ and $G_{n}^{D}.$ Moreover $B(v_{\lambda}^{B})$ and $B(v_{\lambda }^{D})$ are isomorphic to $B^{B}(\lambda)$ and $B^{D}(\lambda)$. A tabloid $\tau$ of type $B$ (resp. $D$) is a Young diagram whose columns are filled to give columns of type $B$ (resp. $D$). If $\tau=C_{1}\cdot\cdot\cdot C_{r}$, we write $(T)=$$(C_{r})\cdot\cdot\cdot$$(C_{1})$ for the reading of $\tau$. \[defKN\]        - Consider $\lambda\in\Omega_{+}^{B}$. A tabloid $T$ of type $B$ is an orthogonal tableau of shape $Y(\lambda)$ and type $B$ if $\mathrm{w(}T\mathrm{)}\in$ $B(v_{\lambda}^{B})$. - Consider $\lambda\in\Omega_{+}^{D}$. A tabloid $T$ of type $D$ is an orthogonal tableau of shape $Y(\lambda)$ and type $D$ if $\mathrm{w(}T\mathrm{)}\in$ $B(v_{\lambda}^{D})$. The orthogonal tableaux of a given shape form a single connected component of $G_{n}$, hence two orthogonal tableaux whose readings occur at the same place in two isomorphic connected components of $G_{n}$ are equal. The shape of an orthogonal tableau $T$ of type $D$ may be regarded as a pair $[O_{T},\varepsilon_{T}]$ where $O_{T}$ is a Young diagram and $\varepsilon_{T}\in\{-,0,+\}$. The $\{-,0,+\}$ part of this shape can be read off directly on $T$. Indeed $\varepsilon=0$ if $T$ does not contain a column of height $n.$ Otherwise write $\mathrm{w(}C_{1})=x_{1}\cdot\cdot\cdot x_{n}$ for the reading of the first column of $T.$ Since it is admissible, $C_{1}$ contains at least a letter, say $x_{k}$ of $\{n,\overline{n}\}.$ Then $\varepsilon$ is given by the parity of $n-k$ according to Proposition \[prop\_KNf\]. Consider $\tau=C_{1}C_{2}\cdot\cdot\cdot C_{r}$ a tabloid whose columns are admissible. The split form of $\tau$ is the tabloid obtained by splitting each column of $\tau$. We write $\mathrm{spl}(\tau)=(lC_{1}rC_{1})(lC_{2}rC_{2})\cdot\cdot\cdot(lC_{r}rC_{r})$. With the notations of Proposition \[prop\_imag\_S2\], we will have $\mathrm{w}(\mathrm{spl}(T))=S_{2}\mathrm{w(}C_{r})\cdot\cdot\cdot S_{2}\mathrm{w(}C_{1}) $. Kashiwara-Nakashima’s combinatorial description [@KN] of an orthogonal tableau $T$ is based on the enumeration of configurations that should not occur in two adjacent columns of $T$. Considering its split form $\mathrm{spl}(T)$, this description becomes more simple because the columns of $\mathrm{spl}(T)$ does not contain any pair $(z,\overline{z})$. \[lem\_split\_tab\]Let $T=C_{1}C_{2}\cdot\cdot\cdot C_{r}$ be a tabloid whose columns are admissible. Then $T$ is an orthogonal tableau if and only if $\mathrm{spl}(T)$ is an orthogonal tableau. Suppose first that $\mathrm{w(}T)$ is a highest weight vertex of weight $\lambda$. Then, by Corollary \[cor\_strech\], $\mathrm{w(spl}(T))$ is a highest weight vertex of weight $2\lambda$. If $T$ is an orthogonal tableau, $\mathrm{w(}T)=v_{\lambda}$ and we have $\mathrm{w(spl}(T))=v_{2\lambda}$. So $\mathrm{spl}(T)$ is an orthogonal tableau. Conversely, if $\mathrm{spl}(T)$ is an orthogonal tableau, $\mathrm{w(spl}(T))=S_{2}\mathrm{w(}C_{r})\cdot \cdot\cdot S_{2}\mathrm{w(}C_{1})$ is a highest weight vertex of weight $2\lambda$ by Corollary \[cor\_strech\]. Hence we have $\mathrm{w(spl}(T))=v_{2\lambda}$ because there exists only one orthogonal tableau of highest weight $2\lambda$. So $\mathrm{w(}T)=v_{\lambda}$. In the general case, denote by $T_{0}$ the tableau such that $\mathrm{w(}T_{0})$ is the highest weight vertex of the connected component of $G_{n}$ containing $\mathrm{w(}T)$. Then $\mathrm{w(spl}(T_{0}))$ is the highest weight vertex of the connected component containing $\mathrm{w(spl}(T))$ and the following assertions are equivalent: $$\begin{aligned} & \text{$\mathrm{(i)}$\textrm{\ }}\mathrm{spl}(T)\text{ is an orthogonal tableau,}\\ & \text{$\mathrm{(ii)}$\textrm{\ }}\mathrm{spl}(T_{0})\text{ is orthogonal tableau,}\\ & \text{$\mathrm{(iii)}$\textrm{\ }}T_{0}\text{ is orthogonal tableau,}\\ & \text{$\mathrm{(iv)}$\textrm{\ }}T\text{ is orthogonal tableau.}$$ Let $\tau=C_{1}C_{2}$ be a tabloid with two admissible columns $C_{1}$ and $C_{2}$. We set: - $C_{1}\preceq C_{2}$ when $h(C_{1})\geq h(C_{2})$ and the rows of $C_{1}C_{2}$ are weakly increasing from left to right, - $C_{1}\trianglelefteq C_{2}$ when $rC_{1}\preceq lC_{2}.$ \[Def\_b\_conf\](Kashiwara-Nakashima) Let $C_{1}=\begin{tabular} [c]{|l|}\hline $x\_[1]{}$\\\hline $$\\\hline $$\\\hline $x\_[N]{}$\\\hline \end{tabular} $ and $C_{2}=\begin{tabular} [c]{|l|}\hline $y\_[1]{}$\\\hline $$\\\hline $$\\\hline $y\_[N]{}$\\\hline \end{tabular} $ be admissible columns of type $D$ and $p,q,r,s$ integers satisfying $1\leq p\leq q<r\leq s\leq M$. $C_{1}C_{2}$ contains an $a$-odd-configuration (with $a\notin\{\overline {n},n\}$) when: - $a=x_{p},\overline{n}=x_{r}$ are letters of $C_{1}$ and $\overline {a}=y_{s},n=y_{q}$ letters of $C_{2}$ such that $r-q+1$ is odd or - $a=x_{p},n=x_{r}$ are letters of $C_{1}$ and $\overline{a}=y_{s},\overline{n}=y_{q}$ letters of $C_{2}$ such that $r-q+1$ is odd $C_{1}C_{2}$ contains an $a$-even-configuration (with $a\notin\{\overline {n},n\}$) when: - $a=x_{p},n=x_{r}$ are letters of $C_{1}$ and $\overline{a}=y_{s},n=y_{q}$ letters of $C_{2}$ such that $r-q+1$ is even or - $a=x_{p},\overline{n}=x_{r}$ are letters of $C_{1}$ and $\overline {a}=y_{s},\overline{n}=y_{q}$ letters of $C_{2}$ such that $r-q+1$ is even Then we denote by $\mu(a)$ the positive integer defined by: $$\mu(a)=s-p$$      \[TH\_KN\] $\mathrm{(i)}$ Consider $C_{1},C_{2},...,C_{r}$ some admissible columns of type $B$. Then the tabloid $T=C_{1}C_{2}\cdot\cdot\cdot C_{r}$ is an orthogonal tableau if and only if $C_{i}\trianglelefteq C_{i+1}$ for $i=1,...,r-1.$ $\mathrm{(ii)}$ Consider $C_{1},C_{2},...,C_{r}$ some admissible columns of type $D$. Then the tabloid $T=C_{1}C_{2}\cdot\cdot\cdot C_{r}$ is an orthogonal tableau if and only if, $C_{i}\trianglelefteq C_{i+1}$ for $i=1,...,r-1$, and $rC_{i}lC_{i+1}$ does not contain an $a$-configuration (even or odd) such that $\mu(a)=n-a$. Kashiwara and Nakashima describe an orthogonal tableau $T$ by listing the configurations that should not occur in two adjacent columns of $T$. If we except the $a$-configurations even or odd, these configurations disappear in $\mathrm{spl}(T)$ because $\mathrm{spl}(T)$ does not contain a column with a pair $(z,\overline{z}).$ Hence the theorem follows from Lemma \[lem\_split\_tab\] and Theorems 5.7.1, 6.7.1 of [@KN]. Suppose $n=4.$ Then $T=\begin{tabular} [c]{|l|ll}\hline $$ & $$ & \multicolumn{1}{|l|}{$\mathtt{4}$}\\\hline $$ & $$ & \multicolumn{1}{|l|}{$\mathtt{\bar{4}}$}\\\hline $$ & $$ & \multicolumn{1}{|l}{}\\\cline{1-2}$$ & & \\\cline{1-1}\end{tabular} $ is an orthogonal tableau of type $B$ because $\mathrm{spl}(T)=\begin{tabular} [c]{|l|l|llll}\hline $$ & $$ & $$ & \multicolumn{1}{|l}{$\mathtt{3}$}& \multicolumn{1}{|l}{$\mathtt{3}$} & \multicolumn{1}{|l|}{$\mathtt{4}$}\\\hline $$ & $$ & $$ & \multicolumn{1}{|l}{$\mathtt{\bar {4}}$} & \multicolumn{1}{|l}{$\mathtt{\bar{4}}$} & \multicolumn{1}{|l|}{$\mathtt{\bar{3}}$}\\\hline $$ & $$ & $$ & \multicolumn{1}{|l}{$\mathtt{\bar{2}}$} & \multicolumn{1}{|l}{} & \\\cline{1-2}\cline{1-4}$$ & $$ & & & & \\\cline{1-2}\end{tabular} \vspace{0.2cm}.$ But \[c\][|l|l|]{}$\mathtt{3}$ & $\mathtt{\bar{4}}$\ $\mathtt{\bar{4}}$ & $\mathtt{\bar{3}}$\ is not orthogonal of type $D$ because it contains a $3$-even configuration with $\mu(3)=1$. \[subsec\_monoids\]Plactic monoids for types $B_{n}$ and $D_{n}$ ---------------------------------------------------------------- \[def\_sam\_plac\]Let $w_{1}$ and $w_{2}$ be two words on $\mathcal{B}_{n}$ (resp. $\mathcal{D}_{n}$) . We write $w_{1}\overset{B}{\sim}w_{2}$ (resp. $w_{1}\overset{D}{\sim}w_{2}$) when these two words occur at the same place in two isomorphic connected components of the crystal $G_{n}^{B}$ (resp. $G_{n}^{D}$). The definition of the orthogonal tableaux implies that for any word $w\in\mathcal{B}_{n}^{\ast}$ (resp. $w\in\mathcal{D}_{n}^{\ast}$) there exists a unique orthogonal tableau $P^{B}(w)$ (resp. $P^{D}(w)$) such that $w\sim\mathrm{w(}P(w))$. So the sets $\mathcal{B}_{n}^{\ast}/\overset{B}{\sim }$ and $\mathcal{D}_{n}^{\ast}/\overset{D}{\sim}$ can be identified respectively with the sets of orthogonal tableaux of type $B$ and $D$. Our aim is now to show that $\overset{B}{\sim}$ and $\overset{D}{\sim}$ are in fact congruencies $\overset{B}{\equiv}$ and $\overset{D}{\equiv}$ so that $\mathcal{B}_{n}^{\ast}/\overset{B}{\sim}$ and $\mathcal{D}_{n}^{\ast }/\overset{D}{\sim}$ are in a natural way endowed with a multiplication. The monoid $Pl(B_{n})$ is the quotient of the free monoid $\mathcal{B}_{n}^{\ast}$ by the relations: $R_{1}^{B}:$ If $x\neq\overline{z}$ and $x\prec y\prec z:$$$yzx\overset{B}{\equiv}yxz\text{ \ and \ }xzy\overset{B}{\equiv}zxy\text{.}$$ $R_{2}^{B}:$ If $x\neq\overline{y}$ and $x\prec y:$$$xyx\overset{B}{\equiv}xxy\text{ for }x\neq0\text{ \ and \ }xyy\overset {B}{\equiv}yxy\text{ for }y\neq0.$$ $R_{3}^{B}:$If $1\prec x\preceq n$ and $x\preceq y\preceq\overline{x}:$ $$y(\overline{x-1})(x-1)\overset{B}{\equiv}yx\overline{x}\text{, \ and \ }x\overline{x}y\overset{B}{\equiv}(\overline{x-1})(x-1)y,$$$$0\overline{n}n\equiv\overline{n}n0.$$ $R_{4}^{B}:$If $x\preceq n:$$$00x\overset{B}{\equiv}0x0\text{ \ and \ }0\overline{x}0\overset{B}{\equiv }\overline{x}00.$$ $R_{5}^{B}:$ Let $w=\mathrm{w}(C)$ be a non admissible column word each strict factor of which is admissible. When $C$ satisfies the assertion $\mathrm{(i)}$ of Remark \[not\_N(z)\], let $z$ be the lowest unbarred letter of $w$ such that the pair $(z,\overline{z})$ occurs in $w$ and $N(z)>z$, otherwise set $z=0$. Then $w\overset{B}{\equiv}\widetilde{w}$ where $\widetilde{w}$ is the column word obtained by erasing the pair $(z,\overline{z})$ in $w$ if $z\preceq n,$ by erasing $0$ otherwise. The monoid $Pl(D_{n})$ is the quotient of the free monoid $\mathcal{D}_{n}^{\ast}$ by the relations: $R_{1}:$ If $x\neq\overline{z}$ $$yzx\overset{D}{\equiv}yxz\text{ for }x\preceq y\prec z\text{ \ and \ }xzy\overset{D}{\equiv}zxy\text{ for }x\prec y\preceq z.$$ $R_{2}:$ If $1\prec x\prec n$ and $x\preceq y\preceq\overline{x}$$$y(\overline{x-1})(x-1)\overset{D}{\equiv}yx\overline{x}\text{ \ and \ }x\overline{x}y\overset{D}{\equiv}(\overline{x-1})(x-1)y.$$ $R_{3}^{D}:$ If $x\preceq n-1:$$$\left\{ \begin{tabular} [c]{l}$\overline{n}\,\overline{x}n\overset{D}{\equiv}\overline{x}\,\overline{n}n$\\ $n\,\overline{x}\,\overline{n}\overset{D}{\equiv}\overline{x}\,n\overline{n}$\end{tabular} \right. \text{ and }\left\{ \begin{tabular} [c]{l}$\overline{n}nx\overset{D}{\equiv}\overline{n}xn$\\ $n\overline{n}x\overset{D}{\equiv}nx\overline{n}$\end{tabular} \right. .$$ $R_{4}^{D}:$$$\left\{ \begin{tabular} [c]{l}$n\overline{n}\,\overline{n}\overset{D}{\equiv}\overline{(n-1)}\,(n-1)\overline{n}$\\ $\overline{n}\,nn\overset{D}{\equiv}\overline{(n-1)}\,(n-1)n$\end{tabular} \right. \text{ and }\left\{ \begin{tabular} [c]{l}$\overline{n}(\overline{n-1})(n-1)\overset{D}{\equiv}\overline{n}\,\overline{n}n$\\ $n(\overline{n-1})(n-1)\overset{D}{\equiv}nn\,\overline{n}$\end{tabular} \right. .$$ $R_{5}^{D}:$ Consider $w$ a non admissible column word each strict factor of which is admissible. Let $z$ be the lowest unbarred letter such that the pair $(z,\overline{z})$ occurs in $w$ and $N(z)>z$ (see Remark \[not\_N(z)\]). Then $w\overset{D}{\equiv}\widetilde{w}$ where $\widetilde{w}$ is the column word obtained by erasing the pair $(z,\overline{z})$ in $w$ if $z\prec n$, by erasing a pair $(n,\overline{n})$ of consecutive letters otherwise. The relations $R_{5}^{B}$ and $R_{5}^{D}$ are called the contraction relations. When the letter $0$ or a pair $(n,\overline{n})$ disappears, we have $l(C)=n+1$ and in $R_{5}^{D}$ the word $\widetilde{w}$ does not depend on the factor $n\overline{n}$ or $\overline{n}n$ erased. Moreover $\widetilde{w}$ is an admissible column word. Note that $w_{1}\equiv w_{2}$ implies $d(w_{1})=d(w_{2})$, that is, $\equiv$ is compatible with the grading given by $d$. \[th\_Psymbol\]Given two words $w_{1}$ and $w_{2}$$$w_{1}\sim w_{2}\Longleftrightarrow w_{1}\equiv w_{2}\Longleftrightarrow P(w_{1})=P(w_{2}) \label{good_rela}$$ This theorem is proved in the same way as in the symplectic case [@Lec], and we will only sketch the arguments. Note first that we have$$w_{1}\sim w_{2}\Longleftrightarrow P(w_{1})=P(w_{2})$$ immediately from the definition of $P.$ For any word $w$ occurring in the left hand side of a relation $R_{1}^{B},...,R_{4}^{B}$ (resp. $R_{1}^{D},...,R_{4}^{D}$), write $\xi^{B}(w)$ (resp. $\xi^{D}(w)$) for the word occurring in the right hand side of this relation. Similarly for $p=1,...,n$ and $w$ a word of length $p+1$ occurring in the left hand side of $R_{5}^{B}$ (resp. $R_{5}^{D}$), denote by $\xi_{p}^{B}(w)$ (resp. $\xi_{p}^{D}(w)$) the word occurring in the right hand side of this relation. By using similar arguments to those of [@Lec], we obtain the followings assertions: - The map $\xi^{B}:w\longmapsto\xi(w)$ is the crystal isomorphism from $B^{B}(121)$ to $B^{B}(112)$. - If $n>2,$ the map $\xi^{D}:w\longmapsto\xi(w)$ is the crystal isomorphism from $B^{D}(121)$ to $B^{D}(112)$ otherwise $\xi^{D}$ is the crystal isomorphism from $B^{D}(121)\cup B^{D}(1\bar{2}1)$ to $B^{D}(112)\cup B^{D}(11\bar{2})$. - For $p=2,...,n-1$, $\xi_{p}:w\longmapsto\xi_{p}(w)$ is the crystal isomorphism from $B(12\cdot\cdot\cdot p\overline{p})$ to $B(12\cdot\cdot\cdot p-1)$. - The map $\xi_{n}^{B}:w\longmapsto\xi_{n}^{B}(w)$ is the crystal isomorphism from $B^{B}(12\cdot\cdot\cdot n\overline{n})\cup B^{B}(12\cdot\cdot\cdot n0)$ to $B^{B}(12\cdot\cdot\cdot n-1)\cup B^{B}(12\cdot\cdot\cdot n)$. - The words $w$ of length $n+1$ occurring in the left hand side of $R_{5}^{D}$ are the vertices of $B^{D}(12\cdot\cdot\cdot n\overline{n})\cup B^{D}(12\cdot\cdot\cdot\overline{n}n)$. Moreover the restriction of the map $\xi_{n}^{D}:w\longmapsto\xi_{n}^{D}(w)$ to $B^{D}(12\cdot\cdot\cdot n\overline{n})$ (resp. to $B^{D}(12\cdot\cdot\cdot\overline{n}n)$) is the crystal isomorphism from $B^{D}(12\cdot\cdot\cdot n\overline{n})$ (resp. $B^{D}(12\cdot\cdot\cdot\overline{n}n)$) to $B^{D}(12\cdot\cdot\cdot n-1)$. $$\begin{gathered} \begin{tabular} [c]{lllllllll}& & & & $121$ & & & & \\ & & & $\overset{\text{1}}{\swarrow}$ & & $\overset{\text{2}}{\searrow}$ & & & \\ & & $122$ & & & & $101$ & & \\ & $\overset{\text{2}}{\swarrow}$ & & & & & {\tiny 1}$\downarrow$ & $\overset{\text{2}}{\searrow}$ & \\ $102$ & & & & & & $201$ & & $1\bar{2}1$\\ {\tiny 2}$\downarrow$ & & & & & $\overset{\text{1}}{\swarrow}$ & {\tiny 2}$\downarrow$ & & {\tiny 1}$\downarrow$\\ $1\bar{2}2$ & & & & $202$ & & $001$ & & $2\bar{2}1$\\ {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\searrow}$ & & & {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\swarrow}$ & {\tiny 2}$\downarrow$ & & {\tiny 1}$\downarrow$\\ $1\bar{2}0$ & & $2\bar{2}2$ & & $002$ & & $0\bar{2}1$ & & $2\bar{1}1$\\ {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\searrow}$ & {\tiny 2}$\downarrow$& & {\tiny 2}$\downarrow$ & & {\tiny 1}$\downarrow$ & $\overset{\text{2}}{\swarrow}$ & {\tiny 1}$\downarrow$\\ $1\bar{2}\bar{2}$ & & $2\bar{2}0$ & & $0\bar{2}2$ & & $0\bar{1}1$ & & $2\bar{1}2$\\ & $\overset{\text{2}}{\swarrow}$ & {\tiny 1}$\downarrow$ & & {\tiny 2}$\downarrow$ & & {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\searrow}$ & {\tiny 2}$\downarrow$\\ $2\bar{2}\bar{2}$ & & $2\bar{1}0$ & & $0\bar{2}0$ & & $\bar{2}\bar{1}1$ & & $0\bar{1}2$\\ {\tiny 1}$\downarrow$ & & {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\swarrow}$ & {\tiny 2}$\downarrow$ & & & $\overset{\text{1}}{\searrow}$ & {\tiny 2}$\downarrow$\\ $2\bar{1}\bar{2}$ & & $0\bar{1}0$ & & $0\bar{2}\bar{2}$ & & & & $\bar {2}\bar{1}2$\\ {\tiny 1}$\downarrow$ & & {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\swarrow}$ & & & & & {\tiny 2}$\downarrow$\\ $2\bar{1}\bar{1}$ & & $0\bar{1}\bar{2}$ & & & & & & $\bar{2}\bar{1}0$\\ & $\overset{\text{2}}{\searrow}$ & {\tiny 1}$\downarrow$ & & & & & $\overset{\text{2}}{\swarrow}$ & \\ & & $0\bar{1}\bar{1}$ & & & & $\bar{2}\bar{1}\bar{2}$ & & \\ & & & $\overset{\text{2}}{\searrow}$ & & $\overset{\text{1}}{\swarrow}$ & & & \\ & & & & $\bar{2}\bar{1}\bar{1}$ & & & & \\ & & & & & & & & \end{tabular} \text{ \ \ \ \ \ } \begin{tabular} [c]{lllllllll}& & & & $112$ & & & & \\ & & & $\overset{\text{1}}{\swarrow}$ & & $\overset{\text{2}}{\searrow}$ & & & \\ & & $212$ & & & & $110$ & & \\ & $\overset{\text{2}}{\swarrow}$ & & & & & {\tiny 1}$\downarrow$ & $\overset{\text{2}}{\searrow}$ & \\ $012$ & & & & & & $210$ & & $11\bar{2}$\\ {\tiny 2}$\downarrow$ & & & & & $\overset{\text{1}}{\swarrow}$ & {\tiny 2}$\downarrow$ & & {\tiny 1}$\downarrow$\\ $\bar{2}12$ & & & & $220$ & & $010$ & & $21\bar{2}$\\ {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\searrow}$ & & & {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\swarrow}$ & {\tiny 2}$\downarrow$ & & {\tiny 1}$\downarrow$\\ $\bar{2}10$ & & $\bar{1}12$ & & $020$ & & $01\bar{2}$ & & $22\bar{2}$\\ {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\searrow}$ & {\tiny 2}$\downarrow$& & {\tiny 2}$\downarrow$ & & {\tiny 1}$\downarrow$ & $\overset{\text{2}}{\swarrow}$ & {\tiny 1}$\downarrow$\\ $\bar{2}1\bar{2}$ & & $\bar{1}10$ & & $\bar{2}20$ & & $02\bar{2}$ & & $22\bar{1}$\\ & $\overset{\text{2}}{\swarrow}$ & {\tiny 1}$\downarrow$ & & {\tiny 2}$\downarrow$ & & {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\searrow}$ & {\tiny 2}$\downarrow$\\ $\bar{1}1\bar{2}$ & & $\bar{1}20$ & & $\bar{2}00$ & & $\bar{2}2\bar{2}$ & & $02\bar{1}$\\ {\tiny 1}$\downarrow$ & & {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\swarrow}$ & {\tiny 2}$\downarrow$ & & & $\overset{\text{1}}{\searrow}$ & {\tiny 2}$\downarrow$\\ $\bar{1}2\bar{2}$ & & $\bar{1}00$ & & $\bar{2}0\bar{2}$ & & & & $\bar {2}2\bar{1}$\\ {\tiny 1}$\downarrow$ & & {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\swarrow}$ & & & & & {\tiny 2}$\downarrow$\\ $\bar{1}2\bar{1}$ & & $\bar{1}0\bar{2}$ & & & & & & $\bar{2}0\bar{1}$\\ & $\overset{\text{2}}{\searrow}$ & {\tiny 1}$\downarrow$ & & & & & $\overset{\text{2}}{\swarrow}$ & \\ & & $\bar{1}0\bar{1}$ & & & & $\bar{2}\bar{2}\bar{1}$ & & \\ & & & $\overset{\text{2}}{\searrow}$ & & $\overset{\text{1}}{\swarrow}$ & & & \\ & & & & $\bar{1}\bar{2}\bar{1}$ & & & & \\ & & & & & & & & \end{tabular} \\ \text{The crystals }B^{B}(121)\text{ and }B^{B}(112)\text{ in }G_{2}^{B}$$ $$\begin{gathered} \begin{tabular} [c]{lll}$121$ & & \\ {\tiny 1}$\downarrow$ & $\overset{\text{2}}{\searrow}$ & \\ $122$ & & $\bar{2}21$\\ {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\swarrow}$ & {\tiny 2}$\downarrow $\\ $\bar{2}22$ & & $\bar{2}\bar{1}1$\\ {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\swarrow}$ & {\tiny 2}$\downarrow $\\ $\bar{2}\bar{1}2$ & & $\bar{2}\bar{1}\bar{2}$\\ {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\swarrow}$ & \\ $\bar{2}\bar{1}\bar{1}$ & & \end{tabular} \text{ \ \ \ \ \ \ \ } \begin{tabular} [c]{lll}$112$ & & \\ {\tiny 1}$\downarrow$ & $\overset{\text{2}}{\searrow}$ & \\ $212$ & & $\bar{2}12$\\ {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\swarrow}$ & {\tiny 2}$\downarrow $\\ $\bar{1}12$ & & $\bar{2}\bar{2}2$\\ {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\swarrow}$ & {\tiny 2}$\downarrow $\\ $\bar{1}\bar{2}2$ & & $\bar{2}\bar{2}\bar{1}$\\ {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\swarrow}$ & \\ $\bar{1}\bar{2}\bar{1}$ & & \end{tabular} \\ \text{The crystals }B^{D}(121)\text{ and }B^{D}(112)\text{ in }G_{2}^{D}$$ $$\begin{gathered} \begin{tabular} [c]{lll}$1\bar{2}1$ & & \\ {\tiny 1}$\downarrow$ & $\overset{\text{2}}{\searrow}$ & \\ $2\bar{2}1$ & & $1\bar{2}\bar{2}$\\ {\tiny 1}$\downarrow$ & $\overset{\text{2}}{\searrow}$ & {\tiny 1}$\downarrow $\\ $2\bar{1}1$ & & $2\bar{2}\bar{2}$\\ {\tiny 1}$\downarrow$ & & {\tiny 1}$\downarrow$\\ $2\bar{1}2$ & & $2\bar{1}\bar{2}$\\ {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\swarrow}$ & \\ $2\bar{1}\bar{1}$ & & \end{tabular} \text{ \ \ \ \ \ \ } \begin{tabular} [c]{lll}$11\bar{2}$ & & \\ {\tiny 1}$\downarrow$ & $\overset{\text{2}}{\searrow}$ & \\ $21\bar{2}$ & & $\bar{2}1\bar{2}$\\ {\tiny 1}$\downarrow$ & $\overset{\text{2}}{\searrow}$ & {\tiny 1}$\downarrow $\\ $22\bar{2}$ & & $\bar{1}1\bar{2}$\\ {\tiny 1}$\downarrow$ & & {\tiny 1}$\downarrow$\\ $22\bar{1}$ & & $\bar{1}2\bar{2}$\\ {\tiny 2}$\downarrow$ & $\overset{\text{1}}{\swarrow}$ & \\ $\bar{1}2\bar{1}$ & & \end{tabular} \\ \text{The crystals }B^{D}(1\bar{2}1)\text{ and }B^{D}(11\bar{2})\text{ in }G_{2}^{D}$$ By (\[TENS1\]) and (\[TENS2\]), this implies that the plactic relations above are compatible with Kashiwara’s operators, that is, for any words $w_{1}$ and $w_{2}$ such that $w_{1}\equiv w_{2}$ one has: $$\left\{ \begin{tabular} [c]{l}$\widetilde{e}_{i}(w_{1})\equiv\widetilde{e}_{i}(w_{2})\text{ and }\varepsilon_{i}(w_{1})=\varepsilon_{i}(w_{2})$\\ $\widetilde{f}_{i}(w_{1})\equiv\widetilde{f}_{i}(w_{2})\text{ and }\varphi _{i}(w_{1})=\varphi_{i}(w_{2}).$\end{tabular} \right. \label{fonda_compatib}$$ Hence: $$w_{1}\equiv w_{2}\Longrightarrow w_{1}\sim w_{2}.$$ To obtain the converse we show that for any highest weight vertex $w^{0}$$$\mathrm{w(}P(w^{0}))\equiv w^{0}. \label{cong_on_HWV}$$ This follows by induction on $\mathrm{l}(w^{0})$. When $\mathrm{l}(w^{0})=1$, $\mathrm{w(}P(w^{0}))=w^{0}$. By writing $w^{0}=v^{0}x^{0}$, it is possible (see the proof of Lemma 3.2.6 in [@Lec]) to show that $\mathrm{w(}P(w^{0}))$ may be obtained from the word $\mathrm{w(}P(v^{0}))x^{0}$ by applying only Knuth relations and contractions relations of type $12\cdot \cdot\cdot r\overline{p}\equiv12\cdot\cdot\cdot\widehat{p}\cdot\cdot\cdot r$ with $p\leq r\leq n$ (the hat means removal the letter $p$). From (\[cong\_on\_HWV\]), we obtain that two highest weight vertices $w_{1}^{0}$ and $w_{2}^{0}$ with the same weight $\lambda$ verify $w_{1}^{0}\equiv w_{2}^{0}$. Indeed there is only one orthogonal tableau whose reading is a highest vertex of weight $\lambda$. Now suppose that $w_{1}\sim w_{2}$ and denote by $w_{1}^{0}$ and $w_{2}^{0}$ the highest weight vertices of $B(w_{1})$ and $B(w_{2})$. We have $w_{1}^{0}\equiv w_{2}^{0}$. Set $w_{1}=\widetilde{F}\,w_{1}^{0}$ where $\widetilde{F}$ is a product of Kashiwara’s operators $\widetilde{f}_{i}$, $i=1,...,n$. Then $w_{2}=\widetilde{F}\,w_{2}^{0}$ because $w_{1}\sim w_{2}$. So by (\[fonda\_compatib\]) we obtain $$w_{1}^{0}\equiv w_{2}^{0}\Longrightarrow\widetilde{F}\,w_{1}^{0}\equiv\widetilde{F}\,w_{2}^{0}\Longrightarrow w_{1}\equiv w_{2}.$$ A bumping algorithm for types $B$ and $D$ ----------------------------------------- Now we are going to see how the orthogonal tableau $P(w)$ may be computed for each vertex $w$ by using an insertion scheme analogous to bumping algorithm for type $A$. As a first step, we describe $P(w)$ when $w=\mathrm{w}(C)x$, where $x$ and $C$ are respectively a letter and an admissible column. This will be called “ the insertion of the letter $x$ in the admissible column $C$ ” and denoted by $x\rightarrow C$. Then we will be able to obtain $P(w)$ when $w=\mathrm{w}(T)x$ with $x$ a letter and $T$ an orthogonal tableau. This will be called “ the insertion of the letter $x$ in the orthogonal tableau $T$ ” and denoted by $x\rightarrow T$. Our construction of $P$ will be recursive, in the sense that if $P(u)=T$ and $x$ is a letter, then $P(ux)=x\rightarrow T$. ### Insertion of a letter in an admissible column\[x\_in\_C\] Consider a word $w=$$C)x$, where $x$ and $C$ are respectively a letter and an admissible column of height $p$. When $w=\mathrm{w}(C^{x})$ is the reading of a column $C^{x}$, we have: $$\begin{aligned} x & \rightarrow C=C^{x}\text{ if }C^{x}\text{ is admissible or}\\ x & \rightarrow C=\widetilde{C^{x}}\text{ where }\widetilde{C^{x}}\text{ is the column whose reading corresponds to }\widetilde{w}\text{ otherwise.}$$ Indeed, $x\rightarrow C$ must be an orthogonal tableau such that $\mathrm{w}(x\rightarrow C)\equiv w$. When $w$ is not a column word, by Lemma \[lem\_phi\_tens\] the highest weight vertex $w^{0}$ of $B(w)\,$may be written $w^{0}=v^{0}\,1$ where $v^{0}\in\{b_{\omega_{p}};p=1,...,n\}\cup\{b_{\overline{\omega}_{n}}\}.$ Then $u^{0}=1\,v^{0}$ is the reading of an orthogonal tableau and $u^{0}\equiv w^{0}$. So $u^{0}$ is the highest weight vertex of the connected component containing $\mathrm{w}(x\rightarrow C)$. Moreover there exists a unique sequence of highest weight vertices $w_{1}^{0},...,w_{p}^{0}$ such that $w_{1}^{0}=w^{0},$ $w_{p}=u^{0}$ and for $i=2,...,p$ $w_{i}^{0}$ differs from $w_{i-1}^{0}$ by applying one relation $R_{1}$ from left to right. This implies that there exists a unique sequence of vertices $w_{1},...,w_{p}$ such that $w_{1}=w$ and for $i=2,...,p-1$ $B(w_{i})=B(w_{i}^{0})$. Each $w_{i}$ differs from $w_{i-1}$ by applying one relation $R_{1},R_{2},R_{3}$ or $R_{4}$ from left to right. The word $w_{p}$ is the reading of an orthogonal tableau and can be factorized as $w_{p}=v^{\prime}\,x^{\prime}$ where $v^{\prime }=\mathrm{w}(C^{\prime})$ is a column word an $x^{\prime}$ a letter. We will have $x\rightarrow C=C^{\prime}x^{\prime}$.      Suppose $n=7$. Let $\mathrm{w}(C)=6700\bar{7}\bar{6}$ be an admissible column word of type $B$. Choose $x=6.$ Then by applying relations $R_{i}^{B}$ $i=1,...,4$ we obtain successively: $6700\mathbf{\bar{7}\bar{6}6}\equiv670\mathbf{0\bar{7}7}\bar{7}\equiv67\mathbf{0\bar{7}7}0\bar{7}\equiv6\mathbf{7\bar{7}7}00\bar{7}\equiv\mathbf{6\bar{6}6}700\bar{7}\equiv\bar{5}56700\bar{7}$ Suppose $n=7$. Let $\mathrm{w}(C)=67\bar{7}7\bar{7}\bar{6}$ be an admissible column word of type $D$. Choose $x=6.$ Then by applying relations $R_{i}^{D}$ $i=1,...,4$ we obtain successively: $67\bar{7}7\mathbf{\bar{7}\bar{6}6}\equiv67\bar{7}\mathbf{7\bar {7}\bar{7}}7\equiv67\mathbf{\bar{7}\bar{6}6}\bar{7}7\equiv6\mathbf{7\bar {7}\bar{7}}7\bar{7}7\equiv\mathbf{6\bar{6}6}\bar{7}7\bar{7}7\equiv\bar {5}567\bar{7}7\bar{7}.$ Hence $$6\rightarrow\begin{tabular} [c]{|l|}\hline $\mathtt{6}$\\\hline $\mathtt{7}$\\\hline $\mathtt{0}$\\\hline $\mathtt{0}$\\\hline $\mathtt{\bar{7}}$\\\hline $\mathtt{\bar{6}}$\\\hline \end{tabular} =\begin{tabular} [c]{|l|l}\hline $\mathtt{5}$ & \multicolumn{1}{|l|}{$\mathtt{\bar{5}}$}\\\hline $\mathtt{6}$ & \\\cline{1-1}$\mathtt{7}$ & \\\cline{1-1}$\mathtt{0}$ & \\\cline{1-1}$\mathtt{0}$ & \\\cline{1-1}$\mathtt{\bar{7}}$ & \\\cline{1-1}\end{tabular} \text{ and }6\rightarrow\begin{tabular} [c]{|l|}\hline $\mathtt{6}$\\\hline $\mathtt{7}$\\\hline $\mathtt{\bar{7}}$\\\hline $\mathtt{7}$\\\hline $\mathtt{\bar{7}}$\\\hline $\mathtt{\bar{6}}$\\\hline \end{tabular} =\begin{tabular} [c]{|l|l}\hline $\mathtt{5}$ & \multicolumn{1}{|l|}{$\mathtt{\bar{5}}$}\\\hline $\mathtt{6}$ & \\\cline{1-1}$\mathtt{7}$ & \\\cline{1-1}$\mathtt{\bar{7}}$ & \\\cline{1-1}$\mathtt{7}$ & \\\cline{1-1}$\mathtt{\bar{7}}$ & \\\cline{1-1}\end{tabular} .$$ ### Insertion of a letter in an orthogonal tableau Consider an orthogonal tableau $T=C_{1}C_{2}\cdot\cdot\cdot C_{r}$. We can prove as in [@Lec] that the insertion $x\rightarrow T$ is characterized as follows: - If $\mathrm{w(}C_{1})\,x$ is an admissible column word , then $x\rightarrow T=C_{1}^{x}C_{2}\cdot\cdot\cdot C_{r}$ where $C_{1}^{x}$ is the column of reading $\mathrm{w(}C_{1})\,x.$ - If $\mathrm{w(}C_{1})\,x$ is a non admissible column word each strict factor of which is admissible and such that $\widetilde{x\mathrm{w(}C_{1})}=x_{1}\cdot\cdot\cdot x_{s},$ then $x\rightarrow T=x_{s}\rightarrow (x_{s-1}\rightarrow(\cdot\cdot\cdot x_{1}\rightarrow T^{\prime}))$ where $T^{\prime}=C_{2}\cdot\cdot\cdot C_{r}$. Moreover the insertion of $x_{1},...,x_{s}$ in $T^{\prime}$ does not cause a new contraction. - If $\mathrm{w(}C_{1})\,x$ is not a column word, the insertion of $x$ in $C_{1}$ gives a column $C_{1}^{\prime}$ and a letter $x^{\prime}$ (with the notation of \[x\_in\_C\]). Then $x\rightarrow T=C_{1}^{\prime}(x^{\prime }\rightarrow T^{\prime})$, that is, $x\rightarrow T$ is the tableau defined by $C_{1}^{\prime}$ and the columns of $x^{\prime}\rightarrow T^{\prime}.$ Notice that the algorithm terminates because in the last two cases we are reduced to the insertion of a letter in a tableau whose number of boxes is strictly less than that of $T$. Finally for any vertex $w\in G_{n}$, we will have: $$\begin{aligned} P(w) & = \begin{tabular} [c]{|l|}\hline $w$\\\hline \end{tabular} \text{ if }w\text{ is a letter,}\\ P(w) & =x\rightarrow P(u)\text{ if }w=ux\text{ with }u\text{ a word and }x\text{ a letter.}$$ Schensted-type Correspondences \[sub\_sec\_Cor\_in\_Gn\] -------------------------------------------------------- In this section a bijection is established between words $w$ of length $l$ on $\mathcal{B}_{n}\ $and pairs $(P^{B}(w),Q^{B}(w))$ where $P^{B}(w)$ is the orthogonal tableau defined above and $Q^{B}(w)$ is an oscillating tableau of type $B$. Similarly we obtain a bijection between words $w$ of length $l$ on $\mathcal{D}_{n}\ $and pairs $(P^{D}(w),Q^{D}(w))$ where $P^{D}(w)$ is an oscillating tableau of type $D$. For type $B$, such a one-to-one correspondence has already been obtained by Sundaram [@SU] using another definition of orthogonal tableaux and an appropriate insertion algorithm. Unfortunately it is not known if this correspondence is compatible with a monoid structure. Our bijection based on the previous insertion algorithm for admissible orthogonal tableaux of type $B$ will be different from Sundaram’s one but compatible with the plactic relations defining $Pl(B_{n})$. \[def\_tab\_osci\]      An oscillating tableau $Q$ of type $B$ and length $l$ is a sequence of Young diagrams $(Q_{1},...,Q_{l})$ whose columns have height $\leq n$ and such that any two consecutive diagrams are equal or differ by exactly one box (i.e. $Q_{k+1}=Q_{k}$, $Q_{k+1}/Q_{k}=(\,\begin{tabular} [c]{|l|}\hline \\\hline \end{tabular} \,)$ or $Q_{k}/Q_{k+1}=(\,\begin{tabular} [c]{|l|}\hline \\\hline \end{tabular} \,)$). An oscillating tableau $Q$ of type $D$ and length $l$ is a sequence $(Q_{1},...,Q_{l})$ of pairs $Q_{k}(O_{k},\varepsilon_{k})$ where $O_{k}$ is a Young diagram whose columns have height $\leq n$ and $\varepsilon_{k}\in\{-,0,+\},$ satisfying for $k=1,...,l$ - $O_{k+1}/O_{k}=(\,\begin{tabular} [c]{|l|}\hline \\\hline \end{tabular} \,)$ or $O_{k}/O_{k+1}=(\,\begin{tabular} [c]{|l|}\hline \\\hline \end{tabular} \,)$, - $\varepsilon_{k+1}\neq0$ and $\varepsilon_{k}\neq0$ implies $\varepsilon_{k+1}=\varepsilon_{k}$. - $\varepsilon_{k}=0$ if and only if $O_{k}$ has no columns of height $n.$ Let $w=x_{1}\cdot\cdot\cdot x_{l}\ $be a word. The construction of $P(w)$ involves the construction of the $l$ orthogonal tableaux defined by $P_{i}=P(x_{1}\cdot\cdot\cdot x_{i}).$ For $w\in\mathcal{B}_{n}^{\ast}$ (resp. $w\in\mathcal{D}_{n}^{\ast}$) we denote by $Q^{B}(w)$ (resp. $Q^{D}(w) $) the sequence of shapes of the orthogonal tableaux $P_{1},...,P_{l}$. \[prop\_Q(w)\_oscill\]$Q_{B}(w)$ and $Q_{D}(w)$ are respectively oscillating tableaux of type $B$ and $D$. Each $Q_{i}$ is the shape of an orthogonal tableau so it suffices to prove that for any letter $x$ and any orthogonal tableau $T$, the shape of $x\rightarrow T$ differs from the shape of $T$ by at most one box according to Definition \[def\_tab\_osci\]. The highest weight vertex of the connected component containing $\mathrm{w(}T)x$ may be written $\mathrm{w(}T^{0})x^{0}$ where $T^{0}$ is an orthogonal tableau. It follows from Lemma \[lem\_coplactic\] $\mathrm{(ii)}$ that $\mathrm{w(}T)\longleftrightarrow\mathrm{w(}T^{0})$. So $\mathrm{wt}(\mathrm{w(}T^{0}))$ is given by the shape of $T$. Then the shape of $x\rightarrow T$ is given by the coordinates of $\mathrm{wt}(\mathrm{w}(T^{0})x^{0})$ on the basis $(\omega_{1}^{B},...,\omega_{n}^{B})$ for type $B$, on the base $(\omega_{1}^{D},...,\omega_{n}^{D})$ or $(\omega_{1}^{D},...,\omega_{n-1}^{D},\overline{\omega}_{n}^{D})$ for type $D$. Suppose that $x\in\mathcal{B}_{n}^{\ast}$ and $T$ is orthogonal of type $B$. Let $(\lambda_{1},...,\lambda_{n})$ be the coordinates of $\mathrm{wt}(T^{0})$ on the basis of the $\omega_{i}^{B}$’s. If $x^{0}=\overline{i}\succ0$ then $\mathrm{wt}(x^{0})=\omega_{i-1}^{B}-\omega_{i}^{B}$. So $\lambda_{i}>0$ and $\mathrm{wt}(\mathrm{w(}T^{0})x^{0})=(\lambda_{1},...,\lambda_{i-1}+1,\lambda_{i}-1,...,\lambda_{n-1}).$ Hence during the insertion of the letter $x$ in $T$, a column of height $i$ (corresponding to the weight $\omega_{i}$) is turned into a column of height $i-1$ (corresponding to the weight $\omega_{i-1}$)$.$ So the shape of $x\rightarrow T$ is obtained by erasing one box to the shape of $T$. If $x^{0}=i\prec0$, then we can prove by similar arguments that the shape of $x\rightarrow T$ is obtained by adding one box to the shape of $T$. When $x^{0}=0,$ $\mathrm{wt}(x^{0})=0$, so $\mathrm{wt}(\mathrm{w(}T^{0})x^{0})=\mathrm{wt}(\mathrm{w(}T^{0}))$. Hence the shapes of $T$ and $x\rightarrow T$ are the same. Suppose $x\in\mathcal{D}_{n}^{\ast}$ and $T$ orthogonal of type $D$. When $\left| x^{0}\right| \neq n,$ the proof is the same as above. If $x^{0}=n,$ $\mathrm{wt}(x^{0})=\Lambda_{n}-\Lambda_{n-1}=\omega_{n}-\omega_{n-1}=\omega_{n-1}-\overline{\omega}_{n}.$ We have to consider three cases, $\mathrm{(i)}$: $\varepsilon_{T}=-$; $\mathrm{(ii)}$: $\varepsilon_{T}=0$ and $\mathrm{(iii)}$: $\varepsilon_{T}=+$. Denote by $(\lambda_{1},...,\lambda _{n})$ the positive decomposition of $\mathrm{wt}(\mathrm{w(}T^{0}))$ on the basis $(\omega_{1}^{D},...,\omega_{n}^{D})$ or on the basis $(\omega_{1}^{D},...,\overline{\omega}_{n}^{D})$. In the first case, $\lambda_{n}>0$ and the positive decomposition of $\mathrm{wt}(x^{0}\mathrm{w(}T^{0}))$ on the basis $(\omega_{1}^{D},...,\overline{\omega}_{n}^{D})$ is $(\lambda_{1},...,\lambda_{n-2},\lambda_{n-1}+1,\lambda_{n}-1)$. It means that during the insertion of $x$ in $T$ a column of height $n$ (corresponding to $\overline{\omega}_{n}$) is turned into a column of height $n-1$ (corresponding to $\omega_{n-1}$). Moreover $\varepsilon_{x\rightarrow T}=\varepsilon_{T}$ if $\lambda_{n}>1$ and $\varepsilon_{x\rightarrow T}=0$ otherwise. In the second case, $\lambda_{n-1}>0,$ $\lambda_{n}=0$ and the positive decomposition of $\mathrm{wt}(x^{0}\mathrm{w(}T^{0}))$ on the base $(\omega_{1}^{D},...,\omega_{n}^{D})$ is $(\lambda_{1},\lambda_{2},...,\lambda_{n-1}-1,1).$ It means that during the insertion of $x$ in $T$ a column of height $n-1$ (corresponding to $\omega_{n-1}$) is turned into a column of height $n$ (corresponding to $\omega_{n}$). Moreover $\varepsilon _{x\rightarrow T}=+$. In the last case, $\lambda_{n-1}>0,$ $\lambda_{n}>0$ and the positive decomposition of $\mathrm{wt}(x^{0}\mathrm{w}(T^{0}))$ on $(\omega_{1}^{D},...,\omega_{n}^{D})$ is $(\lambda_{1},\lambda_{2},...,\lambda_{n-1}-1,\lambda_{n}+1)$. It means that during the insertion of $x$ in $T$ a column of height $n-1$ (corresponding to $\omega_{n-1}$) is turned into a column of height $n$ (corresponding to $\omega_{n}$). Moreover $\varepsilon_{x\rightarrow T}=\varepsilon_{T}$. When $x^{0}=\overline{n},$ the proof is similar. \[th\_Q\_symbol\]For any vertices $w_{1}$ and $w_{2}$ of $G_{n}$: $$w_{1}\longleftrightarrow w_{2}\Leftrightarrow Q(w_{1})=Q(w_{2})\text{.}$$ The proof is analogous to that of Proposition 5.2.1 in [@Lec]. Let $\mathcal{B}_{n,l}^{\ast}$ and $\mathcal{O}_{l}^{B}$ (resp. $\mathcal{D}_{n,l}^{\ast}$ and $\mathcal{O}_{l}^{D}$) be the set of words of length $l$ on $\mathcal{B}_{n}$ (resp. $\mathcal{D}_{n}$) and the set of pairs $(P,Q)$ where $P$ is an orthogonal tableau of type $B$ (resp. $D$) and $Q$ an oscillating tableau of type $B$ (resp. $D$) and length $l$ such that $P$ has shape $Q_{l}$ ($Q_{l}$ is the last shape of $Q$). Then the maps: $$\mathcal{\begin{tabular} [c]{l}$\Psi^{B}:B_{n,l}^{\ast}\rightarrow O_{l}^{B}$\\ $w\mapsto(P^{B}(w),Q^{B}(w))$\end{tabular} }\text{ and }\mathcal{\begin{tabular} [c]{l}$\Psi^{D}:\mathcal{D}_{n,l}^{\ast}\rightarrow\mathcal{O}_{l}^{D}$\\ $w\mapsto(P^{D}(w),Q^{D}(w))$\end{tabular} }$$ are bijections. For type $\Psi^{B}$ the proof is analogous to that of Theorem 5.2.2 in [@Lec]. By Theorems \[th\_Psymbol\] and \[th\_Q\_symbol\], we obtain that $\Psi^{D}$ is injective. Consider an oscillating tableau $Q$ of length $l$ and type $D$. Set $x_{1}=1$ and for $i=2,...,l$ - $x_{i}=k$ if $O_{i}$ differs from $O_{i-1}$ by adding a box in row $k$ of height $<n,$ - $x_{i}=\overline{k}$ if $Q_{i}$ differs from $Q_{i-1}$ by removing a box in row $k.$of height $<n,$ - $x_{i}=n$ if $O_{i}$ differs from $O_{i-1}$ by adding a box in row $n$ and $\varepsilon_{i}=+,$ - $x_{i}=\overline{n}$ if $Q_{i}$ differs from $Q_{i-1}$ by adding a box in row $n$ and $\varepsilon_{i}=-,$ - $x_{i}=\overline{n}$ if $O_{i}$ differs from $O_{i-1}$ by removing a box in row $n$ and $\varepsilon_{i}=+,$ - $x_{i}=n$ if $O_{i}$ differs from $O_{i-1}$ by removing a box in row $n$ and $\varepsilon_{i}=-,$ - Consider $w_{Q}=x_{l}\cdot\cdot\cdot x_{2}1$. Then $Q(w_{Q})=Q$. By Theorem \[TH\_KN\], the image of $B(w_{Q})$ by $\Psi^{D}$ consists in the pairs $(P,Q)$ where $P$ is a symplectic tableau of shape $Q_{l}$. We deduce immediately that $\Psi$ is surjective. \[subsecJDT\]Jeu de Taquin for type B ------------------------------------- In [@SH], J T Sheats has developed a sliding algorithm for type $C$ acting on the skew admissible symplectic tableaux. This algorithm is analogous to the classical Jeu de Taquin of Lascoux and Schützenberger for type A [@LS]. Each inner corner of the skew tableau considered is turned into an outside corner by applying vertical and horizontal moves. We have shown in [@Lec] how to extend it to take into account the contraction relation of the plactic monoid $Pl(C_{n})$ (analogous to $Pl(B_{n})$ and $Pl(D_{n})$ for type $C$). Then we have proved that the tableau obtained does not depend on the way the inner corners disappear. In this section we propose a sliding algorithm for type $B$. The main idea is that the split form of any skew orthogonal tableau $T$ of type $B$ may be regarded as a symplectic skew tableau. Set $\mathcal{C}_{n}=\{1\prec\cdot\cdot\cdot\prec n\prec\overline{n}\prec \cdot\cdot\cdot\prec\overline{1}\}\subset\mathcal{B}_{n}$. The symplectic tableaux are, for type $C,$ the combinatorial objects analogous to the orthogonal tableaux. They can be regarded as orthogonal tableaux of type $B$ on the alphabet $\mathcal{C}_{n}$ instead of $\mathcal{B}_{n}$. The plactic monoid $Pl(C_{n})$ is the quotient of the free monoid $\mathcal{C}_{n}^{\ast} $ by relations $R_{1}^{B},$ $R_{2}^{B}$ and $R_{5}^{B}$. We denote by $\overset{C}{\equiv}$ the congruence relation in $Pl(C_{n})$. Then for $w_{1}$ and $w_{2}$ two words of $\mathcal{C}_{n}^{\ast}$ we have: $$w_{1}\overset{C}{\equiv}w_{2}\Longrightarrow w_{1}\overset{B}{\equiv}w_{2}\text{.}$$ A skew orthogonal tableau of type $B$ is a skew Young diagram filled by letters of $\mathcal{B}_{n}$ whose columns are admissible of type $B$ and such that the rows of its split form (obtained by splitting its columns) are weakly increasing from left to right. Skew orthogonal tableaux are the combinatorial objects analogous to the admissible skew tableaux introduced by Sheats in [@SH] for type $C$. Note that two different skew tableaux may have the same reading. For $n=3,$ $T=\begin{tabular} [c]{lll}\cline{3-3}& & \multicolumn{1}{|l|}{$\mathtt{2}$}\\\cline{2-3}& \multicolumn{1}{|l}{$\mathtt{3}$} & \multicolumn{1}{|l|}{$\mathtt{0}$}\\\hline \multicolumn{1}{|l}{$\mathtt{0}$} & \multicolumn{1}{|l}{$\mathtt{\bar{3}}$} & \multicolumn{1}{|l|}{$\mathtt{\bar{1}}$}\\\hline \multicolumn{1}{|l}{$\mathtt{0}$} & \multicolumn{1}{|l}{} & \\\cline{1-1}\end{tabular} $ is a skew orthogonal tableau of type $B$ because $\mathrm{spl}(T)=\vspace{0.2cm}\begin{tabular} [c]{llllll}\cline{5-6}& & & & \multicolumn{1}{|l}{$\mathtt{2}$} & \multicolumn{1}{|l|}{$\mathtt{2}$}\\\cline{3-6}& & \multicolumn{1}{|l}{$\mathtt{2}$} & \multicolumn{1}{|l}{$\mathtt{3}$} & \multicolumn{1}{|l}{$\mathtt{3}$} & \multicolumn{1}{|l|}{$\mathtt{\bar{3}}$}\\\hline \multicolumn{1}{|l}{$\mathtt{2}$} & \multicolumn{1}{|l}{$\mathtt{\bar{3}}$} & \multicolumn{1}{|l}{$\mathtt{\bar{3}}$} & \multicolumn{1}{|l}{$\mathtt{\bar {2}}$} & \multicolumn{1}{|l}{$\mathtt{\bar{1}}$} & \multicolumn{1}{|l|}{$\mathtt{\bar{1}}$}\\\hline \multicolumn{1}{|l}{$\mathtt{3}$} & \multicolumn{1}{|l}{$\mathtt{\bar{2}}$} & \multicolumn{1}{|l}{} & & & \\\cline{1-2}\end{tabular} $. The relation $0\overline{n}n\equiv\overline{n}n0$ has no natural interpretation in terms of horizontal or vertical slidings in skew orthogonal tableaux. To overcome this problem we are going to work on the split form of the skew tableaux instead of the skew tableaux themselves that is , we are going to obtain a Jeu de Taquin for type $B$ by applying the symplectic Jeu de Taquin on the split form of the skew orthogonal tableaux of type $B$. Let $T$ and $T^{\prime}$ be two skew orthogonal tableaux of type $B$. Then: $$\mathrm{w}(T)\overset{B}{\equiv}\mathrm{w}(T^{\prime})\Longleftrightarrow \mathrm{w}[\mathrm{spl}(T)]\overset{B}{\equiv}\mathrm{w}[\mathrm{spl}(T^{\prime})].$$ We can write $\mathrm{w}(T)=\mathrm{w}(C_{1})\cdot\cdot\cdot\mathrm{w}(C_{r})$ and $\mathrm{w}(T^{\prime})=\mathrm{w}(C_{1}^{\prime})\cdot\cdot \cdot\mathrm{w}(C_{s}^{\prime})$ where $C_{k}$ and $C_{k}^{\prime},$ $k=1,...,r$ are admissible columns. All the vertices $w\in B(\mathrm{w}(T))$ and $w^{\prime}\in B(\mathrm{w}(T^{\prime}))$ can be respectively written on the form $w=c_{r}\cdot\cdot\cdot c_{1}$ and $w^{\prime}=c_{s}^{\prime}\cdot\cdot\cdot c_{1}^{\prime}$ where $c_{i},$ $i=1,..,r$ and $c_{j}^{\prime}$, $j=1,...,s$ are readings of admissible columns of type $B$. Consider the maps: $$\theta_{2}:\left\{ \begin{tabular} [c]{c}$B(\mathrm{w}(T))\rightarrow B(\mathrm{spl}(\mathrm{w}(T))$\\ $w=c_{r}\cdot\cdot\cdot c_{1}\longmapsto S_{2}(c_{r})\cdot\cdot\cdot S_{2}(c_{1})$\end{tabular} \right. \text{ and }\theta_{2}^{\prime}:\left\{ \begin{tabular} [c]{c}$B(\mathrm{w}(T^{\prime}))\rightarrow B(\mathrm{spl}(\mathrm{w}(T))$\\ $w^{\prime}=c_{s}^{\prime}\cdot\cdot\cdot c_{1}\longmapsto S_{2}(c_{r}^{\prime})\cdot\cdot\cdot S_{2}(c_{1}^{\prime})$\end{tabular} \right.$$ where $S_{2}$ is the map defined in Proposition \[prop\_imag\_S2\]. We have $\mathrm{w}[\mathrm{spl}(T)]=\theta_{2}(\mathrm{w}(T))$ and $\mathrm{w}[\mathrm{spl}(T^{\prime})]=\theta_{2}^{\prime}(\mathrm{w}(T^{\prime})).$ By using  Corollary \[cor\_strech\] we obtain $$\mathrm{w}(T)\overset{B}{\equiv}\mathrm{w}(T^{\prime})\Longleftrightarrow \mathrm{w}(T)\overset{B}{\sim}\mathrm{w}(T^{\prime})\Longleftrightarrow \mathrm{w}[\mathrm{spl}(T)]\overset{B}{\sim}\mathrm{w}[\mathrm{spl}(T^{\prime })]\Longleftrightarrow\mathrm{w}[\mathrm{spl}(T)]\overset{B}{\equiv}\mathrm{w}[\mathrm{spl}(T^{\prime})].$$ If $T$ is a skew orthogonal tableau of type $B$ with $r$ columns, then $\mathrm{spl}(T)$ is a symplectic skew tableau with $2r$ columns. We can apply the symplectic Jeu de taquin to $\mathrm{spl}(T)$ to obtain a symplectic tableau $\mathrm{spl}(T)^{\prime}$. We will have $\mathrm{w}[\mathrm{spl}(T)^{\prime}]\overset{C}{\equiv}\mathrm{w}[\mathrm{spl}(T)]$ so $\mathrm{w}[\mathrm{spl}(T)^{\prime}]\overset{B}{\equiv}\mathrm{w}[\mathrm{spl}(T)]$. $\mathrm{spl}(T)^{\prime}$ is the split form of the orthogonal tableau $P^{B}(T)$. It follows from $\mathrm{w}(T)\overset{B}{\equiv}\mathrm{w}(P_{B}(T))$ and the lemma above that $\mathrm{w}[\mathrm{spl}(T)]\overset{B}{\equiv}\mathrm{w}[\mathrm{spl}(P^{B}(T))].$ So we obtain $\mathrm{w}[\mathrm{spl}(T)^{\prime}]\overset{B}{\equiv}\mathrm{w}[\mathrm{spl}(P^{B}(T))]$. But $\mathrm{spl}(T^{\prime})$ and $\mathrm{spl}(P^{B}(T))$ are orthogonal tableaux, hence $\mathrm{spl}(T)^{\prime}=\mathrm{spl}(P^{B}(T)).$ The columns of the split form of a skew orthogonal tableau $T$ of type $B$ contain no letters $0$ and no pairs of letters $(x,\overline{x})$ with $x\preceq n.$ In this particular case most of the elementary steps of the symplectic Jeu de Taquin applied on $T$ are simple slidings identical to those of the original Jeu de Taquin of Lascoux and Schützenberger (that is complications of the symplectic Jeu de taquin are not needed in these slidings). From $\mathrm{spl}\left( \begin{tabular} [c]{l|l|l|}\cline{2-3}\cline{3-3}& $$ & $$\\\hline \multicolumn{1}{|l|}{$\mathtt{1}$} & $$ & $$\\\hline \multicolumn{1}{|l|}{$\mathtt{3}$} & $$ & $$\\\hline \end{tabular} \right) =\begin{tabular} [c]{|l|l|l|l|l|l|}\hline $$ & $$ & $$ & $$ & $$& $$\\\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline \end{tabular} $ we compute successively: $\begin{tabular} [c]{|l|l|l|l|l|l|}\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline \end{tabular} ,$ $\begin{tabular} [c]{|l|l|l|l|l|l|}\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline \end{tabular} ,$ $\begin{tabular} [c]{|l|l|l|l|l|l|}\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline \end{tabular} $ $\begin{tabular} [c]{|l|l|l|l|l|l|}\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline \end{tabular} ,$ $\begin{tabular} [c]{|l|l|l|l|l|l|}\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline \end{tabular} ,$ $\begin{tabular} [c]{|l|l|l|l|l|l|}\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline \end{tabular} ,\vspace{0.5cm}$ $\begin{tabular} [c]{|l|l|l|l|l|l|}\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline \end{tabular} ,$ $\begin{tabular} [c]{|l|l|l|l|l|l|}\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline $$ & $$ & $$ & $$ & $$ & $$\\\hline \end{tabular} =\mathrm{spl}\left( \begin{tabular} [c]{|l|l|l}\hline $$ & $$ & \multicolumn{1}{|l|}{$\mathtt{2}$}\\\hline $$ & $$ & \multicolumn{1}{|l|}{$\mathtt{\bar{3}}$}\\\hline $$ & $$ & \\\cline{1-2}\end{tabular} \right) .$ Note that the sliding applied in the fourth duplicated tableau above is the unique sliding which is not identical to an original Jeu de taquin step. The split form of a skew orthogonal tableau of type $D$ (defined in the same way than for type $B$) is still a symplectic skew tableau. But $$w_{1}\overset{C}{\equiv}w_{2}\not \Longrightarrow w_{1}\overset{D}{\equiv }w_{2}$$ so we can not use the same idea to obtain an Jeu de Taquin for type $D$. Moreover the examples (computed by using $P^{D}$ with $n=3$) $$\begin{tabular} [c]{|l|l|}\hline 1 & $\mathtt{3}$\\\hline $\mathtt{\bar{3}}$ & $\mathtt{\bar{2}}$\\\hline $\mathtt{\ast}$ & $\mathtt{\bar{1}}$\\\hline \end{tabular} \equiv\begin{tabular} [c]{|l|l}\hline $\mathtt{2}$ & \multicolumn{1}{|l|}{$\mathtt{3}$}\\\hline $\mathtt{\bar{3}}$ & \multicolumn{1}{|l|}{$\mathtt{\bar{2}}$}\\\hline $\mathtt{\bar{2}}$ & \\\cline{1-1}\end{tabular} \text{ and } \begin{tabular} [c]{|l|l|}\hline 1 & $\mathtt{\bar{3}}$\\\hline $\mathtt{\bar{3}}$ & $\mathtt{\bar{2}}$\\\hline $\mathtt{\ast}$ & $\mathtt{\bar{1}}$\\\hline \end{tabular} \equiv\begin{tabular} [c]{|l|l}\hline $\mathtt{\bar{3}}$ & \multicolumn{1}{|l|}{$\mathtt{\bar{3}}$}\\\hline $\mathtt{3}$ & \multicolumn{1}{|l|}{$\mathtt{\bar{2}}$}\\\hline $\mathtt{\bar{3}}$ & \\\cline{1-1}\end{tabular}$$ show that it is not enough to know what letter $x$ slides from the second column $C_{2}$ to the first $C_{1}$ to be able to compute an horizontal sliding. Indeed the result depends on the whole column $C_{2}$. Thus, to give a combinatorial description of a sliding algorithm for type $D$ would probably be very complicated. Plactic monoid for $\frak{G}_{n}\label{last_part}$ ================================================== Write $\frak{G}_{n}^{B}$ and $\frak{G}_{n}^{D}$ for the crystal graphs of the direct sums $$\underset{l\geq0}{\bigoplus}(V(\Lambda_{1}^{B}){\textstyle\bigoplus} V(\Lambda_{n}^{B}))^{\otimes l}\text{ and}\underset{l\geq0}{\text{ }\bigoplus }(V(\Lambda_{1}^{D}){\textstyle\bigoplus} V(\Lambda_{n}^{D}){\textstyle\bigoplus} V(\Lambda_{n-1}^{D}))^{\otimes l}.$$ We call $\frak{B}_{n}=\mathcal{B}_{n}\cup SP_{n}$ and $\frak{D}_{n}=\mathcal{D}_{n}\cup SP_{n}$ the sets of generalized letters of type $B$ and $D$. Then we identify the vertices of $\frak{G}_{n}^{B}$ and $\frak{G}_{n}^{D}$ respectively with the words of the free monoid $\frak{B}_{n}^{\ast}$ and $\frak{D}_{n}^{\ast}$. If $w$ is a vertex of $\frak{G}_{n},$ we write $\mathrm{wt}(w)$ for the weight of $w$. The spin representations are minuscule, hence every spin column is determined by its weight. We can extend the Definition \[def\_sam\_plac\] to vertices of $\frak{G}_{n}$. Consider two vertices $\frak{b}_{1}$ and $\frak{b}_{2}$ of $\frak{G}_{n}^{B}$ (resp. $\frak{G}_{n}^{D}$). We write $\frak{b}_{1}\overset{B}{\sim}\frak{b}_{2}$ (resp. $\frak{b}_{1}\overset{D}{\sim}\frak{b}_{2}$) when these vertices occur at the same place in two isomorphic connected components of $\frak{G}_{n}^{B}$ (resp. $\frak{G}_{n}^{D}$). Our aim is now to extend the results of Section \[subsec\_monoids\] to the vertices of $\frak{G}_{n}$. Tensor products of spin representations --------------------------------------- Write $B(0)$ for the connected component of $\frak{G}_{n}$ containing only the empty word. Let $\frak{C}_{0}$ be the spin column containing only barred letters. For $p=1,...,n$, denote by $\frak{C}_{p}$ the spin column containing exactly the unbarred letters $x\preceq p$. For any admissible column $C$, set $\left| C\right| =\{x\preceq n,$ $x\in lC$ or $\overline{x}\in lC\}=\{x\preceq n,$ $x\in rC$ or $\overline{x}\in rC\}.$ \[Lem\_S\_B\]     1. There exists a unique crystal isomorphism $S^{B}$$$B(0)\cup B(v_{\omega_{n}^{B}})\cup\left( \overset{n-1}{\underset{i=1}{\cup}}B(v_{\omega_{i}^{B}})\right) \overset{S^{B}}{\rightarrow}B(v_{\Lambda _{n}^{B}})^{\otimes2}.$$ 2. Let $w$ be the reading of an admissible column $C$ of type $B$. Write - $l\frak{C}$ for the spin column of height $n$ obtained by adding to $lC$ the barred letters $\overline{x}$ such that $x\notin\left| C\right| $, - $r\frak{C}$ for the spin column of height $n$ obtained by adding to $rC$ the unbarred letters $x$ such that $x\notin\left| C\right| $. Then $$S^{B}(w)=r\frak{C}\otimes l\frak{C.}$$ $1:$ From Lemma \[lem\_phi\_tens\] we obtain that the highest weight vertices of $B(v_{\Lambda_{n}^{B}})^{\otimes2}$ are the vertices $v_{p}^{B}=\frak{C}_{n}\otimes\frak{C}_{p}$ with $p=0,...,n$. We have $\mathrm{wt}(v_{p}^{B})=\omega_{p}^{B}$ for $p=1,...,n$ and $\mathrm{wt}(v_{0}^{D})=0.\;$Hence $S^{B}$ is the crystal isomorphism which sends $B(v_{\omega _{p}^{B}})$ on $B(v_{p}^{B})$ for $p=1,...,n$ and $B(0)$ on $B(v_{0}^{B}).$ $2:$When $w=v_{\omega_{p}^{B}}$, the equality $S^{B}(w)=r\frak{C}\otimes l\frak{C}$ is true. Consider $w\in B(v_{\omega_{p}^{B}})$ and $i=1,...,n$ such that $w^{\prime}=\widetilde{f}_{i}(w)\neq0$. Write $w=\mathrm{w}(C)$ and $w^{\prime}=\mathrm{w}(C^{\prime})$ where $C$ and $C^{\prime}$ are two admissible columns of height $p$. The lemma will be proved if we show the implication $$S^{B}(w)=r\frak{C}\otimes l\frak{C\Longrightarrow} S^{B}(w^{\prime})=r\frak{C}^{\prime}\otimes l\frak{C}^{\prime}$$ where $r\frak{C}^{\prime}$ and $l\frak{C}^{\prime}$ are defined from $C^{\prime}$ in the same manner than $r\frak{C}$ and $l\frak{C}$ from $C$. This is equivalent to $$\widetilde{f}_{i}(r\frak{C}\otimes l\frak{C)=}r\frak{C}^{\prime}\otimes l\frak{C}^{\prime}. \label{f_S}$$ Suppose $i\neq n$. Set $E_{i}=\{i,i+1,\overline{i+1},\overline{i}\}$. $\mathrm{(i):}$ If $\{i,i+1\}\subset\left| C\right| $, $lC$ and $l\frak{C}$ coincide on $E_{i}$. Similarly $rC$ and $r\frak{C}$, $lC^{\prime}$ and $l\frak{C}^{\prime}$, $rC^{\prime}$ and $l\frak{C}$’ coincide on $E_{i}$. By Proposition \[prop\_imag\_S2\], we know that $$\widetilde{f}_{i}^{2}(rC\otimes lC\frak{)=}rC^{\prime}\otimes lC^{\prime}.$$ The action of $\widetilde{f}_{i}^{2}$ on $rC\otimes lC$ is analogous to that of $\widetilde{f}_{i}$ on $r\frak{C}\otimes l\frak{C}$. It means that $\widetilde{f}_{i}$ changes a pair $(i,\overline{i+1})$ of $r\frak{C}$ (resp $l\frak{C})$ into a pair $(i+1,\overline{i})$ if and only if $\widetilde {f}_{i}^{2}$ changes a pair $(i,\overline{i+1})$ of $rC$ (resp. $lC)$ into a pair $(i+1,\overline{i})$. So (\[f\_S\]) is true because only the letters of $E_{i}$ may be modified when we apply $\widetilde{f}_{i}.$ $\mathrm{(ii):}$ If $\{i,i+1\}\cap\left| C\right| =\{i+1\},$ we have $[rC]_{i}=[lC]_{i}=\overline{i+1}$ with the notation of the proof of Proposition \[prop\_imag\_S2\]. Then $r\frak{C}\cap E_{i}=\{\overline {i+1},i\}$ and $l\frak{C}\cap E_{i}=\{\overline{i+1},\overline{i}\}$. Moreover $[C^{\prime}]_{i}=\overline{i}$, $r\frak{C}^{\prime}\cap E_{i}=\{\overline{i},i+1\}$ and $l\frak{C}^{\prime}\cap E_{i}=\{\overline {i+1},\overline{i}\}$. Hence $\widetilde{f}_{i}(r\frak{C}\otimes l\frak{C)}$ and $r\frak{C}^{\prime}\otimes l\frak{C}^{\prime}$ coincide on $E_{i}$. So they are equal because $\widetilde{f}_{i}$ does not modify the letters $x\notin E_{i}$. $\mathrm{(iii):}$ If $\{i,i+1\}\cap\left| C\right| =\{i\},$ the proof is analogous to case $\mathrm{(ii).}$ Suppose $i=n$. Set $E_{n}=\{n,\overline{n}\}$. Then $n\in\left| C\right| $ because $\widetilde{f}_{i}(w)\neq0$. We obtain (\[f\_S\]) by using similar arguments to those of $\mathrm{(i)}$. \[Lem\_S\_D\]     1. There exists two crystal isomorphisms $S_{n}^{D}$ and $S_{n-1}^{D}$$$\begin{aligned} & B(0)\cup B(v_{\omega_{n}^{D}})\cup\left( \overset{n-1}{\underset{i=1}{\cup}}B(v_{\omega_{i}^{D}})\right) \overset{S_{n}^{D}}{\rightarrow }B(v_{\Lambda_{n}^{D}})\otimes(B(v_{\Lambda_{n}^{D}})\cup B(v_{\Lambda _{n-1}^{D}})),\\ & B(0)\cup B(v_{\overline{\omega}_{n}^{D}})\cup\left( \overset {n-1}{\underset{i=1}{\cup}}B(v_{\omega_{i}^{D}})\right) \overset{S_{n-1}^{D}}{\rightarrow}B(v_{\Lambda_{n-1}^{D}})\otimes(B(v_{\Lambda_{n-1}^{D}})\cup B(v_{\Lambda_{n}^{D}})).\end{aligned}$$ 2. Let $w$ be the reading of an admissible column $C$ of type $D$. If $h(C)\prec n,$ denote by $t\ $the greatest unbarred letter such that $t\notin\left| C\right| $. Write - $l\frak{C}$ for the spin column of height $n$ obtained by adding to $lC$ the barred letters $\overline{x}$ such that $x\notin\left| C\right| $. - $r\frak{C}$ for the spin column of height $n$ obtained by adding to $rC$ the unbarred letters $x$ such that $x\notin\left| C\right| $. - $l_{t}\frak{C}$ for the spin column of height $n$ obtained by adding to $lC$ the letter $t$ and the barred letters $\overline{x}$ such that $x\notin\left| C\right| \cup\{t\}$. - $r_{t}\frak{C}$ for the spin column of height $n$ obtained by adding to $rC$ the letter $\overline{t}$ and the unbarred letters $x$ such that $x\notin\left| C\right| \cup\lbrack t\}$. Then we have $$\mathrm{(i)}:\left\{ \begin{tabular} [c]{l}$S_{n}^{D}(w)=r\frak{C}\otimes l\frak{C}$ if $r\frak{C}\in B(v_{\Lambda _{n}^{D}})$\\ $S_{n}^{D}(w)=r_{t}\frak{C}\otimes l_{t}\frak{C}$ otherwise \end{tabular} \right. \text{ and }\mathrm{(ii)}:\left\{ \begin{tabular} [c]{l}$S_{n-1}^{D}(w)=r\frak{C}\otimes l\frak{C}$ if $r\frak{C}\in B(v_{\Lambda _{n-1}^{D}})$\\ $S_{n-1}^{D}(w)=r_{t}\frak{C}\otimes l_{t}\frak{C}$ otherwise \end{tabular} \right.$$ (recall that $r\frak{C}\in B(v_{\Lambda_{n}^{D}})$ if and only if it contains an even number of barred letters). We only sketch the proof for $S_{n}^{D},$ the arguments are analogous for $S_{n-1}^{D}.$ $1:$ The highest weight vertices of $B(v_{\Lambda_{n}^{D}})\otimes (B(v_{\Lambda_{n}^{D}})\cup B(v_{\Lambda_{n-1}^{D}}))$ are the vertices $v_{p}^{D}=\frak{C}_{n}\otimes\frak{C}_{p}$ with $p=0,...,n$. We have $\mathrm{wt}(v_{p}^{D})=\omega_{p}^{D}$ for $p=1,...,n$ and $\mathrm{wt}(v_{0}^{D})=0.\;$Hence $S_{n}^{D}$ is the crystal isomorphism which sends $B(v_{\omega_{p}^{D}})$ on $B(v_{p}^{D})$ for $p=1,...,n$ and $B(0)$ on $v_{0}^{D}.$ $2:$When $w=v_{\omega_{p}^{D}}$, the equality $S_{n}^{D}(w)=r\frak{C}\otimes l\frak{C}$ is true. Consider $w\in B(v_{\omega_{p}^{D}})$ and $i=1,...,n$ such that $w^{\prime}=\widetilde{f}_{i}(w)\neq0$. Write $w=\mathrm{w}(C)$ and $w^{\prime}=\mathrm{w}(C^{\prime})$ where $C$ and $C^{\prime}$ are two admissible columns of height $p$. Let $t^{\prime}$ be the greatest unbarred letter such that $t^{\prime}\notin\left| C^{\prime}\right| $. If the number of barred letters of $C$ is equal to that of $C^{\prime},$ $r\frak{C}$ and $r\frak{C}^{\prime}$ belongs together in $B(v_{\Lambda_{n}^{D}})$ or in $B(v_{\Lambda_{n-1}^{D}}).$ In these cases we can prove that $$S_{n}^{D}(w)=r\frak{C}\otimes l\frak{C\Longrightarrow} S_{n}^{D}(w^{\prime })=r\frak{C}^{\prime}\otimes l\frak{C}^{\prime}\text{ and }S_{n}^{D}(w)=r_{t}\frak{C}\otimes l_{t}\frak{C\Longrightarrow} S_{n}^{D}(w^{\prime })=r_{t^{\prime}}\frak{C}^{\prime}\otimes l_{t^{\prime}}\frak{C}^{\prime} \label{f_SD}$$ as we have done for $S^{B}.$ Otherwise we have $i=n$ and $rC\cap E_{n}=(n-1\}$ or $rC\cap E_{n}=(n\}.$ Suppose $i=n$ and $n\in\left| C\right| $. Then $n-1$ is the unique letter of $E_{n}=\{n-1,n,\overline{n},\overline{n-1}\}$ that occurs in $C$. We have $t=n$ and $t^{\prime}=n-1$ because $lC^{\prime}\cap E_{n}=\overline{n}$. So $r\frak{C}\cap E_{n}=\{n,n-1\},$ $r_{t}\frak{C}\cap E_{n}=\{\overline{n},n-1\},$ $l\frak{C}\cap E_{n}=\{\overline{n},n-1\}$ and $l_{t}\frak{C}\cap E_{n}=\{n,n-1\}$. Similarly $r\frak{C}^{\prime}\cap E_{n}=\{\overline{n},n-1\},$ $r_{t}\frak{C}^{\prime}\cap E_{n}=\{\overline {n},\overline{n-1}\},$ $l\frak{C}^{\prime}\cap E_{n}=\{\overline{n},\overline{n-1}\}$ and $l_{t}\frak{C}^{\prime}\cap E_{n}=\{\overline{n},n-1\}$. Hence $\widetilde{f}_{i}(r\frak{C}\otimes l\frak{C)=}r_{t^{\prime}}\frak{C}^{\prime}\otimes l_{t^{\prime}}\frak{C}^{\prime}$ and $\widetilde {f}_{i}(r_{t}\frak{C}\otimes l_{t}\frak{C)=}r\frak{C}^{\prime}\otimes l\frak{C}^{\prime}$. We have $$S_{n}^{D}(w)=r\frak{C}\otimes l\frak{C\Longrightarrow} S_{n}^{D}(w^{\prime })=r_{t^{\prime}}\frak{C}^{\prime}\otimes l_{t^{\prime}}\frak{C}^{\prime }\text{ and }S_{n}^{D}(w)=r_{t}\frak{C}\otimes l_{t}\frak{C\Longrightarrow}S_{n}^{D}(w^{\prime})=r\frak{C}^{\prime}\otimes l\frak{C}^{\prime}. \label{f_SD1}$$ When $i=n$ and $n-1\in\left| C\right| $, we obtain (\[f\_SD1\]) by similar arguments. Finally $\mathrm{(i)}$ follows from (\[f\_SD\]) and (\[f\_SD1\]). Suppose $n=7$ and consider the admissible column $C$ of type $D$ such that $\mathrm{w}(C)=67\overline{7}7\overline{6}.$ Then $\mathrm{w(}lC)=3457\bar{6}$, $\mathrm{w(}rC)=67\bar{5}\bar{4}\bar{3}.$ So $(t,\overline{t})=(2,\overline{2})$ and, by identifying the spin columns with the set of letters that they contain, we have $l\frak{C}=\{3457\bar{6}\bar{2}\bar{1}\},$ $r\frak{C}=\{1267\bar{5}\bar{4}\bar{3}\},$ $l_{t}\frak{C}=\{23457\bar{6}\bar{1}\}$, $r_{t}\frak{C}=\{167\bar{5}\bar{4}\bar{3}\bar{2}\}$. We have $S_{n}^{D}(\mathrm{w}(C))=r_{t}\frak{C}\otimes l_{t}\frak{C}$ and $S_{n-1}^{D}(\mathrm{w}(C))=r\frak{C}\otimes l\frak{C}$ for $r\frak{C}\not \in B(v_{\Lambda_{n}^{D}})$. Although $C$ must be the empty column in Lemmas \[Lem\_S\_B\] and \[Lem\_S\_D\], we only use these Lemmas with $h(C)\geq1$ in the sequel.. Figure (\[FIG2\]) below describe the connected components of $V(\Lambda_{3}^{D})^{\otimes2}$ and $V(\Lambda_{2}^{D})^{\otimes2}$ isomorphic to the vector representation $V(\Lambda_{1}^{D})$ of $U_{q}(so_{6})$ (see also (\[vect\_B\])). \[ptb\] [Figure2.eps]{} Note that it is possible to describe explicitly the isomorphisms $(S^{B})^{-1}$, $(S_{n}^{D})^{-1}$ and $(S_{n-1}^{D})^{-1}.\;$The reader interested by this subject is referred to [@Lec2]. Plactic monoid for $\frak{G}_{n}\label{sub_sec_mon_spin}$ --------------------------------------------------------- Let $\lambda$ be a dominant weight such that $\lambda\notin\Omega_{+}$. If $\lambda\in P_{+}^{B}$ then $\lambda$ has a unique decomposition $\lambda=\Lambda_{n}^{B}+\lambda^{^{\prime}}$ with $\lambda^{\prime}\in \Omega_{+}^{B}$. We set $v_{\lambda}^{B}=v_{\lambda^{\prime}}\otimes v_{\Lambda_{n}^{B}}$. Then $v_{\lambda}^{B}$ is the highest weight vector of $B(v_{\lambda}^{B}),$ a connected component of $\frak{G}_{n}^{B}$ isomorphic to $B^{B}(\lambda)$. Denote by $Y(\lambda)$ the diagram obtained by adding a K.N-diagram of height $n$ to $Y(\lambda^{\prime})$. When$\ \lambda\in P_{+}^{D}$, $\lambda$ has a unique decomposition of type $\lambda=\Lambda_{n}^{D}+\lambda^{\prime}$ with $\lambda^{\prime}\in\Omega _{+}^{D}$ and $\overline{\omega}_{n}^{D}$ not appearing in $\lambda^{\prime}$ or $\lambda=\Lambda_{n-1}^{D}+\lambda^{\prime}$ with $\lambda^{\prime}\in\Omega_{+}^{D}$ and $\omega_{n}^{D}$ not appearing in $\lambda^{\prime}$. According to this decomposition we set $v_{\lambda}^{D}=v_{\lambda^{\prime}}\otimes v_{\Lambda_{n}^{D}}$ or $v_{\lambda}=v_{\lambda^{\prime}}\otimes v_{\Lambda_{n-1}^{D}}$. Then $v_{\lambda}^{D}$ is the highest weight vector of $B(v_{\lambda}^{D}),$ a connected component of $\frak{G}_{n}^{D}$ isomorphic to $B^{D}(\lambda)$. If $Y(\lambda^{\prime})=(Y^{\prime},\varepsilon)$ (see \[Def\_Y(lambda)\]) with $\varepsilon\in\{-,0,+\},$ we set $Y(\lambda )=(Y,\varepsilon)$ where $Y$ is the diagram obtained by adding a K.N diagram of height $n$ to $Y^{\prime}$. Given a tabloid $\tau$ and a spin column $\frak{C}$, the spin tabloid $[\frak{C},T]$ is obtained by adding $\frak{C}$ in front of $\tau$. The reading of the spin tabloid $[\frak{C},\tau]$ is $[\frak{C},\tau])=\mathrm{w(}\tau)\otimes\frak{C=}\mathrm{w(}\tau)\frak{C}$. Note that the vertices of $B(v_{\lambda})$ are readings of spin tabloids. \[def\_tab\_spin\]      - Let $\lambda\in P_{+}^{B}$ such that $\lambda\notin\Omega_{+}^{B}.$ A spin tabloid is a spin tableau of type $B$ and shape $Y(\lambda)$ if its reading is a vertex of $B(v_{\lambda}^{B})$. - Let $\lambda\in P_{+}^{D}$ such that $\lambda\notin\Omega_{+}^{D}.$ A spin tabloid is a spin tableau of type $D$ and shape $Y(\lambda)$ if its reading is a vertex of $B(v_{\lambda}^{D})$. It follows from this definition that for $\frak{T}_{1}$ and $\frak{T}_{2}$ two spin tableaux $\frak{T}_{1}\sim\frak{T}_{2}\Longleftrightarrow\frak{T}_{1}=\frak{T}_{2}$. It is possible to extend Definition \[Def\_b\_conf\] to a spin tableau $[\frak{C,}C]$ of type $D$ with $C$ an admissible column of type $D$. We will say that $[\frak{C,}C]$ contains an $a$-configuration even or odd when this configuration appears in the tableau of two columns $C_{\frak{C}}C$ where $C_{\frak{C}}$ is the admissible column of type $D$ and height $n$ containing the letters of $\frak{C}.$ Kashiwara and Nakashima have obtained in [@KN] a combinatorial description of the orthogonal spin tableaux equivalent to the following: \[TH\_KNS\]       - $\frak{T=}[\frak{C,}T]$ is a spin tableau of type $B$ if and only if $T$ is a tableau of type $B$ and the rows of $[\frak{C,}lC_{1}]$ weakly increase from left to right. - $\frak{T=}[\frak{C,}T]$ is a spin tableau of type $D$ if and only if $T$ is a tableau of type $D$, the rows of $[\frak{C,}lC_{1}]$ weakly increase from left to right and $[\frak{C,}lC_{1}]$ does not contain an $a$-configuration (even or odd) with $q(a)=n-a$. It follows from the definition above that for any spin tableau $[\frak{C,}T]$ of type $D$ $$\begin{aligned} \frak{C} & \in B(\Lambda_{n}^{D})\text{ implies that the shape of }T\text{ is }(Y,\varepsilon)\text{ with }\varepsilon\neq-,\\ \frak{C} & \in B(\Lambda_{n-1}^{D})\text{ implies that the shape of }T\text{ is }(Y,\varepsilon)\text{ with }\varepsilon\neq+\text{.}$$ A generalized tableau is an orthogonal tableau or a spin orthogonal tableau. Similarly to subsection \[subsec\_monoids\], the quotient sets $\frak{G}_{n}/\overset{B}{\sim}$ and $\frak{G}_{n}/\overset{D}{\sim}$ can be respectively identified with the sets of generalized tableaux of type $B$ and $D$. For $x$ a letter of $\mathcal{B}_{n}$ or $\mathcal{D}_{n}$ and $\frak{C}$ a spin column of height $n$ whose greatest letter is $z$, we write $x\vartriangle\frak{C}$ when $x\nleq z.$ \[def\_mono\_spinB\]The monoid $\frak{Pl(}B_{n}\frak{)}$ is the quotient set of $\frak{B}_{n}^{\ast}$ by the relations: - $R_{i}^{B},$ $i=1,...,5$ defining $Pl(B_{n})$, - $R_{6}^{B}$: for $x\in\mathcal{B}_{n}$ and $\frak{C}$ a spin column such that $x\vartriangle\frak{C;}$ $\frak{C}x\equiv\frak{C}^{\prime}$ where $\frak{C}^{\prime}$ is the spin column such that $\mathrm{wt(}\frak{C}^{\prime})=\mathrm{wt(}\frak{C})+\mathrm{wt(}x),$ - $R_{7}^{B}$: for $x\in\mathcal{B}_{n}$ and $\frak{C}$ a spin column such that $x\not \vartriangle\frak{C;}$ $\frak{C}x\equiv x^{\prime}\frak{C}^{\prime}$ where $$\left\{ \begin{tabular} [c]{l}$x^{\prime}=\min\{t\in\frak{C};$ $t\succeq x\}$ if $x\succeq0$\\ $x^{\prime}=\min\{t\in\frak{C};$ $t\succeq x\}\cup\{0\}$ if $x\preceq n$\end{tabular} \right.$$ and $\frak{C}^{\prime}$ is the spin column such that $\mathrm{wt}(\frak{C}^{\prime})=\mathrm{wt(}\frak{C})+\mathrm{wt(}x)-\mathrm{wt(}x^{\prime})$, - $R_{8}^{B}$:for $C$ an admissible column of type $B,$ $S^{B}(\mathrm{w}(C))\equiv\mathrm{w}(C)$. Lemma \[lem\_phi\_tens\] implies that the highest weight vertex of the connected component containing a word $\frak{C}x$ with $x\in\mathcal{B}_{n}$ and $\frak{C}$ a spin column may be written $\frak{C}_{n}x_{0}$ where $x_{0}\in\{0,1\}$. So $\frak{C}x\in B(v_{\Lambda_{n}^{B}}\otimes0)$ or $\frak{C}x\in B(v_{\Lambda_{n}^{B}}\otimes1).\;$The following lemma gives the interpretation of relations $R_{6}^{B}$ and $R_{7}^{B}$ in terms of crystal isomorphisms. \[Lem\_iso\_spin\]            1. The vertices of $B(v_{\Lambda_{n}^{B}}\otimes0)$ are the words of the form $\frak{C}x$ where $\frak{C}$ is a spin column and $x\in\mathcal{B}_{n}$ such that $x\vartriangle\frak{C.}$ 2. The vertices of $B(v_{\Lambda_{n}^{B}}\otimes1)$ are the words of the form $\frak{C}x$ where $\frak{C}$ is a spin column and $x\in\mathcal{B}_{n}$ such that $x\not \vartriangle\frak{C.}$ 3. Denote by $\Psi$ and $\Psi^{\prime}$ the crystal isomorphisms: $$\text{\begin{tabular} [c]{l}$\Psi:B(v_{\Lambda_{n}^{B}}\otimes0)\rightarrow B(v_{\Lambda_{n}^{B}})$\\ $\Psi^{\prime}:B(v_{\Lambda_{n}^{B}}\otimes1)\rightarrow B(1\otimes v_{\Lambda_{n}^{B}})$\end{tabular} .}$$ Then if the word $\frak{C}x$ occur in the left hand side a relation $R_{6}^{B}$ (resp. of $R_{7}^{B}),$ $\Psi(\frak{C}x)$ (resp. $\Psi^{\prime}(\frak{C}x))$ is the word occurring in the right hand side of this relation. $1$ Consider a word $\frak{C}x$ such that $x\vartriangle\frak{C}$ and $\widetilde{f}_{i}(\frak{C}x)\neq0$. Let $y$ be the greatest letter of $\frak{C}$. Set $\widetilde{f}_{i}(\frak{C}x)=\frak{U}t$ where $\frak{U}$ is a spin column and $t$ a letter of $\mathcal{B}_{n}$. We are going to show that $t\vartriangle\frak{U}$. If $y$ is the greatest letter of $\frak{U}$ then $t\succeq x\succ y,$ hence $t\vartriangle\frak{U}$. Otherwise $\widetilde {f}_{i}(\frak{C}x)=\widetilde{f}_{i}(\frak{C)}x$ thus $\varepsilon_{i}(x)=0$ by (\[TENS1\])$.\;$When $i\neq n,$ we must have $y=\overline{i+1}$, $x\succ y$ and $x\notin\{\overline{i},i+1\}$ because $\varepsilon_{i}(x)=0$. Hence $x\succ\overline{i}$ and $x=t\vartriangle\frak{U}$ for $\overline{i}$ is the greatest letter of $\frak{U}$. When $i=n,$ $y=n$ and $x\succ\overline{n}$ because $\varepsilon_{n}(x)=0$. We obtain similarly $t\vartriangle\frak{U}$.. Hence the set of words $\frak{C}x$ such that $x\vartriangle\frak{C}$ is closed under the action of the $\widetilde{f}_{i}.$ By similar arguments we can prove that this set is also closed under the action of the $\widetilde {e}_{i}.$ Moreover $v_{\Lambda_{n}^{B}}\otimes0$ is the unique highest weight vertex among these words $\frak{C}x$. Hence $B(v_{\Lambda_{n}^{B}}\otimes0)$ contains exactly the words of the form $\frak{C}x$ such that $x\vartriangle \frak{C}$.  $2$ Follows immediately from $1.$ $3$ If $x\vartriangle\frak{C}$, $\Psi(\frak{C}x)$ is the unique spin column of weight $\mathrm{wt(}\frak{C}x)$, that is $\Psi(\frak{C}x)=\frak{C}^{\prime}$ with the notation of $R_{6}^{B}$. When $x\not \vartriangle\frak{C,}$ we consider the following cases: $\mathrm{(i)}$: $x\in\frak{C}$. Set $\Psi(\frak{C}x)=y\frak{D}$. Then we deduce from the equality $\mathrm{wt(}y\frak{D})=\mathrm{wt(}\frak{C}x)$ that $y=x$ and $\frak{D=C}$. Indeed $x\frak{C}$ is the unique vertex of $B(1)\otimes B(v_{\Lambda_{n}^{B}})$ of weight $\mathrm{wt}(\frak{C}x).$ Hence $y=x=t$ and $\frak{D=C}^{\prime}$ with the notation of $R_{6}^{B}$. $\mathrm{(ii)}$: $x\notin\frak{C}$ .$\;$When $x\succ0,$ set $x=\overline{p}$ and $\overline{k}=\min\{t\in\frak{C};$ $t\succeq x\}$. Then $\{p,p-1,...,k+1\}\subset\frak{C}$. By using the formulas (\[TENS1\]) and (\[TENS2\]) we obtain $$\widetilde{f}_{k}\cdot\cdot\cdot\widetilde{f}_{p-2}\widetilde{f}_{p-1}(\frak{C}\overline{p})=\frak{C}\overline{k}$$ So, by $\mathrm{(i)}$, $\frak{C}\overline{k}\sim\overline{k}\frak{C}$ which implies $$\frak{C}\overline{p}\sim\widetilde{e}_{p-1}\cdot\cdot\cdot\widetilde{e}_{k}(\overline{k}\frak{C})=\overline{k}\widetilde{e}_{p-1}\cdot\cdot \cdot\widetilde{e}_{k}(\frak{C})=\overline{k}\frak{C}^{\prime}$$ with the notation of $R_{7}^{B}$. It means that $\Psi(\frak{C}x)=\overline {k}\frak{C}^{\prime}$. When $x=0,$ we have $\widetilde{f}_{x^{\prime}-1}\cdot\cdot\cdot\widetilde{f}_{1}\widetilde{f}_{n}(\frak{C}0)=\frak{C}\overline{k}$.because $\{n,n-1,...,k+1\}\subset\frak{C}$ and we terminate as above. When $x=p\prec0$ and $\min\{t\in\frak{C};$ $t\succeq x\}\cup \{0\}=k\prec0$, we have $\{\overline{p},\overline{p+1},...,\overline {k-1}\}\subset\frak{C}.$ So $\widetilde{f}_{k-1}\cdot\cdot\cdot\widetilde {f}_{p+1}\widetilde{f}_{p}(\frak{C}p)=\frak{C}k$ and the proof is similar. If $\min\{t\in\frak{C};$ $t\succeq p\}\cup\{0\}=0$, $\{\overline{p},\overline{p+1},...,\overline{n}\}\subset\frak{C}$. Then $\widetilde{f}_{n}\cdot\cdot\cdot\widetilde{f}_{p+1}\widetilde{f}_{p}(\frak{C}p)=\frak{C}0\sim\overline{n}\frak{C}^{{{}^\circ}}$ with $\frak{C}^{{{}^\circ}}=\frak{C}-\{\overline{n}\}+\{n\}$ by the case $x=0$. So formulas (\[TENS1\]) and (\[TENS2\]) imply that $\frak{C}x\sim\widetilde{e}_{p}\cdot\cdot\cdot\widetilde{e}_{n}(\overline{n}\frak{C}^{{{}^\circ}})=\widetilde{e}_{n}(\overline{n})\widetilde{e}_{p}\cdot\cdot\cdot \widetilde{e}_{n-1}(\frak{C}^{{{}^\circ}})=0\frak{C}^{\prime}$ with the notation of $R_{7}^{B}$. It means that $\Psi(\frak{C}x)=0\frak{C}^{\prime}$. \[def\_mono\_spinD\]The monoid $\frak{Pl(}D_{n}\frak{)}$ is the quotient set of $\frak{D}_{n}^{\ast}$ by the relations: - $R_{i}^{D},$ $i=1,...,5$ defining $Pl(D_{n})$, - $R_{6}^{D}$: for $x\in\mathcal{D}_{n}$ and $\frak{C}$ a spin column such that $x\vartriangle\frak{C;}$ $\frak{C}x\equiv\frak{C}^{\prime}$ where $\frak{C}^{\prime}$ is the spin column such that $\mathrm{wt(}\frak{C}^{\prime})=\mathrm{wt(}\frak{C})+\mathrm{wt(}x)$, - $R_{7}^{D}$: for $x\in\mathcal{D}_{n}$ and $\frak{C}$ a spin column such that $x\not \vartriangle\frak{C;}$ $\frak{C}x\equiv x^{\prime}\frak{C}^{\prime}$ where $x^{\prime}=\min\{t\in\frak{C};$ $t\succeq x\}$ and $\frak{C}^{\prime}$ is the spin column such that $\mathrm{wt(}\frak{C}^{\prime})=\mathrm{wt(}\frak{C})+\mathrm{wt(}x)-\mathrm{wt(}x^{\prime})$, - $R_{8}^{D}$: for $C$ an admissible column of type $D,$ $S_{n}^{D}(\mathrm{w}(C))\equiv\mathrm{w}(C)$ and $S_{n-1}^{D}(\mathrm{w}(C))\equiv\mathrm{w}(C).$ We can prove by using similar arguments to those of Lemma \[Lem\_iso\_spin\] that the relations $R_{6}^{D}$ and $R_{7}^{D}$ read from left to right describe respectively the crystal isomorphisms $$\begin{tabular} [c]{l}$\left\{ \begin{tabular} [c]{l}$B(v_{\Lambda_{n}^{D}}\otimes\overline{n})\rightarrow B(v_{\Lambda_{n-1}^{D}})$\\ $B(v_{\Lambda_{n-1}^{D}}\otimes n)\rightarrow B(v_{\Lambda_{n}^{D}})$\end{tabular} \right. $\\ and\\ $\left\{ \begin{tabular} [c]{l}$B(v_{\Lambda_{n}^{D}}\otimes1)\rightarrow B(1\otimes v_{\Lambda_{n}^{D}})$\\ $B(v_{\Lambda_{n-1}^{D}}\otimes1)\rightarrow B(1\otimes v_{\Lambda_{n-1}^{D}})$\end{tabular} \right. .$\end{tabular} \label{interpret_R7D}$$ \[lem\_copat\_cry\_op\]Let $w_{1}$ and $w_{2}$ be two vertices of $\frak{G}_{n}$ such that $w_{1}\equiv w_{2}$. Then for $i=1,...,n$: $$\begin{aligned} \widetilde{e}_{i}(w_{1}) & \equiv\widetilde{e}_{i}(w_{2})\text{ and }\varepsilon_{i}(w_{1})=\varepsilon_{i}(w_{2}),\\ \widetilde{f}_{i}(w_{1}) & \equiv\widetilde{f}_{i}(w_{2})\text{ and }\varphi_{i}(w_{1})=\varphi_{i}(w_{2}).\end{aligned}$$ By induction we can suppose that $w_{2}$ is obtained from $w_{1}$ by applying only one plactic relation. In this case we write $w_{1}=u\widehat{w}_{1}v$ and $w_{2}=u\widehat{w}_{2}v$ where $u,v,\widehat{w}_{1},\widehat{w}_{2}$ are factors of $w_{1}\ $and $w_{2}$ such that $\widehat{w}_{1}\equiv\widehat {w}_{2}$ by one of the relations $R_{i}$. Formulas (\[TENS1\]) and (\[TENS2\]) imply that it is enough to prove the lemma for $\widehat{w}_{1} $ and $\widehat{w}_{2}$. This last point is immediate because we have seen that each plactic relation may be interpreted in terms of a crystal isomorphism. So we obtain $w_{1}\equiv w_{2}\Longrightarrow w_{1}\sim w_{2}.$ To establish the implication $w_{1}\sim w_{2}\Longrightarrow w_{1}\equiv w_{2}$, it suffices, as in subsection \[subsec\_monoids\] to prove that two highest weight vertices of $\frak{G}_{n}^{B}$ (resp. $\frak{G}_{n}^{D}$) with the same weight are congruent in $\frak{Pl(}B_{n}\frak{)}$ (resp. $\frak{Pl(}D_{n}\frak{)}$). Given a vertex $w\in\frak{G}_{n},$ we know by Theorems \[TH\_KNS\] and \[TH\_KN\] that there exists a unique generalized tableau $\frak{P}(w)$ such that $$\mathrm{w(}\frak{P}(w))\sim w.$$ Let $w$ be a highest weight vertex of $\frak{G}_{n}$. Then $\mathrm{w(}\frak{P}(w))\equiv w$. By using relations $R_{6}$ and $R_{7}$, $w$ is congruent to a word $u\frak{U} $ such that $u\in G_{n}$ and $\frak{U\in G}_{n}$. Relation $R_{8}$ implies that any word consisting in an even number of spin columns is congruent to a vertex of $G_{n}$. If $\frak{U}$ contains an even number of spin columns, there exists $v\in G_{n}$ such that $w\equiv v$. We have $\frak{P}(w)=P(v)$ because $w\equiv v\Longrightarrow w\sim v.$ Thus $\mathrm{w(}\frak{P}(w))=\mathrm{w(}P(v))\equiv v\equiv w$ and the lemma is proved. If $w$ contains an odd number of spin columns, there exists a vertex $v\in G_{n}$ and a spin column $\frak{C}$ such that $w\equiv v\frak{C}$. Set $P(v)=T$. Then $w\equiv\mathrm{w}(T)\frak{C}$. Write $T=C\widehat{T}$ where $C$ is the first column of $T$ and $\widehat{T}$ the tableau obtained by erasing $C$ in $T$. By Lemma \[lem\_phi\_tens\], $\mathrm{w}(T)$ is a highest weight vertex because $w$ is a highest weight vertex of $\frak{G}_{n}$. In particular, $\mathrm{w}(C)$ is a highest weight vertex. Set $p=h(C)$. Suppose first $w\in\frak{G}_{n}^{B}$. We have $S^{B}(\mathrm{w}(C))=\frak{C}_{n}\frak{C}_{p}$ (see Lemma \[Lem\_S\_B\]). So $w\equiv \mathrm{w}(\widehat{T})\frak{C}_{n}\frak{C}_{p}\frak{C}$. By Lemma \[lem\_phi\_tens\] we must have $\varepsilon_{i}(\frak{C})=0$ for $i=p+1,...,n$. This implies that the letters of $\{\overline{p+1},...,\overline{n}\}$ do not appear in $\frak{C}$. Indeed $\overline{n}\notin\frak{C}$ otherwise $\varepsilon_{n}(\frak{C})\neq0$ and if $\overline{q}\succ\overline{n}$ is the lowest barred letter of $\{\overline {p+1},...,\overline{n}\}$ appearing in $\frak{C} $ we obtain $\varepsilon _{q}(\frak{C})=1\neq0$ because $q+1\in\frak{C}$. So $\frak{C}$ contains the letters of $\{p+1,...,n\}$. Let $\{x_{1}\prec\cdot\cdot\cdot\prec x_{s}\}$ be the set of unbarred letters $\preceq p$ that occur in $\frak{C}$. By Lemma \[Lem\_S\_B\], we have $$S^{B}(x_{1}\cdot\cdot\cdot x_{s}\underset{n-p\text{ times}}{\underbrace {0\cdot\cdot\cdot0}})=\frak{C}_{p}\frak{C.}$$ Hence $$w\equiv\mathrm{w(}\widehat{T})\frak{C}_{n}(x_{1}\cdot\cdot\cdot x_{s}\underset{n-p\text{ times}}{\underbrace{0\cdot\cdot\cdot0}})$$ and by applying relations $R_{6}^{B}$ and $R_{7}^{B}$ we have $w\equiv \mathrm{w(}\widehat{T})(x_{1}\cdot\cdot\cdot x_{s})\frak{C}_{n}$. Write $T^{\prime}=x_{s}\rightarrow(\rightarrow\cdot\cdot\cdot x_{1}\rightarrow \widehat{T})$. Then $[\frak{C}_{n},T^{\prime}]$ is a spin orthogonal tableau and $\mathrm{w(}T^{\prime})\frak{C}_{n}\equiv w$. So $T^{\prime}=\frak{P}(w)$ and the lemma is true. Suppose now $w\in\frak{G}_{n}^{D}$. If the shape of $\widehat{T}$ is $(Y,\varepsilon)$ with $\varepsilon\not =-,$ we consider $S_{n}^{D}(\mathrm{w}(C))=\frak{C}_{n}\frak{C}_{p}$. Then $[\frak{C}_{n},\widehat{T}]$ is a spin tableau and the proof is similar to that of the type $B$ case. If the shape of $\widehat{T}$ is $(Y,\varepsilon)$ with $\varepsilon=-,$ it suffices to consider $S_{n-1}^{D}(\mathrm{w}(C))=\frak{C}_{n-1}\frak{C}_{n-1}$ where instead of $S_{n}^{D}(\mathrm{w}(C))$. Now if $w_{1}$ and $w_{2}$ are two highest weight vertices of $\frak{G}_{n}$ with the same weight $\lambda$, we have $\frak{P}(w_{1})=\frak{P}(w_{2})$ because there is only one orthogonal tableau of highest weight $\lambda $. Then the lemma above implies that $w_{1}\equiv w_{2}$. We can state the Let $w_{1}$ and $w_{2}$ two vertices of $\frak{G}_{n}\frak{.}$ Then $w_{1}\sim w_{2}$ if and only if $w_{1}\equiv w_{2}$. For any vertex $w\in\frak{G}_{n}$, it is possible to obtain $\frak{P}(w)\frak{\ }$by using an insertion algorithm analogous to that describe in Section \[sec\_in\_vect\]. Considering the sequence of shape of the intermediate generalized tableaux appearing during the computation of $\frak{P}(w)$, we obtain a $\frak{Q}$-symbol $\frak{Q}(w)$. Then for $w_{1}$ and $w_{2}$ two vertices of $\frak{G}_{n}$ we have: $$w_{1}\longleftrightarrow w_{2}\Longleftrightarrow\frak{Q}(w_{1})=\frak{Q}(w_{1})$$ where $w_{1}\longleftrightarrow w_{2}$ means that $w_{1}$ and $w_{2}$ occur in the same connected component of $\frak{G}_{n}$. The reader interested by this subject is referred to [@Lec2]. [99]{} De Concini, C.: *Symplectic standard tableaux*, Adv. in Math*.* **34**,1-27 (1979). Date, E., Jimbo, M. and Miwa, T.: *Representations of* $U_{q}(gl(n,C))$* at* $q=0$* and the Robinson-Schensted correspondence*, in L. Brink, D. Friedman and A.M. Polyakov (eds), Physics and Mathematic of Strings*,* Word Scientific, Teaneck, NJ, 1990, pp. 185-211. Fulton, W.: *Young tableaux*, London Mathematical Society, Student Text **35**. Kashiwara, M. and Nakashima T.: *Crystal graphs for representations of the q-analogue of classical Lie algebras*, J. Algebra, **165** (1994), 295-345. Kashiwara, M.: *On crystal bases*: Canadian Mathematical Society, Conference Proceedings, Volume **16** (1995). Kashiwara, M.: *Similarity of crystal bases: AMS Contemporary Math. **194**, 177-186 (1996).* King, R. C.: *Weight Multiplicities for the Classical Groups*, Lectures Notes in Physics **50** (New York; Springer 1975) 490-499. Lascoux, A., Leclerc, B., Thibon, J.Y.: *Crystal graph and q-analogues of weight multiplicities for the roots system* $A_{n}^{\ast} $, Lett. Math. Phys. **35**: 359-374, 1994. Lascoux, A. and Schützenberger, M. P.: *Le mono*$\mathit{\ddot{\imath}}$*de plaxique*, in non commutative structures in algebra and geometric combinatorics A. de Luca Ed., Quaderni della Ricerca Scientifica del C.N.R., Roma, 1981*.* Lecouvey, C.: *Schensted-type correspondence, plactic monoid and Jeu de Taquin for type* $C_{n}$: J. Algebra (to appear) Lecouvey, C.: *Algorithmique et combinatoire des algèbres enveloppantes quantiques de type classique,* Thèse, Université de caen, 2001. Littelmann P.: *A plactic algebra for semisimple Lie algebras*, Adv. in Math*.* **124**,312-331 (1996). Littelmann P.: *Crystal graph and Young tableaux*, *J.* Algebra, **175**, 65-87 (1995). Littelmann P.: *A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras*, Inv. Math., **116**, 329-346 (1994). Nakashima T.: *Crystal base and a generalization of the Littelwood-Richardson rule for the classical Lie algebras,* Comm. Math. Phys. , **154**. 215-243 (1993). Sheats J.T.: *A symplectic jeu de taquin bijection between the tableaux of King and of De Concini*, Trans. A.M.S. ,**351**, n${{}^\circ}$7, 3569-3607 (1999). Sundaran S.: *Orthogonal tableaux and an insertion scheme for* $SO_{2n+1}$, J. Combin. Theory, ser*.* A 53*, 239-256 (1990).* Sundaram S.: *Tableaux in the representation theory of the classical groups*, IMA. Volumes in Mathematics and its Applications, volume **19** (Springer-Verlag 1990).
--- abstract: 'We propose a top-down model for cash CLO. This model can consistently price cash CLO tranches both within the same deal and across different deals. Meaningful risk measures for cash CLO tranches can also be defined and computed. This method is self-consistent, easy to implement and computationally efficient. It has the potential to bring much pricing transparency to the cash CLO markets; and it could also greatly improve the risk management of cash instruments.' author: - 'Yadong Li and Ziyu Zheng [^1]' bibliography: - 'creditref.bib' title: 'A Top-down Model for Cash CLO' --- Introduction ============ Throughout the credit crisis since 2007, cash structured finance instruments such as cash CDO, CLO, MBS and CMBS have suffered much larger losses and write-downs than the synthetic instruments such as synthetic CDO/CLOs. The cause of the disparity in losses is mainly due to the difference in their risk management practices. In the synthetic CDO/CLO market, the market participants are much more accustomed to hedging the market risks via credit indices, index tranches and single name CDS/LCDS contracts. The base correlation model is the market standard model for the synthetic CDO/CLOs, which have played a key role in the risk management of synthetic CDO/CLOs. Having a market standard model like base correlation has nurtured and encouraged proper and sophisticated risk management practice across both investors and market makers in the synthetic market. In comparison, the risk management in the cash CDO/CLO market is much less sophisticated, which is mainly due to the limited availability of hedging instruments as well as the lack of a standard risk-neutral model. The model development for cash instruments has lagged far behind comparing to the modeling capabilities of the synthetic CDO/CLOs. The modeling efforts of cash instruments have been mostly focused on the cashflow modeling, and there are few attempts to price cash CDO/CLOs under the proper risk-neutral framework. Risk management of cash instruments without a proper risk-neutral model is inevitably difficult. In this paper, we present a simple and practical risk-neutral model for cash CLO (Collateralized loan obligation), which is one of the most important type of cash CDOs. The total outstanding notional amount of cash CLO assets has grown to over \$400B in 2007 right before the credit crisis. The same methodology can also be applied to other types of cash CDOs. The proposed methodology brings the modeling capability of cash instruments much closer to the modeling capability of the synthetic instruments. This paper is organized as follows, we first give a brief overview of the important differences between the cash and synthetic CLO markets in section \[intro\], then we review the existing pricing method for cash CLOs and its limitations in section \[existing\]. In section \[topdown\], we introduce the top-down method for CLOs. In the rest of this paper, the unqualified term “CLO” always refers to the cash CLO, while the “synthetic CLO” is always fully qualified to avoid confusion. Cash vs. Synthetic CLO\[intro\] =============================== A cash CLO and a synthetic CLO share very little in common besides their names. A cash CLO deal is usually managed by a manager who can actively trade and hedge the underlying loan portfolio, while a synthetic CLO usually trades in single tranche format referencing a static portfolio. A cash CLO often has complex cashflow waterfall structure so that the cashflow of its tranches are highly path-dependent, while the cashflows of synthetic CLO tranche are very simple and not path-dependent. The cash CLO typically attracts institutional buy-and-hold investors whose investment decisions are mainly based on ratings and yields, while the synthetic CLO index (i.e.: the LCDX) and tranches are mostly traded by the correlation desks of hedge funds and banks who are required to mark their positions to the market. The market dynamics are totally different between the cash and synthetic CLO. In the synthetic CLO market, the participants can easily take long or short positions in the credit indices (e.g. LCDX), index tranches and the underlying single name LCDS, therefore there are strong arbitrage relationships among them. It is easy to construct a profitable basis trade if any of the basis becomes very large. The basis between indices, tranches or single names in the synthetic market therefore tends to stay within a reasonable range. It is important for a synthetic CDO model to maintain the consistency between the tranches and its underlying portfolio because of this strong arbitrage relationship. The cashflows of synthetic CLO tranches are very simple and not path-dependent; therefore it is not only necessary but also feasible to model synthetic CLOs as derivatives of the underlying LCDS contracts. This approach is commonly called “bottom-up” because the model drills down to the individual constituents in the collateral portfolio. Base correlation is the most common synthetic CDO/CLOs model in practice, readers are referred to [@bcexplained] for a description of the base correlation model. By modeling the synthetic CDO/CLO tranches as derivatives of the underlying CDS/LCDS, the total protection value of the tranches automatically adds up to the total protection value of the underlying CDS/LCDS. In contrast, the cash CLOs behave very differently. A cash CLO is very similar to a regular company as it has a manager, some assets (the loan collateral) that are funded with various classes of debt (tranches) and equity (tranche). Unlike the synthetic market, it is almost impossible to construct a basis trade for cash CLOs even if there exists large basis between the market value of CLO assets (underlying collateral loans) and CLO liabilities (including tranches, management fees and other expenses). There are several reasons for this: 1) cash CLO managers typically only report the underlying loan positions once a month, thus the exact loan positions are unknown between reports since the manager may trade the underlying loans at any time 2) the individual loans are often illiquid and rarely traded 3) taking short positions in underlying loans or cash CLO tranches is very difficult in the cash market. Therefore the basis trades for cash CLOs have rarely (if ever) been attempted in practice. As a result, there is no market force that brings the values of cash CLO assets and liabilities together quickly. The values of a cash CLO’s assets and liabilities can diverge significantly for an extended time even though they eventually have to converge[^2]. The cash CLO tranches and the underlying loans can behave like unrelated instruments in short term even though the cashflows of cash CLO tranches are derived from the cashflows of the underlying loans. It is not uncommon for the underlying loan prices and the cash CLO tranche prices to move in opposite direction during the same trading session. The cash CLO market dynamics therefore impose certain practical restrictions on the hedging strategies of cash CLO tranches: 1. It is impractical and almost useless to hedge cash CLO tranches by trading individual loans. 2. It could be feasible and cost effective to macro hedge the overall market movements of the underlying loans using instruments such as TRS or LCDX index. The macro-hedges may not work well if the basis between CLO tranches and the underlying loan changes significantly, therefore one must have a view on the basis before putting on a macro hedge. 3. Even though the synthetic LCDX tranches were originally created to hedge the cash CLO tranches, the historical price movements of cash CLO tranches and synthetic LCDX tranches have shown very little correlation due to the fundamental differences in market dynamics. Therefore, the LCDX tranches have been very poor hedging instruments for the cash CLO tranches despite its original intention. These practical hedging restrictions are important considerations when building a cash CLO model. Review of Cash CLO Pricing Method\[existing\] ============================================= Even though the cash CLO tranches trade in the upfront price format just like ordinary corporate bonds, a more structured quoting/pricing convention is required for cash CLO investors to compare relative values among different cash CLO tranches. The current market standard for quoting and pricing cash CLO tranches is based on a single pricing scenario where the overall underlying loan collateral is assumed to have a constant annualized default rate (CADR), constant annualized prepayment rate (CAPR), and a constant recovery rate (CRR) over the whole life of the CLO. The CADR of the single pricing scenario is usually very low, for example 3% CADR is often used. The CLO tranche cashflows from this single pricing scenario of CADR, CAPR and CRR are then discounted using the riskfree rate plus a discount margin (DM) spread to produce the tranche PVs. Different DMs are required for different tranches to re-produce their market prices from the single pricing scenario. Intex[^3] is a standard software package used by most market participants to compute the cashflows from CLO tranches. Intex has modeled the majority of outstanding cash CLO deals in the market and it can compute the cashflows of (almost) any CLO tranches under any CADR, CAPR and CRR scenarios. The wide adoption of Intex tool is important for the transparency of the cash CLO market because it provides a consensus on the CLO tranche cashflow among market participants. Different market participants thus can reach the same conversion between the DM and CLO tranche prices as long as they all use Intex for the cashflow calculation. The implied DM from the market tranche prices can then be used by the investors to compare the relative values of different cash CLO tranches. We can view the DM as a similar measure as the credit spreads for corporate bonds: the higher the DM, the more likely the cashflow from the single low-CADR pricing scenario would not be paid due to the increase in loan default rates. The cash CLO market makers often maintain a matrix of DMs based on recent market transactions of various ratings, vintages and deal types; and the DM matrix is used to price similar cash CLO tranches whose prices are not observable in the market. Due to the lack of liquidity, the DM matrix is only updated infrequently, often once a month. This DM method is widely used because of its simplicity, but it is really a quoting convention rather than a pricing model. The following is a list of limitations of the DM method: 1. Some important cash CLO structural features, such as IC/OC triggers, are not priced in since they are not relevant under the single low-CADR pricing scenario. However in reality, these features provide valuable protection to the senior and mezzanine tranches under the high default rate environment, therefore they should affect CLO tranche pricing. 2. There is no pricing consistency across different tranches of the same cash CLO deal since their cashflows are discounted by different DMs. There is no concept of correlation of loan defaults and value shift between different parts of the CLO capital structure. 3. The collateral loan prices do not enter the cash CLO tranche pricing at all. Even though the values of cash CLO assets and liabilities do not necessarily move together as discussed in section \[intro\], the underlying loan prices is a key piece of information for comparing the relative values between different cash CLO deals since it reflects the expected loan default rates. 4. There is no meaningful risk measures from the DM method as the only risk factor of a cash CLO tranche is its DM by construction, which does not provide any useful guidance for risk management and hedging. Simple questions like: “how much a cash CLO tranche price would change if the underlying loan prices move by 1 point?” cannot be answered. 5. The DM matrix is usually hand marked by the trading desk, who can also adjust the DM of individual deals in the book. There is very little control in place on how a DM can be marked due to the illiquid nature of the cash CLO market. Often the tranches from certain CLO types and vintages do not transact in the market for weeks (or even months). The DM marks and the resulting CLO tranche prices often lack consistency and transparency; and they are easy targets for mis-marks and manipulations. A new modeling paradigm is therefore urgently needed for the cash CLO market. In this paper, we propose a top-down model for CLO that addresses all the limitations above. A Top-down Cash CLO Model\[topdown\] ==================================== It is more difficult to construct proper models for cash CLOs than for synthetic CLOs due to the opaqueness of the underlying collateral and the complex cashflow waterfall features. There have been prior attempts to price cash CLOs tranches by computing their cashflows from a Monte Carlo simulation of the default times and recovery rates of underlying loans. The default time and recovery simulation could be driven by a default time copula (e.g.: Gaussian Copula). This bottom-up approach achieved little success because of the uncertainties in the underlying loan positions and prices, as well as the prohibitive computational cost to obtain the tranche cashflows from a large number of simulated scenarios. Custom implementations of the cash CLO’s cashflow waterfalls are often required to achieve reasonable simulation speeds as the standard Intex tool may not be fast enough to support a large number of simulated scenarios. In our view, it is not only a costly, but also an almost useless exercise to build a bottom-up cash CLO model in practice. The main benefit of a bottom-up model is the ability to produce risks to individual underlying loans, but it is not feasible to hedge cash CLO tranches by trading individual underlying loans anyway (as discussed in section \[intro\]). Recognizing the drawbacks of the current DM based method and the practical hedging restrictions in the cash CLO market, we hereby propose a simple but practical top-down model for cash CLO. Top-down models were originally developed for exotic synthetic instruments. In a top-down model, the collateral portfolio is modeled as a whole instead of drilling down to individual constituents. The benefit of a top-down approach is its simplicity as a result of not having to model the individual constituents of the underlying portfolio. The adoption of top-down models in synthetic CDO/CLOs has been limited so far because the prices of synthetic tranches do move with its underlying CDS/LCDS due to the strong arbitrage relationship. Ignoring the single name risk is considered a drawback for synthetic instruments since it is critical to hedge synthetic tranches by trading the underlying single name CDS/LCDS contracts. However, a top-down approach is ideal for cash CLOs since there is no strong arbitrage relationship between the CLO assets and liabilities in the cash market, and the cash CLO tranche prices do not necessarily move consistently with individual underlying loans. Therefore, ignoring the individual single name information, being a vice in the synthetic CLOs modeling, becomes a virtue in the cash CLO modeling because it is closer to the market reality and it greatly simplifies the model setup. Note that a top-down model can produce risk measures to the overall average loan price in the cash CLO portfolio, thus it is possible to macro-hedge the overall loan price movements using a top-down model. It is just not possible to produce hedge ratios to the individual loans with a top-down approach, which is useless in practice anyway. Calibrate to “Index” CLO ------------------------ Fundamentally, the objective of a pricing model is to find the prices of less liquid instruments from the prices of liquid instruments. For example, when pricing a bespoke synthetic CDO using the base correlation model, we first extracts the correlation information from the liquid index tranches by calibrating a base correlation surface, which is then mapped to the bespoke portfolios to produce the bespoke tranche prices. The pricing consistency among different bespoke tranches are maintained because all of them are priced from the same set of liquid index tranches. This procedure allows us to compute risk sensitivities of bespoke tranches to the liquid index tranches; and we can hedge the illiquid bespoke tranches by trading the liquid index tranches. Given the success of synthetic CDO/CLO models, it is a natural idea trying to apply the same pricing method to cash CLOs. However, a practical challenge is that there is no standard liquid index for cash CLOs. As discussed before, the LCDX tranches can’t be used to price cash CLOs because of the fundamental differences between the cash and synthetic markets. To get around this, we have to assume that there is a representative cash CLO deal whose market price is somewhat transparent, which can be used as an “index” to price other cash CLO deals. In practice, cash CLO market participants can choose a representative CLO that are reasonably liquid as the CLO “index”. Once we identified a cash CLO index and its tranche prices, we then can carry out the calibration and mapping procedure for cash CLOs in a similar manner as in the synthetic CLO models. Figure \[idx1\] showed the deal information and tranche prices of an actual cash CLO deal, whose price marks are provided by the Lehman CLO trading desk as of Aug. 12, 2008. We will use this CLO deal as the “index” for the following discussion. This “index” deal is subsequently referred as CLO-IDX. [**Class**]{} [**Coupon Rate**]{} [**Notional**]{} [**OC Trigger(%)**]{} [**IC Trigger(%)**]{} [**S&P Rating**]{} [**Prices (%)**]{} --------------- --------------------- ------------------ ----------------------- ----------------------- -------------------- -------------------- A Libor+69.5bp 506,250,000 AAA 92.97 B Libor+110.0bp 61,875,000 AA 82.16 C Libor+200.0bp 43,125,000 111.2 112.5 A 78.83 D Libor+3.25% 30,000,000 107.2 107.5 BBB 72.13 E Libor+5.00% 33,750,000 104.4 100.1 BB 63.77 SUBORD - 75,000,000 - - NA 44.89 The traditional DM method only uses a single pricing scenario of CADR, CAPR and CRR for all the tranches, which is an overly simplified assumption since the future default, prepay and recovery rates are by no means deterministic. It is much more realistic to assume that there are a set of possible market scenarios of (CADR, CAPR and CRR), and each scenario has certain risk-neutral probability of realization. Figure \[cadrs\] is a set of representative scenarios that are provided by the Lehman CLO research based on the market condition as of mid 2008. The CAPR and CRR are chosen to be decreasing with the CADR based on historical observations. Even though the default, prepay and recovery rates can be time dependent, we kept them constant in this study for simplicity. It is sensible to add time varying default scenarios if default is likely to be front or back loaded. The distribution of these market scenarios can be calibrated to the market prices of the cash CLO tranches. More formally, we use $S_i$ to represent the i-th (CADR, CAPR, CRR) scenario in Figure \[cadrs\], and we use $v_j(S_i)$ to represent the PV of the j-th CLO tranche under the scenario $S_i$, which can be computed by simply discounting the cashflow from Intex using the risk free rates [**without**]{} any additional DM. We also use $V_j$ to represent the market price of the j-th CLO tranche as shown in Figure \[idx1\], then the calibration reduces to a problem of finding a discrete distribution of $\{p_i\}$ for the given set of scenarios so that for every CLO tranche $j$: $$\label{calib} \sum_i p_i v_j(S_i) = V_j$$ We call the $\{p_i\}$ that solves the market implied scenario distribution (MISD). This approach is similar in spirit to the [@brigogpl] for the synthetic CDOs, the main difference here is that there is no constraints from the underlying collateral loan prices, which is a conscious choice because there is no strong arbitrage relationship between the cash CLO assets and liabilities. The tranche prices across the full capital structures have to be used in the MISD calibration otherwise the overall risk of the underlying portfolio cannot be determined. In this approach, the CLO tranche cashflows are computed by Intex using only the aggregated CADR, CAPR and CRR of the whole loan collateral portfolio, this is effectively a top-down approach since it does not drill down to the individual collateral loans. [**CADR (%)**]{} [**CAPR (%)**]{} [**CRR (%)**]{} [**CADR (%)**]{} [**CAPR (%)**]{} [**CRR (%)**]{} ------------------ ------------------ ----------------- ------------------ ------------------ ----------------- 0 15 84 16 0 36 1 14 81 17 0 33 2 13 78 18 0 30 3 12 75 19 0 27 4 11 72 20 0 24 5 10 69 22 0 18 6 9 66 24 0 12 7 8 63 26 0 6 8 7 60 28 0 0 9 6 57 30 0 0 10 5 54 35 0 0 11 4 51 40 0 0 12 3 48 45 0 0 13 2 45 50 0 0 14 1 42 60 0 0 15 0 39 90 0 0 Since the number of scenarios in Figure \[cadrs\] is much greater than the number of tranches in Figure \[idx1\], there are infinitely many distributions that can reprice all the index CLO tranches. Therefore, certain objective function has to be exogenously chosen so that we can find a unique distribution using an optimization method. The maximum entropy method is well suited for such under-determined optimization problems in derivative pricing as it finds a distribution with the most uncertainty and the least bias. Readers are referred to [@wmc] for an introduction to the maximum entropy optimization method. [**CADR**]{} [**A**]{} [**B**]{} [**C**]{} [**D**]{} [**E**]{} [**SUB**]{} [**COL**]{} -------------- ----------- ----------- ----------- ----------- ----------- ------------- ------------- 0 103.92 107.69 114.67 124.66 139.01 174.62 109.73 1 103.93 107.74 114.81 124.83 139.49 162.27 109.07 2 103.95 107.83 114.95 125.15 139.81 147.21 108.22 3 103.97 107.91 115.16 125.53 140.37 127.51 107.18 4 103.99 108.03 115.38 125.89 141.12 101.18 105.93 5 104.01 108.09 115.52 126.12 141.33 73.03 104.49 6 104.03 108.23 115.80 126.64 136.62 43.27 102.86 7 103.96 108.21 115.88 126.86 129.57 17.97 100.98 8 103.46 107.61 115.77 127.04 112.86 13.46 98.94 9 102.86 106.23 112.14 125.50 80.36 10.26 96.66 10 102.68 105.95 111.33 120.69 35.28 7.12 93.99 11 102.62 105.98 111.63 61.10 29.66 6.22 91.33 12 102.57 105.99 105.77 8.91 25.43 4.30 88.48 13 102.54 106.15 62.05 7.26 18.14 3.16 85.44 14 102.61 105.57 5.81 5.67 18.55 3.09 82.21 15 102.61 70.31 5.80 5.67 13.53 3.01 78.78 16 102.07 36.85 4.53 4.11 13.29 2.94 75.67 17 98.08 33.64 3.28 4.11 8.68 2.26 72.50 18 93.76 32.90 3.28 4.11 12.42 2.10 69.28 19 89.02 32.90 3.28 2.58 12.31 0.31 66.02 20 84.03 32.90 2.40 2.58 12.14 0.20 62.72 22 73.89 31.80 2.05 2.58 11.80 0.16 56.02 24 63.67 31.74 2.05 2.58 7.60 0.13 49.22 26 54.89 29.79 2.05 2.58 1.31 0.09 42.35 28 44.48 29.38 2.05 1.02 2.39 0.00 35.43 30 40.43 29.39 0.80 1.02 9.22 0.00 32.76 35 33.23 17.47 0.80 1.02 8.78 0.00 27.08 40 27.53 10.79 0.80 1.02 3.70 0.00 22.60 45 36.50 7.98 0.80 1.02 0.00 0.00 19.09 50 33.40 4.77 0.00 0.00 0.00 0.00 16.32 60 28.28 3.71 0.00 0.00 0.00 0.00 12.39 90 20.74 1.68 0.00 0.00 0.00 0.00 6.55 Market scenarios are only indexed by its CADR, the full scenario definition is listed in Figure \[cadrs\]. Figure \[intex\] showed the CLO-IDX tranche prices computed by Intex for every scenario in Figure \[cadrs\]. Note that these tranche prices are computed [**without**]{} any additional DM above the risk free rate. Given the data in Figure \[intex\], we can easily find the MISD that reproduces the market CLO tranche prices in Figure \[idx1\] via the maximum entropy method. The calibrated distribution is shown in the left side of Figure \[dist0\]. The fitting quality of the MISD is excellent, the market tranche prices are matched almost exactly, which is not surprising because the number of tranches is much less than the number of market scenarios. By finding the MISD, we have moved from the traditional pricing method of a single pricing scenario with different DMs for different tranches, to a more consistent pricing method of a single MISD and a single set of risk-free discount factors for all tranches. It seems to be a small step to replace multiple DMs with multiple market scenarios in the MISD, however this is a significant step forward since it is not only more realistic, but also addresses the first two limitations of the traditional DM method listed in section \[existing\]. Since the market scenarios in Figure \[cadrs\] cover a wide range of CADR from 0% to 90%, every structural feature in a cash CLO deal is expected to be triggered under some of the scenarios; thus they are fully priced in by the MISD method. Also, it is obvious that all the tranches from the same CLO deal are priced consistently to each other because the same MISD and risk-free discount factors are used. For comparison purposes, we also calibrated the MISD to the synthetic LCDX10 tranches, the results are shown on the right side of Figure \[dist0\]. The tranche PVs of LCDX10 under each market scenario can be directly computed without using Intex since the LCDX tranche cashflow is a simple function of the aggregated portfolio loss. At first glance, the MISD from the CLO-IDX and the LCDX10 are quite different: the MISD from LCDX10 is roughly uni-modal while the MISD from CLO-IDX is obviously multi-modal. We have calibrated the MISD to many cash CLO deals and found that the multi-modality of MISD is a common feature among almost all cash CLOs in mid 2008, whereas it is not present in the MISD from synthetic CLOs such as LCDX10. The multi-modality of CLOs is caused by the strong market demand for the very safe AAA rated assets during the severe market stress of mid 2008. As shown in Figure \[idx1\], the AA tranche is priced more than 10 points cheaper than the AAA tranche, such a steep price drop from the AAA tranche to AA tranche is mostly caused by market technicals instead of fundamentals. If we bump the AA tranche price in Figure \[idx1\] up by 5 points and re-calibrate the MISD to the bumped CLO prices, the resulting MISD (Figure \[aa\]) becomes closer to the uni-modal MISD from the LCDX10. This exercise showed that the MISD method allows us to meaningfully compare and identify discrepancies between cash and synthetic CLO markets, it also demonstrated the huge differences in dynamics and technicalities between cash and synthetic CLO markets. --------------------------------------- -------- [**CADR**]{} 13.94% [**CAPR**]{} 5.77% [**CRR**]{} 49.94% [**Average Collateral Loan Price**]{} 84.42 --------------------------------------- -------- From the calibrated MISD, we can easily compute expectations of various quantities, such as expected CADR, CAPR and CRR rates, as shown in Figure \[implied\]. Since the MISD are calibrated to cash CLO tranche prices, we call them the tranche implied CADR, CAPR and CRR. The tranche implied CRR is much lower than the historical loan recovery rates (usually above 70%), which is a sign of stress in the cash CLO tranche market. Using the average underlying collateral loan price (last column of Figure \[intex\]) computed by Intex, we can also obtain the tranche implied average collateral loan price, which is the average collateral loan price that makes the total asset value equals the total liability value for the given cash CLO deal. Comparing the tranche implied average loan price against the average of actual market loan prices[^4] gives us the basis between the cash CLO tranche market and the underlying loan market. This basis cannot be obtained by simply comparing the notional weighted average of the CLO tranche prices against the average price of the underlying loans since 1) this does not take into account the cash CLO manager’s fee and other expenses that are taken from the collateral loan cashflows 2) the total notional amount of the cash CLO tranches is usually not the same as the total notional amount of the underlying loans. The COL column in Figure \[intex\] reported by Intex, on the other hand, is the PV of the total underlying collateral loan (inclusive of any fees and expenses) normalized by the outstanding notional of the loan collateral, therefore it is the right quantity for computing the tranche implied average loan prices. In the case of CLO-IDX, the average market price of its underlying loans is around 89.51 and the tranche implied average collateral loan price is 84.42 as shown in Figure \[implied\], i.e., there is a very large negative basis of -5 points, which implies that a investor would lose 5 points instantly if he creates a cash CLO out of a pool of loans. At the mid of 2008, almost all cash CLO deals showed large negative basis between the tranche implied loan price and the average market loan price. In a normal market environment, this basis should be positive otherwise there is no economic incentive to package individual loans to CLOs in the first place[^5]. However, during market stress of mid 2008, the cash CLO tranches are severely depressed amid the wide-spread fear of complex structured finance products, therefore the cash CLO tranches traded at deep discounts comparing to the underlying loans. It is another important advantage for the top-down MISD method to be able to compute the tranche implied quantities and compare them against those from the underlying loan market. This offers a meaningful relative value comparison between the cash CLO market and the underlying loan market. These tranche implied quantities cannot be computed from the traditional DM method, nor can they be obtained from any bottom-up models because bottom-up models enforces the value equality between assets and liabilities. Therefore, the top-down MISD approach is actually more useful and closer to market reality than the bottom-up approach for cash CLOs. [**Class**]{} [**Coupon Rate**]{} [**Notional**]{} [**OC Trigger(%)**]{} [**IC Trigger(%)**]{} [**S&P Rating**]{} --------------- --------------------- ------------------ ----------------------- ----------------------- -------------------- A Libor+25.0bp 375,000,000 AAA B Libor+37.0bp 22,500,000 AA C Libor+65.0bp 17,500,000 108.9 108.9 A D Libor+140.0bp 30,000,000 103.5 103.5 BBB E Libor+3.650% 15,000,000 102.1 - BB SUBORD - 40,000,000 - - NA Map to “Bespoke” CLO -------------------- After calibrating the MISD to an “index” CLO, we now investigate how to price tranches of other CLO deals. Borrowing the terminology from synthetic CLO, we refer the cash CLO deal we want to price as the “bespoke” CLO. In this section, we use another actual CLO as a sample bespoke CLO, which we subsequently refer to as CLO-BSPK. The CLO-BSPK is of the same vintage and has similar structural features as the CLO-IDX, therefore it is quite sensible to price CLO-BSPK from CLO-IDX. Figure \[bspkinfo\] shows some basic information on the CLO-BSPK deal, and Figure \[bspk12cf\] shows the tranche prices of CLO-BSPK calculated by Intex under the market scenarios of Figure \[cadrs\]. [**CADR**]{} [**A**]{} [**B**]{} [**C**]{} [**D**]{} [**E**]{} [**SUB**]{} [**COL**]{} -------------- ----------- ----------- ----------- ----------- ----------- ------------- ------------- 0 101.30 102.48 104.50 110.08 127.30 185.91 106.74 1 101.31 102.50 104.54 110.20 127.55 167.50 105.97 2 101.31 102.53 104.60 110.32 127.87 145.65 105.03 3 101.32 102.56 104.67 110.48 128.21 120.74 103.92 4 101.32 102.58 104.70 110.53 128.32 100.83 102.64 5 101.33 102.62 104.79 110.69 122.26 71.86 101.19 6 101.08 102.12 103.90 108.98 112.89 44.05 99.57 7 100.98 101.99 103.60 108.20 111.50 17.45 97.78 8 100.92 101.95 103.50 107.90 86.04 11.69 95.80 9 100.90 101.94 103.49 96.37 39.64 9.07 93.66 10 100.88 101.96 103.54 61.12 38.19 6.03 91.34 11 100.86 101.98 102.66 23.67 37.34 3.57 88.85 12 100.88 102.02 63.94 5.27 33.25 2.31 86.18 13 100.86 92.13 6.31 4.39 32.66 2.29 83.34 14 100.88 36.86 5.27 3.98 16.62 2.26 80.33 15 97.40 27.70 4.25 2.88 16.37 2.24 77.14 16 93.60 27.42 3.30 2.88 16.15 2.22 74.20 17 89.72 27.42 3.30 2.88 15.84 2.20 71.21 18 86.13 27.42 2.38 2.88 9.10 1.46 68.17 19 82.26 27.42 2.38 1.80 9.33 0.62 65.08 20 78.11 27.28 2.38 1.80 13.74 0.00 61.94 22 69.84 26.39 1.49 1.80 15.52 0.00 55.56 24 61.31 26.39 1.49 1.80 15.30 0.00 49.06 26 52.80 25.34 1.49 0.70 15.07 0.00 42.46 28 44.06 25.34 1.49 0.70 14.84 0.00 35.79 30 40.70 24.28 1.49 0.70 14.60 0.00 33.03 35 35.03 22.17 0.58 0.70 0.00 0.00 27.09 40 29.53 10.57 0.58 0.70 0.00 0.00 22.28 45 24.79 9.08 0.58 0.00 0.00 0.00 18.46 50 21.15 4.87 0.58 0.00 0.00 0.00 15.44 60 15.71 3.93 0.00 0.00 0.00 0.00 11.15 90 8.48 2.19 0.00 0.00 0.00 0.00 5.38 Market scenarios are only indexed by its CADR, the full scenario definition is listed in Figure \[cadrs\]. The key to price a “bespoke” CLO is to perturb the index MISD to incorporate the bespoke specific information. Using terminology from synthetic CLO, we need to find a mapping methodology between the index MISD and the bespoke MISD. The cross entropy method is an ideal method for such mapping operation between two distributions since it gives a distribution that is closest to the prior distribution (i.e., the calibrated index MISD) while satisfying additional linear constraints that accounts for important bespoke specific features. Readers are referred to [@wmc] for an introduction to the cross entropy (also known as Kullback-Leibler relative entropy) method. Among all the features of the “bespoke” CLO, two of them are the most important: 1. The average price of the underlying loan collateral: This is important because it adjusts for the loan quality difference between the index and bespoke CLO. It also allows us to compute tranche sensitivities to the average loan prices, which can be used for macro-hedging. 2. The AAA-rated tranche: Since the AAA tranches are the most liquid, and all the AAA CLO tranches are priced very similarly to each other with minor adjustments for coupons and the underlying loan quality. The market participants can accurately determine the bespoke AAA tranche prices from similar AAA transactions on the market. Both of these features can be easily added as linear constraints in the cross entropy optimization. To incorporate the underlying loan price, we need an assumption on the size of the basis between the tranche implied and market loan prices for the bespoke CLO. Here we assume that the basis is constant and we use $K^I$ to denote the basis between the tranche implied loan price and average market loan price for the index CLO and $M^B$ to denote the average market price of the underlying loans in the bespoke portfolio, then the linear constraint for the market loan price of the bespoke CLO becomes: $$\label{basis} \sum_i q_i C(S_i) = K^I + M^B$$ where $q_i$ is the MISD of the bespoke cash CLO we want to price; $C_j(S_i)$ is the average collateral loan prices from Intex as shown in the last column of Figure \[bspk12cf\], which already accounts for the average underlying loan features such as coupon rate, payment schedule and amortization etc. Similarly, the constraints for the AAA tranche can be expressed as: $$\label{aaa} \sum_i q_i v^B_\textup{AAA}(S_i) = V^B_\textup{AAA}$$ where $v^B_\textup{AAA}$ is the tranche PVs for the AAA tranche in Figure \[bspk12cf\], and the $V^B_\textup{AAA}$ is the expected bespoke AAA tranche price. Other adjustments can also be included in the cross entropy mapping method. For example, CLO deals managed by a reputable manager often command a sizable premium comparing to those managed by a mediocre manager. This management quality factor can also be included as an adjustment to . In this example, we assume there is no difference in management quality between the CLO-BSPK and CLO-IDX. [**Class**]{} [**S&P Rating**]{} [**Model Price**]{} [**Model Delta**]{} ----------------- -------------------- --------------------- --------------------- CLO-BSPK A AAA 89.35 0.30 CLO-BSPK B AA 78.24 1.52 CLO-BSPK C A 70.78 2.38 CLO-BSPK D BBB 55.90 3.55 CLO-BSPK E BB 60.95 3.77 CLO-BSPK SUBORD - 47.63 5.21 With these two linear constraints in and , it is easy to find the MISD for the bespoke cash CLO via the cross entropy method. Figure \[bspk\] showed both the MISD of the index cash CLO and the mapped MISD of the bespoke CLO. The bespoke cash CLO tranche prices are easy to compute from the mapped MISD and the PV scenarios in Figure \[bspk12cf\]. Figure \[bspkprice\] showed the resulting bespoke cash CLO tranche prices. It is interesting to note that the E tranche is priced higher than the D tranche for CLO-BSPK as shown in Figure \[bspkprice\]. A careful examination of the tranche PVs in Figure \[bspk12cf\] reveals the reason being that E tranche worths much more than the D tranche under most high CADR scenarios due to a structural features called CERT trigger in the CLO-BSPK deal, which diverts the cashflow to the E tranche instead of the more senior C and D tranches under certain high default rate scenarios. As shown in the cashflow table in Figure \[intex\], a similar but less prominent CERT trigger also exists in the CLO-IDX deal. The purpose of the CERT trigger was to boost the rating of the E tranche. If we use the traditional DM method to price the CLO-BSPK, the E tranche would certainly be priced less than the D tranche since the DM method only uses the single pricing scenario of 3% CADR, under which the D and E tranches of CLO-BSPK behave very similarly to the D and E tranches of CLO-IDX since the CERT trigger is only active under high default rate scenarios. This example showed that the top-down MISD method automatically prices in the structural differences between the index and bespoke cash CLO tranches, thus being able to identify potential mis-prices in the traditional DM method. The proposed calibration and mapping procedure is a one-period model without any term structure. Unlike synthetic CDO/CLOs which can trade at multiple maturities, each cash CLO only have a single pre-determined maturity, therefore a one-period model is adequate if the index and bespoke CLOs are from similar vintage and have similar reinvestment period. Risk Measures ------------- Under the traditional DM method, it is very difficult to quantify the risk of a cash CLO book since there are no meaningful risk measures. The most common view of the risk is the aggregated tranche notional for each rating bucket, which is a very crude estimate of the overall risk as there is no indication of the relative riskiness between different rating buckets. No concrete hedging strategy can be devised from the aggregated cash CLO notional amounts by the rating bucket. With the cross-entropy mapping method between the MISD of the index and bespoke CLO, we can easily define and compute a set of consistent risk measures for the CLO tranches. For example, it is easy to compute the cash CLO tranche sensitivities to the underlying loan price movements via a simple bump-remap-reprice procedure. As shown in Figure \[bspkprice\], the tranche deltas computed this way are quite reasonable as the deltas are positive and decreasing with the tranche seniority. The aggregated tranche deltas can be used to predict the P&L change of the whole book for a given movement of the average underlying loan price. This is a very precise risk measure which can be used to macro-hedge the cash CLO book. [**Bespoke Tranche**]{} [**AAA**]{} [**AA**]{} [**A**]{} [**BBB**]{} [**BB**]{} [**NA**]{} ------------------------- -------------- -------------- -------------- -------------- -------------- -------------- CLO-BSPK A [**0.89**]{} [**0.24**]{} -0.20 0.01 -0.01 0.00 CLO-BSPK B 0.11 [**0.43**]{} [**0.31**]{} 0.00 -0.01 0.00 CLO-BSPK C 0.25 -0.49 [**1.07**]{} [**0.27**]{} -0.08 0.01 CLO-BSPK D -0.02 -0.01 -0.02 [**0.50**]{} [**0.54**]{} -0.06 CLO-BSPK E 0.04 0.12 0.12 -0.08 [**0.72**]{} 0.15 CLO-BSPK SUBORD 0.03 -0.05 0.14 0.04 -0.04 [**1.08**]{} Similarly, we can define and compute the sensitivities to the index cash CLO tranches via the bump-remap-reprice procedure, as shown in Figure \[tranche01\]. This sensitivity is commonly called tranche01 in the synthetic CLO terminology. The tranche01 risk of CLO-BSPK showed that the CLO-BSPK tranches are the most sensitive to the index tranches of the same rating with some spillover to the next junior tranche. The tranche01 risk is a measure of correlation risk, which is very useful in practice because we can break down the risk of a cash CLO trading book into corresponding tranches of the index CLO, thus allowing us to understand and manage the risk exposure to different parts of the capital structure. The tranche01 risk is a much better measure than the aggregated tranche notional by rating bucket. For example, Figure \[tranche01\] showed that the BBB-rated D tranche of the CLO-BSPK behaves like a 50-50 mix of the D(rated BBB) and E(rated BB) tranches of the CLO-IDX; which is mainly due to the fact that the D tranche of CLO-BSPK has 3.5% less subordination than the D tranche of CLO-IDX even though they are both BBB rated. These structural differences between cash CLO deals are automatically captured by the tranche01 risk from the top-down MISD method, thus providing a much more coherent view of the correlation risk of a cash CLO book. Other risk measures such as interest rate risk and theta risk can be similarly defined and computed from the top-down method. Conclusion ========== The proposed top-down method is ideal for cash CLOs as it produces consistent prices and risks across cash CLO deals while being very simple, intuitive and computationally efficient. Drilling down to the individual collateral loans provides very little practical benefits because of the lack of liquidity in individual loans and the lack of strong arbitrage relationship between cash CLO assets and liabilities. In practice, this top-down model allows a cash CLO trading desk to only mark the prices of a few representative cash CLO deals as the indices for different vintages and deal types, then the rest of the cash CLO tranches in the book can be automatically priced via the cross entropy mapping method. This allows the cash CLO tranches to be priced consistently using the same calibrate-and-mapping procedure as in synthetic CDO/CLOs, making it much more difficult to manipulate the price marks and book P&L. All the structural features of a cash CLO are automatically taken into consideration by the proposed top-down method, which is a big improvement over the traditional DM based method. Although the consistency between the value of cash CLO assets and liability is not enforced during the calibration because of the lack of strong arbitrage relationship, the average loan price is a valuable piece of market information and it is used by the cross entropy mapping to adjust for the underlying loan quality difference between cash CLO deals. Therefore this top-down MISD method can be a very effective method in finding relative value trading opportunities between cash CLO tranches, especially when most market participants are still using the traditional DM method. This top-down method also produces a full set of risk measures. It is feasible to attribute the P&L movement of a cash CLO trading book using the change of the average underlying loan prices and the index cash CLO tranche prices. Even though it is not feasible to hedge CLO tranches by trading individual loans, it is certainly possible to macro hedges the risk of overall loan market movement based on the cash CLO deltas from the top-down model. If the market develops and certain “index” cash CLO deals becomes easier to short (for example, via TRS), then it is also possible to hedge the correlation risk of a cash CLO book via the tranche01 risk from the model. Being able to meaningfully define risk measures and devise their hedging strategies for a cash CLO book is certainly another big improvement over the DM based method. This method is also computationally efficient since there is only a limited number of scenarios (Figure \[cadrs\]) to run for each deal. The calibration, pricing and risk measures of cash CLO tranches can be computed very efficiently using the the standard Intex tool, and this is no need to build any custom cashflow waterfall engine. This top-down method is very easy to implement and operate in practice as most cash CLO market participants already use the Intex tool. Using this top-down method, different market participants will reach the same CLO tranche prices if they can agree on a standard set of market scenarios like those listed in Figure \[cadrs\], and if they can establish a poll to determine the prices of a small set of representative “index” cash CLO tranches. Both of these two steps are well within reach therefore this method has the potential to bring much more pricing transparency to the cash CLO market. [^1]: Li: Barclays Capital, yadong.li@gmail.com. Zheng: Morgan Stanley, ziyu.zheng@ms.com. This paper is based the authors’ research between May and Sep 2008 while both were employed by Lehman Brothers. The views expressed in this paper are the authors’ own and do not necessarily reflect those of Lehman Brothers or their current employers. The authors thank Greg Xue for Intex analytics support, Lorraine Fan, Marco Naldi, Ariye Shater, Gaurav Tejwani for many helpful comments and discussions, and the Lehman CLO trading desk for many valuable inputs. [^2]: At the maturity of the CLO deal to the latest since all the loans will either mature or liquidate at maturity. [^3]: A product of Intex Solutions, http://www.intex.com [^4]: Since some of the underlying loans are illiquid, the average loan prices are actually computed only from loans with observable market prices [^5]: Release of capital is another reason to create cash CLO from loans.
--- author: - 'M. Gullieuszik' - 'L. Greggio' - 'E. V. Held' - 'A. Moretti' - 'C. Arcidiacono' - 'P. Bagnara' - 'A. Baruffolo' - 'E. Diolaiti' - 'R. Falomo' - 'J. Farinato' - 'M. Lombini' - 'R. Ragazzoni' - 'R. Brast' - 'R. Donaldson' - 'J. Kolb' - 'E. Marchetti' - 'S. Tordo' bibliography: - 'bibliouks.bib' date: 'Received ... ; accepted ...' title: 'Resolving Stellar Populations outside the Local Group: MAD observations of [UKS2323-326]{} [^1]' --- [We present a study aimed at deriving constraints on star formation at intermediate ages from the evolved stellar populations in the dwarf irregular galaxy [UKS2323-326]{}. These observations were also intended to demonstrate the scientific capabilities of the multi-conjugated adaptive optics demonstrator (MAD) implemented at the ESO Very Large Telescope as a test-bench of adaptive optics (AO) techniques.]{} [We perform accurate, deep photometry of the field using $J$ and $K_s$ band AO images of the central region of the galaxy.]{} [The near-infrared (IR) colour-magnitude diagrams clearly show the sequences of asymptotic giant branch (AGB) stars, red supergiants, and red giant branch (RGB) stars down to $\sim$1 mag below the RGB tip. Optical–near-IR diagrams, obtained by combining our data with Hubble Space Telescope observations, provide the best separation of stars in the various evolutionary stages. The counts of AGB stars brighter than the RGB tip allow us to estimate the star formation at intermediate ages. Assuming a Salpeter initial mass function, we find that the star formation episode at intermediate ages produced $\sim 6\times 10 ^5$ $M_\odot$ of stars in the observed region. ]{} Introduction ============ The study of the resolved stellar populations in external galaxies has developed greatly in the last decade to become arguably the most accurate tool to investigate star formation history in stellar systems. However, with standard instrumentation at ground-based telescopes this study is limited to the nearest galaxies, due to the severe crowding of stars. High-precision photometry for the most distant galaxies in the Local Group (LG) and beyond can be obtained only with the Hubble Space Telescope (HST). New opportunities in this field are foreseen with the realisation of imagers equipped with adaptive optics (AO), on the largest aperture telescopes. The use of AO systems is mandatory for the future larger ($>$ 10m ) telescopes, but it can also significantly improve the performances of telescopes already in operation. A relevant example is given by the multi-conjugated adaptive optics demonstrator (MAD) recently developed by ESO [[(see next section)]{}]{} that allows us to test AO capabilities for stellar photometry on the sky. In this context, as part of a Guaranteed Time Observations program, we obtained MAD near-infrared (IR) images of the dwarf irregular galaxy [UKS2323-326]{}[[ (UGCA438)]{}]{}. We chose this galaxy from a list of targets selected according to various criteria: favourable position on the sky with respect to the availability of stars to perform the AO correction; [[low Galactic latitude ($b=-70\fdg9$), to minimise the contamination by foreground Galactic stars]{}]{}; location slightly beyond the boundary of the LG, so as to maximise the [[physical]{}]{} area sampled within the $1\arcmin$ [[field-of-view]{}]{} (FoV) while still detecting stars at the tip of the red giant branch (TRGB) with an adequate S/N; existence of images of the same field in HST and/or ESO archives; and the presence of a relatively strong intermediate age component. Currently, AO imagers operate only at near-IR wavelengths, which are best suited to studying evolved stellar populations, in particular, cool stars on the asymptotic giant branch (AGB). This evolutionary stage of low and intermediate mass stars is difficult to model because of its sensitivity to uncertain input physics, like mass loss and convection. However, AGB stars provide a major contribution to the integrated light of galaxies with intermediate-age stellar populations [@renzbuzz1986], therefore, it is very important to derive information on the productivity of these stars. This can be done by analysing the stellar content of galaxies with a strong intermediate age component, which is the motivation for our near-IR [[study]{}]{} of LG galaxies [@held+2007; @gull+2007for; @gull+2007sagdig]. Ground-based optical photometry of [UKS2323-326]{} was first presented by [@lee+1999]. The colour-magnitude diagram (CMD) exhibits a well-defined RGB, and a number of AGB stars. From the [[TRGB]{}]{} magnitude and from the colour of these stars, [@lee+1999] derive the distance modulus and average metallicity of the galaxy as $(m-M)_0=26.59\pm0.12$ and [[\[Fe/H\]]{}]{}$=-1.98$. More recently, from photometry obtained with the WFPC2 on board of the HST, [@kara+2002] measured $I^{\text{TRGB}}=22.72\pm0.12$, from which they obtain $(m-M)_0=26.74\pm0.15$, corresponding to $2.23\pm 0.15$ Mpc. Thus, this galaxy is likely a member of the Sculptor Group. Since the HST photometry is more accurate and since the distance determination by [@kara+2002] is based on a more modern calibration, in this paper, we will adopt the [@kara+2002] value. This implies that the absolute magnitude of the galaxy is $M_V = -13.24 $. [[ Although [UKS2323-326]{} contains a young stellar component, there is no evidence of significant emission [@mill1996; @kais+2007], which suggests a very low rate of ongoing star formation. No mid-infrared emission from hot dust nor polycyclic aromatic hydrocarbon is detected at 8${\mu}m$ [@jack+2006] but it is embedded in a neutral hydrogen cloud that asymmetrically covers the whole galaxy [e.g., @buyl+2006]. The mass is $\sim 6 \times 10^6 M_\odot$, while for CO emission only an upper limit on the molecular gas mass of $1.4 \times 10^5 M_\odot$ is available [@buyl+2006 ans refs. therein]]{}]{}. The data {#s:reduc} ======== MAD observations ----------------- MAD is a project [@marc+2007] mainly developed by ESO to test the multi-conjugated adaptive optics (MCAO) capabilities on the sky in the framework of the design of the European Extremely Large Telescope (ELT). MAD was mounted on the UT-3 of the Very Large Telescope (VLT) to realise the first MCAO observation on the sky [@bouy+2008]. The instrument accommodates two wavefront sensors (WFS): a star-oriented multi-Shack–Hartmann and a layer-oriented [LO, @raga+2000; @viar+2005] multi-pyramid [@raga1996]. Both WFS use reference stars on a $2\arcmin$ technical FoV. MAD is complemented with the CAMCAO scientific IR camera, with a $2k\times 2k$ Hawaii[ii]{} IR detector that can be moved across the $2\arcmin$ corrected circular FoV. The pixel scale is $0\farcs028$ pixel$^{-1}$, yielding a $57\arcsec \times 57\arcsec$ square FoV on the detector. We took observations of [UKS2323-326]{} on Sept. 27, 2007 with the LO wavefront sensor option, with the aim of testing single pyramid AO observations in the bright-end regime. The reference star has $V\approx 11.5$ and is located at $\sim 24 \arcsec$ from the centre of the FoV. This is the very first pyramid WFS AO-assisted science observation [see @raga+1999 for a discussion of the advantages of this technique] on an 8m-class telescope. Results from other observations with full multi-pyramid MCAO capabilities will be presented elsewhere. Our data set consists of 21 $J$ frames and 15 $K_s$ frames centred on [UKS2323-326]{}, at $\alpha$(J2000)$= 23^h26^m27^s$, $\delta$(J2000)$=-32\degr23\arcmin16\arcsec$. The total integration time is 37 and 30 min in $J$ and $K_s$ band, respectively. Reduction and photometric calibration ------------------------------------- We calibrated our near-IR photometry by comparing stars in common with the 2MASS point-source catalogue [@stru+2006] and by using $J$ and $K_s$ archive images obtained with the Son of ISAAC (SOFI) camera mounted at the ESO New Technology Telescope (NTT) to define secondary photometric standards. We reduced both SOFI and MAD raw images following the standard procedure for IR data, as described by @gull+2007for. We paid careful attention to image alignment, allowing a correction for possible field rotation. We limited the area used in the final analysis to a $45\arcsec\times45\arcsec$ region because of stray light affecting one edge of the MAD images. [From the integration of the $R$-band surface brightness profile [@lee+1999] of the galaxy, we estimate that the luminosity fraction observed is $\sim 36$%.]{} The point spread function (PSF) of stellar objects on the combined MAD images is fairly uniform across the whole frame, with deviations of $\lesssim 10 \%$ of the full width at half maximum (FWHM). The mean FWHM measured on the $J$ an $K_s$ images is, respectively, $0\farcs15$ and $0\farcs11$. This is a good result, considering that [[the seeing was $0\farcs52$ and $0\farcs42$ in $J$ and $K_s$, respectively (estimated from the ESO DIMM Monitor measurements in the $V$ band)]{}]{} and that the diffraction limit for the VLT is $0\farcs04$ for the $J$ band and $0\farcs07$ for the $K_s$ band. The Strehl ratio measured on the $J$ and $K_s$ frames is 7.6% and 21.4%, respectively. The ellipticity of stellar images is small (9% and 10% in $J$ and $K_s$), with an r.m.s. variation $\lesssim 5\%$ over the whole FoV. We performed stellar photometry on MAD and SOFI images using DAOPHOT/ALLSTAR programs [@stet1987], adopting a Penny model for the PSF, with a quadratic dependence on the position on the frame. The astrometric and photometric calibration of the SOFI data was obtained using the USNO-A2.0 [@mone+1998] and the 2MASS [@stru+2006] databases. We then used the resulting catalogue as a reference for the astrometric and photometric calibration of MAD frames. Considering the uncertainty of our two-steps calibration, the final error on the zero-point of MAD photometry resulted $\simeq$ 0.15 mag. Finally, the near-IR data were complemented with optical WFPC2/HST data from [@holt+2006]. This allows us to test the spatial resolution of MAD images, as well as take advantage of a wide colour baseline to study the stellar population. Figure \[f:maps\] [[compares]{}]{} the 3 images from SOFI, MAD, and WFPC2 for the same region in [UKS2323-326]{}. The improvement in resolution between MAD and SOFI image is clearly apparent. This implies a significantly better photometry of faint objects and, in particular, the possibility of obtaining accurate photometry for faint stars that are embedded in the halo of brighter stars. ![image](fig1a.ps){width=".30\textwidth"} ![image](fig1b.ps){width=".30\textwidth"} ![image](fig1c.ps){width=".30\textwidth"} In order to evaluate the completeness and photometric errors of our catalogue, we performed an extensive set of 160 artificial star experiments using $\sim$1000 stars for each run. Input magnitudes were randomly generated to reproduce an uniform distribution over the colour and magnitude range of real stars in our image ($0<J-K_s<2$ and $15<K_s<22.5$). We found that our photometry is complete at the 50% level down to $K_S\simeq20.7$. At this magnitude level, the photometric error is $\sim0.1$ mag in the $K_s$ band, and a factor of 2 lower in the $J$ band. These results are illustrated in Fig. \[f:cmd\_mad\_sofi\]. The evolved stellar populations of [UKS2323-326]{} ================================================== The CMD of [UKS2323-326]{} obtained from MAD images is shown in Fig. \[f:cmd\_mad\_sofi\] and compared to that obtained from SOFI data in the same FoV. The MAD CMD is about 1 magnitude deeper; more importantly, the photometric accuracy is higher, leading to a much better definition of the sequences on the CMD. This is mostly due to the higher spatial resolution of MAD, which allows us to resolve stars that are blended on SOFI images. The red tail of bright stars in the MAD CMD, extending up to $J-K_S\simeq1.8$ and $K_s\simeq18.5$, is consistent with the locus of carbon-rich AGB stars [c.f. other recent near-IR studies, e.g., @gull+2007for; @gull+2007sagdig; @menzies+2008]. It is tempting to locate the TRGB at $K_s\simeq 20.5$, where a discontinuity is apparent in the stellar magnitude distribution. However, [[a formal measurement]{}]{} of the TRGB from the luminosity function cannot be derived because of the incompleteness of our photometry at these magnitudes. Since the distance modulus is known from optical observations, we can estimate the expected level of the TRGB: according to [@vale+2004] calibration, the absolute $K_s$ magnitude of the TRGB depends on metallicity, being $-5.82$, $-6.11$, and $-6.40$ for ${{\rm [Fe/H]}}=-2,-1.5$, and $-1$, respectively. Assuming a reddening ${\mbox{$E_{B\!-\!V}$}}=0.015$ [@schl+1998], we get $K_s^\text{TRGB}=20.93,20.64$, and $20.35$, as the metallicity increases. These three values are indicated with arrows in Fig. \[f:cmd\_mad\_sofi\]: for the metallicity determined by [@lee+1999] (${{\rm [Fe/H]}}\simeq -2$), the TRGB should be close to the limit of our photometry. The @lee+1999 determination is based on a relatively shallow CMD. Comparing the deeper $(V,I)$ HST CMD by [@holt+2006] to the fiducial lines of Galactic globular clusters by [@dacoarma1990], we obtain a mean metallicity ${{\rm [Fe/H]}}\simeq -1.7$ for the RGB stars in this galaxy. This has to be regarded as a lower limit, since the bulk of [UKS2323-326]{}stellar population is younger than Galactic globular clusters. As an example, [@savi+2000] estimated that for a $\sim5$ Gyr stellar population, the age correction to be applied to the metallicity obtained with our method is $+0.4$ dex. The location of the TRGB would then come close to the discontinuity of the star’s distribution mentioned above. In the following, we adopt ${{\rm [Fe/H]}}= -1.5$, which yields a TRGB magnitude of $K_s=20.65$; we verified that this metallicity is compatible with the mean $V-K_s$ colour of the RGB in [UKS2323-326]{}. Figure \[f:cmdvkk\] shows the $V-K_s$ [*vs*]{} $K_s$ CMD obtained by combining the HST and MAD photometry. This colour combination is particularly well suited to distinguish the different evolutionary sequences. [[ The contamination by foreground stars in our fields is negligible. In fact, using simulations of the Milky Way population performed with the TRILEGAL code [@gira+2005], we expect only three foreground stars in the magnitude and colour range of our CMD. ]{}]{} Guided by the optical CMD, on this figure we draw the lines bordering the areas occupied by blue supergiants, red supergiants, AGB, and RGB stars. The blue and red supergiants are core Helium burning stars with masses down to $\sim 5 M_\odot$, resulting from the star formation activity occurred over the last $\sim 100$ Myr. [[ The main-sequence progenitors of this young population are sampled in the optical CMD by [@holt+2006], in which the blue plume contains stars with ages from $\simeq 80$ to $\simeq 10$ Myr old. A quantitative interpretation of this component needs detailed simulations that take into account photometric errors and completeness of the HST data. Here we concentrate on the intermediate age component for which the near-IR CMD offers a better diagnostic than the optical diagram.]{}]{} The bright portion of the red sequence, extending from $K_s \simeq 20.1$ up to $K_s=18.5$ and with very red colours (up to $V-K_s=6.5$) hosts bright AGB, mostly carbon (C) stars, while below the TRGB (at $K_s=20.65$) we sample the oldest stars. The stars in the intermediate region ([*open squares*]{}, $20.65>K_s>20.1$) [[are probably AGB stars, but some of them could actually be high-metallicity RGB stars. The uncertainty stems from the dependence of the TRGB $K_S$-magnitude on the metallicity discussed above, and on these stars being located (on the optical CMD) just below the TRGB in the $I$ band.]{}]{} [[Our observations can be used to derive a rough estimate of the star formation that occurred in [UKS2323-326]{}at intermediate ages by considering the number of AGB stars brighter than the TRGB, which is proportional to the gas mass converted into stars between $\sim 0.1$ and a few Gyrs [e.g., @greg2002]. The [*specific production*]{} of bright AGB stars (i.e., $\delta n_{\rm AGB,b}$, the number of stars per unit mass of the parent stellar population that fall in this region of the CMD) depends on age and metallicity. We have determined $\delta n_{\rm{AGB,b}}$ as a function of age, for a sample of globular clusters in the Large Magellanic Cloud for which near-IR CMDs, ages, and total luminosities are known from the literature. From these, we found that it reaches a maximum ($\sim 2 \times 10^{-4} M_\odot^{-1}$) at ages $\sim 1$ Gyr, to drop significantly at older ages, down to $\sim 3 \times 10^{-5} M_\odot^{-1}$ at $\sim$ 3 Gyr (close to the limit of our calibration). The stellar distribution in our CMD is suggestive of an extended episode of star formation; by averaging our empirical calibration, we obtain a specific production of 0.15 or 0.08 stars per $10^3 M_\odot$, if this episode started 1.5 or 3 Gyr ago, respectively. Since we count 59 objects on the red sequence at magnitudes brighter than the TRGB, we bracket the stellar mass produced at intermediate ages in the range between 4 and 7.5 $\times 10^{5} M_\odot$. We note that these estimates are based on a straight Salpeter initial mass function (IMF). If a [@chab2005] IMF were assumed, the masses would be $\sim 0.65$ times smaller.]{}]{} Using the classification obtained from the ($V,K_s$) CMD, we construct the ($J,K_s$) diagram to see how the various sequences can be identified when only IR data are available (see Fig. \[f:cmdjkk\]). Although the sequences here are less well traced, the various evolutionary stages are relatively well separated. In particular, this CMD shows that all stars in the red tail are located in the position expected for C-stars, since they are compatible with the main locus of C-stars in nearby dwarf galaxies defined by [@tott+2000]. [[ These results confirm that near-IR CMDs are a very powerful tool to clearly detect C-stars, as indicated by our previous studies of LG galaxies [@gull+2007for; @gull+2007sagdig]. ]{}]{} To summarise our work, we obtained a complete and accurate census of the bright evolved stellar population (red supergiants and AGB stars) in [UKS2323-326]{}, a dwarf irregular galaxy at a distance of 2.23 Mpc. We have shown that with near-IR AO images at 8m class telescopes it is possible to investigate the SFH in galaxies well beyond the LG. Considering the technical limitation of the [*demonstrator*]{}, we believe that these results forecast very promising opportunities for this kind of studies with advanced AO at ELT. [^1]: Based on observations collected at the European Southern Observatory, Chile, as part of MAD Guaranteed Time Observations, on ESO archival observations (Programme 71.D-0560), and on NASA/ESA Hubble Space Telescope observations (Proposal ID 8192).
--- abstract: 'The subject of this work is analysis of hadronic decays of exotic meson $\pi_{1}$ in a fully relativistic formalism, and comparison with the nonrelativistic results. The relativistic spin wave functions of mesons and hybrids are constructed based on unitary representations of the Lorentz group. The radial wave functions are obtained from phenomenological considerations of the mass operator. We find that decay channels $\pi_{1}\rightarrow\pi b_{1}$ and $\pi_{1}\rightarrow\pi f_{1}$ are favored, in agreement with results obtained using other models, thus indicating some model independence of the $S+P$ selection rules. We will also report on effects of meson final state interactions in exotic channels.' author: - | Nikodem Popławski[^1]\ \ \ \ title: 'A relativistic description of hadronic decays of the meson $\pi_{1}$' --- Introduction ============ In a region around 2 GeV a new form of hadronic matter is expected to exist in which the gluonic degrees of freedom are excited. In mesons these can result in resonances with exotic $J^{PC}$ quantum numbers. The adiabatic potential calculations show $\pi_{1}$ ($1^{-+}$) as the lowest energy excited gluonic configuration [@1]. The present models of hybrid decays (for instance [@2; @3]) are nonrelativistic and therefore one should investigate corrections arising from fully relativistic treatment. The case of $\pi_{1}$ is of a special interest also because its evidence has been reported by the E852 collaboration and new experimental searches are planned for JLab and GSI. Relativistic spin wave function for mesons and hybrids ====================================================== For a system of non-interacting particles the spin wave function is constructed as an element of an irreducible representation of the Poincare group. We will assume $m_{u}=m_{d}=m$. In the rest frame of a quark-antiquark pair $$l^{\mu}_{q}=(E(m_{q},{\bf q}),{\bf q}),\,\,l^{\mu}_{\bar{q}}=(E(m_{\bar{q}},-{\bf q}),-{\bf q}),$$ the normalized spin-1 wave function ($J^{PC}=1^{--}$) is given by the Clebsch-Gordan coefficients and can be written in terms of Dirac spinors as $$\Psi^{\lambda_{q\bar{q}}}_{q\bar{q}}({\bf q},{\bf l}_{q\bar{q}}=0,\sigma_{q},\sigma_{\bar{q}})=\frac{1}{\sqrt{2}m_{q\bar{q}}}\bar{u}({\bf q},\sigma_{q})\Bigl[\gamma^{i}-\frac{2q^{i}}{m_{q\bar{q}}+2m}\Bigr]v(-{\bf q},\sigma_{\bar{q}})\epsilon^{i}(\lambda_{q\bar{q}}),$$ where $m_{q\bar{q}}$ is the invariant mass and $\epsilon^{i}(\lambda_{q\bar{q}})$ are polarization vectors corresponding to spin $1$ quantized along the z-axis. The wave function of a $q\bar{q}$ system moving with a total momentum ${\bf l}_{q\bar{q}}={\bf l}_{q}+{\bf l}_{\bar{q}}$ is given by $$\Psi^{\lambda_{q\bar{q}}}_{q\bar{q}}({\bf q},{\bf l}_{q\bar{q}},\lambda_{q},\lambda_{\bar{q}})=\sum_{\sigma_{q},\sigma_{\bar{q}}}\Psi^{\lambda_{q\bar{q}}}_{q\bar{q}}({\bf q},{\bf l}_{\bar{q}}=0,\sigma_{q},\sigma_{\bar{q}})D^{\ast(1/2)}_{\lambda_{q}\sigma_{q}}({\bf q},{\bf l}_{q\bar{q}})D^{(1/2)}_{\lambda_{\bar{q}}\sigma_{\bar{q}}}(-{\bf q},{\bf l}_{q\bar{q}}),$$ where the Wigner rotation matrix $$D^{(1/2)}_{\lambda\lambda'}({\bf q},{\bf P})=\Bigl[\frac{(E(m,{\bf q})+m)(E(M,{\bf P})+M)+{\bf P}\cdot{\bf q}+i{\bf \sigma}\cdot({\bf P}\times{\bf q})}{\sqrt{2(E(m,{\bf q})+m)(E(M,{\bf P})+M)(E(m,{\bf q})E(M,{\bf P})+{\bf P}\cdot{\bf q}+mM)}}\Bigr]_{\lambda\lambda'}$$ corresponds to a boost with ${\bf \beta}\gamma={\bf P}/M$. One can show $$\Psi^{\lambda_{q\bar{q}}}_{q\bar{q}}({\bf q},{\bf l}_{q\bar{q}},\lambda_{q},\lambda_{\bar{q}})=-\frac{1}{\sqrt{2}m_{q\bar{q}}}\bar{u}({\bf l}_{q},\lambda_{q})\Bigl[\gamma^{\mu}-\frac{l^{\mu}_{q}-l^{\mu}_{\bar{q}}}{m_{q\bar{q}}+2m}\Bigr]v({\bf l}_{\bar{q}},\lambda_{\bar{q}})\epsilon_{\mu}({\bf l}_{q\bar{q}},\lambda_{q\bar{q}}),$$ where $\epsilon^{\mu}({\bf l}_{q\bar{q}},\lambda_{q\bar{q}})$ are obtained from $(0,\epsilon^{i}(\lambda_{q\bar{q}}))$ through a boost with ${\bf \beta}\gamma={\bf l}_{q\bar{q}}/m_{q\bar{q}}$. Similarly the normalized wave function for the spin-0 quark-antiquark pair ($J^{PC}=0^{-+}$) is given by $$\Psi_{q\bar{q}}({\bf q},{\bf l}_{q\bar{q}},\lambda_{q},\lambda_{\bar{q}})=\Psi_{q\bar{q}}({\bf l}_{q},{\bf l}_{\bar{q}},\lambda_{q},\lambda_{\bar{q}})=\frac{1}{\sqrt{2}m_{q\bar{q}}}\bar{u}({\bf l}_{q},\lambda_{q})\gamma^{5}v({\bf l}_{\bar{q}},\lambda_{\bar{q}}).$$ By coupling (3) or (4) for ${\bf l}_{q\bar{q}}=0$ with one unit of the orbital angular momentum $L=1$ and then making a boost (2), one obtains respectively the spin wave function for a quark-antiquark pair with quantum numbers $J^{PC}=1^{+-}$ $$\Psi^{\lambda_{q\bar{q}}}_{q\bar{q}}({\bf l}_{q},{\bf l}_{\bar{q}},\lambda_{q},\lambda_{\bar{q}})=\frac{1}{\sqrt{2}m_{q\bar{q}}({\bf l}_{q},{\bf l}_{\bar{q}})}\bar{u}({\bf l}_{q},\lambda_{q})\gamma^{5}v({\bf l}_{\bar{q}},\lambda_{\bar{q}})Y_{1\lambda_{q\bar{q}}}(\bar{{\bf q}}),$$ or $0^{++}$, $1^{++}$ and $2^{++}$ $$\Psi^{\lambda_{q\bar{q}}}_{q\bar{q}}({\bf l}_{q},{\bf l}_{\bar{q}},\lambda_{q},\lambda_{\bar{q}})=-\sum_{\lambda,l}\frac{1}{\sqrt{2}m_{q\bar{q}}}\bar{u}({\bf l}_{q},\lambda_{q})\Bigl[\gamma^{\mu}-\frac{l^{\mu}_{q}-l^{\mu}_{\bar{q}}}{m_{q\bar{q}}+2m}\Bigr]v({\bf l}_{\bar{q}},\lambda_{\bar{q}})\epsilon_{\mu}({\bf l}_{q\bar{q}},\lambda)$$ $$\cdot Y_{1l}(\bar{{\bf q}})<1,\lambda;1,l|J,\lambda_{q\bar{q}}>,$$ with ${\bf q}=\Lambda({\bf l}_{q\bar{q}}\rightarrow0){\bf l}_{q}$. In order to construct meson spin wave functions for higher orbital angular momenta $L$ one need only to replace $Y_{1l}$ with $Y_{Ll}$. In the rest frame of the 3-body system corresponding to a $q\bar{q}$ pair with momentum $-{\bf Q}$ and transverse gluon with momentum ${\bf Q}$, the total spin wave function of the hybrid is obtained by coupling the $q\bar{q}$ spin-1 wave function (3) and the gluon wave function ($J^{PC}=1^{--}$) to a total spin $S=0,1,2$ and $J^{PC}=0^{++},1^{++},2^{++}$ states respectively, and then with one unit of the orbital angular momentum to the exotic state $1^{-+}$: $$\Psi^{\lambda_{ex}}_{q\bar{q}g(S)}(\lambda_{q},\lambda_{\bar{q}},\lambda_{g})=\sum_{\lambda_{q\bar{q}},\sigma=\pm1,M,l}\Psi^{\lambda_{q\bar{q}}}_{q\bar{q}}({\bf q},{\bf l}_{q\bar{q}}=-{\bf Q},\lambda_{q},\lambda_{\bar{q}})<1,\lambda_{q\bar{q}};1,\sigma|S,M>$$ $$\cdot D^{(1)}_{\sigma\lambda_{g}}(\bar{{\bf Q}})Y_{1l}(\bar{{\bf Q}})<S,M;1,l|1,\lambda_{ex}>.$$ The spin-1 Wigner rotation matrix $D^{(1)}$ relates the gluon helicity $\sigma$ to its spin $\lambda_{g}$ quantized along the z-axis. The corresponding normalized wave functions are then given by: $$\Psi^{\lambda_{ex}}_{q\bar{q}g(S=0)}=\sqrt{\frac{3}{8\pi}}\sum_{\lambda_{q\bar{q}}}\Psi^{\lambda_{q\bar{q}}}_{q\bar{q}}({\bf q},-{\bf Q},\lambda_{q},\lambda_{\bar{q}})[{\bf \epsilon}^{\ast}(\lambda_{q\bar{q}})\cdot{\bf \epsilon}_{c}^{\ast}({\bf Q},\lambda_{g})][\bar{{\bf Q}}\cdot{\bf \epsilon}(\lambda_{ex})],$$ $$\Psi^{\lambda_{ex}}_{q\bar{q}g(S=1)}=\sqrt{\frac{3}{8\pi}}\sum_{\lambda_{q\bar{q}}}\Psi^{\lambda_{q\bar{q}}}_{q\bar{q}}({\bf q},-{\bf Q},\lambda_{q},\lambda_{\bar{q}})[{\bf \epsilon}^{\ast}(\lambda_{q\bar{q}})\times{\bf \epsilon}_{c}^{\ast}({\bf Q},\lambda_{g})]\cdot[\bar{{\bf Q}}\times{\bf \epsilon}(\lambda_{ex})],$$ $$\Psi^{\lambda_{ex}}_{q\bar{q}g(S=2)}=\sqrt{\frac{27}{104\pi}}\sum_{\lambda_{q\bar{q}}}\Psi^{\lambda_{q\bar{q}}}_{q\bar{q}}(-{\bf Q},\lambda_{q},\lambda_{\bar{q}})\,\bar{{\bf Q}}\cdot[{\bf \epsilon}^{\ast}(\lambda_{q\bar{q}})\otimes{\bf \epsilon}_{c}^{\ast}({\bf Q},\lambda_{g})]\cdot{\bf \epsilon}(\lambda_{ex}),$$ where $\bar{{\bf Q}}={\bf Q}/|{\bf Q}|,\,\,(A\otimes B)_{ij}=2A_{(i}B_{j)}-\frac{2}{3}\delta_{ij}({\bf A}\cdot{\bf B})$ and $\epsilon^{i}_{c}({\bf Q},\lambda_{g})=\epsilon^{j}(\lambda_{g})(\delta^{ij}-\bar{Q}^{i}\bar{Q}^{j})$. Meson and hybrid states ======================= The $\pi\,(I=1)$ and $\eta\,(I=0)$ states ($J^{PC}=0^{-+}$) are constructed in terms of annihilation and creation operators: $$|M({\bf P},I,I_{3},\lambda)>\,\,=\sum_{all\,\,\lambda,c,f}\int\frac{d^{3}{\bf p}_{q}}{(2\pi)^{3}2E(m,{\bf p}_{q})}\frac{d^{3}{\bf p}_{\bar{q}}}{(2\pi)^{3}2E(m,{\bf p}_{\bar{q}})}2(E(m,{\bf p}_{q})+E(m,{\bf p}_{\bar{q}}))$$ $$\cdot(2\pi)^{3}\delta^{3}({\bf p}_{q}+{\bf p}_{\bar{q}}-{\bf P})\frac{\delta_{c_{q}c_{\bar{q}}}}{\sqrt{3}}\frac{F(I,I_{3})_{f_{q}f_{\bar{q}}}}{\sqrt{2}}\Psi_{q\bar{q}}({\bf p}_{q},{\bf p}_{\bar{q}},\lambda_{q},\lambda_{\bar{q}})\delta_{\lambda 0}\frac{1}{N(P)}$$ $$\cdot\psi_{L}(\frac{m_{q\bar{q}}({\bf p}_{q},{\bf p}_{\bar{q}})}{\mu})\,b^{\dag}_{{\bf p}_{q}\lambda_{q}f_{q}c_{q}}d^{\dag}_{{\bf p}_{\bar{q}}\lambda_{\bar{q}}f_{\bar{q}}c_{\bar{q}}}|0>.$$ In the above $\Psi_{q\bar{q}}$ is the spin-0 wave function (4), $I$ denotes isospin and $I_{3}$ is its third component, $c$ is color and $f$ is flavor. $\psi_{L}$ represents the orbital wave function resulting from the interaction between quarks that leads to a bound state (meson). Such a function depends (due to covariance) only on the invariant mass $m_{q\bar{q}}({\bf p}_{q},{\bf p}_{\bar{q}})$ [@4]. Normalization constants are denoted by $N$ (with $P=|{\bf P}|$) and $\mu$’s are free parameters, being scalar functions of meson quantum numbers. Finally, $F(I,I_{3})$ is 2x2 isospin matrix ($f=1$ for $u$ and $f=2$ for $d$) $$F(0,0)=1,\,\,\,F(1,I_{3})=\sigma^{i}\epsilon^{i}(I_{3}).$$ The flavor structure of the $\eta$ state (as well as of all meson states with isospin $0$) was chosen as a linear combination $\frac{1}{\sqrt{2}}(|u\bar{u}>+\,|d\bar{d}>)$. Strictly speaking, those states are rather linear combinations $\,a|u\bar{u}>+\,b|d\bar{d}>+\,c|s\bar{s}>$. But $|s\bar{s}>$ does not contribute to the amplitude of the decay of $\pi_{1}$ and therefore may be neglected in calculations, provided this amplitude is multiplied by a factor $\sqrt{1-|c|^{2}}$. Similarly the $\rho\,(I=1)$ and $\omega,\,\phi\,(I=0)$ states ($J^{PC}=1^{--}$) are given by (8), but instead of $\Psi_{q\bar{q}}^{\lambda}\delta_{\lambda 0}$ one must use (3). The $b_{1}\,(I=1)$ and $h_{1}\,(I=0)$ states ($J^{PC}=1^{+-}$) contain the wave function (5) with ${\bf q}=\Lambda({\bf P}\rightarrow0){\bf p}_{q}$. Finally, the $a\,(I=1)$ and $f\,(I=0)$ states ($J^{PC}=0,1,2^{++}$) correspond to (6). The hybrid state in its rest frame is given by $$|\pi_{1}(I_{3},\lambda_{ex})>\,\,=\sum_{all\,\,\lambda,c,f}\frac{1}{N_{ex}}\int\frac{d^{3}{\bf p}_{q}}{(2\pi)^{3}2E(m,{\bf p}_{q})}\frac{d^{3}{\bf p}_{\bar{q}}}{(2\pi)^{3}2E(m,{\bf p}_{\bar{q}})}\frac{d^{3}{\bf Q}}{(2\pi)^{3}2E(m_{g},{\bf Q})}$$ $$\cdot(2\pi)^{3}2(E(m,{\bf p}_{q})+E(m,{\bf p}_{\bar{q}})+E(m_{g},{\bf Q}))\delta^{3}({\bf p}_{q}+{\bf p}_{\bar{q}}+{\bf Q})\frac{\lambda^{c_{g}}_{c_{q}c_{\bar{q}}}}{2}\frac{\sigma^{i}_{f_{q}f_{\bar{q}}}\epsilon^{i}(I_{3})}{\sqrt{2}}$$ $$\cdot\Psi^{\lambda_{ex}}_{q\bar{q}g}({\bf p}_{q},{\bf p}_{\bar{q}},\lambda_{q},\lambda_{\bar{q}},\lambda_{g})\psi_{L}'(\frac{m_{q\bar{q}}({\bf p}_{q},{\bf p}_{\bar{q}})}{\mu_{ex}},\frac{m_{q\bar{q}g}({\bf p}_{q},{\bf p}_{\bar{q}},{\bf Q})}{\mu_{ex'}})\,b^{\dag}_{{\bf p}_{q}\lambda_{q}f_{q}c_{q}}d^{\dag}_{{\bf p}_{\bar{q}}\lambda_{\bar{q}}f_{\bar{q}}c_{\bar{q}}}a^{\dag}_{{\bf Q}\lambda_{g}c_g}|0>,$$ where the spin wave function $\Psi_{q\bar{q}g}$ was given in (7) for $S=0,1,2$ and the orbital wave function $\psi_{L}'$ depends only on $m_{q\bar{q}}$ and the invariant mass of the three-body system. Here $m_{g}$ denotes the effective mass of the gluon coming from its interaction with virtual partcles, and $\lambda^{c_{g}}_{c_{q}c_{\bar{q}}}$ are the Gell-Mann matrices. Constants $N$ are fixed by normalization ($m_{M}$ is mass of meson) $$<{\bf P},\lambda,I_{3}|{\bf P}',\lambda',I'_{3}>\,\,=(2\pi)^{3}2E(m_{M},{\bf P})\delta^{3}({\bf P}-{\bf P}')\delta_{\lambda\lambda'}\delta_{I_{3}I_{3}'}.$$ Decays of $\pi_{1}$ and nonrelativistic limit ============================================= We will assume that a transverse gluon in $\pi_{1}$ creates a quark-antiquark pair and therefore the hybrid decays into two mesons. The Hamiltonian of this process in the Coulomb gauge is given by $$H=\sum_{all\,\,c,f}\int d^{3}{\bf x}\,\bar{\psi}_{c_{1}f_{1}}({\bf x})(g{\bf \gamma}\cdot{\bf A}^{c_{g}}({\bf x}))\psi_{c_{2}f_{2}}({\bf x})\delta_{f_{1}f_{2}}\frac{1}{2}\lambda^{c_{g}}_{c_{1}c_{2}},$$ where $$\psi_{cf}({\bf x})=\sum_{\lambda}\int\frac{d^{3}{\bf k}}{(2\pi)^{3}2E(m,{\bf k})}[u({\bf k},\lambda)b_{{\bf k}\lambda cf}+v(-{\bf k},\lambda)d^{\dag}_{-{\bf k}\lambda cf}]e^{i{\bf k}\cdot{\bf x}}$$ and $${\bf A}^{c_{g}}({\bf x})=\sum_{\lambda}\int\frac{d^{3}{\bf k}}{(2\pi)^{3}2E(m_{g},{\bf k})}[{\bf \epsilon}_{c}({\bf k},\lambda)a^{c_{g}}_{{\bf k}\lambda}+{\bf \epsilon}_{c}^{\ast}(-{\bf k},\lambda)a^{\dag c_{g}}_{-{\bf k}\lambda}]e^{i{\bf k}\cdot{\bf x}}.$$ Here $g$ is the strong coupling constant. The matrix element $<M_{1}({\bf P}_{1})|<M_{2}({\bf P}_{2})|H|\pi_{1}>$ ($M$ stands for meson) will be a sum of two terms because pairs $b,b^{\dag}$ and $d,d^{\dag}$ appear twice and one can show they are equal. If $\mu_{\eta}=\mu_{\pi}$ then $A_{\pi\eta}=0$ and the hybrid will not decay into $\pi$ and $\eta$. The same occurs for $\rho+\omega$. Neither can it decay into two pions because of a relative minus sign from isospin that makes both terms cancel out. However, $\mu_{b_{1}}=\mu_{\pi}$ does not imply $A_{\pi b_{1}}=0$ because the orbital wave functions of these mesons are different. Since $\pi$ and $\eta$ have the same quantum numbers (except isospin) and therefore $\mu_{\pi}$ and $\mu_{\eta}$ should be almost equal (not exactly because $SU(3)_{f}$ is only an approximate symmetry and there is a contribution of $s\bar{s}$ in $\eta$), out of two channels $\pi\eta$, $\pi b_{1}$ the latter will be favored. However, free parameters $\mu$ need not to be close to each other for two mesons with different radial quantum numbers, making corresponding channels significant. A nonrelativistic limit is obtained by ignoring Wigner rotation and using nonrelativistic phase space. For $m$ large compared to $\mu$’s and $P$ this limit should be approached by relativistic results. In this case for decays of $\pi_{1}$ into $\pi$ and the S-meson $S_{q\bar{q}g}\not=1$, whereas for those into $\pi$ and the P-meson $S_{q\bar{q}g}\not=0$. The difference in amplitudes coming from the spin wave function can be clearly seen, assuming $m_{\eta}=m_{\rho},\,\mu_{\eta}=\mu_{\rho}$ and $\,m_{b_{1}}=m_{f_{1}}=m_{f_{2}},\,\mu_{b_{1}}=\mu_{f_{1}}=\mu_{f_{2}}$ (the second condition is satisfied with a good approximation by masses of particles): $$\Gamma_{\pi\rho}=\frac{1}{2}\Gamma_{\pi\eta},\,\,\,\Gamma_{\pi f_{1}}=\frac{1}{8}\Gamma_{\pi b_{1}},$$ where $A_{\pi\eta}$, $A_{\pi b_{1}}$ are taken for $S=0$ and $A_{\pi\rho}$, $A_{\pi f_{1,2}}$ for $S=1$. Relations $A_{\pi\eta}$ vers. $A_{\pi b_{1}}$ and $A_{\pi\rho}$ vers. $A_{\pi f_{1,2}}$ depend on the orbital angular momentum wave functions $\psi_{L}$ and $\psi'_{L}$. If $\mu_{\rho}=\mu_{\pi}$ then in a nonrelativistic limit $\pi_{1}$ would not decay into $\pi+\rho$. Therefore the width rate for this process is expected to be much smaller than that of $\pi b_{1}$. Orbital wave function ===================== The orbital angular momentum wave function for a meson or a hybrid depends on the potential between quark and antiquark or for a $q\bar{q}g$ system. An explicit form of such a potential is not known exactly and such a function must be modeled. Because of the Lorentz invariance it may depend on momenta only through the invariant mass of particles. Moreover, it must tend to zero for large momenta fast enough to make the amplitude convergent. The most natural choice is the exponential function $$\psi_{L}(m_{q\bar{q}}({\bf p}_{q},{\bf p}_{\bar{q}})/\mu)=e^{-m^{2}_{q\bar{q}}({\bf p}_{q},{\bf p}_{\bar{q}})/8\mu^{2}}$$ for a meson, and $$\psi'_{L}(m_{q\bar{q}}({\bf p}_{q},{\bf p}_{\bar{q}})/\mu_{ex},m_{q\bar{q}g}({\bf p}_{q},{\bf p}_{\bar{q}},{\bf Q})/\mu'_{ex})=e^{-m^{2}_{q\bar{q}}({\bf p}_{q},{\bf p}_{\bar{q}})/8\mu^{2}_{ex}}e^{-m^{2}_{q\bar{q}g}({\bf p}_{q},{\bf p}_{\bar{q}},{\bf Q})/8\mu'^{2}_{ex}}$$ for a hybrid. The integrals for the decay amplitude are not elementary and must be computed numerically. In a nonrelativistic limit, however, they can be expressed in terms of the error function. The free parameters of the presented model are $m$, $m_{g}$, $\mu$’s and $g$. The pion form factor constants $f_{\pi}$ and $F_{\pi}$ (whose behaviour is experimentally known) defined by $$<0|A^{\mu,i}({\bf 0})|\pi^{k}({\bf p})>=f_{\pi}p^{\mu}\delta_{ik},\,\,\,<\pi^{i}({\bf p}')|V^{\mu,j}({\bf 0})|\pi^{k}({\bf p})>=F_{\pi}(p^{\mu}+p'^{\mu})i\epsilon_{ijk},$$ allow us to fit $m$ and $\mu_{\pi}$ (with the $\pi$ state given by (8)). The axial and the vector currents are defined by $$A^{\mu,i}({\bf 0})=\bar{\psi}_{cf}({\bf 0})\gamma^{\mu}\gamma_{5}\frac{\sigma^{i}}{2}\psi_{cf}({\bf 0}),\,\,\,V^{\mu,j}({\bf 0})=\bar{\psi}_{cf}({\bf 0})\gamma^{\mu}\frac{\sigma^{j}}{2}\psi_{cf}({\bf 0}),$$ with $\psi_{cf}({\bf x})$ given in (10). By virtue of the Lorentz invariance $f_{\pi}$ is a constant, whereas $F_{\pi}$ is a function of $Q^{2}=-({\bf p}-{\bf p}')^{2}$. Impossibility of finding the generators $H$ and $M_{0i}$ of the Poincare group in presence of interaction together with normalization of states violate the Lorentz covariance between spatial and time components but do not break a rotational symmetry. The resulting form factors will depend on the frame of reference. Results and summary =================== Taking $m_{\pi}=(3*140+770)/4=612MeV$ (in normalization) and $m=306MeV$ gives $\mu_{\pi}=220MeV$. Assuming also $g^{2}=10$, $m_{g}=500MeV$ [@5], $m_{ex}=1.6GeV$ and equality of all parameters $\mu$ leads to the following values: $\Gamma_{\pi b_{1}}=$150MeV, $\Gamma_{\pi f_{1}}=$20MeV, $\Gamma_{\pi\rho}=$3MeV. In the nonrelativistic limit one obtains respectively: 230, 31 and 0 MeV. Two important conclusions come from this work. Firstly, numerical results show that relativistic corrections arising from a Wigner rotation are significant. Therefore, models with no Wigner rotation are either nonrelativistic or inconsistent. These corrections decrease in general (for reasonable values of $m$) the width rates for $\pi_{1}$ or make them different from zero if they vanished for the NR case. Secondly, the $\pi_{1}$ prefers to decay into two mesons, one of which has no orbital angular momentum and the other has $L=1$ ($S+P$ selection rule). This is in agreement with other models, for instance [@6]. Calculations involving decay rates of $\pi_{1}$ into strange mesons and final state interactions ($b_{1}\rightarrow\pi+\omega,\,\,\omega+\pi\rightarrow\rho$) are in preparation. [7]{} K.J.Juge, J.Kuti, C.J.Morningstar, Nucl.Phys.Proc.Suppl. [**63**]{}, 326 (1998) N.Isgur, R.Kokoski, J.Paton, Phys.Rev.Lett. [**54**]{}, 869 (1985) F.E.Close, P.R.Page, Nucl.Phys. B[**443**]{}, 233 (1995) B.Bakamjian, L.H.Thomas, Phys.Rev. [**92**]{}, 1300 (1953) A.P.Szczepaniak, E.S.Swanson, arXiv:hep-ph/0308268 (2003) P.R.Page, E.S.Swanson, A.P.Szczepaniak, Phys.Rev. D[**59**]{}, 034016 (1999) [^1]: ©2004 American Institute of Physics
--- abstract: 'We consider the development of anisotropic flow in an expanding system of particles undergoing very few rescatterings, using a kinetic-theoretical description with a nonlinear collision term. We derive the scaling behaviors of the harmonic coefficients $v_n$ with the initial-state eccentricities and the mean number of rescatterings, and argue that hexagonal flow $v_6$ should follow a nontrivial behavior, different from that of the lower harmonics. Our findings should be observable in experimental data for small systems.' author: - Nicolas Borghini - Steffen Feld - Nina Kersting title: Scaling behavior of anisotropic flow harmonics in the far from equilibrium regime --- Introduction {#s:intro} ============ A highlight of the results from the ongoing experimental programs with heavy nuclei at the CERN LHC or the Brookhaven RHIC consists of the measurements of various azimuthal correlations between outgoing particles. In particular, the measured values of Fourier coefficients $v_n$ that quantify the anisotropies of the transverse emission pattern are interpreted as footprints of a strongly collective behavior [@Heinz:2013th], hinting at the creation of a medium close to local thermodynamic equilibrium. According to the most widely accepted theoretical picture, the final-state anisotropies in momentum space reflect asymmetries of the spatial geometry of the “initial state”—i.e. of the distribution of entropy density in the transverse plane after the nuclei have passed through each other [@Ollitrault:1992bk; @Alver:2010gr]—, characterized for instance by “eccentricities” [@Teaney:2010vd; @Gardim:2011xv] $$\label{eccentricities} \epsilon_n{{\rm e}}^{{{\rm i}}n\Phi_n} \equiv -\frac{{\left\langle r^{n\,}{{\rm e}}^{{{\rm i}}n\theta} \right\rangle}}{{\left\langle r^n \right\rangle}}$$ for $n\geq 2$, and by generalizations thereof.[^1] In this definition, $r$ and $\theta$ are centered polar coordinates in the transverse plane, while the angular brackets denote an average over the entropy density. To be more specific, let us briefly quote a number of model findings, without any attempt at exhaustiveness (see also Ref. [@Liu:2018hjh] for a compilation of experimental results from the LHC): - To a good approximation, the (integrated) elliptic flow $v_2$ and triangular flow $v_3$ scale linearly with the corresponding eccentricities [@Ollitrault:1992bk; @Alver:2010gr; @Teaney:2010vd; @Gardim:2011xv; @Niemi:2012aj; @Plumari:2015cfa]: $$\label{v2,v3_vs_eps2,eps3} v_2 \simeq {\cal K}_{2,2}\epsilon_2\quad,\quad v_3 \simeq {\cal K}_{3,3}\epsilon_3.$$ In collisions with a large impact parameter, i.e. a larger $\epsilon_2$, a cubic deviation to this behavior for $v_2$ has been reported [@Noronha-Hostler:2015dbi]. - The quadrangular flow $v_4$ receives two kinds of contributions: a linear scaling with $\epsilon_4$ and a quadratic dependence on $\epsilon_2$ [@Borghini:2005kd; @Gardim:2011xv; @Teaney:2012ke; @Niemi:2012aj; @Lang:2013oba]: $$\label{v4_vs_eps2,4} v_4 \simeq {\cal K}_{4,4}\epsilon_4 + {\cal K}_{4,22}\epsilon_2^2.$$ Similarly, the pentagonal flow $v_5$ depends linearly on $\epsilon_5$ and nonlinearly on the second- and third-harmonic eccentricities [@Gardim:2011xv; @Teaney:2012ke; @Lang:2013oba]: $$\label{v5_vs_eps2,3,5} v_5 \simeq {\cal K}_{5,5}\epsilon_5 + {\cal K}_{5,23}\epsilon_2\epsilon_3.$$ In both cases, the linear term is only visible in ultra-central collisions [@Plumari:2015cfa], while the nonlinear contribution dominates at larger impact parameters. - Starting with hexagonal flow $v_6$, there appear more than one nonlinear terms at the “leading order” that mostly contributes in noncentral collisions [@Bravina:2013ora; @Qian:2016fpi; @Giacalone:2018wpp]: $$\label{v6_vs_eps2,3,6} v_6 \simeq {\cal K}_{6,6}\epsilon_6 + {\cal K}_{6,33}\epsilon_3^2 + {\cal K}_{6,24}\epsilon_2\epsilon_4 + {\cal K}_{6,222}\epsilon_2^3.$$ Note already that there are both quadratic and cubic contributions to $v_6$, which will be of relevance for one of the findings of this paper. The leading candidate theory for describing the evolution of the medium created in collisions of heavy nuclei at ultrarelativistic energies is nowadays relativistic fluid dynamics, and accordingly most of the studies quoted above were performed within that framework. The only exception is Ref. [@Plumari:2015cfa], which relies on a kinetic transport approach, however pushed into a regime where it “mimics” dissipative fluid dynamics. In the corresponding language, one finds that the linear response coefficients ${\cal K}_{n,n}$ depend on the transport properties—mostly, the shear and bulk viscosity—of the expanding medium. This also holds for the nonlinear response coefficients ${\cal K}_{4,22}$, ${\cal K}_{5,23}$, ${\cal K}_{6,222}$, ${\cal K}_{6,33}$…, yet it was argued in Ref. [@Teaney:2012ke] that they are less damped by viscous effects than the linear coefficients. The collective-behavior picture underlying the fluid-dynamical interpretation of the anisotropic flow data has however been challenged by the observations in the past few years of similar signals in collisions of “smaller systems” like proton-lead, deuteron-gold or even proton-proton, in events with a relatively large number of particles in the final state (see Ref. [@Nagle:2018nvi] for a recent review). Accordingly, there has been a revival of models with “weak final-state collectivity” aiming at investigating generic behaviors of the anisotropic flow coefficients $v_n$ in a regime where the outgoing particles undergo in average very few rescatterings [@Heiselberg:1998es; @Borghini:2010hy; @He:2015hfa; @Romatschke:2018wgi; @Kurkela:2018ygx]. The present paper, which will remain at a semi-qualitative yet general level, is a further step in that direction, identifying a scaling behavior of the higher harmonics, in particular of the hexagonal flow $v_6$, which to our knowledge has not been noted before.[^2] Anisotropic flow far from equilibrium {#s:main-section} ===================================== Since our goal is to model a situation in which the final-state particles have only rescattered very little, a natural framework is to treat them as particles all along the system evolution—they never form a continuous medium—and to resort to a kinetic theory framework, as we now detail. Without loss of generality for our argumentation, we consider a single particle type and introduce its on-shell phase space distribution $f(t,{\bf x},{\bf p})$, where three-vectors are denoted in boldface. The evolution of this distribution will generically be described by a kinetic equation of the form [@DeGroot:1980dk] $$\label{Boltzmann-eq} p^\mu\partial_\mu f(t,{\bf x},{\bf p}) = -{\cal C}[f],$$ with ${\cal C}[f]$ the collision term modeling the effect of rescatterings.[^3] As initial condition for the evolution, we consider the following phase space distribution at some time $t_0$, in which the dependences on the space and momentum variables factorize: $$\begin{aligned} f(t_0,{\bf x},{\bf p}) \sim F({\bf p})_{}G(r)\bigg[1 &+ \tilde{\epsilon}_2\bigg(\!\frac{r}{R}\!\bigg)^{\!\!2}\cos[2(\theta\!-\!\Phi_2)] \cr & + \tilde{\epsilon}_3\bigg(\!\frac{r}{R}\!\bigg)^{\!\!3}\cos[3(\theta\!-\!\Phi_3)] + \cdots\bigg], \cr \label{f(t0,x,p)}\end{aligned}$$ where we again use polar coordinates $(r,\theta)$ for the projection of ${\bf x}$ onto the transverse plane, and discard the dependence on the longitudinal coordinate, which is irrelevant in the following. As hinted at by the notations, $F({\bf p})$ is a position-independent momentum distribution, $G(r)$ depends only on the radial coordinate, while $R$ is a length scale ensuring that $\tilde{\epsilon}_2$ and $\tilde{\epsilon}_3$ are dimensionless. As noted by Teaney and Yan [@Teaney:2010vd], the successive powers of $r$ in the square brackets should actually be regulated by some cutoff function to ensure that the distribution $f$ remains positive, yet we did not denote that regulator since it can remain unspecified for our purposes. Obviously, computing the eccentricities  with distribution  (instead of the corresponding entropy density, yet at the present stage this is only a matter of convention) gives $\epsilon_n\propto\tilde{\epsilon}_n$ for every $n\geq 2$. Letting now the initial distribution  evolve, we first consider the collisionless case, ${\cal C}[f]=0$. The corresponding solutions of the equation of motion  are free-streaming solutions $$\label{free-streaming_sol} f^{(0)}(t,{\bf x},{\bf p}) = f^{(0)}(t_0,{\bf x}\!-\!{\bf v}(t\!-\!t_0),{\bf p})$$ with ${\bf v}$ the velocity corresponding to momentum ${\bf p}$. The resulting anisotropic flow coefficients $v_n$,[^4] whose calculation involve integrations over the whole position space as well as over the momentum azimuthal angle $\phi_{\bf p}$, are easily shown to be entirely determined by the initial transverse anisotropies of $F({\bf p})$, and in particular independent of the eccentricities $\epsilon_n$. Thus, if there is no anisotropic flow initially, as we shall from now on assume, there is none in the final state if the particles do not rescatter, as has long been known. To turn on a small amount of rescatterings, we apply the idea whose various incarnations in the heavy-ion physics community went through the successive appellations “low density limit” [@Heiselberg:1998es; @Kolb:2000fha], “far from equilibrium regime” [@Borghini:2010hy] and more recently “eremitic expansion” [@Romatschke:2018wgi] or “one-hit dynamics” [@Kurkela:2018ygx], and write a solution of the kinetic equation  with collision term as $$\label{eremitic-Ansatz} f(t,{\bf x},{\bf p}) = f^{(0)}(t,{\bf x},{\bf p}) + f^{(1)}(t,{\bf x},{\bf p})$$ where $f^{(0)}$ is the free-streaming solution  and $f^{(1)}$ a small correction term. Inserting this ansatz in Eq.  yields at once $$\label{Boltzmann-eq_with_eremitic-Ansatz} p^\mu\partial_\mu f^{(1)}(t,{\bf x},{\bf p}) = -{\cal C}\big[f^{(0)}\!+\!f^{(1)}\big].$$ Multiplied by $\cos(n\phi_{\bf p})$ and integrated over ${\bf x}$ and $\phi_{\bf p}$, the term on the left hand side of this equation yields (up to a normalization factor) the negative of the time derivative $\partial_t v_n$ [@Borghini:2010hy; @NB-NK_inprep]. Integrating over time will then give the final $v_n$. That is, specific integrals of the collision term ${\cal C}\big[f^{(0)}\!+\!f^{(1)}\big]$ yield the $n$-th anisotropic flow harmonic $v_n$. Since we are interested in the development of the higher harmonics, and in particular in the nonlinear contributions, a natural choice for the collision term is Boltzmann’s collision integral for elastic two-to-two scatterings, which we symbolically write in the form $$\begin{aligned} \label{collision-integral_2->2} {\cal C}[f({\bf 1})] = \int_{{\bf p_2},{\bf p_3},{\bf p_4}}\!\big[& f({\bf 3})f({\bf 4})_{}w({\bf 3}\!+\!{\bf 4}\to{\bf 1}\!+\!{\bf 2}) \cr &- f({\bf 1})f({\bf 2})_{}w({\bf 1}\!+\!{\bf 2}\to{\bf 3}\!+\!{\bf 4})\big],\qquad\end{aligned}$$ where the shorthand notation $f(\bm{j})$ stands for $f(t,{\bf x},{\bf p}_j)$ while the terms $w(\bm{i}+\bm{j}\to \bm{k}+\bm{l})$ involve transition probabilities and the necessary $\delta$-distributions implementing energy-momentum conservation. Note that in contrast to Ref. [@Borghini:2010hy] we need not assume that the initial geometry is invariant under the ${\bf x}\to -{\bf x}$ transformation, nor that the involved interactions are parity non-violating. Given a model for the interaction, $w$ will be proportional to some typical cross section $\sigma$. In turn, if one computes the total number of rescatterings taking place over the whole system evolution, it will also be approximately proportional to $\sigma$, as will be the average number of rescatterings per particle $\bar{N}_{\rm resc.}$. The latter constitutes the dimensionless parameter that quantifies the smallness of $f^{(1)}$ relative to $f^{(0)}$. When substituting $f$ by $f^{(0)}+f^{(1)}$ in this collision integral, we make use of the fact that $f^{(1)}$ is assumed to be a small correction and approximate [@Heiselberg:1998es; @Borghini:2010hy; @NB-NK_inprep] $$\label{approximate_collision-term} {\cal C}\big[f^{(0)}\!+\!f^{(1)}\big] \simeq {\cal C}\big[f^{(0)}\big].$$ That is, the free-streaming solution  fully determines the collision term and thereby the flow coefficients. Performing the necessary calculations requires specific models for the as yet unspecified functions $F({\bf p})$ and $G(r)$ in the initial-state distribution  and for the interaction. General scalings can however already be predicted irrespective of any specific choice, which we now list. - The multiplication of the isotropic term in one of the factors $f^{(0)}(\bm{i})$ in the integrand of the collision integral  with the term in $\bar{\epsilon}_n\cos[n(\theta-\Phi_n)]$ in the associated $f^{(0)}(\bm{j})$ yields a contribution to $v_n$ proportional to $\sigma_{}\epsilon_n$, i.e. approximately proportional to $\bar{N}_{\rm resc.}\epsilon_n$. With the values of the eccentricities relevant for heavy ion collisions, this is the dominant contribution to $v_2$ and $v_3$, resulting in linear scalings of the form $$\label{v2,v3_vs_eps2,eps3,sigma} v_2 \sim \bar{N}_{\rm resc.}\kappa_{2,2\,}\epsilon_2\quad,\quad v_3 \sim \bar{N}_{\rm resc.}\kappa_{3,3\,}\epsilon_3.$$ For $n\geq 4$, other contributions to $v_n$ are likely to be as important, which we now discuss. - Besides the linear term in $\sigma_{}\epsilon_4$, another contribution to $v_4$ is generated by multiplying together the terms in $\epsilon_2\cos[2(\theta-\Phi_2)]$ in both distributions $f^{(0)}(\bm{i})$, $f^{(0)}(\bm{j})$ of one of the products $f^{(0)}(\bm{i})f^{(0)}(\bm{j})$ in the integrand of Eq. . Thus, one obtains $$\label{v4_vs_eps2,sigma} v_4 \sim \bar{N}_{\rm resc.}\kappa_{4,4\,}\epsilon_4 + \bar{N}_{\rm resc.}\kappa_{4,22\,}\epsilon_2^2,$$ with a term quadratic in $\epsilon_2$ yet linear in the mean number of rescatterings. Similarly, one finds $$\label{v5_vs_eps2,3,sigma} v_5 \sim \bar{N}_{\rm resc.}\kappa_{5,5\,}\epsilon_5 + \bar{N}_{\rm resc.}\kappa_{5,23\,}\epsilon_2\epsilon_3,$$ again with a contribution nonlinear in the initial-state eccentricities and linear in $\bar{N}_{\rm resc.}$. - Coming to $v_6$, one quickly sees that the approximation  will yield a contribution in $\bar{N}_{\rm resc.}\epsilon_3^2$ and one in $\bar{N}_{\rm resc.}\epsilon_2\epsilon_4$. However it cannot yield the term in $\epsilon_2^3$ observed in fluid-dynamical studies. To recover the latter, one must consider products $f^{(0)}(\bm{i})f^{(1)}(\bm{j})$ in the integrand of the collision term, since $f^{(1)}$ contains a term in $\sigma_{}\epsilon_2^2\cos(4\theta)$—which is reflected in the quadrangular flow . This will indeed yield a term in $\epsilon_2^3$, but the latter is also proportional to $\sigma^2$: $$\begin{aligned} \label{v6_vs_eps2,3,sigma} v_6 \sim &\ \bar{N}_{\rm resc.}\big(\kappa_{6,6\,}\epsilon_6 + \kappa_{6,33\,}\epsilon_3^2 + \kappa_{6,24\,}\epsilon_2\epsilon_4\big) \cr & \ + \bar{N}_{\rm resc.}^2\kappa_{6,222\,}\epsilon_2^3. \end{aligned}$$ At very low $\bar{N}_{\rm resc.}$ the linear and quadratic contributions will dominate, while the cubic term in the ellipticity $\epsilon_2$ will only become meaningful for a larger number of rescatterings. - Similarly, one can easily convince oneself that the setup consisting of the initial distribution , evolved with the Boltzmann equation  with collision term  does not generate any contribution to the harmonics $v_{n\geq 7}$ involving only $\epsilon_2$ and $\epsilon_3$ at linear order in $\bar{N}_{\rm resc.}$: the $\epsilon_2^2\epsilon_3$ contribution to $v_7$ is quadratic in $\bar{N}_{\rm resc.}$; $v_8$ will receive a term in $\epsilon_2\epsilon_3^2$ at order $\bar{N}_{\rm resc.}^2$ and a contribution $\epsilon_2^4$ at order $\bar{N}_{\rm resc.}^3$, and so on. - Eventually, the same reasoning shows that the subleading contribution in $\epsilon_2^3$ to elliptic flow $v_2$, which is observed in fluid dynamical simulations, can also be recovered within our model, at order $\bar{N}_{\rm resc.}^2$. More generally, the model predicts a contribution in $\bar{N}_{\rm resc.}^2\epsilon_n^3$ to $v_n$ for any $n$. Summarizing our findings, we find that our model of a system of self-diffusing particles with an initially asymmetric transverse geometry is able to generate anisotropic flow coefficients $v_n$ with the same scaling dependence on the initial-state eccentricities as within a fluid-dynamical description, as seen from comparing Eqs. – with Eqs. –. This behavior was already known in the literature [@Borghini:2010hy; @Kurkela:2018ygx; @Borghini:2011qc]. What to our knowledge was never mentioned before regards the scaling of the generated flow harmonics with the rescattering cross section, or equivalently with the mean number of rescatterings per particle.[^5] Thus, the “leading contributions”, i.e. those stemming from the largest eccentricities $\epsilon_2$ and $\epsilon_3$, to the successive Fourier coefficients $v_n$ scale with different powers of $\bar{N}_{\rm resc.}$. And, perhaps more interesting, starting with hexagonal flow $v_6$ there can be two or more such leading contributions to $v_n$, which necessarily scale differently with $\bar{N}_{\rm resc.}$: $$\label{v6_vs_eps2,3,Nscat} v_6 \sim {\cal O}(\bar{N}_{\rm resc.})_{}\epsilon_3^2 + {\cal O}(\bar{N}_{\rm resc.}^2)_{}\epsilon_2^3,$$ where the terms in $\epsilon_2\epsilon_4$ or $\epsilon_6$ are assumed to be smaller. That is, the development of the contribution to $v_6$ from the ellipticity $\epsilon_2$ necessitates more rescatterings than that of the triangularity $\epsilon_3$. Similarly, one easily finds $v_8 \sim {\cal O}(\bar{N}_{\rm resc.}^2)_{}\epsilon_2\epsilon_3^2 + {\cal O}(\bar{N}_{\rm resc.}^3)_{}\epsilon_2^4$, again assuming that the contributions to $v_8$ involving eccentricities $\epsilon_p$ with $p\geq 4$ are small. Discussion {#s:discussion} ========== In this last section, we address two issues: first, are our results on the scalings with the average number of rescatterings $\bar{N}_{\rm resc.}$, in particular that of Eq. , robust against changes of the setup which we considered? And second, is there a possibility to evidence these behaviors in experimental data? If anisotropic flow is not present initially, but generated by rescatterings, then the corresponding harmonics will depend on the initial-state eccentricities $\epsilon_n$ and on $\bar{N}_{\rm resc.}$. Regarding the linear response of $v_n$ to $\epsilon_n$, we cannot think of a plausible scenario in which it would not already be generated at linear order in $\bar{N}_{\rm resc.}$, i.e. $v_n = {\cal O}(\bar{N}_{\rm resc.})_{}\epsilon_n$, generalizing Eq. . Less straightforward are the nonlinear response behaviors, which we now discuss at length. The emergence of scalings $v_{n+p} \propto \epsilon_n\epsilon_p$ at linear order in $\bar{N}_{\rm resc.}$, as in Eqs. –, is also straightforward in a description in which the collision term ${\cal C}[f]$ is at least quadratic in the single-particle distribution $f$, which is a natural feature in a picture in which the momentum anisotropies are generated by rescatterings of at least two partners. The less trivial scaling behavior is that of Eq. , in particular the ${\cal O}(\bar{N}_{\rm resc.}^2)$ dependence of the term in $\epsilon_2^3$. To investigate whether it is an artifact of our model or more general, we note that only two types of modifications are possible as long as one remains in a kinetic-theoretical framework with particle scatterings: changes in the initial-state distribution $f(t_0,{\bf x},{\bf p})$, which entirely determines the free-streaming solution $f^{(0)}(t,{\bf x},{\bf p})$, or of the collision term ${\cal C}[f]$. In both cases, we want to see how a term in $\epsilon_2^3$ might arise at first order in $\bar{N}_{\rm resc.}$ or somewhat equivalently in the interaction cross section $\sigma$, thereby invalidating Eq. . Changing the initial-state distribution so as to spoil Eq.  is mathematically feasible, by assuming that $F({\bf p})$ contains a term in $\epsilon_2$. This would however mean that the initial momentum distribution already knows about the global geometry of the collision zone, which is problematic from the physics point of view. In turn, if the isotropic term $G(r)$ contains a term in $\epsilon_2$, then the latter will multiply both terms of Eq. , which is also not what we wish. Accordingly, the only viable modifications to be considered are changes of the collision term ${\cal C}[f]$. Sticking to the general structure of a collision integral involving $f$,[^6] one quickly sees that the generalization of Boltzmann’s ansatz  necessary to obtain a term $\epsilon_2^3$ at first order in $\bar{N}_{\rm resc.}$ is to include a contribution of (at least) cubic order in $f$ in the integrand. Two kinds of physical causes justify such contributions. On the one hand, one may include rescatterings with at least three particles in the initial state, in particular three-to-two scatterings, as can be found e.g. in Eq. (12) of Ref. [@Xu:2004mz]. Here, one should note that including two-to-three scatterings only would not help. In addition, three-to-two rescatterings will in fact generate the desired term in $\epsilon_2^3$ at first order in the corresponding cross section $\sigma_{3\to 2}$: whether the latter yields the leading contribution to $\bar{N}_{\rm resc.}$, so that the term is indeed of order ${\cal O}(\bar{N}_{\rm resc.})_{}\epsilon_2^3$, or else $\bar{N}_{\rm resc.}$ is rather dominated by two-to-two rescatterings becomes a partly model-dependent issue. On the other hand, even restricting oneself to elastic two-to-two rescatterings, one may still consider the generalization of the integrand accounting for Bose–Einstein enhancement or Pauli blocking, i.e. with factors of the form $f(\bm{i})f(\bm{j})[1\pm f(\bm{k})][1\pm f(\bm{l})]$ in lieu of $f(\bm{i})f(\bm{j})$. The inclusion of three-to-two scatterings and/or quantum-mechanical phase-space occupancy effects seems at first face to be relevant only if the initial state is that of a dense system. Naturally, when the latter expands, it becomes more dilute, and the terms beyond quadratic order in $f$ in the collision integral become less important. Nevertheless, it is not clear to us whether a system created in high-energy collisions could be in a regime such that the emitted particles rescatter only very little, while at the same time being initially dense and interacting enough to lead to a breakdown of Eq. . “Small systems” are the most likely candidates for such a departure, provided the initial density is big. In any case, the relative importance of more-than-quadratic terms and their influence on the scaling behavior could be tested in numerical simulations with transport codes in which they can be switched on or off at will, like $2\leftrightarrow 3$ scatterings in BAMPS [@Xu:2004mz] or quantum effects in other codes [@Zhang:2017esm]. Let us now discuss where in experimental data the scaling behaviors – could possibly be at play and measurable. Surprising though it may seem, let us first deal with larger systems, in which fluid dynamics is routinely applied to describe the evolution. On the one hand, the single-collision regime might be applicable to particles in given regions of phase space, e.g. at high transverse momentum [@Romatschke:2018wgi], or to specific particle types, like bottomonia—for which the picture is rather a negative one: a collision means destruction. On the other hand, Eqs. – may also be relevant in phenomenological analyses of the bulk of particles. More accurately, these behaviors play a role in the pre-hydrodynamized stage, which in modern hybrid descriptions is often modeled by a transport cascade [@Liu:2015nwa]. Indeed, this short kinetic period, which only involves rather few rescatterings, will transform “pre-early transport eccentricities”, taken from a model for initial conditions, into some early anisotropic flow, which becomes part of the initial condition for the fluid-dynamical stage of the hybrid description. Following Eq. , the early generated $v_6$ will suffer from a deficit in second-order eccentricity $\epsilon_2$, which will be propagated by the subsequent evolution until the final state. This could then affect attempts at evidencing the scaling  and at interpreting it within a purely fluid-dynamical framework, since the coefficients ${\cal K}_{6,33}$ and ${\cal K}_{6,222}$ will contain a pre-hydrodynamization component, whose relative size possibly varies across centralities. This possibility is a further incentive to investigate the scaling behaviors – in transport models. Eventually, the natural place where Eqs. – are to be looked for is in small systems, in which the applicability of fluid dynamics is most questionable. The biggest issue is of course that the anisotropic flow coefficients in such systems are small. We believe that the more trivial scaling behaviors – should be rather “easily” observable. Note in particular that the unknown mean number of rescatterings cancels in the ratio of two different harmonics $v_2$–$v_5$, so that one can separate the influences of eccentricities and $\bar{N}_{\rm resc.}$, where one can expect that the latter should scale like the cubic root of the charged particle multiplicity $({{\rm d}}N^{\rm ch.\!}/{{\rm d}}\eta)^{1/3}$. In turn, we are aware that in small systems $v_6$ will be at the border of what is measurable with reasonable uncertainties, so that whether measurements allowing to test Eq.  are feasible is not warranted. 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--- abstract: 'Polar codes are a class of linear block codes that provably achieves channel capacity. They have been selected as a coding scheme for the control channel of enhanced mobile broadband (eMBB) scenario for $5^{\text{th}}$ generation wireless communication networks (5G) and are being considered for additional use scenarios. As a result, fast decoding techniques for polar codes are essential. Previous works targeting improved throughput for successive-cancellation (SC) decoding of polar codes are semi-parallel implementations that exploit special maximum-likelihood (ML) nodes. In this work, we present a new fast simplified SC (Fast-SSC) decoder architecture. Compared to a baseline Fast-SSC decoder, our solution is able to reduce the memory requirements. We achieve this through a more efficient memory utilization, which also enables to execute multiple operations in a single clock cycle. Finally, we propose new special node merging techniques that improve the throughput further, and detail a new Fast-SSC-based decoder architecture to support merged operations. The proposed decoder reduces the operation sequence requirement by up to $39\%$, which enables to reduce the number of time steps to decode a codeword by $35\%$. ASIC implementation results with 65 nm TSMC technology show that the proposed decoder has a throughput improvement of up to $31\%$ compared to previous Fast-SSC decoder architectures.' author: - 'Furkan Ercan, Thibaud Tonnellier, Carlo Condo, and Warren J. Gross[^1]' title: Operation Merging for Hardware Implementations of Fast Polar Decoders --- Polar codes, wireless communications, successive cancellation decoding, throughput, 5G Introduction ============ Polar codes, introduced by Ar[i]{}kan [@arikan09], are a class of linear block codes that provably achieves channel capacity. They have been selected as a coding scheme for enhanced mobile broadband (eMBB) scenario under $5^{\text{th}}$ generation wireless communication standards (5G) [@38.212; @hashemi2017asilomar], and are also being considered for ultra reliable low-latency communication (URLLC) and massive machine-type communication (mMTC) in 5G networks [@sharma2017polar; @sybis2016channel]. Successive cancellation (SC) decoding of polar codes is the original decoding scheme proposed in [@arikan09], and can be represented as a binary tree search. However, this approach suffers from long decoding latency due to its sequential nature, and mediocre error-correction performance at moderate to short code lengths. In order to reduce the latency of SC decoding, SSC [@SSC2011] and Fast-SSC [@sarkis14] decoders proposed efficient decoding techniques for particular information and frozen bit patterns, called special nodes, without affecting the error-correction performance. Compared to conventional SC decoder implementations [@TPSC13], Fast-SSC decoding is shown to improve the throughput by an order of magnitude. Further identification and use of special nodes were carried out in both SC-based [@giard16; @giardJSPS; @fastssc-sips17] and SC-List based [@fastSSCL-TCAS-I; @fastSSCL-TSP] decoding techniques. In [@sarkis14], a number of parallel processing elements ($P_e$) allows to achieve high throughput. However, there are two problems regarding the use of parallel processing elements in Fast-SSC decoding. The first issue is that the memory utilization factor of the decoder decreases with increasing $P_e$. Secondly, for nodes with sizes smaller than $P_e$, a single operation is performed where the architecture is able to support multiple operations. In this work, we present a new Fast-SSC decoder architecture that substantially increases the memory utilization. Unlike the previous Fast-SSC-based architectures, the new memory utilization is regardless of the parallelization factor. The new configuration allows an opportunity to perform multiple operations at a single step. By observing the distribution of frozen bits, we identify two categories of operation merging scenarios. The first category includes merging branch-type operations, where a leaf node estimation is not included. The second category includes merging of special nodes at the bottom of the SC tree. A subset of them is selected for a new SC-based decoder implementation to improve the decoder throughput. Results show that, our proposed decoder reduces the number of operations by up to $ 39\%$, which enables to reduce the number of time steps to decode a codeword by $35\%$. Results in 65 nm TSMC CMOS show that the proposed decoder has a throughput improvement of up to $31\%$ compared to previous Fast-SSC decoder architectures, while increasing the memory utilization to $99.6\%$. This paper is an extension of our previous works in [@fastssc-sips17; @hashemi2017jetcas], where memory reduction and operation merging schemes were first introduced, followed by an FGPA implementation. In this paper, we generalize operation merging scenarios and implement a novel decoder architecture with significantly improved throughput. The rest of this paper is organized as follows: In Section \[sec:background\], preliminaries for polar code encoding and decoding are reviewed. A new memory design to improve utilization for Fast-SSC decoding is described in Section \[sec:proposed-memory\]. Section \[sec:proposed-merging\] describes operation merging scenarios for Fast-SSC decoding. In Section \[sec:architecture\], a new Fast-SSC decoder architecture is described. ASIC synthesis results for the new decoder are presented and compared against state-of-the-art decoder implementations in Section \[sec:results\], and finally concluding remarks are addressed in Section \[sec:conclusion\]. Preliminaries {#sec:background} ============= Polar Codes {#sec:polarcodes} ----------- Polar codes are able to achieve channel capacity through channel polarization, that splits $N$ channel utilizations into $K$ reliable ones, through which information bits are sent, and $N-K$ unreliable ones, used for frozen bits. A polar code, represented as $PC(N,K)$, is a linear block code of length $N = 2^n$ and rate $R = K/N$. Encoding of a polar code can be represented by a matrix multiplication: $$\label{eq:enc} \boldsymbol{x_0^{N-1}} = \boldsymbol{u_0^{N-1}}G^{\otimes n}\text{,}$$ where $\boldsymbol{u_0^{N-1}} = \{u_0,u_1,\ldots,u_{N-1}\}$ is the input vector, $\boldsymbol{x_0^{N-1}} = \{x_0,x_1,\ldots,x_{N-1}\}$ is the encoded vector, and the generator matrix $G^{\otimes n}$ is the $n$-th Kronecker product of the polar code matrix $G = \left[\begin{smallmatrix} 1&0\\ 1&1 \end{smallmatrix} \right]$. A polar code of length $N$ is composed of two concatenated polar codes of length $N/2$; Fig. \[fig:polarencode\] depicts the encoding process for $PC(8,5)$. Successive Cancellation Decoding {#sec:scdecoding} -------------------------------- SC decoding [@arikan09] can be interpreted as a binary tree search and is explored depth-first, with priority to the left branch. An example to SC decoder tree for $PC(16,10)$ is shown in Fig. \[fig:polartree\]. The root of the tree consists of the information obtained by the channel, which is expressed in terms of log-likelihood ratio (LLR) for this work. At each stage of the tree, the LLR values $\boldsymbol{\alpha} = \{\alpha_{0},\alpha_{1},..\alpha_{2^S-1}\}$ are passed from a parent node to its child nodes, and hard decision estimates $\boldsymbol{\beta}=\{\beta_{0},\beta_{1},..\beta_{2^S-1}\}$ are passed from a child node to its parent node. The soft information passed to left child $\boldsymbol{\alpha^l}$ and right child $\boldsymbol{\alpha^r}$ are approximated as $$\label{eqn:alphaleft} {\alpha}^l_i = \text{~sgn}(\alpha_{i})\text{~sgn}(\alpha_{i+2^{S-1}}) \text{~min}(|\alpha_{i}|,|\alpha_{i+2^{S-1}}|)$$ $$\label{eqn:alpharight} {\alpha}^r_i = \alpha_{i+2^{S-1}} + (1-2\beta^{l}_{i})\alpha_{i}$$ where $0 \leq i < 2^{S-1}$ for stage $S$, and the root node is at stage $S=\log_2(N)$. The hard decision estimates, $\boldsymbol{\beta}$ for stage $S$, are calculated via the left and right messages from child nodes, $\boldsymbol{\beta^{l}}$ and $\boldsymbol{\beta^{r}}$, as $$\label{eqn:beta} \beta_i=\left\{ \begin{array}{@{}ll@{}} \beta^{l}_{i} \oplus \beta^{r}_{i}, & \text{if}~ i \leq 2^{S-1} \\ \beta^{r}_{i-2^{S-1}}, & \text{otherwise.} \end{array}\right.$$ where $\boldsymbol{\oplus}$ denotes bitwise XOR operation, and where $0 \leq i < 2^S$. At the leaf nodes, $\beta$ values are hard decisions computed by observing the sign bit of their soft information, as $$\label{eqn:beta-leaf} \beta_{i}^{leaf}=\left\{ \begin{array}{@{}ll@{}} 0, & \text{if}~ \alpha_i^{leaf} \geq 0 \text{ } \text{or } i \in \Phi; \\ 1, & \text{otherwise.} \end{array}\right.$$ where $i$ represents the node index and $\Phi$ denotes the set of frozen indices. Fast-SSC Decoding {#sec:fastssc-old} ----------------- Simplified successive cancellation (SSC) decoding [@SSC2011] showed that the SC tree can be pruned, avoiding the descent in case of nodes whose leaf nodes are either all information bits (Rate-1) or all frozen bits (Rate-0). The Fast-SSC decoding algorithm [@sarkis14] evolves SSC by identifying special patterns and presenting efficient decoding techniques for such nodes. ### Algorithm {#subsec:fastssc_alg} If we denote an information bit with I and a frozen bit with F, in addition to Rate-0 and Rate-1 nodes, special nodes can occur in three other different forms in a polar code: repetition (Rep) (FF$\cdot\cdot\cdot$FI), single parity check (SPC) (FI$\cdot\cdot\cdot$II) and a pattern (FFII) which is referred as ML node in [@sarkis14]. In this work, we follow the same naming conventions for simplicity. A repetition node contains a single information bit; all other nodes are frozen. An information node encoded with frozen bits contain the same information bit in all the nodes. The hard decision is made by adding the LLR values together and extracting the sign bit of the result: $$\label{eqn:rep} {\beta}_i=\left\{ \begin{array}{@{}ll@{}} 0, & \text{if}~\sum_{i=0}^{N_v-1} \alpha_i \geq 0 \\ 1, & \text{otherwise.} \end{array}\right.$$ In SPC nodes, due to the nature of the polar code construction, the frozen bit represents the parity of all the information bits of the node. Consequently, the parity check for all hard decisions (HDs) (Eq. \[eqn:SPC-HD\]) of an SPC node must be zero (Eq. \[eqn:SPC-parity\]). $$\label{eqn:SPC-HD} \text{HD}_i=\left\{ \begin{array}{@{}ll@{}} 0, & \text{if}~ \alpha_i \geq 0 \\ 1, & \text{otherwise.} \end{array}\right.$$ $$\label{eqn:SPC-parity} parity = \mathlarger{\mathlarger{\mathlarger{\oplus}}}_{i=0}^{N_S-1} \text{HD}_i$$ If the parity constraint is satisfied, the decoding of the SPC node is assumed successful. If the parity is not satisfied, it means that there is at least one error. To satisfy the parity check constraint, the bit with the least reliable LLR is found (Eq. \[eqn:SPC-argmin\]) and flipped (Eq. \[eqn:SPC-final\]): $$\label{eqn:SPC-argmin} j = \arg\min(|\alpha_i|); ~0 \leq i < N_S,$$ $$\label{eqn:SPC-final} {\beta}_i=\left\{ \begin{array}{@{}ll@{}} \text{HD}_i \oplus parity, & \text{when}~ i = j \text{,} \\ \text{HD}_i, & \text{otherwise.} \end{array}\right.$$ Merging schemes for these special nodes were also proposed in the Fast-SSC decoder to improve the throughput further. A complete list of operations are detailed in Table \[tab:fast-instr\], including their merged operations. ### Decoder Architecture {#sec:fastssc_arch} The Fast-SSC architecture described in [@sarkis14] contains separate memory units for channel LLR values, intrinsic LLR values $\alpha$, partial sums $\beta$, decoding instructions, and the final codeword. Words from channel, $\alpha$ and $\beta$ memory units are routed to an ALU unit, where the identified operations listed in Table \[tab:fast-instr\] are performed. Left operation (F), right operation (G) and combine operation (C) are adopted from the SC decoding of [@arikan09]. The notations 0, 1 and R represent child nodes with Rate-0, Rate-1 and Rate-R (0 $<$ R $<$ 1). The operations with P- notation represent the operations performed without explicitly visiting the right child node. Routing of the memory and configuration of the datapath are controlled based on the instruction list that is compiled offline. The datapath for the Fast-SSC decoder is depicted in Fig. \[fig:fastssc\]. Execution of a single operation is called a step, and each step may take one or more clock cycles, based on the tree stage and the number of physical processing units dedicated to perform the operation. If $P_e$ is too small, operations to decode a codeword takes too many clock cycles which results in reduced throughput. The number of cycles to decode a single frame decreases with increasing $P_e$, which helps increase the throughput. For example, for $PC(1024,512)$, number of clock cycles to decode a codeword is $571$ when $P_e = 16$, and is $217$ when $P_e = 256$. On the other hand, with increasing $P_e$, the idle time of the computational resources increase, decreasing the resource utilization. Fast-SSC decoder is a special decoder that is based on semi-parallel SC decoder family [@sarkis14]. According to [@SPSC13], the utilization rate ($\theta_{SP}$) of a semi-parallel decoder is given by $$\theta_{SP} =\frac{\log_2N}{4\times P_e + \log_2\frac{N}{4P_e}},$$ which leads to $\theta_{SP} = 13.8$ when $P_e = 16$ and to $\theta_{SP} = 0.9$ when $P_e = 256$, respectively. Fig. \[fig:pe\] plots the number of cycles to decode a codeword ($T_\text{latency}$) and resource utilization ($\theta_{SP}$) as a function of the number of processing elements. It can be seen that the utilization rate decreases significantly with increasing $P_e$, whereas after $P_e = 64$, the latency improvement is marginal. Thus, while keeping a reasonable resource utilization and maintaining low decoding latency, $P_e = 2^6$ is a reasonable choice for N = 1024. coordinates [(16 ,13.9) (32 ,7.5) (64 ,3.8) (128,1.9 ) (256, 0.97) ]{}; \[s\] coordinates [(16 ,571 ) (32 ,356 ) (64 ,268 ) (128,229 ) (256,217 ) ]{}; \[f\] The SC tree for a $PC(1024,512)$ with nodes from Table \[tab:fast-instr\] is presented in Fig. \[fig:1ktree\]. The polar code construction is obtained from the 5G standard [@38.212]. The Fast-SSC from [@sarkis14] executes a total number of 212 steps in 268 clock cycles to decode a single frame of Fig. \[fig:1ktree\] when $P_e = 2^6$. ### Memory {#sec:BG-mem} In the SC tree there are $\log_2(N)+1$ stages, with the highest stage $\log_2(N)$ corresponding to the root node, and the lowest stage $\log_2(1) = 0$ to the leaf nodes. For each stage of the SC tree, both $\alpha$ and $\beta$ are stored in designated memory modules. The $\alpha$ memory is comprised of two banks, each of which is $P_e$ LLRs wide. The partial sum $\beta$ memory is composed of two memory units, each of which has two banks, and each bank is $P_e$ bits wide. A full word is defined as the content from an index of a memory unit. At each clock cycle, a full word can be read from each memory. The F and G modules can process $2 \times P_e$ $\alpha$ and $P_e \beta$ values at once to generate $P_e$ $\alpha$ outputs, thus a half word can be written back to the $\alpha$ memory. Combine (C) unit takes a half word from each $\beta$ memory bank, to produce a full word that should be written to either of the banks. Consequently, a half word can be read from each $\beta$ memories, and a full word can be written back to one $\beta$ memory. In terms of both $\alpha$ and $\beta$ memories, at least one word is reserved for each stage of the tree. Fig. \[fig:memarch-old\] represents the memory architecture for both $\alpha$ and $\beta$ memories for $PC(1024,K)$ code. The root node (S=10) is stored explicitly in the channel memory, because it has a different quantization scheme than the internal LLRs. Due to special node decoding techniques mentioned in Section \[sec:fastssc-old\], the lower limit stage is S=2, thus no additional memory is required after stage S=2. In general, given that each word holds $2 \times P_e$ elements for $\alpha$ and $\beta$ memories, the number of words for each memory module is: $$\sum_{S=2}^{\log_2(N-1)} \left\lceil \frac{2^{S-1}}{P_e}\right\rceil.$$ Improving Memory Utilization {#sec:proposed-memory} ============================ As it was previously mentioned in Section \[sec:fastssc-old\], one or more words are reserved in both $\alpha$ and $\beta$ memories per decoding stage in the tree. The word size is decided by the number of processing elements $P_e$, that acts as a parallelization factor. $P_e$ can also be interpreted as a threshold on the SC tree (dashed line in Fig. \[fig:1ktree\]), where the node size $N_v=P_e$. The stages above this parallelization threshold are collectively called high-stage, and each stage in high-stage fully utilizes the dedicated memory words for that level; below the threshold (low-stage), only a portion of the memory is used. In fact, the total number of variables used at low-stage can be expressed as $$\sum_{\text{S}=2}^{\log_2 P_e} 2^\text{S} < 2 \times P_e$$ which can fit into a single memory word. We improve the memory utilization by storing all the variables relative to the low-stage into a single memory word in both $\alpha$ and $\beta$ memory units. Fig. \[fig:memarch-new\] describes the new memory configuration for the low-stage, using a polar code $PC(1024,K)$ and $P_e=64$ as an example. With the proposed solution, the number of $2\times Pe$-sized words for each memory becomes $$\sum_{S=\log_2(P_e)}^{\log_2(N-1)} \left\lceil \frac{2^{S-1}}{P_e}\right\rceil$$ Considering a polar code $PC(1024,K)$ and $P_e=64$, the memory utilization increases from $66\%$ to $99.6\%$. With these modifications to the memory structure, when a node at the low-stage is to be processed, the entire last word is fetched from the memory. Since the content of the memory is relative to multiple stages, executing multiple operations per clock cycle becomes possible. Operation Merging {#sec:proposed-merging} ================= In this work, we define an operation as a *leaf operation* if it involves a leaf node estimation, and as a *branch operation* if it does not include any bit estimations. According to this classification, hard decision (\[eqn:beta-leaf\]), Rate-0, Rate-1, Rep (\[eqn:rep\]) and SPC (\[eqn:SPC-parity\])-(\[eqn:SPC-final\]) calculations are leaf node operations, whereas F (\[eqn:alphaleft\]), G (\[eqn:alpharight\]) and C (\[eqn:beta\]) are branch operations for SC decoding. Note that proposed merged operations do not affect the error-correction performance. Merging Branch Operations {#subsec:proposed-branchmerging} ------------------------- If the memory configuration in Fig. \[fig:memarch-new\] is used within the Fast-SSC decoder architecture, all $\alpha$ and $\beta$ variables below the parallelization threshold becomes available for processing at the same time. This enables the Fast-SSC processor to perform multiple operations at a single cycle. In other words, operations below the parallelization threshold are available for merging. However, the impact of operation merging on system critical path should be minimized and thus the original critical path should be considered as an upper delay bound while performing multiple low-stage operations in a single cycle. It was observed that the critical path of the original Fast-SSC architecture is determined by the SPC node. Compared to SPC-related operations, branch operations introduce a significantly lower delay; this provides the opportunity to merge them without increasing the system critical path. Consequently, we exploit the branch operation merging opportunities at low-stage. Based on the data dependencies while decoding, the following merging scenarios are possible for operations of the same kind: - Multiple F operations: The traversal of the SC tree has a left branch priority, which enables to perform multiple F operations consecutively. - Multiple G0 operations: Tree traversal allows consecutive G operations only when the left node is a Rate-0 node and needs not to be traversed. - Multiple C/C0 operations: A sequence of Combine operations is possible when the operation ascends from a right branch, i.e. $\beta$ values of the left children are already available. This constraint does not apply to the last C operation in the sequence. These four different merging branch operations of the same kind are visualized in Fig. \[fig:merged\_f\_g\_c\]. ![Required conditions to perform multiple F, G0, C and C0 operations on a polar code decoder tree.[]{data-label="fig:merged_f_g_c"}](merged_f_g_c.pdf){width="\columnwidth"} The combination of different branch operations at low-stage is also feasible. It was observed that a G operation is often followed by an F operation, which can be merged together to form a new operation called G-F. Similar observations were made for F-G0, C-G and C0-G. A complete list of merged branch operations and their associated potential step reduction is presented in Table \[tab:potentials-branch\] for $PC(1024,512)$ [@38.212]. According to Table \[tab:potentials-branch\], the amount of time step reduction increases with $P_e$. It can also be observed that G-F merging scenario returns the most amount of reduction. Note that, the merging scenarios in Table \[tab:potentials-branch\] are computed independently, without considering any conflicts between the merging scenarios. A set of guidelines for how to merge operations are detailed in Section \[subsec:guidelines\]. Merging Special Nodes {#subsec:proposed-leafmerging} --------------------- As mentioned in Section \[sec:fastssc-old\], the Fast-SSC architecture from [@sarkis14] uses merging of special nodes in order to improve the throughput. For example, the SPC and Sign operators in between G and C modules in Fig. \[fig:fastssc\] enable the datapath to execute P-RSPC, P-0SPC, P-R1, P-01, G0 and C0 operations from Table \[tab:fast-instr\]. On the other hand, a separate module for RepSPC node is instantiated to avoid the critical path. In this Section, based on the observations made from the polar code tree in Fig. \[fig:polartree\], we identify all possible special node merging scenarios. It should be noted that some of the special node merging scenarios in this Section fall within the generalized nodes from [@ardakani-sc], where however no hardware implementation were proposed. Table \[tab:potentials-leaf\] describes a number of possible leaf node merging scenarios along with their node sizes, breakdown of operations (translations), and amount of time step reduction with respect to $P_e$. It is important that the $P_e$ threshold must be larger than $16$ to support the new node sizes in Table \[tab:potentials-leaf\]. If $P_e \leq 16$, described merging operations fall above the parallelization threshold in the polar code tree, and they cannot be merged. Rep-Rate1 and Rate0-RepSPC nodes were previously identified in [@giardJSPS]; we include them in this work in order to compare them with newly identified nodes in terms of time step reduction. Similar to Table \[tab:potentials-branch\], the time step reduction calculations are done independently from each other. According to Table \[tab:potentials-leaf\], merged special node operation Rep-RepSPC returns the most potential time step reduction, followed by F-Rep. Note that the instances of F-Rep operation occurs within Rep-RepSPC nodes, which can be observed in their translations. The selection of merging scenarios for special nodes should not only be performed with respect to their independent contribution on time step reduction, but also with respect to their impact on the maximum operating frequency. Merging a special node with a Rate-0 node is more favorable than merging with Rate-1 nodes since their calculation takes less time and area. The impact of merged operations on maximum operating frequency should be taken into account when compiling a new instruction set. Guidelines for Operation Merging {#subsec:guidelines} -------------------------------- The time step reduction amounts in Table \[tab:potentials-branch\] and in Table \[tab:potentials-leaf\] are observed independently: not all of the merging can be exploited at the same time. For example, if a series of operations such as (G-F-F) is present at low-stage, the potential time step reduction consider both {G-F;F} and {G-F$^{\times 2}$}, but only one scheme can be implemented at once. In order to minimize the number of conflicts between different operation merging schemes and to maximize the time step reduction, we developed a merging algorithm based on the following observations. Note that a tail operation refers to the last operation, and a head operation refers to the first operation in an operation sequence or subsequence. - G-F vs. F$^{\times\{2,3,4\}}$: When a series of F operations are observed in the original instruction list, they are merged starting from the tail F operation, in order to minimize conflict with the possible merging of a G-F operation. - (C/C0)-G-F sequence: It was observed that in the polar code decoding sequence, the operation flow for C-G or C0-G operation is always followed by an F operation, if not followed by a special node. This is because the node after C-G (C0-G) sequence is always the root node of an an unexploited subtree. In addition, the G-F operation occurs more frequently than C-G/C0-G operations. Finally, the C/C0 operation can be merged within another operation such as C$^{\times 2}$. As a result, although C-G/C0-G is a possible operation merging scenario, it is not used in our approach. - C-G vs. C$^{\times\{2,3,4\}}$ (C0-G vs. C0$^{\times\{2,3\}}$): If C-G/C0-G operation will be used as a merging scenario, the merging of consecutive C (C0) operations is advised to begin from the head operation towards the tail operation, to maximize the chances of merging with a G operation that follows it, leading to a C-G (C0-G) operation. - F-G0 vs. G0$^{\times\{2,3\}}$ and F$^{\times\{2,3,4\}}$: Merging of consecutive G0 operations starts from the tail G0 operation towards the head operation, to minimize the conflict with potential F-G0 operation. To further minimize conflicts, F operations should be merged starting from the head operation; however this decision would contradict the decision made for the sake of maximizing G-F operations. Based on observations, the number of occurrences of G-F is much higher than that of F-G0, thus F is decided to be merged starting from the tail F operation. - F-Rep vs. F$^{\times\{2,3,4\}}$: To minimize the conflict between F-Rep and consecutive F operations, merging F is advised to begin from the head operation. However, similar to the previous case, this conflicts with the G-F operation merging scenario. Although F-Rep is one of the leaf node operations that returns a favorable amount of time step reduction, if Rep-RepSPC node is used, most of F-Rep operations will be included within it. Because of this, F-Rep operation loses priority against G-F operation. Consequently, merging F operations begin from their tail operation, and F-Rep should be merged from the remaining ones. - Merged special nodes vs. F/G0/C/C0: As described in Table \[tab:potentials-leaf\], merging special nodes include branch operations. Additionally, merging special nodes returns more time step reduction compared to branch operation merging. Thus, in order to minimize merging conflict and maximize savings, special nodes must be merged before merging branch operations. - Merging multiple branch operations: It was observed that if the leaf nodes are merged first, followed by merging branch operations of different kinds (e.g. G-F), the impact of merging operations of the same kind on time step savings reduces dramatically; on the other hand, the benefits of merging leaf nodes and branch operations of different kinds are maximized. Hence, given an instruction sequence derived from Table \[tab:fast-instr\], leaf nodes are merged first, followed by merging branch operations of different kinds, and finally merging branch operations of the same kind. Based on the observations above, a new instruction set is derived for our proposed Fast-SSC decoder. The proposed algorithm uses the original Fast-SSC instructions of Table \[tab:fast-instr\] along with the newly identified instructions in Table \[tab:instr:new\]. In order to maintain a reasonable operating frequency, the delay of the critical path should be maintained as much as possible. Our studies show that up to three C/C0 operations, and up to two F/G0 operations can be merged with minimum effect on the critical path delay. Note that some of the identified merged operations from Table \[tab:potentials-branch\] and Table \[tab:potentials-leaf\] are not used in Table \[tab:instr:new\] since either they did not occur after following the described merging guidelines, or their impact on reducing the number of time steps is negligible. The amount of savings in terms of the number of operations and the number of time steps with the new operation set listed in Table \[tab:instr:new\] is detailed in Table \[tab:savings\] for polar codes of length $N=1024$, $R \in \{\frac{1}{4},\frac{1}{2},\frac{3}{4}\}$ and $P_e \in \{32,64,128\}$. Note that the 5G standard allows a wide range of code rates with binary granularity. Thus, we limit our exploration within three selected rates. It can be seen that the amount of reduction in terms of both number of operations and number of time steps increases with $P_e$. It can also be observed that with increasing $P_e$, the amount of time step reduction increases at lower rate codes, since the new instruction list in Table \[tab:instr:new\] favors Rate-0/Rep nodes more than Rate-1/SPC nodes. [l l rrr]{} Polar Code & $R$ & $P_e$ & Oper. Savings & Time Step Savings\ (r)[1-2]{} (l)[3-3]{} (l)[4-4]{}(l)[5-5]{} & & $32$ & $27.43\%$ & $15.14\%$\ & & $64$ & $34.86\%$ & $26.87\%$\ & & $128$ & $38.86\%$ & $35.42\%$\ (r)[1-2]{} (l)[3-3]{} (l)[4-4]{}(l)[5-5]{} & & $32$ & $26.54\%$ & $15.56\%$\ & & $64$ & $32.70\%$ & $25.75\%$\ & & $128$ & $35.55\%$ & $32.47\%$\ (r)[1-2]{} (l)[3-3]{} (l)[4-4]{}(l)[5-5]{} & & $32$ & $22.54\%$ & $12.42\%$\ & & $64$ & $26.01\%$ & $19.91\%$\ & & $128$ & $30.06\%$ & $27.23\%$\ Decoder Architecture {#sec:architecture} ==================== Architecture Overview --------------------- In order to evaluate the impact of merged operations on the throughput, a new Fast-SSC architecture has been implemented. The architecture supports the fast node decoding techniques from Table \[tab:fast-instr\] as well as the new instruction set from Table \[tab:instr:new\]. The high-level description of the architecture is depicted in Fig. \[fig:arch\_new\_highlevel\]. Decoding sequence begins after loading the instructions into the instruction RAM, and when Channel LLRs are present in the Channel RAM. Note that loading the instruction sequence has to be done only once. The LLR values obtained from the channel are stored in the channel memory, which has a different quantization scheme than the internal LLR $\alpha$ memory. The controller tracks the stage size for each instruction and routes the correct words to and from the $\alpha$ and $\beta$ memory units. A codeword RAM is separately instantiated from the $\beta$-RAM and stores the estimated codeword. From $\alpha$-RAM, $2\times P_e$ LLR values are fetched and $P_e$ LLRs are stored at a time. In case of $\beta$-RAM, $2\times P_e$ partial sum values can be read and stored in a single cycle. For high-stage LLR and partial sum computations, the information is processed $2 \times P_e$ elements at a time. Hence, for $S > \log_2P_e$, $2^S/2 \times P_e$ time steps are required. For low-stage operations, a single time step is required. In merged branch operations, the output of a prior sub-operation is immediately used by the operation that follows it, and storage can often be avoided. For example, the output of the first sub-operation of G0$^{\times 2}$ is used only by its following G0 operation and is never used again. Consequently, only the output of last sub-operation is stored into memory. On the other hand, the output of each F sub-operation of F$^{\times 2}$ will be used by another operation in the future, which makes it mandatory to store the output of the first F sub-operation. By avoiding the storage of intermediate values that will not be used in future instructions, it is possible to save memory bandwidth and increase the number of parallel operations that can be performed by merged operations. The complete list of the intermediate data storing choices, and maximum parallel operations for merged operations are listed in Table \[tab:newinstr-details\]. (0.00,0.00) rectangle (2.00,4.00); at (1.00,2.00) [G]{}; (3.00,0.00) rectangle (5.00,1.50); at (4.00,0.75) [SPC]{}; (3.00,2.50) rectangle (5.00,4.00); at (4.00,3.25) [Sign]{}; (2.0,3.25) – (3.0,3.25); (2.5,3.25) – (2.5,0.75); (2.5,0.75) – (3.0,0.75); (2.5,3.25) circle \[radius=.1\]; (5.0,3.25) – (5.5,3.25); (5.0,0.75) – (5.5,0.75); (5.5,-1.50) – (5.5,4.00); (5.5,4.00) – (6.5,3.50); (6.5,3.50) – (6.5,-1.00); (6.5,-1.00) – (5.5,-1.50); (4.5,-0.75) – (5.5,-0.75); at (4.0,-0.75) [$\boldsymbol{\beta_1}$]{}; (6.5,2.00) – (7.5,2.00); (-1.0,4.50) – (7.5,4.50); (1.0,4.5) circle \[radius=.1\]; (1.0,4.50) – (1.0,4.00); (7.50,1.50) rectangle (9.50,5.50); at (8.50,3.50) [C/C0]{}; (9.5,2.00) – (10.5,2.00); at (11.00,2.00) [$\boldsymbol{\beta_1}'$]{}; (9.5,4.50) – (10.0,4.50); (0.00,5.00) rectangle (7.0,8.00); at (2.00,7.25) [Rep-RepSPC]{}; at (2.00,6.50) [Rep-Rate1]{}; at (2.00,5.75) [Rate0-ML]{}; (4.0,5.5) – (4.00,7.5); at (5.50,7.25) [RepSPC]{}; at (5.50,6.50) [Rep]{}; at (5.50,5.75) [ML]{}; (7.0,6.50) – (10.0,6.5); (0.00,9.50) rectangle (2.00,11.00); at (1.00,10.25) [G-F]{}; (2.00,10.25) – (7.50,10.25); (0.00,11.50) rectangle (2.00,13.00); at (1.00,12.25) [G0]{}; (2.00,12.25) – (7.50,12.25); (3.00,10.50) rectangle (5.00,12.00); at (4.00,11.25) [F-G0]{}; (5.00,11.25) – (7.50,11.25); (3.00,12.50) rectangle (5.00,14.00); at (4.00,13.25) [F]{}; (5.00,13.25) – (7.50,13.25); (3.00,8.25) rectangle (6.00,10.00); at (4.50,9.125) [F-Rep]{}; (6.0,8.75) – (10.0,8.75); (10.00,4.00) – (10.00,9.50); (10.00,9.50) – (11.00,9.00); (11.00,9.00) – (11.00,4.50); (11.00,4.50) – (10.00,4.00); (11.00,6.75) – (12.00,6.75); at (12.50,6.75) [$\boldsymbol{\beta_0}'$]{}; (7.50,9.00) – (7.50,13.75); (7.50,13.75) – (8.50,13.25); (8.50,13.25) – (8.50,9.50); (8.50,9.50) – (7.50,9.00); (8.5,11.375) – (9.5,11.375); at (10,11.375) [$\boldsymbol{\alpha}'$]{}; (-1.5,13.25) – (-0.5,13.25); at (-2.00,13.25) [$\boldsymbol{\alpha}$]{}; (-0.5,2.00) – (-0.5,13.25); (-0.5,13.25) circle \[radius=.1\]; (-0.5,13.25) – (3.0,13.25); (-0.5,11.25) circle \[radius=.1\]; (-0.5,11.25) – (3.0,11.25); (-0.5,12.25) circle \[radius=.1\]; (-0.5,12.25) – (0.0,12.25); (-0.5,10.50) circle \[radius=.1\]; (-0.5,10.50) – (0.0,10.50); (-0.5,9.25) circle \[radius=.1\]; (-0.5,9.25) – (3.0,9.25); (-0.5,6.5) circle \[radius=.1\]; (-0.5,6.5) – (0.0,6.50); (-0.5,2.00) – (0.0,2.00); (-1,5.50) – (-1,3.50); (-1,3.50) – (-1.5,3.25); (-1.5,3.25) – (-1.5,5.75); (-1.5,5.75) – (-1,5.50); (-1.5,4.00) – (-2,4.00); (-1.5,5.00) – (-2,5.00); at (-2.5,4.00) [$0$]{}; at (-2.5,5.00) [$\boldsymbol{\beta}_0$]{}; (-1.75,5.00) circle \[radius=.1\]; (-1.75,5.00) – (-1.75,10.00); (-1.75,10.00) – (0.00,10.00); Datapath -------- Fig. \[fig:fasterssc\] shows the datapath architecture for the proposed Fast-SSC decoder, based on the implementation in [@sarkis14]. It supports all the operations listed in Table \[tab:fast-instr\] and Table \[tab:instr:new\]. The original critical path (in both Fig. \[fig:fastssc\] and Fig. \[fig:fasterssc\]) lies on the path G-SPC-C. To support multiple operations in a single cycle, the new G0 and C modules require more hardware complexity than the original G and C modules. Consequently, to avoid lengthening the critical path, modules including G and C operations are instantiated separately, and the old G and C modules are only used for P- prefixed operations from Table I. Modules F-G0 and G-F, and F-Rep are used only on the low-stage. Processing units for special nodes and their merged versions are clustered together in Fig \[fig:fasterssc\]; they receive LLR values from the $\alpha$ memory and output their hard decision estimates to $\beta$ memory. (0.00,0.00) rectangle (1.00,15.00); (0.0,3.0) – (1.0,3.0); (0.0,6.0) – (1.0,6.0); (0.0,9.0) – (1.0,9.0); (0.0,12.0) – (1.0,12.0); (-0.25,0.0) – (-0.25,3.0); (-0.25,3.0) – (-0.25,6.0); (-0.25,6.0) – (-0.25,9.0); (-0.25,9.0) – (-0.25,12.0); (-0.25,12.0) – (-0.25,15.0); at (-0.75,13.50) [$P_e$]{}; at (-0.75,10.50) [$\frac{P_e}{2}$]{}; at (-0.75,7.50) [$\frac{P_e}{4}$]{}; at (-0.75,4.50) [$\frac{P_e}{8}$]{}; at (-0.75,1.50) [$\frac{P_e}{8}$]{}; (10.00,0.00) rectangle (11.00,15.00); (10.0,3.0) – (11.0,3.0); (10.0,6.0) – (11.0,6.0); (10.0,9.0) – (11.0,9.0); (10.0,12.0) – (11.0,12.0); (11.25,0.0) – (11.25,3.0); (11.25,3.0) – (11.25,6.0); (11.25,6.0) – (11.25,9.0); (11.25,9.0) – (11.25,12.0); (11.25,12.0) – (11.25,15.0); at (11.75,13.50) [$\frac{P_e}{2}$]{}; at (11.75,10.50) [$\frac{P_e}{4}$]{}; at (11.75,7.50) [$\frac{P_e}{8}$]{}; at (11.75,4.50) [$\frac{P_e}{16}$]{}; at (11.75,1.50) [$\frac{P_e}{16}$]{}; (4.00,0.50) rectangle (7.00,2.50); (4.00,3.50) rectangle (7.00,5.50); (4.00,6.50) rectangle (7.00,8.50); (4.00,9.50) rectangle (7.00,11.50); (4.00,12.50) rectangle (7.00,14.50); at (5.50,13.50) [$\text{F}_{\frac{P_e}{2}}$]{}; at (5.50,10.50) [$\text{F}_{\frac{P_e}{4}}$]{}; at (5.50,7.50) [$\text{F}_{\frac{P_e}{8}}$]{}; at (5.50,4.50) [$\text{F}_{\frac{P_e}{16}}$]{}; at (5.50,1.50) [$\text{F}_{\frac{P_e}{16}}$]{}; (7.0,1.5) – (10.0,1.5); (7.0,4.5) – (10.0,4.5); (7.0,7.5) – (10.0,7.5); (7.0,10.5) – (10.0,10.5); (7.0,13.5) – (10.0,13.5); (8.5,13.5) circle \[radius=.1\]; (8.5,10.5) circle \[radius=.1\]; (8.5,7.5) circle \[radius=.1\]; (2.5,3.5) – (2.5,5.5); (2.5,5.5) – (3.0,5.0); (3.0,5.0) – (3.0,4.0); (3.0,4.0) – (2.5,3.5); (2.5,6.5) – (2.5,8.5); (2.5,8.5) – (3.0,8.0); (3.0,8.0) – (3.0,7.0); (3.0,7.0) – (2.5,6.5); (2.5,9.5) – (2.5,11.5); (2.5,11.5) – (3.0,11.0); (3.0,11.0) – (3.0,10.0); (3.0,10.0) – (2.5,9.5); (1.0,1.5) – (4.0,1.5); (1.0,13.5) – (4.0,13.5); (8.5,13.5) – (8.5,12.0); (8.5,12.0) – (2.0,12.0); (2.0,12.0) – (2.0,11.00); (2.0,11.00) – (2.5,11.00); (1.0,10.0) – (2.5,10.00); (3.0,10.5) – (4.0,10.5); (8.5,10.5) – (8.5,9.0); (8.5,9.0) – (2.0,9.0); (2.0,9.0) – (2.0,8.00); (2.0,8.00) – (2.5,8.00); (1.0,7.0) – (2.5,7.00); (3.0,7.5) – (4.0,7.5); (8.5,7.5) – (8.5,6.0); (8.5,6.0) – (2.0,6.0); (2.0,6.0) – (2.0,5.00); (2.0,5.00) – (2.5,5.00); (1.0,4.0) – (2.5,4.00); (3.0,4.5) – (4.0,4.5); In order to support single operations of high-stage as well as single and multiple operations of low-stage, the F processing unit has been redesigned and is shown in Fig. \[fig:arch\_mergedF\]. The new F processing unit is capable of performing high-stage operations in multiple clock cycles, as well as performing single and multiple operations at low-stage. The subscripts in Fig. \[fig:arch\_mergedF\] correspond to the number of parallel F processing elements, which is $P_e$ in total. At high-stage operations, the inputs of all F processing units in Fig. \[fig:arch\_mergedF\] are provided from the $\alpha$ memory. For multiple operations at low-stage, the multiplexers are configured by the controller to cascade the processing units. The configuration of multiple operations are established through the multiplexers in the design: if a merged F operation is performed, following operations within the merged F operation take their inputs from the output of a previous F module. Similar architectures have been implemented for G0 and C/C0 modules. table [ 1.0 2.49332e-01 1.25 1.43858e-01 1.5 7.73779e-02 1.75 3.65304e-02 2.0 1.57029e-02 2.25 6.01196e-03 2.5 2.40279e-03 2.75 9.79432e-04 3.0 2.78552e-04 3.25 8.14067e-05 3.5 2.64725e-05 3.75 7.35078e-06 4.0 2.65076e-06 ]{}; table [ 1.0 2.87737e-01 1.25 1.73599e-01 1.5 9.28116e-02 1.75 4.50947e-02 2.0 1.91027e-02 2.25 7.35793e-03 2.5 3.12064e-03 2.75 9.16590e-04 3.0 3.75235e-04 3.25 1.00060e-04 3.5 3.55619e-05 3.75 9.78521e-06 4.0 3.60719e-06 ]{}; table [ 1.0 4.11403e-02 1.25 2.17823e-02 1.5 1.06580e-02 1.75 4.59874e-03 2.0 1.75224e-03 2.25 5.78849e-04 2.5 2.11319e-04 2.75 7.14297e-05 3.0 1.64628e-05 3.25 4.05126e-06 3.5 1.16024e-06 3.75 2.64743e-07 4.0 1.05823e-07 ]{}; table [ 1.0 4.89273e-02 1.25 2.69796e-02 1.5 1.32302e-02 1.75 5.88418e-03 2.0 2.23085e-03 2.25 7.56130e-04 2.5 2.78617e-04 2.75 6.88159e-05 3.0 2.54163e-05 3.25 5.25315e-06 3.5 1.54750e-06 3.75 4.31543e-07 4.0 1.26252e-07 ]{}; (spypoint) at (axis cs:3.25,1e-4); (magnifyglass) at (axis cs:1.5,5e-7); on (spypoint) in node\[fill=white\] at (magnifyglass); at (3.0,5.7) [$PC(1024,256)$]{} ; \ table [ 1.0 7.56730e-01 1.25 5.66600e-01 1.5 3.68275e-01 1.75 2.03978e-01 2.0 9.75773e-02 2.25 4.02692e-02 2.5 1.53739e-02 2.75 5.29412e-03 3.0 1.60518e-03 3.25 4.70146e-04 3.5 1.71028e-04 3.75 5.79643e-05 4.0 1.94382e-05 4.25 7.62091e-06 4.5 1.76080e-06 ]{}; table [ 1.0 7.79212e-01 1.25 5.96570e-01 1.5 3.93240e-01 1.75 2.21705e-01 2.0 1.07507e-01 2.25 4.58425e-02 2.5 1.60518e-02 2.75 5.74277e-03 3.0 1.86441e-03 3.25 5.87199e-04 3.5 2.03832e-04 3.75 7.83269e-05 4.0 2.03182e-05 4.25 7.87879e-06 4.5 2.52682e-06 ]{}; table [ 1.0 1.05130e-01 1.25 6.86705e-02 1.5 3.86167e-02 1.75 1.85484e-02 2.0 7.64556e-03 2.25 2.75605e-03 2.5 8.66210e-04 2.75 2.55269e-04 3.0 5.59260e-05 3.25 1.20750e-05 3.5 3.39383e-06 3.75 1.18419e-06 4.0 3.19288e-07 4.25 1.24733e-07 4.5 2.77189e-08 ]{}; table [ 1.0 1.11173e-01 1.25 7.39820e-02 1.5 4.21431e-02 1.75 2.07406e-02 2.0 8.75619e-03 2.25 3.14954e-03 2.5 9.19916e-04 2.75 2.52348e-04 3.0 6.18457e-05 3.25 1.32808e-05 3.5 4.57826e-06 3.75 1.39826e-06 4.0 3.44457e-07 4.25 1.28646e-07 4.5 4.10608e-08 ]{}; (spypoint) at (axis cs:3.55,1e-4); (magnifyglass) at (axis cs:1.5,3e-7); on (spypoint) in node\[fill=white\] at (magnifyglass); at (3.0,5.7) [$PC(1024,512)$]{} ; table [ 1.0 9.99860e-01 1.5 9.91755e-01 2.0 8.74337e-01 2.5 4.87637e-01 3.0 1.34826e-01 3.25 5.55234e-02 3.5 2.18644e-02 3.75 7.74676e-03 4.0 2.98106e-03 4.25 1.27617e-03 4.5 3.55492e-04 4.75 1.37155e-04 5.0 4.83699e-05 5.25 1.79814e-05 5.5 5.54723e-06 5.75 1.66317e-06 ]{}; table [ 1.0 9.99801e-01 1.5 9.92911e-01 2.0 8.85723e-01 2.5 5.02881e-01 3.0 1.41406e-01 3.25 5.99402e-02 3.5 2.25922e-02 3.75 8.30508e-03 4.0 3.00100e-03 4.25 1.16650e-03 4.5 4.43853e-04 4.75 1.98295e-04 5.0 6.46663e-05 5.25 2.13945e-05 5.5 5.55960e-06 5.75 2.26087e-06 ]{}; table [ 1.0 1.17354e-01 1.5 9.90862e-02 2.0 6.57489e-02 2.5 2.53016e-02 3.0 4.68549e-03 3.25 1.49902e-03 3.5 4.54067e-04 3.75 1.17356e-04 4.0 3.34803e-05 4.25 1.39945e-05 4.5 3.43920e-06 4.75 1.11260e-06 5.0 4.04972e-07 5.25 1.40480e-07 5.5 4.39878e-08 5.75 1.27986e-08 ]{}; table [ 1.0 1.19024e-01 1.5 1.01005e-01 2.0 6.79553e-02 2.5 2.65637e-02 3.0 4.87498e-03 3.25 1.61332e-03 3.5 4.59637e-04 3.75 1.25736e-04 4.0 3.49927e-05 4.25 1.11385e-05 4.5 3.97040e-06 4.75 1.68086e-06 5.0 5.43096e-07 5.25 1.68816e-07 5.5 4.36516e-08 5.75 1.76631e-08 ]{}; (spypoint) at (axis cs:4.75,1e-4); (magnifyglass) at (axis cs:1.75,2e-7); on (spypoint) in node\[fill=white\] at (magnifyglass); at (3.0,5.7) [$PC(1024,768)$]{} ; \ \[perf-ber\] Fig. \[fig:arch:RepRepSPC\] presents the Rep-RepSPC processing unit from Table \[tab:instr:new\]. The input and the output size of the Rep-RepSPC has a fixed length of $16$. The units enclosed with dashed lines are instances of the RepSPC processing unit. The RepSPC modules assume the output of the Rep node as $0$ and $1$ and are processed in parallel with the Rep module. Modules G0 and G1 are G operations that assumes $\beta$ as $0$ and $1$, respectively. Based on the hard decision estimate of Rep node, the output of the RepSPC is selected from the final multiplexer and is stored in the $\beta$ memory. Results {#sec:results} ======= Error-Correction Performance ---------------------------- To validate the error-correction performance for the proposed decoder, a quantization scheme $Q(6,5,1)$ has been used, where $Q(Q_i,Q_c,Q_f)$ are quantization bit size for internal LLRs, channel LLRs, and fraction bit size for both internal and channel LLRs, respectively. Fig. \[fig:BER\] depicts the error correction performance of the proposed decoder in terms of bit error rate (BER) and frame error rate (FER). The polar code construction is obtained from [@38.212] for $N=1024$, and rates are selected as $R \in \{\frac{1}{4},\frac{1}{2},\frac{3}{4}\}$. The selected quantization values result in less than $0.03$ dB loss at FER$=10^{-4}$ compared to floating-point precision. The introduced operation merging techniques do not change the error-correction performance of SC decoding, as they map thoroughly to SC decoding schedule. ASIC Synthesis Results ---------------------- The architecture for the proposed Fast-SSC decoder has been implemented in VHDL and synthesized in TSMC 65 nm CMOS technology using Cadence Genus RTL compiler. For a fair comparison scheme, three other Fast-SSC-based decoders from [@sarkis14; @giardJSPS; @fastssc-sips17] have also been implemented using the same technology node, quantization, voltage supply and $P_e$. Table \[tab:asicresults-fastssc\] presents the ASIC implementation results for code rates $R \in \{\frac{1}{4},\frac{1}{2},\frac{3}{4}\}$. According to Table \[tab:asicresults-fastssc\], the proposed Fast-SSC decoder has a throughput improvement of up to $31\%$ and $26\%$ compared to the earlier implementations from [@sarkis14] and [@giardJSPS], respectively. Compared to our previous work in [@fastssc-sips17], the throughput improvement is up to $16\%$. The power consumption of the proposed decoder has increased by $18\%$ compared to the baseline Fast-SSC decoder from [@sarkis14], which is due to the new decoding nodes introduced in Section \[sec:architecture\]. On the other hand, due to increased throughput, energy consumption and area efficiency of the proposed decoder compared to [@sarkis14] has been improved by up to $10\%$ and $23\%$ despite the increased power consumption. On the other hand, compared to [@giardJSPS], the proposed decoder implementation consumes $32\%$ more energy per bit and has $13\%$ less area efficiency. It can be observed in Table \[tab:asicresults-fastssc\] that, for all Fast-SSC-based implementations, the latency and the coded throughput are the lowest when $R = 1/2$. This is due to the fact that the occurrence of Rate-0 nodes increase when rate becomes lower, and Rate-1 nodes increase when rate becomes higher. Around $R \approx 1/2$, Rep, SPC and ML nodes occur more frequently, which in general takes more time for decoding. As a result, the area efficiency has the same trend with latency and coded throughput with respect to code rate. On the other hand, the information throughput increases with the rate for all Fast-SSC-based implementations since the number of information bits in the codeword increases linearly with the rate. Finally, the energy dissipation per decoded bit is calculated using the number of information bits. As a result, energy per bit reduces with increasing rate for all Fast-SSC decoders in Table \[tab:asicresults-fastssc\]. Table \[tab:asicresults-others\] presents a comparison scheme for the proposed Fast-SSC decoder against other SC-based architectures including tree, semi-parallel (SP) and combinational approaches. The throughput values for each implementation in Table \[tab:asicresults-others\] are scaled for 65 nm for a fair comparison. In [@dizdar16], a combinational approach is used to decode the polar code, which results in low operating frequency, large throughput and increased area. Although the reported throughput is $2.2 \times$ higher than our decoder, the area is $2.62 \times$ larger, which results in $39\%$ less area efficiency compared to the proposed architecture. In fact, compared with any Fast-SSC based architecture from Table \[tab:asicresults-fastssc\], it can be observed that the area efficiency of the combinational decoder is very low due to its excessive area overhead. Compared to semi-parallel decoder implementations with equal $P_e$, our work has up to $9\times$ larger throughput and $8.8\times$ better area efficiency. Finally, compared with tree-based decoder approaches, the proposed decoder has $3.18\times$ larger throughput. Conclusion {#sec:conclusion} ========== In this work, we proposed a new Fast-SSC polar code decoder implementation. The proposed decoder increases the memory utilization by storing the variables relative to the stages below the parallelization threshold into a single memory word, which also enables the decoder to perform multiple operations within a single time step. A generalization of operation merging scenarios with their guidelines are presented for branch and special node operations, and a subset of the merging scenarios are selected to be implemented in hardware. With the proposed technique, the memory utilization is risen from $66\%$ to $99.6\%$, while the proposed operation set reduces the number of operations to decode a codeword by up to $35\%$ compared to the baseline Fast-SSC decoder. Proposed decoder has been implemented in TSMC 65 nm technology node and compared against other Fast-SSC-based implementations. Results show that our decoder implementation has a throughput improvement of up to $31\%$ and $26\%$ compared to the earlier Fast-SSC-based decoder implementations, with a slight increase in power consumption. Energy dissipation per decoded information bit and the area efficiency for the proposed decoder has been improved by $10\%$ and $23\%$ compared to the baseline Fast-SSC decoder. 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[](http://dx.doi.org/10.1109/IPEC.2012.6522661). O. Dizdar, E. Arıkan, A high-throughput energy-efficient implementation of successive cancellation decoder for polar codes using combinational logic, IEEE Transactions on Circuits and Systems I: Regular Papers 63 (3) (2016) 436–447. [](http://dx.doi.org/10.1109/TCSI.2016.2525020). Y. Fan, C. y. Tsui, An efficient partial-sum network architecture for semi-parallel polar codes decoder implementation, IEEE Transactions on Signal Processing 62 (12) (2014) 3165–3179. [](http://dx.doi.org/10.1109/TSP.2014.2319773). , [NR; Multiplexing and Channel Coding]{}, Tech. Rep. TS 38.212 V15.2.0, [Release 15]{} (January 2018). S. A. Hashemi, C. Condo, F. Ercan, W. J. Gross, On the performance of polar codes for 5G eMBB control channel, in: 51st Asilomar Conference on Signals, Systems, and Computers, 2017, pp. 1764–1768. A. Sharma, M. Salim, Polar code: The channel code contender for 5G scenarios, in: International Conference on Computer, Communications and Electronics (Comptelix), 2017, pp. 676–682. M. Sybis, K. Wesolowski, K. Jayasinghe, V. Venkatasubramanian, V.Vukadinovic, Channel coding for ultra-reliable low-latency communication in 5G systems, in: IEEE 84th Vehicular Technology Conference (VTC-Fall), 2016, pp. 1–5. [^1]: F. Ercan, T. Tonnellier, C. Condo and W. J. Gross are with the Department of Electrical and Computer Engineering, McGill University, Montréal, Québec, Canada. e-mail: furkan.ercan@mail.mcgill.ca, thibaud.tonnellier@mail.mcgill.ca, carlo.condo@mail.mcgill.ca, warren.gross @mcgill.ca. This article is published in Journal of Signal Processing Systems (JSPS), vol. 91, pp.995-1007 on November 3, 2018. DOI:10.1007/s11265-018-1413-4
--- abstract: 'In this paper we define two infinite families of graphs called C-$\delta$ graphs and $\delta$- graph and prove that $\delta$-graphs satisfy $\delta$ conjecture. Also we introduce a family of C-$\delta$ graphs from which we can identify $\delta$ graphs as their complements. Finally we give a list of C-$\delta$ graphs and the relationship with their minimum semidefinite rank.' author: - 'Pedro Díaz Navarro[^1]' date: 'August, 2016' title: ' On $\delta$-Graphs and Delta Conjecture' --- Introduction ============ A [*graph*]{} $G$ consists of a set of vertices $V(G)=\{1,2,\dots,n\}$ and a set of edges $E(G)$, where an edge is defined to be an unordered pair of vertices. The [*order*]{} of $G$, denoted $\vert G\vert $, is the cardinality of $V(G)$. A graph is [*simple*]{} if it has no multiple edges or loops. The [*complement* ]{} of a graph $G(V,E)$ is the graph $\overline{G}=(V,\overline{E})$, where $\overline{E}$ consists of all those edges of the complete graph $K_{\vert G\vert}$ that are not in $E$. A matrix $A=[a_{ij}]$ is [*combinatorially symmetric*]{} when $a_{ij}=0$ if and only if $a_{ji}=0$. We say that $G(A)$ is the graph of a combinatorially symmetric matrix $A=[a_{ij}]$ if $V=\{1,2,\dots,n\}$ and $E=\{\{i,j\}: a_{ij}\ne0\}$ . The main diagonal entries of $A$ play no role in determining $G$. Define $S(G,\F)$ as the set of all $n\times n$ matrices that are real symmetric if $\F=\Re$ or complex Hermitian if $\F=\C$ whose graph is $G$. The sets $S_+(G,\F)$ are the corresponding subsets of positive semidefinite (psd) matrices. The smallest possible rank of any matrix $A\in S(G,\F)$ is the [*minimum rank*]{} of $G$, denoted $\mr(G,\F)$, and the smallest possible rank of any matrix $A\in S_+(G,\F)$ is the [*minimum semidefinite rank*]{} of $G$, denoted $\mr_+(G)$ or $\msr(G)$. In 1996, the minimum rank among real symmetric matrices with a given graph was studied by Nylen [@PN]. It gave rise to the area of minimum rank problems which led to the study of minimum rank among complex Hermitian matrices and positive semidefinite matrices associated with a given graph. Many results can be found for example in [@FW2; @VH; @YL; @LM; @PN]. During the AIM workshop of 2006 in Palo Alto, CA, it was conjectured that for any graph $G$ and infinite field $F$, $\mr(G,\F)\le |G|-\delta(G)$ where $\delta(G)$ is the minimum degree of $G$. It was shown that for if $\delta(G)\le 3$ or $\delta(G)\ge |G|-2$ this inequality holds. Also it can be verified that if $|G|\le 6$ then $\mr(G,F)\le |G|-\delta(G)$. Also it was proven that any bipartite graph satisfies this conjecture. This conjecture is called the [*Delta Conjecture*]{}. If we restrict the study to consider matrices in $S_+(G,\F)$ then delta conjecture is written as $\msr(G)\le |G|-\delta(G)$. Some results on delta conjecture can be found in [@AB; @RB1; @SY1; @SY] but the general problem remains unsolved. Graph Theory Preliminaries ========================== In this section we give definitions and results from graph theory which will be used in the remaining chapters. Further details can be found in [@BO; @BM; @CH]. A [**graph**]{} [$G(V,E)$]{} is a pair [$(V(G),E(G)),$]{} where [$V(G)$]{} is the set of vertices and [$E(G)$]{} is the set of edges together with an [**incidence function**]{} $\psi(G)$ that associate with each edge of $G$ an unordered pair of (not necessarily distinct) vertices of $G$. The [**order**]{} of [$G$]{}, denoted [$|G|$]{}, is the number of vertices in [$G.$]{} A graph is said to be [**simple**]{} if it has no loops or multiple edges. The [**complement**]{} of a graph [$G(V,E)$]{} is the graph [$\overline{G}=(V,\overline{E}),$]{} where [$\overline{E}$]{} consists of all the edges that are not in [$E$]{}. A [**subgraph**]{} [$H=(V(H),E(H))$]{} of [$G=(V,E)$]{} is a graph with [$V(H)\subseteq V(G)$]{} and [$E(H)\subseteq E(G).$]{} An [**induced subgraph**]{} [$H$]{} of [$G$]{}, denoted G\[V(H)\], is a subgraph with [$V(H)\subseteq V(G)$]{} and [$E(H)=\{\{i,j\} \in E(G):i,j\in V(H)\}$]{}. Sometimes we denote the edge $\{i,j\}$ as $ij$. We say that two vertices of a graph $G$ are [**adjacent**]{}, denoted $v_i\sim v_j$, if there is an edge $\{v_i,v_j\}$ in $G$. Otherwise we say that the two vertices $v_i$ and $v_j$ are [**non-adjacent**]{} and we denote this by $v_i \not\sim v_j$. Let [$N(v)$]{} denote the set of vertices that are adjacent to the vertex [$v$]{} and let [$N[v]=\{v\}\cup N(v)$]{}. The [**degree**]{} of a vertex [$v$]{} in [$G,$]{} denoted [$\d_G(v),$]{} is the cardinality of [$N(v).$]{} If [$\d_G(v)=1,$]{} then [$v$]{} is said to be a [**pendant**]{} vertex of [$G.$]{} We use [$\delta(G)$]{} to denote the minimum degree of the vertices in [$G$]{}, whereas [$\Delta(G)$]{} will denote the maximum degree of the vertices in [$G$]{}. Two graphs $G(V,E)$ and $H(V',E')$ are identical denoted $G=H$, if $V=V', E=E'$, and $\psi_G=\psi_H$ . Two graphs $G(V,E)$ and $H(V',E')$ are [**isomorphic**]{}, denoted by $G\cong H$, if there exist bijections $\theta:V\to V'$ and $\phi: E\to E' $ such that $\psi_G(e)=\{u,v\}$ if and only if $\psi_H(\phi(e))= \{\theta(u), \theta(v)\}$. A [**complete graph**]{} is a simple graph in which the vertices are pairwise adjacent. We will use [$nG$]{} to denote [$n$]{} copies of a graph [$G$]{}. For example, $3K_1$ denotes three isolated vertices $K_1$ while [$2K_2$]{} is the graph given by two disconnected copies of $K_2$. A [**path**]{} is a list of distinct vertices in which successive vertices are connected by edges. A path on [$n$]{} vertices is denoted by [$P_n.$]{} A graph [$G$]{} is said to be [**connected**]{} if there is a path between any two vertices of [$G$]{}. A [**cycle**]{} on [$n$]{} vertices, denoted [$C_n,$]{} is a path such that the beginning vertex and the end vertex are the same. A [**tree**]{} is a connected graph with no cycles. A graph $G(V,E)$ is said to be [**chordal**]{} if it has no induced cycles $C_n$ with $n\ge 4$. A [**component**]{} of a graph $G(V,E)$ is a maximal connected subgraph. A [**cut vertex**]{} is a vertex whose deletion increases the number of components. The [**union**]{} $G\cup G_2$ of two graphs $G_1(V_1,E_1)$ and $G_2(V_2,G_2)$ is the union of their vertex set and edge set, that is $G\cup G_2(V_1\cup V_2,E_1\cup E_2$. When $V_1$ and $V_2$ are disjoint their union is called [**disjoint union**]{} and denoted $G_1\sqcup G_2$. The Minimum Semidefinite Rank of a Graph ======================================== In this section we will establish some of the results for the minimum semidefinite rank ($\msr$)of a graph $G$ that we will be using in the subsequent chapters. A [**positive definite**]{} matrix $A$ is an Hermitian $n\times n$ matrix such that $x^\star A x>0$ for all nonzero $x\in \C^n$. Equivalently, $A$ is a $n\times n$ Hermitian positive definite matrix if and only if all the eigenvalues of $A$ are positive ([@RC], p.250). A $n\times n$ Hermitian matrix $A$ such that $x^\star A x\ge 0$ for all $x\in \C^n$ is said to be [**positive semidefinite (psd)**]{}. Equivalently, $A$ is a $n\times n$ Hemitian positive semidefinite matrix if and only if $A$ has all eigenvalues nonnegative ([@RC], p.182). If $\overrightarrow{V}=\{\overrightarrow{v_1},\overrightarrow{v_2},\dots, \overrightarrow{v_n}\}\subset \Re^m$ is a set of column vectors then the matrix $ A^T A$, where $A= \left[\begin{array}{cccc} \overrightarrow{v_1} & \overrightarrow{v_2} &\dots& \overrightarrow{v_n} \end{array}\right]$ and $A^T$ represents the transpose matrix of $A$, is a psd matrix called the [**Gram matrix**]{} of $\overrightarrow{V}$. Let $G(V,E)$ be a graph associated with this Gram matrix. Then $V_G=\{v_1,\dots, v_n\}$ correspond to the set of vectors in $\overrightarrow{V}$ and E(G) correspond to the nonzero inner products among the vectors in $\overrightarrow{V}$. In this case $\overrightarrow{V}$ is called an [**orthogonal representation**]{} of $G(V,E)$ in $\Re^m$. If such an orthogonal representation exists for $G$ then $\msr(G)\le m$. The [**maximum positive semidefinite nullity** ]{} of a graph $G$, denoted $M_+(G)$ is defined by $ M_+(G) =\max\{\nulli(A): A\ \hbox{ is symmetric and positive semidefinite and }\ G(A) =G \} $, where $G(A)$ is the graph obtained from the matrix $A$. From the rank-nullity theorem we get $ \msr(G)+ M_+(G)=|G|$. Some of the most common results about the minimum semidefinite rank of a graph are the following: [@VH]\[msrtree\] If $T $ is a tree then $\msr(T)= |T|-1$. [@MP3]\[msrcycle\] The cycle $C_n$ has minimum semidefinite rank $n-2$. \[res2\] [@MP3]  If a connected graph $G$ has a pendant vertex $v$, then $\msr(G)=\msr(G-v)+1$ where $G-v$ is obtained as an induced subgraph of $G$ by deleting $v$. [@PB] \[OS2\] If [$G$]{} is a connected, chordal graph, then [$\msr(G)=\cc(G).$]{} \[res1\] [@MP2] If a graph $G(V,E)$ has a cut vertex, so that $G=G_1\cdot G_2$, then $\msr(G)= \msr(G_1)+\msr(G_2)$. $\mathbf \delta$-Graphs and the Delta Conjecture ================================================ In this section we define a new family of graphs called $\delta$-graphs and show that they satisfy the delta conjecture. \[ccpg\] Suppose that $G=(V,E)$ with $|G|=n \ge 4$ is simple and connected such that $\overline{G}=(V,\overline{E})$ is also simple and connected. We say that $G$ is a [**$\mathbf{\delta}$-graph**]{} if we can label the vertices of $G$ in such a way that 1. the induced graph of the vertices $v_1,v_2,v_3$ in $G$ is either $3K_1$ or $K_2 \sqcup K_1$, and 2. for $m\ge 4$, the vertex $v_m$ is adjacent to all the prior vertices $v_1,v_2,\dots,v_{m-1}$ except for at most $\dis{\left\lfloor\frac{m}{2}-1\right\rfloor}$ vertices. Suppose that a graph $G(V,E)$ with $|G|=n \ge 4$ is simple and connected such that $\overline{G}=(V,\overline{E})$ is also simple and connected. We say that $G(V,E)$ is a [**C-$\mathbf{\delta}$ graph**]{} if $\overline{G}$ is a $\delta$-graph. In other words, $G$ is a [**C-$\mathbf{\delta}$ graph**]{} if we can label the vertices of $G$ in such a way that 1. the induced graph of the vertices $v_1,v_2,v_3$ in $G$ is either $K_3$ or $P_3$, and 2. for $m\ge 4$, the vertex $v_m$ is adjacent to at most $\dis{\left\lfloor\frac{m}{2}-1\right\rfloor}$ of the prior vertices $v_1,v_2,\dots,v_{m-1}$. \[examplecp\] The cycle $C_n, n\ge 6$ is a C-$\delta$ graph and its complement is a $\delta$-graph. ![image](c63prism.png){height="25mm"} Note that we can label the vertices of $C_6$ clockwise $v_1,v_2,v_3,v_4,v_5,v_6$. The graph induced by $v_1,v_2,v_3$ is $P_3$. The vertex $v_4$ is adjacent to a prior vertex which is $v_3$. Also, the vertex $v_5$ is adjacent to vertex $v_4$ and the vertex $v_6$ is adjacent to two prior vertices $v_1$ and $v_5$. Hence, $C_6$ is C-$\delta$ graph. The $3$-prism which is isomorphic to the complement of $C_6$, is a $\delta$-graph. \[lem2\] Let [$G(V,E)$]{} be a $\delta$-graph. Then the induced graph of [$\{v_1,v_2,v_3\}$]{} in [$G$]{} denoted by $H$ has an orthogonal representation in [$\Re^{\Delta(\overline{G})+1}$]{} satisfying the following conditions: 1. the vectors in the orthogonal representation of $H$ can be chosen with nonzero coordinates, and 2. \[L1\]$\overrightarrow{v}\not\in \sp(\overrightarrow{u})$ for each pair of distinct vertices $u,v$ in $H$. Let [$G(V,E)$]{} be a $\delta$-graph. Label the vertices of $G$ in such a way that the labeling satisfies the conditions (1) and (2) for $\delta$-graphs. Let [$H$]{} be the induced graph in $G$ of $\{v_1,v_2,v_3\}\subseteq V$. Then $H$ is either $3K_1$ or $K_2\sqcup K_1$. Since [$G$]{} and [$\overline{G}$]{} are simple and connected it follows that \[ine1\] $2\le \Delta(\overline{G})\le n-2$ Let $\{\overrightarrow{e}_j\}, j=1,2,\dots,\Delta(\overline{G})+1$ be the standard orthonormal basis for $\Re^{\Delta(\overline{G})+1}$.\ [*Case 1.*]{} Suppose the induced graph $H$ of [$\{v_1,v_2,v_3\}\subseteq V$]{} in $G$ is $3K_1$ which is disconnected. Choose $\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}$ in $\Re^{\Delta(\overline{G})+1}$ corresponding to [$v_1,v_2,v_3$]{} respectively such that: $$\begin{aligned} \overrightarrow{v_1} &=& \dis{\sum_{j=1}^{\Delta(\overline{G})+1}k_{1,j}\overrightarrow{e}_j}\\ \overrightarrow{v_2} &=& \dis{\sum_{j=1}^{\Delta(\overline{G})+1}k_{2,j}\overrightarrow{e}_j} \\ \overrightarrow{v_3} &=& \dis{\sum_{j=1}^{\Delta(\overline{G})+1}k_{3,j}\overrightarrow{e}_j}\end{aligned}$$ where the scalars $k_{1,j}, j=1,2,\dots,\Delta(\overline{G})+1$, and $k_{2,s}, s=1,2,\dots, \Delta(\overline{G})$ are chosen not zero from different field extensions in the following way: - $k_{1,1}\not \in \Q$, - $k_{1,2}\not \in\Q[k_{1,1}]$, - $k_{1,3}\not \in \Q[k_{1,1}, k_{1,2}]$, - $\vdots$ - $ k_{1,\Delta(\overline{G})+1}\not \in \Q[k_{1,1},k_{1,2},\dots,k_{1 ,\Delta(\overline{G})}]$, - $ k_{2,1}\not \in \Q[k_{1,1},k_{1,2},\dots,k_{1,\Delta(\overline{G})+1}]$, - $k_{2,2}\not \in \Q[k_{1,1},k_{1,2},\dots,k_{1,\Delta(\overline{G})+1},k_{2,1}]$, - $\vdots$ - $k_{2,\Delta(\overline{G})}\not \in \Q[k_{1,1},k_{1,2},\dots,k_{1,\Delta(\overline{G})+1},k_{2,1},\dots,k_{2,\Delta(\overline{G})-1}]$. Now choose $$k_{2,\Delta(\overline{G})+1}=\frac{-1}{k_{1,\Delta(\overline{G})+1}}\dis{\sum_{j=1}^{\Delta(\overline{G})}k_{1,j}k_{2,j}}$$ As a consequence $\langle \overrightarrow{v}_1,\overrightarrow{v}_2\rangle=0$. In order to find a vector $\overrightarrow{v}_3$, we need to solve the $ 2\times (\Delta(\overline{G})+1)$ system satisfying $$\begin{aligned} \langle\overrightarrow{v}_1,\overrightarrow{v}_3\rangle &=& 0 \\ \langle\overrightarrow{v}_2,\overrightarrow{v}_3\rangle &=& 0\end{aligned}$$ in the variables $k_{3,j}, j=1,2,\dots,\Delta(\overline{G})+1$. The homogeneous system has infinitely many solutions because $\Delta(\overline{G})+1\ge3$. Reducing the matrix of this system to echelon form we get $$\left( \begin{array}{ccccccc} k_{1,1} & k_{1,2}& k_{1,3}& \dots& k_{1,\Delta(\overline{G})+1} \\ k_{2,1} & k_{2,2}& k_{2,3}& \dots& k_{2,\Delta(\overline{G})+1} \\ \end{array} \right)\sim$$ $$\left( \begin{array}{ccccccc} 1 & \frac{k_{1,2}}{k_{1,1}}&\frac{ k_{1,3}}{k_{1,1}}& \dots& \frac{k_{1,\Delta(\overline{G})+1}}{k_{1,1}} \\ k_{2,1} & k_{2,2}& k_{2,3}& \dots& k_{2,\Delta(\overline{G})+1} \\ \end{array} \right)\sim$$ $$\left( \begin{array}{ccccccc} 1 & \frac{k_{1,2}}{k_{1,1}}&\frac{ k_{1,3}}{k_{1,1}}& \dots& \frac{k_{1,\Delta(\overline{G})+1}}{k_{1,1}} \\ 0 & k_{2,2}-\frac{k_{2,1}k_{1,2}}{k_{1,1}}& k_{2,3}-\frac{k_{2,1}k_{1,3}}{k_{1,1}}& \dots& k_{2,\Delta(\overline{G})+1}-\frac{k_{2,1}k_{1,\Delta(\overline{G})+1}}{k_{1,1}} \\ \end{array} \right)$$ Let $\alpha = k_{2,2}-\frac{k_{2,1}k_{1,2}}{k_{1,1}}$. Since $\alpha \ne 0$ because $k_{2,2}\not\in \Q[k_{1,1},\dots,k_{1,\Delta(\overline{G})+1},k_{2,1}]$. We can get the echelon form of the matrix by multiplying the second row by $\frac{1}{\alpha}$. $$\left( \begin{array}{ccccccc} 1 & \frac{k_{1,2}}{k_{1,1}}&\frac{ k_{1,3}}{k_{1,1}}& \dots& \frac{k_{1,\Delta(\overline{G})+1}}{k_{1,1}} \\ 0 & 1&\frac{1}{ \alpha}(k_{2,3}-\frac{k_{2,1}k_{1,3}}{ k_{1,1}})& \dots&\frac{1}{ \alpha} (k_{2,\Delta(\overline{G})+1}-\frac{k_{2,1}k_{1,\Delta(\overline{G})+1}}{ k_{1,1}}) \\ \end{array} \right)$$ Since the system has infinitely many solutions, $k_{3,j}, j= 3,\dots,\Delta(\overline{G})+1$ are free parameters. We can choose them from different field extensions in the following way, - $ k_{3,3}\not \in \Q[k_{1,1},k_{1,2},\dots,k_{1,\Delta(\overline{G})+1},k_{2,1},\dots,k_{2,\Delta(\overline{G})+1}]$, - $ k_{3,4}\not \in \Q[k_{1,1},k_{1,2},\dots,k_{1,\Delta(\overline{G})+1},k_{2,1},\dots,k_{2,\Delta(\overline{G})+1},k_{3,1}]$, - $ \vdots$ - $ k_{3,\Delta(\overline{G})+1}\not \in \Q[k_{1,1},k_{1,2},\dots,k_{1,\Delta(\overline{G})+1},k_{2,1},\dots,k_{2,\Delta(\overline{G})+1},k_{3,3},\dots,k_{3,\Delta(\overline{G})}]$. Since $k_{2,j}-\frac{k_{2,1}k_{1,j}}{ k_{1,1}}\ne 0, j=3,\dots,\Delta(\overline{G})+1$, we can choose these parameters such that $k_{3,1}$ and $k_{3,2}$ are also nonzero. Therefore we get $\langle\overrightarrow{v}_1,\overrightarrow{v}_3\rangle=\langle\overrightarrow{v}_2,\overrightarrow{v}_3\rangle=0$. As a result $\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}$ is an orthogonal representation of the induced graph $H=3K_1$ satisfying conditions (i) and (ii).\ [*Case 2.*]{} Suppose the induced graph $H$ is $K_2\sqcup K_1$. Let us assume $v_1 \not\sim v_2,v_2 \not\sim v_3$, and $v_1 \sim v_3 $.\ The vectors $\overrightarrow{v}_1$ and $\overrightarrow{v}_2$ for the vertices $v_1$ and $v_2$ respectively can be chosen in the same way as in case 1 to get $\langle\overrightarrow{v}_1,\overrightarrow{v}_2\rangle=0$. In order to find a vector $\overrightarrow{v_3}=k_{3,1}\overrightarrow{e_1}+k_{3,2}\overrightarrow{e_2}+\dots +k_{3,\Delta(\overline{G})+1}\overrightarrow{e}_{\Delta(\overline{G})+1}\in \Re^{\Delta(\overline{G})+1}$ with nonzero components for the vertex $v_3$ we know that the vector $\overrightarrow{v}_3$ should satisfy the system $$\begin{aligned} \langle\overrightarrow{v_1},\overrightarrow{v_3}\rangle&=&g_1 , \ \ g_1\ne 0 \label{s11} \\ \langle\overrightarrow{v_2},\overrightarrow{v_3}\rangle &=& 0 \label {s12} \end{aligned}$$ in the variables $k_{3,j}, j=1,2,\dots,\Delta(\overline{G})+1$ because $v_1\sim v_3$ and $v_2\not\sim v_3$ in $G$. Therefore, rewriting the system in the form $$\begin{aligned} k_{1,1}k_{3,1}+k_{1,2}k_{3,2}+\dots+k_{1,\Delta(\overline{G})+1}k_{3,\Delta(\overline{G})+1}&= g_1, \label{esys1}\\ k_{2,1}k_{3,1}+k_{2,2}k_{3,2}+\dots+k_{2,\Delta(\overline{G})+1}k_{3,\Delta(\overline{G})+1} &= 0 \label{esys2} \end{aligned}$$ the augmented matrix of the non-homogeneous system becomes $$\left( \begin{array}{ccccccc} k_{1,1} & k_{1,2}& k_{1,3}& \dots& k_{1,\Delta(\overline{G})+1} & | & g_1 \label{ms1} \\ k_{2,1} & k_{2,2}& k_{2,3}& \dots& k_{2,\Delta(\overline{G})+1} & | & 0 \label {ms2}\\ \end{array} \right)$$ Since $\Delta(\overline{G})+1\ge 3$, the system has infinitely many solutions depending on one or more free parameters if the system is consistent. Since $k_{1,1}$ is nonzero, we can divide the first row by $k_{1,1}$ to obtain $$\left( \begin{array}{ccccccc} 1 &\frac{ k_{1,2}}{k_{1,1}}&\frac{k_{1,3}}{k_{1,1}}& \dots& \frac{k_{1,\Delta(\overline{G})+1} }{k_{1,1}}& | & \frac{g_1}{k_{1,1}} \label{ms3} \\ k_{2,1} & k_{2,2}& k_{2,3}& \dots& k_{2,\Delta(\overline{G})+1} & | & 0 \label {ms4}\\ \end{array} \right)\sim$$ $$\left( \begin{array}{ccccccc} 1 &\frac{ k_{1,2}}{k_{1,1}}&\frac{k_{1,3}}{k_{1,1}}& \dots& \frac{k_{1,\Delta(\overline{G})+1} }{k_{1,1}}& | & \frac{g_1}{k_{1,1}} \label{ms3} \\ 0 & k_{2,2}-\frac{k_{1,2}k_{2,1}}{k_{1,1}}& k_{2,3}-\frac{k_{1,3}k_{2,1}}{k_{1,1}}& \dots& k_{2,\Delta(\overline{G})+1}-\frac{k_{1,\Delta(\overline{G})+1}k_{2,1}}{k_{1,1}} & | &-\frac{g_1k_{2,1}}{k_{1,1}} \label {ms5}\\ \end{array} \right)$$ and since $k_{2,2}\not\in \Q[k_{1,2},k_{2,1},k_{1,1}]$, $k_{2,2}-\frac{k_{1,2}k_{2,1}}{k_{1,1}}\ne 0$. Hence we obtain the echelon form of the matrix dividing by $\alpha=k_{2,2}-\frac{k_{1,2}k_{2,1}}{k_{1,1}}$ $$\left( \begin{array}{ccccccc} 1 &\frac{ k_{1,2}}{k_{1,1}}&\frac{k_{1,3}}{k_{1,1}}& \dots& \frac{k_{1,\Delta(\overline{G})+1} }{k_{1,1}}& | & \frac{g_1}{k_{1,1}} \label{ms3} \\ 0 & 1&\frac{1}{\alpha}( k_{2,3}-\frac{k_{1,3}k_{2,1}}{k_{1,1}})& \dots&\frac{1}{\alpha}( k_{2,\Delta(\overline{G})+1}-\frac{k_{1,\Delta(\overline{G})+1}k_{2,1}}{k_{1,1}}) & | &-\frac{g_1k_{2,1}}{\alpha k_{1,1}} \label {ms6}\\ \end{array} \right)$$ Choose $k_{3,3}\ne 0, k_{3,4}\ne 0,\dots,k_{3,\Delta(\overline{G})+1}\ne0$ one by one from different field extensions $\Q[\gamma_3],\dots,\\ \Q[\gamma_{\Delta(\overline{G})+1}]$ such that $\Q[\gamma_i], i=3,\dots,\Delta(\overline{G})+1$ is not a field extension in the lattice $L(k_{1,1},\dots,\\ k_{1,\Delta(\overline{G})+1},k_{2,1},\dots,k_{2,\Delta(\overline{G})+1},\gamma_3,\gamma_4,\dots,\gamma_{i-1})$. Therefore, since $g_1$ is nonzero, we can choose $g_1\in \Re$ in such a way that $g_1$ does not belong to any of the prior field extensions used so far and satisfy $$\begin{aligned} \frac{-g_1 k_{2,1}}{\alpha k_{1,1}}-\frac{1}{\alpha}\left((k_{2,3}-\frac{k_{1,3}k_{2,1}}{k_{1,1}})k_{3,3}+\dots+(k_{2,\Delta(\overline{G})+1}-\frac{k_{1,\Delta(\overline{G})+1}k_{2,1}}{k_{1,1}})k_{3,\Delta(\overline{G})+1}\right) &\ne&0\ \ \end{aligned}$$ which implies $$\begin{aligned} g_1 &\ne&\frac{-k_{1,1}}{k_{2,1}}\left( (k_{2,3}-\frac{k_{1,3}k_{2,1}}{k_{1,1}})k_{3,2}+\dots+(k_{2,\Delta(\overline{G})+1}-\frac{k_{1,\Delta(\overline{G})+1}k_{2,1}}{k_{1,1}})k_{3,\Delta(\overline{G})+1}\right).\label{cond1}\end{aligned}$$ Also $$\begin{aligned} \frac{g_1}{k_{1,1}}-\left(\frac{k_{1,2}}{k_{1,1}}k_{3,2}+\frac{k_{1,3}}{k_{1,1}}k_{3,3}+\dots+\frac{k_{1,\Delta(\overline{G})+1}}{k_{1,1}}k_{3,\Delta(\overline{G})+1}\right)&\ne&0\end{aligned}$$ which implies $$\begin{aligned} g_1&\ne&k_{1,1}\left(\frac{k_{1,2}}{k_{1,1}}k_{3,2}+\frac{k_{1,3}}{k_{1,1}}k_{3,3}+\dots+\frac{k_{1,\Delta(\overline{G})+1}}{k_{1,1}}k_{3,\Delta(\overline{G})+1}\right).\label{cond2}\end{aligned}$$ Thus, choosing $g_1\ne 0$ satisfying \[cond1\] and \[cond2\] we get that the system \[esys1\] and \[esys2\] is consistent and at least one of its solutions satisfies the adjacency condition and the orthogonal condition for $\overrightarrow{v}_3$ such that none of the coordinates of the vectors $\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3} \in \Re^{\Delta(\overline{G})+1}$ are zero. Therefore $\{\overrightarrow{v}_1,\overrightarrow{v}_2,\overrightarrow{v}_3\}$ is an orthogonal representation in $\Re^{\Delta(\overline{G})+1}$ for the induced graph $H=K_2\sqcup K_1$ satisfying condition (i). Note that if $\overrightarrow{v}_3=a \overrightarrow{v}_1, a\in \Re$ then $k_{3,1}= a k_{1,1}$ and $k_{3,\Delta(\overline{G})+1}= a k_{1,\Delta(\overline{G})+1}$ which implies that $a =\frac{k_{3,1}}{ k_{1,1}}$ and $a =\frac{k_{3,\Delta(\overline{G})+1}}{ k_{1,\Delta(\overline{G})+1}}$. As a consequence $k_{3,\Delta(\overline{G})+1}=\frac{k_{3,1}}{ k_{1,1}} k_{1,\Delta(\overline{G})+1}$. Hence $k_{3,\Delta(\overline{G})+1}\in \Q[k_{1,1},k_{3,1}, k_{1,\Delta(\overline{G})+1}]$ which is a contradiction because $k_{3,\Delta(\overline{G})+1}$ was chosen from a different field extension. Hence, $\overrightarrow{v}_3$ and $\overrightarrow{v}_1$ are linearly independent and the vectors $\overrightarrow{v}_1,\overrightarrow{v}_2,\overrightarrow{v}_3$ are pairwise linearly independent satisfying condition (ii). $\Box$ \[main\] Let $G(V,E)$ be a $\delta$-graph then $$\msr(G)\le\Delta(\overline{G})+1=|G|-\delta(G)\label{mrsineq1}$$ Let $G(V,E)$ be a $\delta$ graph. Let $Y_3(\{v_1,v_2,v_3\},E_{Y_3})$ be the graph induced by the vertices $v_1,v_2$, and $v_3$ which is either $3K_1$ or $K_2\sqcup K_1$. By Lemma \[lem2\], $Y_3$ has an orthogonal representation in $\Re^{\Delta(\overline{G})+1}$. Also, from Lemma \[lem2\] we have 1. \[tcond2\] The components of vectors in the orthogonal representation of $Y_3$ are all nonzero. 2. \[tcon1\] $\overrightarrow{v}\not\in \sp(\overrightarrow{u})$ for each pair of distinct vertices $u,v$ in $V_{Y_3}$. From the definition of $\delta$-graph we have 1. \[tcond3\] $G(V,E)$ can be constructed starting with $Y_3(V_{Y_3},E_{Y_3})$ and adding one vertex at a time such that the newly added vertex $v_m, m\ge 4$ is adjacent to all prior vertices $v_1,v_2,\dots,v_{m-1}$ except for at most $\dis{\left\lfloor\frac{m}{2}-1\right\rfloor}$ vertices. Applying condition (3) we get a sequence of subgraphs $ Y_3,Y_4,\dots,Y_m,\dots,Y_{|G|}\label{seqinG} $ in $G$ induced by $ \{v_1,v_2,v_3\},\{v_1,v_2,v_3,v_4\},\dots,\{v_1,v_2,v_3,v_4,\dots,v_m\},$ $\dots, \{v_1,v_2,v_3,v_4,\dots,v_{|G|}\}\label{seqvert} $ respectively. We will prove that $Y_j=(V_j,E_j)$ has an orthogonal representation in $\Re^{\Delta(\overline{G})+1}$ for all $j= 3,4,\dots,|G|$, satisfying conditions (1) and (2) above. For that purpose, consider the orthogonal representation of $Y_3$ satisfying conditions (i) and (ii) given by Lemma \[lem2\]. $$\begin{aligned} \overrightarrow{v_1} &=& k_{1,1}\overrightarrow{e_1}+k_{1,2}\overrightarrow{e_2}+\dots+k_{1,\Delta(\overline{G})+1}\overrightarrow{e}_{\Delta(\overline{G})+1} \label{tm1}\\ \overrightarrow{v_2} &=& k_{2,1}\overrightarrow{e_1}+k_{2,2}\overrightarrow{e_2}+\dots+k_{2,\Delta(\overline{G})+1}\overrightarrow{e}_{\Delta(\overline{G})+1} \label{tm2} \\ \overrightarrow{v_3} &=& k_{3,1}\overrightarrow{e_1}+k_{3,2}\overrightarrow{e_2}+\dots+k_{3,\Delta(\overline{G})+1} \overrightarrow{e}_{\Delta(\overline{G})+1}\label{tm3}\end{aligned}$$ where all $k_{i,j}, j=1,2,\dots,\Delta(\overline{G})+1, i=1,2,3$, are nonzero and are chosen from different field extensions as in the proof of Lemma \[lem2\]. Let $v_4$ be a vertex of $G$ such that $v_4$ is adjacent to all of $v_1,v_2,v_3$ except at most $\left \lfloor\frac{4}{2}-1\right\rfloor=1$ vertex. Since $G$ and $\overline{G}$ are simple and connected $2\le \Delta(\overline{G})$ and therefore $\Delta(\overline{G}) + 1 \ge 3$. \[claimMainT\] $Y_4$ induced by $\{v_1,v_2,v_3,v_4\}$ has an orthogonal representation in $\Re^{\Delta(\overline{G}) + 1}$ satisfying conditions (1) and (2) above. We know that $$d_{\overline{G}}(v_4)\le\Delta(\overline{G})<\Delta(\overline{G})+1$$ Since $V_{Y_3}=\{v_1,v_2,v_3\}$ from Lemma \[lem2\] we have an orthogonal representation for the induced subgraph of $V_{Y_3}$ of $G$ in $\Re^{\Delta(\overline{G}) + 1}$ satisfying conditions (1) and (2). We need to find a vector $\overrightarrow{v_4}$ for the vertex $v_4$ where $$\overrightarrow{v_4} = k_{4,1}\overrightarrow{e_1}+k_{4,2}\overrightarrow{e_2}+\dots+k_{4,\Delta(\overline{G})+1} \overrightarrow{e}_{\Delta(\overline{G})+1}\label{tm4}$$ satisfying conditions (1) and (2). Since $v_4$ is adjacent with all prior vertices except for at most one of them we have four cases: 1. \[case1\] $v_4$ is adjacent to $v_1,v_2$ and $v_3$ in $G$. 2. \[case2\] $v_4$ is adjacent to only $v_1$ and $v_2$ in $G$. 3. \[case3\] $v_4$ is adjacent to only $v_1$ and $v_3$ in $G$. 4. \[case4\] $v_4$ is adjacent to only $v_2$ and $v_3$ in $G$. [*Case 1.   $v_4\sim v_1,v_4\sim v_2,v_4\sim v_3$ in $G$*]{}.\ Choose $k_{4,j}, j=1,\dots,\Delta(\overline{G})+1$ as follows: 1. $k_{4,1} =\gamma_{4,1}$ does not belong to any of the field extensions in the lattice of fields $L[\Q[k_{i,j}]],\\ i=1,2,3,j=1,2,\dots,\Delta(\overline{G})+1$ . 2. $k_{4,2} =\gamma_{4,2}$ does not belong to any of the field extensions in the lattice of field extensions $L[\Q[k_{i,j},\gamma_{4,1}] , i=1,2,3, j=1,2,\dots,\Delta(\overline{G})+1$. 3. $k_{4,3} =\gamma_{4,3}$ does not belong to any of the field extensions in the lattice of field extensions $L[\Q[k_{i,j},\gamma_{4,1},\gamma_{4,2}]] , i=1,2,3, j=1,2,\dots,\Delta(\overline{G})+1$. 4. Continuing the process until $k_{4,\Delta(\overline{G})+1}$ to obtain 5. $k_{4,\Delta(\overline{G})+1} =\gamma_{4,\Delta(\overline{G})+1}$ does not belong to any of the field extensions in the lattice of field extensions $L[\Q[k_{i,j},\gamma_{4,1},\gamma_{4,2},\dots,\gamma_{4,\Delta(\overline{G})}]] , i=1,2,3, j=1,2,\dots,\Delta(\overline{G})+1$. Then $k_{4,j} \ne0$ for all $j= 1,2,\dots, \Delta(\overline{G})+1$ and $\overrightarrow{v_4}\not \in \sp(\overrightarrow{v_i}), i=1,2,3$. As a consequence $\{\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3},\overrightarrow{v_4}\}$ is an orthogonal representation of $Y_4$ at $\Re^{\Delta(\overline{G})+1}$. Note that if $\langle\overrightarrow{v}_4,\overrightarrow{v}_1\rangle=0$ then we can solve this equation for $k_{4,\Delta(\overline{G})+1}$ which implies that $k_{4,\Delta(\overline{G})+1}\in L[\Q[k_{i,j},\gamma_{4,1},\gamma_{4,2},\dots,\\ \gamma_{4,\Delta(\overline{G})}]] , i=1,2,3, j=1,2,\dots,\Delta(\overline{G})+1$ which is a contradiction. Therefore, $\langle\overrightarrow{v}_4,\overrightarrow{v}_1\rangle\ne0$. In the same way we can prove that $\langle\overrightarrow{v}_4,\overrightarrow{v}_2\rangle\ne0$ and $\langle\overrightarrow{v}_4,\overrightarrow{v}_3\rangle\ne0$. [*Case 2.   $v_4\sim v_1,v_4\sim v_2,v_4\not \sim v_3$ in $G$*]{}. Since $v_4\sim v_1$ and $v_4\sim v_2$ and $v_4\not \sim v_3$ then $$\begin{aligned} \langle\overrightarrow{v_4},\overrightarrow{v_1}\rangle &=& g_{4,1}, \ g_{4,1}\ne 0 \\ \langle\overrightarrow{v_4},\overrightarrow{v_2}\rangle &=& g_{4,2}, \ g_{4,2}\ne 0\\ \langle\overrightarrow{v_4},\overrightarrow{v_3}\rangle &=& 0.\end{aligned}$$ From these conditions the system $S$ in the variables $k_{4,j}, j=1,2,\dots,\Delta(\overline{G})+1$ becomes, $$\begin{aligned} k_{1,1}k_{4,1}+k_{1,2}k_{4,2}+\dots+k_{1,\Delta(\overline{G})+1}k_{4,\Delta(\overline{G})+1} &=&g_{4,1}, g_{4,1} \ne 0\\ k_{2,1}k_{4,1}+k_{2,2}k_{4,2}+\dots+k_{2,\Delta(\overline{G})+1}k_{4,\Delta(\overline{G})+1} &=& g_{4,2}, g_{4,2} \ne 0 \\ k_{3,1}k_{4,1}+k_{3,2}k_{4,2}+\dots+k_{3,\Delta(\overline{G})+1}k_{4,\Delta(\overline{G})+1}&=& 0 \end{aligned}$$ where $k_{i,j}, i=1,2,3, j=1,2,\dots,\Delta(\overline{G})+1$ were chosen from different field extensions as in the proof of Lemma \[lem2\]. Since $$d_{\overline{G}}(v_4)\le\Delta(\overline{G})<\Delta(\overline{G})+1$$ in $\overline{G}$, the number of equations from the orthogonal conditions in the systems are at most $\Delta(\overline{G})<\Delta(\overline{G})+1$, which means that if the non-homogeneous system is consistent then the system will have infinitely many solutions because the system will have at least one free variable. The augmented matrix of the system becomes $ \left( \begin{array}{cccccc} k_{1,1} &k_{1,2} &\dots & k_{1,\Delta(\overline{G})+1} & | & g_{4,1} \\ k_{2,1} &k_{2,2} &\dots & k_{2,\Delta(\overline{G})+1} & | & g_{4,2}\\ k_{3,1} &k_{3,2} &\dots & k_{3,\Delta(\overline{G})+1} & | &0 \end{array} \right) $ where $g_{4,1}\ne0, g_{4,2}\ne0$ are nonzero real numbers. In order to guarantee that all adjacency conditions are satisfied consider the matrix $3\times(\Delta(\overline{G})+3)$ below in the variables $k_{4,1},k_{4,2},\dots,k_{4,\Delta(\overline{G})+1},-g_{4,1},-g_{4,2}$. It is possible to consider $-g_{4,1}$ and $-g_{4,2}$ as variables because we only need them to be nonzero. So we can consider them as two additional variables of the homogeneous system $S_H$ which has the following augmented matrix: $ \left( \begin{array}{cccccc} k_{1,1} &k_{1,2} &\dots & k_{1,\Delta(\overline{G})+1} & 1 & 0 \\ k_{2,1} &k_{2,2} &\dots & k_{2,\Delta(\overline{G})+1} & 0 & 1\\ k_{3,1} &k_{3,2} &\dots & k_{3,\Delta(\overline{G})+1} & 0&0 \end{array} \right). $ Multiplying the first row by $\frac{1}{k_{1,1}}, k_{1,1}\ne 0$ we get $ \left( \begin{array}{cccccc} 1 &\frac{k_{1,2}}{ k_{1,1}} &\dots & \frac{k_{1,\Delta(\overline{G})+1}}{ k_{1,1}} & \frac{1}{ k_{1,1}} & 0 \\ k_{2,1} &k_{2,2} &\dots & k_{2,\Delta(\overline{G})+1} & 0 & 1\\ k_{3,1} &k_{3,2} &\dots & k_{3,\Delta(\overline{G})+1} & 0&0 \end{array} \right). $ Multiplying the first row by $-k_{2,1}$ and adding the result to the second row and multiplying the first row by $-k_{3,1}$ and adding to the third row we get $ \left( \begin{array}{cccccc} 1 &\frac{k_{1,2}}{ k_{1,1}} &\dots & \frac{k_{1,\Delta(\overline{G})+1}}{ k_{1,1}} & \frac{1}{ k_{1,1}} & 0 \\ 0 &k_{2,2}-\frac{k_{1,2}k_{2,1}}{ k_{1,1}} &\dots & k_{2,\Delta(\overline{G})+1}- \frac{k_{1,\Delta(\overline{G})+1}k_{2,1}}{ k_{1,1}} & -\frac{k_{2,1}}{k_{1,1}}& 1\\ 0 &k_{3,2}-\frac{k_{1,2}k_{3,1}}{ k_{1,1}} &\dots & k_{3,\Delta(\overline{G})+1}- \frac{k_{1,\Delta(\overline{G})+1}k_{3,1}}{ k_{1,1}} & 0&0 \end{array} \right). $ Let $\alpha =k_{2,2}-\frac{k_{1,2}k_{2,1}}{ k_{1,1}}$. Since $k_{2,2}\not \in \Q[k_{1,1},k_{1,2},k_{2,1}]$ by construction, $\alpha\ne0$ and we can continue reducing the matrix to echelon form. Then multiplying the second row by $\frac{1}{\alpha}$ we obtain $ \left( \begin{array}{ccccccc} 1 &\frac{k_{1,2}}{ k_{1,1}}&\frac{k_{1,3}}{ k_{1,1}} &\dots & \frac{k_{1,\Delta(\overline{G})+1}}{ k_{1,1}} & \frac{1}{ k_{1,1}} & 0 \\ 0 &1 &\frac{1}{\alpha}(k_{2,3}- \frac{k_{1,3}k_{2,1}}{ k_{1,1}})&\dots &\frac{1}{\alpha}( k_{2,\Delta(\overline{G})+1}- \frac{k_{1,\Delta(\overline{G})+1}k_{2,1}}{ k_{1,1}} )& -\frac{k_{2,1}}{\alpha k_{1,1}}&\frac{1}{\alpha}\\ 0 &k_{3,2}-\frac{k_{1,2}k_{3,1}}{ k_{1,1}} & k_{3,3}- \frac{k_{1,3}k_{3,1}}{ k_{1,1}} &\dots & k_{3,\Delta(\overline{G})+1}- \frac{k_{1,\Delta(\overline{G})+1}k_{3,1}}{ k_{1,1}} & 0&0 \end{array} \right). $ Let $\beta=k_{3,2}-\frac{k_{1,2}k_{3,1}}{ k_{1,1}}$. Multiplying the second row by $-\beta$ and adding the result to the third row we get $ \left( \begin{array}{ccccccc} 1 &\frac{k_{1,2}}{ k_{1,1}}&\frac{k_{1,3}}{ k_{1,1}} &\dots & \frac{k_{1,\Delta(\overline{G})+1}}{ k_{1,1}} & \frac{1}{ k_{1,1}} & 0 \\ 0 &1 &\frac{1}{\alpha}(k_{2,3}- \frac{k_{1,3}k_{2,1}}{ k_{1,1}})&\dots &\frac{1}{\alpha}( k_{2,\Delta(\overline{G})+1}- \frac{k_{1,\Delta(\overline{G})+1}k_{2,1}}{ k_{1,1}} )& -\frac{k_{2,1}}{\alpha k_{1,1}}&\frac{1}{\alpha}\\ 0 &0 & \rho_{3,3}&\dots &\rho_{3,\Delta(\overline{G})+1} & \frac{-k_{3,1}}{k_{1,1}}+\frac{k_{2,1}\beta}{\alpha k_{1,1}}&\frac{-\beta}{\alpha} \end{array} \right) $ where $\rho_{3,3}=k_{3,3}- \frac{k_{1,3}k_{3,1}}{ k_{1,1}}-\frac{\beta}{\alpha}(k_{2,3}- \frac{k_{1,3}k_{2,1}}{ k_{1,1}}) ,\dots,\rho_{3,\Delta(\overline{G})+1}=k_{3,\Delta(\overline{G})+1}- \frac{k_{1,\Delta(\overline{G})+1}k_{3,1}}{ k_{1,1}}-\frac{\beta}{\alpha}( k_{2,\Delta(\overline{G})+1}- \frac{k_{1,\Delta(\overline{G})+1}k_{2,1}}{ k_{1,1}} )$. Note that $\rho_{3,3}\ne 0$ otherwise $k_{3,3}$ belongs to a field extension of the lattice $ L[\Q[k_{1,1},\dots,k_{1,\Delta(\overline{G})+1},k_{2,1},\dots,k_{2,\Delta(\overline{G})+1},k_{3,1},k_{3,2}]] $ which is a contradiction. Thus, multiplying the third row by $\frac{1}{\rho_{3,3}}$ we obtain the echelon form of the homogeneous system $$\left( \begin{array}{cccccccc} 1 &\frac{k_{1,2}}{ k_{1,1}}&\frac{k_{1,3}}{ k_{1,1}} &\frac{k_{1,4}}{k_{1,1}}&\dots & \frac{k_{1,\Delta(\overline{G})+1}}{ k_{1,1}} & \frac{1}{ k_{1,1}} & 0 \\ 0 &1 &\frac{1}{\alpha}(k_{2,3}- \frac{k_{1,3}k_{2,1}}{ k_{1,1}})&\frac{1}{\alpha}(k_{2,4}-\frac{k_{1,4}k_{2,1}}{k_{1,1}})&\dots &\frac{1}{\alpha}( k_{2,\Delta(\overline{G})+1}- \frac{k_{1,\Delta(\overline{G})+1}k_{2,1}}{ k_{1,1}} )& -\frac{k_{2,1}}{\alpha k_{1,1}}&\frac{1}{\alpha}\\ 0 &0 & 1&\frac{\rho_{3,4}}{\rho_{3,3}}&\dots &\frac{\rho_{3,\Delta(\overline{G})+1}}{\rho_{3,3}} &\zeta&\frac{-\beta}{\rho_{3,3}\alpha} \end{array} \right)$$ where all the values $\rho_{3,j}\ne0 , j=4,\dots,\Delta(\overline{G})+1$ as well as $\zeta=\frac{-k_{3,1}}{\rho_{3,3}k_{1,1}}+\frac{k_{2,1}\beta}{\rho_{3,3}\alpha k_{1,1}}$ and $\frac{-\beta}{\rho_{3,3}\alpha}$ are nonzero. Then we can choose values $k_{4,4},\dots,k_{4,\Delta(\overline{G})+1}, g_{4,1}, g_{4,2}$ nonzero and chosen one by one from different field extensions not in the lattice $$L[\Q[k_{1,1}],\dots, \Q[k_{1,\Delta(\overline{G})+1}], \Q[k_{2,1}],\dots,\Q[k_{2,\Delta(\overline{G})+1}],\Q[k_{3,1}],\\ \dots, \Q[k_{3,\Delta(\overline{G})+1}]$$ in the following way. 1. $k_{4,4}\not \in \Q[k_{i,j}],i=1,2,3, j=1,\dots,\Delta(\overline{G})+1$, 2. $k_{4,3}\not \in \Q[k_{i,j},k_{4,4}],i=1,2,3, j=1,\dots,\Delta(\overline{G})+1$, 3. $\vdots$ 4. $k_{4,\Delta(\overline{G})+1}\not \in \Q[k_{i,j},k_{4,4},\dots,k_{4,\Delta(\overline{G})}],i=1,2,3, j=1,\dots,\Delta(\overline{G})+1$, 5. $g_{4,1}\not \in \Q[k_{i,j},k_{4,4},\dots,k_{4,\Delta(\overline{G})+1}],i=1,2,3, j=1,\dots,\Delta(\overline{G})+1$, 6. $g_{4,2}\not \in \Q[k_{i,j},k_{4,4},\dots,k_{4,\Delta(\overline{G})+1}, g_{4,1}],i=1,2,3, j=1,\dots,\Delta(\overline{G})+1$, Therefore $k_{4,3},k_{4,2}$, and $k_{4,1}$ become $$\begin{aligned} k_{4,3} &=&-\frac{\rho_{3,4}k_{4,4}}{\rho_{3,3}} +\dots- \frac{\rho_{3, \Delta(\overline{G})+1}k_{4, \Delta(\overline{G})+1}}{\rho_{3,3}}+ \frac{k_{3,1}g_{4,1}}{\rho_{3, 3}k_{1,1}}-\frac{k_{2,1}\beta g_{4,1}}{\rho_{3,3}\alpha k_{1,1}} +\frac{\beta g_{4,2}}{\rho_{3,3}\alpha}.\\ k_{4,2} &=&-(\frac{1}{\alpha}(k_{2,3}- \frac{k_{1,3}k_{2,1}}{ k_{1,1}}))k_{4,3}-\dots-(\frac{1}{\alpha}( k_{2,\Delta(\overline{G})+1}- \frac{k_{1,\Delta(\overline{G})+1}k_{2,1}}{ k_{1,1}} ))k_{4,\Delta(\overline{G})+1}\\ && +\frac{k_{2,1}g_{4,1}}{\alpha k_{1,1}}-\frac{g_{4,2}}{\alpha}\\ k_{4,1} &=&-\frac{k_{1,2} k_{4,2}}{k_{1,1}}-\dots- \frac{k_{1, \Delta(\overline{G})+1}k_{4, \Delta(\overline{G})+1}}{k_{1,1}}- \frac{g_{4,1}}{k_{1,1}} \label{k41}\end{aligned}$$ Note that $k_{4,1}$ depends on $k_{4,2}$ and $k_{4,3}$. Similarly $k_{4,2}$ depends on $k_{4,3}$. Therefore we should choose $k_{4,3}$ first and then back substitute. But $k_{4,3}$ depends on $g_{4,1}$ and $g_{4,2}$ which are free variables with the restriction that they cannot be zero. Thus, we can choose $g_{4,1}$ in a field extension not in the lattice of $$Q[k_{1,1},\dots,k_{1,\Delta(\overline{G})+1},k_{2,1},\dots,k_{2,\Delta(\overline{G})+1}, k_{3,1},\dots, k_{3,\Delta(\overline{G})+1}, k_{4,4}, \dots,k_{4,\Delta(\overline{G})+1}]$$ and $g_{4,2}$ in some field extension not in the lattice of $$Q[k_{1,1},\dots,k_{1,\Delta(\overline{G})+1},k_{2,1},\dots,k_{2,\Delta(\overline{G})+1}, k_{3,1},\dots, k_{3,\Delta(\overline{G})+1},\\ k_{4,4},\dots, k_{4,\Delta(\overline{G})+1},g_{4,1}]$$ in such a way that $k_{4,3},k_{4,2}$ and $k_{4,1}$ are nonzero. As a consequence, the system $S$ is consistent and there exist a solution of values $k_{4, j}\ne 0 , \ j=1,2,\dots, \Delta(\overline{G})+1$ chosen from different field extensions such that the vector $\overrightarrow{v_4}$ satisfies all adjacency conditions and orthogonal conditions with the vectors $\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}$. Then the vector $\overrightarrow{v_4}=\sum_{j=1}^{\Delta(\overline{G})+1} k_{4,j} \overrightarrow{e_j}$ satisfies conditions (1) and (2) and therefore $\{\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3},\overrightarrow{v_4}\}$ is an orthogonal representation of $Y_4$ in $\Re^{\Delta(\overline{G})+1}$.\ [*Case 3 .   $v_4\sim v_1,v_4\not \sim v_2,v_4 \sim v_3$ in $G$*]{}.\ From the adjacency conditions $v_4\sim v_1,v_4 \sim v_3$ and orthogonal condition $v_4\not \sim v_2$ we get the equations: $$\begin{aligned} \langle\overrightarrow{v_1},\overrightarrow{v_4}\rangle &=& g_{4,1}, g_{4,1}\ne 0 \\ \langle\overrightarrow{v_2},\overrightarrow{v_4}\rangle &=& 0\\ \langle\overrightarrow{v_3},\overrightarrow{v_4}\rangle&=& g_{4,2}, g_{4,2}\ne 0\end{aligned}$$ Interchanging the second and third equations we get a system $S$ similar to case 2. Since all the scalars $k_{i,j}, i= 1,2,3 , j=1,2,\dots,\Delta(\overline{G})+1 $ are not zero and were chosen from different field extensions the same reasoning as in case 2 applies and the conclusion holds for case 3.\ [*Case 4.   $v_4\not \sim v_1,v_4\sim v_2,v_4 \sim v_3$ in $G$*]{}. From the adjacency conditions $v_4\sim v_2,v_4 \sim v_3$ and orthogonal condition $v_4\not \sim v_1$ we get the equations: $$\begin{aligned} \langle\overrightarrow{v_1},\overrightarrow{v_4}\rangle &=& 0 \\ \langle\overrightarrow{v_2},\overrightarrow{v_4}\rangle &=& g_{4,2}, g_{4,2}\ne 0\\ \langle\overrightarrow{v_3},\overrightarrow{v_4}\rangle&=&g_{4,3}, g_{4,3}\ne 0.\end{aligned}$$ Interchanging the first and the third equations we get a system $S$ similar to case 2. Since all the scalars $k_{i,j}, i= 1,2,3 , j=1,2,\dots,\Delta(\overline{G})+1 $ are not zero and were chosen from different field extensions the same reasoning as in case 2 applies and the conclusion holds for case 4. As a consequence, in all of the cases we get an orthogonal representation for $Y_4$ in $\Re^{\Delta(\overline{G})+1}$ satisfying the conditions (1) and (2). This completes of the proof of the claim \[claimMainT\]. Assume that for any $Y_{m-1}=(V_{Y_{m-1}},E_{Y_{m-1}}), V_{Y_{m-1}}= \{v_1,v_2,\dots,v_{m-1}\}$ it is possible to get an orthogonal representation of $Y_{m-1}$ in $\Re^{\Delta(\overline{G})+1}$. Let $\overrightarrow{v}_1,\overrightarrow{v}_2,\dots,\overrightarrow{v}_{m-1}$ be of the form $$\overrightarrow{v}_i=\dis{\sum_{j=1}^{\Delta(\overline{G})+1}k_{i,j}\overrightarrow{e}_j}$$ satisfying conditions (1) and (2) where $k_{i,j}\ne0 $ for all $i=1,2,\dots,m , j=1,2,\dots,\Delta(\overline{G})+1$ , chosen from different field extensions. We need to prove that if $v_{m}$ is adjoined to $Y_{m-1}$ to get $Y_m$ such that $v_{m}$ is adjacent to all prior vertices except at most $\left\lfloor\frac{m}{2}-1\right\rfloor$ vertices then $Y_{m}$ has an orthogonal representation of vectors $ \overrightarrow{v}_1,\overrightarrow{v}_2,\dots,\overrightarrow{v}_{m}$ in $\Re^{\Delta(G)+1}$ satisfying conditions (1) and (2). Assume that $v_{m}$ has an associated vector $\overrightarrow{v}_m$ such that $$\overrightarrow{v}_{m}= k_1^{m}\overrightarrow{e_1}+ k_2^{m}\overrightarrow{e_2}+\dots+ k_{\Delta(\overline{G})+1}^{m}\overrightarrow{e}_{\Delta(\overline{G})+1}.$$ The vertex $v_{m}$ is adjacent to all prior vertices $ v_1,v_2,\dots,v_{m-1}$ except at most $t\le\left\lfloor\frac{m}{2}-1\right\rfloor$ vertices in $G$. Then we see that $\overrightarrow{v}_{m}$ satisfies at least $m-1-t$ adjacency conditions and $t$ orthogonal conditions. Let $\rho$ be a permutation of $(1,2,\dots,m-1)$. Suppose $v_{\rho(1)},v_{\rho(2)},\dots,v_{\rho(m-1-t)}$ are adjacent to $v_m$ and $v_{\rho(m-t)}, v_{\rho(m-t+1)}, \dots,v_{\rho(m-2)},v_{\rho(m-1)}$ are not adjacent to $v_{m}$. The vectors $\overrightarrow{v}_{\rho(1)},\overrightarrow{v}_{\rho(2)},\dots,\\ \overrightarrow{v}_{\rho(m-1-t)}, \overrightarrow{v}_{\rho(m-t)},\overrightarrow{v}_{\rho(m-t+1)},\dots,\overrightarrow{v}_{\rho(m-1)}$ and $\overrightarrow{v}_{m}$ satisfy the system $S$ given by: $$\begin{aligned} \langle\overrightarrow{v}_{\rho(1)},\overrightarrow{v}_{m}\rangle &=&g_{m,1},\ \ g_{m,1}\ne 0\\ \langle\overrightarrow{v}_{\rho(2)},\overrightarrow{v}_{m}\rangle &=&g_{m,2},\ \ g_{m,2}\ne 0\\ \vdots&\vdots&\vdots \\ \langle\overrightarrow{v}_{\rho(m-1-t)},\overrightarrow{v}_{m}\rangle &=&g_{m,m-1-t},\ \ g_{m,m-1-t}\ne 0\\ \langle\overrightarrow{v}_{\rho(m-t)},\overrightarrow{v}_{m}\rangle&=& 0\\ \langle\overrightarrow{v}_{\rho(m-t+1)},\overrightarrow{v}_{m}\rangle&=& 0 \\ \vdots &\vdots& \vdots\\ \langle\overrightarrow{v}_{\rho(m-1)},\overrightarrow{v}_{m}\rangle &=&0\end{aligned}$$ containing $m-1-t$ equations from the adjacency conditions and $t$ equations from the orthogonal conditions. Since the vector $\overrightarrow{v}_{\rho(i)}, i=1,2,\dots,m-1$ has the form $$\label{vectorm} \overrightarrow{v}_{\rho(i)}= k_{\rho(i),1}\overrightarrow{e}_1+k_{\rho(i),2}\overrightarrow{e}_2+\dots+k_{\rho(i),\Delta(\overline{G})+1}\overrightarrow{e}_{\Delta(\overline{G})+1}$$ where all $k_{\rho(i),j}, i=1,2,\dots,m-1 , j= 1,2,\dots,\Delta(\overline{G})+1$ are not zero and chosen from different field extensions, the system $S$ has the form: $$\begin{aligned} k_{\rho(1),1}k_{m,1}+ k_{\rho(1),2}k_{m,2}+\dots+ k_{\rho(1),\Delta(\overline{G})+1}k_{m,\Delta(\overline{G})+1} &=& g_{m,1} \\ k_{\rho(2),1}k_{m,1}+ k_{\rho(2),2}k_{m,2}+\dots+ k_{\rho(2),\Delta(\overline{G})+1}k_{m,\Delta(\overline{G})+1} &=&g_{m,2} \\ \vdots &\vdots& \vdots \\ k_{\rho(m-1-t),1}k_{m,1}+ k_{\rho(m-1-t),2}k_{m,2}+\dots+ k_{\rho(m-1-t),\Delta(\overline{G})+1}k_{m,\Delta(\overline{G})+1} &=&g_{m,m-1-t} \\ k_{\rho(m-t),1}k_{m,1}+ k_{\rho(m-t),2}k_{m,2}+\dots+ k_{\rho(m-t),\Delta(\overline{G})+1}k_{m,\Delta(\overline{G})+1}&=& 0\\ k_{\rho(m-t+1),1}k_{m,1}+ k_{\rho(m-t+1),2}k_{m,2}+\dots+ k_{\rho(m-t+1),\Delta(\overline{G})+1}k_{m,\Delta(\overline{G})+1} &=& 0\\ \vdots &\vdots& \vdots \\ k_{\rho(m-1),1}k_{m,1}+ k_{\rho(m-1),2}k_{m,2}+\dots+ k_{\rho(m-1),\Delta(\overline{G})+1}k_{m,\Delta(\overline{G})+1}&=&0\end{aligned}$$ where $g_{m,1}\ne0,g_{m,2}\ne0, \dots,g_{m,m-1-t}\ne0$. Since $g_{m,1},g_{m,2}, \dots,g_{m,m-1-t}$ could be any nonzero real numbers satisfying the adjacency conditions we can consider them as additional $m-1-t$ variables under the restriction that they cannot be zero. Therefore we can consider a homogeneous system $S_H$ of $m-1$ equations in $m-t+\Delta(\overline{G})$ variables $k_1^{m},k_2^{m}, \dots,k_{\Delta(\overline{G})+1}^{m}, -g_{m,1},-g_{m,2},\\ \dots,-g_{m,m-1-t}$. Now, by hypothesis $t\le \left\lfloor\frac{m}{2}-1\right\rfloor$. Since $t\le d_{\overline{G}}(v_{m})\le \Delta(\overline{G})<\Delta(\overline{G})+1$ the homogeneous system $S_H$ contains at least one more variable than the number of equations. Hence the system $S_H$ given by $$\begin{aligned} k_{\rho(1),1}k_{m,1}+ k_{\rho(1),2}k_{m,2}+\dots+ k_{\rho(1),\Delta(\overline{G})+1}k_{m,\Delta(\overline{G})+1}+(-g_{m,1}) &=&0 \\ k_{\rho(2),1}k_{m,1}+ k_{\rho(2),2}k_{m,2}+\dots+ k_{\rho(2),\Delta(\overline{G})+1}k_{m,\Delta(\overline{G})+1}+(-g_{m,2})&=&0 \\ \vdots &\vdots& \vdots\\ k_{\rho(m-1-t),1}k_{m,1}+ k_{\rho(m-1-t),2}k_{m,2}+\dots+ k_{\rho(m-1-t),\Delta(\overline{G})+1}k_{m,\Delta(\overline{G})+1}+(- g_{m,m-1-t})&=&0 \\ k_{\rho(m-t+),1}k_{m,1}+ k_{\rho(m-t),2}k_{m,2}+\dots+ k_{\rho(m-t),\Delta(\overline{G})+1}k_{m,\Delta(\overline{G})+1}&=& 0 \\ k_{\rho(m-t+1),1}k_{m,1}+ k_{\rho(m-t+1),2}k_{m,2}+\dots+ k_{\rho(m-t+1),\Delta(\overline{G})+1}k_{m,\Delta(\overline{G})+1} &=& 0\\ \vdots &\vdots& \vdots \\ k_{\rho(m-1),1}k_{m,1}+ k_{\rho(m-1),2}k_{m,2}+\dots+ k_{\rho(m-1),\Delta(\overline{G})+1}k_{m,\Delta(\overline{G})+1}&=&0\end{aligned}$$ has infinitely many solutions. Therefore, it is enough to show that there exist at least one solution for $S_H$ satisfying the condition that none of the $k_{m,1},k_{m,2},\dots,k_{m,\Delta(\overline{G})+1},g_{m,1},\\ \dots,g_{m,m-1-t}$ are zero. This implies that the system $S$ has a solution which satisfies all adjacency conditions, all orthogonal conditions, and conditions (1) and (2). For that purpose consider the $(m-1)\times (m-t+\Delta(\overline{G}))$ matrix $A$ of the homogeneous system given on the next page. Let $\overrightarrow{g}=(-g_{m,1},-g_{m,2},\dots,-g_{m,m-1-t})^T$. We consider the two cases where $m-1\le \Delta(\overline{G})+1$ and $m-1> \Delta(\overline{G})+1 $. [*Case 1.*]{}   $m-1\le \Delta(\overline{G})+1$ In this case the number of equations in the non-homogeneous system $S$ is at most the number of unknowns $k_{m,1},k_{m,2},\dots,k_{m,\Delta(\overline{G})+1}$. -0.5cm $$A=\left(\begin{array}{ccccccccccc} k_{\rho(1),1}& k_{\rho(1),2} & k_{\rho(1),3} &\dots & k_{\rho(1),\Delta(\overline{G})+1} &|& 1 & 0 & \dots& 0 &0 \\ k_{\rho(2),1} &k_{\rho(2),2} &k_{\rho(2),3} & \dots & k_{\rho(2),\Delta(\overline{G})+1} &|&0 &1 & \dots&0& 0 \\ k_{\rho(3),1} & k_{\rho(3),2} & k_{\rho(3),3} & \dots &k_{\rho(3),\Delta(\overline{G})+1} &|&0 & 0 & 1 &\vdots &\vdots \\ \vdots& \vdots & \vdots & \vdots &\vdots & \vdots & \vdots &\vdots&\vdots &\vdots& \vdots \\ k_{\rho(m-t-2),1} & k_{\rho(m-t-2),2} & k_{\rho(m-t-2),3} & \dots& k_{\rho(m-t-2),\Delta(\overline{G})+1} &|& 0 & 0 &0 & 1 &0 \\ k_{\rho(m-t-1),1} & k_{\rho(m-t-1),2}& k_{\rho(m-t-1),3} & \dots& k_{\rho(m-t-1),\Delta(\overline{G})+1}&|&0 & 0 &0 &0 & 1 \\ k_{\rho(m-t),1} & k_{\rho(m-t),2}& k_{\rho(m-t),3} & \dots& k_{\rho(m-t),\Delta(\overline{G})+1} &|&0 & 0 &0 &0 & 0 \\ \vdots & \vdots &\vdots& \vdots & \vdots & \vdots & \vdots & \vdots & \vdots &\vdots& \vdots \\ k_{\rho(m-1),1} & k_{\rho(m-1),2} &k_{\rho(m-1),3} & \dots & k_{\rho(m-1),\Delta(\overline{G})+1}&| &0 & 0 & 0 & 0 & 0 \end{array}\right)$$ Thus $A$ can be row reduced to one of the following two echelon form written in block form: 1. $$B= \left(B_1|B_2\right)={\tiny \left(\begin{array}{cccccccccccccccccc} 1&\ast & \ast &\dots &\ast&\ast&\ast&\dots&\ast&\ast&\dots&\ast&| & \delta_1 & 0 & \dots& 0 &0 \\ 0 &1 &\ast & \dots &\ast&\ast&\ast&\dots&\ast&\ast&\dots& \ast&| &\ast &\delta_2 & \dots&0& 0 \\ 0 & 0 & 1 & \dots&\ast&\ast &\ast&\dots&\ast&\ast&\dots& \ast&| &\ast&\ast& \delta_3 &0&0 \\ \vdots& \vdots & \vdots &\dots& \vdots &\vdots& \vdots &\dots&\vdots&\vdots&\dots& \vdots&|&\vdots&\vdots&\vdots &\vdots& \vdots \\ 0 & 0 &0 & \dots&1&\ast&\ast&\dots&\ast&\ast&\dots&\ast&| &\ast &\ast &\dots& \delta_{m-2-t} &0 \\ 0 &0& 0&\dots &0&1&\ast&\dots&\ast&\ast&\dots&\ast&| &\ast & \ast &\dots &\ast& \delta_{m-1-t} \\ 0 &0& 0&\dots &0&0&1&\dots&\ast&\ast&\dots&\ast&| &\ast & \ast &\dots &\ast& \ast \\ \vdots& \vdots & \vdots &\dots& \vdots &\vdots& \vdots &\dots&\vdots&\vdots&\dots& \vdots&|&\vdots&\vdots&\vdots &\vdots& \vdots \\ 0 &0& 0&\dots &0&0&0&\dots&1&\ast&\dots&\ast&| &\ast & \ast &\dots &\ast& \ast \\ \end{array}\right)}$$ 2. $$B=\left(B_3|B_2\right)={\tiny \left(\begin{array}{cccccccccccc} 1&\ast & \ast &\dots &\ast&\ast&| & \delta_1 & 0 & \dots& 0 &0 \\ 0 &1 &\ast & \dots &\ast&\ast&| &\ast &\delta_2 & \dots&0& 0 \\ 0 & 0 & 1 & \dots&\ast&\ast &| &\ast&\ast& \delta_3 &0&0 \\ \vdots& \vdots & \vdots &\dots& \vdots &\vdots&|&\vdots&\vdots&\vdots &\vdots& \vdots \\ 0 & 0 &0 & \dots&1&\ast&| &\ast &\ast &\dots& \delta_{m-t-1} &0 \\ 0 &0& 0&\dots &0&1&| &\ast & \ast &\dots &\ast& \delta_{m-t} \\ \end{array}\right)}.$$ The matrix $B_1$ is a block of size $(m-1)\times (\Delta(\overline{G})+1)$ where $m-1< \Delta(\overline{G})+1$. The matrix $B_2$ is a block matrix of size $(m-1)\times (m-1-t)$. Matrix $B_3$ is a square matrix of size $m-1 (=\Delta(\overline{G})+1)$. In these blocks $\ast$ denotes a nonzero entry. Suppose matrix $B$ is of type I. For each vector $\overrightarrow{v}_i, i=4,5,\dots,m-1$ the entries are found in field extensions which are not in the lattice of the previous field extensions. In the block matrix $B_2$ we have: - $[B_2]_{1,1}=\delta_1= \frac{1}{k_{\rho(1),1}}\ne 0$, - $[B_2]_{2,2}=\delta_2= \left(k_{\rho(2),2}-\frac{k_{\rho(1),2}\cdot k_{\rho(2),1}}{k_{\rho(1),1}}\right)^{-1}\ne 0$. - $[B_2]_{3,3}=\delta_3=\frac{1}{\alpha}\ne 0, \alpha\in Q[k_{\rho(1),1},k_{\rho(1),2},k_{\rho(1),3},k_{\rho(2),1},k_{\rho(2),2},k_{\rho(2),3},k_{\rho(3),1},k_{\rho(3),2},k_{\rho(3),3}]$. Continuing this process we get that all the entries on the diagonal of $B_2$ to be nonzero. So all the rows of $B_2$ have at least one entry nonzero.Thus, - $\left[B_2\overrightarrow{g}\right]_1=-\delta_1\cdot g_{m,1}= \frac{-g_{m,1}}{k_{1,1}}$. Choosing $g_{m,1}$ not in the lattice generated by the previous field extensions for $k_{i,j}, i=1,2,\dots,m-1 , j=1,2,\dots,\Delta(\overline{G})+1$, we get $\left[B_2\overrightarrow{g}\right]_1\ne 0$. - $\left[B_2\overrightarrow{g}\right]_2= \alpha_{2,1}g_{m,1}+\delta_2 g_{m,2}, \alpha_{2,1}\ne 0, \delta_2\ne 0$. Then we can choose a value for $g_{m,2}$ from a field extension not in the lattice of fields generated by the previous values $k_{i,j}, i=1,2,\dots,m-1, j=1,2,\dots, \Delta(G)+1,g_{m,1}$ so that $\left[B_2\overrightarrow{g}\right]_2\ne 0$. - $\left[B_2\overrightarrow{g}\right]_3= \alpha_{3,1}g_{m,1}+\alpha_{3,2}g_{m,2}+\delta_3 g_{m,3}, \alpha_{3,1}\ne 0, \delta_3\ne 0$. As above we can choose $g_{m,3}\ne 0$ and such that $\left[B_2\overrightarrow{g}\right]_3\ne 0$ by taking $g_{m,3}$ neither in $Q[\frac{\alpha_{3,1}g_{m,1}+\alpha_{3,2}g_{m,2}}{-\delta_3}]$ nor in any of the previous field extensions. Continuing this process and applying similar choices we see that $g_{m,4},g_{m,5}\dots,g_{m,m-1}$ can be chosen nonzero. Moreover, matrix $B_1$ shows that there is at least one free variable for the solution of $k_{m,j}, j=1,2,\dots,\Delta(\overline{G})+1$. All of these free variables can be chosen in different field extensions such that all other unknowns are not zero. Otherwise it is possible to show that the last choice belongs to the field containing all the previous chosen values which is a contradiction. Now, suppose that $k_{m,1},k_{m,2},\\ \dots, k_{m,r}, r<\Delta(\overline{G})+1$ can be written in terms of $k_{m,r+1},\dots,k_{m,\Delta(\overline{G}+1)}$ as $$\label{formofks} k_{m,i}= \alpha_{r+1}k_{m,r+1}+\dots+\alpha_{\Delta(\overline{G})+1}k_{m,\Delta(\overline{G})+1}+ \varphi_i(g_{m,1},g_{m,2},\dots, g_{m,m-1}) , i=1,2,\dots,r$$ where $ \alpha_{r+1},\dots,\alpha_{\Delta(\overline{G})+1}$ are all nonzero and $\varphi_i$ is a linear combination of $g_{m,1},\dots,g_{m,m-1}$. Also $\varphi_i(g_{m,1},g_{m,2},\dots, g_{m,m-1})\ne 0 ,i=1,2,\dots,r $. Thus,by choosing values\ $k_{m,r+1},\dots,k_{m,\Delta(\overline{G})+1}$ in different field extensions and substituting them in \[formofks\], we obtain that $k_{m,i}\ne 0, i=1,2,\dots,r$. As a consequence, the vector $\overrightarrow{v}_m$ exists and all of its entries are nonzero. If matrix $B$ is of type II we apply same process as in case of type I. Again, we can obtain the vector $v_m$ having all its entries nonzero and $g_{m,i}\ne 0$ for $i=1,\dots,m-1$. [*Case 2.*]{}  $m-1 > \Delta(\overline{G})+1$ In this case the number of equations in the non-homogeneous system $S$ is more than the number of unknowns $k_{m,1},k_{m,2},\dots,k_{m,\Delta(\overline{G})+1}$. We need to analyze three possible subcases where $m-1-t< \Delta(\overline{G})+1, m-1-t=\Delta(\overline{G})+1$ or $m-1-t>\delta(\overline{G})+1$. 1. If $m-1-t< \Delta(\overline{G})+1$ then matrix $B$ has the form $$B= {\tiny \left(\begin{array}{cccccccccccccccccc} 1&\ast & \ast &\dots &\ast&|&\ast&\dots&\ast&| &\ast &0&0&0& 0 & \dots& 0 &0 \\ 0 &1 &\ast & \dots &\ast&|&\ast&\dots&\ast&| &\ast &\ast& 0&0& 0& \dots&0& 0 \\ 0 & 0 & 1 & \dots&\ast&|&\ast&\dots&\ast&| &\ast&\ast& \ast &0&0&\dots&0&0 \\ \vdots& \vdots & \vdots&\dots& \vdots &|&\vdots&\dots&\vdots&|& \vdots &\dots& \vdots&\vdots&\vdots&\vdots &\vdots& \vdots \\ 0 & 0 &0 & \dots&1&|&\ast&\dots&\ast&|&\ast&\dots&\ast&\ast&\ast &\dots& \ast &\ast \\ -&-&-&-&-&-&-&\dots&-&|&-&-&-&-&-&\dots&-&- \\ 0 & 0 &0 & \dots&0&|&1&\dots&\ast&|&\ast&\dots&\ast &\ast &\ast &\dots& \ast &\ast \\ \vdots & \vdots &\vdots&\vdots&\vdots&|& \vdots&\dots & \vdots &|&\vdots& \dots&\vdots & \vdots & \vdots & \vdots &\vdots& \vdots \\ 0 & 0 &0 & \dots&0&|&0&\dots&\ast&|&\ast&\dots&\ast &\ast &\ast &\dots& \ast &\ast \\ 0 &0& 0&\dots &0&|&0&\dots&1&|&\ast&\dots&\ast &\ast & \ast &\dots &\ast& \ast \\ 0 & 0&0 & \dots&0&|&0&\dots& 0&|&\ast&\dots&\ast &\ast & \ast &\dots &\ast & \ast \\ \vdots & \vdots &\vdots&\vdots&\vdots&|& \vdots&\dots & \vdots &|&\vdots& \dots&\vdots & \vdots & \vdots & \vdots &\vdots& \vdots \\ 0 & 0 &0 & \dots &0&|&0&\dots&0&|&\ast&\dots&\ast&\ast &\ast & \dots& \ast & \ast \end{array}\right)}$$ $$B=\left( \begin{array}{ccc} B_1 & B_2 & B_3 \\ B_4& B_5& B_6 \\ \end{array} \right)$$ The matrix $B_1$ is a square matrix $(m-1-t)\times (m-1-t)$, matrix $B_2$ has size $(m-1-t)\times (\Delta(\overline{G})+ 2 + t-m)$, matrix $B_3$ is a square matrix of size $m-1-t$. Matrix $B_4$ is a zero matrix of size $t\times m-1-t$. The central blocks $B_2, B_5$ form a block of size $(m-1)\times (\Delta(\overline{G})+2+ t-m)$ and corresponds to the columns of free variables $k_{m,m-t}, \dots, k_{m,\Delta(\overline{G})+1}$ of the system $S$. The block $B_6$ has size $t\times (m-1-t)$ . Consider the block matrix $\left(B_5\ B_6\right)$ of size $t\times(\Delta(\overline{G})+1)$. Recalling that the value $t$ is the number of orthogonal conditions for $v_m$ in $G$ which is equivalent to $\d_{\overline{G}}(v_m)$ we get $t\le\Delta(\overline{G})<\Delta(\overline{G})+1$. As a consequence, the homogeneous system $\left(B_5\ B_6\right)\overrightarrow{w}=0$ where $\overrightarrow{w}$ is a vector of size $(\Delta(\overline{G})+1)\times 1$ in the variables $k_{m,m-t},\dots, k_{m,\Delta(\overline{G})+1},(-g_{m,1}),\dots,\\ (-g_{m,m-1-t})$, has infinitely many solutions depending on at least one free variable. Choosing these free variables in different field extensions as we did previously, we get nonzero values for $k_{m,m-t},\\ \dots,k_{m,\Delta(\overline{G})+1},g_{m,1},\dots,g_{m,m-1-t}$. We get the values of the remaining unknowns $k_{m,i}, i=1,2,\dots,m-1-t$ of the system $S$ applying back substitution. Since all the entries with $\ast$ in the block $B_1$ are nonzero and belong to different field extensions, the values $k_{m,i}, i=1,2,\dots,m-1-t$ are also nonzero. As a consequence, the non-homogeneous system $S$ is consistent and the vector $v_m$ with no zero entries exists. 2. If $m-1-t= \Delta(\overline{G})+1$ then the matrix $B$ has the form $$\begin{aligned} B&=& \left( \begin{array}{ccc} B_1&| & B_2 \\ -&&-\\ 0 &|& R \\ \end{array} \right)\\ &=&{\tiny \left(\begin{array}{cccccccccccccccc} 1&\ast & \ast &\dots &\ast&\dots&\ast&| &\delta_1 &0&0&0& 0 & \dots& 0 &0 \\ 0 &1 &\ast & \dots &\ast&\dots&\ast&| &\ast &\delta_2& 0&0& 0& \dots&0& 0 \\ 0 & 0 & 1 & \dots&\ast&\dots&\ast&| &\ast&\ast&\delta_3 &0&0&\dots&0&0 \\ \vdots& \vdots & \vdots&\dots& \vdots &\dots&\vdots&|& \vdots &\dots& \vdots&\vdots&\vdots&\vdots &\vdots& \vdots \\ 0 & 0 &0 & \dots&1&\dots&\ast&|&\ast&\dots&\ast&\ast&\ast&\dots& 0 &0 \\ 0 & 0 &0 & \dots&0&\dots&\ast&|&\ast&\dots&\ast &\ast &\ast &\dots& \delta_{m-t-2} &0 \\ 0 &0& 0&\dots &0&\dots&1&|&\ast&\dots&\ast &\ast & \ast &\dots &\ast& \delta_{m-1-t} \\ -&-&-&-&-&\dots&-&|&-&-&-&-&-&\dots&-&- \\ 0 & 0&0 & \dots&0&\dots& 0&|&1&\dots&\ast &\ast & \ast &\dots &\ast & \ast \\ \vdots & \vdots &\vdots&\vdots& \vdots&\dots & \vdots &\vdots&\vdots& \dots&\vdots & \vdots & \vdots & \vdots &\vdots& \vdots \\ 0 & 0 &0 & \dots &0&\dots&0&|&0&\dots&1&\ast &\ast & \dots& \ast & \ast \end{array}\right)}.\end{aligned}$$ In this case the matrices $B_1$ and $B_2$ are square matrices of size $m-1-t (=\Delta(\overline{G})+1)$. The matrix $R$ has size $t\times m-1-t$. Since $t < \left \lfloor\frac{m}{2}-1\right \rfloor$ we get that $2t<m-2< m-1$ which implies that $t<m-1-t$. Therefore the system $R\overrightarrow{g}=0$ has infinitely many solutions with $m-1-2t$ free variables. Taking the free variables from different field extensions we get all the values $g_{m,1}\dots,g_{m,m-1-t}$ nonzero. Substituting $g_{m,1}\dots,g_{m,m-1-t}$ in the equations of the system $B_2\overrightarrow{g}=0$ we get $[B_2\overrightarrow{g}]_i \ne 0$ for all $i=1,\dots,m-1-t$. Otherwise, it is possible to show that the last choice belongs to the field containing all previous chosen values which is a contradiction. This implies that $k_{m,\Delta(\overline{G})+1}=[B_2\overrightarrow{g}]_{m-1-t}\ne 0$ from the last row of $(B_1\ B_2)$. Applying back substitution and similar argument with $g_{m,1}\dots,g_{m,m-1-t}$ we conclude that $k_{m,1}, \dots,k_{m,\Delta(\overline{G})}$ are also nonzero. Thus the system $S$ has a solution with nonzero values for the unknowns. As a consequence there exists a vector $\overrightarrow{v}_m$ satisfying all the adjacency conditions and orthogonal conditions. 3. $m-1-t> \Delta(\overline{G})+1$ then matrix $B$ has the form $$B= \left( \begin{array}{ccc} B_1&| & B_2 \\ -&&-\\ 0 &|& R \\ \end{array} \right)={\tiny \left(\begin{array}{cccccccccccccccc} 1&\ast & \ast &\dots &\ast&\dots&\ast&| &\ast &0&0&0& 0 & \dots& 0 &0 \\ 0 &1 &\ast & \dots &\ast&\dots&\ast&| &\ast &\ast& 0&0& 0& \dots&0& 0 \\ 0 & 0 & 1 & \dots&\ast&\dots&\ast&| &\ast&\ast& \ast &0&0&\dots&0&0 \\ \vdots& \vdots & \vdots&\dots& \vdots &\dots&\vdots&|& \vdots &\dots& \vdots&\vdots&\vdots&\vdots &\vdots& \vdots \\ 0 & 0 &0 & \dots&1&\dots&\ast&|&\ast&\dots&\ast&\ast&\ast &\dots& 0 &0 \\ \vdots & \vdots &\vdots&\vdots& \vdots&\dots & \vdots &\vdots&\vdots& \dots&\vdots & \vdots & \vdots & \vdots &\vdots& \vdots \\ 0 & 0 &0 & \dots&0&\dots&\ast&|&\ast&\dots&\ast &\ast &\ast &\dots& \ast &0 \\ 0 &0& 0&\dots &0&\dots&1&|&\ast&\dots&\ast &\ast & \ast &\dots &\ast& \ast \\ -&-&-&-&-&\dots&-&|&-&-&-&-&-&\dots&-&- \\ 0 & 0&0 & \dots&0&\dots& 0&|&1&\dots&\ast &\ast & \ast &\dots &\ast & \ast \\ \vdots & \vdots &\vdots&\vdots& \vdots&\dots & \vdots &\vdots&\vdots& \dots&\vdots & \vdots & \vdots & \vdots &\vdots& \vdots \\ 0 & 0 &0 & \dots &0&\dots&0&|&0&\dots&1&\ast &\ast & \dots& \ast & \ast \end{array}\right)}.$$ The matrix $B_1$ has size $(m-1-t)\times (\Delta(\overline{G})+1+r), 0< r<t$  where $m-1-t=\Delta(\overline{G})+1+r$. This matrix contains the columns of the unknowns $k_{m,1},\dots,k_{m,\Delta(\overline{G})+1},\\ (-g_{m,1}),\dots,(-g_{m,r})$.The matrix $B_2$ has size $(m-1-t)\times (\Delta(\overline{G})+1)$. The $0$ matrix has size $t\times (\Delta(\overline{G})+1+r)$. The matrix $R$ has size $t\times (\Delta(\overline{G})+1)$. Since $t\le\Delta(\overline{G})<\Delta(\overline{G})+1$ the system $R\overrightarrow{g}=0$ has infinitely many solutions with at least one free variable. By the same argument as in case 2 we get that the system $S$ is consistent and the solution with nonzero values for $k_{m,1},\dots k_{m,\Delta(\overline{G})+1}$ gives a vector $\overrightarrow{v}_m$ which satisfies all the adjacency conditions and orthogonal conditions. Hence, $Y_{m}$ has an orthogonal representation of vectors in $\Re^{\Delta(\overline{G})+1}$ satisfying the conditions (1) and (2). Thus $Y_{|G|}=G$ has an orthogonal representation in $\Re^{\Delta(\overline{G})+1}$ satisfying conditions (1) and (2). Using the same argument in the construction of vector $v_4$, we prove that $v_1,\dots,v_m$ is pairwise linearly independent set of vectors in $\Re^{\Delta(\overline{G})+1}$. Finally since $\msr(G)$ is the smallest dimension in which $G$ has an orthogonal representation $ \msr(G)\le \Delta(\overline{G})+1. $ Since $\delta(G)+\Delta(\overline{G})= |G|-1$ we conclude that $G$ satisfies the delta conjecture, namely, $ \msr(G)\le |G|- \delta(G). $ $\Box$ In the construction of orthogonal representation of the induced graph $Y_m$ it is sufficient to consider $t\le\left\lfloor\frac{m}{2}-1\right\rfloor $ if $m$ is even and $t<\left\lfloor\frac{m-1}{2}\right\rfloor $ if $m$ is odd. In both cases we obtain the condition $t< (m-1-t)$ that we need to get infinitely many solutions for the system $R\overrightarrow{g}=0$. This difference in the upper bounds for $t$ is important for small values of $m$ but for larger values of $m$ these upper bounds are asymptotically equivalent. However, it means that we could get an orthogonal representation of pairwise linearly independent vectors in $\Re^{\Delta(\overline{G})+1}$ for some graphs which are not necessarily $\delta$-graphs. Reducing the matrix $A$ to an echelon form needs a finite number of operations as well as reducing the matrix $R$ to an echelon form. It means that all the values $k_{i,j}, i=1,2,\dots,m-1, j=1,2,\dots,\Delta(\overline{G})+1$ can be chosen from different field extensions in such a way that all the values $\ast$ in the reduced echelon form of $A$ are nonzero and belong to different field extensions. The condition of choosing values $k_{i,j}, i=1,2,\dots,|G|, j=1,2,\dots,\\ \Delta(\overline{G})+1$ from different field extensions was imposed to guarantee the consistency of the non-homogeneous system $S$. Also, we use this nonzero entries of the vectors $\overrightarrow{v}_1,\dots, \overrightarrow{v}_{m-1}$ to guarantee the adjacency conditions and orthogonal conditions of the vector $\overrightarrow{v}_m$ corresponding to the newly added vertex. But calculating the orthogonal representation using this approach could be time consuming. Since we know that it is possible to get an orthogonal representation of $\delta$-graph $G$ in $\Re^{\Delta(\overline{G})+1}$ and since the representation is not unique, it may be possible to calculate the orthogonal representation using integers or rational numbers. However, calculating the orthogonal representation of a $\delta$-graph $G$ in this way could also be tedious because we may need to apply a backtracking procedure during the calculation due to some adjacency conditions of the vector corresponding to the newly added vertex may not be satisfied. When that happens, we may need to go back to some of the previous vectors and recalculate them until we fix the adjacency conditions. Examples of $\delta$-graphs and their $\msr$ ============================================ The result proved above give us a huge family of graph which satisfies delta conjecture. Since, the complement of a C-$delta$ graphs is a $\delta$-graph, it is enough to identify a C-$\delta$-graph and therefore we know that its complement is a $\delta$-graph satisfying delta conjecture. Some examples of C-$\delta$ graphs that we can find in [@PD] are the Cartesian Product $K_n\square P_m,n\ge 3, m\ge 4$, Mobiüs Lader $ML_{2n}, n\ge 3$, Supertriangles $Tn, n\ge 4$, Coronas $S_n\circ P_m, n\ge 2 , m\ge 1$ where $S_n$ is a star and $P_m$ a path, Cages like Tutte’s (3,8) cage, Headwood’s (3,6) cage and many others, Blanusa Snarks of type $1$ and $2$ with $26, 34$, and $42$ vértices, Generalized Petersen Graphs $Gp1$ to $Gp16$, and many others. In order to show the technique used in the proved result consider the following example \[upper2\] If $G$ is the Robertson’s (4,5)-cage on 19 vertices then it is a 4-regular C-$\delta$ graph. Since $\Delta(G)=4$, the $\msr(G)\le 5$. To see this is a C-$\delta$ graph it is enough to label its vertices in the way shown in the next figure: ![image](cage45.png){height="50mm"} [Figure B.2 Robertson’s (4,5)-cage (19 vertices)]{} \[figA.1.2\] Conclusion ========== The result proved above give us a a tool to identify a wide range of families of graphs which satisfy $\delta$ conjecture. The techniques used in the proof could be used in future research as a new approach to solve delta conjecture. However, it is clear that the main problem is still open. Acknowledment ============= I would like to thanks to my advisor Dr. Sivaram Narayan for his guidance and suggestions of this research. Also I want to thank to the math department of University of Costa Rica and Universidad Nacional Estatal a Distancia because their sponsorship during my dissertation research and specially thanks to the math department of Central Michigan University where I did the researh for this paper. [^1]: Escuela de Matemática, Universidad de Costa Rica
--- abstract: 'A systematic numerical framework based on Integral Equations and Generalized Sheet Transition Conditions (IE-GSTCs) is presented in 2D to synthesize closed metasurface holograms and skins for creating electromagnetic illusions of specified objects and as a special case, to camouflaging them against their backgrounds. The versatile hologram surface is modeled using a zero-thickness sheet model of a generalized metasurface expressed in terms of its surface susceptibilities, which is further integrated into the GSTCs and the IE current-field propagation operators. To estimate the effectiveness of the illusions, the notion of a scene constructed by an observer is developed from first principles and a simple mathematical model, referred to as a Structured Field Observation (SFO), based on spatial Fourier transform is proposed. Using numerical examples, it is shown that to recreate the reference desired fields everywhere in space using a closed metasurface hologram/skin, an internal illumination must be applied inside the hologram, in addition to the applied external illumination fields. Finally, several numerical examples are presented for simple, angle-dependent and dynamic illusions. Finally, a dynamic camouflaged region of space, which can freely move inside a given complex scene without being detected by the observer is demonstrated.' address: 'Department of Electronics (DoE), Carleton University, Ottawa, Ontario, Canada, K1S 5B6' author: - 'and ,' bibliography: - '2020\_Metasurface\_Closed\_Illusions\_TAP\_Smy.bib' title: 'Surface Susceptibility Synthesis of Metasurface Skins/Holograms for Electromagnetic Camouflage/Illusions' --- Electromagnetic Metasurfaces, Metasurface Holograms, Effective Surface Susceptibilities, Boundary Element Method (BEM), Generalized Sheet Transition Conditions (GSTCs), Method of Moments (MoM), Field Scattering, Electromagnetic Illusions, Electromagnetic Camouflage. =-15pt Introduction ============ Electromagnetic (EM) invisibility has gathered an immense interest in the past two decades as a result of a rapid development in the general area of electromagnetic metamaterials. Metamaterials led the way to the realization of cloaking devices based on transformation optics to enclose an object which “bends” the light around the object so that there are no scattered fields that reach the observer, i.e. minimizing the Radar Cross Section (RCS) of the object. The object appears invisible to the observer. However, such cloaks are based on volumetric shells with extremely challenging nonuniform, anisotropic and active material requirements [@CloakingReview; @YAN2009261; @ALITALO200922], which has led to investigation of alternate routes to make an object invisible or undetectable. An alternative to electromagnetic cloaking is an *Electromagnetic Camouflage* which results in an *effective* electromagnetic invisibility making an object hard to detect. The object inherits the scattering property of the background it is in, and “blends" into the background. Consequently, an observer cannot distinguish the fields scattered off from the object and the background. While there is an apparent lack of electromagnetic cloaks in nature, camouflage appears in nature in myriad of ways ranging from static conditions to dynamic camouflage of cephalopods. This suggests that camouflage is a better practical alternative compared to cloaks in the evolutionary sense. In spite of sharing the same goal, cloaking and camouflaging have fundamentally different operating mechanism: where cloaking aims at minimizing the RCS, the camouflage has a non-zero RCS which is engineered to match its background. Electromagnetic camouflage is thus a form of illusion which is similar to what is produced using *Holograms*. Holograms are well-known in optics where the spatial (and possibly temporal) information of an arbitrary object is encoded onto the surface (typically photographic plates) [@Goodman_Fourier_Optics; @Saleh_Teich_FP]. This is a two step process, where the information about scattering properties of an object of interest is first recorded using a given reference beam (i.e. incident fields) and modulated onto a given surface. Once the information is recorded, the encoded surface, when illuminated with a reconstructing beam (i.e. illumination fields)[^1], projects an illusion of the object. With increasing sophistication of encoding capability, more complex illusions can naturally be created. Typically, holograms are encoded with object information that one wants to project as an illusion. However if the object itself is a hologram or is enclosed inside a holographic *skin* which is encoded with the scattering information of the entire environment including the background, the object is camouflaged against the background. Camouflaging thus can be seen as a special case of electromagnetic holograms. Creating holograms naturally demands a versatile and a flexible surface that can be engineered to project myriad of illusions, including camouflage. To realize such holograms, *Electromagnetic Metasurfaces* represent a powerful platform due to their complete control over the scattered fields with respect to both complex amplitude and polarization. They are 2D arrays of sub-wavelength resonating particles, where control of the spatial distribution and EM properties of the individual particles allows the scattered fields to be engineered with unprecedented control of both reflection and transmission, and with complete polarization control [@meta2; @MS_review_Yu]. Consequently, these surfaces have been used to create *metasurface holograms*, due to their advanced information encoding capability [@MSHologram_Review; @MShologram_Review2]. To enable a general treatment of the problem, practical metasurfaces can conveniently be modeled as zero thickness sheets characterized using frequency dependent electromagnetic surface susceptibility tensors $\bar{\bar{\chi}}(\omega)$ [@Chi_Review; @MS_Synthesis; @Chi_extraction_Macrodmodel; @TBC_vs_GSTC_Caloz]. The EM fields around the metasurface then can be described using Generalized Sheet Transition Conditions (GSTCs) [@KuesterGSTC]. The spatial distribution of surface susceptibilities of the metasurface $\bar{\bar{\chi}}(\mathbf{r})$ dictates the scattered (and thus total) fields produced by the metasurface when illuminated by an incident field. Therefore, the key design objective in creating metasurface based illusions (and object camouflage) is to synthesize the spatially varying surface susceptibilities, $\bar{\bar{\chi}}(\mathbf{r})$. A systematic description of *open* metasurface holograms based on GSTCs and surface susceptibility description was recently presented in [@smy2020surface] where various classifications and rigorous procedures were defined to design and synthesize these metasurfaces for achieving a desired EM illusion. Due to the open nature of the metasurface, the illusion could only be created in a half-space. In this work, the surface susceptibility synthesis of metasurface holograms is extended to *closed metasurfaces* recreating the desired fields, so that an observer can move around the metasurface and perceive the illusion from different directions. Moreover, in defining an [*illusion*]{} we extend this field recreation and place it in a [*scene*]{} consisting of an incident field and other objects not part of the intrinsic illusion, as opposed to a standalone hologram used in [@smy2020surface]. This also enables camouflaging an object against its background. Transition from an open to a closed metasurface also has important implications for the illumination fields. As will be shown later, an external illumination of the closed metasurface is not sufficient to fully reconstruct the desired reference fields everywhere in the region. Consequently, an internal illumination in addition to the external illumination is proposed to be a potential solution and it will be shown that a proper choice of internal illumination enables a complete reconstruction of the fields everywhere in space. Many metasurface synthesis and analysis problems using surface susceptibilities have been reported in the literature, where for planar surfaces, metasurface susceptibilities can be analytically computed, for instance [@MS_Synthesis; @Caloz_EM_inversion]. On the other hand, metasurface analysis typically involves integrating GSTCs into bulk Maxwell’s equations using a variety of standard numerical techniques based on Finite-Difference and Finite Element methods [@Caloz_MS_Siijm; @Caloz_Spectral; @Smy_Metasurface_Space_Time], and Integral-Equation (IE) techniques [@stewart2019scattering; @FE_BEM_Impedance; @Caloz_MS_IE; @AppBEMEM; @Smy_EuCap_BEM_2020; @Caloz_EM_inversion]. Given that the field scattering from a metasurface hologram may need to be evaluated for electrically large domains, IE-GSTC methods are a computationally efficient choice and, as will be shown later and in [@smy2020surface], are well suited for analysis of both closed and open metasurface holograms. Given this context, a general methodology of synthesizing and designing closed metasurface holograms, which may be located within a complex environment with variety of objects and backgrounds, is presented in this work using the IE-GSTC method. The 2D IE-GSTC based numerical platform is further developed to synthesize metasurface susceptibilities with an integrated approach, where the desired fields, specified anywhere in space and not necessarily at the metasurface, are generated using a system level description and fed-back into the metasurface design procedure. A novel proposal of utilizing both external and internal illumination fields inside closed metasurface holograms is presented to rigorously reconstruct the desired fields everywhere in space. Next, a simple but powerful method is proposed to model the observer and how it measures the total and scattered fields to construct a given scene of the environment. Finally, using various numerical examples, several metasurface holograms are synthesized to project complex angle dependent illusions and dynamic illusions where the surrounding objects may be moving with respect to the hologram. An example of an object camouflage is further presented, where a closed metasurface is navigating through a complex environment without being detected by the observer. The paper is structured as follows. Sec. II presents the general problem of illusion formation and object camouflage using metasurfaces, and how an observer constructs the image of the environment. The general goal of the metasurface synthesis is formulated and important aspects related to illumination fields are described. Sec. III presents the IE-GSTC architecture, and presents a systematic way of designing an illusion including the object camouflage as its special case. A simple and elegant technique to model the observer and how it constructs the scene is presented in Sec. IV. Several numerical demonstrations to illustrate the presented metasurface synthesis approach is given in Sec. V, for cases of complex merged illusions and that in dynamic environments. Finally, a summary and concluding discussions are provided in Sec. VI. Problem Formulation {#Sec:II} =================== The Observer & Scenes --------------------- [MS\_Illusions.pdf]{} (0,62) (0,20) (50,20) (50,62) (58, 79) (58, 37) (79,79) (79,37) (0, 82) (0, 40) (50, 82) (50, 40) (2, 50) (2, 8) (21, 68) (21, 22) (30, 50) (30, 7) (90, 77) (90, 35) (45, 55) (86, 15) (60, 56) (60, 14) (78, 50) (74, 10) (30, 81)[<span style="font-variant:small-caps;"></span>]{} (30, 39)[<span style="font-variant:small-caps;"></span>]{} (2, 26)[<span style="font-variant:small-caps;"></span>]{} (3, 33)[<span style="font-variant:small-caps;"></span>]{} (2, 68)[<span style="font-variant:small-caps;"></span>]{} (3, 76)[<span style="font-variant:small-caps;"></span>]{} Consider a field scattering problem illustrated in Fig. \[Fig:Illusion\_Cases\](a), where a specified incident wave, $\psi^\text{inc.}(\mathbf{r},\omega)$ is illuminating an object of interest. The object may be placed in a certain environment or a *Scene*, such as in front of a background which could be either a textured reflective/opaque surface or with partial reflection or transmission i.e. semi-transparent background. The incident field strikes different structures in the environment and various scattered fields from the object and the background are generated. Within this scene an *Observer* can be placed in a location that views the object from a point-of-view (POV) over a prescribed field-of-view (FOV). We will refer to the Observer [*rendering*]{} the scene – by which we mean the scene is scanned over the FOV and the intensity of the field determined as a function of the scanning angle; creating an image of the scene. The observer thus detects/measures the total fields $\psi^\text{tot.}(\mathbf{r}_1,\omega)$, in general, which can be decomposed into the incident field and the scattered fields. An intelligent observer can now identify the presence and features of the target object in that scene by processing the information contained in the total fields, in comparison to that of an isolated background. Let us call this scene as the *Reference Scene*. If the object is illuminated from the left and the Observer is placed on the left (a front-lit configuration) then the scattered fields traveling towards the Observer will be detected and the incident field will not, as it is traveling past and away from the Observer. On the other hand if the Observer is placed on the right side of the object the fields detected will be the total fields as both the incident and scattered fields are traveling towards the Observer \[such as in $\psi^\text{tot.}(\mathbf{r}_2,\omega)$\]. Due to this distinction the Observer needs to be characterized in terms of the *directionality* and structure of the fields in the region nearby the Observer. For example, in the illustration of Fig. \[Fig:Illusion\_Cases\](a), there is an incident field present in the region to the left of the background. Both the object and the background surface produce scattered fields which in turn will scatter off each other. The net result of this is a complicated field pattern created by the interference of source field and the scattered field components. An observer placed within the region of interest will detect the EM waves propagating toward its POV within a FOV. If the POV is swept over a range of angles (e.g. by $360^\circ$) defined by the FOV, the scene will be rendered revealing the scattered object, reflected images, shadows on the background and complicated multipath effects, for instance. Principle of Illusion & Camouflage ---------------------------------- We are now interested in artificially engineering the scene so that the observer either perceives a different object in the presence of the original object, or the same object in the absence of the original object. They correspond to the following two different, but closely related physical effects: 1. *Electromagnetic Illusion using a Metasurface Hologram:* Let us remove the object of interest from the scene of Fig. \[Fig:Illusion\_Cases\](a), and replace with an artificial closed surface at ${\mathbf{r}}= {\mathbf{r}}_m$ which is completely encompassing the object, as shown in Fig. \[Fig:Illusion\_Cases\](b). The objective is to engineer this surface so that the new scattered (and thus total) fields are identical to the reference scene at the observer location, when the object of interest was present. Since the observer measures the same fields, it falsely perceives the closed surface as a real object, while in reality, it is a virtual object, i.e. *an electromagnetic illusion*. Such a closed surface, as will be shown later and throughout the paper, will be realized using an electromagnetic metasurface, and will now be referred to as a *Metasurface Hologram*. 2. *Electromagnetic Camouflage using a Metasurface Skin:* Now consider an alternate scenario where the object is first removed from the reference scene as shown in Fig. \[Fig:Illusion\_Cases\](c), so that the observer measures the fields corresponding to the background only in the presence of other possible scattering objects. We now introduce the object into the scene. Naturally, the introduction of this new object into the blank reference scene will perturb the fields. However, if the object is enclosed inside a closed engineered surface at ${\mathbf{r}}= {\mathbf{r}}_m$, as shown in Fig. \[Fig:Illusion\_Cases\](d), such that the newly generated fields are identical to the reference scene of Fig. \[Fig:Illusion\_Cases\](c), the observer will still perceive the background. Therefore, the object while being physically present in the scene is still undetected by the observer, where it has effectively blended into the background, i.e. *an electromagnetic camouflage*. Such a closed surface, as will be shown later, will be realized using an electromagnetic metasurface, and will now be referred to as a *Metasurface Skin*. Electromagnetic illusion and Camouflage, thus represent two practically important phenomena where the wave engineering capabilities of the metasurface becomes crucially useful. In spite of their apparent physical difference, there is a fundamental similarity between the two: while in the case of an electromagnetic illusion, the metasurface hologram mimics an object, the metasurface skin mimics the background in the case of an electromagnetic camouflage. In both cases, metasurface projects false information to the observer. Thus, the objective here now is to synthesize these metasurfaces enabling these operations. We will discuss the methods and procedures pertaining to a metasurface hologram only, while remembering that it is closely related to the camouflage operation. Metasurface Description {#Sec:II^-B} ----------------------- An electromagnetic metasurface as remarked in the introduction, can be rigorously described using a zero thickness sheet model, using Generalized Sheet Transition Conditions (GSTCs) with specific electric and magnetic surface susceptibility densities $\bar{\bar{\chi}}_\text{ee}({\mathbf{r}}_m, \omega)$ and $\bar{\bar{\chi}}_\text{mm}({\mathbf{r}}_m, \omega)$. The problem of creating electromagnetic illusions and camouflage thus becomes the problem of determining the surface susceptibilities of closed metasurface holograms and skins. This formulation captures the general wave transformation capability of physical EM metasurfaces by expressing them as mathematical space discontinuities of zero thickness[@IdemenDiscont; @GSTC_Holloway; @KuesterGSTC]. The GSTCs relate the tangential EM fields around the metasurface to the tangential and normal surface polarization response, rigorously modeling the EM interaction with the metasurface capturing the field transformation capabilities via 36 variables inside the susceptibility tensors. In this paper for simplicity we will limit the analysis to tangential terms and assume scalar susceptibilities and in the frequency domain we have for the GSTCs: [@smy2020surface; @Chi_Review] \[Eq:GSTC\] $$\begin{aligned} {\mathbf{\hat n}}\times \Delta {\mathbf{E}}_T &= -j\omega\mu_0 (\epsilon \chi_\text{mm} {\mathbf{H}}_{T,av} + \chi_\text{me} \sqrt{\epsilon/\mu}\; {\mathbf{E}}_{T,av})\\ {\mathbf{\hat n}}\times \Delta {\mathbf{H}}_T &= j\omega (\epsilon \chi_\text{ee} {\mathbf{E}}_{T,av} + \chi_\text{em} \sqrt{\mu \epsilon}\; {\mathbf{H}}_{T,av}),\end{aligned}$$ where $\Delta \psi_T = \psi({\mathbf{r}}_{m+}) - \psi({\mathbf{r}}_{m-})$, and $\psi_\text{av} = \{\psi({\mathbf{r}}_{m+}) + \psi({\mathbf{r}}_{m-})\}/2$, are expressed in terms of total fields just before and after the metasurface (this implies for a closed object that $^-$ indicates the region or field external to the object and $^+$ internal quantities). These equations can be incorporated into the IE infrastructure [@stewart2019scattering] to provide a complete simulation environment for creating metasurface based illusion and camouflage as will be described in following sections. The Illumination Fields ----------------------- In both cases of an electromagnetic illusion and camouflage, the closed metasurface may be excited with an external (to the metasurface) illumination field $\psi^{\text{ill}-}(\mathbf{r},\omega)$. This external illumination could either be the same as the incident field of the reference scene i.e. $\psi^{\text{ill}-}(\mathbf{r},\omega) = \psi^\text{inc.}(\mathbf{r},\omega)$ or be an entirely independent field, as is the case in Fig. \[Fig:Illusion\_Cases\](b). This figure shows the two possible sources of illumination external and internal. Initially we will consider the case of external illumination only as it would appear to a simple and obvious choice. This illumination configuration however, presents a serious problem for the metasurface synthesis in order to recreate the fields of the reference scene, as the far side of the metasurface is in shadow and has effectively no field incident on it. However, for almost all illusions, this region of the metasurface will need to produce a scattered field (even if producing an illusory shadow, a scattered field needs to be produced to “shape” the shadow to match the recreated object). As we will see the synthesis process presented in Sec. \[Sec:Arch\], will create the required susceptibilities to produce these scattered fields from the small incident fields present, however, these susceptibilities will be active, numerically difficult to handle and physically difficult to implement. A possible solution to this issue is to use *only* an internal illumination (as shown in Fig. \[Fig:Illusion\_Cases\]b), where $\psi^{\text{ill}+}(\mathbf{r},\omega)$ is present inside the closed metasurface region. For this case we can achieve a uniform illumination of a transparent metasurface that modulates the internal source to produce the outgoing scattered fields recreating the illusion. Although this configuration allows for the synthesis of surface susceptibilities to recreate the scattered fields of the object, it can not recreate with any generality the previously present incident field used in the reference scene. This is an issue as from many view points the observer will register the effect of not only the scattered fields from the object but also the original incident field. Therefore, in the general case, both external and internal illumination will be required and it will be shown that such a configuration can optimally recreate the reference scene. Although the external illumination could be different from the incident field (with implications on the effectiveness of the illusion) the natural choice is to use an external illumination that is identical to the original incident field. For this case the entire metasurface is well illuminated (internally), and only has to recreate the original scattered fields propagating out from the object. This procedure allows for passive and well-characterized susceptibilities to be synthesized which have both reflective and transmissive aspects, as will be demonstrated later. Illusion Design Architecture {#Sec:Arch} ============================ The primary requirements for the design of an illusion system are threefold: 1) the field distribution of the reference scene to be recreated (the object within its environment), 2) the specification of the illusion illumination, and 3) the synthesis of the metasurface used to create the illusion. To determine the fields present in the scene to be recreated a full wave EM simulator capable of providing detailed and accurate fields for electrically large complex regions with curvilinear surfaces is needed. The simulation architecture will be used to simulate the original scene and the final illusion within a complex environment. It should also further provide a means to address the problem of specifying the surface characteristics of the final illusion object, and compute the corresponding total fields. Although numerical EM simulation techniques exist in many forms including volumetric methods such as finite difference time and frequency domain approaches [@taflove2000computational], it is integral equation approaches that are the most suitable for this problem[@chew2009integral]. These approaches are based on a discretization of the surfaces present in the problem and the determination of surface currents (virtual or real) that produce the appropriate fields in the simulation domain. Particularly appropriate to scattering problems with curvilinear surfaces in homogeneous regions, they are ideally placed to solve the initial object scattering problem, provide an infrastructure for the synthesis of the metasurface illusion object and confirm the effectiveness of the final illusion. [SimSetup.pdf]{} (5, 67)[$x$]{}(17, 56)[$y$]{} (-3, 36) (10, 2) (32, 47) (23, 65) (47, 7) (50,37) [SimSetup2.pdf]{} (11, 53) (5,75) (-1,15) (57,9.5) (58,75) (12,2) (53,40) (-8,31.5) \[c\]\[c\]\[0.7\][$x$]{} \[c\]\[c\]\[0.7\][$y$]{} Discretized IE Formulation {#Sec:III-A Discretized IE Formulation} -------------------------- Consider a complex scene consisting of various type of objects and surfaces, for which the fields need to be solved, as shown in Fig. \[Fig:SimSetup\](a). To formulate the problem as a solution to the integral formulation of Maxwell’s equations the solution domain is broken into homogeneous regions with a constant index of refraction, separated by surfaces – which can be real or fictitious. Each surface is characterized by a relationship between the tangential surface fields on both sides and an incident (forcing field) can be placed on appropriate surfaces. A complete solution will involve the incident fields at the surfaces (${\mathbf{E}}^i/{\mathbf{H}}^i$) and the scattered fields (${\mathbf{E}}/{\mathbf{H}}$) from each surface determined by surface currents ${\mathbf{J}}$ and ${\mathbf{K}}$. Coupling between the regions is captured by the surface characterization. To make the problem numerically tractable the surfaces are discretized using elements of length $\delta \ell$. Defining for a particular surface ($S$) we have, $$\begin{aligned} {\mathbbm{r}}_S = {\begin{bmatrix}}{\mathbf{r}}_{S,1} & \dots & {\mathbf{r}}_{S,m} {\end{bmatrix}}, \quad {\mathbb{\bar N}}_S = {\begin{bmatrix}}{\mathbf{\hat n}}_{S,1} & \dots & {\mathbf{\hat n}}_{S,m} {\end{bmatrix}}\end{aligned}$$ where ${\mathbbm{r}}_S$ denotes the center of a set of line segments and ${\mathbb{\bar N}}_S$ the element surface normals. On these surface elements we can define surface currents, $$\begin{aligned} {\mathbb{J}}_S = {\begin{bmatrix}}{\mathbf{J}}_{S,1} & \dots & {\mathbf{J}}_{S,m} {\end{bmatrix}}, \quad {\mathbb{K}}_S = {\begin{bmatrix}}{\mathbf{K}}_{S,1} & \dots & {\mathbf{K}}_{S,m} {\end{bmatrix}}\end{aligned}$$ These surface current vectors can be grouped for convenience into the vector ${\mathbb{C}}$, $$\begin{aligned} {\mathbb{C}}_S = {\begin{bmatrix}}{\mathbb{J}}_S & {\mathbb{K}}_S {\end{bmatrix}}^\top. \end{aligned}$$ Once this discretization has been defined, the modeling problem is composed of three related aspects; 1) Field propagation through the regions, 2) surface characterization and 3) coupling between the regions through the surfaces, which now will be treated separately as follows: *1. Field Propagation* The EM fields radiated from each surface due to electric and magnetic surface currents can be generally expressed using a propagation matrix ${\overline{\mathbb{P}}}_{S,p}$ as [@chew2009integral; @Method_Moments]: $$\begin{aligned} \label{eq:EMProp} {\begin{bmatrix}}{\mathbb{E}}({\mathbbm{r}}_p) \\{\mathbb{H}}({\mathbbm{r}}_p) {\end{bmatrix}}& = {\begin{bmatrix}}- j\omega \mu {\mathbb{L}}({\mathbbm{r}}_p, {\mathbbm{r}}_S) - {\mathbb{R}}({\mathbbm{r}}_p, {\mathbbm{r}}_S) \\ - j\omega \epsilon {\mathbb{L}}({\mathbbm{r}}_p, {\mathbbm{r}}_S) + {\mathbb{R}}({\mathbbm{r}}_p, {\mathbbm{r}}_S) {\end{bmatrix}}{\begin{bmatrix}}{\mathbb{J}}_S \\ {\mathbb{K}}_S{\end{bmatrix}}\notag \\ {\mathbb{F}}_{p,S} & = {\overline{\mathbb{P}}}_{S,p} {\mathbb{C}}_S\end{aligned}$$ where ${\mathbbm{r}}_p = {\begin{bmatrix}}{\mathbf{r}}_{p,1} & \dots & {\mathbf{r}}_{p,n} {\end{bmatrix}}$ is a set of points at which the fields are desired, and ${\mathbbm{r}}_S$ are the surfaces where the source currents are defined. The discretized field propagation matrices ${\mathbb{L}}$ and ${\mathbb{R}}$ are given by, $$\begin{aligned} {\mathbb{L}}({\mathbbm{r}}_p, {\mathbbm{r}}_S) &= {\begin{bmatrix}}{\mathbf{\mathcal{L}}}_{1,1} & {\mathbf{\mathcal{L}}}_{1,2} & \dots &{\mathbf{\mathcal{L}}}_{1,m} \\ {\mathbf{\mathcal{L}}}_{2,1} & {\mathbf{\mathcal{L}}}_{2,2} & \dots &{\mathbf{\mathcal{L}}}_{2,m} \\ \vdots & \vdots & \ddots & \vdots\\ {\mathbf{\mathcal{L}}}_{n,1} & {\mathbf{\mathcal{L}}}_{n,2} & \dots &{\mathbf{\mathcal{L}}}_{n,m} \\ {\end{bmatrix}}\\ {\mathbb{R}}({\mathbbm{r}}_p, {\mathbbm{r}}_S) &= {\begin{bmatrix}}{\mathbf{\mathcal{R} }}_{1,1} & {\mathbf{\mathcal{R} }}_{1,2} & \dots &{\mathbf{\mathcal{R} }}_{1,m} \\ {\mathbf{\mathcal{R} }}_{2,1} & {\mathbf{\mathcal{R} }}_{2,2} & \dots &{\mathbf{\mathcal{R} }}_{2,m} \\ \vdots & \vdots & \ddots & \vdots\\ {\mathbf{\mathcal{R} }}_{n,1} & {\mathbf{\mathcal{R} }}_{n,2} & \dots &{\mathbf{\mathcal{R} }}_{n,m} \\ {\end{bmatrix}}\end{aligned}$$ with $$\begin{aligned} {\mathbf{\mathcal{L}}}_{i,j} &= {\mathbf{\mathcal{L}}}({\boldsymbol{{\mathbf{r}}}}_{p,i},{\boldsymbol{{\mathbf{r}}}}_{S,j}) \\ &= \int_{\delta\ell_j}[1+\frac{1}{k^2}\nabla_{{\boldsymbol{{\mathbf{r}}}}_p}\nabla_{{\boldsymbol{{\mathbf{r}}}}_p}\cdotp] [G({\mathbf{r}}_{p,i},{\mathbf{r}}_{S,j})] \,d{\mathbf{r}}_{S,j}\\ {\mathbf{\mathcal{R} }}_{i,j} &= {\mathbf{\mathcal{R} }}({\boldsymbol{{\mathbf{r}}}}_{p,i},{\boldsymbol{{\mathbf{r}}}}_{S,j}) = \int_{\delta\ell_j}\nabla_{{\boldsymbol{{\mathbf{r}}}}_p} \times [G({\mathbf{r}}_{p,i},{\mathbf{r}}_{S,j})] \,d{\mathbf{r}}_{S,j}\end{aligned}$$ where $G(\cdot)$ represents the Green’s function. For a 2D case, the Green’s function is given by the 2$^\text{nd}$ Hankel function, $$\begin{aligned} G({\mathbf{r}}_S, {\mathbf{r}}_p)= H_0^{(2)}({\mathbf{r}}_S, {\mathbf{r}}_p) = J_0({\mathbf{r}}_S, {\mathbf{r}}_p) - i Y_0({\mathbf{r}}_S, {\mathbf{r}}_p),\end{aligned}$$ where $J_0(\cdot)$ and $Y_0(\cdot)$ are the Bessel functions of the 1$^\text{st}$ and 2$^\text{nd}$ kind representing outwardly propagating radial waves. It will be useful to define propagation matrices for when fields are needed on the surface itself so that ${\mathbbm{r}}_p = {\mathbbm{r}}_S$. In such a case, we have for the two sides of the surface, $$\begin{aligned} {\mathbb{F}}_{S^+} = {\overline{\mathbb{P}}}_{S,S^+} {\mathbb{C}}_S, \quad {\mathbb{F}}_{S^-} = {\overline{\mathbb{P}}}_{S,S^-} {\mathbb{C}}_S\end{aligned}$$ where $\{\cdot\}^+$ and $\{\cdot\}^-$ respectively, denote the right and left sides of an open surface or interior and exterior sides for a closed surface. Defining a surface field configuration ${\mathbb{S}}_S = {\begin{bmatrix}}{\mathbb{F}}_{S^+} & {\mathbb{F}}_{S^-} {\end{bmatrix}}^T$ and a surface propagator ${\overline{\mathbb{P}}}_S = {\begin{bmatrix}}{\overline{\mathbb{P}}}_{S^+} & {\overline{\mathbb{P}}}_{S^-} {\end{bmatrix}}$, the above equation can be written in a compact matrix form as: $$\begin{aligned} \label{eq:SurfPropBoth} {\mathbb{S}}_S = {\overline{\mathbb{P}}}_S {\mathbb{C}}_S.\end{aligned}$$ *2. Surface characterization* For each surface (open or closed) present in the domain, it is needed to formulate the surface field relationships that relate the total tangential fields on the two sides of the surface. Although a number of surface formulations can be accommodated within the framework, we will limit ourselves to *dielectric boundaries, perfect electrical conductors (PEC) and metasurfaces described by the GTSCs*. Due to the flexibility of the GSTCs, this set of surfaces can describe a very wide range of possibilities in practice. The simplest surface is the PEC where the tangential electric field on both sides is equal to zero. This can be described by, $$\begin{aligned} \left[ \begin{array}{cccc} {\mathbb{\bar N}}_{T} & {\varnothing}& {\varnothing}& {\varnothing}\\ {\varnothing}&{\varnothing}&{\mathbb{\bar N}}_{T} & {\varnothing}\end{array}\right] \left[ \begin{array}{c}{\mathbb{E}}_{S^+}\\{\mathbb{H}}_{S^+}\\{\mathbb{E}}_{S^-}\\{\mathbb{H}}_{S^-}\end{array}\right] &= \left[ \begin{array}{cccc} {\varnothing}\\ {\varnothing}\end{array}\right]\notag\\ {\mathbb{T}}_{S} {\mathbb{S}}_S &= {\varnothing}\end{aligned}$$ and the matrix operator ${\mathbb{\bar N}}_T$ performs the operation of extracting the two tangential fields at the surface (one in the $xy$ plane and the other with respect to $z$) obtaining ${\mathbf{E}}_T$ from ${\mathbf{E}}$ for every surface element. For a dielectric surface the tangential fields on both sides are equal and we have, $$\begin{aligned} \left[ \begin{array}{cccc} {\mathbb{\bar N}}_{T} & {\varnothing}& -{\mathbb{\bar N}}_{T} & {\varnothing}\\ {\varnothing}&{\mathbb{\bar N}}_{T} &{\varnothing}& -{\mathbb{\bar N}}_{T} \end{array}\right] \left[ \begin{array}{c}{\mathbb{E}}_{S^+}\\{\mathbb{H}}_{S^+}\\{\mathbb{E}}_{S^-}\\{\mathbb{H}}_{S^-}\end{array}\right] &= \left[ \begin{array}{cccc} {\varnothing}\\{\varnothing}\end{array}\right]\notag \\ {\mathbb{\bar D}}_{S} {\mathbb{S}}_S &= {\varnothing}\end{aligned}$$ where ${\mathbb{\bar D}}_{S}$ takes the difference of the tangential fields for each surface element. Finally, the most general surface description we use is for the metasurface. Putting into a discrete form we obtain, $$\begin{aligned} &\left[ \begin{array}{cccc} {\mathbb{\bar N}}_{T\times} & {\varnothing}& -{\mathbb{\bar N}}_{T\times} & {\varnothing}\\ {\varnothing}&{\mathbb{\bar N}}_{T\times} &{\varnothing}& -{\mathbb{\bar N}}_{T\times} \end{array}\right] \left[ \begin{array}{c}{\mathbb{E}}_{S^+}\\{\mathbb{H}}_{S^+}\\{\mathbb{E}}_{S^-}\\{\mathbb{H}}_{S^-}\end{array}\right] = \notag\\ & \left[ \begin{array}{cccc} \gamma_\text{me}{\mathbb{\bar N}}_{T} & \gamma_\text{mm}{\mathbb{\bar N}}_{T} & \gamma_\text{me}{\mathbb{\bar N}}_{T} & \gamma_\text{mm}{\mathbb{\bar N}}_{T}\\ \gamma_\text{ee}{\mathbb{\bar N}}_{T} & \gamma_\text{em}{\mathbb{\bar N}}_{T} & \gamma_\text{ee}{\mathbb{\bar N}}_{T} & \gamma_\text{em}{\mathbb{\bar N}}_{T}\\ \end{array}\right] \left[ \begin{array}{c}{\mathbb{E}}_{S^+}\\{\mathbb{H}}_{S^+}\\{\mathbb{E}}_{S^+}\\{\mathbb{H}}_{S^+}\end{array}\right] \notag\\ &{\mathbb{\bar D}}^\times_{S} {\mathbb{S}}_S = {\mathbb{\bar G}}_{S} {\mathbb{S}}_S \label{Eq:gstcd}\end{aligned}$$ where the surface susceptibility terms are expressed using auxiliary variables as, $$\begin{aligned} \gamma_\text{ee} = \frac{j\chi_\text{ee}\omega\epsilon}{2},~\gamma_\text{me/em} = \mp \frac{j\chi_\text{me/em}\omega\sqrt{\mu\epsilon}}{2},~\gamma_\text{mm} = -\frac{j\chi_\text{mm}\omega\mu}{2}.\end{aligned}$$ The operator ${\mathbb{\bar N}}_{T\times}$ extracts the total tangent field and then rotates these two fields to implement the ${\mathbf{\hat n}}\times\{\cdot\}_T$ operation on every element and ${\mathbb{\bar D}}^\times_{S}$ takes the difference of the rotated tangential fields. *3. System Level Simulation (Surface Couplings)* The illustration example of Fig. \[Fig:SimSetup\](a) consists of three types of objects in the incident field region: a dielectric inclusion (D), a closed PEC object (O) and a background metasurface (B)[^2]. We therefore have three current vectors $C_\text{D}$, $C_\text{O}$ and $C_\text{B}$ which we can group into a single system level current vector, $$\begin{aligned} {\mathbb{C}}= {\begin{bmatrix}}{\mathbb{C}}_\text{D} &{\mathbb{C}}_{O} &{\mathbb{C}}_\text{B} {\end{bmatrix}}^T\end{aligned}$$ For each surface we have propagators to other surfaces – such as ${\overline{\mathbb{P}}}_{\text{B,D}^-}$ which would determine the fields on the exterior side of the dielectric inclusion surface (D$^-$) due to the background surface (B). Using these propagators, we can formulate expressions for the fields at a surface as sums of propagated fields. For the dielectric inclusion, for instance, the internal fields (D$^+$) are due to currents on the dielectric surface, while the external fields (D$^-$) are due to the currents on the dielectric itself, the PEC object (O) and the background surface (B), so that: $$\begin{aligned} {\begin{bmatrix}}{\mathbb{F}}_{D^+} \\ {\mathbb{F}}_{D^-}{\end{bmatrix}}&= {\begin{bmatrix}}{\overline{\mathbb{P}}}_{D,D^+} {\mathbb{C}}_D \\ {\overline{\mathbb{P}}}_{D,D^-} {\mathbb{C}}_D + {\overline{\mathbb{P}}}_{O,D^-} {\mathbb{C}}_O + {\overline{\mathbb{P}}}_{B,D^-} {\mathbb{C}}_B \\ {\end{bmatrix}}\end{aligned}$$ We can then define a system level propagator for each surface which relates the surface fields to all the currents on various objects, such as: $$\begin{aligned} {\mathbb{S}}_D &= {\begin{bmatrix}}{\overline{\mathbb{P}}}_{D,D^+} &{\varnothing}&{\varnothing}\\ {\overline{\mathbb{P}}}_{D,D^-} & {\overline{\mathbb{P}}}_{O,D^-} & {\overline{\mathbb{P}}}_{B,D^-} \\ {\end{bmatrix}}{\begin{bmatrix}}{\mathbb{C}}_D\\{\mathbb{C}}_E\\{\mathbb{C}}_B {\end{bmatrix}}= {\overline{\mathbb{P}}}_D {\mathbb{C}}.\end{aligned}$$ In a similar manner we can define the system level propagators, ${\overline{\mathbb{P}}}_O$ and ${\overline{\mathbb{P}}}_B$, for the PEC object and the background, respectively. For each surface we must also define the surface relationships for the fields i.e, $$\begin{aligned} \begin{array}{ll} {\mathbb{T}}_O {\mathbb{S}}_O = {\varnothing}& \text{(PEC Object)}\\ {\mathbb{\bar D}}_D {\mathbb{S}}_D = {\varnothing}& \text{(Dielectric Inclusion)}\\ {\mathbb{\bar D}}^\times_B {\mathbb{S}}_B = {\mathbb{\bar G}}_B {\mathbb{S}}_B & \text{(Metasurface Background)} \end{array}\end{aligned}$$ Finally, to complete the physical specification of the region equations, we must ensure that the current only flows on the surface by taking the dot product of the surface normal with the currents on the surface: $$\begin{aligned} \{ {\overline{\mathbb{N}}}_D {\mathbb{C}}_O = {\varnothing}, \; {\overline{\mathbb{N}}}_D {\mathbb{C}}_D = {\varnothing}, \; {\overline{\mathbb{N}}}_D {\mathbb{C}}_B = {\varnothing}\} \; \Rightarrow \;{\overline{\mathbb{N}}}_D {\mathbb{C}}= {\varnothing}, $$ where the operator ${\overline{\mathbb{N}}}_D$ performs a dot product for currents on every element. All of these equations can now be assembled into a matrix equation where the unknowns are the surface currents and the fields: $$\begin{aligned} \label{Eq:SystemMatrix} {\begin{bmatrix}}{\overline{\mathbb{P}}}_D & {\varnothing}& {\mathbb{I}}& {\varnothing}& {\varnothing}\\ {\varnothing}& {\varnothing}& {\mathbb{\bar D}}_D & {\varnothing}& {\varnothing}\\ {\overline{\mathbb{P}}}_O & {\varnothing}& {\varnothing}& {\mathbb{I}}& {\varnothing}\\ {\varnothing}& {\varnothing}& {\mathbb{T}}_O & {\varnothing}& {\varnothing}\\ {\overline{\mathbb{P}}}_B & {\varnothing}& {\varnothing}& {\varnothing}& {\mathbb{I}}\\ {\varnothing}& {\varnothing}& {\varnothing}& {\mathbb{\bar D}}^\times_B - {\mathbb{\bar G}}^\times_B \\ {\overline{\mathbb{N}}}_D & {\varnothing}& {\varnothing}& {\varnothing}& {\varnothing}\\ {\end{bmatrix}}{\begin{bmatrix}}{\mathbb{C}}\\ {\mathbb{S}}_D \\ {\mathbb{S}}_O \\ {\mathbb{S}}_B {\end{bmatrix}}= {\begin{bmatrix}}{\varnothing}\\ -{\mathbb{\bar D}}_D{\mathbb{S}}^\text{inc.}_D \\ {\mathbb{T}}_O{\mathbb{S}}^\text{inc.}_B\\ -({\mathbb{\bar D}}_B-{\mathbb{\bar G}}_B){\mathbb{S}}^\text{inc.}_B {\end{bmatrix}}\end{aligned}$$ where an applied incident field ${\mathbb{S}}^\text{inc.}$ is present on all surfaces. This system of equations can now be solved and then the fields anywhere in the region due to newly found currents are computed using various propagation matrices. Illusion Design {#Sec-III-B:Illusion Design} --------------- Let us consider the objective where we wish to remove the PEC object of interest by a metasurface hologram as shown in Fig. \[Fig:SimSetup\](b), which is designed in such a way that the metasurface creates identical reference fields, \[incident $\psi^\text{inc.}({\mathbf{r}},\omega)$ + scattered $\psi_s^\text{ref.}({\mathbf{r}},\omega)$, in general\], at the observer compared to that when the object was present, i.e. object illusion. The first step in the illusion design is to compute the desired fields (i.e. reference fields/scene) to be recreated by the closed metasurface at and beyond its location. The surface of the closed metasurface hologram (also referred to as the illusion surface $I$, here) can be discretized as: $$\begin{aligned} {\mathbbm{r}}_I = {\begin{bmatrix}}{\mathbf{r}}_{I,1} & \dots & {\mathbf{r}}_{I,m} {\end{bmatrix}}\end{aligned}$$ We can create a propagation matrix ${\overline{\mathbb{P}}}_I$ that calculates the *scattered fields* ${\mathbb{F}}_I$ present in the reference scene (without the metasurface hologram) at the illusion surface, $$\begin{aligned} {\mathbb{F}}_I^\text{ref} = {\overline{\mathbb{P}}}_{I}{\mathbb{C}}\label{Eq:ReferenceFields}\end{aligned}$$ where ${\mathbb{C}}$ represents all the currents present in the original reference scene. Next step is to remove the PEC object and introduce the metasurface hologram producing the illusion. To synthesize the illusion surface we need to determine the total fields present on both sides of the surface. Let us consider the general case, where both internal (${\mathbb{F}}_{I}^{\text{ill}+}$) and external (${\mathbb{F}}_{I}^{\text{ill}-}$) illumination fields are present and the illumination field is different from the incident field ${\mathbb{F}}_{I}^\text{inc.}$. The fields in the external region consists of scattered fields we wish to recreate along with the original incident field to which we must add the external illumination, so that $$\begin{aligned} {\mathbb{F}}_I^- = {\mathbb{F}}_I^\text{ref} + {\mathbb{F}}_{I}^\text{inc.} + {\mathbb{F}}_{I}^{\text{ill}-}\label{Eq:ExF_ill_ne_Inc}\end{aligned}$$ If the external illumination is chosen to be the same as the incident fields in the reference simulation, then the total external fields simply becomes $ {\mathbb{F}}_I^- = {\mathbb{F}}_I^\text{ref} + {\mathbb{F}}_{I}^\text{inc.}$. $$\begin{aligned} {\mathbb{F}}_I^- = {\mathbb{F}}_I^\text{ref} + {\mathbb{F}}_{I}^\text{inc.}\label{Eq:ExF_ill_e_Inc}\end{aligned}$$ With respect to the scattered fields interior to the surface, on the other hand, have some design flexibility. While the exact choice will not impact the reconstruction of the desired scattered fields, it will have a significant effect on the required surface susceptibilities of the metasurface hologram. For simplicity, let us set all the internal scattered fields from the metasurface to zero, so that the total internal field is simply given by the internal illumination as $$\begin{aligned} {\mathbb{F}}_I^- = {\mathbb{F}}_{I}^{\text{ill}+}\end{aligned}$$ We can then form the surface field vector as $$\begin{aligned} {\mathbb{S}}_I = {\begin{bmatrix}}{\mathbb{F}}_I^\text{ref} + {\mathbb{F}}_{I}^\text{inc.} + {\mathbb{F}}_{I}^{\text{ill}-}\\ {\mathbb{F}}_{I}^{\text{ill}+} {\end{bmatrix}}.\end{aligned}$$ The internal illumination may be seen as an extra degree of control available to the system designer. In the case of creating an electromagnetic illusion, the metasurface hologram is hollow, and the internal source may be placed anywhere inside the structure. It could be as simple as a radially propagating wave from the center, for instance, which can be modeled as a Henkel function of the $2^\text{nd}$ kind producing a uniform field amplitude at the interior surface of the metasurface shield, i.e. $$\begin{aligned} E_z({\mathbf{r}}_S, {\mathbf{r}}_m^+) = H_0^{(2)}({\mathbf{r}}_S, {\mathbf{r}}_m^+),\end{aligned}$$ where ${\mathbf{r}}_S$ is the source location of the internal illumination, typically at the center of the circular metasurface hologram assumed here, and ${\mathbf{r}}_m^+$ are the locations at the metasurface internal to it where the fields are computed. In the case of an electromagnetic camouflage, the metasurface will enclose the target object, and the internal source must now be engineered to produce the same uniform internal illumination at the interior surface of the metasurface, but in the presence of the object this time. We will hereafter assume that such an internal illumination is available, for either of the application cases of illusion or camouflage to simplify the subsequent synthesis procedure. The metasurface surface susceptibility synthesis next rests on solving the GSTCs matrix equation of Eq. \[Eq:gstcd\] for the illusion surface as described by the matrices ${\mathbb{\bar D}}^\times_{I}$ and ${\mathbb{\bar G}}_{I}$ for the prescribed field ${\mathbb{S}}_I$. To extract the unknown surface susceptibilities we rearrange ${\mathbb{\bar G}}_{I}$ as, $$\begin{aligned} &{\mathbb{\bar G}}_{I} = \left[ \begin{array}{cccc} \chi_\text{me}{\mathbb{A}}_\text{me} & \chi_\text{mm}{\mathbb{A}}_\text{mm} & \chi_\text{me}{\mathbb{A}}_\text{me} & \chi_\text{mm}{\mathbb{A}}_\text{mm}\\ \chi_\text{ee}{\mathbb{A}}_\text{ee} & \chi_\text{em}{\mathbb{A}}_\text{em} & \chi_\text{ee}{\mathbb{A}}_\text{ee} & \chi_\text{em}{\mathbb{A}}_\text{em}\\ \end{array}\right]\end{aligned}$$ with $$\begin{aligned} & {\mathbb{A}}_\text{ee} = \frac{j{\overline{\mathbb{N}}}_{T}\omega\epsilon}{2},~ {\mathbb{A}}_\text{em/me} = \pm \frac{j{\overline{\mathbb{N}}}_{T}\omega\sqrt{\mu\epsilon}}{2},~{\mathbb{A}}_\text{mm} = -\frac{j{\overline{\mathbb{N}}}_{T}\omega\mu}{2} \end{aligned}$$ Considering that the metasurface susceptibilities are discretized over the surface as $\chi({\mathbf{r}}_m)$ and thus become vectors of localized susceptibilities (${\mathbb{X}}$), we can express the right hand side of Eq. \[Eq:gstcd\], as $$\begin{aligned} {\mathbb{\bar G}}_{I}{\mathbb{S}}_{I} &= \left[ \begin{array}{l} {\mathbb{X}}_\text{me}\circ\left({\mathbb{A}}_\text{me}({\mathbb{E}}_1 + {\mathbb{E}}_2)\right) \; + \\ \quad \quad {\mathbb{X}}_\text{mm}\circ\left({\mathbb{A}}_\text{mm}({\mathbb{H}}_1 + {\mathbb{H}}_2)\right)\\ {\mathbb{X}}_\text{ee}\circ\left({\mathbb{A}}_\text{ee}({\mathbb{E}}_1 + {\mathbb{E}}_2)\right) \; + \\ \quad \quad {\mathbb{X}}_\text{em}\circ\left({\mathbb{A}}_\text{em}({\mathbb{H}}_1 + {\mathbb{H}}_2)\right) \end{array}\right],\end{aligned}$$ where $\circ$ is the point-wise Hagamard product. Each of the terms, $$\begin{aligned} {\mathbb{A}}_\text{me}({\mathbb{E}}_1 + {\mathbb{E}}_2) = {\begin{bmatrix}}{\mathbb{B}}_{\text{me},xy}\\{\mathbb{B}}_{\text{me},z} {\end{bmatrix}}= {\mathbb{B}}_\text{me},\end{aligned}$$ is a column vector of one component of tangent fields ($xy$ and $z$). If we wish to create a distributed $\chi$ vector we can form (for example) products such as, $$\begin{aligned} {\mathbb{\bar G}}_\text{me} {\mathbb{X}}_\text{me} &= {\mathbb{X}}_\text{me} \circ {\mathbb{A}}_\text{me}({\mathbb{E}}_1 + {\mathbb{E}}_2) \end{aligned}$$ where we define a diagonal matrix, $$\begin{aligned} {\mathbb{\bar G}}_\text{me} = {\begin{bmatrix}}B_\text{me,1} & 0 & \dots & 0\\ 0 &B_\text{me,2} & \dots & 0 \\ 0 & 0 & \ddots & 0 \\ 0 & 0 & 0 & B_\text{me,2N} {\end{bmatrix}}\end{aligned}$$ This form allows a very convenient expression of RHS of Eq. \[Eq:gstcd\](a), where the susceptibility matrix term is explicitly extracted as $$\begin{aligned} {\mathbb{\bar G}}_{I}{\mathbb{S}}_{I} &= {\begin{bmatrix}}{\mathbb{\bar G}}_{me} & {\mathbb{\bar G}}_{mm} &{\varnothing}& {\varnothing}\\ {\varnothing}& {\varnothing}& {\mathbb{\bar G}}_{ee} & {\mathbb{\bar G}}_{me} {\end{bmatrix}}{\begin{bmatrix}}{\mathbb{X}}_{em}\\{\mathbb{X}}_{mm}\\{\mathbb{X}}_{ee} \\ {\mathbb{X}}_{me}{\end{bmatrix}}_I \nonumber\\ &= {\mathbb{\bar Q}}{\mathbb{X}}_I.\end{aligned}$$ Finally using Eq. \[Eq:gstcd\], we now have the explicit relationship for the spatially varying surface susceptibility matrix as $$\begin{aligned} {\mathbb{X}}_I = {\mathbb{\bar Q}}^{-1} {\mathbb{\bar D}}_{I}{\mathbb{S}}_{I},\label{Eq:Ms_Synth}\end{aligned}$$ which can be used directly for metasurface synthesis for a given ${\mathbb{S}}_{I}$. This finally completes the synthesis of the metasurface shield. This entire design flow is illustrated in the flow chart of Fig. \[Fig:Flow\]. It should be noted that there is no unique surface susceptibility distribution that can generate the specified illusion. There are in total 36 complex unknowns in the susceptibility tensors, and the number of possible solutions generating the same illusion is virtually unlimited. Certain constraints may be applied such as reciprocity and lossless-ness, which may limit the number of combinations and may provide susceptibility distributions which may be more convenient to practically implement, i.e. $$\begin{aligned} \begin{array}{lccc} \text{Reciprocity:} & ~\overline{\overline{\chi}}_\text{ee}^\text{T} = \overline{\overline{\chi}}_\text{ee}, & \overline{\overline{\chi}}_\text{mm}^\text{T} = \overline{\overline{\chi}}_\text{mm}, & \overline{\overline{\chi}}_\text{me}^\text{T} = -\overline{\overline{\chi}}_\text{em}\\ \text{Losslessness:} & ~\overline{\overline{\chi}}_\text{ee}^\text{T} = \overline{\overline{\chi}}_\text{ee}^\ast, & \overline{\overline{\chi}}_\text{mm}^\text{T} = \overline{\overline{\chi}}_\text{mm}^\ast,& \overline{\overline{\chi}}_\text{me}^\text{T} = \overline{\overline{\chi}}_\text{em}^\ast. \end{array} \end{aligned}$$ In many practical cases, a fully general bi-anisotropic metasurface may not be desired (i.e. $\overline{\overline{\chi}}_\text{me} = \overline{\overline{\chi}}_\text{em}=0$), which will reduce the unknowns to 18. If further simplicity is desired, tensors maybe reduced to scalars ($\overline{\overline{\chi}}_\text{ee} = \chi_\text{ee}$ and $\overline{\overline{\chi}}_\text{mm} = \chi_\text{ee}$), with only 2 unknowns, as will be assumed here, which is the minimum number of unknowns to have unique solution. If material bounds must be respected, such as synthesizing lossless metasurfaces, scalar surface susceptibilities are not be sufficient, and extra elements of the tensorial susceptibilities must instead be used, which can easily be incorporated inside the proposed method. Given that we can now synthesize an illusion surface, we turn to a more precise definition of the *Observer*, before providing numerical demonstration of the proposed synthesis method. [SFO.pdf]{} (17,36.5) (1,50) (10,66) (25.5,55.5) (25.5,53) (22,42.5) (6,49) (49,36.5) (33,50) (39,65) (51.5,56.5) (55,43) (81,36) (66,49) (75,66) (82.5,55) (80,50) (89,41) (88,52.5) (17,3) (1,17.5) (7,32.5) (47,3) (33,15.5) (40.5,32.5) (40,8) (43,22.5) (39,28.5) (77,3) (65,17.5) (84,20) Modeling an Observer {#Sec-IV:Modeling An Observer} ==================== In order to evaluate the accuracy of an illusion or its effectiveness under various circumstances, such as environmental changes or illumination variation – it is needed to compare the electromagnetic characteristics of the reference scene and the illusion scene. Two approaches are used in this paper: 1) a direct comparison of scattered or total fields at an observation position or a set of observation positions defined by a line or arc, and 2) a more sophisticated sampling of the field within an observation region allowing for the incorporation of a field of view and the directionality of the scattered and total field components. The second method can be used to, in essence, render an image of the scene. Direct Field Comparison ----------------------- A straight-forward method of evaluating the effectiveness of an illusion is to compare, at prescribed locations, the field distributions within the scene to the original baseline simulation. We will refer to this as Direct Field Observation (DFO). If wished, the complex vectorial field components can be compared or simply the magnitude of the various fields. Although this comparison is easy to accomplish there are a number of complications in interpretation. As discussed previously, if the observer is viewing a front-lit object from the front only the scattered field will be observed as the illuminating (or incident) field is propagating from behind the observer. On the other hand if viewing from the far side of a front lit object the observer will collect contributions from both the scattered field and the illuminating field. For an illusion created from a vertical metasurface as discussed in [@smy2020surface] these distinctions are easily handled by delineating illusions as front-lit or back-lit and subject to front or back illumination, however, for closed surface illusions, these distinctions are not clear. It is not obvious from inspection if total or scattered fields should be compared at a particular observation position. Another significant issue with this method is the lack of any ability to obtain information about the directional structure of the field in the region nearby the observation position. Without this information, the Observer cannot be characterized by a Point-of-View (POV) and a Field-of-View (FOV) – the direction in which the observer is looking and the range of angles over which the observation is obtained. This limitation is a direct consequence of sampling the field with a delta function and only obtaining field information at a point. As a result of this limitation there is no ability to reconstruct the scene from the view point of the observer. [HornVsSFO.pdf]{} (17,32) (2.5,45) (14.5,55.2) (14.5,36.9) (48,32) (34,44) (53,43.5) (46.8,49.5) (56,37) (83,32) (67,43) (90.5,49.5) (90.5,51.5) (77.3,53.8) (17,3) (2.5,16) (14.5,26) (14.5,7.6) (48,3) (34,15) (53,14) (46.8,20) (56,8) (83,3) (67,12) (72,22.7) (92,22.7) (76.4,8) (83,20.5) (10,60.3) (36,60.3) (0,38) (0,7) Distributed Observer -------------------- To obtain a more effective realization of an Observer we sample the field within a local region using a spatial weighting function, $w({\mathbf{r}})$, and then use a defined angular response of the observer to weight the incident fields. This method we will refer to as the Structured Field Observation (SFO). The weighting function is needed to sample the field in the region such that the field is attenuated at the edges to essentially zero but that the field at the center of the region is smoothly modulated. This weighting operation can be written as $$\begin{aligned} \label{Eq:RectSample} \psi^\text{obs.}({\mathbf{r}}) = \psi^\text{tot.}({\mathbf{r}})w({\mathbf{r}}),\end{aligned}$$ where $\psi^\text{tot.}$ is the total fields in the vicinity of the observer, $w({\mathbf{r}})$ is the weighting function used to sample the fields, and $\psi^\text{obs.}({\mathbf{r}})$ is the resulting conditioned fields. This sampling of the field allows for a spatial Fourier transform to be used to obtain the directional structure of the field in the local region around the Observer. We will use a Gaussian weighting function here, as it is simple and well suited to the spatial Fourier transform approach used. To illustrate this approach, in Fig. \[fig:ObsDef\](a), a complex field consisting of a Gaussian beam incident field (propagating left to right along the $x-$axis) and back scattered components off a number of objects. An Observer position is specified in the region and the weighting function is defined as a square area with an extent of $5\lambda$ around this point is defined as the observation area. In this region a high resolution field description is obtained using propagation operators and the currents solved for on the surface of the objects. This field is presented in Fig. \[fig:ObsDef\](b). To obtain the directional structure of this field, a 2D Fast Fourier Transform (FFT) can be performed as $$\begin{aligned} \label{Eq:RectSampleFFT} \tilde{\psi}^\text{obs.}(k_x, k_y)& = \mathcal{F}\{\psi^\text{tot.}(x,y)w(x,y)\} \\ \notag &= \tilde{\psi}^\text{tot.}(k_x,k_y)\circledast\tilde{w}(k_x,k_y)\end{aligned}$$ which is shown in Fig. \[fig:ObsDef\](c), where the sign of the spatial frequencies $k_x$ and $k_y$ provides the directional information of the EM power flow. Although the directional components of the field can be seen in this plot, the field in Fig. \[fig:ObsDef\](a) has effectively been sampled with a 2D pulse function and the resultant FFT has a Sinc function response convoluted ($\circledast$) with the actual field leading to undesirable artificial diffraction and a smeared response. This effect can be alleviated by sampling and conditioning the local field with a Gaussian function instead, with a width of $w_0 = 2.5\lambda$, for instance, given by $$\begin{aligned} \label{Eq:GaussWeightFctn} w(x,y) = \exp\left(\frac{x^2 + y^2}{w_0^2}\right),\end{aligned}$$ where this width $w_0$ is considered one of the characteristic of the observer measurement capability. The resulting sampled field is presented in Fig. \[fig:ObsDef\](d) and a 2D-FFT of the conditioned field is shown in \[fig:ObsDef\](e). The FFT clearly shows the scattered components of the field (left hand side of the plane and negative $k_x$) and the single incident wave on right hand side of the plane (positive $k_x$). Moreover, the spatial asymmetry of the scene with respect to the $x-$axis is also captured in the different strength of $\tilde{\psi}^\text{obs.}$ in the $k_x-k_y$ plane about the $k_x=0$ axis. Note the FFT was performed using zero padding to extend the size of the Observation region by 4 times to smooth the response. The field in Fig. \[fig:ObsDef\](e) can be used to define an Observer with a POV and FOV and allow for a rendering of the scene from this position. A POV is first defined in the $x-y$ plane using an angle $\theta$ and the objective is to determine the total power measured by the observer, $\mathcal{P}(\theta)$, i.e. the scene, taking into account its directional measurement capability, $p(\phi)$ (akin to radiation patterns of electromagnetic antennas). The Fourier-transformed fields of Fig. \[fig:ObsDef\](e) represents the power traveling through the physical region shown in Fig. \[fig:ObsDef\](d) decomposed into plane-waves described by the wave-vectors $k_x$ and $k_y$ in the form of $e^{j\omega t + k_x x + k_yy}$. The angular receive function $p(\phi)$ of the observer is thus defined naturally in the $k_x-k_y$ domain, where it represents the power received by the observer when excited with a plane-wave incoming at an angle $\phi$. The receive function is assumed to be *maximized* at $\phi = -\theta$ which represents a plane-wave incoming at normal incidence to the observer POV. As the simulation is at a fixed $\omega$, all the waves will be present on the circle with a radius of $k_0$. In order to determine the total field incident on the observer from a particular direction POV, $\theta$, we first extract the fields present on the $k_0$ circle. The response function $p(\phi)$ is next orientated such that it points in the direction of the POV, i.e. maxima at $\phi = -\theta$. The fields on the circle are then weighted by the receive function and integrated over the circle. To obtain the total power $P(\theta)$, we then square the absolute magnitude of the field. This operation can be expressed as: $$\begin{aligned} \label{Eq:PowerScan} \mathcal{P}(\theta) = \left|\int_{\theta - \pi}^{\theta + \pi} p(\phi)\tilde{\psi}^\text{obs.}(\rho = k_0, \phi) d\phi \right|^2\end{aligned}$$ To finally render a scene over the entire FOV, $\theta$ is simply scanned over the angular extent of interest and $P(\theta)$ determined. Such a response is shown in Fig. \[fig:ObsDef\](f), where $\mathcal{P}(\theta)$ is shown for the angular range of $\theta$ covering the second half of the $x-y$ region, for instance. [Horn\_SFO\_2\_Case.pdf]{} (8,32.5) (16.5,3) (1.5,17) (8,25) (22,15) (7,16) (40,32.5) (48.5,3) (32.5,17) (40,25) (38,16) (51,12) (82,4.5) (65.5,10.5) (73,28.6) (73,27.6) (82,18) (65.5,25.2) (72.5,14.6) (72.5,13.6) Horn Antenna vs SFO ------------------- A direct comparison of this distributed method of observation and an actual observation device can be made to test the efficacy of the above procedure in modeling the observer. For a radio frequency signals, an appropriate observing device is a *Horn antenna*. A horn antenna is directional with an angular response defined by the input mode distribution at its front radiating aperture. Let us consider two obliquely propagating plane-waves ($\pm 15^\circ$) which are being received by a horn antenna as shown in Fig. \[fig:HornvsSFO\](a). Since, the two waves form a standing wave along $y$, the exact placement of the horn is important. This can clearly be seen by setting the POV to $\theta = 180^\circ$ and using two specific conditions of in-phase and out-of-phase plane-waves at the aperture of the horn – with a corresponding high and low reception of the waves as evident from the field distribution at the end of the waveguide section. If such a horn antenna is to be used to render a scene, the horn must be rotated about $(0,0)$, for instance, and for every angle $\theta$, the full-wave simulation must be performed to compute the total power received by the horn to construct $\mathcal{P}(\theta)$. This will correspond to a Direct Field Observation (DFO) with the horn in place. While it provides accurate and rigorous description of the scattered fields around the horn, it is computationally expensive, and lacks deeper physical insight. Let us apply the Structure Field Observation (SFO) to reconstruct the scene. In this case, the horn is *removed from the scene*, and the total fields are computed. In the region of interest around the horn, the field is conditioned by the spatial Gaussian function and the 2D-FFT taken, as shown in Fig. \[fig:HornvsSFO\](b) for the two cases of in-phase and out-of-phase conditions. For each case, the two plane-waves are clearly identified lying on the $k_0$ circle, with correct phase relationship between the two, evident in the symmetric and antisymmetric field magnitudes. Using this $k_x-k_y$ domain field representation, the scene can now be rendered following the methodology of Fig. \[fig:ObsDef\]. Before proceeding further, the angular receive function, $p(\phi)$ of the horn must be determined. Fig. \[fig:HornvsSFO\](c) shows the co-polarized radiation pattern (power gain) of the horn obtained using HFSS, showing a 3-dB beamwidth of $18^\circ$ at 60 GHz. For convenience, this receive function can be approximated by a Gaussian function, as shown in Fig. \[fig:HornvsSFO\](c). Clearly, the Gaussian function approximates the horn pattern well, particularly the main beam, while failing to capture the side-lobes. Although, in principle the FEM-HFSS pattern may be directly used as the $p(\phi)$, the Gaussian function represents a simple and a generic form of the receive function that may be applicable to wide variety of observers. Next, the scene is rendered using with the assumed Gaussian receive function, and the computed power scan is shown in Fig. \[fig:HornvsSFO\](d). As a sanity check, the SFO scene of determined using single plane-waves, one at a time were compared with those computed using FEM-HFSS (solid vs dashed curves). A near-perfect match is obtained between the two, thus providing confidence that the SFO captures the basic behavior of a directionally sensitive observer. Next, both plane-waves are present and two computed scenes are compared with those using full-wave HFSS in Fig. \[fig:HornvsSFO\](d) for cases of in and out-of-phase plane-wave conditions, respectively. A good agreement is obtained between the two, where the two plane-waves are seen to be resolved with correct magnitudes at $\theta=0^\circ$. While the overall power distribution is correctly captured by the SFO, the discrepancies with the full-wave results can be attributed to the actual scattering by the horn which are naturally not present in the SFO method. This is particularly attractive, since the use of SFO is very efficient as it provides the complete angular response with a single simulation and does not require a set of simulations as does a direct simulation of a Horn in either HFSS or using the IE method. To further confirm the appropriateness of the use of the SFO, Fig. \[fig:hornvw\] shows a comparison between direct Horn simulations using the IE framework and the SFO for two different scenes with more complexity. These two examples scenes are: 1) a single highly reflective cylindrical object; and 2) two interacting reflective cylinders (Fig. \[fig:hornvw\](a) and \[fig:hornvw\](b) respectively). A comparison of rendering the two scenes is presented in Fig. \[fig:hornvw\](c). A very good agreement is observed between the brute force IE method and the SFO, thereby justifying the usability of the SFO for constructing an Observer scene. Next, several examples will be presented using the proposed IE-GSTC infrastructure and the SFO model of the observer, to demonstrate the camouflaging and illusion forming capabilities of a closed metasurface structure. Examples: Illusion & Camouflaging {#Sec-V} ================================= To illustrate the synthesis procedure for metasurface based illusions and object camouflaging, we will first consider some basic examples and then progress to more complicated cases involving merged illusions and dynamic illusions. The simulation setup follows the illustration of Fig. \[Fig:SimSetup\], in the $x-y$ plane, and the frequency of operation is chosen to be $60$ GHz ($\lambda = 5$ mm). The surface meshing is set to $\lambda/10$ based on proper convergence. The region over which fields were plotted was 40$\lambda$ in both the $x$ and $y$ directions. The objects from which the illusions were created are of the order of 10’s of $\lambda$ is size, with a variety of shapes chosen to illustrate the methods ability to capture the full wave diffraction present. Objects with sharp corners produce large diffractive effects, but smoother objects can produce lens like effects. Although these objects were kept moderate in size to illustrate effects. It should be noted that for the IE-BEM method the fields are plotted post-simulation and the region to be plotted is an arbitrary choice. Only the scalar electric and magnetic surface susceptibilities are used which were found sufficient to create the illusions. More complex susceptibility distributions utilizing other tensorial elements may be used to satisfy possible practical constraints such as lossless-ness and reciprocity as mentioned in Sec. III. As will be seen, all the surface susceptibilities are found to be passive and physical with appropriate choices of the illumination fields. Simple Illusions ---------------- Let us consider a reference case consisting of 4 circular cylinders made up of two different materials (PEC and dielectric), which are excited with a wide Gaussian beam. The scattered and the total fields produced as a result are shown in Fig. \[Fig:SimpleIllusion\_Ref\]. This collection of objects, produce a large shadow behind them. An observation circle is defined at $r_0$ for DFO. In addition, two observers are placed in the upper and lower halves, respectively, to construct the SFO. We now wish to reconstruct the same fields and scenes by removing the objects and replacing them with a single metasurface hologram, as shown in Fig. \[Fig:Simple\_Illusion\_Circle\]. [Simple\_Illusions\_Reference.pdf]{} (26,1) (1,25) (10,48) (32,42.5) (25,15) (9,41) (76,1) (50,25) (62,48) (89,29) (89,36) [Simple\_Illusion\_Ext\_Int.pdf]{} (15,75) (3,86.5) (8,94.5) (19,77.7) (7,98) (42.2,75) (30,86.5) (35,98) (70,87) (56,90) (64.5,91.8) (64.5,90.5) (71,75) (56,79) (15,49.5) (3,61) (18.5,70) (14.5,56.2) (42.2,49.5) (30,61) (44,70) (70,61) (57,64) (71,49.5) (57,53) (15,3) (3,14) (18.5,23.3) (14.5,9.5) (8,22.2) (42.2,3) (30,14) (70,14.8) (57,18) (71,2.8) (57,6.6) (85,97) (85,72) (85,26) (16,28.5) (3,35) (44,28.5) (31,35) (36,44) (72,28.5) (58.6,37) (80,32) (75,44) (75,41) (75,38.5) [Simple\_Illusion\_Square.pdf]{} (16,1.2) (0.5,15.3) (5.5,26) (21,5) (8,31) (48.5,1.2) (32.5,15.3) (42,31) (78,16) (64,21) (81,22) (81,20.3) (78,1.2) (64,6) Let us consider a natural choice of incident fields as the illumination fields *external* to the metasurface. Fig. \[Fig:Simple\_Illusion\_Circle\](a) shows the computed scattered and total fields everywhere in the region around the surface. Comparison with the reference fields of Fig. \[Fig:SimpleIllusion\_Ref\] shows a clear and a large discrepancy in the reconstructed fields, where the metasurface synthesis essentially fails. The method produces susceptibilities that are large, active and disjointed, where the simulation becomes very sensitive to illumination intensity. This poor match may be attributed to a non-uniform one-sided illumination of surface, where the surface fails to re-create the fields to “shape” the shadow to match the illusion. Noting that the previous simulation failed due to non-uniform illumination a natural modification is to use a uniform internal illumination. We thus chose an internal illumination that is a radially propagating wave from center of the metasurface hologram with $|E_z|= 2$ at the surface. The resulting total and scattered fields are shown in Fig. \[Fig:Simple\_Illusion\_Circle\](b), where this time the illusions is better formed and reproduces the reflected scattered fields in the reference scene well. The corresponding surface susceptibilities are well-behaved and are passive as also shown in Fig. \[Fig:Simple\_Illusion\_Circle\](b). However, the illusion can not reproduce the *total fields* throughout the scene as the fields are restricted to outwardly propagating waves. Only within the shadow does it reasonably produce the total fields. This is clearly evident in the DFO and SFO results shown in Fig. \[Fig:Simple\_Illusion\_Circle\](c), compared to the reference case. While the POV\#2 lying in the shadow region (with mostly scattered fields) is reconstructing the scene well to some extent, the POV\#1 fails and measures significantly reduced fields. We thus conclude that the internal illumination only is also not sufficient to correctly produce the illusion. Thus the only way to recreate the total fields throughout the region is to have the incident field also present in the illusion scene, so that we have both an internal illumination (radially propagating internal wave) and an external illumination identical to the original incident fields. Fig. \[Fig:Simple\_Illusion\_Circle\](d) shows the resulting scattered and total fields along with the synthesized surface susceptibilities for this case. This time a perfect reconstruction of the reference fields of Fig. \[Fig:SimpleIllusion\_Ref\] is observed proving that its critical to have both internal and external illumination fields to be present. Internal illumination creates the fields on the right side of the illusion surface, as in this region, it only needs to create the outward propagating scattered fields that form the shadow. On the left side it can not create the total field on the surface due to the need to create inwardly propagating waves. Adding an external illumination assists in recreating this part of the fields and the match between the reference and reconstructed fields is prefect. Consequently, the DFO and SFO accurately agree with the reference scene as shown in Fig. \[Fig:Simple\_Illusion\_Circle\](c). Finally, to show the versatility of the method and the metasurface hologram, the same reference scene of Fig. \[Fig:SimpleIllusion\_Ref\] can be recreated using a shape different than a circular surface. Fig. \[Fig:Simple\_Illusion\_Square\] shows one such example, where a rectangular shaped metasurface hologram is used. As expected, the fields recreated perfectly similar to that of Fig. \[Fig:Simple\_Illusion\_Circle\], except this time, the surface susceptibilities across the surface are different, and showing larger spatial variation around the two corner regions of the square. Angle-dependent Metasurface Illusions ------------------------------------- In the previous example, an observer moving about the metasurface (and thus around the virtual object), will measure identical fields compared to that of the reference scene, and thus perceive the same object from all angles of observations. However, the metasurface hologram maybe engineered to reproduce almost arbitrary fields around it so that different illusions may be projected and the observer perceives different objects when viewing the surface from different directions. Such a design capability is illustrated in Fig. \[Fig:MergedIlusion\], where four different object configurations are shown in Fig. \[Fig:MergedIlusion\](a), with variety of objects producing significantly different scattered fields. We now wish to a synthesize a metasurface hologram that can recreate the total fields produced by these four configurations in four quadrants simultaneously. To measure the effectiveness of this illusion, four observers with their own POVs are placed across all quadrants. [Merged\_Illusion.pdf]{} (14.5,67) (3,76.5) (9.5,84.5) (9,88) (37,67) (26,76.5) (41.5,84.5) (28,88) (60,67) (48,76.5) (53.5,71) (51,88) (81,67) (70.7,76.5) (85.5,71) (75,88) (15,34) (0.5,47) (9.5,57) (22,57) (9.5,39.5) (22,39.5) (10,61.5) (49,34) (33.5,47) (41.5,57) (54,57) (41.5,39.5) (54,39.5) (38,61.5) (45,59) (47,39) (80,34) (65.5,47) (71,61.5) (14,16) (0.5,20) (15,21.6) (15,20.2) (48,16) (32,20) (40.5,26) (40.5,24.5) (83,16) (65.8,20) (14,3) (0.5,6.5) (48,3) (32,6.5) (83,3) (65.8,6.5) The first step in this synthesis is to spatially filter each of the reference configurations and combine them into a single scene. A 2D Sigmoid function with transition width $w = 0.5\lambda$ used here as a spatial filter, for instance, which is given by $$\begin{aligned} f(x,y) = \left(\frac{1}{1+e^{-x/w}}\frac{1}{1+e^{-y/w}}\right).\end{aligned}$$ The resulting combined scattered fields are shown in Fig. \[Fig:MergedIlusion\](b), along with an intended location of the metasurface hologram. Next, the metasurface is synthesized with both internal and external illumination along with the desired scattered fields of Fig. \[Fig:MergedIlusion\](b). The resulting total and scattered fields are shown in Fig. \[Fig:MergedIlusion\](c). An excellent recreation of the fields is observed all around the metasurface, whose surface susceptibilities are shown in Fig. \[Fig:MergedIlusion\](d). The scene measured by the four observers in each quadrant is shown further in Fig. \[Fig:MergedIlusion\](e) in comparison with the reference scene, which are computed using the SFO method. A very good agreement is observed in all cases. The slight discrepancies are due to the finite and smooth field transitions introduced by the Sigmoid function when moving from one quadrant to the other. The four reconstructed scenes are thus specific to the observer in their respective quadrants, and an observer moving around the metasurface hologram will sequential perceive these four object configurations, and experience an angle dependent illusion. [Dynamic.pdf]{} (15,84) (1.5,91) (10.1,97) (10.1,95.5) (45,84) (32,91) (41,97) (41,95.5) (41,93.8)[ ]{} (78,84) (62.5,91) (72,97) (72,95.5) (72,93.8)[ ]{} (0.5,66.5) (7,79.5) (22,56.6) (22.4,72) (24,67.4) (7,75) (31,66.5) (38,79.5) (53,56.6) (62,66.5) (69,79.5) (84,56.6) (16,20.5) (0.5,34) (8,49) (46,20.5) (31,34) (38,49) (77.5,20.5) (62,34) (69,49) (16,3) (2,8) (21,44) (47,3) (32,8) (51,25) (78,3) (63,8) Dynamic Illusions ----------------- So far, all the scenes were considered to be static with no time-variation. In many practical scenarios, the scene may be changing with time due to spatial movement of objects or addition or removal of some scene elements. In such cases, to maintain the illusion of a specified object, the metasurface hologram must reconfigure itself in response to the changes in the scene. There have been several works in the literature on devising novel reconfigurable metasurfaces across the electromagnetic spectrum where the basic unit cell building the metasurface, is either loaded with external tuning elements like PIN and varactor diodes, common at microwave frequencies or using exotic and advanced materials whose electronic properties may be tuned using external controls [@Reconfigr_MS; @Reconfigr_MS2; @Optical_MS_Reconfig]. While majority of the works on reconfigurable metasurfaces features slow time-variation, so that the modulation frequency is very small compared to the operation frequency of the surface, i.e. $\omega_p \ll \omega_0$, there has been a several research activities in the area of space-time modulation, where the modulations are comparable to operation frequencies of the surface [@STmod]. Let us consider the scenario of a slowly changing environment to illustrate the case of a dynamic illusion assuming reconfigurable surface susceptibilities of the metasurface hologram. [Cloak.pdf]{} (16,4.5)[$x$ (m)]{} (16,38.5)[$x$ (m)]{} (49,4.5)[$x$ (m)]{} (81,4.5)[$x$ (m)]{} (1.5, 52) (1.5, 18) (34.5, 18) (66, 18) (47,38.5)[$\theta$ (deg)]{} (80,38.5)[$\theta$ (deg)]{} (31, 50) (64, 52) (22.5,43)[<span style="font-variant:small-caps;"></span> ]{} (9,43)[<span style="font-variant:small-caps;"></span>]{} (8,62)[<span style="font-variant:small-caps;">Observer</span>]{} (86,59)[ ]{} (8,28)[<span style="font-variant:small-caps;">Observer</span>]{} (41,28)[<span style="font-variant:small-caps;">Observer</span>]{} (73,28)[<span style="font-variant:small-caps;">Observer</span>]{} (8,32) (40,32) (72,32) (8,66.5) (13,24)[<span style="font-variant:small-caps;"></span>]{} (45,23)[<span style="font-variant:small-caps;"></span>]{} (9,9)[<span style="font-variant:small-caps;"></span>]{} (49,46.25)[$t_0$]{} (49,44.75)[$t_1$]{} (49,43)[$t_2$]{} Consider the reference scene of Fig. \[Fig:Dynamic\](a) to be recreated. It consists of a square dielectric inclusion (DI) and a PEC triangular object in front of a perfectly opaque background. They are excited from the left using a Gaussian beam, and an observer located in the scene measures the scattered fields to image the scene. The DI moves through region on parameterized trajectory to simulate a changing environment, while the PEC object and the background remain fixed with respect to the observer. At time instant $t_0$, the scattered fields are computed and used to synthesize a metasurface with $\chi_\text{ee}(\theta, t_0)$. The PEC object is then removed and replaced by a circular metasurface hologram, successfully recreating the fields, as seen from the field and the SFO plots shown in Fig. \[Fig:Dynamic\](a). Next, the DI is allowed to move along a specified trajectory, which perturbs the originals scattered fields and the new reference scene is shown in Fig. \[Fig:Dynamic\](b). The scattered fields are recomputed and a new metasurface is synthesized to recreate the scene at $t = t_1$, resulting in $\chi_\text{ee}(\theta, t_1)$. Consequently, the observer measures the new scene and deduce only the motion of the DI while falsely perceiving the presence the PEC object as before. Had the metasurface susceptibilities been constant, the scene would have changed as a result of the DI motion, which is clearly observed in the static SFO plot of Fig. \[Fig:Dynamic\](b). This would have enabled the observer to detect a different and a distorted non-triangular PEC object, thereby breaking the illusion. This process of computing the scattered fields and synthesizing the metasurface hologram is repeated periodically, as long as the DI is in motion, as further illustrated for a third time instant $t=t_2$ in Fig. \[Fig:Dynamic\](c)[^3]. Such a dynamic illusion naturally rests on the reconfigurable capability of the metasurface, and the time-scale at which such changes happen on the device level [@Nature_Cloak_AI]. An important step in this case is the sensing and prediction of the changes in the environment by either the metasurface itself or the software control unit configuring the surface, so that the scattered fields may be efficiently computed to synthesize the surface susceptibilities. The total time to reconfigure a metasurface hologram thus essentially depends on this intermediate step, where the actual reconfiguration of the surface is usually fast, typically in the order of few microseconds or less [@Reconfigr_MS; @Reconfigr_MS2; @Optical_MS_Reconfig]. Electromagnetic Camouflage -------------------------- As illustrated in Sec. II, while electromagnetic camouflage is a sub-set of a general holograms, where the holograms projects its background to the observer to deceive it, it is an important practical application on its own. Thus let us consider an example of electromagnetically camouflaging a region in a complex scene. Fig. \[Fig:Cloak\](a) shows a reference scene consisting of triangular PEC object placed in front of a patterned background. The background is modeled using a metasurface with alternating reflective and absorbing region to generate complex scattering fields. An incident Gaussian beam from the left is next applied which generate the scattered fields from the PEC object and the pattern background, in the entire region of interest. The observer simply detects the object and the background. Now, we wish to have an arbitrary object move and navigate through this region without being detected by the observer, thereby effectively camouflaging it at all times. To achieve this, a metasurface skin could enclose the object of interest, and with internal illumination to mimic its composite background consisting of the PEC object and the patterned wall. Knowing the desired reference fields of Fig. \[Fig:Cloak\](a), the required surface susceptibilities of the metasurface are computed and applied to the region. Fig. \[Fig:Cloak\](a) shows the scene with the region included, and as expected, the reference fields are accurately reproduced in the entire region. Since, the observer measures the same fields, it does not detect the camouflaged region. As the region moves inside the scene, the metasurface susceptibilities must be recomputed at each time similar to that in Fig. \[Fig:Dynamic\]. Fig. \[Fig:Cloak\](b) shows three time instants corresponding to three different positions of the camouflaged region along with slightly different spatial distributions of the surface susceptibilities shown in Fig. \[Fig:Cloak\](c). In all cases, the SFO measured by the observer is the same, as shown in Fig. \[Fig:Cloak\](d), and thus the camouflaged region remains undetected at all times[^4]. Conclusions =========== A systematic numerical framework based on IE-GSTCs has been presented in 2D to synthesize closed metasurface holograms and skins for creating electromagnetic illusions of specified objects. The general hologram surface has been modeled using a zero-thickness sheet model of a generalized metasurface expressed in terms of its surface susceptibilities, which has been further integrated into the GSTCs. In combination with the IE current-field propagation operators, a simple yet powerful numerical framework of IE-GSTC has been developed. It has been further shown that the phenomenon of electromagnetic camouflage can be considered a special case of a general hologram principle, where instead of projecting the illusion of a specific object, the object wrapped in a metasurface skin may simply mimic its background by projecting that to the observer as the illusion. To estimate the effectiveness of the illusions, the notion of a scene constructed by an observer is developed from first principles and a simple mathematical model, termed as SFO, based on spatial Fourier transform has been proposed to construct the angular scene measured by an observer. The SFO method has been found to be more insightful and computationally superior to an other-wise brute-force simulation of the scene with a real detector. Next, using numerical examples, it has been shown that to recreate the reference desired fields everywhere in space using a closed metasurface hologram/skin, an internal illumination must be applied inside the hologram, in addition to the applied external illumination fields. Finally, several numerical examples have been presented of simple, angle-dependent and dynamic illusions, along with one example showing a dynamic camouflaged region of space, where it can freely move inside a given complex scene without being detected by the observer. This work represents a continuation of the metasurface hologram synthesis of [@smy2020surface] focussing on open metasurfaces, which made it possible to have a clear delineation of the two half-spaces resulting in a simple classification of Front/back-lit Posterior/Anterior Illusion with Front/Back-illumination. The Hologram synthesis has been now extended to closed metasurfaces where the previous classification doesn’t apply. Furthermore a requirement of two kinds of illuminations (external and internal) become a must for field recreation instead of a single front or back illumination of an open surface. Although implemented in 2D, the extension of the method to 3D is a straightforward task. A wide variety of 3D electromagnetic BEM code has been reported in the literature with even free software codes [@BEMpp; @PUMA-EMS; @Puma-EM], for instance. The integration of the GSTC interface equations with wide variety of BEM methods poses no fundamental issues and scaling the method to large 3D problems is simply a matter of implementation. Finally, the number of possible configurations and situations for creating EM illusions using both open and closed metasurface holograms are virtually unlimited, and so are the number of surface susceptibility solutions. While the few selected examples chosen here have been to highlight and illustrate various features of the hologram synthesis, the proposed framework represents a flexible test-bed to explore a wider variety of illusion scenarios useful for hologram designers before attempting practical demonstrations. Naturally, the unprecedented capabilities of EM metasurfaces in achieving very complex wave transformations is the core of this procedure, and this presented work in conjunction with [@smy2020surface], represents a comprehensive set of tools to design and implement versatile metasurface based holographic surfaces and camouflaging skins. Acknowledgements {#acknowledgements .unnumbered} ================ The authors acknowledge funding from the Department of National Defence’s Innovation for Defence Excellence and Security (IDEaS) Program in support of this work. [^1]: In typical holograms the illumination fields are the same as the incident fields used in the first stage of information encoding on the plates. However, a distinction is made here between the two, where the information maybe encoded in such a way, that the object may still be recreated with a different illumination field. [^2]: Note that the background is modeled using GSTCs describing an arbitrarily complex surface, as a way to introduce sophisticated texture effects to be used in later examples. [^3]: A continuous animation of this process is provided in the supplementary information as a separate multimedia file. [^4]: A continuous animation of this camouflaging process is provided in the supplementary information as a separate multimedia file.
--- abstract: | We use the action-angle variables to describe the geodesic motions in the $5$-dimensional Sasaki-Einstein spaces $Y^{p,q}$. This formulation allows us to study thoroughly the complete integrability of the system. We find that the Hamiltonian involves a reduced number of action variables. Therefore one of the fundamental frequency is zero indicating a chaotic behavior when the system is perturbed.   [*Keywords:*]{} Sasaki-Einstein spaces, complete integrability, action-angle variables.   [*PACS Nos:*]{} 11.30-j; 11.30.Ly; 02.40.Tt   author: - 'Mihai Visinescu[^1]' title: 'Action-angle variables for geodesic motions in Sasaki-Einstein spaces $Y^{p,q}$' --- Introduction ============ There has been considerable interest recently in Sasaki-Einstein (SE) geometry [@JS]. In dimension five, an infinite family of explicit SE metrics $Y^{p,q}$ on $S^2 \times S^3$ has been constructed, where $p$ and $q$ are two coprime positive integers, with $q < p$ [@GMSW2]. A $(2n-1)$-dimensional manifold $M$ is a *contact manifold* if there exists a $1$-form $\eta$ (called a contact $1$-form) on $M$ such that $$\eta \wedge (d \eta)^{n-1} \neq 0\,.$$ The *Reeb vector field* $\xi$ dual to $\eta$ satisfies: $$\eta (\xi) = 1 \quad \text{and} \quad \xi {\raisebox{-0.35ex}{\makebox[0.6em][r] {\scriptsize $-$}}\hspace{-0.15em}\raisebox{0.25ex} {\makebox[0.4em][l]{\tiny $|$}}}d\eta = 0\,,$$ where ${\raisebox{-0.35ex}{\makebox[0.6em][r] {\scriptsize $-$}}\hspace{-0.15em}\raisebox{0.25ex} {\makebox[0.4em][l]{\tiny $|$}}}$ is the operator dual to the wedge product. A contact Riemannian manifold $(Y_{2n-1}, g_{Y_{2n-1}})$ is Sasakian if its metric cone $C(Y_{2n-1}) = Y_{2n-1} \times \mathbb{R}_+$ with the metric $$ds^2( C(Y_{2n-1})) = dr^2 + r^2 ds^2 (Y_{2n-1})\,,$$ is Kähler [@BG]. Here $r\in (0,\infty)$ may be considered as a coordinate on the positive real line $\mathbb{R}_+$. If the Sasakian manifold is Einstein, the metric cone is Ricci-flat and Kähler, i.e. Calabi-Yau. The orbits of the Reeb vector field $\xi$ may or may not close. If the orbits of the Reeb vector field $\xi$ are all closed, then $\xi$ integrates to an isometric $U(1)$ action on $(Y_{2n-1}, g_{Y_{2n-1}})$. Since $\xi$ is nowhere zero this action is locally free. If the $U(1)$ action is in fact free, the Sasakian structure is said to be *regular*. Otherwise it is said to be *quasi-regular*. If the orbits of $\xi$ are not all closed, the Sasakian structure is said to be *irregular* and the closure of the $1$-parameter subgroup of the isometry group of $(Y_{2n-1}, g_{Y_{2n-1}})$ is isomorphic to a torus $\mathbb{T}^n$ [@JS]. The homogeneous SE metric on $S^2 \times S^3$, known as $T^{1,1}$, represents an example of regular Sasakian strucure with $SU(2) \times SU(2) \times U(1)$ isometry. The $Y^{p,q}$ spaces have isometry $SU(2) \times U(1) \times U(1)$ and for $4p^2 - 3 q^2$ a square they are examples of quasi-regular SE manifolds. The geometries $Y^{p,q}$ with $4p^2 - 3 q^2$ not a square are irregular SE spaces. In a recent paper [@BV] the constants of motion for geodesic motions in the five-dimensional spaces $Y^{p,q}$ have been explicitly constructed. This task was achieved using the complete set of Killing vectors and Killing-Yano tensors of these toric SE spaces. A multitude of constants of motion have been generated, but only five of them are functionally independent implying the complete integrability of geodesic flow on $Y^{p,q}$ spaces. The complete integrability of geodesics permits us to construct explicitly the action-angle variables. The formulation of an integrable system in these variables represents a useful tool for developing perturbation theory. The action-angle variables define an $n$-dimensional surface which is a topological torus (Kolmogorov-Arnold-Moser (KAM) tori) [@VIA]. Our motivation for studying the action-angle parametrization of the phase space for geodesic motions in SE spaces comes from recent studies of non-integrability and chaotic behavior of some classical configuration of strings in the context of AdS/CFT correspondence. It was shown that certain classical string configurations in $AdS_5 \times T^{1,1}$ [@BZ1] or $AdS_5 \times Y^{p,q}$ [@BZ2] are chaotic. There were used numerical simulations or an analytic approach through the Kovacic’s algorithm [@JJK] The purpose of this paper is to describe the geodesic motions in the SE spaces $Y^{p,q}$ in the action-angle formulation. We find that the Hamiltonian (energy) involves only four action variables which have the corresponding frequencies different of zero. One of the fundamental frequency is zero foreshadowing a chaotic behavior when the system is perturbed. The paper is organized as follows. In the next Section we give the necessary preliminaries regarding the metric and the constants of motion for geodesics on $Y^{p,q}$ spaces. In Sec. 3 we perform the separation of variables and give an action-angle parametrization of the phase space. The paper ends with conclusions in Sec. 4. $Y^{p,q}$ spaces ================ The AdS/CFT correspondence represents an important advancement in string theory. A large class of examples consists of type $IIB$ string theory on the background $AdS_5 \times Y_5$ with $Y_5$ a $5$-dimensional SE space. In the frame of AdS/CFT correspondence $Y^{p,q}$ spaces have played a central role as they provide an infinite class of dualities. We write the metric of the 5-dimensional $Y^{p,q}$ spaces [@GMSW2; @GMSW1; @BK] as $$\label{Ypq} \begin{split} ds^2_{Y^{p,q}} & = \frac{1-c\, y}{6}( d \theta^2 + \sin^2 \theta\, d \phi^2) + \frac{1}{w(y)q(y)} dy^2 + \frac{q(y)}{9} ( d\psi - \cos \theta \, d \phi)^2 \\ & \quad + w(y)\left[ d\alpha + \frac{ac -2y+ c\, y^2}{6(a-y^2)} (d\psi - \cos\theta \, d\phi)\right]^2\,, \end{split}$$ where $$w(y) = \frac{2(a-y^2)}{1-cy} \,, \quad q(y) = \frac{a-3y^2 + 2c y^3}{a-y^2}\,.$$ This metric is Einstein with $\Ric g_{Y^{p,q}} = 4 g_{Y^{p,q}}$ for all values of the constants $a,c$. For $c=0$ the metric takes the local form of the standard homogeneous metric on $T^{1,1}$ [@MS]. Otherwise the constant $c$ can be rescaled by a diffeomorphism and in what follows we assume $c=1$. A detailed analysis of the SE metric $Y^{p,q}$ [@GMSW2] showed that for $0 \leq \theta \leq \pi$ and $0 \leq \phi \leq 2\pi$ the first two terms of give the metric on a round two-sphere. The two-dimensional $(y, \psi)$-space defined by fixing $\theta$ and $\phi$ is fibred over this two-sphere. The range of $y$ is fixed so that $1-y >0\,,\, a-y^2 >0$ which implies $w(y) > 0$. Also it is demanded that $q(y) \geq 0$ and that $y$ lies between two zeros of $q(y)$, i.e. $y_1 \leq y \leq y_2$ with $q(y_i)=0$. To be more specific, the roots $y_i$ of the cubic equation $$a-3y^2 + 2 y^3 = 0 \,,$$ are real, one negative $(y_1)$ and two positive, the smallest being $y_2$. All of these conditions are satisfied if the range of $a$ is $$0 < a < 1 \,.$$ Taking $\psi$ to be periodic with period $2 \pi$, the $(y,\psi)$-fibre at fixed $\theta$ and $\phi$ is topologically a two-sphere. Finally, the period of $\alpha$ is chosen so as to describe a principal $S^1$ bundle over $B_4 = S^2 \times S^2$. For any $p$ and $q$ coprime, the space $Y^{p,q}$ is topologically $S^2 \times S^3$ and one may take [@MS; @GMSW2] $$0 \leq \alpha \leq 2 \pi \ell\,,$$ where $$\ell = \frac{q}{ 3q^2 - 2 p^2 + p(4 p^2 - 3 q^2 )^{1/2}}\,.$$ To put the formulas in a simpler forms, in that follows we introduce also $$f(y)= \frac{a-2y +y^2}{6(a-y^2)}\,,$$ $$p(y)= \frac{w(y) q(y)}{6} =\frac{a-3y^2 + 2y^3}{3(1-y)}\,.$$ The conjugate momenta to the coordinates $(\theta,\phi, y, \alpha, \psi)$ are: $$\label{momenta} \begin{split} &P_{\theta} = \frac{1-y}{6} \dot{\theta}\,,\\ &P_y = \frac{1}{6 p(y)} \dot{y}\,,\\ &P_{\alpha}=w(y) \left(\dot{\alpha} + f(y) \left(\dot{\psi} - \cos\theta \dot{\phi}\right)\right) \,,\\ &P_{\psi} = w(y) f(y) \dot{\alpha} + \left[ \frac{q(y)}{9} + w(y) f^2(y)\right]\left(\dot{\psi} - \cos\theta \dot{\phi}\right)\,,\\ &P_{\phi} = \frac{1-y}{6} \sin^2\theta \dot{\phi} - \cos\theta P_{\psi}\\ &~~~~= \frac{1-y}{6} \sin^2\theta \dot{\phi} - \cos\theta w(y) f(y) \dot{\alpha} - \cos\theta\left[\frac{q(y)}{9} +w(y)f^2(y) \right]\dot{\psi}\\ &~~~~~~~+\cos^2\theta\left[ \frac{q(y)}{9} + w(y) f^2(y) \right] \dot{\phi}\,, \end{split}$$ with overdot denoting proper time derivative. The Hamiltonian describing the motion of a free particle is $$\label{freeHam} H = \frac12 g^{\mu\nu} P_\mu P_\nu \,,$$ which for the $Y^{p,q}$ metric and using the momenta has the form: $$\label{HYpq} \begin{split} H=&\frac12 \Biggl\{ 6 p(y) P_y^2 + \frac{6}{1-y}\biggl(P_\theta^2 + \frac{1}{\sin^2 \theta}(P_\phi + \cos\theta P_\psi)^2\biggr) + \frac{1-y}{2(a-y^2)}P^2_\alpha\Biggr.\\ & \Biggl.+ \frac{9(a-y^2)}{a-3y^2 +2 y^3}\biggl(P_\psi - \frac{a -2y +y^2}{6(a-y^2)} P_\alpha \biggr)^2\Biggr\}\,. \end{split}$$ Starting with the complete set of Killing vectors and Killing-Yano tensors of the SE spaces $Y^{p,q}$ it is possible to find quite a lot of integrals of motions [@BV; @SVV1; @SVV2]. However the number of functionally independent constants of motion is only five implying the complete integrability of geodesic flow on $Y^{p,q}$ spaces. For example we can choose as independent conserved quantities the energy $$\label{E} E=H\,,$$ the momenta corresponding to the cyclic coordinates $(\phi\,,\, \psi\,,\,\alpha)$ $$\label{Pcyc} \begin{split} &P_{\phi} = c_{\phi}\,,\\ &P_{\psi} = c_{\psi}\,,\\ &P_{\alpha} = c_{\alpha}\,, \end{split}$$ where $(c_{\phi}\,,\,c_{\psi}\,,c_{\alpha})$ are some constants, and the total $SU(2)$ angular momentum $$\label{J2} \vec{J}^{~2} =P_{\theta}^2 + \frac{1}{\sin^2\theta} \left(P_{\phi}+ \cos\theta P_{\psi}\right)^2 + P_{\psi}^2 \,.$$ Action-angle variables ====================== The connection between completely integrable systems and toric geometry in the symplectic setting is described by the classical Liouville-Arnold theorem [@VIA; @GPS]. A dynamical system defined by a given Hamiltonian $H$ on a $2n$-dimensional symplectic manifold $(M^{2n},\omega)$ is called Liouville integrable if it admits $n$ functionally independent first integrals in involution. In other words, there are $n$ functions $\mathbf{F} = (f_1 =H, f_2,\dots,f_n)$ such that $df_1\wedge \dots \wedge f_n \neq 0$ almost everywhere and $$\{f_i,f_j\} = 0 \quad, \quad \forall i,j \,.$$ Let $\mathbf{F}_\mathbf{c} = (f_1 =E, f_2= c_2,\dots,f_n= c_n)$ by a common invariant level set. If $\mathbf{F}_\mathbf{c}$ is regular, compact and connected, then it is diffeomorphic to the $n$-dimensional Lagrangian torus. For $n$ degrees of freedom the motion is confined to an $n$-torus $$\label{it} \mathbb{T}^n = \underbrace{S^1 \times S^1 \times \cdots \times S^1}_{\text{n~times}} \,.$$ These are called *invariant tori* and never intersects taking into account the uniqueness of the solution to the dynamical system, expressed as a set of coupled ordinary differential equations. In a neighborhood of $\mathbf{F}_\mathbf{c}$ there are action-angle variables $\mathbf{J},\mathbf{w}$ mod $2\pi$, such that the symplectic form becomes $$\omega = \sum_{i=1}^{n} d J_i \wedge d w_i\,,$$ and the Hamiltonian $H$ depends only on actions $J_1,\dots,J_n$. An action variable $J_i$ specifies a particular $n$-torus $\mathbb{T}^n$ and is constant since the tori are invariant. The location on the torus is specified by $n$ angle variables $w_i$. Even the system is integrable, the dynamics on the singular set (where the differentials of the integrals $f_1,\dots,f_n$ are dependent) can be quite complicated [@JJD]. In the case of the geodesic motions on $Y^{p,q}$, for the beginning, we fix a level surface $\mathbf{F}=(H, P_\phi,P_\psi,P_\alpha,\vec{J}^{~2})=\mathbf{c}$ of the mutually commuting constants of motion –. The differentials of the chosen first integrals are real analytic [@BV]. Then it suffices to require their functional independence at least at one point [@BJ] to apply the Liouville-Arnold theorem. Further we introduce the generating function for the canonical transformation from the coordinates $(\mathbf{p},\mathbf{q})$, where $\mathbf{p}$ are the conjugate momenta to the coordinates $\mathbf{q}=(\theta,\phi,y,\alpha,\psi)$, to the action-angle variables $(\mathbf{J},\mathbf{w})$ as the indefinite integral $$S(\mathbf{q},\mathbf{c}) = \int_{\mathbf{F}=\mathbf{c}} \mathbf{p}\cdot d\mathbf{q}\,.$$ Since the Hamiltonian has no explicit time dependence, we can write $$S(\mathbf{q},\mathbf{c}) = W(\mathbf{q},\mathbf{c}) - Et\,,$$ with the Hamilton’s characteristic function $$\label{Hcf} W =\sum_i \int p_i d q_i\,.$$ In the case of geodesic motions in SE spaces $Y^{p,q}$ the variables in the Hamilton-Jacobi equation are separable and consequently we seek a solution of the Hamilton’s characteristic function of the form $$\label{W} W(y,\theta,\phi, \psi, \alpha)= W_{y}(y) + W_{\theta}(\theta) + W_{\phi}(\phi) + W_{\psi}(\psi) + W_{\alpha}(\alpha)\,.$$ The *action variables* $\mathbf{J}$ are defined as integrals over complete period of the orbit in the $(p_i,q_i)$ plane $$J_i = \oint p_i d q_i = \oint \frac{\partial W_i(q_i;c)}{\partial q_i} dq_i \qquad \mbox{(no summation)}\,.$$ $J_i$’s form $n$ independent functions of $c_i$’s and can be taken as a set of new constant momenta. Conjugate *angle variables* $w_i$ are defined by the equations: $$\label{av} w_i = \frac{\partial W}{\partial J_i} = \sum_{j=1}^n \frac{\partial W_j(q_j;J_1,\cdots,J_n)}{\partial J_i}$$ having a linear evolution in time $$\label{ff} w_i = \omega_i t + \beta_i$$ with $\beta_i$ other constants of integration and $\omega_i$ are frequencies associated with the periodic motion of $q_i$. Hamilton characteristic functions associated with cyclic variables are $$\begin{split} &W_{\phi} = P_{\phi} \phi = c_{\phi} \phi \,,\\ &W_{\psi} = P_{\psi} \psi = c_{\psi} \psi \,,\\ &W_{\alpha} = P_{\alpha} \alpha = c_{\alpha} \alpha \,, \end{split}$$ where $c_\phi,c_\psi,c_\alpha$ are the constants introduced in . The corresponding action variables are $$\begin{split} &J_{\phi} = 2 \pi c_{\phi} \,,\\ &J_{\psi} = 2 \pi c_{\psi} \,,\\ &J_{\alpha} = 2 \pi \ell c_{\alpha} \,. \end{split}$$ Taking into account and , the Hamilton-Jacobi equation becomes $$\label{EJYpq} \begin{split} E=&\frac12 \Biggl\{ 6 p(y) \left(\frac{\partial W_y}{\partial y}\right)^2 + \frac{6}{1-y}\biggl[\left(\frac{\partial W_\theta}{\partial \theta}\right)^2 + \frac{1}{\sin^2 \theta}(c_\phi + \cos\theta c_\psi)^2\biggr] \Biggr.\\ & \Biggl.\frac{1-y}{2(a-y^2)}c^2_\alpha + \frac{9(a-y^2)}{a-3y^2 +2 y^3}\biggl[c_\psi - \frac{a -2y +y^2}{6(a-y^2)} c_\alpha \biggr]^2\Biggr\}\,. \end{split}$$ This equation can be written as follows $$\label{sepJYpq} \begin{split} &\left(\frac{\partial W_\theta}{\partial \theta}\right)^2 + \frac{1}{\sin^2 \theta}(c_\phi + \cos\theta _\psi)^2\\ &~~=\frac{1-y}{3}E - p(y)(1-y) \left(\frac{\partial W_y}{\partial y}\right)^2 - \frac{(1-y)^2}{12(a-y^2)}c^2_\alpha\\ &~~~~-\frac{3(a-y^2)(1-y)}{2(a-3y^2 +2 y^3)}\biggl[c_\psi - \frac{a -2y +y^2}{6(a-y^2)} c_\alpha \biggr]^2\,. \end{split}$$ We observe that the LHS of this equation depends only $\theta$ and independent of $y$. Therefore we may set $$\left(\frac{\partial W_\theta}{\partial \theta}\right)^2 + \frac{1}{\sin^2 \theta}(c_\phi + \cos\theta c_\psi)^2 = c_{\theta}^2\,,$$ with $c_{\theta}$ another constant. From the last equation we can evaluate the action variable $$\label{J_} J_{\theta} = \oint d\theta \sqrt{c^2_{\theta} - \frac{(c_{\phi} + c_\psi \cos\theta )^2} {\sin^2\theta}}\,.$$ The limits of integrations are defined by the roots $\theta_{-}$ and $\theta_{+}$ of the expressions in the square root sign and a complete cycle of $\theta$ involves going from $\theta_{-}$ to $\theta_{+}$ and back to $\theta_{-}$. This integral can be evaluated by elementary means or using the complex integration method of residues which turns out to be more efficient [@GPS; @MV2016]. For the evaluation of the integral we put $\cos \theta = t$, extend $t$ to a complex variable $z$ and interpret the integral as a closed contour integral in the complex $z$-plane. Consider the integrand in $$\label{integr} \frac{\sqrt{-(c^2_\theta + c^2_\psi)z^2 - 2c_\phi c_\psi z + c^2_\theta- c^2_\phi}}{z^2-1} = \frac{\sqrt{-(c^2_\theta + c^2_\psi)}}{z^2-1} \sqrt{(z-t_+) (z-t_-)}\,,$$ where the roots $$t_{\pm} = \frac{-c_{\phi}c_\psi \pm c_{\theta}\sqrt{c^2_{\theta} + c^2_\psi -c^2_{\phi}}}{c^2_{\theta} + c^2_\psi}\,,$$ are the turning points of the $t$-motion. They are real for $$\label{constr} c^2_{\theta} + c^2_\psi -c^2_{\phi}\geq 0 \,,$$ and situated in the interval $(-1,+1)$. For $z>t_+$ we specify the right side of the square root from as positive. We cut the complex $z$-plane from $t_{-}$ to $t_{+}$ and the closed contour integral of the integrand is a loop enclosing the cut in a clockwise sense. The contour can be deformed to a large circular contour plus two contour integrals about the poles at $z= \pm 1$. After simple evaluation of the residues and the contribution of the large contour integral we finally get: $$J_{\theta} = 2\pi\Biggl[\sqrt{c^2_{\theta} + c^2_\psi} - c_{\phi} \Biggr] \,.$$ For the action variable corresponding to $y$ coordinate we have from $$\label{JyYpq} \begin{split} \frac{\partial W_y}{\partial y} = & \Biggl\{\frac{1-y}{a - 3y^2 +2 y^3} E - \frac{3}{a-3y^2 +2y^3} c_{\theta}^2 \Biggr.\\ & \Biggl. ~- \frac{9(a-y^2)(1-y)}{2(a-3y^2+2 y^3)^2} c_{\psi}^2 + \frac{3(a-2y +y^2)(1-y)}{2 (a-3y^2 +2y^3)^2} c_{\psi}c_{\alpha} \Biggr.\\ &\Biggl. ~- \frac{(1-y)(2a+a^2 -6ay -2y^2 +2ay^2 +6y^3 -3y^4)}{8(a-3y^2 +2y^3)^2(a-y^2)} c_{\alpha}^2 \Biggr\}^{\frac12}\,. \end{split}$$ It is harder to evaluate the action variable $J_y$ in a closed analytic form taking into account the complicated expression . In fact the closed-form of $J_y$ is not at all illuminating. More important is the fact that $J_y$ depends only of four constants of motion: $E, J_\theta, J_\alpha, J_\psi$. In consequence the energy depends only on four action variables $J_y, J_\theta, J_\alpha, J_\psi$ representing a reduction of the number of action variables entering the expression of the energy of the system. For the angular variable $w_{\phi}$ we have $$w_{\phi} = \frac{1}{2\pi} J_{\phi} + \frac{\partial W_\theta}{\partial J_\phi}\,.$$ Putting $\cos\theta = t$ the second term is $$\label{wphi} \begin{split} \frac{\partial W_\theta}{\partial J_\phi} &=- \frac{1}{2\pi} \int dt\frac{(J_\phi + J_\theta) t^2 + J_\psi t} {(1-t^2)\sqrt{-(J_\phi + J_\theta)^2 t^2- 2 J_\phi J_\psi t + (J^2_\theta + 2 J_\theta J_\phi - J^2_\psi)}}\\ &= \frac{1}{2\pi}\int \frac{dt}{1- t^2} \frac{\mathfrak{d} t^2 + \mathfrak{e} t} {\sqrt{\mathfrak{a} + \mathfrak{b}t + \mathfrak{c}t^2}}\,, \end{split}$$ where $$\begin{split} \mathfrak{a}= & J^2_\theta + 2 J_\theta J_\phi - J^2_\psi\,,\\ \mathfrak{b}= & - 2 J_\theta J_\psi \,,\\ \mathfrak{c}= & - (J_\theta + J_\phi)^2\,,\\ \mathfrak{d}= & J_\theta + J_\phi \,,\\ \mathfrak{e}= & J_\psi\,. \end{split}$$ We necessitate the following integrals [@GR]: $$\begin{split} I_1(\mathfrak{a},\mathfrak{b},\mathfrak{c};t) = &\int \frac{dt}{\sqrt{\mathfrak{a} + \mathfrak{b}t + \mathfrak{c}t^2}}\\ =&\frac{-1}{\sqrt{-\mathfrak{c}}}\arcsin \Biggl(\frac{2 \mathfrak{c} t + \mathfrak{b}}{\sqrt{-\Delta}}\Biggr) \end{split}$$ evaluated for $\mathfrak{c} <0\,,\,\Delta = 4\mathfrak{a}\mathfrak{c} - \mathfrak{b}^2 <0$, and $$\begin{split} I_2(\mathfrak{a},\mathfrak{b},\mathfrak{c};t) +&\int \frac{dt}{t\sqrt{\mathfrak{a} + \mathfrak{b}t + \mathfrak{c}t^2}}\\ =&\frac{1}{\sqrt{-\mathfrak{a}}}\arctan \Biggl(\frac{2\mathfrak{a}+ \mathfrak{b}t}{2\sqrt{-\mathfrak{a}} \sqrt{\mathfrak{a} +\mathfrak{b}t +\mathfrak{c}t^2}}\Biggr) \end{split}$$ evaluated for $\mathfrak{a}<0$. That is the case of the constants $\mathfrak{a},\mathfrak{b},\mathfrak{c}$ taking into account the constraint . Using these integrals we get for the angular variable $w_\phi$ $$\begin{split} w_{\phi} =& \frac{1}{2\pi} J_{\phi} -\frac{\mathfrak{d}}{2\pi} I_1 (\mathfrak{a},\mathfrak{b},\mathfrak{c};\cos \theta)\\ &- \frac{\mathfrak{d}+\mathfrak{e}}{4 \pi} I_2(\mathfrak{a}+\mathfrak{b}+\mathfrak{c}, \mathfrak{b} + 2\mathfrak{c},\mathfrak{c};\cos\theta -1)\\ &- \frac{\mathfrak{e}-\mathfrak{d}}{4 \pi}I_2(\mathfrak{a}-\mathfrak{b}+\mathfrak{c}, \mathfrak{b}-2\mathfrak{c},\mathfrak{c};\cos\theta +1)\,. \end{split}$$ The explicit evaluation of the angular variables $w_\theta, w_\psi, w_\alpha, w_y$ is again intricate due to the absence of a simple closed-form for the action variable $J_y$. However, it is remarkable the fact that one of the fundamental frequencies $$\omega_i = \frac{\partial H}{\partial J_i}\,,$$ is zero, namely $$\label{fzero} \omega_\phi = \frac{\partial H}{\partial J_\phi} =0\,,$$ since the action $J_\phi$ does not enter the expression of the energy. The topological nature of the flow of each invariant torus depends on the properties of the frequencies $\omega_i$ . There are essentially two cases [@JP]: 1. The frequencies $\omega_i$ are nonresonant $$k_i \omega_i \neq 0 \quad \text{for all} \quad 0\neq k_i \in \mathbb{Z}^n \,.$$ Then, on this torus each orbit is dense and the flow is ergodic. 2. The frequencies $\omega_i$ are resonant or rational dependent $$k_i \omega_i = 0 \quad \text{for some} \quad 0\neq k_i \in \mathbb{Z}^n \,.$$ The prototype is $\mathbf{\omega} = (\omega_1, \cdots , \omega_{n-m}, 0,\cdots,0)$ with $1\leq m\leq n-1$ zero frequencies and $(\omega_1, \cdots , \omega_{n-m})$ nonresonant frequencies. The KAM theorem [@VIA] describes how an integrable system reacts to small non-integrable deformations. The KAM theorem states that for nearly integrable systems, i.e. integrable systems plus sufficiently small conservative Hamiltonian perturbations, most tori survive, but suffer a small deformation. However the resonant tori which have rational ratios of frequencies get destroyed and motion on them becomes chaotic. In the case of geodesics on $Y^{p,q}$ space, the frequencies are resonant giving way to chaotic behavior when the system is perturbed. The analysis performed in [@BZ2] confirms the present results produced in the action-angle approach. Conclusions =========== The action-angle formulation for $Y^{p,q}$ spaces gives us a better understanding of the dynamics of the geodesic motions in these spaces. In spite of the complexity of the evaluation of some variables, we are able to prove that the energy of the system depends on a reduced number of action variables signaling a degeneracy of the system. This fact corroborates a similar result obtained in the case of geodesic motions in the homogeneous SE space $T^{1,1}$ [@MV2016]. The metric on $T^{1,1}$ may be written by utilizing the fact that it is a $U(1)$ bundle over $S^2 \times S^2$. The evaluations of all action and angle variables was completely done putting them in closed analytic forms. In the case of the space $T^{1,1}$ the isometry is $SU(2)^2 \times U(1)$ and there are two pairs of fundamental frequencies which are resonant. The degeneracy of these two pairs of frequencies may be removed by a canonical transformation to new action-angle variables. Finally the Hamiltonian governing the motions on $T^{1,1}$ can be written in terms of only *three* action variables for which the corresponding frequencies are different from zero. In conclusion, the action-angle approach offers a strong support for the assertion that certain classical string configurations in $AdS_5 \times Y_5$ with $Y_5$ in a large class of Einstein spaces is non-integrable [@BDG; @ZE]. It would be interesting to extend the action-angle formulations to other five-dimensional SE spaces as well as to their higher dimensional generalizations relevant for the predictions of the AdS/CFT correspondence. Acknowledgements {#acknowledgements .unnumbered} ================ The author would like to thank the referee for valuable comments and suggestions which helped to improve the manuscript. This work has been partly supported by the project [*CNCS-UEFISCDI PN–II–ID–PCE–2011–3–0137*]{} and partly by the project [*NUCLEU 16 42 01 01/2016*]{}. [99]{} J. 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--- abstract: 'To gain some sense about the likelihood of measuring the Higgs boson quartic coupling, we calculate the contribution to the triple Higgs production cross section from the subprocesses $q\bar{q}\to ZHHH$ and $q\bar{q}''\to WHHH$. Our results illustrate that determining this coupling, or even providing experimental evidence that it exists, will be very difficult.' author: - 'Duane A. Dicus' - Chung Kao - 'Wayne W. Repko' title: 'Self Coupling of the Higgs boson in the processes $p\,p\,\rightarrow\,ZHHH+X$ and $p\,p\,\rightarrow\,WHHH+X$' --- Introduction ============ The Standard Model (SM) has been very successful in explaining almost all experimental data to date, culminating in the discovery of the long awaited Higgs boson at the CERN Large Hadron Collider (LHC) [@Aad:2012tfa; @Chatrchyan:2012xdj]. The most important experimental goals of Run 2 at the Large Hadron Collider are the investigation of Higgs properties and the search for new physics beyond the Standard Model. Thus far the results from the LHC indicate that the couplings of the Higgs boson to other particles are consistent with the Standard Model. However the ultimate test as to whether this particle is the SM Higgs boson will be the trilinear Higgs coupling that appears in Higgs pair production and the quartic Higgs coupling that shows up in triple Higgs production. The self interaction of the Higgs field, $H$, is $$\label{V(H)} V(H)\,=\,\lambda\,v^2\,H^2\,+\,\kappa_3\lambda\,v\,H^3\,+\,\frac{1}{4}\kappa_4\,\lambda\,H^4$$ where $\lambda\,v^2\,=\,\frac{1}{2}m_H^2$ and $v$ is the vacuum expectation value given by the $Z$ mass, $M_Z$, the weak mixing angle $\theta_W$, and the fine structure constant $\al$ as $v\,=\,M_Z\,\cos\theta_W\sin\theta_W/\sqrt{\pi\alpha}$. $\kappa_3$ and $\kappa_4$ are one in the standard model; these are what we would like to measure. To get a feeling for the relative strengths of the terms in Eq.(\[V(H)\]) above we consider here the contribution of the subprocesses $q\bar{q}\,\rightarrow\,ZHHH$ to $p\,p\,\rightarrow\,ZHHH+X$ and $q\bar{q}\,\rightarrow\,W^{+}HHH$ to $p\,p\,\rightarrow\,W^{+}HHH+X$. Typical diagrams for this process are shown in Fig.(\[HHHdiag\]). ![Typical diagrams for the process $q\bar{q}\to ZHHH$ are shown. The same set applies to $q\bar{q}\to W^\pm HHH$ . \[HHHdiag\]](4H.eps){height="1.25in"} ![Typical diagrams for the process $q\bar{q}\to ZHHH$ are shown. The same set applies to $q\bar{q}\to W^\pm HHH$ . \[HHHdiag\]](H2H.eps){height="1.25in"} ![Typical diagrams for the process $q\bar{q}\to ZHHH$ are shown. The same set applies to $q\bar{q}\to W^\pm HHH$ . \[HHHdiag\]](HQHH.eps){height="1.25in"} \ \[10pt\] ![Typical diagrams for the process $q\bar{q}\to ZHHH$ are shown. The same set applies to $q\bar{q}\to W^\pm HHH$ . \[HHHdiag\]](3H.eps){height="1.25in"} ![Typical diagrams for the process $q\bar{q}\to ZHHH$ are shown. The same set applies to $q\bar{q}\to W^\pm HHH$ . \[HHHdiag\]](HQH.eps){height="1.25in"} ![Typical diagrams for the process $q\bar{q}\to ZHHH$ are shown. The same set applies to $q\bar{q}\to W^\pm HHH$ . \[HHHdiag\]](2H2H.eps){height="1.25in"} Contributions from Trilinear and Quartic Couplings ================================================== The matrix element from the Feynman diagrams above has terms of the form $$\label{M} \mathcal{M}\,\sim\,A\kappa_4+B\kappa_3+C+D\kappa_3^2\,,$$ where $A$ comes from diagram (a), $B$ from diagrams (b) and (c), $C$ from diagrams (d) and (e), and $D$ from diagram (f). The total cross section is given by $$\label{sigma} \sigma\,=\,\kappa_4^2\,\sigma_{44}\,+\,\kappa_3^2\,(\sigma_{33}+\sigma_{330})\,+\,\sigma_0\,+\,\kappa_4\kappa_3\sigma_{43}\,+\,\kappa_4\sigma_{40}\, +\,\kappa_3\sigma_{30}\,+\,\kappa_3^4\sigma_{3333}\,+\,\kappa_4\kappa_3^2\sigma_{433}\,+\,\kappa_3^3\sigma_{333}\,,$$ where $$\begin{array}{lcl} \sigma_{44}\,\,\,\sim\,|A|^2\, &\; & \sigma_{33}\,\,\,\,\,\,\sim\,|B|^2\,\\ \sigma_{330}\,\sim\,C\,D^*+C^*\,D\,&\; & \sigma_0\,\,\,\,\,\,\,\,\sim\,|C|^2\,\\ \sigma_{43}\,\,\,\sim\,A\,B^*+A^*\,B\, &\; & \sigma_{40}\,\,\,\,\,\,\sim\,A\,C^*+A^*\,C\,\\ \sigma_{30}\,\,\,\sim\,B\,C^*+B^*\,C\, &\; & \sigma_{3333}\,\sim\,|D|^2\,\\ \sigma_{433}\,\sim\,A\,D^*+A^*\,D\, &\;& \sigma_{333}\,\,\,\sim\,B\,D^*+B^*\,D \end{array}$$ These separate cross sections for the various terms in Eq.(\[sigma\]), in femtobarns, for several center of mass energies, are given in Table I for $Z$ and Table II for $W^+$. These were derived using CTEQ6L1 distribution functions [@CTEQ6L1] with scale $\sqrt{\hat{s}}$. We do not include any contribution from $gq$ or $gg$ initial states. A $K$ factor of [@Han] $$\label{K} K\,=\,1+\frac{\alpha_s}{2\pi}\pi^2\frac{16}{9}\,\approx\,1.3$$ was included where $$\alpha_s^{-1}\,=\,\frac{1}{0.130}+\frac{21}{12\pi}\log(\frac{\hat{s}}{M_t^2})+\frac{46}{12\pi}\log{\frac{M_t}{M_Z}}\,.$$ $\sqrt{s}$ $\sigma_{44}$ $\sigma_{3333}$ $\sigma_{433}$ $\sigma_{40}$ $\sigma_{330}$ $\sigma_{43}$ $\sigma_{0}$ $\sigma_{333}$ $\sigma_{30}$ $\sigma_{33}$ $\sigma_{TOT}$ ------------ ---------------- ----------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- 8 4.72 $10^{-7}$ 1.20 $10^{-6}$ 1.43 $10^{-6}$ 2.38 $10^{-6}$ 3.38 $10^{-6}$ 6.03 $10^{-6}$ 7.69 $10^{-6}$ 9.01 $10^{-6}$ 3.05 $10^{-5}$ 3.60 $10^{-5}$ 9.80 $10^{-5}$ 13 1.57 $10^{-6}$ 3.61 $10^{-6}$ 4.47 $10^{-6}$ 6.94 $10^{-6}$ 9.31 $10^{-6}$ 1.80 $10^{-5}$ 2.32 $10^{-5}$ 2.55 $10^{-5}$ 9.05 $10^{-5}$ 1.09 $10^{-4}$ 2.92 $10^{-4}$ 14 1.85 $10^{-6}$ 4.21 $10^{-6}$ 5.22 $10^{-6}$ 8.01 $10^{-6}$ 1.07 $10^{-5}$ 2.08 $10^{-5}$ 2.70 $10^{-5}$ 2.94 $10^{-5}$ 1.05 $10^{-4}$ 1.27 $10^{-4}$ 3.39 $10^{-4}$ 33 9.37 $10^{-6}$ 1.90 $10^{-5}$ 2.46 $10^{-5}$ 3.47 $10^{-5}$ 4.38 $10^{-5}$ 9.24 $10^{-5}$ 1.23 $10^{-4}$ 1.24 $10^{-4}$ 4.64 $10^{-4}$ 5.84 $10^{-4}$ 1.52 $10^{-3}$ 60 2.43 $10^{-5}$ 4.67 $10^{-5}$ 6.16 $10^{-5}$ 8.41 $10^{-5}$ 1.04 $10^{-4}$ 2.26 $10^{-4}$ 3.06 $10^{-4}$ 2.97 $10^{-4}$ 1.14 $10^{-3}$ 1.45 $10^{-3}$ 3.74 $10^{-3}$ 100 5.15 $10^{-5}$ 9.55 $10^{-5}$ 1.27 $10^{-4}$ 1.70 $10^{-4}$ 2.07 $10^{-4}$ 4.61 $10^{-4}$ 6.26 $10^{-4}$ 5.96 $10^{-4}$ 2.31 $10^{-3}$ 2.97 $10^{-3}$ 7.62 $10^{-3}$ $\sqrt{s}$ $\sigma_{44}$ $\sigma_{3333}$ $\sigma_{433}$ $\sigma_{40}$ $\sigma_{330}$ $\sigma_{43}$ $\sigma_{0}$ $\sigma_{333}$ $\sigma_{30}$ $\sigma_{33}$ $\sigma_{TOT}$ ------------ ---------------- ----------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- 8 6.58 $10^{-7}$ 1.63 $10^{-6}$ 1.96 $10^{-6}$ 2.27 $10^{-6}$ 3.17 $10^{-6}$ 7.26 $10^{-6}$ 6.14 $10^{-6}$ 1.07 $10^{-5}$ 2.87 $10^{-5}$ 4.16 $10^{-5}$ 1.04 $10^{-4}$ 13 2.03 $10^{-6}$ 4.53 $10^{-6}$ 5.65 $10^{-6}$ 6.00 $10^{-6}$ 7.96 $10^{-6}$ 1.99 $10^{-5}$ 1.70 $10^{-5}$ 2.78 $10^{-5}$ 7.79 $10^{-5}$ 1.17 $10^{-4}$ 2.85 $10^{-4}$ 14 2.36 $10^{-6}$ 5.19 $10^{-6}$ 6.52 $10^{-6}$ 6.58 $10^{-6}$ 9.02 $10^{-6}$ 2.28 $10^{-5}$ 1.96 $10^{-5}$ 3.16 $10^{-5}$ 8.93 $10^{-5}$ 1.34 $10^{-4}$ 3.27 $10^{-4}$ 33 1.08 $10^{-5}$ 2.12 $10^{-5}$ 2.78 $10^{-5}$ 2.66 $10^{-5}$ 3.34 $10^{-5}$ 9.15 $10^{-5}$ 8.08 $10^{-5}$ 1.21 $10^{-4}$ 3.55 $10^{-4}$ 5.54 $10^{-4}$ 1.32 $10^{-3}$ 60 2.66 $10^{-5}$ 5.01 $10^{-5}$ 6.65 $10^{-5}$ 6.13 $10^{-5}$ 7.56 $10^{-5}$ 2.13 $10^{-4}$ 1.91 $10^{-4}$ 2.77 $10^{-4}$ 8.29 $10^{-4}$ 1.31 $10^{-3}$ 3.10 $10^{-3}$ 100 5.43 $10^{-5}$ 9.95 $10^{-5}$ 1.33 $10^{-4}$ 1.20 $10^{-4}$ 1.64 $10^{-4}$ 4.21 $10^{-4}$ 3.81 $10^{-4}$ 5.40 $10^{-4}$ 1.63 $10^{-3}$ 2.62 $10^{-3}$ 6.14 $10^{-3}$ For the process $pp\,\rightarrow\,ZHHH+X$, the contents of Table \[ZHHH\] are illustrated in Fig. \[ZHHH\_Fig\]. The figure for $pp\,\rightarrow\,W^{+}HHH+X$ is similar. ![The various contributions to the total cross section for $pp\,\rightarrow\,ZHHH+X$ are shown as a function of $\sqrt{s}$. The ordering of the curves corresponds to the ordering of the columns in Table \[ZHHH\]. $\sigma_{44}$ is the lowest curve and $\sigma_{TOT}$ is the highest. The contributions involving the quartic coupling are indicated by dashed lines. \[ZHHH\_Fig\]](ZHHH_PLOT.eps){height="2.5in"} The amplitude $C$ in Eq.(\[M\]) comes from the $ZZH$ and $ZZHH$ couplings (diagrams (d) and (e)). Superficially these diagrams grow faster with energy than the diagrams that involve Higgs propagators. However, the largest energy behavior cancels between the diagrams with only $ZHH$ couplings and those that involve a $ZZH$ and a $ZZHH$ coupling. Explicitly implementing this cancellation of the large energy behavior seems essential for the calculation of $\sigma_0$; depending on the phase space integral to find the cancellation does not work for large center of mass energies. A similar high energy behavior occurs in the $W^+$ cross section and requires the same analytic cancellation. A detailed description of how this cancellation occurs is given in the next section. Cancellation of the leading high energy behavior ================================================ If we label the momenta as $$q(p_1)+\bar{q}(p_2)\,\rightarrow\,H(k_1)+H(k_2)+H(k_3)+Z(P)$$ then the matrix element for $C$ can be written $$\label{M0} M\,\sim\,\bar{v}(p_2)\gamma_{\mu}(g_V-\gamma_5)u(p_1)X^{\mu\lambda}\epsilon_{\lambda}(P)$$ The spinor factor goes as $E^1$ at large energy $E$. $X^{\mu\lambda}$ has two or three $Z$ propagators depending on the diagram. The propagator which couples to the spinor factor goes as $E^{-2}$ because the momentum in the $p^{\mu}p^{\nu}/M_Z^2$ term is $p_1+p_2$ which is zero when dotted into spinor factor. The other one or two propagators do not have this cancellation and thus go as $E^{0}$. The $Z$ polarization vector can be longitudinal and thus go as $E^{1}$. So these diagrams go as $E^{0}$ for large $E$. The diagrams for contributions other than $C$ go as $E^{-2}$ or faster because they have Higgs propagators. To see this $E^{0}$ behavior cancel we need to write out $X^{\mu\lambda}$ $$\begin{aligned} X^{\mu\lambda}\,&=&\,\frac{1}{2}A_1^{\mu\rho}(B_{2\rho}^{\lambda}+B_{3\rho}^{\lambda}+g_{\rho}^{\lambda}) +\frac{1}{2}A_2^{\mu\rho}(B_{1\rho}^{\lambda}+B_{3\rho}^{\lambda}+g_{\rho}^{\lambda}) +\frac{1}{2}A_3^{\mu\rho}(B_{1\rho}^{\lambda}+B_{2\rho}^{\lambda}+g_{\rho}^{\lambda})\,\nonumber\\ &+&\,\frac{1}{2}(A_1^{\mu\rho}+A_2^{\mu\rho}+g^{\mu\rho})B^{\lambda}_{3\rho} +\frac{1}{2}(A_2^{\mu\rho}+A_3^{\mu\rho}+g^{\mu\rho})B^{\lambda}_{1\rho} +\frac{1}{2}(A_1^{\mu\rho}+A_3^{\mu\rho}+g^{\mu\rho})B^{\lambda}_{2\rho}\end{aligned}$$ where $$\begin{aligned} A_i^{\mu\lambda}\,&=&\,C_i(M_Z^2g^{\mu\lambda}+k_i^{\mu}Q_i^{\lambda})\,\,\,\,\,\,\,\,{\rm no\,\,sum\,\,on\,\,i} \\ B_i^{\mu\lambda}\,&=&\,D_i(M_Z^2g^{\mu\lambda}-R_i^{\mu}k_i^{\lambda})\,\,\,\,\,\,\,\,{\rm no\,\,sum\,\,on\,\,i}\end{aligned}$$ for $i=1,2,3$ with $$\begin{aligned} Q_i^{\mu}\,&=&\,p_1^{\mu}+p_2^{\mu}-k_i^{\mu} \\ R_i^{\mu}\,&=&\,P^{\mu}+k_i^{\mu}\end{aligned}$$ and $$\begin{aligned} C_i\,=\,\frac{1}{Q_i^2-M_Z^2} \\ D_i\,=\,\frac{1}{R_i^2-M_Z^2}\end{aligned}$$ The large $E$ behavior comes from the $P^\mu P^\nu/M_Z^2$ part of the sum over $Z$ polarizations, so replace the polarization vector $\epsilon_{\lambda}(P)$ by $P_\lambda$ and dot $P_{\lambda}$ into $X^{\mu\lambda}$. Then use $$D_i^{-1} = 2P\Dot k_i+\frac{1}{2}m_H^2$$ to eliminate $P\Dot k_i$ factors in favor of mass factors or the cancellation of $D_i$ terms. The remaining large $E$ terms will occur in the combination $C_i(p_1+p_2-k_i)^2$, which can be replaced by $M_Z^2$ and terms that vanish when contracted with the lepton factor. In particular if we define $$F_i^{\rho}\,=\,D_i[M_Z^2P^{\rho}+\frac{1}{2}M_H^2(P+k_i)^{\rho}]$$ then $$\begin{aligned} \label{XMU} X^{\mu\rho}P_{\rho}\,&=&\,\frac{1}{2}A_i^{\mu\rho}(F_{j\rho}+F_{k\rho})+\frac{1}{2}[A_i^{\mu\rho}+A_j^{\mu\rho}+g^{\mu\rho}]F_{k\rho} \nonumber \\ &\equiv&\,X^{\mu}\end{aligned}$$ where $$i,j,k\,=\,(1,2,3)\,+\,(2,3,1)\,+\,(3,1,2)$$ and $F_i^{\rho}$ is smaller than $B_i^{\rho\lambda}P_{\lambda}$ by two factors of mass rather than momenta. (The terms with the additional factors of momenta are proportional to $P^{\mu}+k_1^{\mu}+k_2^{\mu}+k_3^{\mu}\,=\,p_1^{\mu}+p_2^{\mu}$ dotted into the spinor factor.) If we call the square of the spinor factor in Eq.(\[M0\]), summed over spins, $L_{\mu\nu}$, then the square of the matrix element for $\sigma_0$, including the transverse polarizations of the $Z$, is $$\sum_{pol}|M|^2\,\sim\,L_{\mu\nu}X^{\mu\lambda}X^{\nu\eta}(-g_{\lambda\eta})\,+\,L_{\mu\nu}X^{\mu}X^{\nu}/M_Z^2$$ where $X^{\mu}$, defined in Eq.(\[XMU\]) above, can be simplified to $$X^{\mu}\,=\,A_1^{\mu\rho}(F_{2\rho}+F_{3\rho})+A_2^{\mu\rho}(F_{1\rho}+F_{3\rho}) +A_3^{\mu\rho}(F_{1\rho}+F_{2\rho})+\frac{1}{2}g^{\mu\rho}(F_{1\rho}+F_{2\rho}+F_{3\rho})\,.$$ By explicitly implementing this cancellation, the integration over phase space is well behaved for all beam energies. Results and Conclusions ======================= The Tables show that the coefficients of $\kappa_4$ in the cross section Eq.(\[sigma\]) are small which makes a value for $\kappa_4$ almost impossible to determine independent of the value of $\kappa_3$. For example Figs.\[ZHHH13\] and \[ZHHH100\] show the cross section for the $Z$ process with $\sqrt{s}\,=\,13$ and $\sqrt{s}\,=\,100$ as a function of $\kappa_3$ for two values of $\kappa_4$. ![The cross section for $pp\,\rightarrow\,ZHHH+X$ from $q\bar{q}\,\rightarrow\,ZHHH$ for $\sqrt{s}=100$ TeV is shown as a function of $\kappa_3$ for $\kappa_4=1$ (solid line) and $\kappa_4=10$ (dashed line).\[ZHHH100\]](s13.eps){height="2.0in"} ![The cross section for $pp\,\rightarrow\,ZHHH+X$ from $q\bar{q}\,\rightarrow\,ZHHH$ for $\sqrt{s}=100$ TeV is shown as a function of $\kappa_3$ for $\kappa_4=1$ (solid line) and $\kappa_4=10$ (dashed line).\[ZHHH100\]](s100.eps){height="2.0in"} ![The variation of the cross section for $pp\,\rightarrow\,ZHHH+X$ from $q\bar{q}\,\rightarrow\,ZHHH$ for $\sqrt{s}=100$ TeV is shown for $\kappa_3=1$ (solid line) and the dashed band $0.5\leq\kappa_3\leq 1.5$ as a function of $\kappa_4$.\[k4100\]](kk4_13.eps){height="2.0in"} ![The variation of the cross section for $pp\,\rightarrow\,ZHHH+X$ from $q\bar{q}\,\rightarrow\,ZHHH$ for $\sqrt{s}=100$ TeV is shown for $\kappa_3=1$ (solid line) and the dashed band $0.5\leq\kappa_3\leq 1.5$ as a function of $\kappa_4$.\[k4100\]](kk4_100.eps){height="2.0in"} At $\sqrt{s}\,=\,13$ TeV with $\kappa_3\,=\,1$ the difference in the cross section between $\kappa_4\,=\,1$ and $\kappa_4\,=\,10$ is $4.2\times10^{-4}$ fb. For $\sqrt{s}\,=\,100$ TeV the same difference is $1.2\times10^{-2}$ fb and if $\kappa_3\,=\,10$ the difference is still less than $0.16$ fb. Figures \[k413\] and \[k4100\] illustrate this further by fixing $\kappa_3$ near $1$ and varying $\kappa_4$ to find that the cross sections change by only small fractions of a femtobarn. For the $W^{+}$ process the contributions that include the quartic Higgs coupling are again too small to measure $\kappa_4$ or even to determine if it is nonzero. This is illustrated in Figs.\[WHHH13\] and \[WHHH100\]. ![The cross section for $pp\,\rightarrow\,W^+HHH+X$ from $q\bar{q}\,\rightarrow\,W^+HHH$ for $\sqrt{s}=100$ TeV is shown as a function of $\kappa_3$ for $\kappa_4=1$ (solid line) and $\kappa_4=10$ (dashed line).\[WHHH100\]](ss13.eps){height="2.0in"} ![The cross section for $pp\,\rightarrow\,W^+HHH+X$ from $q\bar{q}\,\rightarrow\,W^+HHH$ for $\sqrt{s}=100$ TeV is shown as a function of $\kappa_3$ for $\kappa_4=1$ (solid line) and $\kappa_4=10$ (dashed line).\[WHHH100\]](ss100.eps){height="2.0in"} The total cross section for the $W^{-}$ process is smaller than that for $W^{+}$ by a factor of $2.66,\, 2.24,\, 2.19,\, 1.75,\, 1.57,\, 1.47$ for $\sqrt{s}\,=\,8,\, 13,\, 14,\, 33,\, 60,\, 100$ TeV. The ratios of the individual cross sections (eg., $\sigma_{44}$) vary from these numbers by less than $10\%$. The parameter $\kappa_3$ can be determined from processes with two Higgs bosons in the final state. For example, the subprocess $gg\,\rightarrow\,HH$ obviously depends on the three Higgs coupling as does $gg\,\rightarrow\,t\bar{t}HH$ [@DKW; @Glover; @Daw; @Dol; @Bag; @Goe; @Barr; @Arh; @deFlorian; @Hes; @Liu; @deFl; @Fred; @dawson; @DKR; @Cao1; @Cao2]. Processes with three Higgs bosons in the final state are necessary to determine $\kappa_4$. We show that the processes considered here are not sufficient at any energy to even verify the existence of a four Higgs coupling. This is most obvious from Figure 2 where the coefficients of $\kappa_4$ (dashed lines) are very small compared to most of the other partial cross sections. In general the problem of determining $\kappa_4$ will be very difficult. Similar conclusions have been reached by Binoth, Karg, Kauer, and Rückl [@BKKR] and others [@PS; @Chen] for the gluon fusion process $gg\to HHH$. [**Acknowledgements**]{}\ D. A. D. was supported in part by the U. S. 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--- abstract: 'Large-capacity Content Addressable Memory (CAM) is a key element in a wide variety of applications. The inevitable complexities of scaling MOS transistors introduce a major challenge in the realization of such systems. Convergence of disparate technologies, which are compatible with CMOS processing, may allow extension of Moore’s Law for a few more years. This paper provides a new approach towards the design and modeling of Memristor (Memory resistor) based Content Addressable Memory (MCAM) using a combination of memristor MOS devices to form the core of a memory/compare logic cell that forms the building block of the CAM architecture. The non-volatile characteristic and the nanoscale geometry together with compatibility of the memristor with CMOS processing technology increases the packing density, provides for new approaches towards power management through disabling CAM blocks without loss of stored data, reduces power dissipation, and has scope for speed improvement as the technology matures.' author: - 'Kamran Eshraghian, Kyoung-Rok Cho,  Omid Kavehei,  Soon-Ku Kang, Derek Abbott,  and Sung-Mo Steve Kang, [^1] [^2][^3][^4]' bibliography: - 'omid\_ieeetvlsi.bib' title: 'Memristor MOS Content Addressable Memory (MCAM): Hybrid Architecture for Future High Performance Search Engines' --- [Eshraghian : Memristor MOS Content Addressable Memory (MCAM): Hybrid Architecture for Future High Performance Search Engines]{} Memristor, Content Addressable Memory, MCAM, Memory, Memristor-MOS Hybrid Architecture, Modeling Introduction {#sec:Introduction} ============ quest for a new hardware paradigm that will attain processing speeds in the order of an exaflop ($10^{18}$ floating point operations per second) and further into the zetaflop regime ($10^{21}$ flops) is a major challenge for both circuit designers and system architects. The evolutionary progress of networks such as the Internet also brings about the need for realization of new components and related circuits that are compatible with CMOS process technology as CMOS scaling begins to slow down [@Bourianoff2007]. As Moore’s Law becomes more difficult to fulfill, integration of significantly different technologies such as spintronics [@Bourianoff2007], carbon nano tube field effect transistors (CNFET) [@Akinwande2008], optical nanocircuits based on metamaterials [@Engheta2007], and more recently the memristor [@Strukov2008], are gaining more focus thus creating new possibilities towards realization of innovative circuits and systems within the [*System on System*]{} (SoS) domain. In this paper we explore conceptualization, design, and modeling of the memory/compare cell as part of a Memristor based Content Addressable Memory (MCAM) architecture using a combination of memristor and n-type MOS devices. A typical Content Addressable Memory (CAM) cell forms a SRAM cell that has 2 n-type and 2 p-type MOS transistors, which requires both $V_{\rm DD}$ and GND connections as well as well-plugs within each cell. Construction of a SRAM cell that exploits memristor technology, which has a non-volatile memory (NVM) behavior and can be fabricated as an extension to a CMOS process technology with nanoscale geometry, addresses the main thread of current CAM research towards reduction of power consumption. The design of the CAM cell is based on the $4^{\rm th}$ passive circuit element, the Memristor (M) predicted by Chua in 1971 [@Chua1971] and generalized by Kang [@Kang1975; @Chua1976]. Chua postulated that a new circuit element defined by the single-valued relationship $d\phi=Mdq$ must exist, whereby current moving through the memristor is proportional to the flux of the magnetic field that flows through the material. In another words, the magnetic flux between the terminals is a function of the amount of charge, $q$, that has passed through the device. This follows from Lenz’s law whereby the single-valued relationship $d\phi=Mdq$ has the equivalence $v=M(q)i$, where $v$ and $i$ are memristor voltage and current, respectively. The memristor behaves as a switch, much like a transistor. However, unlike the transistor, it is a 2-terminal rather than a 3-terminal device and does not require power to retain either of its two states. Note that a memristor changes its resistance between two values and this is achieved via the movement of mobile ionic charge within an oxide layer, furthermore, these resistive states are non-volatile. This behavior is an important property that influences the architecture of CAM systems, where the power supply of CAM blocks can be disabled without loss of stored data. Therefore, memristor-based CAM cells have the potential for significant saving in power dissipation. This paper has the following structure: Section \[sec:Characterization-and-Modeling-Behavior-of-Memristor\] is an introductory section and reviews the properties of the memristor and then explores various options available in the modeling of this device. In Section \[sec:Memristor-MOS-Memory\], circuit options for realization of MCAM is investigated whereby the two disparate technologies converge to create a new CMOS-based design platform. Section \[sec:simresult\] provides simulation results of a basic MCAM cell to be implemented as part of a future search engine. The details of our proposed layout and preliminary CMOS overlay fabrication approach are also presented in Section \[sec:layfab\]. The concluding comments are provided in Section \[sec:Conclusions\]. Characterization and Modeling Behavior of Memristor {#sec:Characterization-and-Modeling-Behavior-of-Memristor} =================================================== @Strukov2008 presented a physical model whereby the memristor is characterized by an equivalent time-dependent resistor whose value at a time $t$ is linearly proportional to the quantity of charge $q$ that has passed through it. They realized a proof-of-concept memristor, which consists of a thin nano layer ($2$ nm) of TiO$_2$ and a second oxygen deficient nano layer of TiO$_{2-x}$ ($8$ nm) sandwiched between two Pt nanowires ($\sim$ 50 nm), shown in Fig. \[fig:Mem-switch-behav\] [@Strukov2008]. Oxygen (O$^{2-}$) vacancies are +2 mobile carriers and are positively charged. A change in distribution of O$^{2-}$ within the TiO$_2$ nano layer changes the resistance. By applying a positive voltage, to the top platinum nanowire, oxygen vacancies drift from the TiO$_{2-x}$ layer to the TiO$_2$ undoped layer, thus changing the boundary between the TiO$_{2-x}$ and TiO$_2$ layers. As a consequence, the overall resistance of the layer is reduced corresponding to an “ON” state. When enough charge passes through the memristor that ions can no longer move, the device enters a hysteresis region and keeps $q$ at an upper bound with fixed memristance, $M$ (memristor resistance). By reversing the process, the oxygen defects diffuse back into the TiO$_{2-x}$ nano layer. The resistance returns to its original state, which corresponds to an “OFF” state. The significant aspect to be noted here is that only ionic charges, namely oxygen vacancies (O$^{2-}$) through the cell, change memristance. The resistance change is non-volatile hence the cell acts as a memory element that remembers past history of ionic charge flow through the cell. -- -- -- -- Simplified Memristor Model {#sec:sub:Simplified-Memristor-Model} -------------------------- The memristor can be modeled in terms of two resistors in series, namely the doped region and undoped region each having vertical width of $w$ and $L-w$, respectively, as shown in Fig. \[fig:Mem-switch-behav\], where $L$ is the TiO$_2$ film thickness [@Strukov2008]. The voltage-current relationship defined as $M(q)$, can be modeled as [@Chua1971] $$\label{Equ:memvolt} v(t) = \Bigg(R_{\rm ON}\frac{w(t)}{L}+R_{\rm OFF}\Big(1-\frac{w(t)}{L}\Big)\Bigg)i(t)~,$$ where $R_{\rm ON}$ is the resistance for completely doped memristor, while $R_{\rm OFF}$ is the resistance for the undoped region. The width of the doped region $w(t)$ is given by, $$\label{Equ:memstate} \frac{dw(t)}{dt} = \mu_v \frac{R_{\rm ON}}{L}i(t)~,$$ where $\mu_v$ represents the average dopant mobility $\sim 10^{-10}~{\rm cm}^2/{\rm s}/{\rm V}$. Taking a normalized variable, $x(t)=w(t)/L$, instead of $w(t)$ assists in tracking memristance, $M(q)=d\phi/dq$, or memductance, $W(\phi)=dq /d\phi$. The new normalized relation is $$\label{Equ:memXstate} \frac{dx(t)}{dt} = \mu_v \frac{R_{\rm ON}}{L^2}i(t)~,$$ where $L^2 / \mu_v$ has the dimensions of magnetic flux($\phi$). Following the calculation steps from @Kavehei2009, a simple memristance model can be defined as $$\label{Equ:ourmodelM} M(t) = R_{\rm OFF}\Bigg(\sqrt{1-\frac{2c(t)}{r}}\Bigg)~,$$ where $c(t)=\mu_v \phi(t)/L^2$, and $r$ is a ratio of $R_{\rm OFF}/R_{\rm ON}$ and $\sqrt{1-\frac{2c(t)}{r}}$ is the [*resistance modulation index*]{}. Here, $x(t)$ can now be rewritten as $$\label{Equ:ourmodelX} x(t) = 1-\Bigg(\sqrt{1-\frac{2\phi(t)}{r\beta}}\Bigg)~,$$ which highlights that the $r\beta$ term (where $\beta=L^2 / \mu_v$) must be made sufficiently large to maintain $2\phi(t) / r\beta$ between the range 0 and 1. The simplified linear ionic drift model facilitates the understanding of the operational characteristics of the memristor. However, for a highly nonlinear [@Yang2008] relationship between electric field and drift velocity that exists at the boundaries, the ratio cannot be maintained. Thus this function is unable to model large nonlinearities close to the boundaries of the memristor characteristics. At the boundaries, i.e. when $x$ approaches 0 or 1, there is a nonlinearity associated with the memristor behavior that is discussed in the following subsection. Modelling the Nonlinear Behavior of Memristor {#sec:sub:More-Complex-Model} --------------------------------------------- The electrical behavior of the memristor as a switch/memory element is determined by the boundary between the two regions in response to an applied voltage. To model this nonlinearity, the memristor state equation Eq. \[Equ:memXstate\] is augmented with a [*window function*]{}, $F(w,i)$ [@Strukov2008; @Strukov2009; @Biolek2009; @Benderli2009], where $w$ and $i$ are the memristor’s state variable and current, respectively. Thus, Eq. \[Equ:memXstate\] can be rewritten as $$\frac{dx(t)}{dt}=\frac{R_{\rm ON}}{\beta}i(t)F(x(t),p)~, \label{equ:nonlinstat}$$ where $p$ is its *control parameter*. The nonlinearity at the boundaries can now be controlled with parameter $p$. The influence of a window function described by Eq. \[equ:nonlinstat\] is illustrated in Fig. \[fig:MemModel\](a) for $2\leq p\leq 10$. ---------------------- --------------------------------- \(a) Window function \(b) Hysteresis characteristics ---------------------- --------------------------------- @Joglekar2009 proposed a modified window function to approximately address linear ionic drift and the nonlinear behaviour at the boundaries when $0<x<1$. For the window function $F(x)=1-(2x-1)^{2p}$, $p$ is a positive integer and $x=w/L$. This model considers a simple boundary condition, $F(0)=F(1)=0$, when $p\geq 4$, the state variable equation is an approximation of the linear drift assumption, $F(0<x<1)\approx 1$. This model is denoted by B-I in Table \[tab:models\]. Based on this model, when a memristor is at the terminal states, no external stimulus can change its state. @Biolek2009 addressed this problem with a new window function, $F(x)=1-(x-{\rm sgn}(-i))^{2p}$, where $i$ is the memristor current, ${\rm sgn}(i)=1$ when $i\geq 0$, and ${\rm sgn}(i)=0$ when $i<0$. When current is positive, the doped region length, $w$, is expanding. This model is denoted by B-II in Table \[tab:models\] and is adopted for the simulations that follow. The hysteresis characteristic using the nonlinear drift assumption is illustrated in Fig. \[fig:MemModel\](b). This hysteresis shows a highly nonlinear relationship between current and voltage at the boundaries as is derived using similar parameters reported by @Strukov2008. To conclude this section Table \[tab:models\] shows a brief comparison between different behavioral memristor models. It is also important to emphasis that the modeling approach in this paper is based on the behavioral characteristics of the solid-state thin film memristor device [@Strukov2008]. @Shin2010 recently proposed compact macromodels for the solid-state thin film memristor device. Even though the assumption is still based on the linear drift model, their approach provides a solution for bypassing current flow at the two boundary resistances. ---------- ----------------- ---------------------------- -------------------------------------- ----------------------------------- Window Function Boundaries Model Ref $F(\cdot)$ $(x\rightarrow 0$, $x\rightarrow 1)$ Problem(s) A-I [@Strukov2008] $w(1-w)/L^2$ ($0$,$\sim 0$) Linear approximation, $x\in[0,1]$ Stuck at the terminal states $F(w\rightarrow L)\neq 0$ A-II [@Benderli2009] $x(1-x)$ ($0$, $0$) Linear approximation, $x\in[0,1]$ Stuck at the terminal states B-I [@Joglekar2009] $1-(2x-1)^{2p}$ ($0$, $0$) Stuck at the terminal states B-II$^*$ [@Biolek2009] $1-(x-{\rm sgn}(-i))^{2p}$ ($0$, $0$) Discontinuity at the boundaries ---------- ----------------- ---------------------------- -------------------------------------- ----------------------------------- \ $^*$ This model is adopted for the simulations. \[tab:models\] Emerging Memory Devices and Technologies {#sec:sub:emrging} ---------------------------------------- Memory processing has been considered as the pace-setter for scaling a technology. A number of performance parameters including capacity (that relate to area utilization), cost, speed (both access time and bandwidth), retention time, and persistence, read/write endurance, active power dissipation, standby power, robustness such as reliability and temperature related issues characterize memories. Recent and emerging technologies such as Phase-Change Random Access Memory (PCRAM), Magnetic RAM (MRAM), Ferroelectric RAM (FeRAM), Resistive RAM (RRAM), and Memristor, have shown promise and some are already being considered for implementation into emerging products. Table \[tab:itrs\] summarizes a range of performance parameters and salient features of each of the technologies that characterize memories [@ITRS2009; @Freitas2008]. A projected plan for 2020 for memories highlight a capacity greater than $1$ TB, read/write access times of less than $100$ ns and endurance in the order of $10^{12}$ or more write cycles. ---------------------------- ----------- ----------- ----------- ----------- ------------- -------------------- ------------------ ----------- DRAM SRAM NOR NAND FeRAM MRAM PCRAM Memristor Knowledge level advanced early stage Cell Elements 1T1C 6T 1T1C 1T1R 1T1R 1M Half pitch ($F$) (nm) $50$ $65$ $90$ $90$ $180$ $130$ $65$ $3$-$10$ Smallest cell area ($F^2$) $6$ $140$ $10$ $5$ $22$ $45$ $16$ $4$ Read time (ns) $< 1$ $< 0.3$ $< 10$ $< 50$ $< 45$ $< 20$ $< 60$ $< 50$ Write/Erase time (ns) $< 0.5$ $< 0.3$ $10^5$ $10^6$ $10$ $20$ $60$ $< 250$ Retention time (years) seconds N/A $> 10$ $> 10$ $> 10$ $> 10$ $> 10$ $> 10$ Write op. voltage (V) $2.5$ $1$ $12$ $15$ $0.9$-$3.3$ $1.5$ $3$ $< 3$ Read op. voltage (V) $1.8$ $1$ $2$ $2$ $0.9$-$3.3$ $1.5$ $3$ $< 3$ Write endurance $10^{16}$ $10^{16}$ $10^5$ $10^5$ $10^{14}$ $10^{16}$ $10^9$ $10^{15}$ Write energy (fJ/bit) $5$ $0.7$ $10$ $10$ $30$ $1.5\times 10^{5}$ $6\times 10^{3}$ $< 50$ Density (Gbit/cm$^2$) $6.67$ $0.17$ $1.23$ $2.47$ $0.14$ $0.13$ $1.48$ $250$ Voltage scaling no poor promising Highly scalable promising promising ---------------------------- ----------- ----------- ----------- ----------- ------------- -------------------- ------------------ ----------- \[tab:itrs\] Flash memories suffer from both a slow write/erase times and low endurance cycles. FeRAMs and MRAMs are poorly scalable. MRAMs and PCRAMs require large programming currents during write cycle, hence an increase in dissipation per bit. Furthermore, voltage scaling becomes more difficult. Memristors, however, have demonstrated promising results in terms of the write operation voltage scaling [@Strukov2009; @Kuekes2005]. Memristor crossbar-based architecture is highly scalable [@Strukov2009b] and shows promise for ultra-high density memories [@Vontobel2009]. For example, a memristor with minimum feature sizes of $10$ nm and $3$ nm yield $250$ Gb/cm$^2$ and $2.5$ Tb/cm$^2$, respectively. In spite of the high density, zero standby power dissipation, and long life time that have been pointed out for the emerging memory technologies, their long write latency has a large negative source of impact on memory bandwidth, power consumption, and the general performance of a memory system. Conventional CAM and the Proposed MCAM Structures {#sec:Memristor-MOS-Memory} ================================================= A content addressable memory illustrated in Fig. \[fig:genarch\] takes a search word and returns the matching memory location. Such an approach can be considered as a mapping of the large space of input search word to that of the smaller space of output match location in a single clock cycle [@Tyshchenko2008]. There are numerous applications including Translation Lookaside Buffers (TLB), image coding [@Kumaki2007], classifiers to forward Internet Protocol (IP) packets in network routers [@Kim2009b], etc. Inclusion of memristors in the architecture ensures that data is retained if the power source is removed enabling new possibilities in system design including the all important issue of power management. ![Generic Content Addressable Memory Architecture for $n\times n$ NAND-type CAM cells. In this structure each data (D) and search (S) bits share one common bus line (D/S) to reduce the interconnection complexity. The architecture is based on the MCAM cell of Fig. \[fig:CAMcells\](d) and the match lines (MLs) composed of nMOS pass transistors.[]{data-label="fig:genarch"}](figgenarch.eps) Conventional Content Addressable Memory {#sec:sub:sub:convcam} --------------------------------------- To better appreciate some of the benefits of our proposed structure we provide a brief overview of the conventional CAM cell using static random access memory (SRAM) as shown in Fig. \[fig:convcam\](a). The two inverters that form the latch use four transistors including two p-type transistors that normally require more silicon area. Problems such as relatively high leakage current particularly for nanoscaled CMOS technology [@Verma2008] and the need for inclusion of both $V_{\rm DD}$ and ground lines in each cell bring further challenges for CAM designers in order to increase the packing density and still maintain sensible power dissipation. Thus, to satisfy the combination of ultra dense designs, low-power (low-leakage), and high-performance, the SRAM cell is the focus of architectural design considerations. For instance, one of the known problems of the conventional 6-T SRAM for ultra low-power applications is its static noise margin (SNM) [@Verma2008]. Fundamentally, the main technique used to design an ultra low-power memory is voltage scaling that brings CMOS operation down to the subthreshold regime. @Verma2008 demonstrated that at very low supply voltages the static noise margin for SRAM will disappear due to process variation. To address the low SNM for subthreshold supply voltage @Verma2008 proposed 8-T SRAM cell shown in Fig. \[fig:convcam\](b). This means, there is a need for significant increase in silicon area to have reduced failure when the supply voltage has been scaled down. Failure is a major issue in designing ultra dense (high capacity) memories. Therefore, a range of fault tolerance techniques are usually applied [@Lu2006]. As long as the defect or failure results from the SRAM structure, a traditional approach such as replication of memory cells can be implemented. Obviously it causes a large overhead in silicon area which, exacerbates the issue of power consumption. Some of the specific CAM cells, for example, ternary content addressable memory (TCAM) normally used for the design of high-speed lookup-intensive applications in network routers, such as packet forwarding and classification two SRAM cells, are required. Thus, the dissipation brought about as the result of leakage becomes a major design challenge in TCAMs [@Mohan2009]. It should be noted that the focus in this paper is to address the design of the store/compare core cell only, leaving out details of CAM’s peripherals such as read/write drivers, encoder, matchline sensing selective precharge, pipelining, matchline segmentation, current saving technique etc., that characterize a CAM architecture [@Pagiamtzis2006]. [cc]{}\ \ \ Generic Memristor-nMOS Circuit {#sec:sub:genericcir} ------------------------------ Fig. \[fig:basicm\] shows the basic structure for a memristor-nMOS storage cell. For writing a logic “1”, the memristor receives a positive bias to maintain an “ON” state. This corresponds to the memristor being programmed as a logic “1”. To program a “0” a reverse bias is applied to the memristor, which makes the memristor resistance high. This corresponds to logic “0” being programmed. -- -- -- -- MCAM Cell {#sec:sub:Memristor-MOS-based-Content-Addressable-Memory-Cell-Architecture} --------- In this subsection, variations of MCAM cells as well as a brief architectural perspective are introduced. The details of read/write operations and their timing issues are also discussed in the next section. A CAM cell serves two basic functions: “bit storage” and “bit comparison”. There are a variety of approaches in the design of basic cell such as NOR based match line, NAND based match line, etc. This part of the paper reviews the properties of conventional SRAM-based CAM and provides a possible approach for the design of content addressable memory based on the memristor. ### MCAM Cell Properties {#sec:sub:sub:mcam} Fig. \[fig:CAMcells\] illustrates several variations of the MCAM core whereby bit-storage is implemented by memristors ME1 and ME2. Bit comparison is performed by either NOR or alternatively NAND based logic as part of the match-line ML$_i$ circuitry. The matching operation is equivalent to logical XORing of the search bit (SB) and stored bit (D). The match-line transistors (ML) in the NOR-type cells can be considered as part of a pull-down path of a pre-charged NOR gate connected at the end of each individual ML$_i$ row. The NAND-type CAM functions in a similar manner forming the pull-down of a pre-charged NAND gate. Although each of the selected cells in Fig. \[fig:CAMcells\] have their relative merits, the approach in Fig. \[fig:CAMcells\](c) where Data bits and Search bits share a common bus is selected for detailed analysis. The structure of the 7-T NAND-type, shown in Fig. \[fig:CAMcells\](d), and the NOR-type are identical except for the position of the ML transistor. In the NOR-type, ML makes a connection between shared ML and ground while in the NAND-type, the ML transistors act as a series of switches between the ML$_{i}$ and ML$_{i+1}$. [cc]{}\ \ \ \ \ \ \ Simulation Results Analysis and Comparison {#sec:simresult} ========================================== Generally, there are the “write” and “read” operations that require consideration. In this section the “write” and “read” operations of the basic MCAM cell for 7-T NOR-type are reported. Simulations of the circuits are based on the following parameters [@Witrisal2009a]: $R_{\rm ON}=100~\Omega$, $R_{\rm OFF}=100~{\rm k}\Omega$, $p=4$, $L=3~{\rm nm}$, and $\mu_v=3\times 10^{-8}~{\rm m^2/s/V}$. Both the conventional CAM and MCAM circuits have been implemented using Dongbu HiTech $0.18~\mu{\rm m}$ technology where $1.8$ Volts is the nominal operating voltage for the CAM. The MCAM cell is implemented using nMOS devices and memristors without the need for $V_{\rm DD}$ voltage source. Using the above memristor parameters, together with the behavioral model B-II of Table \[tab:models\], satisfactory operation of the MCAM cell is achieved at $3.0$ Volts. We have referred to this voltage as the nominal voltage for the MCAM cell. Furthermore, the initial state of the memristors (“ON”, “OFF”, or in between) is determined by initial resistance, $R_{\rm INIT}$. Write operation {#sec:sub:write} --------------- At the write phase, the memristor ME1 is programmed based on the data bit on the D line. The complementary data is also stored in ME2. During the write operation, the select line is zero and an appropriate write voltage is applied on VL. The magnitude of this voltage is half of supply voltage, that corresponds to $V_{\rm DD}/2$. The pulse width is determined by the time required for the memristor to change its state from logic “1” ($R_{\rm ON}$) to logic “0” ($R_{\rm OFF}$) or vice versa. Waveforms in Fig. \[fig:writeph\] illustrate the write operation. In this case $R_{\rm INIT}=40~{\rm k}\Omega$ and the initial state is around $0.6$. The diagrams show two write operations, for both when D is “1” and when it is “0”. By applying $V_{\rm DD}/2$ to VL line, there will be a $-V_{\rm DD}/2$ potential across the memristor ME2 and $V_{\rm DD}-V_{\rm th,M1}$ across the memristor ME1. The highlighted area in Fig. \[fig:writeph\](b) shows the difference in the write operation between ME1 and ME2. When ${\rm D}=0$ and $\overline{\rm D}=V_{\rm DD}$, there is a threshold voltage ($V_{\rm th}$) drop at the $\overline{\rm SB}$ node. Thus, the potential across the memristor would be $V_{\rm DD}/2-V_{\rm th,M2}$. At the same time, $-V_{\rm DD}/2$ is the voltage across the ME1, so the change in state in ME1 occurs faster than memristor ME2. The time for a state change is approximately $75$ ns for ME1 and $220$ ns for ME2. Therefore, $145$ ns delay is imposed because of the voltage drop across the ME2. Fig. \[fig:writeph\](b) illustrates simulation results carried out using a behavioral SPICE macro-model. [cc]{}\ \ \ Read operation {#sec:sub:read} -------------- Let us assume that ME1 and ME2 were programmed as a logic “1” and logic “0”, respectively. Therefore, ME1 and ME2 are in the “ON” and “OFF” states and $R_{\rm INIT,ME1}=200~\Omega$ and $R_{\rm INIT,ME2}=99~{\rm k}\Omega$. In this case, the search line, S, is activated first. At the same time search select signal, SS, is activated to turn on the two select transistors, M5 and M6. The word select (WS) is disabled during the read operation. Fig. \[fig:readph\] shows the waveforms for a complete read cycle. Read operation requires higher voltage for a short period of time. The VL pulse width (PW) for read operation is $12~{\rm ns}$ as illustrated in Fig. \[fig:readph\](b) which is the “minimum” pulse width necessary to retain memristor’s state. For a matching “1” (when S=$V_{\rm DD}$), the sequence of operations are as follows: (i) match line, ML, is pre-charged, (ii) SS is activated, and (iii) VL is enabled as is shown in Fig. \[fig:readph\](a)-(c). A logic “1” is transferred to the bit-match node, which discharges the match line, ML$_i$, through transistor ML. At this point $x_{\rm ME1}$ commences to decrease its state from $1$ to $0.84$ and $x_{\rm ME2}$ increases its state from $0$ to $0.05$. Thus, there is a match between stored Data and Search Data. The following read operation for S=“0” follows a similar pattern as shown in Fig. \[fig:readph\](c). The simulation results confirm the functionality of proposed MCAM circuitry. -- -- -- -- Simulation results analysis {#sec:sub:compari} --------------------------- Table \[tab:comp\] provides a comparison between the various MCAM cells that are proposed in Fig. \[fig:CAMcells\]. It is worth noting that simulations are based on a single cell. Therefore there are no differences in characteristics between 7-T NAND and 7-T NOR cells. The difference in minimum VL pulse width for read operation (VL$_{\rm min.PW,R}$), between different MCAM cells, is relatively significant and is brought about as the result of pass-transistors in the path from search line to the bit-match node. One important issue in the design of MCAM cells is endurance. For instance, DRAM cells must be refreshed at least every $16~{\rm ms}$, which corresponds to at least $10^{10}$ write cycles in their life cycle [@Lewis2009]. Analysing a write operation followed by two serial read operations shows that 5-T, 6-T, and 7-T NOR/NAND cells deliver a promising result. After two serial read operations the memristor state values for $x_{\rm ME1}$ and $x_{\rm ME2}$ are, $0.74$ and $0.06$, and $0.71$ and $0.09$, for 5-T, 6-T, and 7-T NOR/NAND cell, respectively. The overall conclusion from the simulation results shows that in terms of speed, the 6-T NOR-type MCAM cell has improved performance, but it uses separate Data and Search lines. The 7-T NOR/NAND cell shares the same line for Data and Search inputs. However, it is slightly slower VL$_{\rm min.PW,R}=12~{\rm ns}$, while the swing on the match-line is reduced by threshold voltage ($V_{\rm th}$) drop. ------------------------------------------- ----------------------------- ---------------------------- ----------------------------- --------------- VL$_{\rm min.PW,W}$ \[ns\] VL$_{\rm min.PW,R}$ \[ns\] V$_{\rm drop}($bit-match$)$ Data & Search VL$_{\rm W}$=$V_{\rm DD}$/2 VL$_{\rm R}$=$V_{\rm DD}$ Voltage \[V\] Buses 6-T NOR (Fig. \[fig:CAMcells\](b)) $223$ $5$ $0$ Separate 5-T NOR (Fig. \[fig:CAMcells\](a)) $219$ $9$ $V_{\rm th}$ Separate 7-T NOR/NAND (Fig. \[fig:CAMcells\](c/d)) $220$ $12$ $V_{\rm th}$ Shared ------------------------------------------- ----------------------------- ---------------------------- ----------------------------- --------------- \[tab:comp\] ### Power Analysis {#sec:sub:sub:power} A behavioral model was used to estimate peak, average, and RMS power dissipation of an MCAM cell compared to the conventional SRAM-based cell. The power consumption is the total value for the static and dynamic power dissipation. A reduction of some $96$% in average power consumption with an MCAM cell was noted. The maximum power dissipation reduction is over $74$% for the memristor-based structure. The Root Mean Square (RMS) value of current, which is sunk from the supply rail for the MCAM, is around $47$ $\mu$A less than the conventional SRAM-based circuitry, which shows over $95$% reduction. To the best of our knowledge this is the first power consumption analysis of a memristor-based structure using a behavioral modeling approach. As the technology matures it is conjectured that a similar power source could be used for the hybrid scaled CMOS/Memristor cell. A $2\times 2$ Structure Verification {#sec:sub:2x2} ------------------------------------ Fig. \[fig:2x2arch\] illustrates implementation of a $2\times 2$ structure whereby the 7-T NAND-type (Fig. \[fig:CAMcells\](d)) is used. As is stated before, in the NOR-type, ML makes a connection between shared ML and ground while in the NAND-type, the ML transistors act as a series of switches between the ML$_{\rm out}$ and ground. The ML$_{1}$ and ML$_{2}$ match signals, illustrated in Fig. \[fig:2x2arch\](a), are these ML$_{\rm out}$ signals. The cells are initially programmed to be “0” or “1” and the search bit vector is “10”. The first row cells are programmed “10”. As the consequence, ML$_1$ is discharged since there is a match between the stored and search bit vectors. Fig. \[fig:2x2arch\](b) and (c) demonstrate the ML$_1$ and ML$_2$ outputs, respectively. Basically, using the ML transistors as an array of pass-transistors in a NAND-type structure imposes a significant delay, but in this case, the timing information shows the delay of matching process is around $12~{\rm ns}$. A large scale co-simulation of crossbar memories can be carried out each junction assumed to be either a diode or a 1D-1R (a parallel structure of one diode and one resistor) or even a linear resistor [@Ziegler2003]. However, the modeling approach should be carefully revisited since large resistor nonlinearity is associated with crosspoint devices [@Vontobel2009]. A co-simulation of crossbar memories, considering the highly nonlinear crosspoint junctions, is underpins our longer term research objective. [c]{}\ \ \ \ \ Physical Layout and Fabrication {#sec:layfab} =============================== Physical Layout {#sec:sub:Physical-Layout-and-Layer-Definitions} --------------- Layout of conventional 10-T NOR-type CAM and 7-T NOR-type MCAM cells are shown in Fig. \[fig:layout\]. The MCAM cell has a dimensions of $4.8\times 4.36~\mu {\rm m}^2$ while the dimensions for the conventional SRAM-based cell is $6.0\times 6.5~\mu {\rm m}^2$. Thus, the reduction in silicon area is in the order of $46$%. The $2\times 2$ structure also shows over a $46$% area reduction. The two memristors, shown in highlighted regions of Fig. \[fig:layout\](b) are implemented between metal-3 and metal-4 layers as part of CMOS post processing. [cc]{}\ \ \ Fabrication and Layer Definitions {#sec:sub:Fabrication} --------------------------------- Fig. \[fig:fab\](a) illustrates a cross-section of Pt, TiO$_2$, and TiO$_{2-x}$ layers over silicon substrate. The TiO$_2$ layer thickness must be restricted below two nanometers, to prevent separate conduction through the individual layers. The n-type MOS devices are patterned onto a silicon wafer using normal CMOS processing techniques, which subsequently is covered with a protective oxide layer. The Pt memristor wires are patterned and connections made to the n-type MOS devices. The upper Pt nanowire is patterned and, electrical connections made by photolithography (to spatially locate the vias) and aluminum metal deposition [@Strukov2008]. [cc]{}\ \ \ Fig. \[fig:fab\](b) demonstrates a TEM microphotograph of a TiO$_{2-x}$ overlay on a silicon substrate in order to explore the controllability of oxygen ions. The device consists of a top gate Pt, TiO$_2$/TiO$_{2-x}$ layer and back gate Pt on SiO$_2$ layer of silicon. TiO$_{2-x}$ thin film with a thickness of $9.4$ nm was deposited on a silicon wafer using sputtering technique. Table \[tab:fab\] is deposition result with sputtering technique. Samples show that 1.85% oxygen (O) vacancy can be achieved keeping within the $2$% tolerance. --- --------- --------- ---------------------------- --------------------------------------- O Ti ${\rm O}-2\times {\rm Ti}$ $({\rm O}-2\times {\rm Ti})/{\rm Ti}$ % % Normalized Normalized 1 $66.46$ $33.54$ $-0.62$ $-1.85$ 2 $66.67$ $33.32$ $0.03$ $0.09$ --- --------- --------- ---------------------------- --------------------------------------- : Deposition results using sputtering technique. \[tab:fab\] Conclusions {#sec:Conclusions} =========== The idea of a circuit element, which relates the charge $q$ and the magnetic flux $\phi$ realizable only at the nanoscale with the ability to remember the past history of charge flow, creates interesting approaches in future CAM-based architectures as we approach the domain of multi-technology hyperintegration where optimization of disparate technologies becomes the new challenge. The scaling of CMOS technology is challenging below $10$ nm and thus nanoscale features of the memristor can be significantly exploited. The memristor is thus a strong candidate for tera-bit memory/compare logic. The non-volatile characteristic and nanoscale geometry of the memristor together with its compatibility with CMOS process technology increases the memory cell packing density, reduces power dissipation and provides for new approaches towards power reduction and management through disabling blocks of MCAM cells without loss of stored data. Our simulation results show that the MCAM approach provides a $45$% reduction in silicon area when compared with the SRAM equivalent cell. The Read operation of the MCAM ranges between $5$ ns to $12$ ns, for various implementations, and is comparable with current SRAM and DRAM approaches. However the Write operation is significantly longer. Simulation results indicate a reduction of some $96$% in average power dissipation with the MCAM cell. The maximum power reduction is over $74$% for the memristor-based structure. The RMS value of current sunk from the supply rail for the MCAM is also approximately $47$ $\mu$A, which correspond to over a $95$% reduction when compared to SRAM-based circuitry. To the best of our knowledge this is the first power consumption analysis of a memristor-based structure that has been presented using a behavioral modeling approach. As the technology is better understood and matures further improvements in performance can be expected Acknowledgement {#sec:Acknowledgement} =============== The support provided by grant No. R33-2008-000-1040-0 from the World Class University (WCU) project of MEST and KOSEF through CBNU is gratefully acknowledged. The authors also note the contribution of iDataMap Pty Ltd for the initial concept and gratefully acknowledge Drs Jeong Woo Kim, Han Heung Kim, and Boung Ju Lee of NanoFab in Korea Advanced Institute Science and Technology (KAIST) for their contribution towards fabrication. \[sec:References\] [^1]: Manuscript received December 31, 2009. This work was supported by grant No. R33-2008-000-1040-0 from the World Class University (WCU) project of MEST and KOSEF through Chungbuk National University (CBNU). [^2]: K. Eshraghian, K. R. Cho, O. Kavehei, and S. K. Kang are with the College of Electrical and Information Engineering, WCU Program, Chungbuk National University, Cheongju, South Korea (e-mails: k.eshraghian@innovationlabs.com.au, krcho@cbnu.ac.kr, omid@hbt.cbnu.ac.kr, and skkang@hbt.cbnu.ac.kr). O. Kavehei is also with the School of Electrical and Electronic Engineering, University of Adelaide, SA 5005, Australia (e-mail: omid@eleceng.adelaide.edu.au). [^3]: D. Abbott is with the School of Electrical and Electronic Engineering, University of Adelaide, SA 5005, Australia (e-mail: dabbott@eleceng.adelaide.edu.au). [^4]: S. M. Kang is with the School of Engineering, University of California, Merced, CA 95343 USA (e-mail: smk123@ucmerced.edu).
--- abstract: 'Unsupervised domain adaptation (UDA) for nuclei instance segmentation is important for digital pathology, as it alleviates the burden of labor-intensive annotation and domain shift across datasets. In this work, we propose a Cycle Consistency Panoptic Domain Adaptive Mask R-CNN (CyC-PDAM) architecture for unsupervised nuclei segmentation in histopathology images, by learning from fluorescence microscopy images. More specifically, we first propose a nuclei inpainting mechanism to remove the auxiliary generated objects in the synthesized images. Secondly, a semantic branch with a domain discriminator is designed to achieve panoptic-level domain adaptation. Thirdly, in order to avoid the influence of the source-biased features, we propose a task re-weighting mechanism to dynamically add trade-off weights for the task-specific loss functions. Experimental results on three datasets indicate that our proposed method outperforms state-of-the-art UDA methods significantly, and demonstrates a similar performance as fully supervised methods.' author: - 'Dongnan Liu$^{1}$' - 'Donghao Zhang$^{1}$' - 'Yang Song$^{2}$' - 'Fan Zhang$^{3}$' - 'Lauren O’Donnell$^{3}$' - 'Heng Huang$^{4}$' - 'Mei Chen$^{5}$' - 'Weidong Cai$^{1}$' - | \ $^{1}$School of Computer Science, University of Sydney, Australia\ $^{2}$School of Computer Science and Engineering, University of New South Wales, Australia\ $^{3}$Brigham and Women’s Hospital, Harvard Medical School, USA\ $^{4}$Department of Electrical and Computer Engineering, University of Pittsburgh, USA\ $^{5}$Microsoft Corporation, USA\ [{dliu5812, dzha9516}@uni.sydney.edu.au, yang.song1@unsw.edu.au]{}\ [{fzhang, odonnell}@bwh.harvard.edu, henghuanghh@gmail.com]{}\ [may4mc@gmail.com, tom.cai@sydney.edu.au]{} bibliography: - 'egbib.bib' title: 'Unsupervised Instance Segmentation in Microscopy Images via Panoptic Domain Adaptation and Task Re-weighting' --- Introduction ============ Nuclei instance segmentation in histopathology images is an important step in the digital pathology workflow. Pathologists are able to diagnose and prognose cancers according to mitosis counts, the morphological structure of each nucleus, and spatial distribution of a group of nuclei [@elston1991pathological; @le1989prognostic; @clayton1991pathologic; @basavanhally2011multi; @nawaz2016computational]. Currently, supervised learning-based methods for nuclei instance segmentation are prevalent as they are efficient while preserving high accuracy [@kumar2017dataset; @naylor2018segmentation; @chen2017dcan; @graham2019hover; @mahmood2018deep; @zhang2018panoptic; @liu2019nuclei; @liu2020cell]. However, their performance heavily relies on large-scale training data, which requires expertise for annotation. This process is time-consuming and labor-intensive due to the complicated cellular structures, as shown in Fig. \[intro\](b), and large image sizes. For example, annotating a histopathology dataset with $50$ images and $12$M pixels costs a pathologist $120$ to $230$ hours [@hou2019robust]. Moreover, in real clinical studies, even one whole slide image in $40\times$ objective magnification contains $1$B pixels [@gutman2013cancer]. Therefore, investigating methods without depending on histopathology annotations is necessary. It can help pathologists to reduce the workload, and tackle the issue of lacking histopathology annotations. ![Example images of our proposed framework. (a) fluorescence microscopy images; (b) real histopathology images; (c) our synthesized histopathology images; (d) nuclei segmentation generated by our proposed UDA method; (e) ground truth.[]{data-label="intro"}](cvpr2020-fig1.png){width="46.00000%"} The recently proposed unsupervised domain adaptation (UDA) methods tackle this issue by conducting supervised learning on the source domain and obtain a good performance model for the target domain without annotations [@pan2009survey; @ganin2014unsupervised; @tzeng2017adversarial]. Currently, UDA reduces distances between the distribution of feature maps of the source and target domains. In addition, some other methods focus on the pixel-to-pixel translation from the source domain images to the target ones, for aligning cross-domain image appearances [@isola2017image; @zhu2017unpaired]. For these methods, there still remain some differences in the distributions between the synthesized and real images, due to the imperfect translations [@hoffman2017cycada; @chen2019synergistic; @kim2019diversify]. To incorporate the benefits of the image translation and the UDA methods, several works have been proposed to learn the domain-invariant features between the target and the synthesized target-like images [@hoffman2017cycada; @kim2019diversify; @chen2019synergistic]. Such methods achieve state-of-the-art performance on UDA classification, object detection, and semantic segmentation tasks. However, currently there is a lack of UDA methods specifically designed for instance segmentation, and directly extending the existing UDA methods on object detection [@chen2018domain; @kim2019diversify; @he2019multi] to the UDA nuclei instance segmentation task still suffers from challenges. First, existing UDA object detection methods focus on alleviating the domain bias at the image level (image contrast, brightness, etc.) and the instance level (object scale, style, etc.) [@kim2019diversify; @chen2018domain; @he2019multi]. They ignore the domain shift at the semantic level, such as the relationship between the foreground and background, and the spatial distribution of the objects. Second, these UDA object detection methods are multi-task learning paradigms, which optimize different loss functions simultaneously. If the feature extractors fail to generate domain-invariant features in some training iterations, then back-propagating the weights according to the task loss functions in these iterations causes the model bias towards the source domain. To solve the aforementioned problems in UDA nuclei instance segmentation tasks in histopathology images, we propose a Cycle-Consistent Panoptic Domain Adaptive Mask R-CNN (CyC-PDAM) model. As none of the previous UDA methods are specially designed for instance segmentation, we extend the CyCADA [@hoffman2017cycada] to an instance segmentation version based on Mask R-CNN [@he2017mask], as our baseline. In our CyC-PDAM, we firstly propose a simple nuclei inpainting mechanism to remove the auxiliary nuclei in the synthesized histopathology images. Second, inspired by the panoptic segmentation architectures [@kirillov2019panoptic; @kirillov2019panopticfpn], we propose a semantic-level adaptation module for domain-invariant features based on the relationship between the foreground and the background. By reconciling the domain-invariant features at the semantic and instance levels, our proposed CyC-PDAM achieves panoptic-level domain adaptation. Furthermore, a task re-weighting mechanism is proposed to reset the importance for each task loss. During training, the specific task losses are down-weighted if the features for task predictions are not domain-invariant and source-biased, and up-weighted if the features are hard to differentiate. To prove the effectiveness of our proposed CyC-PDAM architecture, experiments have been conducted on three public datasets for unsupervised nuclei instance segmentation of histopathology images on two different datasets by unsupervised domain adaptation from a fluorescence microscopy image dataset. Unlike histopathology images, no structures are similar to the nuclei in the background of fluorescence microscopy images, due to the differences between image acquisition techniques, as shown in Fig. \[intro\](a). It is much easier to obtain manual annotation for the fluorescence microscopy images compared with histopathology images, therefore it is chosen as our source domain. Our contribution is summarized as follows: (1) We propose a CyC-PDAM model for UDA nuclei instance segmentation in histopathology images. To our best knowledge, this is the first UDA instance segmentation method. (2) A simple nuclei inpainting mechanism is proposed to remove false-positive objects in the synthesized images. (3) Our CyC-PDAM produces domain-invariant features at the panoptic level, by integrating the instance-level adaptation with a newly proposed semantic-level adaptation module. (4) A task re-weighting mechanism is proposed to alleviate the domain bias towards the source domain. (5) Compared with state-of-the-art UDA methods, our proposed CyC-PDAM paradigm outperforms them by a large margin. Moreover, it achieves competitive performance compared with state-of-the-art fully supervised methods for nuclei segmentation. Related Work ============ ![image](overall3){width="85.00000%"} Domain Adaptation for Natural Images ------------------------------------ Domain adaptation aims at transferring the knowledge learned from one labeled domain to another without annotation [@pan2009survey]. Recently, UDA methods have reduced the cross-domain discrepancies based on the content in the feature level and the appearance in the pixel level. For the feature-level adaptation, adversarial learning for domain-invariant features [@ganin2014unsupervised; @tzeng2017adversarial], Maximum Mean Discrepancy minimization (MMD) [@long2015learning], local pattern alignment [@wen2019exploiting], and cross-domain covariance alignment [@sun2016return] are widely employed for classification tasks. In addition, domain adaptation is further employed for other tasks such as semantic segmentation [@vu2019advent; @li2019bidirectional] and object detection [@chen2018domain; @kim2019diversify; @inoue2018cross; @wang2019few]. In semantic segmentation tasks, the segmentation results are forced to be domain-invariant, together with intermediate feature maps [@li2019bidirectional; @vu2019advent; @tsai2018learning]. Additionally, ADVENT [@vu2019advent] further minimized the Shannon entropy for the semantic segmentation predictions in source and target domains to alleviating the cross-domain discrepancy. For object detection, a domain adaptive Faster R-CNN [@ren2015faster], consisting of the image- and instance-level adaptions, was usually proposed for domain-invariant features of the whole image and each object [@chen2018domain; @kim2019diversify; @he2019multi]. On the other hand, image-to-image translation addresses the domain adaptation problems in the pixel level by generating target-like images and training task-specific fully supervised models on them [@liu2017unsupervised; @huang2018multimodal; @isola2017image; @zhu2017unpaired; @mahmood2018deep; @park2019semantic]. However, domain bias still exists because of imperfect translation. Moreover, several methods have been proposed to align the feature-level adaptation with the pixel-level one, by learning domain-invariant features between the target images and the synthesized images [@hoffman2017cycada; @kim2019diversify; @chen2019synergistic]. Domain Adaptation for Medical Images ------------------------------------ Unsupervised domain adaptation for medical image analysis has rarely been explored [@ren2018adversarial; @zhang2018task; @chen2019synergistic; @huang2017epithelium; @hou2019robust]. [@ren2018adversarial] and [@huang2017epithelium] solve the UDA histopathology images classification problems with GAN based architectures. In addition, DAM [@dou2018unsupervised] is proposed to generate domain-invariant intermediate features and model predictions, for UDA semantic segmentation in CT images. With the help of cycle-consistency reconstruction, TD-GAN [@zhang2018task] and SIFA [@chen2019synergistic] are proposed for semantic segmentation on different medical images, with both pixel- and feature-level adaptations. However, none of them is designed for UDA nuclei instance segmentation. Even though Hou  [@hou2019robust] proposed to train a GAN based refiner and a nuclei segmentation model with the synthesized histopathology images for unsupervised nuclei instance segmentation, their paradigm only contains pixel-level adaptation and is still not capable for minimizing the domain gap in the feature level. In this work, we therefore propose a CyC-PDAM paradigm for UDA nuclei instance segmentation, which alleviates the domain bias issue in the pixel and feature levels. Methods ======= Our proposed architecture is based on CyCADA and we fuse CyCADA with the instance segmentation framework Mask R-CNN. Furthermore, we improve it with nuclei inpainting mechanism, panoptic-level domain adaptation, and task re-weighting mechanism. Fig. \[overall\] illustrates the overall architecture of our approach. CyCADA with Mask R-CNN \[baseline-sec\] --------------------------------------- As there is no UDA architectures targeting instance-level segmentation, we firstly design a domain adaptive Mask R-CNN. The backbone of the Mask R-CNN in this work is constructed with ResNet101 [@he2016deep] and Feature Pyramid Network (FPN) [@lin2017feature]. Inspired by the previous UDA methods for object detection [@chen2018domain; @kim2019diversify], we add one discriminator after FPN for the image-level adaptation, and the other after the instance branch for instance-level adaptation, as shown in Fig. \[pda\]. For the image-level adaptation, the multi-resolution feature maps of the FPN output are firstly down-sampled to the size $8 \times 8$ with average pooling, and then summed together for the image-level discriminator. The image-level discriminator consists of $4$ convolutional layers (details in Table \[dimg\]) and a gradient reversal layer (GRL) for adversarial learning. In the instance-level adaptation, the $14 \times 14 \times 256$ feature map in the mask branch is down-scaled to the size $2 \times 2 \times 256$ with average pooling and then resized to $1024 \times 1$, to sum with the $1024 \times 1$ feature from the bounding box branch. The instance-level discriminator consists of $3$ fully connected layers and a GRL, whose input is the summation of features mentioned above. ![image](pdam2){width="80.00000%" height="30.00000%"} ![image](ada1){width="80.00000%" height="10.00000%"} Nuclei Inpainting Mechanism --------------------------- Even though CycleGAN is effective for synthesizing histopathology-like images, due to the large domain gap and nuclei number incompatibility between the source and target domains, the label space for the generated images sometimes changes after transferring from the source domain. For example, there are redundant and undesired nuclei in the synthesized images shown in Fig. \[dsig-vis\]. If these images are directly used to train the task-specific CNN with the original labels, the model is forced to regard redundant nuclei as background, even though they appear as real nuclei. ![Visual results for the effectiveness of nuclei inpainting mechanism. (a) original fluorescence microscopy patches; (b) corresponding nuclei annotations; (c) initial synthesized images from CycleGAN; (d) final synthesized images after nuclei inpainting mechanism.[]{data-label="dsig-vis"}](dsig-vis-1){width="30.00000%"} Therefore, we propose an auxiliary nuclei inpainting mechanism to remove the nuclei which only appear in the synthesized images without corresponding annotations. Denoting a raw synthesized histopathology image by CycleGAN as $S_{raw}$ and its corresponding mask as $M$, we first obtain the mask predictions $M_{aux}$ of all the auxiliary generated nuclei, formulated as: $$\begin{aligned} M_{aux} = (otsu(S_{raw}) \cup M) - M \end{aligned}$$ where $ostu(S_{raw})$ represents a binary segmentation method for $S_{raw}$ based on Otsu threshold. In $M_{aux}$, only auxiliary nuclei without annotation is labeled. Then, we get the newly synthesized image $S_{inp}$ after removing these nuclei, which can be represented as: $$\begin{aligned} S_{inp} =inp(S_{raw}, M_{aux}) \end{aligned}$$ where $inp$ is a fast marching based method for inpainting objects [@telea2004image], by replacing the pixel values for the auxiliary nuclei labeled in $M_{aux}$ with them for the unlabeled background. Fig. \[dsig-vis\] illustrates the visual effectiveness of our proposed nuclei inpainting mechanism. However, some background materials are labeled as false positive predictions in $M_{aux}$. Directly inpainting them makes the texture and appearance of synthesized images unrealistic, and enlarges the domain gap between the synthesized and real images. However, the image-level adaptation is able to address this issue by alleviating the domain bias on global visual information, such as curve, texture, and illumination. Our nuclei inpainting mechanism is time-efficient, which takes $0.09$ second to process one single $256 \times 256$ synthesized histopathology patch, on average. Panoptic Level Domain Adaptation -------------------------------- We define the semantic-level features of an image as the relationship between its foreground and background. In addition to the image- and feature-level domain bias, the domain shift at the semantic level also exists. Due to the differences in the nuclei objects and background between the synthesized and real histopathology images, domain adaptive Mask R-CNN mentioned in Sec. \[baseline-sec\] suffers from domain bias in the semantic-level features, as the Mask R-CNN only focuses on the local features for each object and lacks a semantic view of the whole image. Inspired by the previous panoptic segmentation architecture, which unified the semantic and instance segmentation to process the global and local features of the images, we propose a semantic-level adaptation to induce the model to learn domain-invariant features based on the relationship between the foreground and background. By incorporating the semantic- and instance-level adaptation, our panoptic domain adaptive method reduces the cross-domain discrepancies in a global and local view. As shown in Fig. \[pda\], a semantic branch for semantic segmentation prediction is added to the output of the FPN. Our semantic branch has the same implementation as [@kirillov2019panopticfpn]. As the fluorescence microscopy images and histopathology images can both be acquired from tissue samples and they can show complementary and correlated information, the semantic segmentation label spaces of the synthesized and real histopathology images have a strong similarity. In addition, aligning the cross-domain entropy distributions helps to minimize the entropy prediction in the target domain, which makes the model suitable for the target images [@vu2019advent]. Therefore, we use the Shannon entropy [@shannon1948mathematical] of the softmax semantic predictions to induce the domain-invariant features to learn at the semantic level. Denoting the softmax semantic prediction as $P$ and $P \in (0,1)$, its Shannon entropy is defined as: $-plog(p)$. Fig. \[pda\] and Table \[dsem\] indicate the detailed structure of the discriminator for semantic level adaptation. We employ residual connected CNN blocks to avoid gradient vanishing [@he2016deep; @he2016identity]. To make the adversarial learning more stable, instead of bilinear interpolation, we use stride convolutional layers for upsampling. Finally, the domain label is predicted as a $16 \times 16$ patch. Due to the small mini-batch size, the patch-based domain label prediction increases the number of training samples, to avoid overfitting. Task Re-weighting Mechanism --------------------------- In the previous UDA methods, the task-specific loss functions (segmentation, classification, and detection) are based on the source domain predictions. Even though several adversarial domain discriminators are employed to ensure the predicted feature maps are domain-invariant, the cross-domain discrepancies of these feature maps are still large in some training iterations, where the features are far from the decision boundaries of the domain discriminators. If the task-specific losses are updated to optimize the models with these easily-distinguished features, the models will bias towards the source images when testing it with the target data. To this end, we propose a task re-weighting mechanism to add a trade-off weight for each task-specific loss function according to the prediction of the domain discriminator. Denote the probability of the feature map before the final task prediction belonging to the source and target domains as $p_{s}$ and $p_{t}$, respectively, and the task-specific loss function as $L$, then the re-weighted task-specific loss $L_{rw}$ is: $$\begin{aligned} L_{rw} = min (\frac{p_{t}}{p_{s}} , \beta)L = min(\frac{1-p_{s}}{p_{s}}, \beta) L \end{aligned} \label{taskrw}$$ where $\beta$ is a threshold value to avoid the $\frac{1-p_{s}}{p_{s}}$ becoming large and making the model collapse, when $p_{s} \to 0$. According to Eq. \[taskrw\], if the feature map deciding the task prediction belongs to the source domain ($p_{s} \to 1$), the loss function is then down-weighted, to alleviate the source-bias feature learning of the model. As illustrated in Fig. \[pda\], the loss function for the region proposal network (RPN), semantic branch, and the instance branch are re-weighted by the prediction at the image-, semantic-, and instance-level domain discriminators, respectively. Network Overview and Training Details ------------------------------------- In our proposed CyC-PDAM, the CycleGAN has the same implementation as its original work [@zhu2017unpaired]. When training the CycleGAN, the initial learning rate was set to $0.0001$ for the first $1/2$ of the total training iterations, and linearly decayed to $0$ for the other $1/2$. The PDAM is trained with a batch size of $1$ and each batch contains $2$ images, one from the source and the other from the target domain. Due to the small batch size, we replace traditional batch normalization layers with group normalization [@wu2018group] layers, with the default group number $32$ as in [@wu2018group]. The overall loss function of PDAM is defined as: $$\begin{aligned} L_{pdam} & = \alpha_{img} L_{rpn} + \alpha_{ins} L_{det} + \alpha_{sem} L_{(sem-seg)} \\ & + \alpha_{da}(L_{(img-da)} + L_{(sem-da)} + L_{(ins-da)}) \end{aligned}$$ where $L_{rpn}$ is the loss function for the RPN, $L_{det}$ is the loss of class, bounding box, and instance mask prediction of Mask R-CNN, $L_{(sem-seg)}$ is the cross entropy loss for semantic segmentation, $L_{(img-da)}$, $L_{(sem-da)}$ and $L_{(ins-da)}$ are cross entropy losses for domain classification at image, semantic and instance levels. $\alpha_{img}$, $\alpha_{ins}$, and $\alpha_{isem}$ are calculated according to Eq. \[taskrw\] for task re-weighting. In our experiment, we set $\beta$ as $2$. $\alpha_{da}$ is updated as: $$\begin{aligned} \alpha_{2} = \frac{2}{1 + exp(-10t)} - 1 \end{aligned}$$ where $t$ is the training progress and $t \in [0, 1]$. Thus $\alpha_{da}$ is gradually changed from $0$ to $1$, to avoid the noise from the unstable domain discriminators in the early training stage. During training, the PDAM is optimized by SGD, with a weight decay of $0.001$ and a momentum of $0.9$. The initial learning rate is $0.002$, with linear warming up in the first $500$ iterations. The learning rate is then decreased to $0.0002$ when it reaches $3/4$ of the total training iteration. During inference, only the original Mask R-CNN architecture is used with the adapted weight and all of the hyperparameters for testing are fine-tuned on the validation set. All of our experiments were implemented with Pytorch [@paszke2017automatic], on two NVIDIA GeForce 1080Ti GPUs. Experiments =========== Datasets Description and Evaluation Metrics ------------------------------------------- Our proposed architecture was validated on three public datasets, referred to as Kumar [@kumar2017dataset], TNBC [@naylor2018segmentation], and BBBC039V1 [@ljosa2012annotated], respectively. Among them, Kumar and TNBC are histopathology datasets, while BBBC039V1 is a fluorescence microscopy dataset. Kumar was acquired from The Cancer Genome Atlas (TCGA) at $40 \times$ magnification, containing $30$ annotated $1000 \times 1000$ patches from $30$ whole slide images of different patients. All these images are from $18$ different hospitals and $7$ different organs (breast, liver, kidney, prostate, bladder, colon, and stomach). In contrast to the disease variability in Kumar, the TNBC dataset especially focuses on Triple-Negative Breast Cancer (TNBC) [@naylor2018segmentation]. In TNBC, there are $50$ annotated $512 \times 512$ patches from $11$ different patients from the Curie Institute at $40 \times$ magnification. BBBC039V1 is about U2OS cells under a high-throughput chemical screen [@ljosa2012annotated]. It contains $200$ $520 \times 696$ images about bioactive compounds, with the DNA channel staining of a single field of view. For evaluation, we employ three commonly used pixel- and object-level metrics. Aggregated Jaccard Index (AJI) is an extended Jaccard Index for object-level evaluation [@kumar2017dataset], and object-level F1 score is the average harmonic mean between the precision and recall for each object. For pixel-level evaluation, we employ pixel-level F1 score for binarization predictions. Experiment Setting ------------------ We conducted our experiments on two nuclei segmentation tasks: adapting from BBBC039V1 to Kumar, and from BBBC039V1 to TNBC. As the source domain in two experiments, $100$ training images and $50$ validation images from BBBC039V1 are used, following the official data split[^1]. The annotations for Kumar and TNBC are not used during training the UDA architecture, only for evaluation. The preprocessing for source fluorescence microscopy images has $3$ steps. First, all images are normalized into range $[0, 255]$. Second, $10K$ patches in size $256 \times 256$ are randomly cropped from the $100$ training images, with data augmentation including rotation, scaling, and flipping to avoid overfitting. Third, the patches with fewer than $3$ objects are removed. For better synthesizing target-like histopathology images, we finally inverse the pixel value of foreground nuclei and background for all source fluorescence microscopy patches. For validation, $50$ images in the BBBC039V1 validation set are transferred to synthesized histopathology images by CycleGAN and nuclei inpainting mechanism. For the Kumar dataset as the target domain, we have the same data split as previous work in [@kumar2017dataset; @naylor2018segmentation], with $16$ images for training, and $14$ for testing. When training the model, totally $10K$ patches in size $256 \times 256$ are randomly cropped from the $16$ training histopathology images, with basic data augmentation including flipping and rotation, to avoid overfitting. As for TNBC, we use $8$ cases with $40$ images for training, and the remaining $3$ cases with $10$ images for testing. To train the model with TNBC, $10K$ $256 \times 256$ patches are randomly extracted from the training images with basic data augmentation including flipping and rotation. Comparison Experiments ---------------------- ### Comparison with Unsupervised Methods \[sec-cmp\] In this section, our proposed CyC-PDAM is compared with several state-of-the-art UDA methods, including CyCADA [@hoffman2017cycada], Chen [@chen2018domain], SIFA [@chen2019synergistic], and DDMRL [@kim2019diversify]. As the original CyCADA focuses on classification and semantic segmentation, we extend it with Mask R-CNN for UDA instance segmentation, as described in Sec. \[baseline-sec\]. Chen [@chen2018domain] are originally for UDA object detection based on Faster R-CNN, by adapting the features at the image and instance levels. For UDA instance segmentation, we replace the original VGG16 based Faster R-CNN with the same Mask R-CNN in our architecture, and the original image- and instance-level adaptation in [@chen2018domain] with ours in Sec. \[baseline-sec\]. SIFA [@chen2019synergistic] is a UDA semantic segmentation architecture for CT and MR images, with a pixel- and feature-level adaptation. In our experiment, we add the watershed algorithm to separate the touching objects in the semantic segmentation prediction of SIFA, for a fair comparison. DDMRL [@kim2019diversify] learns multi-domain-invariant features from various generated domains for UDA object detection and it is extended for instance segmentation, in a similar way as CyCADA [@hoffman2017cycada] and Chen [@chen2018domain]. In addition, we also compared with Hou [@hou2019robust], which is particularly designed for unsupervised nuclei segmentation in histopathology images. They trained a multi-task (segmentation, detection, and refinement) CNN architecture with their synthesized histopathology images from randomly generated binary nuclei masks. ![Visualization result for the comparison experiments experiment. The first $3$ rows are from Kumar dataset, and the last $3$ rows are from TNBC. []{data-label="cmp-vis"}](vis-cmp5.png){width="49.00000%"} Table \[cmp-exp\] shows that our proposed method outperforms all the comparison methods by a large margin, on different histopathology datasets. In addition, the one-tailed paired t-test is employed to prove that all of our improvements are statistically significant, with all the p-values under $0.05$. Chen [@chen2018domain] learns the domain-invariant features at the image and instance levels. However, due to the large differences between the fluorescence microscopy and real histopathology images, feature-level adaptation only is not enough to reduce the domain gap. With pixel-level adaptation on appearance, all the other methods achieve better performance. Compared with the baseline method CyCADA [@hoffman2017cycada], our CyC-PDAM has a large improvement of $6 - 12 \%$, due to the effectiveness of our proposed nuclei inpainting mechanism, panoptic-level adaptation, and task re-weighting mechanism. SIFA [@chen2019synergistic] focuses on domain-invariant features in the image and semantic levels, with a UDA semantic segmentation structure. As there exists a large number of nuclei objects in the histopathology images, the effectiveness of SIFA is still limited without any instance-level learning or adaptation. Although DDMRL [@kim2019diversify] only adapts the features at the image level, its performance is still at the same level as CyCADA, by adapting knowledge across various domains. Among all the comparison methods, Hou [@hou2019robust] achieves the second-best performance. Due to the effectiveness of panoptic-level feature adaptation and task re-weighting mechanism, our method still outperforms it under all three metrics, in both two experiments. Fig. \[cmp-vis\] are visualization examples of all the comparison methods. ### Ablation Study In order to test the effectiveness of each component in our proposed CyC-PDAM, ablation experiments are conducted on the Kumar dataset. Based on our CyC-PDAM, we remove the nuclei inpainting mechanism, task re-weighting mechanism, and semantic branch for panoptic-level adaptation and train the ablated models with the same setting and dataset as Sec. \[sec-cmp\]. Table \[tcga-abl\] and Fig. \[abl-vis\] show the detailed results of the ablation experiment. As shown in Fig. \[abl-vis\], the method without nuclei inpainting mechanism (w/o NI) tends to ignore some nuclei, which increases the false-negative predictions. Moreover, we notice that there are also false split and merged predictions for w/o NI model. It is because the increasing false negative predictions are harmful to the spatial distribution of all the objects, which further affects the effectiveness of the semantic-level adaptation. Among the predictions of the method without task re-weighting mechanism (w/o TR), there exist some objects with irregular sizes. The task re-weighting mechanism prevents the model from being influenced by the domain-specific features in the source domain, and removing it, therefore, incurs source-biased predictions. Compared with our method, the model without semantic-branch (w/o SEM) is not able to learn domain-invariant features at the semantic level, including the spatial distribution of the nuclei objects and the detailed information in the background. Therefore, there not only remain falsely split and merged predictions, but also false-positive and imperfect segmentation results. As shown in Table \[tcga-abl\], the segmentation accuracy under three metrics decreases by $4 - 6\%$ after removing each module. In addition, the one-tailed paired t-test is employed to calculate the p-value between our proposed method and the other ablated methods. After adding each of the three modules, the improvements are statistically significant ($P < 0.05$), which further demonstrates the effectiveness of our proposed method. ![Visualization results for the ablation experiment. NI: nuclei inpainting mechanism; TR: task re-weighting mechanism; SEM: semantic branch.[]{data-label="abl-vis"}](abl-vis2.png){width="45.00000%"} ### Comparison with Fully Supervised Methods As our data split in Kumar dataset is the same as several state-of-the-art methods for fully supervised nuclei segmentation, we compare their original reported results with ours. Table \[tcga-sup\] illustrates the comparison results between our proposed UDA architecture and other fully supervised methods. CNN3 [@kumar2017dataset] is a contour-based nuclei segmentation architecture, which considers nuclei boundaries as the third class, in addition to the foreground and background classes. DIST [@naylor2018segmentation] is a regression model based on the distance map. For Panoptic FPN [@kirillov2019panopticfpn], we directly train it using the same set of $16$ real histopathology patches as CNN3 and DIST and it is employed as the upper bound of our unsupervised method. The testing images for Kumar are divided into two subsets: one contains $8$ images from $4$ organs known to training set, referred to as seen, and the other contains $6$ images from $3$ organs unknown to the training set, referred to as unseen. As shown in Table \[tcga-sup\], the performance of our proposed UDA architecture is superior to the fully supervised CNN3 and DIST. It is because our proposed method is able to process each ROI on the local level, while CNN3 and DIST only process the image at a global semantic level. By adapting the semantic-level features of the foreground and the background, the performance of our method is at the same level as the fully supervised Panoptic FPN for the pixel-level F1-score. Even though our AJI is slight lower than the fully supervised Panoptic FPN, we notice that our method works better when tested on the unseen testing set. This is because our proposed CyC-PDAM focuses on learning the domain-invariant features and avoids being influenced by the domain bias of testing images from unseen organs. These results show that, although there remains large differences between the fluorescence microscopy images and histopathology images, our proposed UDA architecture still successfully narrows the domain gap between them, and achieves even better performance compared with fully supervised methods requiring histopathology nuclei annotations. Conclusion ========== In this work, we propose a CyC-PDAM architecture for UDA nuclei segmentation in histopathology images. We firstly design a baseline architecture for UDA instance segmentation, including appearance-, image-, and instance-level adaptation. Next, a nuclei inpainting mechanism is designed to remove the auxiliary objects in the synthesized images, to further avoid false-negative predictions. In the feature-level adaptation, a semantic branch is proposed to adapt the features with respect to the foreground and background, and incorporating semantic- and instance-level adaptation enables the model to learn domain-invariant features at the panoptic level. In addition, a task re-weighting mechanism is proposed to reduce the bias. Extensive experiments on three public datasets indicate our proposed method outperforms the state-of-the-art UDA methods by a large margin and reaches the same level as the fully supervised methods. From a larger perspective, the UDA instance segmentation problems are not limited to histopathology image analysis. With the promising performance close to fully supervised methods in this work, we suggest that our proposed method can also contribute to other general image analysis applications. [^1]:  <https://data.broadinstitute.org/bbbc/BBBC039/>
--- abstract: 'The ABSTRACT is to be in fully-justified italicized text, at the top of the left-hand column, below the author and affiliation information. Use the word “Abstract” as the title, in 12-point Times, boldface type, centered relative to the column, initially capitalized. The abstract is to be in 10-point, single-spaced type. Leave two blank lines after the Abstract, then begin the main text. Look at previous CVPR abstracts to get a feel for style and length.' author: - | First Author\ Institution1\ Institution1 address\ [firstauthor@i1.org]{} - | Second Author\ Institution2\ First line of institution2 address\ [secondauthor@i2.org]{} bibliography: - 'egbib.bib' title: LaTeX Author Guidelines for CVPR Proceedings --- Introduction ============ Please follow the steps outlined below when submitting your manuscript to the IEEE Computer Society Press. 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Sometimes your paper is about a problem which you tested using a tool which is widely known to be restricted to a single institution. For example, let’s say it’s 1969, you have solved a key problem on the Apollo lander, and you believe that the CVPR70 audience would like to hear about your solution. The work is a development of your celebrated 1968 paper entitled “Zero-g frobnication: How being the only people in the world with access to the Apollo lander source code makes us a wow at parties”, by Zeus . You can handle this paper like any other. Don’t write “We show how to improve our previous work \[Anonymous, 1968\]. This time we tested the algorithm on a lunar lander \[name of lander removed for blind review\]”. That would be silly, and would immediately identify the authors. Instead write the following: > We describe a system for zero-g frobnication. This system is new because it handles the following cases: A, B. Previous systems \[Zeus et al. 1968\] didn’t handle case B properly. Ours handles it by including a foo term in the bar integral. > > ... > > The proposed system was integrated with the Apollo lunar lander, and went all the way to the moon, don’t you know. It displayed the following behaviours which show how well we solved cases A and B: ... As you can see, the above text follows standard scientific convention, reads better than the first version, and does not explicitly name you as the authors. A reviewer might think it likely that the new paper was written by Zeus , but cannot make any decision based on that guess. He or she would have to be sure that no other authors could have been contracted to solve problem B. FAQ\ [**Q:**]{} Are acknowledgements OK?\ [**A:**]{} No. Leave them for the final copy.\ [**Q:**]{} How do I cite my results reported in open challenges? 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--- abstract: 'We prove two types of functional equations for double series of Euler type with complex coefficients. The first one is a generalization of the functional equation for the Euler double zeta-function, proved in a former work of the second-named author. The second one is more specific, which is proved when the coefficients are Fourier coefficients of cusp forms and the modular relation is essentially used in the course of the proof. As a consequence of functional equation we are able to determine trivial zero divisors.' address: - | Department of Mathematics and PMI\ Pohang University of Science and Technology\ Pohang, 790–784, Korea - | Graduate School of Mathematics\ Nagoya University\ Chikusa-ku, Nagoya 464-8602, Japan author: - YoungJu Choie - Kohji Matsumoto title: Functional equations for double series of Euler type with coefficients --- [^1] [^2] [^3] **[Introduction]{}** {#sec-0} ==================== Inspired by two source, namely, the theory of multiple zeta values on the one hand, and the theory of modular symbols and periods of cusp forms on the other, Manin in [@M-iterate], [@M-S] extended the theory of periods of modular forms replacing integration along geodesics in the complex upper half plane by iterated integrations to set up the foundation of the theory of “iterated noncommutative modular symbols”. In particular Manin[@M-iterate; @M-S] considered the following iterated Mellin transform $$I_{i\infty}^{0}(\omega_{s_{\ell}},..,\omega_{s_{1}}): =\int_{i\infty}^0 \omega_{s_{\ell}}( \tau_{\ell})\int_{i\infty}^{\tau_{\ell}} \omega_{s_{\ell-1}}(\tau_{\ell-1})..\int_{i\infty}^{\tau_{2}}\omega_{s_{1}}(\tau_{1})$$ of a finite sequence of cusp forms $f_1,...,f_{\ell}$ of weight $k_j \in \mathbb{N}$ with respect to a congruence subgroup $\Gamma$ of $ SL_2(\mathbb{Z})$ and $\omega_{s_j}(\tau):=f_j(\tau)\tau^{s_j-1}d\tau , s_j\in \mathbb{C}, j=1,...,\ell.$ When $\ell=1$ then $$I_{i\infty}^0(\omega_{s}) =\int_{i\infty}^0 f(\tau)\tau^{s-1} d\tau$$ is the classical Mellin transform of a cusp form $f\in S_k(\Gamma)$ satisfying the following functional equation: $$I_{i\infty}^0(\omega_{s}) = -\epsilon_f e^{\pi i s} N^{\frac{k}{2}-s}I_{i\infty}^{0}(\omega_{ k-s})$$ if $f$ is an eigenform with eigenvalue $\epsilon_f = \pm 1$ with respect to the involution $\omega_N={\left(\begin{smallmatrix}}0 & -1 \\ N & 0 {\end{smallmatrix}\right)}$ (see [@Miy]). However it seems that it is not anymore true to expect a simple functional equation if $\ell \geq 2.$ Manin [@M-S] said “since a neat functional equation can be written not for these individual integrals but for their generating series..,” so the functional equation of the “total Mellin transform” associated to the finite family $\{f_j|j=1,..,\ell,..\}$ of cusp forms was derived.\ Now consider the case when $\ell=2 $: for $ f_j(\tau)=\sum_{n\geq 1}a_j(n)e^{2\pi i n \tau},j=1,2$, we have $$I_{i\infty}^{0}(\omega_{s_{2}}, \omega_{s_{1}})=\int_{i\infty}^{0} f_2(\tau_2){\tau_2}^{s_2-1} \int_{i\infty}^{\tau_2}f_1(\tau_1) {\tau_1}^{s_1-1}d\tau_1 d\tau_2$$ $$=\int_{i\infty}^{0} f_2(\tau_2){\tau_2}^{s_2-1}d\tau_2 \int_{i\infty}^{0}f_1(\tau_1+\tau_2) (\tau_1+\tau_2)^{s_1-1}d\tau_1.$$ If $s_1\geq 2$ is a positive integer, then $I_{i\infty}^{0}(\omega_{s_2}, \omega_{s_1})$ is a finite linear combination of the following multiple Dirichlet series:$$\sum_{n,m\geq 1} \frac{a_1(n)a_2(m)}{(n+m)^{s_2}m^{r}}, \quad 0\leq r\leq s_1,$$ where $s_2\in\mathbb{C}$ whose real part is sufficiently large. In this paper we study more general type of multiple Dirichlet series, motivated by the above iterated Mellin transform of Manin. Take ${\frak A}=\{a(n)\}_{n\geq 1}$ be a sequence of complex numbers, and define $$\label{A-zeta} L_2(s_1,s_2;{\frak A})=\sum_{m,n\geq 1} \frac{a(n)}{m^{s_1}(m+n)^{s_2}},$$ where $s_j=\sigma_j+it_j$ ($j=1,2$) be two complex variables. The purpose of the present paper is to prove two types of functional equations for this double series. This is more general situation since we take $a_1(n)$ be an arbitrary complex number and allow $a_2(n)=1$ for any $n\geq 1.$ Before stating our main results, we recall functional equations for classical zeta-functions, and for the double zeta-function without coefficients. It is well-known that the Riemann zeta-function $\zeta(s)$ ($s=\sigma+it\in\mathbb{C}$) has the beautiful symmetric functional equation $$\begin{aligned} \label{0-riemann-fe} \pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)= \pi^{-(1-s)/2}\Gamma\left(\frac{1-s}{2}\right)\zeta(1-s).\end{aligned}$$ For Hurwitz zeta-functions $\zeta(s,\alpha)=\sum_{n\geq 0}(n+\alpha)^{-s}$ ($\alpha>0$), however, this symmetricity is no longer valid in general. In fact, the functional equation for $\zeta(s,\alpha)$ is of the form $$\begin{aligned} \label{0-hurwitz-fe} \zeta(s,\alpha)=\frac{\Gamma(1-s)}{i(2\pi)^{1-s}}\left\{e^{\pi is/2} \phi(1-s,\alpha)-e^{-\pi is/2}\phi(1-s,-\alpha)\right\},\end{aligned}$$ where $\phi(s,\alpha)=\sum_{n\geq 1}e^{2\pi in\alpha}n^{-s}$ is the Lerch zeta-function (Titchmarsh [@Tit (2.17.3)]). See also a recent work of Lagarias and Li [@LL]. Those functional equations are very fundamental in the study of $\zeta(s)$ and $\zeta(s,\alpha)$. The theory of multiple zeta-functions has been studied very actively in recent decades. Searching for some kind of functional equations is a quite natural problem in order to develop further analytic studies of multiple zeta-functions. In the simplest double zeta case, the following functional equation is already known: consider the case ${\frak{A}_1}:=\{a(n)=1\; {\rm for\;all}\; n\}$ in the definition , that is $$\label{orig-zeta} \zeta_2(s_1,s_2)=\sum_{m,n\geq 1} \frac{1}{m^{s_1}(m+n)^{s_2}}.$$ This is called the Euler double zeta-function, and satisfies the following functional equation: $$\begin{aligned} \label{0-double-fe} \zeta_2(s_1,s_2)=\frac{\Gamma(1-s_1)\Gamma(s_1+s_2-1)}{\Gamma(s_2)}\zeta(s_1+s_2-1)\\ +\Gamma(1-s_1)\left\{F_+(1-s_2,1-s_1; \frak{A}_1)+F_-(1-s_2,1-s_1; \frak{A}_1)\right\},\notag\end{aligned}$$ where $$\label{0-Fpm} F_{\pm}(s_1,s_2; \frak{A}_1)=\sum_{k\geq 1}\sigma_{s_1+s_2-1}(k)\Psi(s_2,s_1+s_2; \pm 2\pi ik),$$ with $\sigma_c(k)=\sum_{0<d|k}d^c$ and $$\begin{aligned} \label{hyp-def} \Psi(a,b;x)=\frac{1}{\Gamma(a)}\int_0^{e^{i\phi}\infty}e^{-xy}y^{a-1} (y+1)^{b-a-1}dy\end{aligned}$$ (the confluent hypergeometric function), where $\Re a>0$, $-\pi<\phi<\pi$, and $|\phi+\arg x|<\pi/2$. Formula may be regarded as a double analogue of . In fact, since the asymptotic expansion $$\begin{aligned} \label{asymp} \Psi(a,b;x)=\sum_{j=0}^{M-1}\frac{(-1)^j(a-b+1)_j(a)_j}{j!}x^{-a-j} +\rho_M(a,b;x)\end{aligned}$$ (where $(a)_j=\Gamma(a+j)/\Gamma(a)$ and $\rho_M(a,b;x)$ is the remainder term; see [@Er formula 6.13.1(1)]) is known, taking the first term of the right-hand side of , we can “approximate” $F_{\pm}(s_1,s_2; \frak{A}_1)$ by the Dirichlet series $\sum_{k\geq 1}\sigma_{s_1+s_2-1}(k)(\pm 2\pi ik)^{-s_2}$. Therefore $F_{\pm}(s_1,s_2; \frak{A}_1)$ is a kind of “generalized Dirichlet series”. Moreover, gives a symmetric form of functional equation, similar to , on some hyperplanes (see Remark \[rmk-symm\] at the end of Section \[sec-4\]). Formula was essentially included in [@Mat04], and first explicitly stated in [@Mat-Sugaku] (in a generalized form with certain shifting parameters). In [@KMT-Debrecen] a generalization of formula to the case where the denominator includes certain complex parameters. Moreover in [@KMT-IJNT] a formula analogous to was shown for double $L$-functions whose numerator includes Dirichlet characters. The purpose of the present paper is to discuss such kind of functional equations in a more general setting. As a consequence we are able to determine trivial zero divisors of double series. We will state the main results in the next section. **[Statement of results]{}** {#sec-0.5} ============================ Our first main result in the present paper is a further generalization of formula . We assume that ${\frak A}=\{a(n)\}_{n\geq 1}$ be a sequence of complex numbers satisfying\ 1. $a(n)\ll n^{(\kappa-1)/2+\varepsilon}$ with a certain constant $\kappa\geq 1$, where $\varepsilon$ is an arbitrarily small positive number, 2. the Dirichlet series $L(s,{\frak A})=\sum_{n\geq 1}a(n)n^{-s}$ (which is absolutely convergent for $\Re(s) >(\kappa+1)/2$ by (i)) can be continued to the whole complex plane as a meromorphic function which has only finitely many poles. Let $\mathcal{H}$ be the complex upper half plane and let $$\label{form} f(\tau)=\sum_{n\geq 1}a(n)q^n,$$ where $q=e^{2\pi i \tau}, \tau \in \mathcal{H}$. It is obvious by (i) that $f(\tau)$ is convergent for $\tau\in\mathcal{H}$ and holomorphic in $\tau$. Moreover, using (i) we find that the right-hand side of is $$\begin{aligned} &\ll \sum_{m,n\geq 1}n^{(\kappa-1)/2+\varepsilon} m^{-\sigma_1}(m+n)^{-\sigma_2}\\ &\leq \sum_{m,n\geq 1}m^{-\sigma_1} (m+n)^{(\kappa-1)/2+\varepsilon-\sigma_2},\end{aligned}$$ so, using [@Mat-Millennial Theorem 3], we see that is convergent absolutely in the region $$\begin{aligned} \label{region1} \sigma_2>\frac{\kappa+1}{2},\quad \sigma_1+\sigma_2>\frac{\kappa+3}{2}.\end{aligned}$$ Under ussumption (ii), using [@MT] we can show that has meromorphic continuation to the whole complex space $\mathbb{C}^2$. Let $$\begin{aligned} \label{A-def2} A_c(l)=\sum_{0<n|l}n^c a(n),\end{aligned}$$ and $$\begin{aligned} \label{def-F2} F_{\pm}(s_1,s_2;{\frak A})= \sum_{l\geq 1}A_{s_1+s_2-1}(l)\Psi(s_2,s_1+s_2;\pm 2\pi il).\end{aligned}$$ Also put $$\begin{aligned} \label{def-G} L_1(s_1,s_2;{\frak A})= \frac{\Gamma(1-s_1)\Gamma(s_1+s_2-1)}{\Gamma(s_2)}L(s_1+s_2-1;{\frak A}).\end{aligned}$$ \[thm-1\] [(]{}The first form of the functional equation[)]{} Under the above assumptions [(i)]{} and [(ii)]{}, The functions $F_{\pm}(s_1,s_2;{\frak A})$ can be continued meromorphically to the whole space $\mathbb{C}^2$, and for any $s_1,s_2\in\mathbb{C}$, except for singularity points, it holds that $$\begin{aligned} \label{thm-formula} &L_2(s_1,s_2;{\frak A})=L_1(s_1,s_2;{\frak A})\\ &\quad+\Gamma(1-s_1)\left\{F_+(1-s_2,1-s_1;{\frak A})+F_-(1-s_2,1-s_1;{\frak A}) \right\}. \notag\end{aligned}$$ We can determine the location of singular locus of $L_2(s_1,s_2;{\frak A})$ from the right-hand side of . In fact, the explicit form of $L_1(s_1,s_2;{\frak A})$ is given by , while the explicit information on the singular locus of $F_{\pm}$ can be obtained from and in Section \[sec-4\]. The proof of Theorem \[thm-1\], which will be described in Sections \[sec-1\] to \[sec-4\], is analogous to that in [@Mat04], the basic idea of which goes back to Motohashi [@Mot] and Katsurada and Matsumoto [@KM]. Since the assumptions (i) and (ii) for ${\frak A}$ is very general, we may discuss various specific examples. For instance, by replacing $\frak{A}$ by $\frak{A}_1$ in (\[thm-1\]) we recover the functional equation (\[0-double-fe\]) of double zeta function. Further let us consider the very special situation that $$\frak{A}_0(n)=\{a(n)=1 \mbox {\, for only one fixed $n$, and $a(n)=0$ for all other $n$}\}.$$ In this case (\[thm-1\]) is reduced to $$\begin{aligned} \label{specialcase} L_2(s_1,s_2;{\frak A}_0(n))=\sum_{m\geq 1}\frac{1}{m^{s_1} (m+n)^{s_2}},\end{aligned}$$ a single series in two variables. This is a special case of the series $$\xi(s_1,s_2;(\alpha,\beta)):=\sum_{m\geq 0}\frac{1} {(m+\alpha)^{s_1}(m+\beta)^{s_2}}\qquad(\beta\geq\alpha>0),$$ which was used in [@Mat03]. From the above theorem we immediately obtain the following “two-variables analogue” of . \[cor\] For any $s_1,s_2\in\mathbb{C}$, except for singularity points, it holds that $$\begin{aligned} \label{cor-formula} L_2(s_1,s_2;{\frak A}_0(n))& =\frac{\Gamma(1-s_1)\Gamma(s_1+s_2-1)}{\Gamma(s_2)} \cdot \frac{1}{ n^{s_1+s_2-1}}\\ &+\Gamma(1-s_1)\{F_+(1-s_2,1-s_1;\frak{A}_0(n))+ F_-(1-s_2,1-s_1;\frak{A}_0(n))\},\notag\end{aligned}$$ where $$F_{\pm}(s_1,s_2;\frak{A}_0(n))=n^{s_1+s_2-1}\sum_{k\geq 1}\Psi(s_2,s_1+s_2;\pm 2\pi ikn).$$ It is to be noted that $$L_2(s_1,s_2; \frak{A})= \sum_{n\geq 1}a(n)L_2(s_1,s_2; \frak{A}_0(n)).$$ Therefore, multiplying the both sides of by $a(n)$ and adding with respect to $n$, we obtain . From this observation we may say that Corollary \[cor\] is a “refinement” or “decomposition” of Theorem \[thm-1\]. Another important example, closely related with our original motivation on periods, is the case that ${\frak A}$ is the set of Fourier coefficients of a certain cusp form. Now assume that is a holomorphic cusp form of weight $\kappa$ with respect to the Hecke congruence subgroup $\Gamma_0(N)$. In this case the assumptios (i) and (ii) are surely satisfied; (i) is Deligne’s estimate and $\kappa$ is the weight. In this case we write $L_2(s_1,s_2;{\frak A})$, $L_1(s_1,s_2;{\frak A})$, $L(s,{\frak A})$ and $F_{\pm}(s_1,s_2;{\frak A})$ by $L_2(s_1,s_2;f)$, $L_1(s_1,s_2;f)$, $L(s,f)$ and $F_{\pm}(s_1,s_2;f)$, respectively. Then can be written as $$\begin{aligned} \label{thm-formula-cusp} &L_2(s_1,s_2;f)=L_1(s_1,s_2;f)\\ &\quad+\Gamma(1-s_1)\left\{F_+(1-s_2,1-s_1;f)+F_-(1-s_2,1-s_1;f)\right\}. \notag\end{aligned}$$ Since this formula is proved under the above very general setting, no property of cusp form is used in the proof. When $f$ is a cusp form, it is natural to expect some different type of results, for which the modularity is essestially used. Our second main result gives such a functional equation. Let $$\begin{aligned} \label{f-tilde} \widetilde{f}(\tau)=\left(f\left|_{\kappa}\; \omega_N\right.\right)(\tau) =\left(\sqrt{N}\tau\right)^{-\kappa}f\left(-\frac{1}{N\tau}\right).\end{aligned}$$ This $\widetilde{f}$ is again a cusp form of weight $\kappa$ with respect to $\Gamma_0(N)$, and especially $\widetilde{f}=f$ when $N=1$. We write the Fourier expansion of $\widetilde{f}$ at $\infty$ as $\widetilde{f}(\tau)=\sum_{n\geq 1}\widetilde{a}(n)q^n$. Define $$\begin{aligned} \label{H-2-def} H_{2,N}^{\pm}(s_1,s_2;\widetilde{f})&=\sum_{m,n\geq 1}m^{-s_1-s_2}\widetilde{a}(n) \Psi(s_1+s_2,s_2;\pm 2\pi in/Nm).\end{aligned}$$ \[thm-2\] (The second form of the functional equation) When $f(\tau)$ is a cusp form of weight $\kappa$ with respect to $\Gamma_0(N)$, the functions $H_{2,N}^{\pm}(s_1,s_2;\widetilde{f})$ can be continued meromorphically to the whole space $\mathbb{C}^2$, and we have $$\begin{aligned} \label{thm2-formula} &L_2(s_1,s_2;f)=L_1(s_1,s_2;f)\\ &\;+\frac{(2\pi)^{s_1+s_2-1}\Gamma(1-s_1)}{\Gamma(s_2)} \Gamma(\kappa-s_1-s_2+1)\notag\\ &\qquad\times N^{-\kappa/2}\left\{e^{\pi i(1-s_1-s_2)/2} H_{2,N}^+(-s_1,\kappa-s_2+1;\widetilde{f})\right.\notag\\ &\;\qquad\left. +e^{\pi i(s_1+s_2-1)/2} H_{2,N}^-(-s_1,\kappa-s_2+1;\widetilde{f})\right\}.\notag\end{aligned}$$ The proof of this theorem will be given in Sections \[sec-5\] and \[sec-6\]. Recall that an important application of the classical functional equation is that from which we can find the “trivial zeros” at negative even integer points of $\zeta(s)$. The above Theorem \[thm-2\] has the same type of application. In fact, as we will see in Section \[sec-7\], we can show the following \[cor-2\] For any non-negative integer $l$, the hyperplane $\Re s_2=-l$ is a zero-divisor of $L_2(s_1,s_2;f)$. Note that this corollary cannot be deduced from Theorem \[thm-1\]. These zero-divisors may be regarded as “trivial zeros” of $L_2(s_1,s_2;f)$. \[rem1.5\] It is to be noted that, when $f(\tau)$ is a cusp form, $L_1(s_1,s_2;f)$ also satisfies a functional equation. Let $$L_1^*(s_1,s_2;f)=(2\pi)^{-s_1-s_2}\Gamma(s_1)\Gamma(s_2)L_1(s_1,s_2;f).$$ Then from the functional equation for $L(s,f)$ (see, e.g., [@Miy Theorem 4.3.6]) we can deduce $$L_1^*(s_1,\kappa-2s_1-s_2+2;\widetilde{f})=(-1)^{\kappa/2}N^{s_1+s_2-\kappa/2-1} L_1^*(s_1,s_2;f).$$ \[rem2\] A very different type of functional equation for certain iterated integrals related with certain multiple Hecke $L$-series has been proved by [@CI]. [**[ Acknowledgement ]{}**]{} The authors express their sincere gratitude to Professor Takashi Taniguchi for his valuable comments. **[An integral expression]{}** {#sec-1} ============================== Now we assume (i) and (ii) in the introduction and start the proof of Theorem \[thm-1\]. In this section we prove the following integral expression of $L_2(s_1,s_2;{\frak A})$. \[prop-2\] In the region $$\begin{aligned} \label{region2} \sigma_1>0,\quad \sigma_2>\frac{\kappa+1}{2},\quad \sigma_1+\sigma_2>\frac{\kappa+3}{2},\end{aligned}$$ the double integral $$\begin{aligned} \label{Lambda} \Lambda(s_1,s_2;{\frak A})=\int_{0}^{\infty} f(iy) \int_{0}^{\infty}\frac{1}{e^{2\pi(x+y)}-1}x^{s_1-1} y^{s_2-1} dxdy\end{aligned}$$ converges, and we have $$\begin{aligned} \label{f-i-e-sp} L_2(s_1,s_2;{\frak A}) =\frac{(2\pi)^{s_1+s_2}}{\Gamma(s_1)\Gamma(s_2)}\Lambda(s_1,s_2;{\frak A}).\end{aligned}$$ Put $$\begin{aligned} \label{g-trivial} g_0(\tau)=\sum_{m\geq 1}q^m=\sum_{m\geq 1}e^{2\pi i\tau m} =\frac{e^{2\pi i\tau}}{1-e^{2\pi i\tau}}=\frac{1}{e^{-2\pi i\tau}-1}.\end{aligned}$$ Let $\delta>0$, and at first assume $y\geq\delta$. Then $$\begin{aligned} \label{f-i-e1} \int_0^{\infty}g_0(i(x+y))x^{s_1-1}dx &=\int_0^{\infty}\sum_{m\geq 1}e^{2\pi i\cdot i(x+y)m}x^{s_1-1}dx\\ &=\sum_{m\geq 1}e^{-2\pi ym}\int_0^{\infty}e^{-2\pi xm}x^{s_1-1}dx\notag\\ &=(2\pi)^{-s_1}\Gamma(s_1)\sum_{m\geq 1}\frac{e^{-2\pi ym}}{m^{s_1}}\notag\end{aligned}$$ if the gamma integral converges, that is, if $\sigma_1>0$ holds. The change of integration and summation in the above can be justified by absolute convergence (because $y>0\;$). Therefore $$\begin{aligned} \label{f-i-e2} &\int_{\delta}^{\infty}f(iy)\int_0^{\infty}g_0(i(x+y))x^{s_1-1}dx\;y^{s_2-1}dy\\ &\;=(2\pi)^{-s_1}\Gamma(s_1)\int_{\delta}^{\infty}\sum_{n\geq 1}a(n)e^{-2\pi yn} \sum_{m\geq 1}\frac{1}{m^{s_1}}e^{-2\pi ym}y^{s_2-1}dy\notag\\ &\;=(2\pi)^{-s_1}\Gamma(s_1)\sum_{m,n\geq 1}\frac{a(n)}{m^{s_1}} \int_{\delta}^{\infty}e^{-2\pi y(m+n)}y^{s_2-1}dy,\notag\end{aligned}$$ if we can again change the integration and summation. The series on the right-hand side is $$\begin{aligned} &\leq \sum_{m,n\geq 1}\frac{|a(n)|}{m^{\sigma_1}} \int_{\delta}^{\infty}e^{-2\pi y(m+n)}y^{\sigma_2-1}dy\\ &\leq \sum_{m,n\geq 1}\frac{|a(n)|}{m^{\sigma_1}} \int_{0}^{\infty}e^{-2\pi y(m+n)}y^{\sigma_2-1}dy\\ &=(2\pi)^{-\sigma_2}\Gamma(\sigma_2)\sum_{m,n\geq 1}\frac{|a(n)|} {m^{\sigma_1}(m+n)^{\sigma_2}}\end{aligned}$$ if $\sigma_2>0$ holds. The resulting infinite series is convergent if holds. Therefore, if (which includes the conditon $\sigma_2>0$) holds, then the change of integration and summation in the course of is justified, and moreover, on the right-hand side of , we can take the limit $\delta\to 0$ termwisely. Then the right-hand side tends to $$\begin{aligned} \label{f-i-e3} (2\pi)^{-s_1-s_2}\Gamma(s_1)\Gamma(s_2)\sum_{m,n\geq 1}\frac{a(n)} {m^{s_1}(m+n)^{s_2}},\end{aligned}$$ while the left-hand side of tends to $\Lambda(s_1,s_2;{\frak A})$. This completes the proof. **[Separating a single series factor]{}** {#sec-2} ========================================= The integrand of the inner integral of the right-hand side of is singular at $x+y=0$. The next step is to “separate” the contribution of this singularity. Let $$\begin{aligned} \label{h-def} h(z)=\frac{1}{e^{2\pi z}-1}-\frac{1}{2\pi z}.\end{aligned}$$ Using this function, we rewrite as follows: $$\begin{aligned} \label{decompose} \Lambda(s_1,s_2;{\frak A})&=\int_{0}^{\infty} f(iy) \int_{0}^{\infty}h(x+y)x^{s_1-1} y^{s_2-1}dxdy\\ &\quad+\int_{0}^{\infty} f(iy) \int_{0}^{\infty}\frac{x^{s_1-1}y^{s_2-1}} {2\pi(x+y)}dxdy\notag\\ &=I_1+I_2,\notag\end{aligned}$$ say. To verify this decomposition, we have to check the absolute convergence of $I_1$ and $I_2$. Consider $I_2$ under the condition $$\begin{aligned} \label{region3} 0<\sigma_1<1.\end{aligned}$$ It is known that $$\begin{aligned} \label{beta} \int_0^{\infty}\frac{x^{s_1-1}}{x+y}dx=y^{s_1-1}\Gamma(s_1)\Gamma(1-s_1)\end{aligned}$$ holds for $0<\sigma_1<1$ and $y>0$. Therefore, under , we have $$\begin{aligned} \label{I-2-1} I_2&=\lim_{\delta\to 0}\int_{\delta}^{\infty}f(iy)y^{s_2-1}\frac{y^{s_1-1}}{2\pi} \Gamma(s_1)\Gamma(1-s_1) dy\\ &=\frac{1}{2\pi}\Gamma(s_1)\Gamma(1-s_1)\int_0^{\infty}f(iy)y^{s_1+s_2-2}dy,\notag\end{aligned}$$ if the last integral is convergent. But the last integral is $$\begin{aligned} \label{I-2-2} &=\int_0^{\infty}\sum_{n\geq 1}a(n)e^{-2\pi yn}y^{s_1+s_2-2}dy\\ &=\sum_{n\geq 1}a(n)\int_0^{\infty}e^{-2\pi yn}y^{s_1+s_2-2}dy\notag\\ &=(2\pi)^{-s_1-s_2+1}\Gamma(s_1+s_2-1)\sum_{n\geq 1}\frac{a(n)}{n^{s_1+s_2-1}}\notag\end{aligned}$$ if $\sigma_1+\sigma_2>1$, and the last sum is absolutely convergent if $\sigma_1+\sigma_2>(\kappa+3)/2$ and is equal to $L(s_1+s_2-1,{\frak A})$. This verifies the change of integration and summation in the course of , and the convergence of the last integral of . Therefore we obtain $$\begin{aligned} \label{I-2-3} I_2=(2\pi)^{-s_1-s_2}\Gamma(s_1)\Gamma(1-s_1)\Gamma(s_1+s_2-1)L(s_1+s_2-1,{\frak A})\end{aligned}$$ in the region $$\begin{aligned} \label{region4} 0<\sigma_1<1,\quad \sigma_1+\sigma_2>\frac{\kappa+3}{2}.\end{aligned}$$ As for $I_1$, we first note that $h(z)$ is holomorphic at $z=0$, so it is $O(1)$ when $|z|$ is small. If the real part of $z$ is large, then clearly $h(z)=O(|z|^{-1})$. Therefore $$\begin{aligned} \label{I-1-1} I_1&\ll\int_0^{\infty}|f(iy)|\left\{\int_0^1 x^{\sigma_1-1}dx + \int_1^{\infty}\frac{x^{\sigma_1-1}}{x+y}dx\right\}y^{\sigma_2-1}dy,\end{aligned}$$ and the quantity in the curly bracket is $O(1)$, uniformly in $y$, if holds. Under the condition (i) at the beginning of Section \[sec-1\], it is known that $$\begin{aligned} \label{I-1-2} f(iy)\ll\left\{ \begin{array}{lll} y^{-(\kappa+1)/2-\varepsilon} & {\rm as} & y\to 0,\\ e^{-2\pi y} & {\rm as} & y\to\infty \end{array}\right.\end{aligned}$$ (see [@Miy Lemma 4.3.3]). Using these estimates we find that the right-hand side of is convergent absolutely if and $\sigma_2>(\kappa+1)/2$ holds. Therefore now we verify the decomposition under the condition . In this region, combining with and , we obtain $$\begin{aligned} \label{basic} L_2(s_1,s_2;{\frak A})=J_2(s_1,s_2;{\frak A})+L_1(s_1,s_2;{\frak A})\end{aligned}$$ where $$\begin{aligned} \label{J-2-def} J_2(s_1,s_2;{\frak A})=\frac{(2\pi)^{s_1+s_2}}{\Gamma(s_1)\Gamma(s_2)} \int_0^{\infty}f(iy)\int_0^{\infty}h(x+y)x^{s_1-1}y^{s_2-1}dxdy.\end{aligned}$$ **[Contour integration]{}** {#sec-3} =========================== In this section we show an infinite series expression of $J_2(s_1,s_2;{\frak A})$, whose terms can be written in terms of confluent hypergeometric functions. Let $\mathcal{C}$ be the contour which starts at $+\infty$, goes along the real axis to a small positive number, rounds the origin counterclockwise, and then goes back to $+\infty$ again along the real axis. At first we assume . Then, since $\sigma_1>0$, we can replace the inner integral of by the integral along $\mathcal{C}$ to obtain $$\begin{aligned} \label{contour} J_2(s_1,s_2;{\frak A})= \frac{(2\pi)^{s_1+s_2}}{\Gamma(s_1)\Gamma(s_2)(e^{2\pi is_1}-1)}I_3,\end{aligned}$$ where $$\begin{aligned} \label{I-3-def} I_3=\int_0^{\infty}f(iy)y^{s_2-1}\int_{\mathcal{C}}h(x+y)x^{s_1-1}dxdy.\end{aligned}$$ The inner integral of is absolutely convergent for any $s_1$ with $\sigma_1<1$, and is $O(1)$ uniformly in $y$. Therefore, using , we find that the double integral on the right-hand side of is absolutely convergent when $$\begin{aligned} \label{region5} \sigma_1<1,\quad \sigma_2>\frac{\kappa+1}{2}.\end{aligned}$$ Our assumption (ii) implies that $L(s,{\frak A})$ is meromorphic in $\mathbb{C}$, and $L_2(s_1,s_2;{\frak A})$ is meromorphic in the whole space $\mathbb{C}^2$, as was mentioned in the introduction. Therefore we can now conclude that formula is valid in the region . Next we replace the contour $\mathcal{C}$ by $$\mathcal{C}_R=\{x=-y+2\pi(R+1/2)e^{i\phi}\;|\;0\leq\phi<2\pi\}$$ ($R\in\mathbb{N}$), and let $R\to\infty$. Since $h(x+y)=O(1)$ on $\mathcal{C}_R$ ([@Mat98 formula (5.2)]), we see that $$\int_{\mathcal{C}_R}h(x+y)x^{s_1-1}dx \to 0$$ (as $R\to\infty$) if $\sigma_1<0$, which we now assume. That is, we are now in the subregion $$\begin{aligned} \label{region6} \sigma_1<0,\quad \sigma_2>\frac{\kappa+1}{2}\end{aligned}$$ of . Then by the residue calculus we have $$\begin{aligned} \label{inner} \int_{\mathcal{C}}h(x+y)x^{s_1-1}dx=-2\pi i\sum_{m\in\mathbb{Z},m\neq 0}{\rm Res} _{x=-y+im}\left(h(x+y)x^{s_1-1}\right),\end{aligned}$$ and the value of the residue at $x=-y+im$ ($m\neq 0$) is given by $$\begin{aligned} \label{residue} &\lim_{\delta\to 0}\delta\left(\frac{1}{e^{2\pi(im+\delta)}-1}-\frac{1} {2\pi(im+\delta)}\right)(-y+im+\delta)^{s_1-1}\\ &=\lim_{\delta\to 0}\frac{1}{2\pi}\frac{2\pi\delta}{e^{2\pi\delta}-1} (-y+im+\delta)^{s_1-1}=\frac{1}{2\pi}(-y+im)^{s_1-1}.\notag\end{aligned}$$ Substituting these results into we obtain $$\begin{aligned} \label{I-3-1} I_3=-i\int_0^{\infty}f(iy)\sum_{m\in\mathbb{Z},m\neq 0}(-y+im)^{s_1-1} y^{s_2-1}dy.\end{aligned}$$ When $m>0$, we see that $$\begin{aligned} \label{m-posi} (-y+im)^{s_1-1}=(e^{\pi i}y+e^{\pi i/2}m)^{s_1-1} =(e^{\pi i/2}m(z+1))^{s_1-1},\end{aligned}$$ where $z$ is defined by $y=me^{-\pi i/2}z$. Similarly, when $m<0$, $$\begin{aligned} \label{m-nega} (-y+im)^{s_1-1}=(e^{\pi i}y+e^{3\pi i/2}|m|)^{s_1-1} =(e^{3\pi i/2}|m|(z+1))^{s_1-1},\end{aligned}$$ where $z$ is defined by $y=|m|e^{\pi i/2}z$. Therefore, if the change of integration and summation is possible, from we obtain $$\begin{aligned} \label{I-3-2} I_3=-i(I_{31}+I_{32}),\end{aligned}$$ where $$\begin{aligned} \label{I-31-def} I_{31}&=\sum_{m\geq 1}(e^{\pi i/2}m)^{s_1-1}\int_0^{i\infty}f(mz) (me^{-\pi i/2}z)^{s_2-1}(z+1)^{s_1-1}me^{-\pi i/2}dz\\ &=\sum_{m\geq 1}e^{\pi i(s_1-s_2-1)/2}m^{s_1+s_2-1}\int_0^{i\infty}f(mz) z^{s_2-1}(z+1)^{s_1-1}dz\notag\end{aligned}$$ and (rewriting $|m|$ as $m$) $$\begin{aligned} \label{I-32-def} I_{32}&=\sum_{m\geq 1}(e^{3\pi i/2}m)^{s_1-1}\int_0^{-i\infty}f(-mz) (me^{\pi i/2}z)^{s_2-1}(z+1)^{s_1-1}me^{\pi i/2}dz\\ &=\sum_{m\geq 1}e^{\pi i(3s_1+s_2-3)/2}m^{s_1+s_2-1}\int_0^{-i\infty}f(-mz) z^{s_2-1}(z+1)^{s_1-1}dz.\notag\end{aligned}$$ Substitute the definition of $f(mz)$ into and change the integration and summation again to obtain $$I_{31}=\sum_{m,n\geq 1}e^{\pi i(s_1-s_2-1)/2}m^{s_1+s_2-1}a(n) \int_0^{i\infty}e^{2\pi imnz}z^{s_2-1}(z+1)^{s_1-1}dz.$$ Putting $mn=l$, this is equal to $$\begin{aligned} \label{I-31-1} e^{\pi i(s_1-s_2-1)/2}\sum_{l\geq 1}A_{s_1+s_2-1}^0(l) \int_0^{i\infty}e^{2\pi ilz}z^{s_2-1}(z+1)^{s_1-1}dz,\end{aligned}$$ where $$\begin{aligned} \label{A-def} A_c^0(l)=\sum_{mn=l}m^c a(n).\end{aligned}$$ Then we see that the integral on the right-hand side of is $\Gamma(s_2)\Psi(s_2,s_1+s_2;-2\pi il)$ (because $\sigma_2>0$ is satisfied by , and $\phi=\pi/2$ so $\phi+\arg(-2\pi il)=0$), hence $$\begin{aligned} \label{I-31-2} I_{31}=e^{\pi i(s_1-s_2-1)/2}\Gamma(s_2)\sum_{l\geq 1}A_{s_1+s_2-1}^0(l) \Psi(s_2,s_1+s_2;-2\pi il).\end{aligned}$$ Similarly we obtain $$\begin{aligned} \label{I-32-2} I_{32}=e^{\pi i(3s_1+s_2-3)/2}\Gamma(s_2)\sum_{l\geq 1}A_{s_1+s_2-1}^0(l) \Psi(s_2,s_1+s_2;2\pi il).\end{aligned}$$ To verify the above changing process (twice) of integration and summation, we check the absolute convergence of the resulting expression. Putting $lz=i\xi$, we see that the integral on the right-hand side of is $$\begin{aligned} \label{I-31-3} &=\int_0^{\infty}e^{-2\pi\xi}\left(\frac{i\xi}{l}\right)^{s_2-1} \left(1+\frac{i\xi}{l}\right)^{s_1-1}\frac{i}{l}d\xi\\ &\ll \int_0^{\infty}e^{-2\pi\xi}\left(\frac{\xi}{l}\right)^{\sigma_2-1} \left|\left(1+\frac{i\xi}{l}\right)^{s_1-1}\right|\frac{d\xi}{l}\notag,\end{aligned}$$ where the implied constant depends on $s_2$. Further, $$\begin{aligned} \label{I-31-3bis} \left|\left(1+\frac{i\xi}{l}\right)^{s_1-1}\right| &=\left|1+\frac{i\xi}{l}\right|^{\sigma_1-1}e^{-t_1\arg(1+i\xi/l)}.\end{aligned}$$ The first factor on the right-hand side is $\leq 1$, because $|1+i\xi/l|\geq 1$ and $\sigma_1<0$ by , while the second factor is $O_{t_1}(1)$ because $|\arg(1+i\xi/l)|\leq\pi/2$. Hence the right-hand side of is $$\ll \int_0^{\infty}e^{-2\pi\xi}\left(\frac{\xi}{l}\right)^{\sigma_2-1} \frac{d\xi}{l} \ll l^{-\sigma_2}.$$ Therefore is $$\begin{aligned} &\ll\sum_{l\geq 1}|A_{s_1+s_2-1}^0(l)|l^{-\sigma_2} \ll \sum_{m,n\geq 1}m^{\sigma_1+\sigma_2-1}|a(n)|(mn)^{-\sigma_2}\\ &\ll\sum_{m\geq 1}m^{\sigma_1-1}\sum_{n\geq 1}n^{(\kappa-1)/2+\varepsilon -\sigma_2},\end{aligned}$$ which is convergent in the region . Therefore the whole step of the above procedure is verified. Define $$\begin{aligned} \label{F-def} F_{\pm}^0(s_1,s_2;{\frak A})=\sum_{l\geq 1}A_{s_1+s_2-1}^0(l)\Psi(s_2,s_1+s_2; \pm 2\pi il).\end{aligned}$$ Using notation , from , and we obtain $$\begin{aligned} \label{I-3-3} I_3&=-i\Gamma(s_2)\left\{e^{\pi i(s_1-s_2-1)/2}F_-^0(s_1,s_2;{\frak A})+ e^{\pi i(3s_1+s_2-3)/2}F_+^0(s_1,s_2;{\frak A})\right\}\\ &=i\Gamma(s_2)\left\{e^{\pi i(s_1-s_2+1)/2}F_-^0(s_1,s_2;{\frak A})+ e^{\pi i(3s_1+s_2-1)/2}F_+^0(s_1,s_2;{\frak A})\right\}\notag\end{aligned}$$ in the region . Using the identity $$\begin{aligned} \label{gamma} \frac{1}{\Gamma(s_1)(e^{2\pi is_1}-1)}=\frac{\Gamma(1-s_1)} {2\pi ie^{\pi i s_1}}\end{aligned}$$ we find that is rewritten as $$\begin{aligned} \label{contour2} J_2(s_1,s_2;{\frak A})= \frac{(2\pi)^{s_1+s_2-1}\Gamma(1-s_1)}{ie^{\pi i s_1}\Gamma(s_2)}I_3.\end{aligned}$$ Substituting into the above, we obtain $$\begin{aligned} \label{J-1} J_2(s_1,s_2;{\frak A}) =&\frac{(2\pi)^{s_1+s_2-1}\Gamma(1-s_1)}{e^{\pi i s_1}}\\ &\qquad \times \left\{e^{\pi i(s_1-s_2+1)/2}F_-^0(s_1,s_2;{\frak A})+ e^{\pi i(3s_1+s_2-1)/2}F_+^0(s_1,s_2;{\frak A})\right\}\notag\\ =&(2\pi)^{s_1+s_2-1}\Gamma(1-s_1)\notag\\ &\qquad \times \left\{e^{\pi i(1-s_1-s_2)/2}F_-^0(s_1,s_2;{\frak A})+ e^{\pi i(s_1+s_2-1)/2}F_+^0(s_1,s_2;{\frak A})\right\}\notag\end{aligned}$$ in the region . **[Completion of the proof of Theorem \[thm-1\]]{}** {#sec-4} ==================================================== The transformation formula $$\begin{aligned} \label{transf} \Psi(a,c;x)=x^{1-c}\Psi(a-c+1,2-c;x)\end{aligned}$$ of the confluent hypergeometric function is well-known ([@Er formula 6.5(6)]). Using , we see that $$\begin{aligned} \label{transf2} &F_{\pm}^0(s_1,s_2;{\frak A})\\ &\quad=(2\pi e^{\pm \pi i/2})^{1-s_1-s_2}\sum_{l\geq 1}A_{s_1+s_2-1}^0(l) l^{1-s_1-s_2}\Psi(1-s_1,2-s_1-s_2;\pm 2\pi il).\notag\end{aligned}$$ Since $$A_c^0(l)l^{-c}=\sum_{mn=l}m^c a(n)l^{-c}=\sum_{mn=l}\left(\frac{l}{m}\right)^{-c} a(n)=\sum_{0<n|l}n^{-c}a(n)$$ which is equal to $A_{-c}(l)$ (recall ), we obtain $$\begin{aligned} \label{F-0-F} F_{\pm}^0(s_1,s_2;{\frak A})=(2\pi e^{\pm \pi i/2})^{1-s_1-s_2} F_{\pm}(1-s_2,1-s_1;{\frak A}).\end{aligned}$$ Substituting this into , we obtain $$\begin{aligned} \label{J-2} J_2(s_1,s_2;{\frak A})=\Gamma(1-s_1)\left\{F_-(1-s_2,1-s_1;{\frak A}) +F_+(1-s_2,1-s_1;{\frak A})\right\}\end{aligned}$$ in the region . Combining with , we now obtain in the region . To complete the proof of the above theorem, it is sufficient to show that $F_{\pm}(s_1,s_2;{\frak A})$ can be continued to the whole space $\mathbb{C}^2$. For this purpose we use . This implies $$\begin{aligned} \label{asymp2} &\Psi(s_2,s_1+s_2;\pm 2\pi il)\\ &\quad=\sum_{j=0}^{M-1}\frac{(-1)^j (1-s_1)_j(s_2)_j}{j!} (\pm 2\pi il)^{-s_2-j}+\rho_M(s_2,s_1+s_2;\pm 2\pi il),\notag\end{aligned}$$ so $$\begin{aligned} \label{asymp3} F_{\pm}(s_1,s_2;{\frak A})&=\sum_{j=0}^{M-1}\frac{(-1)^j (1-s_1)_j(s_2)_j}{j!} \sum_{l\geq 1}A_{s_1+s_2-1}(l)(\pm 2\pi il)^{-s_2-j}\\ &+\sum_{l\geq 1}A_{s_1+s_2-1}(l)\rho_M(s_2,s_1+s_2;\pm 2\pi il).\notag\end{aligned}$$ We see that $$\begin{aligned} \label{asymp4} &\sum_{l\geq 1}A_{s_1+s_2-1}(l)(\pm 2\pi il)^{-s_2-j}\\ &\quad=(\pm 2\pi i)^{-s_2-j}\sum_{l\geq 1}\sum_{n|l}n^{s_1+s_2-1}a(n)l^{-s_2-j} \notag\\ &\quad=(\pm 2\pi i)^{-s_2-j}\sum_{m,n\geq 1}n^{s_1+s_2-1}a(n)(mn)^{-s_2-j} \notag\\ &\quad=(\pm 2\pi i)^{-s_2-j}\sum_{m\geq 1}m^{-s_2-j}\sum_{n\geq 1}a(n) n^{s_1-1-j},\notag\end{aligned}$$ whose last two sums are convergent when $\sigma_2>1-j$ and $\sigma_1<j-(\kappa-1)/2$, and so the above is equal to $(\pm 2\pi i)^{-s_2-j}\zeta(s_2+j)L(1-s_1+j,{\frak A})$. Therefore $$\begin{aligned} \label{asymp5} &F_{\pm}(s_1,s_2;{\frak A})\\ &\;=\sum_{j=0}^{M-1}\frac{(-1)^j (1-s_1)_j(s_2)_j}{j!}(\pm 2\pi i)^{-s_2-j} \zeta(s_2+j)L(1-s_1+j,{\frak A})\notag\\ &\quad+\sum_{l\geq 1}A_{s_1+s_2-1}(l)\rho_M(s_2,s_1+s_2;\pm 2\pi il).\notag\end{aligned}$$ The explicit form of $\rho_M$ is $$\begin{aligned} \label{rho} \rho_M(a,c;x)&=\frac{(-1)^M(a-c+1)_M}{\Gamma(a)}\int_0^{e^{i\phi}\infty} e^{-xy}y^{a+M-1}\\ &\quad\times\int_0^1 \frac{(1-\tau)^{M-1}}{(M-1)!}(1+\tau y)^{c-a-M-1}d\tau dy \notag\end{aligned}$$ (see [@Mat04 (3.3)]). In the present situation, $\phi=\mp\pi/2$. Therefore, if the change of integration and summation is possible, we have $$\begin{aligned} \label{rho-2} &\sum_{l\geq 1}A_{s_1+s_2-1}(l)\rho_M(s_2,s_1+s_2;\pm 2\pi il)\\ &\;=\frac{(-1)^M(1-s_1)_M}{\Gamma(s_2)}\int_0^{\mp i\infty} \sum_{l\geq 1}A_{s_1+s_2-1}(l)e^{\mp 2\pi ily}y^{s_2+M-1}\notag\\ &\quad\times \int_0^1 \frac{(1-\tau)^{M-1}}{(M-1)!}(1+\tau y)^{s_1-M-1}d\tau dy \notag\\ &\;=\frac{(-1)^M(1-s_1)_M}{\Gamma(s_2)}\int_0^{\infty} \sum_{l\geq 1}A_{s_1+s_2-1}(l)e^{-\eta}\left(\frac{\mp i\eta}{2\pi l}\right) ^{s_2+M-1}\notag\\ &\quad\times \int_0^1 \frac{(1-\tau)^{M-1}}{(M-1)!}\left(1\mp \frac{i\tau\eta} {2\pi l}\right)^{s_1-M-1}d\tau\frac{\mp i}{2\pi l}d\eta\notag\\ &\;=\frac{(-1)^M(1-s_1)_M}{\Gamma(s_2)}\int_0^{\infty} \sum_{l\geq 1}A_{s_1+s_2-1}(l)l^{-s_2-M}e^{-\eta} \left(\frac{\mp i\eta}{2\pi}\right)^{s_2+M-1}\notag\\ &\quad\times\int_0^1 \frac{(1-\tau)^{M-1}}{(M-1)!}\left(1\mp \frac{i\tau\eta} {2\pi l}\right)^{s_1-M-1}d\tau\frac{\mp i}{2\pi}d\eta.\notag\end{aligned}$$ Similarly to , the sum with respect to $l$ is $\zeta(s_2+M)L(1-s_1+M,{\frak A})$, if $$\begin{aligned} \label{region7} \sigma_1<M+1-\frac{\kappa+1}{2},\quad \sigma_2>1-M.\end{aligned}$$ Also, similarly to the argument around , we see that $$\left(1\mp \frac{i\tau\eta}{2\pi l}\right)^{s_1-M-1}=O(1)$$ if $\sigma_1<M+1$. Therefore the integral on the right-hand side of is $$\ll_{s_1,s_2,M}\int_0^{\infty}e^{-\eta} \left|\left(\frac{\mp i\eta}{2\pi}\right)^{s_2+M-1}\right|d\eta$$ which is convergent if $\sigma_2>-M$. Hence the above change is verified, and we can now conclude that the second sum on the right-hand side of is convergent in the region . This implies that gives the meromorphic continuation of $F_{\pm}(s_1,s_2;{\frak A})$ to the region . Since $M$ is arbitrary, $F_{\pm}(s_1,s_2;{\frak A})$ can be continued meromorphically to the whole space $\mathbb{C}^2$. This completes the proof of Theorem \[thm-1\]. \[rmk-symm\] In [@KMT-Debrecen] it was pointed out that Theorem \[thm-1\] in the special case ${\frak A}={\frak A}_1$ gives a symmetric form of the functional equation on some hyperplanes ([@KMT-Debrecen Theorem 2.2]). It is desirable to deduce such a symmetric form of the functional equation for general ${\frak A}$. However there is a difficulty, caused by the fact that $F_{\pm}^0$ on the left-hand side of is different from $F_{\pm}$ on the right-hand side. **[Modularity comes into play]{}** {#sec-5} ================================== Now we proceed to the proof of Theorem \[thm-2\]. Therefore, hereafter, we assume that $\kappa$ is an even positive integer and $f(\tau)$ is a cusp form of weight $\kappa$ with respect to $\Gamma_0(N)$. In Section \[sec-3\], we only used the estimates to show that the double integral is convergent in the region . However, since now we assume that $f(\tau)$ is a cusp form, we see that $f(iy)$ is also of exponential decay when $y\to 0 $. Therefore $I_3$ is convergent for any $s_2\in\mathbb{C}$, and so we can remove the second condition $\sigma_2>(\kappa+1)/2$ from , and also from . This implies that, if $f$ is a cusp form, the whole argument in Section \[sec-3\] is valid in the region $$\label{region6bis} \sigma_1<0$$ instead of . (This is important, because the former region has no intersection with below.) We begin with , in the region . Using , we can rewrite as $$\begin{aligned} \label{I-31-mod} I_{31}&=\sum_{m\geq 1}e^{\pi i(s_1-s_2-1)/2}m^{s_1+s_2-1}\int_0^{i\infty} (-\sqrt{N}mz)^{-\kappa}\widetilde{f}\left(-\frac{1}{Nmz}\right)z^{s_2-1} (z+1)^{s_1-1}dz\\ &=N^{-\kappa/2}\sum_{m\geq 1}e^{\pi i(s_1-s_2-1)/2}m^{s_1+s_2-1-\kappa} \int_0^{i\infty}\widetilde{f}\left(-\frac{1}{Nmz}\right)z^{-\kappa+s_2-1} (z+1)^{s_1-1}dz,\notag\end{aligned}$$ because $(-mz)^{-\kappa}=(mz)^{-\kappa}$ since $\kappa$ is even. Putting $z=1/w$, we see that the above integral is $$\begin{aligned} &=\int_{-i\infty}^0 \widetilde{f}\left(-\frac{w}{Nm}\right)w^{\kappa-s_2+1} \left(\frac{1}{w}+1\right)^{s_1-1}\left(-\frac{dw}{w^2}\right)\\ &=\int_0^{-i\infty}\sum_{n\geq 1}\widetilde{a}(n)e^{2\pi in(-w/Nm)}w^{\kappa-s_1-s_2} (w+1)^{s_1-1}dw,\end{aligned}$$ so, if we can change the integration and summation, we have $$\begin{aligned} \label{I-31-mod2} I_{31}&=N^{-\kappa/2}e^{\pi i(s_1-s_2-1)/2}\sum_{m,n\geq 1}m^{s_1+s_2-1-\kappa} \widetilde{a}(n)\\ &\quad\times\int_0^{-i\infty}e^{-2\pi i(n/Nm)w}w^{\kappa-s_1-s_2} (w+1)^{s_1-1}dw.\notag\end{aligned}$$ If $$\begin{aligned} \label{region8} \sigma_1+\sigma_2<\kappa+1\end{aligned}$$ holds, then the last integral is expressed by the confluent hypergeometric function (with $\phi=-\pi/2$ here); that is, $$\begin{aligned} \label{I-31-mod3} I_{31}&=N^{-\kappa/2}e^{\pi i(s_1-s_2-1)/2}\Gamma(\kappa-s_1-s_2+1)\sum_{m,n\geq 1} m^{s_1+s_2-1-\kappa}\widetilde{a}(n)\\ &\quad\times \Psi(\kappa-s_1-s_2+1,\kappa-s_2+1;2\pi in/Nm)\notag\\ &=N^{-\kappa/2}e^{\pi i(s_1-s_2-1)/2}\Gamma(\kappa-s_1-s_2+1) H_{2,N}^+(-s_1,\kappa-s_2+1;\widetilde{f}).\notag\end{aligned}$$ We have to check the convergence of in order to verify the above interchange of integration and summation. Put $2\pi i(n/Nm)w=y$. Then the integral on the right-hand side of is $$\begin{aligned} &=\int_0^{\infty}e^{-y}\left(\frac{Nmy}{2\pi in}\right)^{\kappa-s_1-s_2} \left(\frac{Nmy}{2\pi in}+1\right)^{s_1-1}\frac{Nm}{2\pi in}dy\\ &=\left(\frac{Nm}{2\pi in}\right)^{\kappa-s_1-s_2+1}\int_0^{\infty}e^{-y} y^{\kappa-s_1-s_2}\left(\frac{Nmy}{2\pi in}+1\right)^{s_1-1}dy.\end{aligned}$$ Under we can show $$\left(\frac{Nmy}{2\pi in}+1\right)^{s_1-1}=O(1)$$ similarly to the argument around . But this is not sufficient to prove the convergence of the series with respect to $m$. We should be more careful here: Using $$\left(\frac{Nmy}{2\pi in}+1\right)^{s_1-1}=\left(\frac{Nmy+2\pi in}{2\pi in}\right) ^{s_1-1},$$ we obtain $$\begin{aligned} \label{I-31-mod4} I_{31}&=N^{-\kappa/2}e^{\pi i(s_1-s_2-1)/2}\sum_{m,n\geq 1} m^{s_1+s_2-1-\kappa}\widetilde{a}(n)\\ &\;\times (Nm)^{\kappa-s_1-s_2+1}(2\pi in)^{-\kappa+s_2}\notag\\ &\;\times\int_0^{\infty}e^{-y}y^{\kappa-s_1-s_2}(Nmy+2\pi in)^{s_1-1}dy.\notag\end{aligned}$$ Denote the last integral by $J_{31}$. Then $$\begin{aligned} \label{I-31-mod5} I_{31}\ll_N \sum_{m,n\geq 1}|\widetilde{a}(n)| n^{-\kappa+\sigma_2}|J_{31}|\end{aligned}$$ (where $\ll_N$ means that the implied constant depends on $N$), and $$\begin{aligned} J_{31}&=\int_0^{n/Nm}+\int_{n/Nm}^{\infty}\\ &\ll\int_0^{n/Nm}e^{-y}y^{\kappa-\sigma_1-\sigma_2}n^{\sigma_1-1}dy +\int_{n/Nm}^{\infty}e^{-y}y^{\kappa-\sigma_1-\sigma_2}(Nmy)^{\sigma_1-1}dy\\ &\ll n^{\sigma_1-1}\int_0^{n/Nm}e^{-y}y^{\kappa-\sigma_1-\sigma_2}dy +(Nm)^{\sigma_1-1}\int_{n/Nm}^{\infty}e^{-y}y^{\kappa-\sigma_2-1}dy\\ &=n^{\sigma_1-1}J_{311}+(Nm)^{\sigma_1-1}J_{312},\end{aligned}$$ say. Note that $J_{311}$ is convergent in the region , and $J_{312}$ is always convergent. As for $J_{312}$, we just use the following simple estimate: $$\begin{aligned} \label{J-312} J_{312}\leq \int_0^{\infty}e^{-y}y^{\kappa-\sigma_2-1}dy,\end{aligned}$$ where the integral on the right-hand side is convergent (hence $O(1)$) if $$\begin{aligned} \label{region9} \sigma_2<\kappa\end{aligned}$$ holds. Consider $J_{311}$. When $Nm\geq n$, we see that $$\begin{aligned} \label{J-311-1} J_{311}\leq \int_0^{n/Nm}y^{\kappa-\sigma_1-\sigma_2}dy \ll \left(\frac{n}{Nm}\right)^{\kappa-\sigma_1-\sigma_2+1}\end{aligned}$$ (under ), while when $Nm<n$ we have $$\begin{aligned} \label{J-311-2} J_{311}\leq \int_0^{\infty}e^{-y}y^{\kappa-\sigma_1-\sigma_2}dy\ll 1\end{aligned}$$ (under ). Therefore, under the conditions and , we have $$\begin{aligned} \label{J-31} J_{31}\ll\left\{ \begin{array}{lll} n^{\sigma_1-1}\left(\frac{n}{Nm}\right)^{\kappa-\sigma_1-\sigma_2+1} +(Nm)^{\sigma_1-1} & {\rm if} & Nm\geq n,\\ n^{\sigma_1-1}+(Nm)^{\sigma_1-1} & {\rm if} & Nm<n. \end{array}\right.\end{aligned}$$ Substituting this into , we obtain $$\begin{aligned} \label{I-31-mod6} I_{31}&\ll_N \sum_{Nm\geq n}|\widetilde{a}(n)|n^{-\kappa+\sigma_2} \left((Nm)^{\sigma_1+\sigma_2-\kappa-1}n^{\kappa-\sigma_2} +(Nm)^{\sigma_1-1}\right)\\ &\;+\sum_{Nm<n}|\widetilde{a}(n)|n^{-\kappa+\sigma_2} \left(n^{\sigma_1-1}+(Nm)^{\sigma_1-1}\right)\notag\\ &\;=I_{31}^*+I_{31}^{**},\notag\end{aligned}$$ say. Using Deligne’s estimate, we find that $$\begin{aligned} \label{mmmmm} I_{31}^*\ll \sum_{m\geq 1}(Nm)^{\sigma_1+\sigma_2-\kappa-1}\sum_{n\leq Nm} n^{(\kappa-1)/2+\varepsilon} +\sum_{m\geq 1}(Nm)^{\sigma_1-1}\sum_{n\leq Nm} n^{(\kappa-1)/2+\varepsilon-\kappa+\sigma_2}.\end{aligned}$$ The first double sum is $$\begin{aligned} \ll \sum_{m\geq 1}(Nm)^{\sigma_1+\sigma_2-\kappa-1+(\kappa+1)/2 +\varepsilon} \ll_N \sum_{m\geq 1}m^{\sigma_1+\sigma_2-(\kappa+1)/2+\varepsilon},\end{aligned}$$ which is convergent if $$\begin{aligned} \label{region9bis} \sigma_1+\sigma_2<\frac{\kappa-1}{2}.\end{aligned}$$ The inner sum of the second double sum of is $O((Nm)^{\sigma_2-(\kappa-1)/2+\varepsilon})$ if $\sigma_2\geq (\kappa-1)/2$, and $O(1)$ otherwise. Therefore the second double sum is convergent if $\sigma_2\geq (\kappa-1)/2$ and holds, or if $\sigma_2< (\kappa-1)/2$ and $\sigma_1<0$. Consequently we find that the right-hand side of is convergent in the region $$\begin{aligned} \label{region9bb} \sigma_1<0,\quad \sigma_1+\sigma_2<\frac{\kappa-1}{2}.\end{aligned}$$ Similarly we find that $$\begin{aligned} \label{nnnnn} I_{31}^{**}&\ll \sum_{n\geq 1}n^{(\kappa-1)/2+\varepsilon-\kappa+\sigma_2+\sigma_1-1} \sum_{Nm< n}1+\sum_{n\geq 1}n^{(\kappa-1)/2+\varepsilon-\kappa+\sigma_2} \sum_{Nm<n}(Nm)^{\sigma_1-1}\end{aligned}$$ whose first double sum is $$\ll \sum_{n\geq 1}n^{\sigma_1+\sigma_2-(\kappa+1)/2+\varepsilon}$$ which is convergent in the region . The second double sum converges in the same region if $\sigma_1\geq 0$, while if $\sigma_1<0$ it is convergent when $\sigma_2<(\kappa-1)/2$. Therefore the right-hand side of is convergent in the region $$\begin{aligned} \label{region9bbb} \sigma_2<\frac{\kappa-1}{2},\quad \sigma_1+\sigma_2<\frac{\kappa-1}{2}.\end{aligned}$$ Therefore by , , and , we now arrive at the conclusion that the right-hand side of is absolutely convergent in the region $$\begin{aligned} \label{region10} \sigma_1<0,\quad \sigma_2<\frac{\kappa-1}{2}.\end{aligned}$$ Next we consider $I_{32}$. Using the modularity again, similarly to , we have $$\begin{aligned} \label{I-32-mod} I_{32}&=\sum_{m\geq 1}e^{\pi i(3s_1+s_2-3)/2}m^{s_1+s_2-1}\int_0^{-i\infty} (\sqrt{N}mz)^{-\kappa}\widetilde{f}\left(\frac{1}{Nmz}\right)z^{s_2-1} (z+1)^{s_1-1}dz\\ &=N^{-\kappa/2}\sum_{m\geq 1}e^{\pi i(3s_1+s_2-3)/2}m^{s_1+s_2-1-\kappa}\int_0^{-i\infty} \widetilde{f}\left(\frac{1}{Nmz}\right)z^{-\kappa+s_2-1}(z+1)^{s_1-1}dz.\notag\end{aligned}$$ Putting $z=1/w$, similarly to , we obtain $$\begin{aligned} \label{I-32-mod2} I_{32}&=N^{-\kappa/2}e^{\pi i(3s_1+s_2-3)/2}\sum_{m,n\geq 1}m^{s_1+s_2-1-\kappa} \widetilde{a}(n)\\ &\;\times\int_0^{i\infty}e^{2\pi i(n/Nm)w}w^{\kappa-s_1-s_2} (w+1)^{s_1-1}dw\notag\\ &=N^{-\kappa/2}e^{\pi i(3s_1+s_2-3)/2}\Gamma(\kappa-s_1-s_2+1)\sum_{m,n\geq 1} m^{s_1+s_2-1-\kappa}\widetilde{a}(n)\notag\\ &\;\times\Psi(\kappa-s_1-s_2+1,\kappa-s_2+1;-2\pi in/Nm)\notag\\ &=N^{-\kappa/2}e^{\pi i(3s_1+s_2-3)/2}\Gamma(\kappa-s_1-s_2+1) H_{2,N}^-(-s_1,\kappa-s_2+1;\widetilde{f}).\notag\end{aligned}$$ The convergence can be discussed exactly the same way as in the case of $I_{31}$. (This time we start with $-2\pi i(n/m)w=y$.) Hence is also valid in the region . Substituting and into , and combining with , we have $$\begin{aligned} \label{J-H} J_2(s_1,s_2;f)&=-\frac{(2\pi)^{s_1+s_2-1}\Gamma(1-s_1)}{e^{\pi is_1}\Gamma(s_2)} \Gamma(\kappa-s_1-s_2+1)\\ &\qquad\times N^{-\kappa/2} \left\{e^{\pi i(s_1-s_2-1)/2}H_{2,N}^+(-s_1,\kappa-s_2+1;\widetilde{f})\right.\notag\\ &\;\qquad \left.+e^{\pi i(3s_1+s_2-3)/2}H_{2,N}^-(-s_1,\kappa-s_2+1;\widetilde{f}) \right\}\notag\\ &=\frac{(2\pi)^{s_1+s_2-1}\Gamma(1-s_1)}{\Gamma(s_2)} \Gamma(\kappa-s_1-s_2+1)\notag\\ &\qquad\times N^{-\kappa/2}\left\{e^{\pi i(1-s_1-s_2)/2} H_{2,N}^+(-s_1,\kappa-s_2+1;\widetilde{f})\right.\notag\\ &\;\qquad \left.+e^{\pi i(s_1+s_2-1)/2} H_{2,N}^-(-s_1,\kappa-s_2+1;\widetilde{f})\right\}.\notag\end{aligned}$$ This with gives in the region . Therefore, to complete the proof of Theorem \[thm-2\], the remaining task is to show the meromorphic continuation of $H_{2,N}^{\pm}$. **[The meromorphic continuation of $H_{2,N}^{\pm}(s_1,s_2;\widetilde{f})$]{}** {#sec-6} ============================================================================== In this section we prove that $H_{2,N}^{\pm}(s_1,s_2;\widetilde{f})$ can be continued meromorphically to the whole space $\mathbb{C}^2$. We first consider the case $H_{2,N}^+$. Applying (with $\phi=-\pi/2$) to the right-hand side of and putting $y=-i\eta$, we obtain $$\begin{aligned} \label{H-7-1} H_{2,N}^+(s_1,s_2;\widetilde{f})&=\frac{-i}{\Gamma(s_1+s_2)}\sum_{m,n\geq 1} m^{-s_1-s_2}\widetilde{a}(n)\int_0^{\infty}e^{-2\pi(n/Nm)\eta}\\ &\times(-i\eta)^{s_1+s_2-1}(-i\eta+1)^{-s_1-1}d\eta.\notag\end{aligned}$$ The argument in the preceding section shows that the double series form of $H_{2,N}^+(-s_1,\kappa-s_2+1;\widetilde{f})$ (that is, ) is absolutely convergent in the region . This implies that the right-hand side of is absolutely convergent in the region $$\begin{aligned} \label{region11} \sigma_1>0,\quad \sigma_2>\frac{\kappa+3}{2}.\end{aligned}$$ Therefore, if we assume , we may change the order of summation and integration on the right-hand side of to obtain $$\begin{aligned} \label{H-7-2} &H_{2,N}^+(s_1,s_2;\widetilde{f})\\ &=\frac{-i}{\Gamma(s_1+s_2)}\int_0^{\infty} \sum_{m\geq 1}m^{-s_1-s_2}\widetilde{f}\left(\frac{i\eta}{Nm}\right) (-i\eta)^{s_1+s_2-1}(-i\eta+1)^{-s_1-1}d\eta\notag\\ &=\frac{-i}{\Gamma(s_1+s_2)}\int_0^{\infty}\widetilde{\mathcal{F}}(i\eta,s_1+s_2) (-i\eta)^{s_1+s_2-1}(-i\eta+1)^{-s_1-1}d\eta,\notag\end{aligned}$$ where $$\begin{aligned} \label{cal-F-def} \widetilde{\mathcal{F}}(\tau,s)=\sum_{m\geq 1}\widetilde{f} \left(\frac{\tau}{Nm}\right)m^{-s}.\end{aligned}$$ \[mellin\] Let $u$ be a complex variable. We have $$\begin{aligned} \label{mellin-formula} \int_0^{\infty}\widetilde{\mathcal{F}}(i\eta,s)\eta^{u-1}d\eta=\Gamma(u) \left(\frac{N}{2\pi}\right)^u\zeta(s-u)L(u,\widetilde{f})\end{aligned}$$ in the region $$\begin{aligned} \label{region12} \Re(s)-1>\Re(u)>\frac{\kappa+1}{2}.\end{aligned}$$ We have $$\begin{aligned} &\int_0^{\infty}\widetilde{\mathcal{F}}(i\eta,s)\eta^{u-1}d\eta\\ &=\sum_{m,n\geq 1}m^{-s}\widetilde{a}(n)\int_0^{\infty}e^{-2\pi(n/Nm)\eta} \eta^{u-1}d\eta\\ &=\sum_{m,n\geq 1}m^{-s}\widetilde{a}(n)\Gamma(u)\left(\frac{Nm}{2\pi n}\right)^u\\ &=\Gamma(u)\left(\frac{N}{2\pi}\right)^u\sum_{m\geq 1}m^{-s+u}\sum_{n\geq 1} \widetilde{a}(n)n^{-u}\\ &=\Gamma(u)\left(\frac{N}{2\pi}\right)^u\zeta(s-u)L(u,\widetilde{f}),\end{aligned}$$ where in the above calculations, changes of summation and integration can be verified because of absolute convergence under condition . Therefore, $\widetilde{\mathcal{F}}(i\eta,s)$ is the inverse Mellin transform of the right-hand side of , and hence $$\begin{aligned} \label{F-mod1} \widetilde{\mathcal{F}}(i\eta,s_1+s_2)=\frac{1}{2\pi i}\int_{(c)} \eta^{-u}\Gamma(u)\left(\frac{N}{2\pi}\right)^u\zeta(s_1+s_2-u)L(u,\widetilde{f})du,\end{aligned}$$ where $c=\Re(u)$ satisfies $$\begin{aligned} \label{condition-c} \sigma_1+\sigma_2-1>c>\frac{\kappa+1}{2}\end{aligned}$$ and the path of integration is the vertical line from $c-i\infty$ to $c+i\infty$. From and we obtain $$\begin{aligned} \label{H-7-3} &H_{2,N}^+(s_1,s_2;\widetilde{f}) =\frac{-1}{2\pi\Gamma(s_1+s_2)}\int_0^{\infty}\int_{(c)} \eta^{-u}\Gamma(u)\left(\frac{N}{2\pi}\right)^u\zeta(s_1+s_2-u)L(u,\widetilde{f})du\\ &\qquad\qquad\times (-i\eta)^{s_1+s_2-1}(-i\eta+1)^{-s_1-1}d\eta\notag\\ &\qquad=\frac{-1}{2\pi\Gamma(s_1+s_2)}\int_{(c)} \Gamma(u)\left(\frac{N}{2\pi}\right)^u\zeta(s_1+s_2-u)L(u,\widetilde{f})J(u)du,\notag\end{aligned}$$ where $$\begin{aligned} \label{J-def} J(u)=\int_0^{\infty}\eta^{-u}(-i\eta)^{s_1+s_2-1}(-i\eta+1)^{-s_1-1}d\eta,\end{aligned}$$ if the change of the order of integration is possible. The integral is absolutely convergent in the region $$\begin{aligned} \label{region13} \sigma_2<c+1,\quad \sigma_1+\sigma_2>c,\end{aligned}$$ and hence the above change of integration is valid by Fubini’s theorem in this region. From and we see that is valid in the region $$\begin{aligned} \label{region13b} \sigma_2<c+1,\quad \sigma_1+\sigma_2-1>c>\frac{\kappa+1}{2}.\end{aligned}$$ Since the intersection of and is non-empty, now we find that $H_{2,N}^+(s_1,s_2;\widetilde{f})$ is continued to the region by the expression . Putting $y=-i\eta$ on the right-hand side of , and rotating the path of integration to the positive real axis (this is possible under condition ), we obtain $$J(u)=e^{\pi i(1-u)/2}\int_0^{\infty}y^{s_1+s_2-1-u}(1+y)^{-s_1-1}dy.$$ Therefore, applying the beta integral formula we obtain $$\begin{aligned} \label{J-mod1} J(u)=e^{\pi i(1-u)/2}\frac{\Gamma(u-s_2+1)\Gamma(s_1+s_2-u)}{\Gamma(s_1+1)}.\end{aligned}$$ Substituting this into , we now arrive at the expression $$\begin{aligned} \label{H-7-4} H_{2,N}^+(s_1,s_2;\widetilde{f})=&\frac{-1} {2\pi \Gamma(s_1+s_2)\Gamma(s_1+1)} \int_{(c)}\Gamma(u)\Gamma(u-s_2+1)\Gamma(s_1+s_2-u)\\ &\times e^{\pi i(1-u)/2}\left(\frac{N}{2\pi}\right)^u \zeta(s_1+s_2-u)L(u,\widetilde{f})du\notag\end{aligned}$$ in the region . We prove that the right-hand side of can be continued meromorphically to the whole space $\mathbb{C}^2$ by suitable modifications of the path of integration. Let $(s_1^0,s_2^0)$ be any point in the space $\mathbb{C}^2$. We choose a point $(s_1^*,s_2^*)$ in the region , which satisfies $\Im s_1^*=\Im s_1^0$, $\Im s_2^*=\Im s_2^0$. Then holds for $(s_1,s_2)=(s_1^*,s_2^*)$. The poles of the integrand on the right-hand side of are \(A) $u=0,-1,-2,-3,\ldots$, \(B) $u=s_2^*-1,s_2^*-2,s_2^*-3,\ldots$, \(C) $u=s_1^*+s_2^*,s_1^*+s_2^*+1,s_1^*+s_2^*+2,\ldots$, \(D) $u=s_1^*+s_2^*-1$. The poles (A) and (B) are on the left of the vertical line $\Re u=c$, while the poles (C) and (D) are on the right of $\Re u=c$. First consider the case when $\Im(s_1^*+s_2^*)\neq\Im s_2^*$ and $\Im(s_1^*+s_2^*)\neq 0$. Let $L_1$ be the line segment joining $s_2^*-1$ and $s_2^0-1$, and $L_2$ the line segment joining $s_1^*+s_2^*-1$ and $s_1^0+s_2^0-1$. We deform the original path $\Re u=c$ to make a new path $\mathcal{D}$, such that $L_1$ is on the left of $\mathcal{D}$ while $L_2$ is on the right of $\mathcal{D}$ (see Fig.1). ![image](aa.eps) Fig.1 Then we move the variables $(s_1,s_2)$ from $(s_1^*,s_2^*)$ to $(s_1^0,s_2^0)$, keeping the values of their imaginary parts. Then the location of poles moves, but during this procedure they do not encounter the new path $\mathcal{D}$. Therefore in this case we can continue $H_{2,N}^+(s_1,s_2;\widetilde{f})$ to the point $(s_1^0,s_2^0)$ holomorphically. Next consider the situation when $\Im(s_1^*+s_2^*)=\Im s_2^*$ or $\Im(s_1^*+s_2^*)= 0$. We discuss the former case, because the latter case can be treated similarly. When $\Im(s_1^*+s_2^*)=\Im s_2^*$, we deform $\Re u=c$ to make $\mathcal{D}'$, which only requires that $L_1$ is on the left of $\mathcal{D}'$. Then, when $s_1^*+s_2^*-1$ is moved to $s_1^0+s_2^0-1$, several poles encounter $\mathcal{D}'$. Therefore at the point $(s_1^0,s_2^0)$, we have to add the residue terms coming from the above poles to the right-hand side of . This new expression of $H_{2,N}^+(s_1,s_2;\widetilde{f})$ gives the meromorphic continuation to the point $(s_1^0,s_2^0)$. We thereby obtain the proof of meromorphic continuation of $H_{2,N}^+(s_1,s_2;\widetilde{f})$ to the whole space $\mathbb{C}^2$. The function $H_{2,N}^-(s_1,s_2;\widetilde{f})$ can be treated quite similarly. We can show the integral expression of $H_{2,N}^-(s_1,s_2;\widetilde{f})$, almost the same as , only the factor $e^{\pi i(1-u)/2}$ is replaced by $e^{\pi i(u-1)/2}$, which can be continued meromorphically as above. The proof of Theorem \[thm-2\] is now complete. \[rmk-sing\] We can discuss the location of singularities of $H_{2,N}^{\pm}(s_1,s_2;\widetilde{f})$ more closely, by using the more sophisticated path (like the path $\mathcal{C}'$ defined in [@Mat06], described in Fig.2 of [@Mat06]) instead of $\mathcal{D}'$. **[Proof of Corollary \[cor-2\]]{}** {#sec-7} ==================================== We conclude this paper with the proof of Corollary \[cor-2\]. The right-hand side of the formula given in Theorem \[thm-2\] consists of two terms, the term $L_1(s_1,s_2;f)$ and the other. First consider the term $L_1(s_1,s_2;f)$. The denominator $\Gamma(s_2)$ on the right-hand side of the definition of $L_1(s_1,s_2;f)$ has poles at $s_2=-l$, and the other factors do not cancel those poles. Therefore $s_2=-l$ are zero-divisors of $L_1(s_1,s_2;f)$. Consider the other term on the right-hand side of . Again the denominator is $\Gamma(s_2)$, so what we have to show is that the other factors do not cancel the poles $s_2=-l$ of $\Gamma(s_2)$. This is obvious except for the terms $H_{2,N}^{\pm}(-s_1,\kappa-s_2+1;\widetilde{f})$. As for $H_{2,N}^{\pm}$, in Section \[sec-6\] we noticed that these are absolutely convergent, hence especially finite, in the region . Since the region includes $s_2=-l$ when $\sigma_1<0$, we now arrive at the conclusion that $s_2=-l$ are zero-divisors of the second term on the right-hand side of . [99]{}\[bib\] Y. Choie and K. Ihara, Iterated integral and multiple Hecke $L$-series, Manuscripta Math. [**142**]{} (2013), 245-255. A. Erd[é]{}lyi (ed.), Higher Transcendental Functions, Vol.1, McGraw-Hill, 1953. M. Katsurada and K. Matsumoto, Asymptotic expansions of the mean values of Dirichlet $L$-functions, Math. Z. [**208**]{} (1991), 23-39. Y. Komori, K. Matsumoto and H. 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Matsumoto, On the analytic continuation of various multiple zeta-functions, in “Number Theory for the Millennium II”, Proc. Millennial Conf. on Number Theory (Urbana-Champaign, May 2000), M. A. Bennett et al. (eds.), A K Peters, 2002, pp.417-440. K. Matsumoto, Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series, Nagoya Math. J. [**172**]{} (2003), 59-102. K. Matsumoto, Functional equation for double zeta-functions, Math. Proc. Camb. Phil. Soc. [**136**]{} (2004), 1-7. K. Matsumoto, Analytic properties of multiple zeta-functions in several variables, in “Number Theory: Tradition and Modernization”, Proc. 3rd China-Japan Seminar on Number Theory (Xi’an, Feb 2004), W. Zhang and Y. Tanigawa (eds.), Springer, 2006, pp.153-173. K. Matsumoto, The analytic theory of multiple zeta-functions and its applications, S[ū]{}gaku [**59**]{} (2007), 24-45 (in Japanese); English transl.: Sugaku Expositions [**23**]{} (2010), 143-167. K. Matsumoto and Y. Tanigawa, The analytic continuation and the order estimate of multiple Dirichlet series, J. Th[é]{}or. Nombr. Bordeaux [**15**]{} (2003), 267-274. T. Miyake, Modular Forms, Springer, 1989. Y. Motohashi, A note on the mean value of the zeta and $L$-functions I, Proc. Japan Acad. [**A61**]{} (1985), 222-224. E. C. Titchmarsh, The Theory of the Riemann Zeta-function, Oxford, 1951. [^1]: Keywords: iterated Integrals of modular forms, modular symbol, double zeta function, functional equation, confluent hypergeometric function, modular relation [^2]: 1991 Mathematics Subject Classification:11F68, 11F32, 11M32 [^3]: This work was partially supported by NRF 2012047640, NRF 2011-0008928 and NRF 2008-0061325
--- abstract: 'We compute corrections to precision electroweak observables in supersymmetry in the limit that scalar superpartners are very massive and decoupled. This leaves charginos and neutralinos and a Standard Model-like Higgs boson as the only states with unknown mass substantially affecting the analysis. We give complete formulas for the chargino and neutralino contributions, derive simple analytic results for the pure gaugino and higgsino cases, and study the general case. We find that in all circumstances, the precision electroweak fit improves when the charginos and neutralinos are near the current direct limits. Larger higgsino and gaugino masses worsen the fit as the theory predictions asymptotically approach those of the Standard Model. Since the Standard Model is considered by most to be an adequate fit to the precision electroweak data, an important corollary to our analysis is that all regions of parameter space allowed by direct collider constraints are also allowed by precision electroweak constraints in split supersymmetry.' --- MSUHEP-041221\ MCTP-04-74\ Virtual effects of light gauginos and higgsinos:\ a precision electroweak analysis of split supersymmetry Stephen P. Martin$^a$, Kazuhiro Tobe$^b$, James D. Wells$^{c}$\ ${}^{(a)}$Physics Department, Northern Illinois University, DeKalb, IL 60115\   [and]{} Fermi National Accelerator Laboratory, PO Box 500, Batavia, IL 60510\ ${}^{(b)}$Department of Physics and Astronomy\ Michigan State University, East Lansing, MI 48824\ ${}^{(c)}$Michigan Center for Theoretical Physics (MCTP)\   University of Michigan, Ann Arbor, MI 48109-1120, USA\ hep-ph/0412424\ December 2004 Introduction ============ Ordinary intuition about finetuning and naturalness applied to supersymmetry implies the existence of light (TeV or less) scalar superpartners, gauginos and higgsinos. Supersymmetry has several other good reasons for its existence beyond its ability to naturally stabilize the weak scale and high scale (e.g., GUT scale or Planck scale). For example, supersymmetry is nicely compatible with gauge coupling unification and dark matter. These two good reasons are arguably less philosophical than the naturalness reason. There has been much discussion recently centered on supplanting [@Arkani-Hamed:2004fb] or suspending [@Wells:2003tf] our ordinary view of naturalness from consideration in supersymmetry model building to allow for very heavy scalar masses. A large hierarchy between scalars and fermions is sometimes called split supersymmetry [@Giudice:2004tc]. The phenomenology of split supersymmetry, where all scalars (even third generation) are significantly heavier than the gaugino masses, has unique features that put it in contrast with other approaches to supersymmetry (see also [@split; @pheno; @1; @split; @pheno; @2]). The ideas of split supersymmetry may have interesting motivations within string theory [@split; @strings]. In this article, we wish to study the effects of light gauginos and higgsinos on precision electroweak analysis. We will demonstrate below that the best fit to the precision electroweak data, when the scalar superpartners are decoupled, is light gauginos and higgsinos near the current direct collider limits. As a corollary to this finding, no combination of gaugino and higgsino masses above the current direct experimental limits are in conflict with the precision electroweak data. This is because as the gauginos and higgsino get heavier, the fit approaches the Standard Model fit, which is known to be compatible with the data as long as the Higgs mass is lighter than about 200 GeV. Such a cap on the Higgs boson is guaranteed in minimal supersymmetry even if the superpartner masses decouple to the grand unification scale. We start our analysis by computing the $\chi^2$ fit to the precision data within the Standard Model using the latest data and theoretical computations of observables. We then review the corrections to the precision observables for general beyond-the-SM contributions to vector boson self-energies. We also provide explicit formulas for these self-energy functions in the general supersymmetric version of the Standard Model [@precision; @ew; @groups; @precision; @ew; @groups; @2]. This is then specialized to our case of light gauginos/higgsinos and heavy scalar superpartners. In the process, we provide some useful analytic results in the pure gaugino and in the pure higgsino limits. The key task within these sections is to justify the claims made above. These results and additional thoughts are summarized in the conclusions section. Standard Model precision electroweak fit ======================================== The Higgs scalar boson has not been found by direct experiment, yet its effects are present in precision electroweak observables by virtue of its contributions at one loop to the self energies of electroweak vector bosons. In general, new particles that affect observables only through their induced oblique corrections, such as the Higgs boson, are best constrained [@Peskin:2001rw] by the three observables $\sin^2\theta_{\rm eff}^l$ ($=s^2_{\rm eff}$), $M_W$ and $\Gamma(Z\to l^+l^-)$ ($=\Gamma_l$). The measurements of these three observables[@:2003ih; @Renton:2004wd] are $$\begin{aligned} s^2_{\rm eff} &=&0.23147\pm 0.00017~[{\rm Average~ of~ all~ results}], \label{s2_ave}\\ M_W&=&80.425\pm 0.034~{\rm GeV},\\ \label{mw_ave} \Gamma_l&=&83.984\pm 0.086~{\rm MeV}. \label{Gammal_ave}\end{aligned}$$ It should be noted that the $s^2_{\rm eff}$ determined by the hadronic asymmetries and the one determined by the leptonic asymmetries have a $2.8\, \sigma$ discrepancy: $$\begin{aligned} s^2_{\rm eff}&=&0.23113\pm 0.00021~[A_{FB}^{0,l},~A_l(P_\tau),~A_l(SLD)], \label{s2_lep} \\ s^2_{\rm eff}&=&0.23213\pm 0.00029~[A_{FB}^{0,b},A_{FB}^{0,c}, \langle Q_{FB}\rangle]. \label{s2_had}\end{aligned}$$ For the Standard Model (SM) analysis, we use the state-of-the-art computations for $s^2_{\rm eff}$ [@Awramik:2004ge], $M_W$ [@Awramik:2003rn] and $\Gamma_l$ [@Ferroglia:2002rg]. In the SM, these are expressed in terms of the SM parameters, $\Delta \alpha_h^{(5)}(M_Z)$, $M_t$, $\alpha_s(M_Z)$ and $M_h$. For the experimental values of the SM parameters, we employ the following values: $$\begin{aligned} \Delta \alpha_h^{(5)}(M_Z)&=&0.02769\pm 0.00035~\cite{Jegerlehner:2003rx}, \label{alpha_ave} \\ M_t&=&178.0\pm 4.3~{\rm GeV}~\cite{Azzi:2004rc},\\ \alpha_s(M_Z)&=&0.1187\pm 0.0020~\cite{RPP}.\end{aligned}$$ In our analysis, $M_Z$ is fixed to be $91.1875~{\rm GeV}$. By searching for the minimum of $\chi^2$ for the electroweak observables, the best-fit Higgs mass can be found. This is carried out by computing $ \chi^2 (M_h, M_t, \alpha_s, \Delta \alpha_h^{(5)}) = \sum_X (X - X_{\rm exp})^2/\sigma_X^2 $ where $X = (M_t,$ $\alpha_s,$ $\Delta \alpha_h^{(5)},$ $s^2_{\rm eff},$ $M_W,$ $\Gamma_l)$, with the last three predicted in terms of the first four by their Standard Model expressions, and $X_{\rm exp}$, $\sigma_X$ the experimental central values and uncertainties. In fig. \[SM\_fit\], the minimum total $\chi^2$ in the SM is shown as a function of $M_h$, with the best-fit values given in Table \[SM\_table\]. In all cases, the best fit is achieved for $M_W$ and $s^2_{\rm eff}$ lower than their experimental central values, and $\Gamma_l$ higher than its experimental central value. ![Minimum total $\chi^2$ as a function of $M_h$ using different values of $s^2_{\rm eff}$: the average $s^2_{\rm eff}=0.23147\pm 0.00017$ (solid line), the one from leptonic processes only $s^2_{\rm eff}=0.23113\pm 0.00021$ (dashed line), and the one from hadronic processes only $s^2_{\rm eff}=0.23213\pm 0.00029$ (dotted line).[]{data-label="SM_fit"}](chi2mh.eps){width="12.0cm"} $\chi^2$ $M_h$ \[GeV\] $M_t$ \[GeV\] $\alpha_S(M_Z)$ $\Delta \alpha_{\rm had}^{(5)}(M_Z)$ $M_W$ \[GeV\] $s^2_{\rm eff}$ $\Gamma_l$ \[MeV\] ---------- --------------- ---------------- ----------------- -------------------------------------- ---------------- ----------------- -------------------- 1.67 120 178.40 0.1187 0.02775 80.391 0.23140 84.043 ($+0.1\sigma$) ($0\sigma$) ($+0.2\sigma$) ($-1.0\sigma$) ($-0.4\sigma$) ($+0.7\sigma$) 0.75 60 176.81 0.1187 0.02772 80.418 0.23111 84.052 ($-0.3\sigma$) ($0\sigma$) ($+0.1\sigma$) ($-0.2\sigma$) ($-0.1\sigma$) ($+0.8\sigma$) 6.38 196 179.46 0.1187 0.02784 80.366 0.23163 84.014 ($+0.3\sigma$) ($0\sigma$) ($+0.4\sigma$) ($-1.7\sigma$) ($-1.7\sigma$) ($+0.4\sigma$) : Results of the best fit for the electroweak observables in the Standard Model. The first, second, and third rows use the averaged, leptonic, and hadronic values for $s^2_{\rm eff}$, respectively.[]{data-label="SM_table"} The minimum of the $\chi^2$ is at $M_h=120$ GeV if the total averaged value of $s^2_{\rm eff}$ is used in the analysis. The 95% confidence level upper bound on the Higgs mass, which requires $\Delta \chi^2< 1.64$, comes to $227\gev$. Note that if we used only the leptonic data to compute $s^2_{\rm eff}$, the precision fit would have given a best fit value for $M_h$ of $60\gev$ and a 95% CL upper bound of $132 \gev$. If we used only the hadronic data to compute $s^2_{\rm eff}$ we obtain a much lower quality fit, with $M_h$ of $196\gev$, and a 95% CL upper bound of $365\gev$. Although these differences in the Higgs mass fit between the leptonic- and hadronic-determined $s^2_{\rm eff}$ are interesting, our view at the present is that we should not differentiate the data, so we take the world averaged $s^2_{\rm eff}$ as the appropriate observable in our analysis of the Standard Model and its extensions. Oblique corrections to electroweak observables ============================================== Throughout this paper we will be working in a theoretical framework where no corrections to electroweak observables are expected through vertex loops. Such an assumption is valid for many theories. No vertex loop corrections are important if all flavor-charged states that would have contributed to the vertex corrections are too massive to be relevant. In this case, all substantive corrections come from loop corrections of the vector boson self-energies, which only requires the presence of light states charged under the symmetries these bosons generate. In our case, the flavor-charged squark and slepton fields are decoupled in split supersymmetry, suppressing vertex corrections. However, the gauge-charged charginos and neutralinos are not decoupled and can contribute substantively to the electroweak boson self-energies. That is why we focus on the oblique corrections. In this section we give the formalism for general oblique corrections. We then apply this formalism to the charginos and neutralinos of minimal supersymmetry. We also briefly describe heavy scalar corrections to the oblique corrections, which will be helpful in characterizing the small corrections to the chargino/neutralino results from the heavy scalars sector. [*General oblique corrections*]{} Our analysis uses the $S,T,U$ parameter expansions of [@Peskin:1991sw], augmented by $Y,V,W$ parameters inspired by [@Maksymyk:1993zm]. The latter take into account the corrections from nonzero momentum that are important when the new physics states have mass near $M_Z$. This is crucial in particular for light charginos and neutralinos in the MSSM, and will become more important in the future when the top-quark mass, the $W$ boson mass, and other electroweak observables become known with better accuracy. The $S,T,U$ Peskin-Takeuchi parameters are defined as $$\begin{aligned} \frac{\alpha S}{4s_W^2 c_W^2}&=& \frac{\Pi_{ZZ}(M_Z^2)-\Pi_{ZZ}(0)}{M_Z^2} -\frac{c_{2W}}{c_W s_W} \frac{\Pi_{Z\gamma}(M_Z^2)}{M_Z^2} -\frac{\Pi_{\gamma \gamma}(M_Z^2)}{M_Z^2},\\ \alpha T&=& \frac{\Pi_{WW}(0)}{M_W^2}-\frac{\Pi_{ZZ}(0)}{M_Z^2},\\ \frac{\alpha U}{4 s_W^2} &=& \frac{\Pi_{WW}(M_W^2)-\Pi_{WW}(0)}{M_W^2} -c_W^2\frac{\Pi_{ZZ}(M_Z^2)-\Pi_{ZZ}(0)}{M_Z^2}\nonumber \\ &&-2s_W c_W \frac{\Pi_{Z \gamma}(M_Z^2)}{M_Z^2} -s_W^2 \frac{\Pi_{\gamma \gamma}(M_Z^2)}{M_Z^2}.\end{aligned}$$ Here we have followed the definitions given, for example, in [@RPP], which differ slightly from those given originally in [@Peskin:1991sw]. We use the notation $c_W$ and $s_W$ to refer to the cosine and sine of the weak mixing angle, and $c_{2W} = c_W^2 - s_W^2$. All of the self-energy functions $\Pi_{XY}$ are taken to contain only the new physics contributions (beyond the Standard Model with a Higgs boson), and follow the sign convention of e.g. ref. [@RPP]. In order to completely describe oblique corrections near the $Z$ pole and at zero momentum, it is necessary to introduce three more parameters, as in [@Maksymyk:1993zm]. These can be written in combinations $V,W,Y$, defined as $$\begin{aligned} \alpha Y &=& \frac{\Pi_{\gamma \gamma}(M_Z^2)}{M_Z^2} -\hat{\Pi}_{\gamma \gamma}(0), \\ \alpha V &=& \Pi_{ZZ}^{'}(M_Z^2)- \left[\frac{\Pi_{ZZ}(M_Z^2)-\Pi_{ZZ}(0)}{M_Z^2} \right], \\ \alpha W &=& \Pi_{WW}^{'}(M_Z^2)- \left[\frac{\Pi_{WW}(M_W^2)-\Pi_{WW}(0)}{M_W^2} \right], \end{aligned}$$ Here $\hat{\Pi}_{\gamma \gamma}(p^2)=\Pi_{\gamma \gamma}(p^2)/p^2$, and $\Pi'(p^2)=d\Pi/dp^2$. The parameter $Y$ we use here is a convenient linear combination of the parameters originally defined in [@Maksymyk:1993zm]. Note also that ref. [@Maksymyk:1993zm] used a slightly different definition of $S,T,U$ than used here or in other references. In terms of the parameters defined above, the observables pertinent to our discussion are expressed as follows: $$\begin{aligned} \frac{M_W^2}{(M_W^2)_{\rm SM}} &=& 1-\frac{\alpha S}{2 c_{2W}} + \frac{c_W^2 \alpha T}{c_{2W}} +\frac{\alpha U}{4 s^2_W} -\frac{s^2_W \alpha Y}{c_{2W}}.\\ \frac{s^2_{\rm eff}}{(s^2_{\rm eff})_{\rm SM}} &=&1+\frac{\alpha S}{4s^2_W c_{2W} } -\frac{c_W^2 \alpha T}{c_{2W}} +\frac{c_W^2 \alpha Y}{c_{2W}}.\\ \frac{\Gamma_l}{(\Gamma_l)_{\rm SM}} &=& 1- d_W \alpha S + (1+ 4 s_W^2c_W^2 d_W ) \alpha T +\alpha V - 4 s_W^2 c_W^2 d_W \alpha Y,\end{aligned}$$ where $X_{\rm SM}~(X=M_W,s^2_{\rm eff}~{\rm and}~\Gamma_l)$ are the SM values, and $d_W= (1-4 s^2_W)/[(1-4 s_W^2+8s_W^4) c_{2W}]$. Note that the quantity $W$ does not contribute at all to these particular observables in this parameterization. [*Oblique corrections in low-energy supersymmetry*]{} The preceding analysis applies to a general theory of new physics in which vertex corrections are small. In order to employ these parameter expansions in supersymmetry, we need to compute the contributions to the vector boson self-energies. These will be given in terms of kinematic functions $B$, $H$, $G$, $F$, which are defined in the Appendix. They are implicitly functions of an external momentum invariant $s = p^2$ (in a signature $+$$-$$-$$-$ metric). In some special cases, it is often convenient to then expand these kinematic functions in $r = s/M^2$, where $M$ is the mass of the heavier particle in the loop, to obtain relatively simple and understandable expressions. The name of a particle stands for its squared mass when appearing as the argument of a kinematic function. The chargino and neutralino contributions to the electroweak vector boson self-energies are \_[WW]{} &=& -[g\^2 16 \^2]{} \_[i=1]{}\^4 \_[j=1]{}\^2\ \_[ZZ]{} &=& -[g\^2 16 \^2 c\_W\^2]{} \_[i,j=1]{}\^4\ && + \_[i,j=1]{}\^2\ \_[Z]{} &=& [g\^2 s\_W16 \^2 c\_W]{} \_[i=1]{}\^2 (O\_[ii]{}\^[L]{} + O\_[ii]{}\^[R]{}) G()\ \_ &=& -[g\^2 s\_W\^2 8 \^2]{} \_[i=1]{}\^2 G() The notation for the chargino and neutralino couplings is the same as in [@conventions; @Martin:2002iu], and can be described as follows. In the $(\stilde B, \stilde W^0, \stilde H_d^0, \stilde H_u^0)$ basis, the neutralino mass matrix is M\_[\^0]{} = , \[neutmass\] where $v_{u,d}=\langle H^0_{u,d}\rangle$ and $v_u/v_d=\tan\beta$, such that $v_u^2+v_d^2\simeq (174 \gev)^2$. The unitary matrix $N$ diagonalizes $M_{\tilde\chi^0}$: N\^\* M\_[\^0]{} N\^[-1]{} &=& [diag]{}( M\_[\^0\_1]{}, M\_[\^0\_2]{}, M\_[\^0\_3]{}, M\_[\^0\_4]{}), \[diagonalizemN\] where the mass eigenvalues $M_{\tilde \chi^0_i}$ are all real and positive (see [@Martin:2002iu] for technique). In the $(\stilde W^\pm,\stilde H^\pm)$ basis, the chargino mass matrix is M\_[\^+]{} = . \[charmass\] The unitary matrices $U$ and $V$ diagonalize the above matrix according to U\^\* M\_[\^+]{} V\^&=& where again $M_{\tilde \chi^+_i}$ are real and positive. One finds $U$ and $V$ by solving VM\_[\^+]{}\^M\_[\^+]{}V\^[-1]{} = U M\_[\^+]{}\^\* M\_[\^+]{}\^T U\^[-1]{} = . The $O_{ij}$-couplings are &&O\^[L]{}\_[ij]{} = N\_[i2]{} V\_[j1]{}\^\* - N\_[i4]{} V\_[j2]{}\^\*/, O\^[R]{}\_[ij]{} = N\^\*\_[i2]{} U\_[j1]{} + N\^\*\_[i3]{} U\_[j2]{}/,\ &&O\^[’L]{}\_[ij]{} = -V\_[i1]{} V\_[j1]{}\^\* - [12]{} V\_[i2]{} V\_[j2]{}\^\* + s\_W\^2\_[ij]{}, O\^[’R]{}\_[ij]{} = -U\_[i1]{}\^\* U\_[j1]{} - [12]{} U\_[i2]{}\^\* U\_[j2]{} + s\_W\^2\_[ij]{},\ &&O\^[”L]{}\_[ij]{} = (-N\_[i3]{} N\_[j3]{}\^\* + N\_[i4]{} N\_[j4]{}\^\*)/2 . In eqs. (\[neutmass\]) and (\[charmass\]), we have assumed that the gaugino couplings to Higgs-higgsino pairs are given by the tree-level supersymmetric relation. We will discuss the merits of this assumption in section \[sec:inofits\]. For completeness, we also compute oblique corrections due to the sfermions and heavy Higgs bosons. As we stated in the introduction, we are assuming that the sfermions and heavy Higgs bosons are decoupled and have no substantive effect on the fits if their masses are above a TeV. The equations below are used to justify that statement. First, for the sfermions, we assume, as is consistently suggested by experimental constraints and theoretical prejudice, that the first two families have negligible sfermion mixing. For the third family sfermions $\tilde t_i$, $\tilde b_i$, and $\tilde \tau_i$ with $i=1,2$, the mixing (including possible CP violating phases) is described by = where $|c_{\tilde f}|^2 + |s_{\tilde f}|^2 = 1$. When there is no CP violation, $c_{\tilde f}$ and $s_{\tilde f}$ are real and are the sine and cosine of a sfermion mixing angle. Then the sfermion contributions to the self energies of the vector bosons are \_[WW]{} &=& [g\^2 32 \^2]{}\ \_[ZZ]{} &=& [g\^2 16 \^2 c\_W\^2]{} \_f N\_f \_[i,j]{} |g\_[Zf\_i f\_j\^\*]{}|\^2 F(f\_i, f\_j) \[PIZZsfermions\]\ \_[Z]{} &=& [g\^2 s\_W 16\^2 c\_W]{} \_[f\_i]{} N\_f Q\_f g\_[Zf\_if\_i\^\*]{} F(f\_i, f\_i) \[PIZgsfermions\]\ \_ &=& [g\^2 s\_W\^2 16\^2 ]{} \_[f\_i]{} N\_f Q\_f\^2 F(f\_i, f\_i) \[PIggsfermions\] Here, $N_f=3,1$ and $Q_f = +2/3, -1/3, -1,0$ in the obvious way. In eq. (\[PIZZsfermions\]) the sum on $f$ is over the 12 symbols $(d,s,b,u,c,t,e,\mu,\tau,\nu_e,\nu_\mu,\nu_\tau)$, and $i,j$ run over 1,2, except for the sneutrinos. In eqs. (\[PIZgsfermions\]) and (\[PIggsfermions\]), the sums are over the 18 charged sfermion mass eigenstates. The $Z$ couplings for the first two families and the tau sneutrino are &&g\_[Zd\_L d\_L\^\*]{} = -[1/2]{} + s\_W\^2/3, g\_[Zu\_L u\_L\^\*]{} = [1/2]{} - 2s\_W\^2/3, g\_[Ze\_L e\_L\^\*]{} = -[1/2]{} + s\_W\^2,\ &&g\_[Z\^\*]{} = [1/2]{}, g\_[Zu\_R u\_R\^\*]{} = -2 s\_W\^2/3, g\_[Zd\_R d\_R\^\*]{} = s\_W\^2/3, g\_[Ze\_R e\_R\^\*]{} = s\_W\^2, and for the third-family sfermions other than the tau sneutrino are &&g\_[Zf\_1 f\_1\^\*]{} = |c\_[f]{}|\^2 g\_[Zf\_L f\_L\^\*]{} + |s\_[f]{}|\^2 g\_[Zf\_R f\_R\^\*]{}, g\_[Zf\_2 f\_2\^\*]{} = |s\_[f]{}|\^2 g\_[Zf\_L f\_L\^\*]{} + |c\_[f]{}|\^2 g\_[Zf\_R f\_R\^\*]{},\ && g\_[Zf\_1 f\_2\^\*]{} = (g\_[Zf\_2 f\_1\^\*]{})\^\* = s\_[f]{} c\_[f]{} (g\_[Zf\_R f\_R\^\*]{} - g\_[Zf\_L f\_L\^\*]{}). The $W$ couplings for the third-family sfermions are &&g\_[Wb\_1t\_1\^\*]{} = c\_[b]{} c\_[t]{}\^\*, g\_[Wb\_2t\_2\^\*]{} = s\_[b]{}\^\* s\_[t]{}, g\_[Wb\_1t\_2\^\*]{} = -c\_[b]{} s\_[t]{}, g\_[Wb\_2t\_1\^\*]{} = -s\_[b]{}\^\* c\_[t]{}\^\*,\ &&g\_[W \_1\_]{} = c\_, g\_[W\_2\_]{} = -s\_\^\*. Finally, we consider the contributions of the Higgs scalar bosons, $h^0$, $H^0$, $A^0$, and $H^\pm$. We assume that the Standard Model result already includes contributions from the lightest Higgs scalar. Therefore, to compensate, in the following we subtract a contribution from $h^0$ with Standard Model couplings, in other words $\sin^2(\beta-\alpha) \rightarrow 1$. This just converts each term involving $h^0$ and $W,Z$ with coefficient $\sin^2(\beta-\alpha)$ into one with coefficient $-\cos^2(\beta-\alpha)$. The results below are therefore the difference between the MSSM and the Standard Model with Higgs mass $m_{h}$: \_[WW]{} &=& [g\^2 64\^2]{} F(A\^0, H\^+) + \^2(-) F(H\^0, H\^+) +\^2(-)\ \_[ZZ]{} &=& [g\^2 64 \^2 c\_W\^2]{} c\_[2W]{}\^2 F(H\^+,H\^+) + \^2(-) F(A\^0,H\^0) + \^2 (-)\ \_[Z]{} &=& [g\^2 s\_W c\_[2W]{} 32 \^2 c\_W]{} F(H\^+,H\^+)\ \_ &=& [g\^2 s\_W\^2 16 \^2]{} F(H\^+,H\^+) . Precision fits with light supersymmetric fermions\[sec:inofits\] ================================================================ We are now in position to compute the effects of light supersymmetric particles and Higgs scalars on precision electroweak observables. In this section we consider the effects of light supersymmetric fermions (charginos and neutralinos). We do this in the light gaugino limit first, then the light higgsino limit. These two limits admit nice analytic results, and are interesting to study in their own right. We then consider results for a general mixed higgsino and gaugino scenario. [*Light gauginos limit*]{} In this section we assume that the $\mu$ term is very heavy along with the squarks and sleptons. Thus, we assume all states in the theory decouple except a light SM-like Higgs boson and light gauginos. We then have pure electroweak bino and winos at the low-energy scale. The gluino can also be light, of course, but it has no effect on the electroweak observables. Thus, all results depend on the unknown $h$ mass and the unknown wino mass. Since the pure bino does not couple to $Z$, $W$, and $\gamma$, it does not contribute to oblique corrections, and so its value with respect to the wino mass is irrelevant to this analysis. The values of the couplings for the light gaugino limiting case are O\^L\_[21]{} = O\^R\_[21]{} = -O\^[L]{}\_[11]{}/c\_W\^2 = -O\^[R]{}\_[11]{}/c\_W\^2 = 1, and all other couplings are either zero or irrelevant. The expressions for the self-energies in this case collapse into a rather convenient form: \_[WW]{} = = = = - [g\^2 8 \^2]{} G(W). \[wino pis\] The function $G(x)$ is defined in the appendix. The $p^2$ argument of $G(x)$, and of other loop functions that we will define later, is not explicitly written for simplicity of notation. We note immediately that $S=T=0$ for this case. Thus, an $S-T$ parameter analysis cannot capture the effects of winos on precision electroweak observables. The non-zero contributions to the precision electroweak observables come from the $U$, $Y$ and $V$ parameters. These parameters can be computed straightforwardly given their definitions and eq. (\[wino pis\]). We use these to obtain a convenient expansion of the observables in powers of $\rwino=M_Z^2/M^2_{2}$: M\_W([GeV]{}) &=& 0.00954 + 0.00157 \^2 + 0.00030 \^3,\ s\^2\_[eff]{} &=& -0.0000549 - 0.0000059\^2 - 0.0000009\^3,\ \_l([MeV]{}) &=& -0.0435- 0.0096\^2 - 0.0022\^3 . We have found numerically that this expansion is quite accurate even when $\rwino$ is near one, provided that $|\mu|$ is large. Note that in the above equations $\Delta M_W >0$, $\Delta s^2_{\rm eff}<0$ and $\Delta \Gamma_l<0$ for the wino corrections. Comparing the SM predictions in Table \[SM\_table\] with the experimental data in eqs. (\[s2\_ave\])-(\[Gammal\_ave\]), the light winos improve $M_W$ and $\Gamma_l$ predictions. To see how light winos can affect these observables, we consider “0.5-$\sigma$ sensitive wino mass” which changes the SM predictions of observables by 0.5-$\sigma$. The current experimental uncertainties for observables are $\delta M_W=0.034$ GeV, $\delta s^2_{\rm eff}=0.00017$ and $\delta \Gamma_l =0.086$ MeV at 1-$\sigma$ level, and hence we can calculate “0.5-$\sigma$ sensitive wino mass” using the above expansions. They are $M_2=77$ GeV from $\Delta M_W=\delta M_W/2$, $M_2=79$ GeV from $\Delta s^2_{\rm eff}=\delta s^2_{\rm eff}/2$ and $M_2=101$ GeV from $\Delta \Gamma_l=\delta \Gamma_l/2$. Therefore winos with $M_2 \sim 100$ GeV can affect the electroweak fit. Note that $\Gamma_l$ is the most sensitive to the light winos and it is improved. ![Contours of $\Delta\chi^2=\chi^2-\chi^2_{SM,min}$ as a function of Higgs mass $M_h$ and the wino mass $M_2$. The region inside (outside) of the solid line produces a better (worse) fit than the best fit point of the Standard Model, which corresponds to the limit of superpartner masses decoupling to infinity and $M_h = 120$ GeV. The best fit point, not taking into account direct searches for the charged wino, is indicated by the point marked $\times$ at $M_h = 141$ GeV, $M_2 = 86$ GeV. This figure was made using $\alpha_s(M_Z)=0.1187$.[]{data-label="wino_fit"}](chi2_wino.eps){width="13.0cm"} In fig. \[wino\_fit\], we show the total $\Delta\chi^2=\chi^2-\chi^2_{SM,min}$ for the electroweak fit as a function of Higgs mass $M_h$ and wino mass $M_2$. As one can see, the $\chi^2$ improves as $M_2$ gets smaller (up to about 90 GeV) in the range of 115 GeV $<M_h<$ 170 GeV. The minimum of the $\chi^2$ is about $0.95$ at $M_h\simeq 140$ GeV and $M_2\simeq 85$ GeV. Note that the minimum of the $\chi^2$ in the SM is about $\chi^2_{SM,min}=1.7$ at $M_h=120$ GeV (see fig. \[SM\_fit\] and Table \[SM\_table\]). Note also that the current wino mass limit is about $90$ GeV when all squarks and higgsinos are heavy [@Heister:2002mn]. [*Light higgsinos limit*]{} Next, we consider a different limit: all gaugino masses are large, but $\mu$ is small. In this case, low-energy charginos and neutralinos are pure higgsinos. All precision electroweak results are functions of the light SM-like Higgs mass and the higgsino mass $\mu$. The light higgsino case corresponds to $O^L_{11} = O^R_{11} = -1/2$, and $O^L_{21} = O^R_{21} = i/2$, $O^{\prime L}_{11} = O^{\prime R}_{11} = s_W^2 - 1/2$, $O^{\prime\prime L}_{11} = O^{\prime\prime L}_{22} = 0$, and $O^{\prime\prime L}_{12} = -O^{\prime\prime L}_{21} = i/2$, and all others are irrelevant. The vector boson self energies reduce to \_[WW]{} = [c\_W\^2 c\_W\^4 + s\_W\^4]{} \_[ZZ]{} = [c\_W s\_W (c\_W\^2 - s\_W\^2)]{} \_[Z]{} = [12 s\_W\^2]{} \_ = - [g\^2 16\^2]{}G(H) Similar to the pure gaugino limit, $S=T=0$ in this limit, and $S-T$ analysis alone cannot capture the effects of light higgsinos on precision electroweak observables. We can also expand the pure higgsino limit analytically as a power series in $\rmu=m^2_Z/\mu^2$: M\_W([GeV]{}) &=& 0.00620+ 0.00094\^2 + 0.00017\^3,\ s\^2\_[eff]{} &=& -0.0000549- 0.0000059\^2 - 0.0000009\^3,\ \_l([MeV]{}) &=& -0.0225- 0.0051\^2 - 0.0012\^3 . The higgsinos also improve $M_W$ and $\Gamma_l$ compared to the SM predictions. Again, we can calculate “0.5-$\sigma$ sensitive higgsino mass”: $\mu=65$ GeV from $\Delta M_W=\delta M_W/2$, $\mu=79$ GeV from $\Delta s^2_{\rm eff}=\delta s^2_{\rm eff}/2$ and $\mu=78$ GeV from $\Delta \Gamma_l=\delta \Gamma_l/2$. (Recall from above that $\delta {\cal O}_i$ is the $1\sigma$ experimental error for observable ${\cal O}_i$.) The total $\Delta\chi^2$ is shown in fig. \[higgsino\_fit\] as a function of $\mu$ and Higgs mass $M_h$. The effect is smaller than what we found in the pure wino case, but the light higgsinos (with $\mu>80$ GeV) also improve the total $\chi^2$ compared to the SM fit. ![Contours of $\Delta\chi^2=\chi^2-\chi^2_{SM,min}$ as a function of Higgs mass $M_h$ and the higgsino mass. The region inside (outside) of the solid line produces a better (worse) fit than the best fit point of the Standard Model, which corresponds to the limit of all superpartners decoupling to infinite mass and $M_h = 120$ GeV. The best fit point, not taking into account direct searches for the charged higgsino, is indicated by the point marked $\times$ at Higgs mass of $147\gev$ and higgsino mass of $73\gev$. This figure was made using $\alpha_s(M_Z)=0.1187$.[]{data-label="higgsino_fit"}](chi2_hino.eps){width="13.0cm"} [*Mixed gauginos and higgsinos*]{} Here we consider the more general case for light charginos and neutralinos. When the $\mu$-term is near in mass to the wino mass term $M_2$, both the higgsino and wino sectors contribute substantially to the oblique corrections. In this case the general mixing matrix angles for the charginos and neutralinos vary over large ranges. Unfortunately, there are no simple analytic equations that capture all the effects succinctly. All results for the mixed case will be numerical results taking into account proper diagonalizations of the mass matrices. Unlike the pure gaugino and the pure higgsino cases, the mixed-case results strongly depend on $\tan\beta$. This is easy to understand since the gaugino/higgsino mass mixing insertions are $\tan\beta$ dependent and not small. Thus, $\tan\beta$ is a crucial parameter to keep track of. Another parameter that we should keep track of is the ratio of the higgsino parameter to the wino parameter, $\mu/M_2$. We will assume that this mixing parameter can be anything along the real line (no complex, CP violating phases). We demonstrate the effect of mixing by plotting contours of shifts in observables in the $\tan\beta$ vs. $\mu/M_2$ plane. We do this in fig. \[d\_obs\_AMSB\] keeping the lightest chargino mass eigenvalue fixed at $M_{\tilde \chi^+_1}=120\gev$. We also assume that $M_1\simeq 3M_2$ according to anomaly mediated supersymmetry [@AMSB], although we have checked that the results depend very mildly on this assumption. Choosing $M_1\simeq M_2$ or $M_1\simeq M_2/2$, for example, generates very similar figures. ![Contours of shifts in electroweak observables in the $\tan\beta$ vs. $\mu/M_2$ plane, with the lightest chargino mass eigenstate fixed at $120\gev$ and $M_1=3M_2$.[]{data-label="d_obs_AMSB"}](dMW_mch120_AMSB.eps "fig:"){width="11.0cm"} ![Contours of shifts in electroweak observables in the $\tan\beta$ vs. $\mu/M_2$ plane, with the lightest chargino mass eigenstate fixed at $120\gev$ and $M_1=3M_2$.[]{data-label="d_obs_AMSB"}](ds2_mch120_AMSB.eps "fig:"){width="11.0cm"} ![Contours of shifts in electroweak observables in the $\tan\beta$ vs. $\mu/M_2$ plane, with the lightest chargino mass eigenstate fixed at $120\gev$ and $M_1=3M_2$.[]{data-label="d_obs_AMSB"}](dGl_mch120_AMSB.eps "fig:"){width="11.0cm"} There are several things to notice in fig. \[d\_obs\_AMSB\]. First, as $\mu/M_2\gg 1$ the precision electroweak analysis asymptotes to that of light wino superpartners. As $\mu/M_2\ll 1$ the precision electroweak analysis asymptotes to that of light higgsinos. In both cases, the $\tan\beta$ dependence disappears and the variation of the corrections to the observables disappears when there is a factor of 10 or higher in the hierarchy of $\mu$ and $M_2$. One finds in fig. \[d\_obs\_AMSB\] a strong $\tan\beta$ dependence when $\mu/M_2\sim 1$, which can induce a correction of either sign for $\Delta M_W$, depending on the value of $\tan\beta$, and only positive (negative) corrections to $s^2_{\rm eff}$ ($\Gamma_l$). The variability in the corrections is large in that region. Having established that the observables change significantly when $\mu \simeq M_2$, we now wish to determine the effect these variations have on $\Delta\chi^2$. To this end we have plotted in fig. \[d\_chi2\_AMSB\] contours of $\Delta \chi^2$ in the plane of $M_{h}$ versus $\mu/M_2$. We have fixed $M_{\tilde \chi^+_1}=120\gev$, and $\tan\beta=2$ in the top graph and $\tan\beta=50$ in the bottom graph. ![$\Delta\chi^2$ contours with $\tan\beta=2,50$ and $M_{\tilde\chi^+_1}=120\gev$ fixed. We have also chosen $M_1=3M_2$, although the contours depend very mildly on this assumption. The results illustrate the general finding that a mixed scenario of light gauginos and higgsinos lead to a better fit to the precision electroweak data. This figure was made using $\alpha_s(M_Z)=0.1187$.[]{data-label="d_chi2_AMSB"}](chi2_tan2.eps "fig:"){width="11.0cm"} ![$\Delta\chi^2$ contours with $\tan\beta=2,50$ and $M_{\tilde\chi^+_1}=120\gev$ fixed. We have also chosen $M_1=3M_2$, although the contours depend very mildly on this assumption. The results illustrate the general finding that a mixed scenario of light gauginos and higgsinos lead to a better fit to the precision electroweak data. This figure was made using $\alpha_s(M_Z)=0.1187$.[]{data-label="d_chi2_AMSB"}](chi2_tan50.eps "fig:"){width="11.0cm"} On the far left of the figure we have the result of the pure higgsino case, where the $\Delta \chi^2$ changes only when the Higgs boson mass changes. The contour lines become parallel to the $x$-axis. On the far right of figure we find the result asymptoting toward the pure gaugino case. Again, $\Delta \chi^2$ changes only due to the Higgs boson mass, and the lines again level horizontally out there. In the center of the contours of fig. \[d\_chi2\_AMSB\] the mixing angles are varying significantly. The variation is causing the changes in the observables that we witnessed in fig. \[d\_obs\_AMSB\], and subsequently affects $\Delta\chi^2$. As we see, the mixing effect reduces $\Delta\chi^2$. Reduction of $\Delta\chi^2$ in the region of parameter space where gauginos and higgsinos are light and heavily mixed is a general result in split supersymmetry. We can understand this result from the graphs. Let us draw our attention to a segment of the graphs at $\mu/M_2\sim 1$. At high $\tan\beta$ the corrections to $s^2_{\rm eff}$ and $\Gamma_l$ are becoming small, whereas the correction to $M_W$ is increasing. The increasing contribution to $M_W$ makes the theory prediction come closer to the experimental prediction, thus reducing the $\Delta\chi^2$. At low $\tan\beta$ the contribution to $M_W$ is negative. Although this goes in the wrong direction, the magnitude is somewhat smaller than in the large $\tan\beta$ case, and more importantly, the contributions to $s^2_{\rm eff}$ are increasingly positive, which goes in the right direction, and the contributions to $\Gamma_l$ are increasingly negative, which also goes in the right direction. Overall, the $\chi^2$ improves. This result is true for all $\tan\beta$ although we have only shown it graphically for $\tan\beta=2$ and $\tan\beta=50$. The improved $\Delta \chi^2$ results also hold similarly for $\mu/M_2\simeq -1$. Therefore, mixed higgsinos and gauginos near the direct experimental limit is the split supersymmetry spectrum most compatible with the precision electroweak data. Let us say a few words about dark matter [@Wells:2003tf; @split; @pheno; @1; @split; @pheno; @2] in relation to our precision electroweak analysis. If we assume that R-parity is conserved, the lightest supersymmetric partner (LSP) will be stable. In the case of a pure wino LSP, the dark matter thermal relic abundance is negligible unless the mass is about $2.3\tev$. Pure higgsino LSP has negligible relic abundance unless its mass is about $1.2\tev$. Higher masses mean overclosure, i.e., cosmological problems. However, these conclusions are applicable for thermal relic abundance calculations. Non-thermal sources, such as gravitino or moduli decay in the early universe can transform what looked to be a negligibly abundant LSP into a good dark matter candidate. Thus, the light winos, light higgsinos and light mixed states are probably not good thermal dark matter candidates, but could be good dark matter candidates when all non-thermal sources are taken into account. If $M_1<M_2$ and the LSP has significant bino fraction, one expects either the LSP to annihilate efficiently through a Higgs boson pole or $\mu$ should be somewhat near $M_1$ to mix with the bino for acceptable dark matter (see Pierce in [@split; @pheno; @1]). This is good for dark matter and good for precision electroweak fits. [*Indirect effects from heavy scalars*]{} In the above analysis we have assumed that the neutralino and chargino matrices used to obtain the mass eigenvalues and mixing angles are valid. However, the $g$ and $g'$ that are in those matrices are gauge couplings because of supersymmetry invariance. In broken supersymmetry those couplings deviate from the gauge couplings, which has been emphasized within the context of split supersymmetry [@Arkani-Hamed:2004fb; @Giudice:2004tc]. Unfortunately, these “gauge-ino couplings” cannot be directly measured by experiment since the scalar masses are likely to not be accessible. However, they can have a subtle effect on the precision electroweak observables. To demonstrate, we introduce the following couplings &a\_u = g\_u/(g),  &a\_d=g\_d/(g)\ &a\_u’=g’\_u/(g’),  &a’\_d=g’\_d/(g’) where $\tilde g_{u,d}$ and $\tilde g'_{u,d}$ are defined in [@Giudice:2004tc]. The $a$-variables are defined such that the usual values taken in the MSSM are $a_u=a_d=a'_u=a'_d=1$. The neutralino and chargino mass matrices at the weak scale in this parameterization are M\_[\^0]{}=( [cccc]{} M\_1 & 0 & -a’\_dg’v\_d/ & a’\_ug’v\_u/\ 0 & M\_2 & a\_dgv\_d/ & -a\_ugv\_u/\ -a’\_dg’v\_d/ & a\_dgv\_d/ & 0 & -\ a’\_ug’v\_u/ & -a\_ugv\_u/ & -& 0 ), M\_[\^+]{} =( [cc]{} M\_2 & a\_u g v\_u\ a\_d g v\_d & ) We have computed the numerical values of the $a$-variables under different assumptions for $\tan\beta$ and the scale of the scalar superpartners. (We have reproduced fig. 5 of [@Giudice:2004tc] and agree with their results.) We then compute the corrections to the precision electroweak observables for various values of the scalar sector mass, which for simplicity we assume is a common scale $M_s$. For PeV-scale sfermion masses none of the $a$-variables deviate from 1 by more than 10% for any value of $\tan\beta$. If the scalar masses are near the GUT scale, the effects can be more sizable and deviations from $a_i=1$ can approach $30\%$ at very high $\tan\beta\sim 50$, but are less significant for lower $\tan\beta$. We note that the effects of various $M_s$ values can cause the magnitude of the oblique correction to change by as much as $30\%$ at special points such as when $M_2\simeq |\mu| \simeq 100\gev$ and $M_s\simeq 10^{16}\gev$. This is especially true for $s^2_{\rm eff}$, which is very sensitive to the chargino and neutralino mixing angles. However, when $\mu$ deviates from $M_2$ one finds the effects on precision electroweak corrections to be much smaller. In all cases, the corrections to the oblique corrections are not discernible by current experiment. This is why we ignored these $a'_{u,d}$ and $a_{u,d}$ parameters for much of our analysis. In the future, these small deviations might be discernible at dedicated next-generation $Z$ factories [@GigaZ]. Conclusion ========== We have found above that the precision electroweak corrections from light charginos and neutralinos generally improve the overall $\chi^2$ fit to the data. This is true in the pure gaugino limit and in the pure higgsino limit. We emphasize that in both of these cases we have merely added one new parameter to the theory and the fit gets better. It did not have to go this direction. In the mixed higgsino and gaugino case, there are more free parameters introduced, and it turns out that the fit to the data gets even better. The case of higgsinos and gauginos both near the direct experimental limit ($\sim 100\gev$) is the split supersymmetry spectrum most compatible with the data. These conclusions are made within the full supersymmetric framework, where we have assumed the scalar superpartner masses are too heavy to have any noticeable effect on the precision electroweak observables. Even for scalar superpartners decoupled up to the grand unification scale, the light Higgs mass is still less than about $170\gev$ [@Giudice:2004tc]. Our best fit results are all compatible with this low range of Higgs mass, and the global fits for the various cases we discussed have global minima with light Higgs boson mass. The global fit to the data approaches that of the SM fit when the gaugino and higgsino masses are dialed to larger values. In that case, the global minimum of the $\chi^2$ fit remains in the low Higgs mass region, but the $\chi^2$ value at that SM minimum is increased somewhat compared to the light gauginos/higgsino case. Throughout we have assumed that the squarks, sleptons and heavy Higgs bosons have no effect on precision electroweak analysis. Our computations demonstrate that we expect less than a $10^{-4}$ effect on all relative corrections of the observables, $\Delta {\cal O}_i/{\cal O}_i$, if the scalar masses are above $1\tev$. In split supersymmetry, we expect the scalar masses to be significantly beyond the TeV scale, justifying our neglect of the scalar masses. As an aside, we have found that lowering the sfermion masses usually does not improve the quality of fit compared to the decoupling limit, for fixed values of other quantities. Although not statistically significant, the precision electroweak data may have a mild preference for decoupled scalars and light gauginos/higgsinos over any other form of supersymmetry breaking patterns. In short, the precision electroweak data is compatible with split supersymmetry spectrum for all values of gaugino and higgsino masses above direct collider limits. Near the direct limits, the overall fit improves by nearly a full unit in $\Delta\chi^2$. A priori, fitting to precision electroweak observables did not have to be favorable to split supersymmetry, and could have been incompatible for light gauginos and/or higgsinos. As it is, the improved fits are mildly encouraging for the scenario. [**Note Added:**]{} Following the appearance of the present work, a recent interesting paper [@Marandella:2005wc] suggests that high-energy LEP2 data can contribute substantively to precision electroweak analysis of light superpartners. Although we have not independently confirmed this analysis, we want to bring it to the reader’s attention. If correct, the conclusion from doing a precision electroweak analysis with that super-set of observables would eliminate the mild preference for light charginos and neutralinos. As noted in ref. [@Marandella:2005wc], the analysis might change again after all $e^+ e^- \rightarrow e^+e^-$ data above the $Z$ pole becomes available. Appendix: Useful functions {#appendixA .unnumbered} ========================== The kinematic loop-integral functions needed above are given in terms of B(x,y) &=& -\_0\^1 dt where $s=p^2$ is the external momentum invariant in a ($+$$-$$-$$-$) metric, and X (X/Q\^2) where $Q$ is the renormalization scale. In the text, the arguments $(x,y)$ are particle names and should be interpreted to substitute $x\to m^2_x$ and $y\to m^2_y$ into the equations. The other functions used in the text are defined in terms of $B(x,y)$: H(x,y) &=& \[2s-x-y-(x-y)\^2/s\] B(x,y)/3\ && + \[2 x x + 2 y y - 2s/3 + (y-x)(x x - x - y y +y)/s\]/3 ,\ F(x,y) &=& H(x,y) + (x+y-s) B(x,y),\ G(x) &=& H(x,x) + 2 x B(x,x) . These can be expanded in powers of $s$ according to: B(x,y) &=& b\_0(x,y) + s b\_1(x,y) + s\^2 b\_2(x,y) + …\ H(x,y) &=& h\_0(x,y) + s h\_1(x,y) + s\^2 h\_2(x,y) + …\ F(x,y) &=& f\_0(x,y) + s f\_1 (x,y) + s\^2 f\_2(x,y) + …\ G(x) &=& s g\_1 (x) + s\^2 g\_2(x) + …with coefficients that follow from b\_0(x,y) &=& \[x - x x -y + yy \]/(x-y) ,\ b\_1(x,y) &=& \[2 x y (y/x) + x\^2 - y\^2\]/2(x-y)\^3 ,\ b\_2(x,y) &=& \[6 x y (x+y) (y/x) + x\^3 - y\^3 + 9 x\^2 y - 9 x y\^2\]/6 (x-y)\^5 ,\ b\_3(x,y) &=& \[12 x y (x\^2 + 3 x y + y\^2) (y/x) + x\^4 - y\^4 + 28 x\^3 y - 28 x y\^3\]/12 (x-y)\^7,\ b\_4 (x,y) &=& \[60 x y (x\^3 + 6 x\^2 y + 6 x y\^2 + y\^3) (y/x) + 3 x\^5 - 3 y\^5 + 175 x y (x\^3 - y\^3)\ && + 300 x\^2 y\^2 (x-y)\]/60(x-y)\^9 and && b\_0(x,x) = -(x),b\_1(x,x) = 1/6x,b\_2(x,x) = 1/60x\^2,\ && b\_3(x,x) = 1/420 x\^3, b\_4(x,x) = 1/2520 x\^4 . The expansions converge provided that $\sqrt{s} < \sqrt{x}+ \sqrt{y}$ (in other words, below the threshold branch cut). 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--- abstract: 'The evaporation of black holes into apparently thermal radiation poses a serious conundrum for theoretical physics: at face value, it appears that in the presence of a black hole quantum evolution is non-unitary and destroys information. This information loss paradox has its seed in the presence of a horizon causally separating the interior and asymptotic regions in a black hole spacetime. A quantitative resolution of the paradox could take several forms: (a) a precise argument that the underlying quantum theory is unitary, and that information loss must be an artifact of approximations in the derivation of black hole evaporation, (b) an explicit construction showing how information can be recovered by the asymptotic observer, (c) a demonstration that the causal disconnection of the black hole interior from infinity is an artifact of the semiclassical approximation. This review summarizes progress on all these fronts.' --- [**Quantitative approaches to**]{}\ [**information recovery from black holes**]{} Vijay Balasubramanian$^{a,}$[^1], Bart[ł]{}omiej Czech$^{b,}$[^2] $^a$*David Rittenhouse Laboratory, University of Pennsylvania,* *209 South 33$^{\rm rd}$ Street, Philadelphia, PA 19104, USA* $^b$*Department of Physics and Astronomy, University of British Columbia,* *6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada* Introduction ============ In 1974 Hawking analyzed quantum mechanical fields propagating in the background of matter collapsing to form a black hole [@Hawking:1974rv; @Hawking:1974sw]. He found that the incipient horizon affects field modes by mixing positive and negative frequencies in such a manner that at late times thermal radiation emerges from the black hole. At least in the semiclassical approximation in which these calculations are done, the radiation at each time is in an exactly thermal density matrix, and leads to slow evaporation of the black hole. Eventually, the black hole disappears, its energy dissipated to infinity by the radiation. This leads to a puzzle. We could have formed the black hole initially from a pure state of matter. But the final state created by radiation in a thermal density matrix appears to be oblivious to all details of the initial state, except for its mass, angular momentum, and global charges. Thus it is not possible for the asymptotic observer to determine the state of the matter that originally formed the black hole. If this conclusion is correct, then quantum mechanics in the presence of a black hole is non-unitary [@Hawking:1976ra], threatening its consistency as a physical theory. There is a related puzzle arising from the large entropy associated to a black hole. Consider, for example, an eternal black hole that exists for all of time in equilibrium with a bath of radiation [@Hawking:1982dh] or a very large black hole that is evaporating very slowly. Such a black hole is thought to have an entropy proportional to its horizon area: $S = A/4G_N \hbar$ [@Bekenstein:1973ur]. In any consistent quantum theory of gravity we expect this entropy to be explained by the existence of $e^S$ microstates that are commensurate with the macroscopic parameters of the black hole (mass, angular momentum, and global charges). Microcanonically, the black hole can be in any one of these microstates. However, the classical observer outside the black hole has no way of determining this microstate, and thus should interact with the black hole as if it were in a density matrix over the underlying states. Indeed, correlation functions computed by an asymptotic observer in the presence of a classical black hole horizon decay at late times as they would in a thermal background [@KeskiVakkuri:1998nw]. Is there any method, even in principle, of “detecting a black hole microstate”, or is that information always lost to the asymptotic observer? The most mysterious property of a black hole, and the one that gives it its name, is the fact that the metric can be smoothly continued past the horizon giving a geometric region that extends all the way to a spacetime singularity. The interior region is causally disconnected from infinity – observers following inertial trajectories can fall in unmolested, but cannot make their way back out because all causal trajectories in the interior point towards the singularity. Thus the doings of the infalling observer, and other events that occur inside the horizon, seem to be inaccessible to the outside. Tracing over the interior degrees of freedom again leads the outside observer to lose information about the quantum state of the universe. There is a subtle point that the initial conditions for all infalling modes can be set outside the horizon, and hence one could imagine that all the information is somehow also represented outside the black hole. To make matters even more confusing, in the time coordinates of an observer who remains outside the black hole, nothing ever falls behind the horizon – all infalling matter gets redshifted and piles up at the horizon. While this raises the question of whether the interior “exists” in a meaningful sense for external observers, naïvely tracing over the interior will cause the external observer to represent the black hole as a density matrix. The preceding paragraphs relate the information loss problem to three questions: (a) what are the properties of the vacuum state in the presence of a horizon? (b) can an asymptotic observer identify a black hole microstate? (c) can the fate of infalling observers be detected at infinity? These questions highlight an important conceptual point, which suggests an avenue for reconciling black holes with unitarity. If the area of the black hole horizon is related to its entropy, how should we think of a black hole that is in a pure microstate, a single member of the underlying statistical ensemble? After all, a pure state has no microscropic entropy except in a coarse grained description. Thus, the relation between horizon area and entropy mandates that a universe in a pure black hole microstate must not have a wavefunction localized on a geometry with a finite area horizon. The latter might emerge after some coarse-graining, but the microscopic picture must be fundamentally different. Perhaps it does not even make sense to talk about the interior of a black hole except, at best, for some kinds of coarse grained observers. In this picture, the causally disconnected black hole interior would be an artifact of the semiclassical approximation and there would simply be no room for information loss. This might sound like an unacceptably drastic departure from the conventional perspective. On the other hand, [*all*]{} approaches to avoiding information loss have to invoke some exotic escape from naïve semiclassical reasoning. For example, it has been argued [@Lowe:1995pu] that interactions between the enormously blueshifted infalling quanta and the enormously redshifted quanta climbing out of a black hole involve very high center of mass energies, which suggests that low energy effective field theory cannot be used near the horizon. This claim has engendered two approaches to the black hole information problem. The idea of complementarity [@Susskind:1993if] states that since the temporal redshift at the horizon prevents asymptotic observers from witnessing anything fall into a black hole, asymptotic and infalling observers give equivalent descriptions of the same physics. Meanwhile, Maldacena proposed that quantum gravity enjoys a dual “holographic” description in terms of degrees of freedom of a lower dimensional field theory [@Maldacena:1997re], which is manifestly unitary. The approach championed in this paper makes extensive use of holographic techniques. These examples illustrate that solving the information problem requires one to be skeptical about semiclassical intuitions. Perhaps the least radical detour from the traditional picture would be to restore unitarity with subtle correlations in Hawking radiation, but as we discuss in the text, it is difficult to produce the requisite correlations sufficiently quickly. This review article emphasizes that the restoration of unitarity involves resolving corrections to the classical results which are of magnitude $\mathcal{O}(e^{-S})= \mathcal{O}(e^{-A/4G_N \hbar})$. In terms of scaling with $\hbar$ and $G_N$ the corrections are evidently not perturbative and become immeasurably small as $\hbar \to 0$, explaining the loss of information in the semiclassical limit. We will further argue that the scale $\mathcal{O}(e^{-S})$ arises from [*statistics*]{}, not dynamics, and that quantum gravity mainly enters the problem in determining the non-perturbatively tiny gap in the spectrum that is necessary to account for the enormous entropy of black holes. In making these arguments, it will frequently be convenient to talk about extremal black holes, and especially those where the horizon is of zero size and coincides with the singularity. These objects are not the conventional black holes that result from gravitational collapse, but they provide a useful laboratory. They are stable, because their temperature vanishes, and they often have a substantial degeneracy, though not generally enough to result in a finite horizon. External probes of these black holes shed light on the conceptual foundations of information recovery from black holes. It will also be convenient to consider black holes in Anti-de Sitter (AdS) space, because in this case we can expect to describe the physics of spacetime using the dual conformal field theory [@Maldacena:1997re; @Witten:1998qj]. In these settings, there is simply no room for information loss, because the dual field theory is manifestly unitary. This allows us to concentrate on a more precise question: what is the mechanism for the recovery of information from black holes? General remarks {#preliminaries} =============== Virtues and features of a laboratory in AdS space {#bhads} ------------------------------------------------- Much of the progress in thinking about black holes and information has been in the context of asymptotically AdS spacetimes. Below we will describe such black holes and explain why they are such a useful laboratory. Any resolution of the information paradox that works in AdS backgrounds should in principle carry over to all black holes. This is because Hawking’s argument for information loss arose from the structure of local quantum field theory in the vicinity of the horizon and does not depend in an essential way on the asymptotics of the spacetime. On the other hand, the global properties of AdS make it a particularly advantageous laboratory for studying the information paradox. This section reviews the many virtues of AdS: it is convenient for calculations and it affords a sharper definition of the information paradox, in which its structure (as well as its resolution) become more apparent. These advantages are traced to the AdS/CFT duality [@Maldacena:1997re; @Witten:1998qj], which posits that gravity in a spacetime asymptotic to AdS space enjoys an equivalent formulation in terms of a conformal field theory on the boundary of AdS. For a review of AdS/CFT, the reader is referred to [@Aharony:1999ti]. Black holes in AdS fall into two classes, small and large [@Hawking:1982dh]. Small black holes have horizon radii, which are small compared to the AdS curvature radius and as a result of that their physics is qualitatively similar to black holes in flat space. Large AdS black holes, on the other hand, are qualitatively different. First, their specific heat is positive, so they are stable minima in the canonical ensemble. Second, because AdS acts effectively as a box, large black holes can be *eternal*, that is they can attain thermal equilibrium with their own, reflected Hawking radiation. These features are convenient for studying the information problem. Consider dropping a quantum into an eternal black hole. There are two sharply defined scenarios for the subsequent evolution of the system: 1. Deviations from the thermal description of the black hole decay to zero and information is lost. 2. Deviations from the thermal description initially decay as the black hole churns the fallen information, but eventually they hover around some small finite value and information is preserved. The facts that large AdS black holes are stationary and well-described in the canonical ensemble are essential for defining and contrasting the two scenarios. In fact, the AdS/CFT correspondence [@Maldacena:1997re] automatically eliminates the non-unitary scenario, because the dual conformal field theory is manifestly unitary. (The only subtlety is that it may be necessary to include separate components of the conformal field theory on causally disconnected segments of the AdS boundary.) This explains away the information paradox in its basic form, but it does not make clear the precise way in which Hawking’s original argument fails. Before declaring victory over the information paradox, one must first provide a satisfactory account of where and how Hawking’s derivation breaks down. This is the essence of the AdS/CFT-motivated version of the information problem. Before launching an attack on the information paradox in AdS, it is useful to contrast the ingredients of holographic duality with those of Hawking’s argument. The AdS/CFT correspondence, which from the viewpoint of string theory is a manifestation of open-closed string duality, arises when we consider the near-horizon limit of a large stack of parallel D-branes. In the low energy regime the asymptotic and near-horizon regions decouple, and one identifies the low energy dynamics of the D-branes with string theory on the near-horizon geometry, which always contains an AdS factor. The duality is of the weak-strong type: weakly coupled gravity in AdS maps into strongly coupled world-volume field theory. Furthermore, the duality interchanges the IR with the UV: local disturbances in the field theory correspond to gravity deformations on a very large scale and near the boundary of spacetime [@Susskind:1998dq]. Thus, though the dual CFT provides in principle a complete definition of quantum gravity of AdS, its manifest unitarity comes at the expense of having a simple description of local physics in spacetime. In particular, it is difficult to use the CFT description to track how information falls into and leaks out of a black hole. On the other hand, Hawking’s derivation uses methods of local effective field theory, with local quantum fields propagating on a fixed classical spacetime containing a horizon. Naïvely, this argument should be valid so long as the Schwarzschild radius of the black hole is large in Planck units, because in that regime the calculation is safe from effects of quantum fluctuations. At least some of the approaches presented in this review challenge the assumption that the concept of a fixed, classical black hole spacetime remains valid in the vicinity of the horizon. The information paradox depends on another important problem in quantum gravity – that of explaining the entropy of a black hole. Information can only be preserved if the internal state (microstate) of a black hole changes uniquely after any specific external state falls in. If one could detect differences between internal states, there would be no information paradox. Conversely, to resolve the information paradox is tantamount to demonstrating that there exist microstates, which are in principle distinguishable, though not necessarily to semiclassical observers. In addition, there should be some microscopic sense in which each microstate is horizonless because a pure state has zero entropy. From this perspective, the causal disconnection of the black hole interior, which results in the thermality of black hole radiation, should be demonstrably a semiclassical arifact. The AdS/CFT correspondence provides a framework in which these words may be translated into actual computations. Different states in asymptotically AdS spacetime map into different states in the dual field theory: empty AdS corresponds to its ground state, AdS with some particles corresponds to a low energy excited state, and a black hole corresponds to a thermal state. If the arguments of the previous paragraph are correct, then the black hole geometry with a horizon could be seen to arise as an effective, thermodynamic description of many (microscopically horizonless) microstates. In gravity it is not clear how to define such microstates and how to enact a thermodynamic averaging over them, but in field theory this is natural. The dual CFT contains a notion of an inner product on its Hilbert space, so one can work with an orthonormal basis of states. This basis is discrete, because the curvature of AdS breaks translational invariance and produces a gap in the spectrum. We shall see explicit examples of such orthonormal, discrete bases in Sec. \[llmetc\]. Furthermore, the CFT is a *background independent* description of gravity in the interior of $AdS_d$: it is defined over the conformal boundary of the spacetime, $\mathbb{R}_t \times S^{d-2}$, where $\mathbb{R}_t$ parameterizes time. (The compactness of $S^{d-2}$ reflects the discrete nature of gravity states.) Importantly, the interior geometry does not enter the definition of the dual CFT and although the geometry of the boundary does, this is not an impediment because a quantum fluctuation altering the boundary would be suppressed by an infinite action. As a result, it becomes meaningful to compare responses to probes in various microstates and to average over them, something which would be difficult to define in gravity alone. In short, with holographic duality we gain access to the toolkit of statistical mechanics. Consider a large black hole in AdS, which is characterized by some set of commuting charges, including a mass $M$. In the dual field theory this black hole is represented in the canonical ensemble by a thermal density matrix: $$\rho_\beta = ({\rm Tr} \exp(-\beta H))^{-1} \exp(-\beta H)$$ The response of the black hole to a probe is computed in the field theory as a thermal correlator $$\label{thermcorr} \langle \mathcal{P} \rangle_\beta = {\rm Tr}\, \rho_\beta\mathcal{P} \,,$$ where the operator $\mathcal{P}$ is dual to the gravity probe according to the standard AdS/CFT dictionary. An intuitive example of $\mathcal{P}$ could be $$\mathcal{P} = T_{\sigma\rho}(t) \, T^\dagger_{\mu\nu}(0)\,,$$ with $T_{\mu\nu}$ being the stress tensor, which emulates dropping a graviton from near the boundary of AdS at time 0 and detecting its return after a time $t$. However, we do not wish to limit attention to probes of the type $\mathcal{P} = \mathcal{P}_2(t) \mathcal{P}^\dagger_1(0)$, so the operator $\mathcal{P}$ should be considered general. Consider instead the microcanonical ensemble. From this perspective, the thermal density matrix (and hence the spacetime description as a conventional black hole) should simply be an effective, slightly coarse-grained description of most of the microstates. In the dual field theory a microstate $i$ is created by acting on the vacuum with a heavy operator $\mathcal{O}^\dagger_i$. A natural object of interest is an analogue of (\[thermcorr\]), evaluated in a microstate: $$\label{statecorr} \langle \mathcal{P} \rangle_i = \langle 0 | \mathcal{O}_i \,\mathcal{P}\, \mathcal{O}_i^\dagger | 0 \rangle$$ The key question is how (\[statecorr\]) varies between microstates. We will argue that the standard properties of black holes – the existence of a horizon and *apparent* loss of information – are artifacts of an effective, semiclassical treatment, which averages over $e^S$ black hole microstates. The individual microstates are states in quantum gravity and may not have classical descriptions in terms of a metric. In the exceptional situation that a microstate has a description in classical gravity, the metric is necessarily horizonless because the entropy of one microstate is zero. On the other hand coarse-grained observables should be unable to tell the microstates apart and thus will effectively see a substantial entropy. In this way, a horizon will be seen as an artifact of our (semiclassical) inability to distinguish gravitational microstates. To verify this picture, our strategy will be to compare expressions (\[thermcorr\]) with (\[statecorr\]). We expect that the thermal correlator (\[thermcorr\]), which corresponds to the semiclassical black hole, is an ensemble average of the correlators (\[statecorr\]) evaluated in individual microstates[^3]: $$\label{bhisaverage} \langle \mathcal{P} \rangle_\beta = \frac{\sum_i e^{-\beta E_i} \langle \mathcal{P} \rangle_i}{\sum_i e^{-\beta E_i}}$$ The key in preserving information is that microstates differ from one another and their plurality has the capacity of preserving information. The differences among microstates, which encode the information fallen into a black hole, are quantified by the variance of the correlators, $\sigma^2(\langle\mathcal{P}\rangle_i)$. A properly normalized quantity is the ratio of the standard deviation in $\langle\mathcal{P}\rangle_i$ to the mean, ${\sigma(\langle \mathcal{P} \rangle_i)}/{\langle \mathcal{P} \rangle_\beta}$. When this is of order unity, information is preserved and readily available and conversely, when it vanishes, information is lost. Based on the argument of the previous subsection, we expect that: $$\label{infloss} \lim_{\hbar \rightarrow \infty} \frac{\sigma(\langle \mathcal{P} \rangle_i)}{\langle \mathcal{P} \rangle_\beta} = 0$$ This is the statement that information is lost in the semiclassical limit. A significant part of this review is devoted to studying when (\[infloss\]) holds. It is useful to spell out the ingredients which are necessary for the proposed resolution of the information paradox: 1. There must exist a set of quantum gravity states (microstates), which are responsible for the entropy of the black hole. When a CFT dual is available, these are best defined and studied in the dual field theory. 2. The behavior of the black hole must be recovered as an average over the ensemble of microstates, as in eq. (\[bhisaverage\]). 3. The microstates must be capable of encoding information fallen into a black hole: $\sigma^2(\langle \mathcal{P} \rangle_i) \neq 0$. 4. Information must be lost in the semiclassical limit, so we may speak of the black hole as a universal description of all microstates as they appear to semiclassical observers. This is the content of eq. (\[infloss\]). The scale $\exp{(-S)}$: distinguishing basis microstates {#eSbasis} -------------------------------------------------------- Point 3 above required that information be preserved and encoded in the different responses of individual microstates to probes. We now present two independent arguments, which show that in order to take advantage of the non-zero variance $\sigma^2(\langle \mathcal{P} \rangle_i) \neq 0$ to recover information from black holes, one must be able to perform measurements with a resolution of order $\exp{(-S)} = \exp({-A/4 G_N \hbar})$. This vanishes very rapidly in the limit $\hbar \rightarrow 0$. Consequently, in the strict semiclassical limit an observer can recover information from a black hole only if she can muster infinite precision or patience, which is in agreement with Point 4. #### Gravity argument: Consider an orthonormal basis of microstates, which are eigenstates of a complete set of commuting observables. In the absence of symmetries other than time translation invariance, this becomes a basis of non-degenerate Hamiltonian eigenstates. In order to account for a degeneracy of $\mathcal{O}(e^{S})$ the energy gaps in the spectrum must be $\mathcal{O}(e^{-S})$. Evidently, all one has to do in order to identify a basis microstate is to measure its energy (and other conserved charges). Now, in any theory of gravity the mass (total energy) can be measured at infinity. Thus an asymptotic observer will be able to distinguish microstates by measuring their mass and need not experience information loss! However, this requires access to energy resolutions of the order $\Delta E \sim \exp{(-S)}$. To see this, note that an apparatus with a resolution $\Delta E$ interacts with $$\frac{d\,\Omega(E)}{dE} \Delta E \approx \frac{d\,e^{S}}{dE} \Delta E = e^S \frac{dS}{dE} \Delta E$$ states [@Balasubramanian:2006iw] and thereby effectively interacting with a system which has entropy: $$\mathcal{S} = S + \log \frac{dS}{dE} + \log{\Delta E}$$ Up to logarithmic corrections, this is equal to the Bekenstein-Hawking entropy of the underlying black hole *unless* $\Delta E$ can be arranged to fall exponentially with $S$. The quantity $\mathcal{S}$ will only vanish, signaling a detection of an individual microstate, if $$\Delta E \sim \exp{(-S)}\,.$$ Such a measurement would necessarily extend over a time scale exceeding the Poincaré recurrence time: $$\Delta T \gtrsim (\Delta E)^{-1} \sim \exp S$$ A more complete version of this argument, which applies to quantum superposition microstates, is given in [@Balasubramanian:2006iw]. Thus, while for a generic black hole in AdS space all information about the microstates is present at infinity in the mass of the system thereby avoiding any fundamental information loss, a semiclassical observer with finite energy resolution simply cannot access this information. #### CFT argument: Suppose you drop something into a thermal black hole and wait for a time $\Delta T$ before attempting to recover it. In the dual CFT, the thermal black hole is described as a density matrix over the accessible microstates. However, the probe is in a pure state and we can ask what it would take to distinguish the small deviations from a thermal density matrix that must be present at late times. In CFT, the result of the experiment we are describing is quantified by a thermal expectation value: $$\label{thermunit} F(\Delta T) \equiv \langle \mathcal{P}(\Delta T) \mathcal{P}^\dagger(0) \rangle_\beta$$ Information is lost if $F(\Delta T)$ vanishes at late times. Thus, the scale at which information is conserved can be read off from the late time behavior of (\[thermunit\]). By general arguments in statistical physics, the function $F(\Delta T)$ initially decays as the contributions of different ensemble states decohere, but eventually it begins to hover around some limiting non-zero value. This happens when the phases in $$\label{ft} F(\Delta T) = \frac{\sum_{ij} e^{-\beta E_i} \langle i | \mathcal{P}(\Delta T) | j \rangle \langle j| \mathcal{P}^\dagger(0) | i \rangle}{\sum_i e^{-\beta E_i}} = \frac{\sum_{ij}e^{-\beta E_i} |\langle j | \mathcal{P}^\dagger | i\rangle|^2 e^{-i(E_j - E_i)\Delta T}}{\sum_i e^{-\beta E_i}}$$ become randomized, that is after the interval $\Delta T$ exceeds the reciprocal of the minimal level spacing $$\label{timescale} \Delta T \gtrsim {\Delta E}/{e^S}\,,$$ where $\Delta E$ denotes the overall energy spread among the microstates. Assuming that $\Delta E \ll \beta^{-1}$ so all the Boltzmann factors are comparable, and further, assuming that the matrix elements of $\mathcal{P}$ are of order unity, at the late times satisfying (\[timescale\]) $F(t)$ reduces to an average of $e^{2S}$ random phases, viz. eq. (\[ft\]). We conclude that information is preserved on scales $$\label{esscale} \lim_{\Delta T \rightarrow \infty} |F(\Delta T)| \approx e^{-S}\,.$$ Thus, again, unitarity of the theory is preserved in corrections of $\mathcal{O}(e^{-S})$ which cannot be accessed by semiclassical observers. The point that restoration of unitarity arises in AdS/CFT at $\mathcal{O}(e^{-S})$ was emphasized in [@Maldacena:2001kr] via consideration of the contributions of different saddlepoints of the Euclidean action (see further discussion in Sec. \[addsaddle\]). The scale $\exp{(-S)}$: distinguishing superposition microstates {#eSsup} ---------------------------------------------------------------- The arguments of Sec. \[eSbasis\] explain why identifying a black hole microstate (decoding information) can only happen by measurement on scales of $\mathcal{O}(\exp{(-S)})$. However, both arguments implicitly assumed that the black hole was in an ensemble of Hamiltonian eigenstates. In particular, the gravity argument posits that an exact measurement of energy at infinity singles out a microstate while on the gauge theory side, eq. (\[ft\]) computes the magnitude of the unitarity-preserving signal from the decoherence behavior of a family of Hamiltonian eigenstates. But there is no reason to assume that a black hole must always be found in a Hamiltonian eigenstate. We shall now see that as a result of considering black hole superposition microstates, information recovery is made more difficult by an additional factor of $\exp{(-S)}$. Consider first that a general microstate of spacetime could have a wavefunction with support on regions of configuration space with many different geometries and topologies, or even without any geometric interpretation all. Thus in general to talk about about the microstate of a black hole spacetime we have to take a quantum cosmological perspective and think in terms of a wavefunction of the universe with appropriate boundary asymptotic boundary conditions. It is only sensible to speak of a semiclassical geometry when the wavefunction is sharply peaked [@Balasubramanian:2007zt] on an appropriate set of configurations. Let us imagine a spacetime which is in a generic superposition of microstates which have the quantum numbers of a black hole. This superposition is to be interpreted in the sense of quantum cosmology – the universe is repeatedly prepared in an identical microstate, which is then repeatedly probed by operators $\mathcal{P}$. A superposition wavefunction is then experimentally characterized by the expected outcome of probe experiments: $$\label{ppsi} \langle \mathcal{P} \rangle_\psi = \langle \psi | \mathcal{P} | \psi \rangle\,,$$ where $|\psi \rangle$ is the microstate wavefuction. Thus, in quantum cosmological settings, the usefulness of a probe for recovering information is properly characterized by the variance in the expectation values (\[ppsi\]), that is $\sigma^2 (\langle \mathcal{P}\rangle_\psi)$. It turns out that the variance in expectation values computed over all superposition states is suppressed relative to the variance over the Hamiltonian eigenbasis [@Balasubramanian:2007qv], and that the suppression factor is again $\exp{(-S)}$: $$\label{2sigmas} \sigma^2 (\langle \mathcal{P}\rangle_\psi) \approx \frac{\sigma^2 (\langle \mathcal{P}\rangle_i)}{e^S}$$ It is easiest to see this in the case, where $\mathcal{P}$ commutes with the Hamiltonian and where the microcanonical ensemble techniques apply ($\Delta E \ll \beta^{-1}$). Expand the superposition state $|\psi\rangle$ in the Hamiltonian eigenbasis $$|\psi\rangle = \sum_i^{\exp{S}} c_i |i\rangle$$ and write down the variance in expectation values as the integral over the space of wavefunctions: $$\label{sigmapsi} \sigma^2 (\langle \mathcal{P}\rangle_\psi) = \frac{\int \mathcal{D}\Psi \langle \mathcal{P}\rangle_\psi^2}{\int \mathcal{D}\Psi} - \left( \frac{\int \mathcal{D}\Psi \langle \mathcal{P}\rangle_\psi}{\int \mathcal{D}\Psi} \right)^2 = \sum_{ij} \langle \mathcal{P} \rangle_i \langle \mathcal{P} \rangle_j \frac{\int \mathcal{D}\Psi |c_i|^2 |c_j|^2}{\int \mathcal{D}\Psi} - \left( \sum_{i} \langle \mathcal{P} \rangle_i \frac{\int \mathcal{D}\Psi |c_i|^2}{\int \mathcal{D}\Psi} \right)^2$$ The two integrals on the right hand side are independent of the detailed dynamics and depend only on the dimensionality of the Hilbert space, which is by definition $\exp{S}$: $$\begin{aligned} \frac{\int \mathcal{D}\Psi |c_i|^2}{\int \mathcal{D}\Psi} & = & \frac{1}{e^S} \\ \frac{\int \mathcal{D}\Psi |c_i|^2 |c_j|^2}{\int \mathcal{D}\Psi} & = & \frac{\delta_{ij}+1}{e^S(e^{S}+1)}\end{aligned}$$ Plugging these into eq. (\[sigmapsi\]), we obtain $$\sigma^2 (\langle \mathcal{P}\rangle_\psi) = \frac{e^S-1}{e^{2S}(e^S+1)}\sum_i \langle \mathcal{P} \rangle_i^2 - \frac{2}{e^{2S}(e^S+1)} \sum_{i<j} \langle \mathcal{P} \rangle_i \langle \mathcal{P} \rangle_j\,,$$ which confirms eq. (\[2sigmas\]) since the variance in Hamiltonian eigenstates is simply: $$\sigma^2 (\langle \mathcal{P}\rangle_i) = \frac{1}{e^S}\sum_i \langle \mathcal{P} \rangle_i^2 - \left( \frac{1}{e^S} \sum_i \langle \mathcal{P} \rangle_i \right)^2 = \frac{e^S-1}{e^{2S}}\sum_i \langle \mathcal{P} \rangle_i^2 - \frac{2}{e^{2S}} \sum_{i<j} \langle \mathcal{P} \rangle_i \langle \mathcal{P} \rangle_j$$ Ref. [@Balasubramanian:2007qv] presents a more complete derivation, which shows that (\[2sigmas\]) applies so long as $\mathcal{P}$ is a finitely local observable. The relation (\[2sigmas\]) represents a limitation of the semiclassical observer’s ability to decode the information held by a black hole that goes beyond the very high precision required to detect specific operator eigenstates as different from the thermal density matrix (Sec. \[eSbasis\]). Here we see that the statistical “de-phasing” that occurs in a generic state in the Hilbert space will further suppress the variance of any observable by $e^S$. The scale $\exp{(-S)}$: a summary {#bhvsthermal} --------------------------------- In order for a gravitational system to be effectively described by a black hole, it must have an exponentially large degeneracy $\sim \exp{S}$ and an exponentially small level spacing $\delta E \sim \exp{(-S)}$, where $S \propto 1/\hbar$ and diverges in the semiclassical limit, precipitating semiclassical information loss. In particular, recovering information from a black hole requires an observer with access to: - resolution of order $\exp{(-S)}$, - time scales of order $\exp{S}$, We arrived at these observations entirely by applying [*statistical*]{} arguments to gravitational states. The AdS/CFT correspondence provides a sharp definition of such states and guarantees that they may be studied using statistical mechanics. However, one may worry that the argument does not use any features unique to black holes and may therefore be applied to any thermal object in gravity, like thermal AdS space. That is indeed correct – it appears that, at least on the CFT side of the AdS/CFT correspondence, black holes are simply conventional thermal enembles and may be treated statistically in exactly the same way. The only exotic thing about the black hole ensemble is the very large density of microscopic states. This is where the dynamics of quantum gravity (or the dual strongly coupled gauge theory) comes into the problem, by setting the nonperturbatively tiny ($\mathcal{O}(e^{-A/4 G_N \hbar})$) energy gaps that are required to account for the entropy. But in classical gravity the [*sine qua non*]{} of a black hole is its horizon which certainly makes it seem very different from a thermal gas. While the statistical arguments that we have outlined explain how semiclassical information loss can be consistent with microscopic information recovery, they do not suffice to explain why black hole entropy is proportional to a geometric quantity – the area of the horizon. Neither do our arguments explain how we should understand the causally disconnected black hole interior that is present in the semiclassical description or the observations of infalling observers traversing this interior region. The problem of understanding horizon formation and the semiclassical black hole interior in the dual CFT, though not logically necessary for resolving the information paradox, is very interesting in its own right. It is difficult, because it is set in the realm of strongly coupled gauge theory, of which little is known quantitatively. Some material related to this question is reviewed in Sec. \[microgeometries\]. Black holes as mirrors {#secmirror} ---------------------- The consequences of the holographic argument for unitarity may be even more profound: it has been argued [@Hayden:2007cs] that if black holes are not information sinks, they must be information mirrors! Specifically, Ref. [@Hayden:2007cs] posits that, as a matter of principle, any $k$ bits of information may be recovered from a black hole whose size (entropy) is $n$ bits after observing $(n+k)/2$ bits expelled from it as Hawking radiation. This implies that from an information theoretic standpoint, the lifeline of a black hole is divided into two stages: 1. Before the black hole has radiated away half of its information content, any additional infalling information gets temporarily trapped. 2. The black hole evaporation process reaches its half-way point, whereafter any additional infalling and all previously infallen information becomes accessible, one bit at a time, with each bit of Hawking radiation released by the black hole. To wit, a black hole past its information theoretic halflife is an information mirror. It is best to first describe the procedure by which an outside observer may recover the black hole-reflected information and then discuss the underlying assumptions. For the sake of clarity, we present the argument in its classical form. The quantum analogue, with the usual substitutions of “bit” with “qubit”, “knows” with “holds a system perfectly entangled with”, and “typical permutation” with “a unitary transformation chosen uniformly with respect to the Haar measure”, is given in Ref. [@Hayden:2007cs]. Think of the internal state of a black hole as a classical bit string of length $n-k$, and model black hole dynamics as a typical ([**Assumption 3**]{}), deterministic permutation of the $2^{n-k}$ bit strings (not of the $n-k$ bits). An outside observer has been collecting information about the black hole since its formation and now knows ([**Assumption 1**]{}) both the internal state of the black hole (one of the $2^{n-k}$ strings) and the dynamical permutation (one of the $(2^{n-k})!$ such permutations). Now $k$ unknown bits are dropped into the black hole and appended to its internal state to form a new string of length $n$. The black hole dynamically processes its information content by applying one of $(2^n)!$ permutations (known to the outside observer by [Assumption 1]{}) *before* releasing one bit of Hawking radiation ([**Assumption 2**]{}). The black hole repeats the permutation-release cycle indefinitely. In this model, the observer will decode the $k$ fallen bits of information after detecting $k+c$ quanta of Hawking radiation, with probability of false detection bounded by $2^{-c}$. To see this, note that after releasing $k+c$ quanta of information, the black hole’s internal state is one of $2^{n-k-c}$ bit strings of length $n-k-c$. Because the black hole dynamical permutation is assumed typical, each of the $2^{n-k-c}$ current internal state strings is equally likely to have been produced by the dynamics of permutation from each the $2^{n-k}$ original black hole internal state strings. Thus, a false positive identification of the $k$ infallen quanta will randomly occur on a $$\frac{2^{n-k-c}}{2^{n-k}} = 2^{-c}$$ fraction of trials. Mechanically, the identification procedure involves selecting the $2^{n-k-c}$ possible precursor internal states consistent with the $k+c$ bits of detected Hawking radiation, and then weeding out those which do not agree with the $n-k$ previously known bits with which the black hole started out. That this procedure can be algorithmically carried out in times parametrically shorter than the black hole evaporation time is the content of [**Assumption 4**]{}. We now comment on the four assumptions. #### Assumption 1 Fundamentally, the argument assumes that one can identify the internal state of a black hole after recording its Hawking radiation for a sufficiently long time. This assumption agrees with the holographic intuition of Sec. \[bhads\], though as seen in Sec. \[eSbasis\]-\[eSsup\] the requisite measurements would of necessity probe the scale $\exp{(-S/\hbar)}$. In addition, the argument requires that the internal dynamics of black holes be knowable and computable, but this is in principle no different from invoking the deterministic nature of physical laws in other contexts. Indeed, there is no reason to doubt that black hole physics and quantum gravity are deterministic and amenable to scientific enquiry, regardless of how little we may understand of these areas at present. Hayden and Preskill state in [@Hayden:2007cs] that [Assumption 1]{} begins to hold after the black hole has radiated away half of its original entropy. This is a consequence of a calculation by Page [@Page:1993df], who computed the entanglement entropy seen by an observer with access to a subset of the degrees of freedom of a total system found in a random pure state. If the total system is divided into two parts, the one visible to the observer ($V$) and the one hidden from her ($H$), the result is that the observed entanglement entropy is generically almost equal to the maximal entropy of the smaller subsystem. In particular, an observer with access to more than half the degrees of freedom of the total system ($|H| \leq |V|$) can generically read off the hidden information from her experiments, which is the statement that the state of $H$ is perfectly entangled with a subsystem of $V$. As pointed out in [@Page:1993up], this finding is directly relevant to black hole physics, in which outside observers have access to only those quanta of the system that have been radiated out by the black hole. Page’s result states that the internal state of the black hole becomes identifiable after $V$ grows to contain at least half the quanta of the total system, that is after the mid-point of the black hole’s evaporation. #### Assumption 2 The second assumption is that the black hole’s thermalization time $t_{\rm t}$ must not exceed the time between the emissions of two successive radiation quanta $t_{\rm r}$. The claim that all information dropped into a black hole is immediately encoded in the subsequent Hawking radiation needs this assumption so that the internal black hole dynamics may incorporate the infalling information in its computation of the next radiation quantum to be emitted. From the viewpoint of the dual CFT, one may expect that $t_{\rm t}$ and $t_{\rm r}$ are both given by $T^{-1}$, since the temperature of the black hole is the only scale in the theory. With reference to the black hole membrane paradigm [@Thorne:1986iy] and the complementarity hypothesis (see Sec. \[complementarity\]), the authors of Ref. [@Hayden:2007cs] arrive at different estimates. In either case, an upper bound on $t_{\rm t}$ is only necessary for the hypothesis that black holes are information mirrors (reflect information without delay). The statement that information is accessible after half a black hole’s entropy has been radiated away holds independently of Assumption 2. #### Assumption 3 The third assumption is that the black hole dynamics acts as a randomizer. Said differently, the effect of black hole dynamics should be typical in its class: for classical bit strings – a random permutation, for quantum memory – a unitary transformation chosen uniformly with respect to the Haar measure. The assumption is needed to ensure that the infalling $k$ bits (qubits) of information get uniformly jumbled up with the rest of the black hole’s memory, so that any $k$ bits (qubits) of the future Hawking radiation are equally good for decoding the dropped message. If Assumption 3 failed, a black hole may expel sudden bursts of specific information as opposed to leaking it at a uniform pace in a maximally scrambled form[^4]. For this reason, Assumption 3 is heuristically appealing, though it is difficult to evaluate its plausibility rigorously without a deeper understanding of quantum gravity. #### Assumption 4 The last assumption is that the complexity of the computational problem faced by an outside observer attempting to retrieve information from a black hole does not somehow invalidate the rest of the argument. Indeed, there may be a tension between Assumption 3 and Assumption 4, because a typical permutation (or a typical unitary transformation) acting on a string of length $2^n$ takes $\mathcal{O}(2^n)$ steps to implement, a number commensurate with the Poincare recurrence time of the model black hole. Without belaboring the philosophical points, we point out that one cannot a priori exclude the possibility that the computational complexity of the decoding procedure may form an integral part of the black hole information puzzle. If so, it would be a manifestation of an exciting interplay between physics and complexity theory. The authors of Ref. [@Hayden:2007cs] attempted to resolve the tension between Assumptions 3 and 4 by conjecturing that black holes dynamics may make preferential use of special classes of transformations, which avoid a conflict with Assumption 4. A less speculative resolution of this problem would be interesting. Quantitative approaches {#examples} ======================= This section reviews several context in which it has been demonstrated quantitatively that an exponentially large density of gravity states is necessary and sufficient for resolving the information paradox. Specifically, we examine families of heavy, exponentially dense gravity states and report dual field theory calculations, which show that their mutual differences vanish in the semiclassical limit as in eq. (\[infloss\]). Extremal black holes in $AdS_5\times S^5$ {#llmetc} ----------------------------------------- Black holes asymptotic to $AdS_5\times S^5$ have vanishing horizon areas if they preserve more than $1/16$ of the supersymmetries. In the following we concentrate on the half-BPS, extremal black holes. These objects are stable and we have complete control over the microstates in gravity and in the dual field theory. While this allows powerful, precise calculations, there is a price to be paid – in this case the horizon coincides with the singularity and has vanishing area, because the statistical degeneracy is not large enough to produce a macroscopic horizon. However, any small excess of energy beyond the extremal limit leads to a finite area horizon, making the 1/2-BPS black holes of AdS$_5$ an extremely useful model system. ### Background information The field theory dual to gravity asymptotic to $AdS_5\times S^5$ is $SU(N)$ super-Yang-Mills theory on $S^3 \times \mathbb{R}_t$. In the half-BPS sector, [@Lin:2004nb] developed a complete map between non-singular type IIB supergravity solutions on $AdS_5\times S^5$ and heavy field theory states. #### Heavy states in field theory $\mathcal{N}=4$ $SU(N)$ super-Yang-Mills theory contains three scalar fields, which are $N\times N$ matrices. In the half-BPS sector, highest weight representatives in each BPS multiplet are created by operators that are gauge invariant polynomials in the zero-modes of a single adjoint scalar field. The s-wave reduction leads to a Lagrangian describing $N$ fermions in a harmonic potential [@Berenstein:2004kk]. A basis of highest weight half-BPS states is specified by sets of increasing integers ${\cal F} =\{f_1,\,f_2,\,\dots ,\,f_N\}$ related to excitation numbers of individual fermions via $E_i = \hbar\left(f_i + \frac{1}{2}\right)$, $i=1,\dots ,N$. In the following, we shall make use of three other ways of representing 1/2-BPS basis states: 1. Individual fermions’ excitations *above the vacuum*: $$r_i = f_i + 1 - i$$ A basis state is graphically represented as a Young diagram with at most $N$ rows, whose row lengths are given by $r_i$. 2. The numbers of columns of length $j$ in the Young diagram: $$\begin{aligned} c_j & = & r_{N-j+1} - r_{N-j} \\ r_i & = & c_{N-i+1} + \ldots + c_N,\end{aligned}$$ where we use the convention $r_0 = 0$. 3. The moments $$M_k = \sum_{i=1}^N f_i^k\,, ~~~~k = 1,\, \ldots,\, N. \label{momentdef}$$ These contain equivalent information, because one may recover the numbers $f_i$ from the set of $M_k$’s. This is done by finding the roots of the characteristic polynomial $$P = \prod_{i=1}^N (x-f_i) = \sum_{p=0}^N (-1)^p \pi_p x^{N-p}\,, ~~~~\pi_p = \sum_{i_1 < i_2 < \cdots i_p} f_{i_1} f_{i_2} \cdots f_{i_p}$$ after rewriting its coefficients in terms of the moments using the Newton-Girard formula $$p \pi_p + \sum_{k=1}^p (-1)^k M_k \pi_{p-k} = 0.$$ A set of $N$ fermions in a harmonic well is integrable and the moments $M_k$ form the tower of commuting, integrable charges. #### Half-BPS supergravity solutions asymptotic to $AdS_5 \times S^5$ They have $S\mathcal{O}(4) \times S\mathcal{O}(4) \times U(1)$ symmetry, a 5-form flux and a constant dilaton, and were found in [@Lin:2004nb]: $$ds^2 = -h^{-2} \, (dt+V_idx^i)^2 + h^2 \, (dy^2 + dx^idx^i) + R^2 \, d\Omega_3^2 + \tilde{R}^2 \, d\tilde{\Omega}_3^2 \Label{llmmetric}$$ The coefficients are given in terms of a function $u(x^1,x^2,y)$ $$R^2 = y \, \sqrt{\frac{1-u}{u}}, \quad \tilde{R}^2 = y \, \sqrt{\frac{u}{1-u}}, \quad h^{-2} = \frac{y}{\sqrt{u(1-u)}} \Label{llmfunctions1}$$ and the one form $V$ is $$V_i(x^1,x^2,y) = -\frac{\epsilon_{ij}}{\pi} \int_{\mathbb{R}^2} \frac{u(v^1,v^2,0) \, (x^j-v^j) \, \, dv^1dv^2}{[(\vec{x}-\vec{v})^2 +y^2]^2} \, . \Label{llmfunctions2}$$ Thus, the solution is entirely specified in terms of $u(x^1,x^2,y)$, which in turn satisfies a harmonic equation in $y$ and, consequently, is fully determined by its boundary condition on the $y=0$ plane: $$u(r,\varphi,y) = \frac{y^2}{\pi} \int_{\mathbb{R}} \frac{u(r',\varphi',0) \,\, d^2\vec{r}'}{[(\vec{r}-\vec{r}')^2+y^2]^2} \Label{llmufunction}$$ The last equation uses polar coordinates for the $x^1x^2$ plane. The expression for the 5-form field strength is given in [@Lin:2004nb]. The metrics (\[llmmetric\]) are regular if and only if the function $u(x^1,x^2,0)$ is restricted to take the values $\pm 1/2$. Functions $u(x^1,x^2,0)$ that fall strictly between $-1/2$ and $+1/2$ anywhere on the $x^1x^2$ plane, when plugged into (\[llmmetric\]), give rise to singular geometries. When $u(x^1,x^2,0)$ falls outside the range $[-1/2,+1/2]$, the metric (\[llmmetric\]) develops closed timelike curves [@Milanesi:2005tp]. A dictionary between these solutions and the field theory states has been established in [@Lin:2004nb]. The $x^1x^2$ plane at $y=0$ is identified with the single particle oscillator phase space in the dual gauge theory while the function $u(x^1,x^2,0)$ – with a phase space distribution $W(p,q)$: $$\begin{aligned} (x^1,x^2) & \leftrightarrow & (p,q) \\ \label{phasespaceident} u(x^1,x^2,0)+1 & \leftrightarrow & W(p,q)\end{aligned}$$ The most well-known phase space distributions to be used on the right hand side are the Wigner [@wignerreview] and the Husimi [@husimi1; @husimi2; @OzoriodeAlmeida:1996sr] distributions, but the choice is immaterial in the semiclassical limit, because it corresponds to the operator ordering prescription on the field theory side. The semiclassical limit is implemented by sending $$\begin{aligned} (l_p^4 \leftrightarrow \hbar) & \rightarrow & 0 \label{lplanck} \\ N & \rightarrow & \infty ~~~~{\rm with}~(N\hbar)\sim L_{AdS}~{\rm fixed.} \label{nsemiclassical}\end{aligned}$$ ### Information loss in an integrable theory {#inflossinteg} In the usual form of the black hole information paradox one must explain the apparent loss of information, such as when a black hole has swallowed an elephant and emits thermal radiation. The half-BPS geometries asymptotic to $AdS_5 \times S^5$ present one with the opposite challenge: because the half-BPS sector is integrable, one expects that information is trivially preserved and the problem is instead to explain *apparent* information loss. This was accomplished in Refs. [@Balasubramanian:2005mg; @Balasubramanian:2006jt]. Recall that half-BPS states of $\mathcal{N}=4$ super-Yang-Mills are in one-to-one correspondence with Young diagrams with at most $N$ rows. We are interested in *heavy* states, whose conformal dimensions (or masses, or the numbers of boxes in their Young diagrams) scale as $\mathcal{O}(N^2)$. This is because in the dual gravity this is the largest possible mass compatible with the asymptotics, so these operators create the heaviest objects in $AdS_5 \times S^5$. Indeed, the scale $\mathcal{O}(N^2)$ agrees with the masses of all known black holes in this background. #### Almost all heavy states look alike Ref. [@vershik] used canonical ensemble techniques to show rigorously that in the limit of large number of boxes, almost all Young diagrams approach a certain limiting shape described by the so-called limit curve. The authors of [@Balasubramanian:2005mg] extended this analysis to studying Young diagrams with $\mathcal{O}(N^2)$ boxes and at most $N$ rows. When expressed in terms of the variables $c_j$ of Sec. \[ads5background\], the partition function factorizes: $$Z = \sum_{c_1,\ldots,c_N=1}^\infty q^{\sum_j j c_j} = \prod_{j=1}^N (1-q^j)^{-1} \label{zsym}$$ A choice of $q = \exp{(-\beta \hbar)}$ sets the expected conformal dimension. When the latter is $\mathcal{O}(N^2)$, $(\beta \hbar)$ is $\mathcal{O}(N^{-1})$ and consequently the entropy is of order $N$. This growth rate is insufficient to produce a macroscopic horizon area in the semiclassical limit, which echoes the statement that half-BPS black holes in $AdS_5 \times S^5$ are horizonless. From eq. (\[zsym\]) one easily calculates the expectations of the numbers of $j$-height columns: $$\label{cjaverage} \langle c_j \rangle = (\exp{\beta j} - 1)^{-1}$$ For $j \ll N$ the occupancies are large since $\beta \propto N^{-1}$. The values (\[cjaverage\]) define a “typical state” represented with a smooth “limit curve”. The significance of the limit curve is that the Young diagrams of almost all states lie close to it. In particular, deviations from (\[cjaverage\]) are exponentially suppressed; for further details, see [@Balasubramanian:2005mg]. #### The typical state represents a black hole Compute the phase space distribution $W(p,q)$ of a Young diagram state and plug it into the metric (\[llmmetric\]) as in (\[phasespaceident\]). The resulting geometry is, up to $\mathcal{O}(\hbar)$ corrections in $W(p,q)$, smooth. However, applying the same procedure to the typical state produces a singular geometry. On the field theory side, the emergence of a singularity may be traced to the fact that a Young diagram with its discrete steps of row-length approaches and, in the semiclassical limit, is effectively being replaced with, a smooth curve (see [@Balasubramanian:2005mg] for details). The loss of information in going from the precise microscopic description of a state to the coarse-grained summary represented by the limit curve translates in gravity into semiclassical information loss, and the appearance of a singularity. In summary, almost all field theory states, each of which can have a good, non-singular gravity dual (up to $\hbar$-corrections), share a common semiclassical description, whose gravity dual is singular. At least in this case the gravitational singularity is an artifact of imposing an effective, semiclassical description on quantum gravity states. #### Integrable charges are encoded in metric multipoles We have reviewed the argument showing that almost all half-BPS states look identical in the semiclassical limit, and argued that this leads to semiclassical information loss in the gravity theory. It is instructive to see how exactly information loss comes about. This is all the more interesting since, naïvely, the integrable structure of the theory should guarantee that information is preserved. The first step is to see where the integrable charges are encoded in gravity. A calculation carried out in [@Balasubramanian:2006jt] showed that they are in one-to-one correspondence with multipole moments of metric components. As an illustration, the function $u(x^1,x^2,y)$ that specifies the metric (\[llmmetric\]) via eqs. (\[llmfunctions1\]) admits the multipole expansion $$u(\rho,\theta,\phi) = \sum_{k=0}^\infty \left( \frac{\hbar}{\rho^2} \right)^{k+1} \!\!\Big(M_k + \big({\rm some~linear~combination~of~}M_1,\ldots,M_{k-1} \big) \Big)\, F_k(\cos^2 \theta), \label{ucharges}$$ where we have used spherical coordinates. Thus, the charges $M_k$ may be systematically extracted in gravity from the multipole moments of the metric. Now take the semiclassical limit (\[lplanck\]-\[nsemiclassical\]) of expression (\[ucharges\]). On dimensional grounds alone, one easily sees that the charges $M_k$ generically scale as $N^{k+1}$. Writing $$M_k = m_k N^{k+1} \left(1 + \mathcal{O}(N^{-1})\right)\,, \label{masymptotic}$$ eq. (\[ucharges\]) reduces in the limit $N \rightarrow \infty$ to: $$u(\rho,\theta,\phi) = \sum_{k=0}^\infty \frac{m_k}{\rho^{2k+2}} F_k(\cos^2 \theta) \label{uasymptotic}$$ Thus, the conserved charges $M_k$ may be directly read off in gravity from the multipole expansion of the metric components. #### Semiclassically the multipoles are universal However, a semiclassical observer will not be able to distinguish microstates or extract information from a black hole. The reason is that $m_k$, the leading order piece of $M_k$ that alone survives the semiclassical limit, is the same for almost all states. The differences among microstates, which are responsible for preserving unitarity, are subleading in $N$ and are not accessible to semiclassical observers. Thus information is preserved in the full theory, but not in the semiclassical regime. Heuristically, one confirms this by noting that in terms of the excitation numbers $f_i$ states differ from one another at $\mathcal{O}(1)$, which the definition (\[momentdef\]) of $M_k$ translates into subleading, $\mathcal{O}(N^k)$ corrections. A more rigorous argument considers two regimes separately. For $k \ll N$, it is possible to rigorously show that $$\lim_{N \rightarrow \infty} \sigma(m_k) / \langle m_k \rangle = 0\,, \label{typicalitymk}$$ which is a version of eq. (\[infloss\]). Meanwhile, a measurement of multipoles beyond $k \sim N^{1/4}$ requires either probing sub-Planckian scales or scales that diverge in the semiclassical limit. The reason is that the $k^{\rm th}$ multipole may be isolated from a $k^{\rm th}$ derivative of some metric component, but measuring that involves partitioning the measurement span into $k$ sub-segments. Using the relation $$\frac{L_{AdS}}{l_P} \propto N^{1/4}\,,$$ which states that even the largest scale in $AdS$ can only fit at most $\mathcal{O}(N^{1/4})$ Planck length segments, one concludes that the higher multipoles are semiclassically unobservable. For more details, consult [@Balasubramanian:2006jt]. Overall, a semiclassical observer interacts with an effective, singular geometry and observes information loss even though the fundamental theory is unitary and even integrable. ### Generalizations with less supersymmetry In the preceding subsection we looked at heavy states in $\mathcal{N}=4$ $SU(N)$ super-Yang-Mills theory. This is the most symmetric theory in the class of toric quiver gauge theories [@Kennaway:2007tq], which have $\mathcal{N}=1$ supersymmetry in four dimensions. Toric quiver gauge theories are dual to type IIB string theory on $AdS_5 \times X^5$ [@Kehagias:1998gn; @Klebanov:1998hh; @Acharya:1998db; @Morrison:1998cs], where $X^5$ is a compact five-dimensional manifold satisfying certain stringent conditions. One may apply the methods of Sec. \[inflossinteg\] to study heavy half-BPS operators in a general $\mathcal{N}=1$ quiver gauge theory. This generalization extends the previous analysis to other theories and to the $1/8$-BPS sector of $\mathcal{N}=4$ super-Yang-Mills. The relevant calculations were published in [@Balasubramanian:2007hu]. The result is that certain generalizations of (\[typicalitymk\]) apply and heavy operators in quiver gauge theories exhibit typicality, too. Consequently, we expect that the corresponding geometries give rise to singularities and information loss when treated semiclassically. However, a detailed map between field theory states and supergravity solutions such as [@Lin:2004nb] has not been developed, so at present it is not possible to verify this claim. In the meantime, a family of candidate black hole solutions has been written down [@Gauntlett:2006ns] as progress towards a map between field theory states and supergravity solutions continues [@Brown:2007xh]. Massless BTZ black hole {#masslessbtz} ----------------------- We now turn to black holes in $AdS_3$ – the BTZ spacetimes [@Banados:1992wn]. To apply our methodology we examine cases in which a known conformal field theory is dual to an asymptotically AdS$_3$ theory of gravity. One such example involves the worldvolume theory of $N_1$ D1-branes wrapped on $S^1$ and $N_5$ D5-branes wrapped on $S^1 \times T^4$, whose dual is $AdS_3 \times S^3 \times T^4$ with the $AdS_3$ radius scaling as $$L_{AdS} \propto (N_1 N_5)^{1/4} \equiv N^{1/4}.$$ As in the previous subsection, the semiclassical limit is $N \rightarrow \infty$. While we could have wrapped the D5-branes on $S^1 \times K3$, choosing $S^1 \times T^4$ is convenient because the field theory is simple. Specifically, it is the $\mathcal{N}=(4,4)$ supersymmetric sigma model on $(T^4)^N/S_N$, deformed by a set of marginal operators [@Strominger:1996sh; @deBoer:1998ip; @Seiberg:1999xz; @Larsen:1999uk]. In the following we consider this theory at the so-called orbifold point, with all the marginal deformations turned off. (Recent papers aimed at extending the material presented in this section away from the orbifold point include [@Avery:2010er; @Avery:2010hs].) This is convenient, because the field theory at the orbifold point is free. For the sector of BPS states moving away from the orbifold point of the field theory will not change counting tasks such as evaluating the degeneracy or checking whether there is a typical state. ### Background information The BTZ family of $AdS_3$ black holes [@Banados:1992wn] is parameterized by the mass $M \geq 0$. In the following we concentrate on the BPS, massless solution: $$\label{btzmetric} ds^2 = -\frac{r^2}{L_{AdS}^2} dt^2 + \frac{L_{AdS}^2}{r^2}dr^2 + r^2 d\phi^2$$ According to the AdS/CFT dictionary, one can recover a CFT two-point function of an operator by considering the bulk-boundary propagator of the dual spacetime field and sending the bulk point to the boundary. For a field of conformal weights $(1, 1)$ this yields [@KeskiVakkuri:1998nw]: $$\label{btzcorr} \langle \mathcal{A}(w) \mathcal{A}(0) \rangle = \sum_{k=-\infty}^{\infty} \frac{1}{(w-2\pi k)^2(\bar{w}-2\pi k)^2}$$ In the above, we expressed the separation between the operator insertions in terms of: $$w = \phi - t / L_{AdS} \qquad \bar{w} = \phi + t / L_{AdS}$$ Eq. (\[btzcorr\]) represents the massless BTZ black hole in the natural parlance of the dual CFT, that is in terms of correlation functions. The BTZ black hole has the boundary conditions and quantum numbers of a ground state in the Ramond sector of the D1-D5 CFT [@Strominger:1997eq]. Such ground states are built of bosonic ($\sigma^\mu_n$) and fermionic ($\tau^\mu_n$) twist operators, which create winding sectors of the worldsheet that wrap $n$ copies of the $T^4$. The superscripts $.^\mu$ correspond to global symmetries of $S^3 \times T^4$ and range from 1 to 8. Physical states are required to carry total twist $N$. Overall, the Ramond ground states are created by composite twist operators $\sigma$ $$\label{sigmadef} \sigma = \prod_{n\mu} (\sigma^\mu_n)^{N_{n\mu}} (\tau^\mu_n)^{N'_{n\mu}}\,,$$ which in turn correspond to partitions of $N$ of the following type: $$\label{d1d5partition} N = \sum_{n=1}^N \sum_{\mu = 1}^8 n(N_{n\mu} + N'_{n\mu}), \qquad N_{n\mu} = 0,1,\ldots, \qquad N'_{n\mu} = 0,1.$$ Canonical ensemble techniques analogous to those used in Sec. \[inflossinteg\] reveal that in the thermodynamic limit $N \rightarrow \infty$, there are $\mathcal{O}(\sqrt{N})$ such partitions. A finite horizon area would require $L_{AdS}^4 \sim N$ (the volume of $S^3$ contributes a factor of $L_{AdS}^3$), so this is consistent with the fact that the massless BTZ black hole is horizonless. ### The typical state reproduces the massless BTZ black hole {#microrepBTZ} In the canonical ensemble the occupation numbers $N_{n\mu},N'_{n\mu}$ are independent and therefore their expectations are given by the usual Bose-Einstein and Fermi-Dirac values $$\langle N_{n\mu} \rangle = (\exp{\beta n} -1)^{-1} \qquad \langle N'_{n\mu} \rangle = (\exp{\beta n} +1)^{-1}\,, \label{befd}$$ where condition (\[d1d5partition\]) sets $\beta = \pi\sqrt{2/N}$. As in the case of $\mathcal{N}=4$ super-Yang-Mills, these values define a typical Ramond ground state, which represents the common structure of almost all Ramond ground states (\[sigmadef\]). Can one distinguish the states $\sigma$? For a non-twist bosonic probe of conformal weights $(1,1)$ the two-point function takes the form [@Balasubramanian:2005qu] $$\label{d1d5corr} \langle \mathcal{A}(w) \mathcal{A}(0) \rangle_\sigma = \frac{1}{N} \sum_n n \sum_\mu N_{n\mu} \sum_{k=0}^{n-1} \frac{1}{[2n \sin (\frac{w-2\pi k}{2n})]^2 [2n \sin (\frac{\bar{w}-2\pi k}{2n})]^2}\,,$$ where the numbers $N_{n\mu}$ define $\sigma$. Ref. [@Balasubramanian:2007qv] analyzed the variance in this expression and showed that eq. (\[infloss\]) holds. Consequently, almost all states $\sigma$ respond in the same way to the probe $\mathcal{A}(w)$ in the limit of large $N$ and there is semiclassical information loss. Furthermore, at time scales $t \ll N^{1/2}$, the universal response of almost all probes reproduces the behavior of the massless BTZ black hole (\[btzcorr\]). To see this, note that the form of $\langle N_{n\mu} \rangle$ implies that the summation over $n$ is dominated by terms of order $n \sim N^{1/2}$, but for those summands the summation over $k$ is a good approximation of eq. (\[btzcorr\]). Ref. [@Balasubramanian:2005qu] contains a similar set of arguments for fermionic probes. At larger times the correlator (\[d1d5corr\]) hovers around a non-zero value, in contrast to the BTZ correlator (\[btzcorr\]). This is in agreement with the intuition spelled out in Sec. \[eSbasis\]. In that section we also anticipated that the non-zero long-time average of (\[d1d5corr\]) would be approximately $e^{-S}$, but this is not the case. One explanation is that the non-twist probes evaluated in the orbifold CFT are insufficient to explore the full set of BTZ microstates [@Balasubramanian:2005qu]. ### Spacetime microstates? {#microgeometries} The D1-D5 system has been widely used in the literature to support and illustrate the fuzzball proposal [@Mathur:2005zp], which states that black hole microstates have spacetime realizations in terms of horizonless bound states of D-branes that have an extended spacetime structure. The proposal further suggests that the transverse size of the underlying bound state is responsible for the apparent presence of a horizon, and that the region behind this apparent horizon is in a topologically complex or perhaps even non-geometric configuration arising from the underlying bound state. This complexity, which is not resolvable by a semiclassical observer, is then responsible for apparent information loss. In this sense the fuzzball proposal is conceptually similar to the point of view described in Sec. \[preliminaries\], although the statistical arguments presented there make no specific claims about the structure of the spacetime microstate at or behind the horizon. Below we briefly review the ideas leading to the fuzzball proposal; for a more complete exposition see [@Mathur:2005zp]. Before delving into the details, we recall a general argument in support of a spacetime realization of microstates given in [@Mathur:2009hf]. Its authors argue that under the assumption that the equivalence principle extends to a classical horizon it is impossible to salvage unitarity by including small corrections to Hawking radiation. If one attempts to do so, one instead finds that the entanglement between the interior and the exterior of the black hole always increases, which implies that the end result of a unitary black hole evaporation process would have to be a remnant (see Sec. \[secremnants\]). The best way to motivate this finding is to highlight the difference between black hole evaporation and ordinary phenomena such as burning paper [@Mathur:2011wg]. In the latter case, the entanglement between the exterior (photons expelled by burning paper) and the interior (the remaining, unburnt paper) is bounded by the size of either subsystem and decreases as the remainder of the paper burns up. In contrast, the process of Hawking radiation proceeds by forming entangled pairs of positive energy particles that escape to the outside and negative energy particles that fall into the interior of the black hole. These negative energy messengers increase the entanglement while decreasing the mass of the black hole, which forces the black hole to eventually form a small mass / high entropy object otherwise known as a remnant. Note that this argument directly contradicts Assumption 1 of [@Hayden:2007cs] (see Sec. \[secmirror\]), which applies to black holes the result that the entropy of a subsystem is bounded by its size [@Page:1993df]. In summary, Ref. [@Mathur:2009hf] claims that if we exclude remnants, the only way to salvage unitarity is to introduce large corrections at the location of a putative horizon, for instance those effected by fuzzballs. #### The Lunin-Mathur geometries To each Ramond ground state (\[sigmadef\]) corresponds a spacetime geometry without horizons [@Lunin:2001jy]. After a U-duality, the D1-D5 system maps to the fundamental string carrying $N_1$ units of momentum and winding $N_5$ times around an $S^1$. One may then U-dualize back the metric of the fundamental string, which is known from the null chiral model [@Tseytlin:1996yb] for an arbitrary classical string profile ${\bf x} = {\bf F}(v)$. The resulting metric is smooth and horizonless and asymptotes to $\mathbb{R}^{4,1} \times S^1 \times T^4$ coordinatized by $({\bf x}, t, y, {\bf z})$. In the string frame it takes the form: $$\begin{aligned} \label{d1d5metric} ds^2 & = & \frac{1}{\sqrt{f_1 f_5}} \Big( -(dt +A)^2 + (dy + B)^2 \Big) + \sqrt{f_1 f_5}\, d{\bf x}^2 + \sqrt{\frac{f_1}{f_5}}\,d{\bf z}^2 \nonumber \\ e^{2\Phi} & = & \frac{f_1}{f_5} \equiv \left({1 + \frac{Q_5}{L}\int_0^{L} \frac{|\dot{\bf F}(v)|^2 \,dv}{|{\bf x} - {\bf F}(v)|^2}} \right) \left/ \vphantom{\frac{Q_5}{L}} \right. \Bigg({1 + \frac{Q_5}{L}\int_0^{L} \frac{dv}{|{\bf x} - {\bf F}(v)|^2}} \Bigg) \nonumber \\ A & = & \frac{Q_5}{L} \int_0^{L} \frac{\dot{\bf F}(v) \,dv}{|{\bf x} - {\bf F}(v)|^2} \qquad \qquad dB = *_4 dA\end{aligned}$$ The classical profile ${\bf F}(v = t - y)$ has periodicity $L \propto N_5$. For bosonic ground states it is the shape of the fundamental string to which the momentum mode $\alpha^\mu_{-n}$ was applied $N_{n\mu}$ times; when fermionic modes are turned on, the expression for ${\bf F}(v)$ becomes more complicated [@Taylor:2005db]. The near-horizon limit of the metric (\[d1d5metric\]) removes the terms $1 + \ldots$ from the definitions of $f_1,f_5$ and the resulting geometries are asymptotic to $AdS_3 \times S^3 \times T^4$. We already know that there are $\mathcal{O}(\sqrt{N})$ geometries (\[d1d5metric\]), because they correspond to the ground states $\sigma$ counted in eq. (\[d1d5partition\]). It has been shown on very general grounds [@Strominger:1996sh; @Callan:1996dv] (see [@Mathur:2005zp] for a more pedestrian account) that bound states of D-branes carrying three different charges, such as the D1-D5-P system, carry entropy that in the large $N$ limit scales appropriately to account for finite horizon areas of black holes. #### The emergence of the $M=0$ BTZ black hole Let $r$ be the radial coordinate in the $\mathbb{R}^4$ parameterized by ${\bf x}$. For most states ${\bf F}(v)$ is so complex that $\dot{{\bf F}}(v)$ is effectively random and as a result for $r \gg |{\bf F}(v)|$ the 1-forms $A, B$ vanish. Consequently, in that regime the metric (\[d1d5metric\]) reduces directly to $$\label{microeffgeom} ds^2 = \frac{r^2}{L_{AdS}^2} \left( -dt^2 + dy^2 \right) + \frac{L_{AdS}^2}{r^2} \left( dr^2 + r^2 d\Omega_3^2 \right) + \frac{L_{AdS}^4}{Q_5^2} d{\bf z}^2\,,$$ where $$\label{q1q5ads} L_{AdS}^4 \equiv Q_1 Q_5 = \frac{Q_5^2}{L} \int_0^L |\dot{{\bf F}}(v)|^2\, dv.$$ This is the product space of $S^3 \times T^4$ and the $M=0$ BTZ black hole (\[btzmetric\]) with $\phi = y / L_{AdS}$. A simple argument [@Balasubramanian:2005qu] confirms that $|{\bf F}(v)| \ll L_{AdS}$ so the regime $r \gg |{\bf F}(v)|$ extends down to scales that are parametrically smaller than the $AdS$ radius. We conclude that outside a small core region the Lunin-Mathur geometries mimic the BTZ black hole. This is a geometric dual of the conclusion of Sec. \[microrepBTZ\]. #### The emergence of an apparent horizon How can bound states of D-branes wrapped on compact dimensions give rise to an apparent horizon? They must develop a transverse size. A transverse size can act as an effective horizon, because it sets the radial scale at which (a) microstates become readily distinguishable and (b) incident particles get absorbed by the D-branes, which from afar looks like falling behind a horizon. To test this conjecture let us estimate of the size of a bound state. In the fundamental string picture the transverse size of a bound state can be retrieved from the typical oscillation of the string. The typical mode has wavenumber $\sqrt{N}$ and wavelength $L / \sqrt{N}$; to estimate the amplitude one multiplies the wavelength by the mean value of $|\dot{\bf F}(v)|$. Observe that the effective length of the string is $L \propto N_5$, the mean profile slope is from eq. (\[q1q5ads\]) $\smash{\overline{|\dot{{\bf F}}(v)|} \propto \sqrt{Q_1 / Q_5} \propto \sqrt{N_1 / N_5}}$ and the typical wavenumber is proportional to $\sqrt{N} \sim \sqrt{N_1 N_5}$. The radial size of the bound state is therefore independent of $N_1, N_5$ and it remains so after U-dualizing back to the D1-D5 system. We could have anticipated this result from the fact that the $M=0$ BTZ black hole is a naked singularity; the microstates of a black hole with a finite horizon area should have transverse sizes that grow with the charges. Nevertheless, we can perform a non-trivial check of the claim that the size of a fuzzball matches the horizon. Looking at eq. (\[d1d5metric\]), the surface area of the eight-dimensional hypersurface enclosing the bound state is $$\Big(f_1 f_5\Big)^{-1/4} \cdot \Big(f_1 f_5\Big)^{3/4} \cdot \left( \frac{f_1}{f_5} \right) \cdot \left( \frac{f_5}{f_1} \right) \propto (N_1 N_5)^{1/2} \sim N^{1/2}\,,$$ where the successive terms denote volumes of the $S^1$, $S^3$ and $T^4$ and the factor of $\exp{(-2\Phi)}$ that takes us to the Einstein frame. Thus, the area enclosing the bound state reproduces the microscopic count of microstates. In fact, for some 2-charge black holes one may obtain a finite horizon by including higher derivative terms in the supergravity Lagrangian [@Dabholkar:2004yr; @Dabholkar:2004dq]; in such cases the horizon size also matches the microscopic count. For a recent review of the fuzzball proposal, consult [@Chowdhury:2010ct]. Ref. [@Mathur:1997wb] argued on general grounds that the transverse size of a 3-charge bound state should match the finite horizon area of the corresponding black hole, which is known to agree with the number of string theory states [@Strominger:1996sh; @Callan:1996dv]. There is an extensive literature on constructing candidate spacetime realizations of such microstates; early works include [@Bena:2006is; @Balasubramanian:2006gi; @Berglund:2005vb; @Bena:2005va] while Sec. 5 of [@Balasubramanian:2008da] contains a review. Interestingly, for some of these one can explicitly see how the transverse size grows from 0 to a finite value as one increases the string coupling [@Denef:2002ru; @Balasubramanian:2006gi]. Another interesting class of geometries, called the scaling solutions [@Bena:2006kb; @Denef:2007vg], has a classical moduli space, which naïvely includes geometries with arbitrarily deep AdS throats. Such solutions are problematic, because in the cases with holographic duals their existence should imply that the spectrum of the CFT is continuous [@Bena:2007qc]. However, as shown in [@deBoer:2008zn], quantization effectively caps the throats at a finite depth, because an infinite throat requires a localization in phase space that is forbidden by the uncertainty principle. This is another example of how quantum effects can cure a distressing aspect of classical geometries. #### A geometric mechanism of information loss Consider a graviton falling towards the throat of a microstate geometry. In the CFT picture, it is absorbed into the bound state and creates one left and one right mover on a component string. The two excitations will collide again and possibly re-emit the graviton after a time that is $\mathcal{O}(\sqrt{N})$. On the gravity side, the graviton enters the (topologically) complex interior of a fuzzball, bounces and escapes. For certain candidate microstate geometries with extra symmetry the return time has been calculated [@Lunin:2001dt] and found to agree precisely with the CFT calculation, including the coefficient. In this way the fuzzball proposal seems to explain the appearance of information loss: a quantum that seems to have disappeared behind a horizon wanders about the labyrinthine interior of a fuzzball. Information is preserved because it will eventually find an exit. Towards dynamics – Matrix models -------------------------------- In the preceding sections we have illustrated the idea that the information paradox is an artifact of the semiclassical description of a quantum system, each of whose microstates evolves without violating unitarity. The examples we looked at involved one parameter $N$, which controlled the size of the system and the relative magnitude of quantum effects. In particular, the examples of Secs. \[llmetc\]-\[masslessbtz\] were free theories at $T=0$. (Note that when we used an inverse “effective temperature” $\beta$ in eqs. (\[zsym\], \[befd\]), it was only as a calculational shortcut to obtaining an effective description of heavy states.) We now turn to calculations that track the effect of a non-zero coupling constant and temperature. To do this, we look at the evolution of a disturbance in a thermal background and ask whether the disturbance remains detectable at late times. In these settings, information loss may still arise from a growing density of states, but now there is an alternative mechanism – the dynamics. Below we present some toy models in which information loss requires the large $N$ limit *and* non-trivial dynamics. The systems of choice are matrix models. They arise frequently as subsectors of field theories with gravitational duals, but for our purposes it suffices to demand from a matrix model that it suffer information loss in the $N \rightarrow \infty$ limit. Any matrix model with this key property is a potential laboratory for studying how information is preserved and why it appears lost. ### Festuccia and Liu’s model {#flmodel} Ref. [@Festuccia:2006sa] considers matrix quantum mechanics with the action: $$\label{class} S = N\, {\rm Tr}\! \int\! dt \sum_\alpha \left( \frac{1}{2} (D_t M_\alpha)^2 - \frac{1}{2} \omega_\alpha^2 M_\alpha^2 \right) - \int dt\, V(M_\alpha; \lambda)$$ Here $M_\alpha$ are $N \times N$ matrices, $D_t = \partial_t - i \[A, \cdot\]$ is a covariant derivative, the frequencies $\omega_\alpha$ are all positive and $V(M_\alpha; \lambda)$ is a linear combination of single trace operators determined by the ’t Hooft coupling $\lambda = g_{\rm YM}^2 N$, which is kept fixed in the large $N$ limit. This class of theories is well motivated, because it contains the bosonic sector of $\mathcal{N} = 4$ super-Yang-Mills theory on $S^3$ and, as we shall see momentarily, it suffers the expected information loss in the large $N$ limit. We follow [@Festuccia:2006sa] and concentrate on the specific model $$\label{exemplar} S = \frac{N}{2} {\rm Tr} \int dt \Big( (D_t M_1)^2 + (D_t M_2)^2 - \omega^2 (M_1^2 + M_2^2) - \lambda\, M_1 M_2 M_1 M_2 \Big)\,,$$ but the results presented below apply generally to the class (\[class\]). The regime relevant to black hole physics is where energy scales as $N^2$. This is no different from the half-BPS sector of $\mathcal{N}=4$ super-Yang-Mills considered in Sec. \[llmetc\], whose Lagrangian is also of the form (\[class\]). However, we now require that $S \propto N^2$, which can be enforced by considering systems with at least two matrices as in eq. (\[exemplar\]). We deviate here from Sec. \[llmetc\], where entropy scaled as $N$ as a consequence of the Lagrangian containing only a single matrix with its off-diagonal entries gauged away. A convenient object to study is the connected Wightman function $$\label{wightman} G_+ (t) = \langle \mathcal{O}(t) \mathcal{O}(0) \rangle_\beta = \frac{1}{Z} {\rm Tr} \left( e^{-\beta H} \mathcal{O}(t) \mathcal{O}(0) \right) - C\,,$$ where $C$ subtracts the contributions of the diagonal part of $\mathcal{O}$, which we take to be a multi-trace operator with $K$ insertions of $M_\alpha$. We keep $K$ fixed in the limit of large $N$; in the models (\[class\]) that have holographic duals, this makes $\mathcal{O}$ dual to a string probe. The quantity (\[wightman\]) is useful because $$\lim_{t \rightarrow \infty} G_+ (t) = 0$$ is a signature of information loss. Looking at the Fourier transform $G_+(\omega)$, we expect that all its singularities are poles, of which the one closest to the real axis controls the decay rate of $G_+(t)$ at large times [@Festuccia:2005pi] (see also Sec. \[excursions\]). Begin with the free theory $\lambda = 0$. Not surprisingly, at finite $N$ $G_+(\omega)$ takes the form [@Festuccia:2005pi] $$G_+(\omega) = 2\pi \sum_{ij} e^{-\beta E_i}\, |\langle i | \mathcal{O}(0) | j \rangle|^2 \,\delta(\omega - E_j + E_i)$$ and there is no information loss. However, even in the large $N$ limit $G_+(\omega)$ remains a weighted sum of delta functions [@Festuccia:2006sa] and the system, once disturbed, does not thermalize. Evidently this theory needs interactions to forget information. In fact, for $\lambda > 0$ a naïve planar calculation of $G_+(\omega)$ continues to yield a discrete spectrum. The caveat is that planar perturbation theory breaks down in the limit of large time. Heuristically, a non-zero perturbation breaks the degeneracy of energy eigenstates, reducing the level spacing to some $N$-dependent value that vanishes in the large $N$ limit. When the level spacing vanishes at infinite $N$, $G(\omega)$ becomes continuous, $G_+(t)$ decays at large times and information is lost. The models (\[class\]) carry several lessons. As discussed throughout this paper, the semiclassical limit (which is here a large $N$ limit) is an essential ingredient in information loss, and arises from the inability for the semiclassical observer to resolve physical data such as energy gaps with a precision of $\mathcal{O}(e^{-S})$. The matrix models of [@Festuccia:2006sa] require a non-zero coupling $\lambda$ to produce such tiny gaps. Indeed the planar perturbative calculation of $G_+(t)$ in [@Festuccia:2006sa], which reveals no information loss, is valid for $t \lesssim 1/\lambda$. Thus, as the coupling increases and the gap in the theory decreases, information becomes ever harder to recover. ### Iizuka and Polchinski’s model {#ipmodel} Another matrix model demonstrates the importance of temperature [@Iizuka:2008hg] in information loss. Consider the following Hamiltonian: $$\label{polchinski} H = \frac{1}{2} {\rm Tr}\, \Pi^2 + \frac{m^2}{2} {\rm Tr} X^2 + M (a_i^\dagger a_i + \bar{a}_i^\dagger \bar{a}_i) + g (a_i^\dagger X a_i + \bar{a}_i^\dagger X^{\rm T} \bar{a}_i)$$ Here $X_{ij}$ is a Hermitian matrix with canonical conjugate momentum $\Pi_{ij}$ and $a^\dagger_i,a_i$ ($\bar{a}^\dagger_i,\bar{a}_i$) are creation and annihilation operators for a complex vector particle $\phi$ (and its conjugate $\phi^\dagger$). The field $X_{ij}$ transforms in the adjoint and $\phi$ ($\phi^\dagger$) in the fundamental (antifundamental) representation of $SU(N)$, so the indices $i,j$ range from 1 to $N$. The Hamiltonian commutes with the number operators $N_\phi = a_i^\dagger a_i$ and $N_{\phi^\dagger} = \bar{a}_i^\dagger \bar{a}_i$, so the spectrum decomposes into independent sectors of definite $(N_\phi,N_{\phi^\dagger})$. While the model (\[polchinski\]) has no ground state, each $(N_\phi,N_{\phi^\dagger})$ sector does. In the following we restrict attention to the $(0,0)$ sector. One can accomplish that without modifying the dynamics by deforming the Hamiltonian with a quadratic term $$H' = H + c (N_\phi+N_{\phi^\dagger}) (N_\phi+N_{\phi^\dagger}-1)$$ with a sufficiently high pre-factor $c$. Several considerations motivate (\[polchinski\]). In comparison with the theory (\[exemplar\]), it substitutes a trilinear fundamental-adjoint-fundamental interaction for a quartic interaction of adjoints. Thus, in terms of the double-line notation for Feynman diagrams, eq. (\[polchinski\]) effectively halves the basic interaction of (\[exemplar\]), which leads to computational simplifications. The model is also closely related to a previously studied description of the fundamental string stretched between a probe D0-brane and a D0-brane black hole [@Iizuka:2001cw]. In addition, further studies of information loss in (\[polchinski\]) give hints of an emergent bulk description [@Iizuka:2008eb], though we shall not pursue this here. Last but not least, it suffers information loss in the limit of large $N$, so it is a useful setting for examining resolutions of the information paradox. Interestingly, as we will see, information loss occurs in the model (\[polchinski\]) only at large temperature. We concentrate on the observable $$e^{iM(t-t')} \langle\, {\rm T}\, a_i(t) a^\dagger_j(t) \,\rangle_T \equiv \delta_{ij} G(T, t-t')\,.$$ The Schwinger-Dyson equation for the Fourier transform $\tilde{G}(T,\omega)$ reads: $$\label{sde} \tilde{G}(T, \omega) = \tilde{G}_0(\omega) - g^2 N \tilde{G}_0(\omega) \tilde{G}(T,\omega) \int_{-\infty}^\infty \frac{d\omega'}{2\pi} \tilde{G}(T,\omega') \tilde{K}_0 (T, \omega-\omega')$$ The above expression involves the free $\phi$-propagator $$\tilde{G}_0(\omega) = \frac{i}{\omega + i\epsilon}\,,$$ because in the $(N_\phi,N_{\phi^\dagger})=(0,0)$ sector the thermal background does not affect the zeroth order evolution of $\phi$. In contrast, the propagation of $X$ is given by the free thermal propagator $$\tilde{K}_0(T,\omega) = \frac{i}{1-e^{-m/T}} \left( \frac{i}{\omega^2 - m^2 + i\epsilon} - \frac{e^{-m/T}}{\omega^2 - m^2 - i\epsilon} \right)\,.$$ The time ordering in the definition of $G(T,t)$ guarantees that $\tilde{G}(T,\omega)$ is non-singular in the upper half-plane. The fact that $g$ has mass dimension $3/2$ suggests that at sufficiently high frequency $\tilde{G}(T,\omega)$ reduces to its free form $1/\omega$. These observations allow us to close the integral in the upper half plane and take the residue. The resulting equation reads: $$\label{sde2} \tilde{G}(T,\omega) = \frac{i}{\omega} \left( 1 - \frac{g^2 N \tilde{G}(T,\omega)}{2 m(1 - e^{-m/T})} \left( \tilde{G}(T,\omega - m) + e^{-m/T} \tilde{G}(T,\omega + m)\right) \right)$$ At $T=0$, this reduces to $$\tilde{G}(\omega) = \frac{i}{\omega} \left( 1 - \frac{g^2 N}{2m} \tilde{G}(\omega) \tilde{G}(\omega - m)\right)\,,$$ which enjoys a closed form solution in terms of Bessel functions: $$\label{zerotemp} \tilde{G}(\omega) = i\, \sqrt{\frac{2\,m}{g^2 N}} \, \frac{J_{-\omega/m}\left(\sqrt{\frac{2g^2 N}{m^3}}\right)} {J_{-1-\omega/m}\left(\sqrt{\frac{2g^2 N}{m^3}}\right)}$$ In particular, the zero temperature solution (\[zerotemp\]) has a set of poles on the real axis, the correlator is quasi-periodic and there is no loss of information. As temperature is increased, however, one may show that the solutions of (\[sde2\]) have no poles and instead gradually take on a smooth form that signals information loss. The authors of [@Iizuka:2008hg] studied numerically how the solutions of (\[sde2\]) vary with temperature and showed that at sufficiently high temperature $G(T,t)$ decays exponentially. In summary, the model (\[polchinski\]) undergoes information loss only when the large $N$ limit is accompanied by a sufficiently large temperature. In the examples we reviewed earlier large $N$ was responsible for the smallness of the gap and, by extension, for information loss. Temperature, on the other hand, controls how uniformly different states are populated in the canonical ensemble. The fact that in the present model information loss requires both large $N$ and a finite temperature is intuitive: in order to lose information, one needs many closely spaced states, all of which participate in the dynamics. Additional approaches {#otherappr} ===================== In this section we review several alternative approaches to the black hole information paradox and contrast them with the view presented in Sec. \[examples\]. Backreaction of Hawking radiation --------------------------------- Ref. [@Parikh:1999mf] highlighted the fact that quanta of Hawking radiation are produced in tunneling events that create pairs of particles astride the horizon, and speculated that the detailed sequence of such tunneling events could carry information out of a black hole. This possibility was recently revisited in [@Zhang:2009td], which argued that there are enough tunneling sequences to account for the full entropy content of a black hole. A series of tunneling events, whose precise sequence is decided by the black hole microstate, is reminiscent of the scenarios discussed above. The difference lies in whether the horizon is to be treated as an emergent, semiclassical concept, as opposed to a bona fide spacetime site where Hawking quanta are produced. Refs. [@Mathur:2009hf; @Mathur:2011wg] discussed in Sec. \[microgeometries\] repudiate the latter possibility. Complementarity --------------- The principle of black hole complementarity posits that black hole physics enjoys two complementary descriptions. On the one hand, no exotic phenomena alert a freely falling observer when she crosses a horizon and falls inside a black hole. On the other hand, the physics seen by static observers is well described by phenomena restricted to the exterior region. From the latter perspective, the effective dynamics of matter near the black hole can be summarized in terms of physical degrees of freedom on a [*stretched horizon*]{} – a surface first studied in the membrane paradigm of black holes [@Thorne:1986iy], defined to extend one Planck unit outward from the event horizon. The principle of complementarity states that these two descriptions of black hole physics are equivalent and cannot both be simultaneoursly accessible to any single observer – they are in this sense complementary. The principle further states that any attempt to confront the two perspectives will necessarily fail, or else involve physics beyond the Planck scale. The authors of [@Susskind:1993mu] considered various thought experiments in which observers attempt to invalidate black hole complementarity by: (1) attempting to establish the non-existence of a physical membrane at the stretched horizon for either static and collapse geometries, or (2) first sampling Hawking radiation and then diving into the black hole in an attempt to duplicate information. In every case, the observer encounters super-Planckian physics somewhere in the process. In this way, [@Susskind:1993mu] argued that despite the small spacetime curvatures encountered at the horizon, a resolution of the information paradox necessarily requires control of short distance physics, i.e. a complete quantum theory of gravity. Indeed, in the controlled context of two-dimensional dilaton gravity [@Callan:1992rs] in which black hole complementarity was originally proposed [@Susskind:1993if], the authors of [@Balasubramanian:1995sm] showed that the outgoing late-time Hawking radiation and low-energy infalling observers cannot simultaneously be described in a low energy effective theory in which all interactions must have center-of-mass energies below a specified cutoff. In this way, a low-energy effective theory can either describe the asymptotic observer and the Hawking radiation flux, or the infalling observer at the horizon, even though there are no large curvatures in the problem until the observer reaches the singularity. This precise form of complementarity, and the conclusions of [@Susskind:1993mu], follow in the end from the enormous relative boost of infalling and outgoing observers near a black hole horizon. The principle of black hole complementarity manifestly preserves unitarity; indeed, it posits that information is present and in principle accessible on or outside the black hole horizon at all times. Although the stretched horizon looks like a hot membrane in a thermodynamic description, on a fundamental level it supports quantum gravitational degrees of freedom, whose dynamics in principle determines the evolution of the system. In particular, it determines the detailed structure of the emitted Hawking radiation, which encodes the information about the initial state. In this sense, the resolution of the black hole information paradox offered by complementarity is not fundamentally different from the statistical view presented in previous sections.[^5] Both views contend that information loss is an artifact of imposing a semiclassical description on an essentially quantum gravitational system, and that departures of Hawking radiation from thermality preserve unitarity. Black hole complementarity makes an additional claim: that the black hole interior enjoys a complementary description in terms of degrees of freedom located outside the horizon, and further, that no observer can combine or confront the two descriptions without relying on super-Planckian physics. This extra claim is not essential to the information paradox, however, and to date there is no precise quantitative realization of black hole complementarity that shows how the physics seen by an infalling observer may related to measurements by an asymptotic observer. It has proved difficult over the years to refute the idea of complementarity. Ref. [@Sekino:2008he] discusses one narrow miss based on the results of [@Hayden:2007cs] (see also Sec. \[secmirror\]). It shows that complementarity gets as close as possible to violating the no-cloning theorem [@Wootters:1982zz] (essentially the linearity of quantum mechanics) without actually violating it. One should stress, however, that such studies do not provide positive evidence for black hole complementarity. The latter would presumably have to arise from physics beyond the Planck scale. Nonlocality ----------- Complementarity states that the degrees of freedom inside a black hole are duplicated outside. In particular, a local field operator from a black hole interior must not commute with its outside-horizon duplicate. Thus, complementarity stipulates that a broad family of pairs of spacelike separated operators must not commute and ergo, requires a large scale violation of locality. Potentially, this is most embarrassing on so-called ‘nice slices’ [@Lowe:1995pu; @Lowe:1995ac], which form a foliation of spacetime that interpolates smoothly between free fall frame on or inside the horizon and the fiducial observer’s frame far away from the black hole. In principle, they contain information about both the outgoing Hawking radiation and any infalling matter (information), and accomplish that without stumbling upon large curvature anywhere except near the singularity. Furthermore, the four-momenta of infalling matter and outgoing Hawking radiation are small when projected onto a nice slice. Consequently, one expects that the physics on nice slices should be aptly described by effective field theory with no violations of locality. As it turns out, effective field theory on nice slices must be supplemented by nonlocal terms. They arise whenever one writes down the full quantum gravity theory on nice slices and then truncates to low energies; the argument of the previous paragraph implicitly performed these operations in the reverse order. The presence of nonlocality on nice slices was shown in [@Lowe:1995pu] using the perturbative S-matrix of string theory and in [@Lowe:1995ac] using commutators of string field theory. Remarkably, the freely falling observer does not detect violations of locality until she is very close to the singularity [@Lowe:2006xm], consistent with the reasoning that motivated the complementarity hypothesis. The quest to exonerate complementarity from charges of unphysical nonlocality has led to a separate line of research, which examines the limits on locality in black hole physics independent of complementarity or string theory. Refs. [@Giddings:2004ud; @Giddings:2006sj] explored the consequences of the observation that the notion of spacelike separation can at best be approximate in a dynamical theory of gravity, especially in the presence of large blueshifts. In particular, the regions of nice slices that intersect the black hole interior and Hawking radiation are necessarily separated by large blueshifts. Consequently, nice slices are unlikely to host a local field theory [@Giddings:2006be], which is a possible loophole in the standard argument for information loss. Interestingly, complementarity seems to demand a stronger violation of locality than does unitarity alone [@Giddings:2009ae]. Thus, complementarity appears to assume enough structure to accommodate two independent, plausible resolutions of the information paradox, a dynamical one based on nonlocality and the statistical one that is the subject of this review. For a brief summary of the locality bound and its relevance to black holes, the reader is referred to [@Giddings:2007pj]. Excursions beyond the horizon {#excursions} ----------------------------- ![The Penrose diagram of Schwarzschild-AdS in $d > 3$. In $d=3$, the diagram is a perfect square. Arrows mark the directions of Schwarzschild time $t$ in each region. The dashed line is fixed under the reflection symmetry $t \leftrightarrow -t$.[]{data-label="schwarzschildads"}](schwarzschild-ads.pdf) The issue of information loss arises fundamentally from the existence of a spacetime horizon, and a geometric region behind it, in the semiclassical theory. Every proposed solution to the information paradox provides some mechanism for the asymptotic observer to have access to physics from behind the horizon, e.g., because it is “complementary" to the exterior physics, or because it is encoded in subtle Hawking radiation correlations, or because physics is nonlocal near a horizon, or because the causal disconnection of the interior is an artifact of the semiclassical limit etc. This question becomes particularly interesting in the context of the description of eternal black holes in the AdS/CFT correspondence, because the unitary CFT is by definition supposed to encode the physics of the entire spacetime including any causally disconnected regions that might lie behind a horizon. The extended Penrose diagram of the Schwarzschild-AdS geometry [@Fidkowski:2003nf] is presented in Fig. \[schwarzschildads\]. It contains two asymptotically AdS regions and standard holographic reasoning posits that each of them contains a conformal field theory on its boundary. The fact that we have two independent copies of the same field theory is reminiscent of the thermofield formalism, which is a way of studying thermal field theory in real time. The idea is that a particular pure state $| \Psi \rangle$ in $\mathcal{H} \otimes \mathcal{H}$, where $\mathcal{H}$ is the Hilbert space of one copy of the field theory, is capable of encoding the thermal information in the entanglement between the two $\mathcal{H}$s. It is easy to see that the correct choice of state is [@Balasubramanian:1998de; @Maldacena:2001kr] $$| \Psi \rangle = \frac{1}{\sqrt{Z(\beta)}} \sum_n e^{-\beta E_n / 2} |n\rangle_1 \otimes |n\rangle_2, \label{hhstate}$$ because thermal expectation values of any field theory operator $\mathcal{O}_1$ living in CFT$_1$ can be obtained as expectations in $| \Psi \rangle$, the usual Boltzmann factor recovered from tracing over the states of CFT$_2$: $$\langle \mathcal{O}_1 \rangle_\beta = \frac{1}{Z(\beta)} \sum_n e^{-\beta E_n}\!\! \phantom{.}_1\langle n | \mathcal{O}_1 | n \rangle_1 = \langle \Psi | \mathcal{O}_1 | \Psi \rangle$$ Standard AdS/CFT reasoning shows that correlation functions of operators inserted on the same copy of the CFT probe the regions outside horizons – the left and right wedges of Fig. \[schwarzschildads\]. Ref. [@Maldacena:2001kr] proposed that that the full tensor product of the two CFTs contains information about the full extended Penrose diagram, including the regions adjacent to the singularities. The latter are probed by *two-sided* correlation functions, $$\langle \Psi | \mathcal{O}_1 \mathcal{O}'_2 | \Psi \rangle\,, \label{excursioncorr}$$ which combine operator insertions in *distinct* CFTs. These are simply a technical tool in the thermofield formalism, but in the holographic context of the Schwarzschild-AdS geometry they are recovered from spacelike geodesics connecting the two boundaries. Such geodesics necessarily cross horizons. We now show that extracting information from behind horizons requires a study of analytically continued correlation functions. The $d$-dimensional Schwarzschild-AdS geometry is given by $$\begin{aligned} ds^2 & = & -f(r) dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_{d-2}^2 \label{schwadscoord} \\ f(r) & = & \frac{r^2}{L_{AdS}^2} + 1 - \frac{r_+^2}{r^2} \left( \frac{r_+^2}{L_{AdS}^2} + 1 \right)\,,\end{aligned}$$ where $r_+$ is the horizon radius. The Lorentzian metric (\[schwadscoord\]) is valid in one region outside the horizon and one region inside the horizon at a time, with the boundary located at $r=\infty$ and the singularity at $r=0$. However, it can be extended to the full Penrose diagram of Fig. \[schwarzschildads\] by complexifying the Schwarzschild time $t$ [@Hemming:2002kd; @Fidkowski:2003nf]. Indeed, the Schwarzschild time in which a radial null geodesic reaches the singularity picks up an imaginary contribution from the pole on the horizon: $$t = \int_\infty^0 \frac{dr}{f(r)} = -\frac{i\beta}{4} + {\rm real} \label{imaginarytime}$$ Consequently, it is consistent to think of (\[schwadscoord\]) as a global metric of Fig. \[schwarzschildads\], with the proviso that each of the four regions has a fixed imaginary part. From eq. (\[imaginarytime\]) we read off that the correct assignment is to pick up an additional $-i\beta/4$ each time one crosses a horizon in the clockwise direction in the sense of Fig. \[schwarzschildads\]. As a matter of convention $t$ is real in the region adjacent to the right boundary. In order to maintain (\[imaginarytime\]) for geodesics departing the left asymptotic boundary, the real part of the Schwarzschild time runs backwards in the left wedge. Overall, in each region the real part of $t$ increases in the clockwise direction, as indicated in the figure. The diagram is $t \leftrightarrow -t$ symmetric as a consequence of the $\beta$-periodicity in the imaginary time direction. These observations define a recipe for computing the two-sided correlation functions (\[excursioncorr\]) that contain behind-the-horizon information. All one has to do is to analytically continue the insertions from CFT$_2$ according to $t \rightarrow -t - i\beta/2$. This prescription, in principle straightforward, turns out to be tricky, because one must avoid singularities in the complex $t$-plane that arise whenever the operator insertions become lightlike separated. For BTZ black holes the correct procedure was carried out in [@Kraus:2002iv]. Depending on the chosen time slicing, the resulting, analytically continued correlators admit two interpretations. According to the first, one integrates interaction vertices over the entire, extended diagram of Fig. \[schwarzschildads\]; in the other, interactions are construed to happen only in the left and right wedges outside horizons, but the contour of the analytically continued path integral contains an additional piece imposing correlated boundary conditions on the horizons, which mimic the effect of (\[hhstate\]). This dual interpretation is reminiscent of the ideas reviewed in Sec. \[complementarity\]. In the following we do not pursue the BTZ story and instead follow [@Fidkowski:2003nf], who studied Schwarzschild-AdS black holes in $d>3$. These black holes are more instructive, because they contain curvature singularities and because they illustrate a further subtlety in extracting behind-the-horizon information, reviewed below. Consider a highly massive probe in AdS space that is dual to some scalar operator $\mathcal{O}$ in the CFT. The AdS path integral for the spacelike two-point correlator of $\mathcal{O}$ will be dominated by geodesic paths in the bulk, so that $\langle \mathcal{O}(x_1) \mathcal{O}(x_2) \rangle \approx \exp{(-m \mathcal{L})}$ where $\mathcal{L}$ is the length of a geodesic joining the two insertions on the AdS boundary [@Balasubramanian:1999zv]. This expression requires regulation to remove the divergence in length near the AdS boundary, and some care is necessary when there are multiple geodesics especially in the case of non-static spacetime [@Louko:2000tp; @Festuccia:2005pi]. Neglecting the latter subtlety for the moment, the two-sided correlator between operators inserted in the CFTs on the two boundaries of the eternal AdS black hole is given by $\exp{(-m \mathcal{L})}$, where $\mathcal{L}$ is the length of a spacelike geodesic joining the two insertions. Thus, a study of correlation functions reduces to understanding spacelike geodesics in the black hole background. To simplify the problem, [@Fidkowski:2003nf] noted that a general two-sided correlator, $\langle \phi_2(t_2) \phi_1(t_1) \rangle$ (where $t_1, t_2$ are Lorentzian times in CFT$_1$ and CFT$_2$), is related to $$\left\langle\!\! \Psi\! \left| \phi_2\!\left(\frac{t_1-t_2}{2}\right)\! \phi_1\!\left(\!-\frac{t_1-t_2}{2}\right)\! \right| \!\Psi\!\! \right\rangle \equiv \langle \Psi | \phi_2(-t_0) \phi_1(t_0) | \Psi \rangle \equiv \langle \phi(t_0-i\beta/2) \phi(t_0) \rangle_\beta \equiv \langle \phi \phi \rangle (t_0)$$ by time translation since time runs backwards in CFT$_2$. The geodesics determining $\langle \phi\phi \rangle(t_0)$ are the simplest to study due to their symmetry, which guarantees that their closest approach to the black hole singularity falls at $t = -i\beta/4$. In the following we concentrate on those and adopt a notation, in which correlators depend on a single parameter, $t_0$, which in the bulk perspective is the time at which a symmetric geodesic reaches a boundary. A simple study of the geodesics shows that there exists a time $t_c<0$ such that correlation functions of opetators inserted before $t_c$ or after $-t_c$ vanish: $$t_0 \not\in \[t_c, -t_c\] \quad \Rightarrow \quad \langle \phi \phi \rangle (t_0) = 0$$ In the parlance of gravity, if a spacelike geodesic departs the boundary outside this interval, it cannot be symmetric and spacelike. In between $t_c$ and $-t_c$, however, the geodesics penetrate some finite distance past the horizon, then reverse direction and continue to the other boundary. Heuristically, one could think of such geodesics as determining the amplitudes for two virtual particles created on the different asymptotic boundaries to annihilate behind the horizon. ![Spacelike geodesics in Schwarzschild-AdS. For initial times $t_0 \in \[t_c, -t_c\]$ symmetric geodesics cross the horizon, reverse direction, and escape to the other asymptotic boundary; outside this interval geodesics cannot be symmetric and spacelike. Points in the bulk at $t=0$ are traversed by three distinct spacelike geodesics, precursors of the three sheets of $\mathcal{L}(t_0)$.[]{data-label="geodesics"}](geodesics.pdf) What is the signature of the black hole singularity in this language? We know we must insert operators on two different boundaries, but within this family of correlators, some will be more useful for probing the black hole singularity than others. To wit, the $t=0$ correlator is given by the $t=0$ radial line, which skirts the horizon at a point and does not approach the singularity. The correlators with $t \sim t_c$ are more promising, because the relevant geodesics get arbitrarily close to the singularity. They also become almost null, so one may anticipate that the signature of the black hole singularity will be a pole in the correlator at $t = t_c$. The bad news for an observer hoping to peek behind the horizon is that such a pole is forbidden: $$\begin{aligned} \langle \phi \phi \rangle(t_0) & = & \sum_{nm} e^{-\beta(E_n+E_m)/2 - 2it(E_n -E_m)} |\langle n | \phi(0) | m \rangle|^2 \nonumber \\ & \leq & \sum_{nm} e^{-\beta(E_n+E_m)/2} |\langle n | \phi(0) | m \rangle|^2 = \langle \phi \phi \rangle(0)\end{aligned}$$ The quantity $\langle \phi \phi \rangle(0)$ is finite and explicitly computable from the $t_0=0$ geodesic, which is simply the dashed line in Fig. \[schwarzschildads\]. Thus, the expected lightcone pole is absent from the two-sided Lorentzian correlator. To understand the correlator as $t_0$ approaches $t_c$, [@Fidkowski:2003nf] compute the action $\mathcal{L}$ of a spacelike geodesic. Near $t_0=0$ its dependence on the boundary insertion time $t_0$ is $$\mathcal{L}(t_0) \sim -t_0^{4/3},$$ which signals that in complexified Schwarzschild-AdS three different geodesics connect a generic pair of points on the two boundaries. A way to visualize this is to consider geodesics connecting points in the bulks of the two outside-the-horizon regions. For example, for a given $s$, the boundary-boundary geodesics with $t_0=-s,0,s$ coalesce at two different spacetime points, as illustrated in Fig. \[geodesics\]. In the complexified Schwarzschild-AdS the triumvirate of geodesics persists across the bulk, with the exception of some non-generic points such as $(r,t)=(\infty, 0)$, where the trio becomes degenerate. The conclusion is that $\mathcal{L}(t_0)$ is a three-sheeted Riemann surface. One of the sheets yields real geodesics in Lorentzian signature; the naïvely anticipated lightcone pole at $t_0 = t_c$ in the correlator would arise from this real sheet. The other two sheets define complexified geodesics in complexified Schwarzschild-AdS. The authors of [@Fidkowski:2003nf] find that in Euclidean signature for $t_0$ not too close to 0 the contributions of the complexified geodesics dominate the contribution from the real sheet. Because the Lorentzian correlators are defined as analytic continuations of their Euclidean counterparts, we conclude that the real sheet does not contribute to the real time correlator, which explains the absence of the lightcone pole. However, near $t=0$ the real sheet is dominant. Consequently, an analytic continuation of the correlator from the neighborhood of $t=0$ allows one to recover the real sheet along with the $t=t_c$ pole, a signature of the black hole singularity, despite its absence from the Lorentzian correlator. Analytic continuation is a ticket for an excursion beyond the horizon. These findings can be used in a variety of ways. Ref. [@Fidkowski:2003nf] includes a discussion of finite $m$, $\alpha'$, and $g_s$ corrections while the authors of [@Festuccia:2005pi] study analytically continued correlators in Fourier space. The authors of [@Levi:2003cx; @Balasubramanian:2004zu] used similar techniques to explore the representation of the [*inner*]{} horizons of rotating black holes, and their instability to collapse, in the dual field theory description From the perspective of the black hole information paradox, a salient feature of these efforts is the treatment of the horizon as being “really there” in spacetime, at least in the description of thermal density matrices in the field theory, and not as an artifact of an effective description of many underlying microstates. Ref. [@Fidkowski:2003nf] found that the scale $\exp{(-S)}$ is not relevant the discussion of excursions beyond the horizon. Meanwhile [@Festuccia:2005pi] pointed out that the analytic continuation does not commute with the large $N$ limit. In the strict large $N$ limit, where the semiclassical description with a horizon is valid, analytic continuation seems to permit us to probe behind the horizon. But this analytic continuation need not have much to do with physics at finite $N$ where, as we have argued, the horizon simply arises in the course of an approximate description of individual microstates. Indeed, the analytic continuation to imaginary time of correlation functions computed in particular microstates will generally disagree markedly from the analytic continuation of correlation functions computed in the strictly thermal density matrix at large N [@Balasubramanian:2007qv]. This suggests that studies of excursions behind the horizon, while certainly informative about the structure of physics in the semiclassical limit and the dual CFT representation of spacetime singularities, will not tell us about the recovery of information from black holes. Additional saddle points {#addsaddle} ------------------------ In the preceding subsection we employed the Hartle-Hawking state (\[hhstate\]) for computing correlation functions in the Schwarzschild-AdS background. In its original derivation [@Hartle:1983ai], one glues along the $t=0$ section the upper part of the Penrose diagram of Fig. \[schwarzschildads\] with its Euclidean continuation. The Euclidean path integral determines the state (\[hhstate\]), which then evolves in Lorentzian time. However, it has been argued [@Maldacena:2001kr; @Hawking:2005kf] that the Euclidean path integral receives contributions from geometries other than the black hole, and that those may restore unitarity. Said differently, this proposal posits that the state (\[hhstate\]) neglects contributions of other saddle points and is only part of the full, unitary story. The general arguments of Sec. \[eSbasis\]-\[eSsup\] indicate that the key to restoring unitarity lies in quantum gravity. Saddle points of the Euclidean path integral pick out semiclassical solutions to the equations of motion, so it is hard to imagine how an extra saddle point may ever repair unitarity. It is not surprising, therefore, that a quantitative examination of CFT correlation functions [@Barbon:2003aq; @Barbon:2004ce; @Kleban:2004rx; @Barbon:2005jr] did not find the Poincare recurrences expected in a unitary theory. Furthermore, one expects that a [*sine qua non*]{} for unitarity is finite $N$, yet the onset of additional saddle points happens via the Hawking-Page transition [@Hawking:1982dh], which persists to infinite $N$. The latter objection was raised and emphasized by the authors of [@Iizuka:2008hg]; their model is reviewed in Sec. \[ipmodel\] above. The black hole final state {#bhfinalstate} -------------------------- It has been suggested that unitarity in black hole physics may be restored by imposing a boundary condition at the singularity of the black hole [@Horowitz:2003he]. Ordinarily, information loss occurs because particles that fall inside a black hole are eaten by the singularity. On the other hand, outgoing Hawking radiation is accompanied by an infalling counterpart, as can be seen from ordinary energy conservation and from the original derivation [@Hawking:1974rv]. One expects that the states of outgoing and infalling Hawking radiation are maximally entangled. The proposal of [@Horowitz:2003he] is to impose an appropriate boundary condition at the singularity, which will effectively entangle infalling matter and infalling Hawking radiation. The chain of two entanglements, one between infalling matter and infalling Hawking radiation, and one between infalling and outgoing Hawking radiation, would effectively ensure that no information is lost. A natural objection to this scenario is that it seems to burden the physics with a teleological ingredient. It has also been pointed out [@Gottesman:2003up] that the feasibility of the scenario depends sensitively on the interactions between infalling matter and Hawking radiation inside the black hole. From the viewpoint of the effective field theory describing the black hole interior away from the singularity, the black hole final state proposal is fine-tuned. Remnants {#secremnants} -------- We have not yet considered the logical possibility that the process of black hole evaporation in fact emits no information to the environment, but instead leaves behind a remnant that stores the information in its internal state. In addition to its arguable lack of parsimony, this idea suffers from an infinite production problem [@Giddings:1994qt]: since the remnant must be able to encode the pedigree of every black hole, its density of states must skyrocket around the Planck scale, which leads to infinite pair production. Evading the problem usually runs into problems with energy conservation, locality, crossing symmetry, or leads to fine-tuned interactions between remnants and ordinary fields. An interesting realization of the idea of remnants involves a third quantized theory of baby universe remnants [@Polchinski:1994zs]. In another scenario, black hole singularities are conjectured to connect two distinct semiclassical spacetimes. It has been argued [@Smolin:1994vb] that this conjecture unifies a resolution of the black hole information paradox with a retrodiction of the parameters of the standard models of particle physics and cosmology. This happens via a selection principle loosely analogous to anthropic reasoning, which favors models that maximize the production of black holes. For a recent account of baby universe remnants, see [@Hossenfelder:2009xq] and references therein. Open problems and outlook {#further} ========================= This paper has reviewed many approaches to information recovery from black holes, especially in the context of the AdS/CFT correspondence where the problem can be posed precisely and sometimes quantitatively. We have emphasized that the key to both information loss and recovery lies in phenomena that correct the observables at $O(e^{-S}) = O(e^{-A/4 G_N \hbar})$. Such non-perturbatively tiny corrections to observables are inevitable in any theory that microscopically realizes the enormous statistical degeneracy associated to a black hole. A central question that we have not discussed is how and why the entropy of black holes gets realized as a geometric quantity in spacetime, i.e., the horizon area. At present there is no answer to this question or even a particularly good approach to answering it, at least within string theory.[^6] Put otherwise, if a system acquires a sufficiently large statistical degeneracy, why does it also develop a distinguished co-dimension one surface in spacetime which reflects the logarithm of the degeneracy? Mathur has suggested that this phenomenon occurs because the underlying bound states that give rise to black hole degeneracy necessarily acquire a large transverse size [@Mathur:1997wb]. Some of the evidence for this was discussed in Sec. 3.2.3 and the references cited there, but even if this is the case we do not have a quantitative argument explaining why the dynamics of string theory produces a horizon proportional to entropy. Another important question is whether the horizon is “really there”. As emphasized in this review, the causal disconnection of an “interior” by a horizon is likely to be an artifact of coarse-graining over quantum gravitational details of the spacetime. Pushing this sort of idea to its logical limits suggests the possibility, raised by Verlinde, that gravity might really be an entropic force [@Verlinde:2010hp]. If this point of view can be quantitatively realized, one would presumably find that semiclassical probes are absorbed into black holes and are unable to get out for phase space reasons – i.e. there are many more configurations with the probe absorbed than released. While there does not appear to be any room in this picture for the destruction of fine-grained information (and plenty of room for the destruction of coarse-grained information), it would be interesting to better understand how our conventional picture of Hawking radiation, the black hole interior, asymptotic observers, and infalling observers could fit within an entropic gravity scenario. Is there, for example, a precise notion of black hole complementarity here? The approaches to information loss discussed in this review largely address the point of view of the asymptotic observer and the recovery of information at infinity. But what about the perspective of the infalling observer? In the semiclassical picture such an observer effectively propagates in a well-defined “interior geometry” before encountering a singularity. Does this picture remain valid in a fully quantum theory that resolves information loss, and if so, how? A recent approach to this question involved the holographic description of D-branes falling through the horizon of an AdS black hole [@Horowitz:2009wm]. The authors found evidence that the fully quantum spacetime should not simply be viewed as a small smoothing out of the semiclassical geometry near the singularity. Rather, they suggested that D-brane probes penetrating the horizon effectively enter a non-geometric phase of some kind, so that concepts like a global event horizon that are well defined in the classical limit simply do not have a meaningful quantum analog. From this perspective, the event horizon and the causal disconnection of an “interior” region appear as artifacts of the classical limit (Fig. 4 of [@Horowitz:2009wm]) very much as advocated in this review. It would be very useful to study such scenarios more carefully. Understanding the black hole interior holographically would certainly shed light on information loss and recovery in these geometries. The effort would also elucidate other basic puzzles in gravity. One of these is whether time can be emergent. Recall that the radial direction of classical black holes becomes timelike behind the horizon and this is why observers penetrating the horizon are necessarily drawn into the the singularity. A holographic understanding of the black hole interior must somehow describe this interior time as an emergent phenomenon. Via the AdS/CFT correspondence we have many precise examples of emergent [*space*]{}, and there have been occasional suggestions that the time in cosmological settings might emerge from the collective dynamics of a Euclidean theory [@Strominger:2001pn; @Balasubramanian:2001nb; @Balasubramanian:2006sg]. However, the latter proposals are on much less solid footing than the standard AdS/CFT correspondence. Related to this is the issue of resolving the spacelike (localized in time) singularities of classical gravity. Such singularities appear inside Schwarzschild black holes and at the Big Bang. String theory has provided beautiful resolutions to many kinds of singularities in gravity, but all of these have been timelike (localized in space) or null. There may be some connection between the challenge of resolving spacelike singularities, the question of emergent time, and the conundrum of information loss in black holes. The approaches to the information problem reviewed in this paper seek to reconcile unitarity on a fundamental level with information loss on the level of an effective description. This implies that a sufficiently powerful observer can recover information fallen into a black hole. However, there is a trade-off between how fast and how easy this recovery process is: as an example, the conclusion that black holes are information mirrors [@Hayden:2007cs] (see Sec. \[secmirror\]) relies on the assumption that they eject information in a maximally homogenized form. It is possible that the two-step process of (1) waiting for information to come out of a black hole and (2) decoding the information from a scrambled signal is subject to a fundamental limitation, with a barrier to step (2) coming from complexity theory. Indeed, it has been suggested [@Aaronson:2005qu] that complexity theory may contain lessons about fundamental laws of nature.[^7] Black holes are perhaps the most likely arena for realizing this possibility. Acknowledgements {#acknowledgements .unnumbered} ================ We thank our collaborators for extensive discussions of the ideas presented in this review. BC is supported by the Natural Sciences and Engineering Research Council of Canada. VB is supported by DOE grant DE-FG02-95ER20893. [..]{} S. W. Hawking, “Black hole explosions,” Nature [**248**]{}, 30 (1974). S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys.  [**43**]{}, 199-220 (1975). 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--- abstract: | We present the general form of potentials with two given energy levels $% E_{1} $, $E_{2}$ and find corresponding wave functions. These entities are expressed in terms of one function $\xi (x)$ and one parameter $\Delta E=E_{2}$-$E_{1}$. We show how the quantum numbers of both levels depend on properties of the function $\xi (x)$. Our approach does not need resorting to the technique of supersymmetric (SUSY) quantum mechanics but generates the expression for the superpotential automatically. address: | [$^{1}$B. Verkin Institute for Low Temperature Physics and Engineering, 47,]{}\ Lenin Prospekt, Kharkov 61164, Ukraine\ E-mail: dolya@ilt.khakrov.ua\ $^{2}$Department of Physics, Kharkov V.N. Karazin’s National University,\ Svoboda\ Sq.4, Kharkov 61077, Ukraine\ E-mail: aptm@kharkov.ua author: - 'S. N. Dolya$^{1}$ and O. B. Zaslavskii$^{2}$' title: General approach to potentials with two known levels --- Introduction ============ The potentials whose spectrum can be found exactly are very rare in quantum mechanics. Meanwhile, the condition of exact solvability can be weakened: one may demand that only for a finite part of the spectrum eigenstates and eigenvalues be found explicitly or from a finite algebraic equation. This opens two different possibilities. First, there exist so-called quasi-exactly solvable (QES) systems, whose Hamiltonian can be expressed in terms of the generators of the algebra having finite-dimensional representation (for one-dimensional potentials the relevant algebra is $% sl_{2}$, the corresponding generators having the meaning of the effective spin operators) [@zu1] - [@z90]. In so doing, the dimension of the finite subspace of the whole Hilbert space is determined by the value of the effective spin that usually enters the QES potential as a parameter. Second, instead of relating the dimension of the finite subspace to an underlying structure of a Lie algebra representation, one may fix the number of known levels ”by hands”. In the simplest case this number is equal to two, so we deal with two-dimensional subspace. Although such a procedure makes the underlying algebraic structure more poor, it extends considerably the set of potentials with the known part of the spectrum. Physical motivation for interest in potentials with two known energy levels stems from the fact that a two-level system represents a very wide class of models often used in solid state and nuclear physics and quantum optics. Let us mention here only few examples: the Dicke model of interaction between atoms and radiation [@dic], Lipkin-Meshkov-Glick model of interacting nucleons [@gil], the phenomenon of macroscopic quantum tunnelling [@leg]. We would like to stress that it is just the potential description of systems with a finite number of energy levels that enabled one to give clear and simple explanation of the phenomenon of spin tunnelling [@rev]. Therefore, finding potentials, that correspond to a fixed numbers of eigenstates, was an important step in calculation of tunneling rates in ferro- and superparamagnets. Meanwhile, there is also the inner motivation that stems from quantum mechanics as such. From general viewpoint, recovering potentials from a known set of eigenvalues is nothing else than the reduced variant of the inverse scattering problem. As is well known, using Darboux transformation, one can get many-soliton solutions of the Schrödinger equation with N energy levels, fixed in advance. Understanding, how the truncation of the scattering data modifies the structure of the theory, could gain further insight into the inverse scattering approach. The first necessary step here is to find the full solution of the problem for N=2. If N=1 (only one level is fixed), it follows from the Schrödinger equation that the potential is $U=E+\psi ^{\prime \prime }/\psi $, where $E$ is the value of energy, $\psi $ is a wave function. Choosing any $\psi (x)$ having no zeros at the real axis, we obtain immediately the corresponding potential $U(x)$, regular on the real axis. We would like to stress, however, that, in contrast to the N=1 case, when the solution of the problem is straightforward, already for N=2 resolving this problem needed the elaboration of different approaches discussed in literature. The existence of exact solutions with two levels for power-like potentials was indicated in [@fl2], [@leach]. The rather powerful technique based on supersymmetric (SUSY) quantum mechanics (see the review [@susy]) was suggested in [@tk1], [@tk2]. It enables one to generate the potentials with known ground and first excited states. The aim of the present paper is to suggest a general approach to the potentials with two known levels valid for any n-th excited states. The corresponding method and results turn out to be surprisingly simple and do not require sophisticated technique (for instance, such as SUSY quantum mechanics). Basic equations =============== Consider the Schrödinger equation with the Hamiltonian $H=-\frac{d^{2}}{% dx^{2}}+U(x)$. Let $\psi _{1}$ and $\psi _{2}$ be wave functions obeying the Schrödinger equation: $$\begin{aligned} H\psi _{1} &=&E_{1}\psi _{1}\text{,} \label{e1} \\ H\psi _{2} &=&E_{2}\psi _{2}\text{.} \label{e2}\end{aligned}$$ Then it follows from (\[e1\]), (\[e2\]) that $$\begin{aligned} U &=&E_{1}+\frac{\psi _{1}^{\prime \prime }}{\psi _{1}}\text{,} \label{wv1} \\ \frac{\psi _{2}^{\prime \prime }}{\psi _{2}} &=&E_{1}-E_{2}+\frac{\psi _{1}^{\prime \prime }}{\psi _{1}}\text{.} \label{wv2}\end{aligned}$$ Let, by definition, $$\psi _{2}=\xi \psi _{1}\text{.} \label{def}$$ Then we have for $\psi _{2}$ from (\[wv2\]): $$\frac{\psi _{1}^{\prime }}{\psi _{1}}\equiv -\chi ^{\prime }=-\frac{(\xi ^{\prime \prime }+\Delta E\xi )}{2\xi ^{\prime }}\text{, } \label{q}$$ where $\Delta E=E_{2}-E_{1}$. By substitution of (\[def\]) and (\[q\]) to (\[wv2\]), we obtain three equivalent forms for the potential: $$\begin{aligned} U &=&E_{1}-\frac{\Delta E}{2}+\frac{3}{4}(\frac{\xi ^{\prime \prime }}{\xi ^{\prime }})^{2}-\frac{1}{2}\frac{\xi ^{\prime \prime \prime }}{\xi ^{\prime }}+\Delta E\frac{\xi \xi ^{\prime \prime }}{\xi ^{\prime 2}}+\frac{1}{4}% (\Delta E)^{2}(\frac{\xi }{\xi ^{^{\prime }}})^{2}\text{,} \label{ux} \\ U &=&E_{1}-\frac{\Delta E}{2}(1-2\frac{\xi \xi ^{\prime \prime }}{\xi ^{\prime 2}})+\frac{1}{4}(\Delta E)^{2}(\frac{\xi }{\xi ^{^{\prime }}})^{2}-% \frac{1}{2}[\xi ]_{x}\text{,} \label{sc} \\ U &=&E_{1}+\chi ^{\prime 2}-\chi ^{\prime \prime }\text{,} \label{su}\end{aligned}$$ where $[\xi ]_{x}\equiv \frac{\xi ^{\prime \prime \prime }}{\xi ^{\prime }}-% \frac{3}{2}(\frac{\xi ^{\prime \prime }}{\xi ^{\prime }})^{2}$ is Schwarzian derivative (see, e.g., Ch. 2.7 of Ref. [@ba]). The wave functions of the states under discussions are $$\psi _{1}=e^{-\chi }\text{, }\psi _{2}=e^{-\chi }\xi \text{.} \label{wave}$$ Eq. (\[ux\]) gives us the general formula for the potential with two given energy levels. It is expressed directly in terms of their values $E_{1}$, $% E_{2}$ as parameters and one function $\xi (x)$, corresponding wave functions are given by (\[q\]), (\[wave\]) and expressed in terms of the same quantities. It is worth noting that the function $\xi (x)$ does not enter the set of known data - rather, the freedom in its choice reflects the fact that for two given eigenvalues there exists an infinite number of potentials having two fixed eigenvalues. The structure of these potentials is not arbitrary but is governed by the form of $\xi (x)$ according to (7). Eqs. (\[ux\]), (\[q\]) and (\[wave\]) constitute the main result of this paper. It is worth stressing that the derivation of eq. (\[ux\]) is very simple, direct and does not need sophisticated technique, such as SUSYmachinery. On the other hand, the potential in terms of the function $\chi ^{\prime }$ has the form (\[su\]), typical for SUSY quantum mechanics, automatically. In so doing, $\chi ^{\prime }$ plays the role of a superpotential. As is well known (see, e.g., [@susy]), one-dimensional quantum mechanics can always be formulated in a SUSY way. However, given a potential, the superpotential cannot be found explicitly for a generic model. Meanwhile, in our case we found not only the potential but the explicit expression for the superpotential (\[q\]) as well. It is worth stressing that the derivation of (\[ux\]) - (\[su\]) relies strongly on the successful choice of the function $\xi (x)$ that parametrizes the family of solutions. The fact that, for given $E_{1}$, $% E_{2}$, the ratio of two eigenfunctions determines the potential completely generalizes the observation made in Refs. [@tk1], [@tk2] for the particular case when the eigenfunctions under consideration refer to the ground and first excited states. The formalism elaborated above for the one-dimensional Schrödinger equation can be also applied to the three-dimensional one for a particle moving in a spherically-symmetrical potential $U(r)$. After the separation of variables, the effective potential entering the radial part of the Schrödinger equation, is equal to $U_{ef}=U+$ $\frac{l(l+1)}{r^{2}}$. Then, repeating calculations step by step, we obtain $$U=U^{(0)}+\lambda ^{2}\frac{\xi ^{2}}{4r^{4}\xi ^{\prime }}-\frac{\lambda \xi }{r^{2}\xi ^{\prime }}(\frac{1}{r}+\frac{\xi ^{\prime \prime }}{\xi }% )-\lambda \frac{\Delta E\xi ^{2}}{2r^{2}\xi ^{\prime 2}}\text{.}+\frac{% \lambda -2l_{1}(l_{1}+1)}{2r^{2}}\text{,} \label{ans}$$ $$\lambda =(l_{2}-l_{1})(1+l_{1}+l_{2}) \label{la}$$ and $U^{(0)}$ is expressed in terms of $E_{1}$, $E_{2}$ and $\xi $ by the same formulas (\[ux\]) - (\[su\]) as in the one-dimensional case. It is worth noting that now a new interesting possibility can arise that is absent in the one-dimensional case: $\Delta E=0$. It becomes possible due to the fact that two quantum states can refer to different effective potentials ($l_{1}\neq l_{2}$): we are faced with degeneracy with respect to the angular momentum. Then the potential acquires the form $$U=E_{1}-\frac{1}{2}[\xi ]_{r}+\lambda ^{2}\frac{\xi ^{2}}{4r^{4}\xi ^{\prime }}-\frac{\lambda \xi }{r^{2}\xi ^{\prime }}(\frac{1}{r}+\frac{\xi ^{\prime \prime }}{\xi })\text{.}+\frac{\lambda -2l_{1}(l_{1}+1)}{2r^{2}}\text{.} \label{deg}$$ We will not discuss the three-dimensional case further and will concentrate on the one-dimensional one. General properties and classification of states =============================================== The potential $U\equiv U(E_{1},E_{2},\xi )$ possesses the symmetries that follow directly from (\[ux\]): $$\begin{aligned} &&U(E_{1},E_{2},a\xi )=U(E_{1},E_{2},\xi )\text{,} \label{sym} \\ &&U(E_{1},E_{2},\xi )=U(E_{2},E_{1},\xi ^{-1})\text{.} \nonumber\end{aligned}$$ Throughout the paper we assume that the potential $U(x)$ is regular everywhere, except, perhaps, infinity. Then all zeros and poles of the function $\xi (x)$ are simple - otherwise the potential $U$ would become singular and the wave function $\psi _{2}$ would cease to be normalizable. If the function $\xi $ has a pole at $x=x_{1}$, $\xi \approx A(x-x_{1})^{-1}$, one gets from (\[q\]) that $\chi ^{\prime }\approx -(x-x_{1})^{-1}$, so $% \psi _{1}(x_{1})=0$, $\psi _{2}(x_{1})=const\neq 0$. Therefore, every zero of $\xi $ generates a node of the wave function $\psi _{2}$ and every pole of $\xi $ generates a node of $\psi _{1}$. The set of possible nodes of wave functions depends also on the behavior of the function $\chi (x)$ in the vicinity of zeros of the function $\xi ^{\prime }(x)$ due to possible zeros of the factor $\exp (-\chi )$ in (\[wave\]). Let $\xi ^{\prime }(x_{0})=0$. Then, according to (\[q\]), if $% x\rightarrow x_{0}$, $\chi ^{\prime }\approx B/(x-x_{0})$, where $B=$ $\frac{% (\xi ^{\prime \prime }+\Delta E\xi )_{\mid x=x_{0}}}{2\xi ^{\prime \prime }(x_{0})}$, and the potential contains the term that behaves like $% B(B+1)(x-x_{0})^{-2}$. The regularity of the potential entails $B=0$ or $% B=-1.$ Consider these two cases separately. Let, first, $B=0.$ Now the condition $$(\xi ^{\prime \prime }+\Delta E\xi )_{\mid x=x_{0}}=0 \label{reg}$$ must hold. In so doing, the function $\chi (x)$ is regular in the vicinity of $x_{0}$ due to the condition (\[reg\]) and the factor $\exp (-\chi )$ cannot vanish. Consider the case $B=-1$. Now we have $$(3\xi ^{\prime \prime }+\Delta E\xi )_{\mid x=x_{0}}=0 \label{reg2}$$ Then $\chi ^{\prime }\approx -(x-x_{0})^{-1}$ and the functions $\psi _{1}$ and $\psi _{2}$ share the common node at $x=x_{0}$ due to the factor $\exp (-\chi )$, as it follows from (\[wave\]). (For example, if the potential is even, $U(-x)=U(x)$, all wave functions of odd states vanish at $x=0$.) As a result, we arrive at the conclusion that, if (i) $\xi (x)$ has $n_{1}$ poles and $n_{2}$ zeros, (ii) $\xi ^{^{\prime }}(x)$ has $m^{(0)}$ zeros such that (\[reg\]) is satisfied ($B=0$) and $m^{(-)}$ zeros such that (\[reg2\]) is satisfied ($B=-1$), the function $\psi _{1}(x)$ describes the state with the number of nodes $N_{1}=n_{1}+m^{(-)}$, while $\psi _{2}(x)$ corresponds to the state with the number of nodes $N_{2}=n_{2}+m^{(-)}$. Therefore, the quantum number that label states is equal to $N_{1}$ for $% \psi _{1}$ and $N_{2}$ for $\psi _{2}$ ($N_{1,2}=0$, $1$, $2...$). It follows directly from the definition (\[def\]): if $\psi _{1}$ has simple zeros at $x_{i}$ and $\psi _{2}$ has simple zeros at $x_{k}$ with $% x_{i}\neq x_{k}$, the function $\xi $ has poles at $x=x_{i}$ and zeros at $% x=x_{k}$. However, if some $x_{i}=x_{k}$, corresponding zeros of both functions compensate each other and this results in the fact that, if some coefficients $B=-1$, the state labels are not determined completely by the numbers $n_{1}$, $n_{2}$. Let us have two fixed energy levels $E_{1}$, $E_{2}$ ($E_{2}>E_{1}$) and the function $\xi (x)$ such that $N_{2}>N_{1}$. Then, it is the potential $% U(E_{1},E_{2},\xi )$ for which the level $E_{1}$ belongs to the $N_{1}$-th state and $E_{2}$ corresponds to the $N_{2}$-th state. If $N_{1}>N_{2}$, the relevant potential is $U(E_{1},$ $E_{2},$ $\xi ^{-1})=U(E_{2},$ $E_{1,}$ $% \xi )$, the level $E_{1}$ belongs to the $N_{2}$-th state, while $E_{2}$ corresponds to the $N_{1}$-th state. If $N_{1}=N_{2}$, the function $\xi (x)$ is not suitable for constructing $U(x)$ with two different fixed levels. Indeed, in this case we would have had two different wave functions with the same number of zeros corresponding to two different levels, in disagreement with the oscillation theorem. Comparison with susy approach and tkachuk’s results =================================================== Let us introduce the function $W_{+}$ according to $$W_{+}=\frac{\delta \xi }{\xi ^{\prime }}\text{,} \label{nw}$$ where $\delta =\Delta E>0$. Then, with (\[q\]) taken into account, we obtain $$\begin{aligned} \chi ^{^{\prime }} &=&\frac{1}{2}(W_{+}-\frac{W_{+}^{^{\prime }}-\delta }{% W_{+}})\equiv W\equiv \frac{W_{+}-W_{-}}{2}\text{, }U=E_{1}+W^{2}-W^{^{% \prime }}\text{,} \label{ss} \\ \psi _{1} &=&e^{-\int dxW}\text{, }\psi _{2}=\xi e^{-\int dxW}=W_{+}\exp [-% \frac{1}{2}\int dx(W_{+}+W_{-})]\text{,} \nonumber\end{aligned}$$ where, by definition, $$W_{-}=\frac{W_{+}^{^{\prime }}-\delta }{W_{+}}\text{. } \label{w-}$$ Since $\psi _{1}$ must be normalizable, sign$(W_{+}(\pm \infty ))=\pm 1$. Let $W_{+}$ have only one zero at $x=x_{0}$. If we want $W_{-}$ to be regular at $x=x_{0}$, $W_{+}^{^{\prime }}(x_{0})=\delta $. The formulas (\[ss\]) (with $E_{1}=0$ and $\delta =2\varepsilon $) were derived in [@tk1] by solving equations for the superpotential which appear in the SUSY approach. In our terms, this approach deals with the function $\xi (x)$ such, that $\xi $ has only one zero (just in the point $% x_{0}$), $\xi ^{\prime }$ changes sign nowhere and $\xi $ does not have poles on a real axis (otherwise they would give rise to additional zeros of $% W_{+}$). Therefore, in the situation considered in [@tk1], [@tk2] $% \psi _{1}$ corresponds to the ground state and $\psi _{2}$ describes the first excited state - in agreement with the conclusion of the previous section of the present article. Thus, in this particular case our approach reproduces the results of [@tk1], [@tk2]. Illustrations. Deformations of potential leaving two levels fixed ================================================================= To illustrate the general results (\[ux\]) - (\[su\]), let us consider the following example: $\xi =x^{4}+2x^{2}x_{0}^{2}-x_{1}^{4}$. The derivative $\xi ^{\prime }=0$ at $x=0$; therefore, as is explained in the preceding section, the corresponding example cannot belong to the set considered in [@tk1]. After straightforward calculations, one obtains $$U=E_{1}+\frac{x^{2}x_{0}^{4}}{4x_{1}^{8}}+\frac{1}{4x_{1}^{4}}[A_{0}+\frac{% A_{1}}{x^{2}+x_{0}^{2}}+\frac{A_{2}}{(x^{2}+x_{0}^{2})^{2}}]\text{,} \label{exu}$$ where $$A_{0}=2x_{0}^{2}(2+\frac{x_{0}^{4}}{x_{1}^{4}})\text{, }% A_{1}=(3x_{1}^{4}+x_{0}^{4})(5-\frac{x_{0}^{4}}{x_{1}^{4}})\text{, }% A_{2}=-(3x_{1}^{4}+x_{0}^{4})x_{0}^{2}(7+\frac{x_{0}^{2}}{x_{1}^{4}})\text{.} \label{ai}$$ The functions are equal to $$\psi _{1}=(x^{2}+x_{0}^{2})^{-\alpha }\exp (-\frac{x^{2}x_{0}^{2}}{2x_{1}^{4}% })\text{, }\alpha =\frac{3x_{1}^{4}+x_{0}^{4}}{4x_{1}^{4}}\text{,} \label{w1}$$ $$\psi _{2}=\psi _{1}(x^{2}-x_{-}^{2})(x^{2}+x_{+}^{2})\text{, }x_{\pm }=\sqrt{% x_{0}^{4}+x_{1}^{4}}\pm x_{0}^{2}\text{.} \label{w2}$$ It is seen from (\[w1\]), (\[w2\]) that $\psi _{1}$ has no nodes at the real axis, while $\psi _{2}$ turns into zero at $x=\pm x_{-}$. Therefore, $% \psi _{1}$ corresponds to the ground state, while $\psi _{2}$ describes the second excited state. As we see from (\[sc\]), the Schwarzian derivative is an essential ingredient of the expression for the potential under discussion. It is known that the Schwarzian derivative is invariant with respect to the linear-fractional transformations. Therefore, it is instructive to apply such a transformation to the potential as a whole and look at the resulting expression. Let us make the substitution $$\xi =\frac{c_{2}\eta +d_{2}}{c_{1}\eta +d_{1}}\text{.} \label{c}$$ We will use it below for generating in an explicit form rather rich families of the potentials, corresponding to two known levels. As $[\eta ]_{x}$ remains invariant, only the part of (\[sc\]) contains the terms proportional to $\Delta E$ and $(\Delta E)^{2}$, changes under this transformation. Then the potential and wave functions of the states under discussion take the form $$\begin{aligned} U &=&E_{1}-\frac{\Delta E}{2}+\frac{2\Delta Ec_{1}(d_{2}+c_{2}\eta )}{% c_{1}d_{2}-c_{2}d_{1}}+\frac{1}{4}\frac{Y^{2}}{\eta ^{\prime 2}}-\frac{\eta ^{\prime \prime }}{\eta ^{\prime 2}}Y-\frac{1}{2}[\eta ]_{x}\text{,} \label{uy} \\ Y &=&\Delta E\frac{(c_{1}\eta +d_{1})(c_{2}\eta +d_{2})}{% c_{1}d_{2}-c_{2}d_{1}}\text{.} \nonumber\end{aligned}$$ $$\begin{aligned} \psi _{1,2} &=&e^{-\rho }\Phi _{1,2}\text{, }\Phi _{1,2}=c_{1,2}\eta +d_{1,2}% \text{,} \label{wy} \\ \rho ^{^{\prime }} &=&\frac{\eta ^{\prime \prime }-Y}{2\eta ^{\prime }}\text{% , }\chi =\rho -\ln (c_{1}\eta +d_{1})\text{.} \nonumber\end{aligned}$$ If $c_{2}=0=d_{1}$, one can see that $Y=\Delta E\eta $ and $% U(E_{1},E_{2},\xi )=U(E_{1},E_{2},\eta ^{-1})=U(E_{2},E_{1},\eta $) in accordance with (\[sym\]). In the limit $$c_{1}=0=d_{2} \label{lim}$$ we obtain the original potential $U(E_{1},E_{2},\xi )=U(E_{1},E_{2},\eta )$. Let us assume first we have some function $\eta (x)$ characterized by the set of numbers ($n_{1}$, $n_{2}$, $m^{(-)}$) introduced in Sec. III. The original potential has the form (\[ux\]) with $\xi =\eta $. Then, let us take $\xi (x)$ according to (\[c\]) with nonzero arbitrary coefficients $% c_{i}$ and $d_{i}$. As the result of the transformation of (\[c\]), each of the aforementioned numbers can change (for example, zeros $x_{k}^{(1)}$ of the combination $c_{1}\eta +d_{1}$ generate poles of $\xi $, zeros $% x_{i}^{(2)}$ of $c_{2}\eta +d_{2}$ correspond to zeros of $\xi $, each zero $% x_{j}^{(0)}\neq x_{k}^{(1)}$ of $\eta ^{\prime }$ generates a zero of $\xi ^{\prime }$). Therefore, the levels $E_{1}$, $E_{2}$ which corresponded to the $N_{1}$-th and $N_{2}$-th levels now can, in principle, correspond to another quantum numbers ($M_{1}$, $M_{2}$). Making the transformation, inverse to (\[c\]), one may restore the form of the potential (\[ux\]), but now $\xi \neq \eta $, with $\xi $ having the form (\[c\]), in which coefficients under discussion play the role of parameters. Thus, we obtain a family of deformations leaving two energy levels $E_{1}$, $E_{2}$ fixed. These deformations can be described, on equal footing, by the deformation of the form of the function $\xi (x)$ or of that of the potential. For definiteness, we will choose the second possibility. If $c_{1}$, $% c_{2}\neq 0$, one can always achieve $c_{1}=c_{2}\equiv c$ by proper rescaling the function $\xi (x)$ that does not affect, according to (\[sym\]), the function $U(x)$. Then, defining $$c=2\beta \text{, }d_{1}=-\bar{\delta}-\Delta E\text{, }d_{2}=-\bar{\delta}% +\Delta E\text{, }\gamma =\frac{(\Delta E)^{2}-\bar{\delta}^{2}}{4\beta }% \text{,} \label{coef}$$ we obtain $$Y=\beta \eta ^{2}-\bar{\delta}\eta -\gamma \text{,} \label{ny}$$ $$U=E_{1}+\frac{\Delta E}{2}-\bar{\delta}+2\beta \eta -\frac{1}{2}[\eta ]_{x}+% \frac{1}{4}\frac{(\beta \eta ^{2}-\bar{\delta}\eta -\gamma )^{2}}{\eta ^{^{\prime }2}}-\frac{\eta ^{^{\prime \prime }}}{\eta ^{^{\prime 2}}}(\beta \eta ^{2}-\bar{\delta}\eta -\gamma )\text{,} \label{1}$$ Below we will see how introducing nonzero parameters $\beta $,$\gamma $ affects the potential and wave functions (\[wy\]). Example 1 --------- Let us choose $\eta $ as a polynomial : $$\eta ^{^{\prime }}=4ax(x_{0}^{2}-x^{2})\text{, }a>0\text{, }\eta =a(V_{0}+x_{0}^{4}-z^{2})\text{, }z=x^{2}-x_{0}^{2}\text{, }V_{0}=const. \label{v'}$$ Demanding that $\rho ^{^{\prime }}$ be regular at $x=0$ and at $x=\pm x_{0}$, we obtain from (\[wy\]) the constraints $$\gamma =\beta a^{2}(V_{0}+x_{0}^{4})^{2}+8ax_{0}^{2}-\delta a(V_{0}+x_{0}^{4})=\beta a^{2}V_{0}^{2}-4ax_{0}^{2}-\bar{\delta}aV_{0}\text{,% } \label{g}$$ whence $$\begin{aligned} V_{0} &=&-\frac{x_{0}^{4}}{2}-\frac{6}{a\beta x_{0}^{2}}+\frac{\bar{\delta}}{% 2\beta a}\text{,} \label{x} \\ \gamma &=&\beta ^{-1}(R-\frac{\bar{\delta}^{2}}{4})\text{, }R=\frac{(\Delta E)^{2}}{4}=\frac{\beta ^{2}a^{2}x_{0}^{8}}{4}+2\beta ax_{0}^{2}+36x_{0}^{-4}% \text{.} \nonumber\end{aligned}$$ The expression for the function $\xi $ reads $$\xi =1+\frac{2\Delta E}{A_{2}x^{4}+A_{1}x^{2}+A_{0}}\text{.} \label{xempl}$$ The potential can be obtained form (\[1\]) or directly from \[ux\]. It has the form $U=\sum_{n=0}^{5}c_{2n}x^{2n}$, $c_{10}=\frac{(\beta a)^{2}}{64} $.It is convenient to rescale the variable in such a way that the coefficient in the potential $U$ at the largest power be equal to $1$. This can be achieved by $x=\lambda y$, $\beta a\lambda ^{6}=8\omega $, where $% \omega =1$ or $-1$. After some manipulations we get the new potential $\bar{U% }=\lambda ^{2}U$, corresponding to levels $\bar{E}_{1,2}=\lambda ^{2}E_{1,2}$: $$\begin{aligned} \bar{U} &=&y^{10}-6\mu y^{8}+(13\mu ^{2}+\frac{3\omega }{\mu })y^{6}-(12\mu ^{3}+22\omega )y^{4}+(4\mu ^{4}+31\mu \omega +\frac{9}{4\mu ^{2}})y^{2} \label{u1} \\ &&+\frac{\bar{E}_{1}+\bar{E}_{2}}{2}-\frac{15}{2\mu }-6\omega \mu ^{2}\text{,% } \nonumber \\ \rho &=&\frac{y^{6}}{6}-\frac{3\mu \omega y^{4}}{4}+y^{2}(\mu ^{2}\omega +% \frac{3}{4\mu })\text{,} \nonumber\end{aligned}$$ $$(\Delta \bar{E})^{2}=16\mu ^{-2}(4\mu ^{6}+4\omega \mu ^{3}+9)\text{.} \label{par}$$ where $\mu =x_{0}^{2}/\lambda ^{2}$. (In fact, the formula (\[u1\]) can be extended to negative $\mu $ as well.) The quantities $\mu $ and $\bar{E}% _{1,2}$ are not independent but connected by eq. (\[par\]) that appeared due to the condition of the regularity of the potential (\[reg\]). It is worth noting that the parameter $\bar{\delta}$ cancels and does not enter the expression (\[u1\]) due to the conditions (\[g\]), (\[x\]). It follows from (\[wy\]), (\[coef\]) and (\[x\]) that, up to the constant factor, the function $\Phi _{1,2}=z^{2}-2\mu z+q_{1,2}=(z-z_{1})(z-z_{2})$, $z\equiv y^{2}$, $z_{1,2}=\mu \pm \sqrt{\mu ^{2}-q_{1,2}}$, $q_{1,2}=\frac{\mu ^{2}}{2}+\frac{3}{4\mu }\omega \pm \frac{% \Delta \bar{E}}{16}\omega $. Let, for definiteness, $E_{2}>E_{1}$. Consider first the case $\omega =1$. Then after some algebra one easily finds that $% 0<q_{2}<\mu ^{2}$ and $q_{1}>\mu ^{2}$. Therefore, the function $\Phi _{1}$ does not have the nodes at the real axis and corresponds to the ground state. The function $\Phi _{2}$ has four zeros and corresponds to the fourth state. In a similar way, we obtain that for $\omega =-1$ the quantities $% q_{1}<0$, $0<q_{2}<\mu ^{2}$, so the wave functions under discussion describe the second and fourth excited states. Example 2 --------- One may exploit the ansatz (\[nw\]) with $\xi =\eta $ for the potential (\[1\]) with $\beta $, $\gamma \neq 0$. Substituting it into (\[wy\]), one obtains: $$\rho ^{^{\prime }}=-\frac{1}{2}[\frac{(W_{+}^{^{\prime }}-\delta )}{W_{+}}-% \frac{\bar{\delta}W_{+}}{\delta }+\frac{\beta W_{+}}{\delta }\exp (\delta \int \frac{dx}{W_{+}})-\frac{\gamma W_{+}}{\delta }\exp (-\delta \int \frac{% dx}{W_{+}})]\text{.} \label{ff}$$ Let us consider the example: $$W_{+}=ax+bx^{3}\text{, }\gamma =0\text{, }\beta \neq 0\text{, }a=\delta =% \bar{\delta}>0\text{, }b>0\text{.} \label{ex}$$ After simple but cumbersome manipulations we get the potential $$\begin{aligned} U &=&\frac{\beta x}{x_{0}}(-\frac{x_{0}^{2}}{r}+3r+\frac{ar^{3}}{2}-\frac{% ar^{5}}{2x_{0}^{2}})+u\text{, }r\equiv \sqrt{x_{0}^{2}+x^{2}\text{,}}\text{ }% x_{0}^{2}\equiv \frac{a}{b}\text{,} \label{u} \\ u &=&\frac{b^{2}}{4}x^{6}+(\frac{ab}{2}+\frac{\beta ^{2}}{4})x^{4}+\frac{% (a^{2}-12b)}{4}x^{2}-\frac{a}{2}+\frac{3}{4(x_{0}^{2}+x^{2})}+\frac{3}{4}% \frac{x_{0}^{2}}{(x^{2}+x_{0}^{2})^{2}}\text{.} \nonumber\end{aligned}$$ and wave functions $$\begin{aligned} \text{ }\psi _{1} &=&\frac{(x^{2}+x_{0}^{2})^{3/4}}{(x+\sqrt{x^{2}+x_{0}^{2}}% )^{\frac{\beta x_{0}^{3}}{16}}}e^{-\alpha }(1-\frac{\beta }{a}\eta )\text{, }% \psi _{2}=\frac{\eta (x^{2}+x_{0}^{2})^{3/4}}{(x+\sqrt{x^{2}+x_{0}^{2}})^{% \frac{\beta x_{0}^{3}}{16}}}e^{-\alpha }\text{, }\eta =\frac{xx_{0}}{\sqrt{% x^{2}+x_{0}^{2}}}\text{,} \label{exf} \\ \alpha &=&\frac{ax^{2}}{4}+\frac{bx^{4}}{8}-\frac{\beta x}{16x_{0}}% (2x^{2}+x_{0}^{2})\sqrt{x^{2}+x_{0}^{2}}\text{.} \nonumber\end{aligned}$$ It is seen from (\[exf\]) that $\eta ^{\prime }>0$ and the potential (\[u\]) is regular for any choice of parameters, so the conditions (\[reg\]), (\[reg2\]) are irrelevant for this case. The functions $\psi _{1}$ and $% \psi _{2}$ are normalizable, provided $\beta ^{2}<ab$. It can be readily seen from (\[exf\]) that the function $\psi _{2}$ has one node at $x=0$, whereas $\psi _{1}$ turns into zero nowhere. Thus, $\psi _{1}$ and $\psi _{2} $ correspond to the ground and the first excited states, respectively. In the limit $\beta =0$ we reproduce the result for the example 3 of [@tk1]. Example 3 --------- Let now $W_{+}=A(shx-shx_{0})$, $\gamma =0$, $\beta \neq 0$, $\delta =\bar{% \delta}$. Then, repeating calculations for this case, we get $$\begin{aligned} U &=&\frac{E_{1}+E_{2}}{2}-\frac{\delta }{2}+U_{0}(x)+\frac{\beta sh\frac{% (x-x_{0})}{2}}{chx_{0}ch\frac{(x+x_{0})}{2}}[chx+chx_{0}-\frac{\delta (shx-shx_{0})^{2}}{2chx_{0}}]+\frac{\beta ^{2}sh^{4}\frac{(x-x_{0})}{2}}{% ch^{2}x_{0}}\text{,} \label{U1} \\ U_{0}(x) &=&\frac{\delta ^{2}}{4ch^{2}x_{0}}(shx-shx_{0})^{2}-\frac{\delta }{% 2chx_{0}}(2chx-chx_{0})+\frac{1}{4}\text{,} \nonumber \\ \psi _{1} &=&ch\frac{(x+x_{0})}{2}e^{-\alpha }(1-\frac{\beta }{\delta }\eta )% \text{, }\psi _{2}=ch\frac{(x+x_{0})}{2}e^{-\alpha }\eta \text{, }\eta =% \frac{sh\frac{(x-x_{0})}{2}}{ch\frac{(x+x_{0})}{2}}\text{,} \nonumber \\ \alpha &=&(2chx_{0})^{-1}[\delta chx-\beta sh(x-x_{0})+x(\beta -\delta shx_{0})]\text{.} \nonumber\end{aligned}$$ Here $U_{0}$ is the potential corresponding to the anisotropic paramagnet of the spin $1/2$ in an oblique magnetic field [@zu1]. The function $\eta ^{\prime }>0$, so $\rho (x)$ is regular in any point for any choice of parameters. The wave function is normalizable provided $-\delta e^{-x_{0}}<\beta <\delta e^{x_{0}}$. One can easily show that it follows from this condition that $\psi _{1}$ does not have nodes and corresponds to the ground state, while $\psi _{2}$ has one node at $x=x_{0}$ and corresponds to the first excited state. In the limit $\beta =0$ the example 1 of [@tk1] is reproduced. Concluding remarks ================== Thus, in a very simple and direct approach we found a rather general solution that gives us the structure of potentials with two known eigenstates $E_{1}$, $E_{2}$ in terms of one function $\xi (x)$ and one parameter coinciding with the energy difference $\Delta E=E_{2}-E_{1}$. Moreover, we get not only the potential itself but, also (in terms of the SUSY language), the superpotential. Depending on properties of the function $\xi (x)$ and the type of the regularity condition of the potential in the vicinity of zeros of $\xi ^{\prime }(x)$ ((\[reg\]) or (\[reg2\])), one can obtain not only the ground or first excited state but, in principle, any pair of levels. The natural question arises whether the approach of the present paper is extendable to the case of three (or more) levels. This problem needs separate treatment. We hope that movement in this direction will promote further understanding links between QES-type systems, SUSY and the inverse scattering approaches. One of authors (O. Z.) thanks Claus Kiefer and Freiburg university for hospitality and acknowledges gratefully finansial support from the German Academic Exchange Service (DAAD). Zaslavskii O B and Ulyanov V V 1984 [*Zh. Eksp. Teor. Phys.*]{}[** 87**]{} 1724 \[[*Sov. Phys. JETP*]{} [**60**]{} 991\] Zaslavskii O B and Ulyanov V V 1987 [*Teor. Mat. Fiz*]{}. [**71**]{} 260 \[[*Theor. 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--- abstract: 'A long-standing conjecture asserts that there exists a constant $c>0$ such that every graph of order $n$ without isolated vertices contains an induced subgraph of order at least $cn$ with all degrees odd. Scott (1992) proved that every graph $G$ has an induced subgraph of order at least $|V(G)|/(2\chi(G))$ with all degrees odd, where $\chi(G)$ is the chromatic number of $G$, this implies the conjecture for graphs with [ bounded]{} chromatic number. But the factor $1/(2\chi(G))$ seems to be not best possible, for example, Radcliffe and Scott (1995) proved $c=\frac 23$ for trees, Berman, Wang and Wargo (1997) showed that $c=\frac 25$ for graphs with maximum degree $3$, so it is interesting to determine the exact value of $c$ for special family of graphs. In this paper, we further confirm the conjecture for graphs with treewidth at most 2 with $c=\frac{2}{5}$, and the bound is best possible.' author: - | Xinmin Hou$^a$, Lei Yu$^b$, Jiaao Li$^c$, Boyuan Liu$^d$\ $^{a,b,d}$ Key Laboratory of Wu Wen-Tsun Mathematics\ School of Mathematical Sciences\ University of Science and Technology of China\ Hefei, Anhui 230026, China.\ $^{c}$Department of Mathematics\ West Virginia University\ Morgantown, WV 26506, USA title: 'Odd induced subgraphs in graphs with treewidth at most two [^1] ' --- Introduction ============ Gallai [@Lovasz-CPE1979] proved that for every graph $G$, the vertex set $V(G)$ can be partitioned into two sets, each of which induces a subgraph with all degrees even. This implies that every graph of order $n$ contains an induced subgraph of order at least $\lceil \frac{n}{2}\rceil$ with all degrees even, and this is best possible by considering paths. This motivates us to consider the problem that how large we can find an induced subgraph with all degrees odd. We call a graph with all degrees odd an [*odd graph*]{}. Let $f(G)$ denote the maximum order of an odd induced subgraph in a graph $G$. The following long-standing conjecture was cited by Caro in [@Caro-DM1994] as “part of the graph theory folklore” and the origin is unclear.  \[CONJ: OSC\] There exists a constant $c>0$ such that for every graph $G$ without isolated vertices, $f(G)\geq c|V(G)|$. The “without isolated vertices” constraint is natural because an odd graph does not contain isolated vertices. Many results related to Conjecture \[CONJ: OSC\] have been obtained in literatures. In particular, Caro [@Caro-DM1994] proved that $f(G)\geq (1-o(1))\sqrt{|V(G)|/6}$, laterly, Scott [@Scott-CPC92] improved the lower bound to $\frac{c|V(G)|}{\log{|V(G)|}}$ for some $c>0$, in the same paper, Scott also proved that every graph $G$ has an odd induced subgraph of order at least $|V(G)|/(2\chi(G))$, where $\chi(G)$ is the chromatic number of $G$, this implies the conjecture for graphs with [ bounded]{} chromatic number. But the factor $1/(2\chi(G))$ seems to be not best possible, for example, Radcliffe and Scott [@RS-DM95] confirmed the conjecture for trees (graphs with treewidth one) with $c=\frac{2}{3}$ and Berman, Wang and Wargo [@BWW-AJC97] proved the conjecture for graphs with maximum degree $3$ with $c=\frac{2}{5}$. In this paper, we further confirm Conjecture \[CONJ: OSC\] for graphs with treewidth at most 2 with $c=\frac 25$, and the value of $c$ is best possible. A [*tree decomposition*]{} of a graph $G$ is a tree $T$, where \(1) Each vertex $i$ of $T$ is labeled by a subset $B_i$ of vertices of $G$. \(2) Each edge of $G$ is in a subgraph induced by at least one of the $B_i$, \(3) For every three vertices $i,j,k$ in $T$ with $j$ lying on the path from $i$ to $k$ in $T$, $B_i\cap B_k\subseteq B_j$.\ The [*tree-width*]{} tw($G$) of $G$ is the minimum integer $p$ such that there exists a tree decomposition of $G$ with all subsets of cardinality at most $p+1$. Tree-decomposition is one of the most general and effective techniques for designing efficient algorithms, and a tree-like structure allows us to solve certain difficult problems. It is well-known that a connected graph has treewidth one if and only if it is tree. In terms of treewidth, the result of Radcliffe and Scott [@RS-DM95] can be restated as follows. \[THM: RSMain\][@RS-DM95] For any connected graph $T$ with $tw(T)=1$, $f(T)\geq 2\lfloor\frac{|V(T)|+1}{3}\rfloor$ . The following theorem is our main result. \[THM: Main\] For every graph $G$ with $tw(G)\le 2$ and without isolated vertices, $f(G)\geq \frac{2}{5}|V(G)|$. The lower bound is sharp by considering the graph of which each component is a cycle of length $5$. [ We remark that graph with treewidth at most two is also known as [*$K_4$-minor-free graph*]{}, see Proposition \[PROP: p1\] in section 3. Some upper and lower bounds on graphs with small treewidth are also discussed in the last section.]{} In this paper, standard notation follows from [@Diestel-GT00]. In particular, for a graph $G$ and a set $S\subseteq V(G)$, let $G[S]$ be the subgraph induced by $S$ and let $N_G(S)$ be the union of neighbors of vertices in $S$, for a vertex $u\in V(G)$, let $N_G^1(u)=\{x\ |\ x\in N_G(u) \mbox{ and } d_G(x)=1\}$ and $N_G^2(u)=\{x\ |\ x\in N_G(u) \mbox{ and } d_G(x)=2\}$, and denote $N_G^2(u,v)=N_G^2(u)\cap N_G^2(v)$. A vertex of degree $k$ is called a [*$k$-vertex*]{}. Define $S_G(u)=\{x\ |\ x\in N_G(u)$ with $d_G(x)\geq 3$ or there exists a vertex $z\in N_G^2(u,x)\}$. Let $D_G(u)=|S_G(u)|$. For two sets $S, T$, we use $S\setminus T$ denote $S-(S\cap T)$. The rest of the paper is arranged as follows. In section 2, we establish structural properties of minimum counterexample of Theorem \[THM: Main\]. Then the proof of Theorem \[THM: Main\] is presented in Section 3, and in the last section, we give some discussions. Properties of minimal counterexample ==================================== Let $G$ be a minimum counterexample of Theorem \[THM: Main\] with respect to the order of $G$. The main idea of the proof is as the following. We first pick some set $V_0\subset V(G)$ so that $G'=G-V_0$ has no isolated vertex, by the minimality of $G$, $G'$ has an odd induced subgraph $H'$ with $|V(H')|/|V(G')|\ge 2/5$. We will find a set $S_0\subset V_0$ with $|S_0|\geq \frac{2}{5}|V_0|$ such that $S_0\cup V(H')$ induces an odd induced subgraph $H$ of $G$. We should be careful to remain the parity of the degrees of the vertices in $N_G(S_0)\cap V(H')$ and $S_0\cap N_G(V(H'))$. Here we allow $V(G')=\emptyset$. \[LEM: P1\] Let $u$ be a vertex of $G$ with $D_G(u)=1$ and let $S_G(u)=\{v\}$. Then $N_G^1(u)\cup N_G^2(u,v)=\emptyset$. Suppose to the contrary that $G$ has a vertex $u$ with $D_G(u)=1$ and $N_G^1(u)\cup N_G^2(u,v)\not=\emptyset$. Let $t_1=|N_G^1(u)|$ and $t_2=|N_G^2(u,v)|$. Then $t_1+t_2>0$. [**Case 1**]{}. $|N_G^1(v)|\le 1$. Set $V_0=N_G^1(u)\cup N_G^2(u,v)\cup \{u,v\}\cup N_G^1(v)$ and $G'=G-V_0$. Then $G'$ has no isolated vertex, so, by the minimality of $G$, $G'$ has an odd induced subgraph $H'$ with $|V(H')|\ge \frac 25 |V(G')|$. Now let $S_0=V_0\setminus (N_G^1(v)\cup\{v\})$. Then $G[S_0]\cong K_{1, t_1+t_2}$, and so $G[S_0]$ contains an odd induced subgraph $K=K_{1,t}$ with $t=t_1+t_2$ if $t_1+t_2$ is odd or $t=t_1+t_2-1$ if $t_1+t_2$ is even. So $(t+1)/|V_0|\ge (t+1)/(t_1+t_2+3)\ge 2/5$. Furthermore, we have $N_G(V(K))\cap V(H')=\emptyset$ and $V(K)\cap N_G(V(H'))=\emptyset$. Hence $H=H'\cup K$ is an odd induced subgraph of $G$ with $|V(H)|\ge \frac 25|V(G)|$, a contradiction. [**Case 2**]{}. $|N_G^1(v)|\ge 2$. Choose a vertex $x\in N_G^1(v)$ and set $V_0=N_G^1(u)\cup N_G^2(u,v)\cup \{u,x\}$ and $G'=G- V_0$. Then $G'$ has no isolated vertex and so, by the minimality of $G$, $G'$ has an odd induced subgraph $H'$ with $|V(H')|/|V(G')|\ge 2/5$. \[CLAIM: l5c1\] $v$ must be in $V(H')$. Suppose to the contrary that $v\notin V(H')$. Set $S_0=V_0\setminus\{x\}$, then $G[S_0]\cong K_{1, t_1+t_2}$, and so $G[S_0]$ contains an odd induced subgraph $K=K_{1, t}$ with $t=t_1+t_2$ or $t=t_1+t_2-1$ with respect to the parity of $t_1+t_2$. Note that $(t+1)/|V_0|=(t+1)/(t_1+t_2+2)> 2/5$, $N_G(V(K))\cap V(H')=\emptyset$ and $V(K)\cap N_G(V(H'))=\emptyset$. Therefore, $H=K\cup H'$ is an odd induced subgraph of $G$ with $|V(H)|/|V(G)|>2/5$, a contradiction. The claim is true. Now suppose $v\in V(H')$. \[l5c2\] We have $t_2\le t_1$. Suppose to the contrary that $t_2\geq t_1+1$. Set $S_0=N_G^2(u,v)\cup \{x\}$ if $t_2$ is odd or $S_0=N_G^2(u,v)$ if $t_2$ is even, then $S_0\cup V(H')$ still induces an odd subgraph $H$ of $G$ with $|V(H)|=|S_0|+|V(H')|\ge \frac 25|V_0|+\frac 25|V(G')|=\frac 25|V(G)|$, a contradiction, where the second inequality holds since $|S_0|/|V_0|\ge |S_0|/(t_1+t_2+2)\ge |S_0|/(2t_2+1)\ge 2/5$. Hence the claim holds. Now suppose $t_2\leq t_1$ and let $T_1$ (resp. $T_2$) be a maximum subset of odd (resp. even) order in $N_G^1(u)$. Set $S_0=T_1\cup\{u\}$ if $uv\notin E(G)$ or $S_0=T_2\cup \{u,x\}$ if $uv\in E(G)$. In both cases, $|S_0|/|V_0|=|S_0|/(t_1+t_2+2)\ge |S_0|/(2t_1+2)\geq 2/5$ unless $t_1=t_2=2$ and $uv\notin E(G)$. Then $S_0\cup V(H')$ induces an odd subgraph $H$ of $G$ with $|V(H)|=|S_0|+|V(H')|\ge 2/5|V_0|+2/5|V(G')|=2/5|V(G)|$, a contradiction. For $t_1=t_2=2$ and $uv\notin E(G)$, reset $V_0=N_G^1(u)\cup N_G^2(u,v)\cup \{u\}=N_G(u)\cup\{u\}$ and let $G'=G-V_0$, then $G'$ has no isolated vertex and so, by the minimality of $G$, $G'$ has an odd induced subgraph $H'$ with $|V(H')|/|V(G')|\ge 2/5$. Let $N_G^1(u)=\{a,b\}$ and set $S_0=\{a,u\}$. Then $S_0\cup V(H')$ induces an odd subgraph $H$ of $G$ with $|V(H)|\ge 2+\frac 25|V(G')|=\frac 25|V(G)|$, a contradiction again. This completes the proof of the lemma. \[LEM: P2\] Let $u$ be a vertex of $G$ with $D_G(u)=2$ and let $S_G(u)=\{v,w\}$. Then $N_G^1(u)\cup N_G^2(u,v)\cup N_G^2(u,w)=\emptyset$. Suppose to the contrary that $G$ has a vertex $u$ with $S_G(u)=\{v,w\}$ and $N_G^1(u)\cup N_G^2(u,v)\cup N_G^2(u,w)\not=\emptyset$. Let $t_1=|N_G^1(u)|$, $t_2=|N_G^2(u,v)|$ and $t_3=|N_G^2(u,w)|$. Then $t_1+t_2+t_3>0$. Let $\bar{N}_G^2(v,w)=N_G^2(v, w)\setminus\{u\}$. \[CLAIM: l6c2\] If $N^1_G(v)=\emptyset$ then $N^1_G(w)\cup \bar{N}_G^2(v,w)\not=\emptyset$; symmetrically, if $N^1_G(w)=\emptyset$ then $N^1_G(v)\cup \bar{N}_G^2(v,w)\not=\emptyset$. We only prove the first statement, the second one can be proved similarly. Suppose to the contrary that $N^1_G(w)\cup \bar{N}_G^2(v,w)=\emptyset$. Set $V_0=N_G(u)\cup \{u,v,w\}$ and $G'=G-V_0$. Then $G'$ has no isolated vertex and so, by the minimality of $G$, $G'$ has an odd induced subgraph $H'$ with $|V(H')|\ge \frac 25 |V(G')|$. Let $S$ be a maximum subset of $N^1_G(u)\cup N^2_G(u,v)\cup N_G^2(u,w)$ so that $s=|S|$ is odd. Then $S_0=S\cup\{u\}$ induces an odd subgraph $K\cong K_{1,s}$ of $G[V_0]$, furthermore $|S_0|/|V_0|\ge (s+1)/(t_1+t_2+t_3+3)\ge 2/5$. Note that $N_G(S_0)\cap V(H')=\emptyset$ and $S_0\cap N_G(V(H'))=\emptyset$. Therefore, $H=K\cup H'$ is an odd induced subgraph of $G$ with $|V(H)|\ge \frac 25|V_0|+\frac 25|V(G')|=\frac 25|V(G)|$, a contradiction. The claim is true. [**Case 1**]{}. $N_G^1(v)=\emptyset$. [**Subcase 1.1**]{}. $|N_G(w)\setminus (N_G^2(u,w)\cup \{u,v\})|\le 1$. Note that $|N_G(w)\setminus (N_G^2(u,w)\cup \{u,v\})|\le 1$ implies that $|N_G^1(w)\cup \bar{N}_G^2(v,w)|\le 1$. By Claim \[CLAIM: l6c2\], $|N_G^1(w)\cup \bar{N}_G^2(v,w)|= 1$ and so $N_G(w)\setminus (N_G^2(u,w)\cup \{u,v\})=N_G^1(w)\cup \bar{N}_G^2(v,w)$. Let $N_G^1(w)\cup \bar{N}_G^2(v,w)=\{x\}$ and set $V_0=N_G(u)\cup \{u,v,w,x\}$ and $G'=G- V_0$. Then $G'$ has no isolated vertex and so, by the minimality of $G$, $G'$ has an odd induced subgraph $H'$ with $|V(H')|\ge \frac 25 |V(G')|$. Let $S$ be a maximum subset of $N^1_G(u)\cup N^2_G(u,v)\cup N^2_G(u,w)$ so that $s=|S|$ is odd. Then $S_0=S\cup\{u\}$ induces an odd subgraph $K\cong K_{1,s}$ of $G[V_0]$, furthermore $|S_0|/|V_0|\ge (s+1)/(t_1+t_2+t_3+4)\ge 2/5$ unless $t_1+t_2+t_3=2$. Note that $N_G(S_0)\cap V(H')=\emptyset$ and $S_0\cap N_G(V(H'))=\emptyset$. Hence $H=K\cup H'$ is an odd induced subgraph of $G$ with $|V(H)|\ge \frac 25|V_0|+\frac 25|V(G')|=\frac 25|V(G)|$ provided that $t_1+t_2+t_3\not=2$, a contradiction. For $t_1+t_2+t_3=2$, notice that $E_G(w, V(G'))=\emptyset$ because $N_G(w)\setminus (N_G^2(u,w)\cup \{u,v\})=N_G^1(w)\cup \bar{N}_G^2(v,w)$. If $E_G(v, V(G'))=\emptyset$ then $G$ is a graph of order six, it can be easily checked that $G$ cannot be a counterexample. If $t_3=2$ then $S_0=N_G^2(u,w)\cup \{w,x\}$ induces an odd subgraph $K\cong K_{1,3}$, and therefore $H=K\cup H'$ is an odd induced subgraph of $G$ with $|V(H)|\ge 4+\frac 25|V(G')|>\frac 25|V(G)|$, a contradiction. Hence suppose $E_G(v, V(G'))\not=\emptyset$ and $t_3<2$. Reset $V_0=(N_G(u)\cup \{u,w, x\})\setminus\{v\}$ and $G'=G-V_0$. Then $G'$ has no isolated vertex and so, by the minimality of $G$, $G'$ has an odd induced subgraph $L'$ with $|V(L')|\ge \frac 25|V(G')|$. If $v\notin V(L')$ or $vw, vx\notin E(G)$, then $\{w,x\}\cup V(L')$ induces an odd subgraph $H$ of $G$ with $|V(H)|\ge 2+\frac 25|V(G')|\ge\frac 25|V(G)|$, a contradiction. So suppose $v\in V(L')$ and $vw\in E(G)$ or $vx\in E(G)$. If $N_G(v)\cap V_0$ has two nonadjacent vertices, say $\{a,b\}$, then $\{a,b\}\cup V(L')$ induces an odd subgraph of $G$ with order at least $\frac 25|V(G)|$, a contradiction. This implies that $N_G^2(u,v)=\emptyset$ (i.e $t_2=0$), $vx\notin E(G)$ (i.e. $x\in N_G^1(w)$) and $vw, uw\in E(G)$ (otherwise, it is easy to choose two nonadjacent vertices from $N_G^2(u,v)\cup\{u,w,x\}$). As $t_1+t_2+t_3=2$, $t_2=0$, and $t_3<2$, we have $t_1>0$. Choose $a\in N_G^1(u)$, then $\{a,u,w,x\}\cup V(L')$ induces an odd subgraph $H$ of $G$ with $|V(H)|\ge 4+\frac 25|V(G')|>\frac 25|V(G)|$, a contradiction. [**Subcase 1.2**]{}. $|N_G(w)\setminus (N_G^2(u,w)\cup \{u,v\})|\geq 2$. Choose $x\in N_G^1(w)\cup \bar{N}_G^2(v,w)$ (this can be done because $N_G^1(w)\cup \bar{N}_G^2(v,w)\not=\emptyset$ by Claim \[CLAIM: l6c2\]) and set $V_0=(N_G(u)\cup \{u,v,x\})\setminus \{w\}$ and $G'=G-V_0$. Then $G'$ has no isolated vertex and so, by the minimality of $G$, $G'$ has an odd induced subgraph $H'$ with $|V(H')|\ge\frac 25|V(G')|$. \[CLAIM: l6c3\] $w\in V(H')$. If $w\notin V(H')$, choose a maximum subset $S$ of $N_G(u)\setminus\{v, w\}$ so that $s=|S|$ is odd, then $S_0=S\cup\{u\}$ induces an odd subgraph $K\cong K_{1,s}$ of $G[V_0]$ such that $|S_0|/|V_0|=(s+1)/(t_1+t_2+t_3+3)\geq 2/5$. Clearly, $N_G(S_0)\cap V(H_0)=\emptyset$ and $S_0\cap N_G(V(H_0))=\emptyset$. Hence $H=K\cup H'$ is an odd induced subgraph of $G$ with $|V(H)|\ge \frac 25|V(G)|$, a contradiction. The claim holds. \[CLAIM: l6c4\] $t_3\le t_1+t_2$. If $t_3\ge t_1+t_2+1$, choose a maximum subset $S_0$ of $N_G^2(u,w)\cup \{x\}$ so that $|S_0|$ is even, then $|S_0|/|V_0|=|S_0|/(t_1+t_2+t_3+3)\geq |S_0|/(2t_3+2)\ge 2/5$ unless $t_3=2$ and $t_1+t_2=1$. Therefore, $S_0\cup V(H')$ induces an odd subgraph $H$ of $G$ with $|V(H)|\ge\frac 25|V(G)|$ unless $t_3=2$ and $t_1+t_2=1$. For $t_3=2$ and $t_1+t_2=1$, reset $V_0=(N_G(u)\cup \{u,v\})\setminus \{w\}$ and $G'=G-V_0$, then, again by the minimality of $G$, $G'$ has an odd induced subgraph $L'$ with $|V(L')|\ge \frac 25|V(G')|$. If $w\in V(L')$, set $S_0=N_G^2(u,w)$, and if $w\notin V(L')$, set $S_0=\{u,y\}$, where $y$ is a vertex in $N_G^2(u,w)$. In both cases, $S_0\cup V(L')$ induces an odd subgraph $H$ of $G$ with $|V(H)|\ge \frac 25|V(G)|$. Therefore, we always obtain a contradiction and so the claim follows. Now let $S$ be a maximum subset of $N_G^1(u)\cup N_G^2(u,v)$ so that $s=|S|$ is even if $uw\in E(G)$, and $s=|S|$ is odd if $uw\notin E(G)$. Set $S_0=S\cup \{u,x\}$ if $uw\in E(G)$ and $S_0=S\cup\{u\}$ if $uw\notin E(G)$. Clearly, $S_0\cup V(H')$ induces an odd subgraph $H$ of $G$ and furthermore, for $uw\in E(G)$, $|S_0|/|V_0|\ge (s+2)/(t_1+t_2+t_3+3)\ge (t_1+t_2+1)/(2t_1+2t_2+3)\geq 2/5$; and for $uw\notin E(G)$, $|S_0|/|V_0|=(s+1)/(t_1+t_2+t_3+3)\geq 2/5$ unless $t_1+t_2=2$, $t_3=1$ or $t_1+t_2=2$, $t_3=2$ or $t_1+t_2=t_3=4$. Therefore, but some exceptions, $H$ is an odd induced subgraph with $|V(H)|\ge\frac 25|V(G)|$, a contradiction. Note that all the exceptions occur under the assumption $uw\notin E(G)$. In the following of the case, we show that each of the three exceptions cannot occur in the minimal counterexample $G$ as well. For $t_1+t_2=2$ and $t_3=1$, reset $V_0=N_G(u)\cup \{u,v\}$ and let $G'=G-V_0$, then, by the minimality of $G$, $G'$ has an odd induced subgraph $L'$ with $|V(L')|\ge \frac 25|V(G')|$. Choose a vertex $a\in N_G^1(u)\cup N_G^2(u,v)$, then $S_0=\{u,a\}$ induces an odd subgraph $K\cong K_{1,1}$ of $G[V_0]$. As $N_G(S_0)\cap V(L')=\emptyset$ and $S_0\cap N_G(V(L'))=\emptyset$, $H=K\cup L'$ is an odd induced subgraph of $G$ with $|V(H)|\ge \frac 25|V(G)|$, a contradiction. For $t_1+t_2=t_3=2$. If $|N_G^1(w)\cup \bar{N}_G^2(v,w)|\le 2$, reset $V_0=N_G(u)\cup N_G^1(w)\cup \bar{N}_G^2(v,w)\cup\{u,v,w,x\}$, then $G'=G-V_0$ has no isolated vertex and so, by the minimality of $G$, $G'$ has an odd induced subgraph $L'$ with $|V(L')|\ge\frac 25|V(G')|$. Let $S$ be a subset of $N_G(u)\setminus\{v\}$ with $s=|S|=3$ (this can be done because $|N_G(u)|\ge t_1+t_2+t_3=4$). Then $S_0=S\cup \{u\}$ induces an odd subgraph $K\cong K_{1,3}$ of $G[V_0]$ and therefore $H=K\cup L'$ is an odd induced subgraph of $G$ with $|V(H)|\ge \frac 25|V(G)|$, a contradiction. Now suppose $|N_G^1(w)\cup \bar{N}_G^2(v,w)|\ge 3$. Choose a vertex $y\in N_G^1(w)\cup \bar{N}_G^2(v,w)$ with $y\not=x$. Reset $V_0=N_G(u)\cup \{u,v,x,y\}$ and $G'=G-V_0$, then, by the minimality of $G$, $G'$ has an odd induced subgraph $L'$ with $|V(L')|\ge\frac 25|V(G')|$. Let $S_0=N_G^2(u,w)\cup \{x,y\}$ if $w\in V(L')$, and let $S_0=S\cup \{u\}$ if $w\notin V(L')$, where $S$ is a maximum subset of $N_G(u)\setminus\{v\}$ with $s=|S|=3$. Clearly, $S_0\cup V(L')$ induces an odd subgraph $H$ of $G$ with $|V(H)|>\frac 25|V(G)|$, a contradiction. For $t_1+t_2=t_3=4$, reset $V_0=N_G(u)\cup \{u,v\}$ and $G'=G-V_0$, then $G'$ has no isolated vertices and so, by the minimality of $G$, $G'$ has an odd induced subgraph $L'$ with $|V(L')|\ge\frac 25|V(G')|$. Let $S_0=N_G^2(u,w)$ if $w\in V(L')$, or let $S_0=N_G^1(u)\cup N_G^2(u,v)\cup \{u\}$ if $w\notin V(L')$. Then $|S_0|/|V_0|\ge 2/5$ and $S_0\cup V(L')$ induces an odd subgraph $H$ of $G$ with $|V(H)|\ge \frac 25|V(G)|$, a contradiction. This proves Case 1. By symmetry, we may also assume $N_G^1(w)\neq \emptyset$ to verify the following remaining case. [**Case 2**]{}. $N_G^1(v)\neq \emptyset$. Choose $x\in N_G^1(v)$ and $y\in N_G^1(w)$, set $V_0=(N_G(u)\cup \{u,x,y\})\setminus \{v,w\}$ and $G'=G-V_0$. \[l6c5’\] $G'$ has no isolated vertex. Suppose to the contrary that $G'$ has isolated vertices. Then $v$ or $w$ must be an isolated vertex of $G'$. Without loss of generality, assume $v$ is an isolated vertex of $G'$. Then $D_G(v)=1$. But $N_G^1(v)\not=\emptyset$, this is a contradiction to Lemma \[LEM: P1\]. Hence $G'$ has no isolated vertex and so, by the minimality of $G$, $G'$ has an odd induced subgraph $H'$ with $|V(H')|\ge\frac 25|V(G')|$. \[CLAIM: l6c5\] $H'$ contains at least one of $\{v, w\}$. Suppose to the contrary that $H'$ contains none of $\{v, w\}$. Let $S$ be a maximum subset of $N_G(u)\setminus\{v,w\}$ so that $s=|S|$ is odd. Then $S_0=S\cup \{u\}$ induces an odd subgraph $K\cong K_{1,s}$ with $|S_0|/|V_0|=(s+1)/(t_1+t_2+t_3+3)\geq 2/5$. Note that $N_G(S_0)\cap V(H')=\emptyset$ and $S_0\cap N_G(V(H'))=\emptyset$. Thus $H=K\cup H'$ is an odd induced subgraph of $G$ with $|V(H)|\ge \frac 25|V(G)|$, a contradiction. \[CLAIM: l6c6\] If $w\in V(H')$ then $t_3\le t_1+t_2$. Symmetrically, if $v\in V(H')$ then $t_2\le t_1+t_3$. We show that $t_3\le t_1+t_2$ when $w\in V(H')$. Suppose to the contrary that $t_3\ge t_1+t_2+1$. Let $S_0$ be a maximum subset of $N_G^2(u,w)\cup \{y\}$ such that $|S_0|$ is even. Then $S_0\cup V(H')$ induces an odd subgraph $H$ of $G$ with $|V(H)|=|S_0|+|V(H')|\ge \frac 25|V_0|+\frac 25|V(G')|=\frac 25 |V(G)|$ unless $t_3=2$ and $t_1+t_2=1$. For $t_3=2$ and $t_1+t_2=1$, reset $V_0=(N_G(u)\cup \{u,x\})\setminus \{v,w\}$ and $G'=G-V_0$, then $G'$ has no isolated vertex and so $G'$ has an odd induced subgraph $L'$ with $|V(L')|\ge \frac 25|V(G')|$ by the minimality of $G$. If $w\in V(L')$, let $S_0= N_G^2(u,w)$, then $S_0\cup V(L')$ induces an odd subgraph $H$ with $|V(H)|\ge \frac 25|V(G)|$. Now suppose $w\notin V(L')$. If $v\notin V(L')$, choose a vertex $z$ from $N^1_G(u)\cup N^2_G(u,v)$, then $\{u, z\}\cup V(L')$ induces an odd subgraph $H$ of $G$ with $|V(H)|\ge \frac 25|V(G)|$. Hence $v\in V(L')$, choose a vertex $z\in N^2_G(u,v)\cup\{u\}$ which is adjacent to $v$, then $\{x,z\}\cup V(L')$ induces an odd subgraph $H$ of $G$ with $|V(H)|\ge \frac 25|V(G)|$. In all cases we get contradictions and so the claim follows. For $t_1+t_2=2$ and $t_3=1$, reset $V_0=(N_G(u)\cup\{u,x\})\setminus\{w, v\}$ and $G'=G-V_0$, then $G'$ has no isolated vertex and so, by the minimality of $G$, $G'$ has an odd induced subgraph $L'$ with $|V(L')|\ge \frac 25|V(G')|$. If $v\in V(L')$, choose a vertex $z\in N^2_G(u,v)\cup\{u\}$ which is adjacent to $v$, note that $uw\notin E(G)$, then $\{x,z\}\cup V(L')$ induces an odd subgraph $H$ of $G$ with $|V(H)|\ge \frac 25|V(G)|$, a contradiction. Hence $v\notin V(L')$, choose a vertex $z$ from $N^1_G(u)\cup N^2_G(u,v)$, then $\{u, z\}\cup V(L')$ induces an odd subgraph $H$ of $G$ with $|V(H)|\ge \frac 25|V(G)|$, a contradiction. For $t_1+t_2=t_3=p$, $p=2,4$, reset $V_0=(N_G(u)\cup\{u\})\setminus\{v,w\}$ and $G'=G-V_0$, then $G'$ has no isolated vertex and so, by the minimality of $G$, $G'$ has an odd induced subgraph $L'$ with $|V(L')|\ge\frac 25|V(G')|$. If $w\in V(L')$, note that $|N^2_G(u,w)|=t_3=p$ is even, $N^2_G(u,w)\cup V(L')$ induces an odd subgraph $H$ of $G$ with $|V(H)|\ge \frac 25|V(G)|$, a contradiction. So suppose $w\notin V(L')$. If $uv\notin E(G)$ or $v\notin V(L')$, choose a subset $S$ of $N^2_G(u,w)$ so that $|S|=p-1$, then $S\cup\{u\}\cup V(L')$ induces an odd subgraph $H$ of $G$ with $|V(H)|\ge\frac 25|V(G)|$, a contradiction. Hence $uv\in E(G)$ and $v\in V(L')$. If $x\in V(L')$, then $N^2_G(u,w)\cup\{u\}\cup (V(L')\setminus\{x\})$ induces an odd subgraph $H$ with $|V(H)|=p+1+|V(L')|-1\ge \frac 25 |V(G)|$, a contradiction. Hence $x\notin V(L')$. Then $N^2_G(u,w)\cup\{u\}\cup V(L')\cup\{x\}$ induces an odd subgraph $H$ of $G$ with $|V(H)|\ge \frac 25|V(G)|$, a contradiction. This proves the claim. By Claim \[CLAIM: l6c5\], we may assume, without loss of generality, $w\in V(H')$. Hence, by Claim \[CLAIM: l6c6\], $t_3\le t_1+t_2$. Now we divide the discussion into two subcases below. [**Subcase 2.1**]{}. $v\notin V(H')$. Let $S$ be a maximum subset of $N_G^1(u)\cup N_G^2(u,v)$ such that $s=|S|$ is odd if $uw\notin E(G)$ or $s=|S|$ is even if $uw\in E(G)$. Set $S_0=S\cup\{u\}$ if $uw\notin E(G)$ or $S_0=S\cup\{u,y\}$ if $uw\in E(G)$. Note that $s=t_1+t_2$ or $t_1+t_2-1$ depending on the parity of $t_1+t_2$ and $|S_0|=s+1$ or $s+2$ depending on $uw\notin E(G)$ or $uw\in E(G)$. Notice that $t_3\le t_1+t_2$, we have $|S_0|/|V_0|=|S_0|/(t_1+t_2+t_3+3)\geq 2/5$ unless $uw\notin E(G)$ and $t_1+t_2=2$, $t_3=1$, or $t_1+t_2=t_3=2$, or $t_1+t_2=t_3=4$. Therefore $S_0\cup V(H')$ induces an odd subgraph $H$ of $G$ with $|V(H)|\ge \frac 25|V(G)|$ but three exceptions. However, none of the exceptions occur in $G$ by Claim \[CLAIM: l6c9\]. This yields a contradiction and verifies Subcase 2.1. [**Subcase 2.2**]{}. $v\in V(H')$. By Claim \[CLAIM: l6c6\], we have $t_3\le t_1+t_2$ and $t_2\le t_1+t_3$. Furthermore, we have the following claim. \[CLAIM: l6c8\] We have $t_2+t_3\leq t_1$. Suppose to the contrary that $t_2+t_3\geq t_1+1$. Let $S_v$ be a maximum subset of $ N_G^2(u,v)\cup\{x\}$ such that $|S_v|$ is even, let $S_w$ be a maximum subset of $ N_G^2(u,w)\cup\{y\}$ such that $|S_w|$ is even, and set $S_0=S_u\cup S_v$. Then $S_0\cup V(H')$ induces an odd subgraph of $G$. By checking the parity of $t_2$ and $t_3$ with certain calculation, we have $|S_0|/|V_0|\ge \frac{2}{5}$ unless $t_1=1$, $t_2+t_3=2$ and $t_i$, $i=2,3$, is even. But this exception cannot occur because $t_3\le t_1+t_2$ and $t_2\le t_1+t_3$, a contradiction. Hence the claim holds. Now, we choose a set $S_0$ according to the following rules: *(i) If $uv\in E(G), uw\in E(G)$, let $S_0=S_u\cup \{u, x, y\}$, where $S_u$ is the maximum subset of $N_G^1(u)$ with size odd;* \(ii) If $uv\in E(G), uw\notin E(G)$, let $S_0=S_u\cup \{u, x\}$, where $S_u$ is the maximum subset of $N_G^1(u)$ with size even; \(iii) If $uv\notin E(G), uw\in E(G)$, let $S_0=S_u\cup \{u, y\}$, where $S_u$ is the maximum subset of $N_G^1(u)$ with size even; \(iv) If $uv\notin E(G), uw\notin E(G)$, let $S_0=S_u\cup \{u\}$, where $S_u$ is the maximum subset of $N_G^1(u)$ with size odd. Then $S_0\cup V(H')$ induces an odd subgraph of $G$ by definition. It remains to compute $|S_0|/|V_0|$. If $t_1$ is odd, we have $|S_0|/|V_0|\ge(t_1+1)/(t_1+t_2+t_3+3)\geq 2/5$ by Claim \[CLAIM: l6c8\] in each of the cases (i)-(iv). If $t_1$ is even, it follows from Claim \[CLAIM: l6c8\] that $|S_0|/|V_0|\ge(t_1+2)/(t_1+t_2+t_3+3)\geq 2/5$ in each of the cases (i)-(iii), and in the case (iv), $|S_0|/|V_0|=t_1/(t_1+t_2+t_3+3)\geq 2/5$ unless $t_1=2$, $t_2+t_3=2$ or $t_1=4$, $t_2+t_3=4$. Therefore, $S_0\cup V(H')$ induces an odd subgraph $H$ of $G$ with $|V(H)|\ge \frac 25|V(G)|$ unless $t_1=t_2+t_3=2$ or $t_1=t_2+t_3=4$. For $t_1=t_2+t_3=p$, $p=2,4$, reset $V_0=N_G(u)\cup\{u\}$ and $G'=G-V_0$, then $G'$ has no isolated vertex and so, by the minimality of $G$, $G'$ has an odd induced subgraph $L'$ with $|V(L')|\ge\frac 25|V(G')|$. Choose a subset $S$ of $N^1_G(u)$ so that $|S|=p-1$, then $S\cup\{u\}\cup V(L')$ induces an odd subgraph $H$ of $G$ with $|V(H)|\ge\frac 25|V(G)|$, a contradiction. The proof of the lemma is completed. The following three structural properties of the minimum counterexample $G$ are direct consequence of Lemmas \[LEM: P1\] and \[LEM: P2\]. \[COR: c1\] Let $V_1$ be the set of all 1-vertices in $G$ and let $P=N_G(V_1)$. Suppose $G_1=G-V_1$, then $d_{G_1}(x)\geq 3$ for any $x\in P$. Suppose to the contrary that there is a vertex $x\in P$ with $d_{G_1}(x)\le 2$. If $d_{G_1}(x)=0$ then $G$ is isomorphic to a star, which cannot be a counterexample. Hence $0<d_{G_1}(x)\le 2$. This implies that $0<D_G(x)\le 2$. But $|N_G^1(x)|\geq 1$, this is a contradiction to Lemmas \[LEM: P1\] or \[LEM: P2\]. \[COR: c2\] $G$ has no adjacent $2$-vertices. Suppose to the contrary that $G$ has two adjacent $2$-vertices $u,v$. Then $D_G(u)\le 2$. Let $v_1=N_G(v)\setminus \{u\}$. Then $v\in N^2_G(u, v_1)$, which is a contradiction to Lemmas \[LEM: P1\] or \[LEM: P2\]. \[COR: c3\] $G$ has no vertex $u$ with $d_G(u)\ge 3$ so that $D_G(u)\le 2$. Suppose to the contrary that $G$ has a vertex $u$ with $d_G(u)\ge 3$ and $D_G(u)\le 2$. By Lemmas \[LEM: P1\] and \[LEM: P2\], $u$ has no neighbor of degree at most 2 since $G$ cannot be isomorphic to a star. This implies $D_G(u)\ge d_G(u)\ge 3$, a contradiction. Proof of Theorem \[THM: Main\] ============================== Before giving the proof, we need some definition and structural properties of graphs with treewidth at most 2. A graph $G$ contains a graph $H$ as a [*minor*]{} if $H$ can be obtained from a subgraph of $G$ by contracting edges, and $G$ is called [*$H$-minor*]{} free if $G$ does not have $H$ as a minor. It is well known that \[PROP: p1\]\[Proposition 12.4.2, [@Diestel-GT00]\] A graph has treewidth at most 2 if and only if it is $K_4$-minor free. For $K_4$-minor free graphs, Lih, Wang, and Zhu ([@LWZ-DM03]) gave a powerful structural property of them. \[LEM: K\_4-free\]\[Lemma 2, [@LWZ-DM03]\] If $G$ is a $K_4$-minor free graph, then one of the following holds: \(a) $\delta(G)\leq 1$; \(b) there exist two adjacent $2$-vertices; \(c) there exists a vertex $u$ with $d_G(u)\geq 3$ such that $D_G(u)\leq 2$. Let $G$ be a minimum counterexample with respect to the order of $G$. By the minimality of $G$, $G$ must be connected. Let $V_1$ be the set of all 1-vertices in $G$ and $P=N_G(V_1)$. Let $G_1=G-V_1$. By Corollaries \[COR: c1\] and \[COR: c2\], $\delta(G_1)\geq 2$ and $G_1$ has no adjacent $2$-vertices. Clearly, $tw(G_1)\le 2$ and hence $G_1$ is $K_4$-minor free. By Lemma \[LEM: K\_4-free\], $G_1$ has a vertex $u$ with $d_{G_1}(u)\geq 3$ and $D_{G_1}(u)\leq 2$. Clearly, $d_G(u)=d_{G_1}(u)+|N^1_G(u)|$ and the adding of the vertices of $N^1_G(u)$ to $G_1$ does not increases the value of $D_{G_1}(u)$. So $D_G(u)=D_{G_1}(u)\leq 2$, this is a contradiction to Corollary \[COR: c3\]. The proof of Theorem \[THM: Main\] is completed. Concluding remarks ================== Let $$\mathcal{G}_k=\{G\colon\, tw(G)\le k \mbox{ and $G$ contains no isolated vertex}\},$$ and $c_k=\min_{G\in \mathcal{G}_k}\frac{f(G)}{|V(G)|}$. [ Since each graph in $\mathcal{G}_k$ has chromatic number at most $k+1$, Scott’s result [@Scott-CPC92] implies $c_k\ge \frac{1}{2k+2}$. The follow graphs $H_k$ in Figure \[FIG: Hk\] gives an upper bound $c_k\le \frac{2}{k+3}$ for $k=1,2,3,4$. Note that the graph $H_4$ is found by Caro [@Caro-DM1994], which is the smallest known ratio of $\frac{f(G)}{|V(G)|}$ for all graphs $G$.]{} As we have known, [Theorem \[THM: RSMain\] of Radcliffe and Scott [@RS-DM95] and the upper bound of $c_k$ implies $c_1=1/2$]{}, and in this paper, we show that $c_2=2/5$ (Theorem \[THM: Main\]). As a far more step, we want ask the question: what is the exact value $c_k$ for graphs in $\mathcal{G}_k$. [ It is plausible that $c_3=\frac{1}{3}$ and $c_4=\frac{2}{7}$.]{} (110,35) (-3, 5)(9, 5)(-3, 15)(9, 15) (0, -4)[$H_1$]{} (-3, 5)(-3, 5)(9, 5)(-3, 5)(-3, 5)(-3, 15)(9, 5)(9, 15)(9, 15) (46.4127, 19.63525)(23.5873, 19.63525) (35, 27)(42.0534, 5.2918)(27.9466, 5.2918) (32, -4)[$H_2$]{} (35, 27)(46.4127, 19.63525)(46.4127, 19.63525) (42.0534, 5.2918)(42.0534, 5.2918)(46.4127, 19.63525) (27.9466, 5.2918)(27.9466, 5.2918)(42.0534, 5.2918) (23.5873, 19.63525)(23.5873, 19.63525)(27.9466, 5.2918) (35,27)(35,27)(23.5873, 19.63525) (63.5, 3.75) (76.5, 3.75) (83, 15) (76.5, 26.25) (63.5, 26.25) (57, 15) (67, -4)[$H_3$]{} (63.5, 3.75)(63.5, 3.75)(76.5, 3.75) (63.5, 3.75)(63.5, 3.75)(57, 15) (57, 15)(57, 15)(63.5, 26.25) (63.5, 26.25)(63.5, 26.25)(76.5, 26.25) (76.5, 26.25)(76.5, 26.25)(83, 15) (83, 15)(83, 15)(76.5, 3.75) (76.5, 26.25)(76.5, 26.25)(76.5, 3.75) (57, 15)(57, 15)(76.5, 3.75) (57, 15)(57, 15)(76.5, 26.25) (83, 15)(83, 15)(63.5, 3.75) (83, 15)(83, 15)(63.5, 26.25) (114.3820, 22.4819) (116.6991, 12.32975) (110.2066, 4.1884) (99.7934, 4.1884) (93.3009, 12.32975) (95.6180, 22.4819) (105, 27) (102, -4)[$H_4$]{} (114.3820, 22.4819)(114.3820, 22.4819)(116.6991, 12.32975) (114.3820, 22.4819)(114.3820, 22.4819)(110.2066, 4.1884) (114.3820, 22.4819)(114.3820, 22.4819)(95.6180, 22.4819) (110.2066, 4.1884)(110.2066, 4.1884)(116.6991, 12.32975) (93.3009, 12.32975)(110.2066, 4.1884)(110.2066, 4.1884) (105, 27)(116.6991, 12.32975)(116.6991, 12.32975) (105, 27)(93.3009, 12.32975)(93.3009, 12.32975) (95.6180, 22.4819)(99.7934, 4.1884)(99.7934, 4.1884) (116.6991, 12.32975)(99.7934, 4.1884)(99.7934, 4.1884) (110.2066, 4.1884)(110.2066, 4.1884)(99.7934, 4.1884) (99.7934, 4.1884)(93.3009, 12.32975)(93.3009, 12.32975) (95.6180, 22.4819)(93.3009, 12.32975)(93.3009, 12.32975) (95.6180, 22.4819)(95.6180, 22.4819)(105, 27) (105, 27)(105, 27)(114.3820, 22.4819) [99]{} D. M. Berman, H. Wang, L. Wargo, Odd induced subgraphs in graphs of maximum degree three. Aust. J. Comb. (1997) 81-85. Y. Caro, On induced subgraphs with odd degrees. Discrete Math., (1994) 23-28. R. Diestel, Graph Theory, Springer-Verlag New York, 2000. K. W. Lih, W. F. Wang, X. D. Zhu, Coloring the square of a $K_4$-minor free graph. Discrete Math. (2003) 303-309. L. Lovász, Combinatorial Problems and Exercises. (North-Holland, Amsterdam, 1979). A. J. Radcliffe, A. D. Scott, Every tree contains a large induced subgraph with all degrees odd. Discrete Math. (1995) 275-279. A. D. Scott, Large induced subgraphs with all degrees odd. Comb. Probab. Comput. 1(1992) 335-349. [^1]: The work was supported by NNSF of China (No. 11671376) and NSF of Anhui Province (No. 1708085MA18) and the Fundamental Research Funds for the Central Universities.
CPHT–S708-0299\ [hep-th/9902055]{} [**Low-Scale Closed Strings and their Duals$^\star$**]{} [I. Antoniadis and B. Pioline ]{} [*Centre de Physique Th[é]{}orique$^\dagger$, Ecole Polytechnique,\ F-91128 Palaiseau\ *]{} CPHT–S708-0299, February 1999, to appear in Nucl. Phys. B. ------------------------------------------------------------------------ width 6.7cm .1mm [$^{\dagger}$[Unit[é]{} mixte CNRS UMR 7644]{}\ $^{\star}$[Research supported in part by the EEC under the TMR contract ERBFMRX-CT96-0090.]{} ]{} Introduction ============ Large dimensions are of particular interest in string theory because of their possible use to explain outstanding physical problems, such as the mechanism of supersymmetry breaking [@ia], the gauge hierarchy [@add; @ab] and the unification of fundamental interactions [@w; @ddg; @cb]. In the context of perturbative heterotic string theory, large TeV dimensions are motivated by supersymmetry breaking which identifies their size to the breaking scale. Although the full theory is strongly coupled in ten dimensions, there are many quantities that can be studied perturbatively, such as gauge couplings that are often protected by non renormalisation theorems [@ia; @ant], as well as all soft breaking terms because of the extreme softness of the supersymmetry breaking mechanism, in close analogy with the situation at finite temperature [@ia; @soft]. An obvious question is whether there is some dual theory that provides a perturbative description of the above models. More recently, it was proposed that the observed gauge hierarchy between the Planck and electroweak scales may be due to the existence of extra large (transverse) dimensions, seen only by gravity which becomes strong at the TeV [@add]. This scenario can be realised [@aadd; @st] in the context of a weakly coupled type I$^\prime$ string theory with a string scale at the TeV [@l] and the Standard Model living on D-branes, transverse to $p$ large dimensions of size ranging from (sub)millimeter (for $p=2$) to a fermi (for $p=6$). As we will discuss below, these models are dual to the heterotic ones with $n=4$, 5 or 6 large dimensions. More precisely, the cases $n=4$ and $n=6$ correspond to $p=2$ and $p=6$, respectively, while the description of $n=5$ uses an anisotropic type I$^\prime$ theory with 5 large and one extra large dimension. In this work, we study large dimensions in the context of weakly coupled type II string compactifications. One of the main characteristics of these theories is that non-abelian gauge symmetries appear non-perturbatively, even at very weak coupling, when the compactification manifold is singular [@wi; @ht2]. In particular, on the type IIA (IIB) side, they can be obtained from even (odd) D-branes wrapped around even (odd) collapsing cycles. As we will show in Section \[secii\], we find two novel possibilities which cannot be realised perturbatively either in the heterotic or in type I constructions. The first case consists of type IIA four-dimensional (4d) compactifications with all internal radii of the order of the string length, that can be as large as TeV$^{-1}$ due to a superweak string coupling. Despite this, Standard Model gauge couplings remain of order unity because their magnitude is determined by the geometry of the internal manifold and not by the value of the string coupling[^1]. This scenario offers an alternative to the brane framework for solving the gauge hierarchy, with very different experimental signals. There are no missing energy events from gravitons escaping into the bulk [@add; @aadd; @grav], while string excitations interact with Planck scale suppressed couplings. As a result, the production of Kaluza-Klein (KK) excitations with gauge interactions remain the only experimental probe [@ia; @abq]. Furthermore, the problem of proton decay and flavor violation becomes in principle much easier to solve than in low energy quantum gravity models. This model will be shown in Section \[sechet\] to be dual to the heterotic string with $n=2$ large dimensions. The second case consists of type IIB theory with two large dimensions at the TeV and the string tension at an intermediate scale of the order of $10^{11}$ GeV. The string coupling is of order unity (but perturbative), while the largeness of the 4d Planck mass is attributed to the large TeV dimensions compared to the string length. Gravity becomes strong at the intermediate scale and the main experimental signal is again the production of KK gauge modes. However, the gauge theory above the compactification scale is very different than in the previous type IIA case. In fact, this model offers the first instance of large radius along non-transverse directions and contains an energy domain where its effective theory becomes six-dimensional below the string scale. This limit corresponds to a non-trivial infrared conformal point described by a tensionless self-dual string [@wi2]. This model is again dual to the heterotic string with a single ($n=1$) large dimension at the TeV. Of course, in the above two examples, one can increase the type IIA string coupling or lower the type IIB string scale by introducing some extra large dimensions transverse to the 5-brane where gauge interactions are localised. In particular, we will show that the heterotic string with $n=3$ large TeV dimensions is described by a type IIA compactification with a string scale and two longitudinal dimensions at the TeV, four transverse dimensions at the fermi scale, and order 1 string coupling. The paper is organised as follows. In Section 2, we study TeV strings and TeV dimensions in type II theories and we describe the first two examples mentioned above. In Section 3, we review briefly the string dualities among heterotic, type I and type II theories, and give the basic relations we use in the sequel; in particular, we give a simple (yet heuristic) derivation of heterotic–type IIA duality in the framework of M-theory. In Section 4, we discuss large dimensions in heterotic compactifications and provide a perturbative description of all cases using heterotic – type I or heterotic – type II dualities. For completeness, in Section 5, we examine large dimensions in type II theories, and show how the heterotic theories of Section 4 are recovered. Finally, Section 6 contains some concluding remarks. Type II theories with low string scale\[secii\] =============================================== The Standard Model gauge interactions can be in principle embedded within three types of four-dimensional string theories, obtained by compactification of the ten-dimensional heterotic, type II and type I theories. On the heterotic side, gauge interactions appear in the perturbative spectrum and, like the gravitational interactions, are controlled by the string coupling ${\ensuremath{g_{\rm H}}}$. In type I theories, gauge interactions are described by open strings and confined on D-branes, whereas gravity propagates in the bulk; both interactions are controlled again by the string coupling ${\ensuremath{g_{\rm I}}}$, although gauge forces are enhanced by a factor $1/\sqrt{{\ensuremath{g_{\rm I}}}}$. In type II theories, the matching condition forbids the existence of non-abelian vector particles in the perturbative spectrum; however, gauge interactions do arise non-perturbatively at singularities of the ${\ensuremath{K_3}}$-fibered Calabi-Yau manifold, from D2-branes (in type IIA) or D3-branes (in type IIB) wrapping the vanishing cycles[^2]. The gauge symmetry is dictated by the intersection matrix of the vanishing two-cycles in the ${\ensuremath{K_3}}$ fiber, whereas extra matter arises from vanishing cycles localised at particular points on the base [@mayr]. As a result, gauge interactions are localised on 5-branes at the singularities with a gauge coupling given by a geometric modulus (the size of the base in type IIA), whereas gravitational effects are still controlled by the string coupling ${\ensuremath{g_{\rm II}}}$. More precisely, the gauge and gravitational kinetic terms in the effective type IIA four-dimensional field theory are, in a self-explanatory notation: $$S_{\rm IIA}= \int d^4x\sqrt{-g}\left( \frac{1}{{\ensuremath{g_{\rm 6IIA}}}^2}\frac{R_5R_6}{{\ensuremath{l_{\rm II}}}^4}{\cal R}+ \frac{R_5R_6}{{\ensuremath{l_{\rm II}}}^2}F^2\right)\, , \label{SII}$$ where ${\ensuremath{l_{\rm II}}}$ is the type II string length, ${\ensuremath{g_{\rm 6IIA}}}$ is the six-dimensional string coupling, and $R_5R_6$ is the two-dimensional volume of the base, along the 5-brane where gauge fields are localised. For simplicity, we consider here the base to be a product of two circles with radii $R_5$ and $R_6$. In eq. (\[SII\]) and henceforth we set all numerical factors to one, although we take them into account in the numerical examples. Identifying the coefficient of $\cal R$ with the inverse Planck length ${\ensuremath{l_{\rm P}}}^2$ and the coefficient of $F^2$ with the inverse gauge coupling ${\ensuremath{g_{\rm YM}}}^2$, one gets: $$\frac{1}{{\ensuremath{g_{\rm YM}}}^2}=\frac{R_5R_6}{{\ensuremath{l_{\rm II}}}^2}\qquad,\qquad {\ensuremath{g_{\rm 6IIA}}}=\frac{1}{{\ensuremath{g_{\rm YM}}}}\frac{{\ensuremath{l_{\rm P}}}}{{\ensuremath{l_{\rm II}}}}\, . \label{relii}$$ Keeping the Yang-Mills coupling of order unity implies that $R_5R_6$ is of order ${\ensuremath{l_{\rm II}}}^2$, while ${\ensuremath{g_{\rm 6IIA}}}$ is a free parameter that allows to move the string length away from the Planck scale. As a result, one can take the type II string scale to be at the TeV, keeping all compactification radii to be at the same order of TeV$^{-1}$, by introducing a tiny string coupling of the order of $10^{-14}$. We stress again that this situation cannot be realised in heterotic or type I theories at weak coupling. This scenario is very different from the TeV strings arising in type I theory, where the string coupling is fixed by the 4d gauge coupling, while the type I string scale is lowered by introducing large transverse dimensions implying that gravity becomes strong at the TeV scale. Here, all internal dimensions have string size (TeV$^{-1}$) and gravitational/string interactions are extremely suppressed by the 4d Planck mass. As a result, Regge excitations cannot be detected in particle accelerators and the main experimental signal is the production of KK excitations of gauge particles, along the $(5,6)$ directions parallel to the 5-brane where gauge interactions are localised, due to the singular character of the compactification manifold; furthermore, these excitations come in multiplets of $N=4$ supersymetry, which is recovered in the six-dimensional limit. Quarks and leptons, on the other hand, do not in general have KK excitations since matter fields are localised at particular points of the base [@mayr]; they look similar to the twisted fields in heterotic orbifold compactifications. Notice the similarity of these predictions with those of heterotic string with large dimensions, despite its strong 10d coupling [@ia]. The requirement of $N=4$ excitations was, there, a way to keep the running of gauge couplings logarithmic above the compactification scale. In fact, as we will show in Section \[sechet\], the above type II models are dual to heterotic compactifications with two large TeV dimensions. An obvious advantage of this scenario is that several potential phenomenological problems, such as proton decay and flavor violations, appear much less dangerous than in type I TeV strings, since they are restricted to the structure of KK gauge modes only, and one does not have to worry about string excitations. Model building, on the other hand, becomes more involved since it requires a deeper understanding of the gauge theory on the 5-brane localised at the singular points of ${\ensuremath{K_3}}$; due to the weakness of the string coupling, the dynamics of the gauge theory is determined by classical string theory in a strongly curved background, which can be analysed, for instance, in the framework of geometric engineering [@mayr]. A related question is the one of gauge hierarchy, which in the present context consists of understanding why the type IIA string coupling is so small. The technical aspect of this problem is whether string interactions decouple from the effective gauge theory on the 5-brane, in the limit of vanishing coupling. Although naively such a decoupling seems obvious, there may be subtleties related to the non-perturbative origin of non-abelian gauge symmetries due to the singular character of the compactification manifold. In fact, we would like to argue that there are in general logarithmic singularities similar to the case of having an effective transverse dimensionality $d_\perp=2$ in the D-brane/type I scenario of TeV strings [@ab]. The argument is based on threshold corrections to gauge and gravitational couplings, that take the form: $$\Delta = b\ln(\mu a) +\Delta_{\rm reg}(t_i)\, ,$$ where the first term corresponds to an infrared (IR) divergent contribution, regularised by an IR momentum cutoff $\mu$, $b$ is a numerical $\beta$-function coefficient, and $\Delta_{\rm reg}$ is a function of the moduli $t_i$, which in $N=2$ compactifications belong to vector supermultiplets. For dimensional reasons, we have also introduced an ultraviolet (UV) scale $a$, which naively should be identified with the type II string length ${\ensuremath{l_{\rm II}}}$. However because of the relation (\[relii\]), in supergravity units, $\Delta$ would acquire a dependence on the 4d string coupling, which is impossible because it belongs to a neutral hypermultiplet that cannot mix with vector multiplets in the low energy theory. This suggests that $a$ should be identified with ${\ensuremath{l_{\rm P}}}$ and thus, in string units, there should be an additional contribution depending logarithmically on the string coupling. This can also be understood either as a result of integration over the massive charged states which have non perturbative origin, implying a UV cutoff proportional to $({\ensuremath{l_{\rm II}}}{\ensuremath{g_{\rm II}}})^{-1}$, or from heterotic–type II duality that we will discuss in Sections 3 and 4. Such a logarithmic dependence on the string coupling has also been observed in gravitational thresholds [@log]. The logarithmic sensitivity on ${\ensuremath{g_{\rm II}}}$ is very welcome because it allows in principle the possible dynamical determination of the hierarchy by minimising the effective potential. Note that the one-loop vacuum energy in non-supersymmetric type II vacua behaves as $\Lambda\sim {\ensuremath{l_{\rm II}}}^{-4}$ and thus is different from a quadratically divergent contribution that would go as $({\ensuremath{l_{\rm P}}}l_{\rm str})^{-2}$. This should be contrasted to the generic case of softly broken supersymmetry, as well as to TeV type I strings with large transverse dimensions, where the cancellation of this quadratic divergence implies a condition on the bulk energy density [@aadd]. Above, we discussed the simplest case of type II compactifications with string scale at the TeV and all internal radii having the string size. In principle, one can allow some of the ${\ensuremath{K_3}}$ transverse directions to be large. From eq. (\[relii\]), it follows that the string coupling ${\ensuremath{g_{\rm II}}}={\ensuremath{g_{\rm 6IIA}}}({\ensuremath{V_{K_3}}}/{\ensuremath{l_{\rm II}}}^4)^{1/2}$, with ${\ensuremath{V_{K_3}}}$ the volume of ${\ensuremath{K_3}}$, increases making gravity strong at distances ${\ensuremath{l_{\rm P}}}({\ensuremath{V_{K_3}}}/{\ensuremath{l_{\rm II}}}^4)^{1/2}$ larger than the Planck length. In particular, it can become strong at the TeV when ${\ensuremath{g_{\rm II}}}$ is of order unity. This corresponds to ${\ensuremath{V_{K_3}}}/{\ensuremath{l_{\rm II}}}^4\sim 10^{26}$. It follows that in the isotropic case there are 4 transverse dimensions at a fermi, while in the anisotropic case ${\ensuremath{V_{K_3}}}\sim R^\ell{\ensuremath{l_{\rm II}}}^{4-\ell}$ the size of the transverse radii $R$ increases with $\ell$ and reaches a micron for $\ell =2$. A more detailed analysis will be given in Section 5. We now turn on type IIB. As in type IIA, non-abelian gauge symmetries arise at singularities of ${\ensuremath{K_3}}$ from D3-branes wrapping around vanishing 2-cycles times a 1-cycle of the base. Therefore, at the level of six dimensions, they correspond to tensionless strings [@wi2]. The gauge and gravitational kinetic terms in the effective type IIB four-dimensional action can be obtained from eq. (\[SII\]) by T-duality with respect, for instance, to the 6th direction: $$R_6\to\frac{{\ensuremath{l_{\rm II}}}^2}{R_6}\qquad\qquad {\ensuremath{g_{\rm 6IIA}}}\to{\ensuremath{g_{\rm 6IIB}}}={\ensuremath{g_{\rm 6IIA}}}\frac{{\ensuremath{l_{\rm II}}}}{R_6}\, . \label{Tdual}$$ One obtains $$S_{\rm IIB}= \int d^4x\sqrt{-g}\left( \frac{1}{{\ensuremath{g_{\rm 6IIB}}}^2}\frac{R_5R_6}{l_{II}^4}{\cal R}+ \frac{R_5}{R_6}F^2\right)\, , \label{SIIB}$$ which leads to the identifications: $$\frac{1}{{\ensuremath{g_{\rm YM}}}^2}=\frac{R_5}{R_6}\qquad,\qquad {\ensuremath{g_{\rm 6IIB}}}=\frac{{\ensuremath{l_{\rm P}}}}{{\ensuremath{l_{\rm II}}}}\sqrt{\frac{R_5R_6}{{\ensuremath{l_{\rm II}}}^2}}= {\ensuremath{g_{\rm YM}}}\frac{R {\ensuremath{l_{\rm P}}}}{{\ensuremath{l_{\rm II}}}^2}\, . \label{reliib}$$ Keeping the Yang-Mills coupling of order unity, now implies that $R_5$ and $R_6$ are of the same order, $R_5=R_6/{\ensuremath{g_{\rm YM}}}^2\equiv R$, while ${\ensuremath{g_{\rm 6IIB}}}$ is a free parameter. Obviously, in order to get a situation different from type IIA, $R$ should be larger than the string length ${\ensuremath{l_{\rm II}}}$[^3]. This corresponds to a type IIA string with large $R_5$ and small $R_6$ so that $R_5R_6\sim{\ensuremath{l_{\rm II}}}^2$. Imposing the condition of weak coupling ${\ensuremath{g_{\rm II}}}={\ensuremath{g_{\rm 6IIB}}}({\ensuremath{V_{K_3}}}/{\ensuremath{l_{\rm II}}}^4)^{1/2}\simlt 1$, we find $${\ensuremath{l_{\rm II}}}\ge\sqrt{{\ensuremath{g_{\rm YM}}}R {\ensuremath{l_{\rm P}}}}\, ,$$ when all ${\ensuremath{K_3}}$ radii have string scale size. As a result, the type IIB string scale can be at intermediate energies $10^{11}$ GeV when the compactification scale $R^{-1}$ is at the TeV. The string coupling is then of order unity and gravity becomes strong at the intermediate string scale. This brings back some worry on proton stability although in a much milder form than in TeV scale quantum gravity models. Of course, the string scale can be lowered by decreasing the string coupling or by introducing large transverse dimensions in the ${\ensuremath{K_3}}$ part. In the latter case, gravitational interactions become strong at lower energies. This result provides the first instance of a weakly coupled string theory with large longitudinal dimensions seen by gauge interactions. In fact, as we will show in Section 4, this theory describes heterotic compactifications with a single large dimension. The existence of such dimension is motivated by supersymmetry breaking in the process of compactification [@ia]. Note that the physics above the compactification scale but below the type IIB string scale is described by an effective six-dimensional theory of a tensionless string [@wi2]. This theory possesses a non-trivial infrared dynamics at a fixed point of the renormalisation group [@sei][^4]. It would be interesting to study the dynamics of such theories in detail from the viewpoint of the reduced four-dimensional gauge theory. Heterotic–type I–type II triality \[trial\] =========================================== Here, we review briefly the basic ingredients of the dualities between the heterotic, type I and type II string theories, that we will use in our subsequent analysis. The reader with less string theoretical background may skip most of equations in this part with the exception of the duality relations (\[heti\]) and (\[hetii\]). As is now well known, these three string theories are related by non-perturbative dualities, which take the simplest form in the $N=4$ supersymmetric case of ($E_8\times E_8$ or $SO(32)$) heterotic theory compactified on $T^6$, type II on ${\ensuremath{K_3}}\times T^2$ and type I on $T^6$. We will restrict our attention to this case, since it already exhibits the main features we want to discuss. Heterotic–type I duality relates the two ten-dimensional string theories with $SO(32)$ gauge group, upon identifying [@pw] $${\ensuremath{l_{\rm I}}}={\ensuremath{g_{\rm H}}}^{1/2}{\ensuremath{l_{\rm H}}}\ \,\qquad {\ensuremath{g_{\rm I}}}=\frac{1}{{\ensuremath{g_{\rm H}}}}\ , \label{heti}$$ where ${\ensuremath{l_{\rm I}}}$ and ${\ensuremath{l_{\rm H}}}$ denote the type I and heterotic string length. The heterotic–type I duality itself holds in lower dimensions as well, and does not affect the physical shape or size of the compactification manifold. The $E_8\times E_8$ heterotic theory can in turn be perturbatively related to the $SO(32)$ heterotic theory upon compactifying to nine dimensions on a circle, since a $SO(1,17)$ boost transforms the two even self-dual lattices into one another. More precisely, the two theories are related by T-duality after breaking the gauge symmetry to $SO(16)\times SO(16)$ on both sides by Wilson lines [@gin][^5]: $$R_{\rm H'}=\frac{{\ensuremath{l_{\rm H}}}^2}{R_{H}} \,\qquad g_{\rm H'} = g_{\rm H}\frac{{\ensuremath{l_{\rm H}}}}{R_{H}}\ . \label{hethet}$$ where the prime refers to the $E_8\times E_8$ theory. On the type I side, T-duality maps the theory with 32 D9-branes to a theory with 32 lower D$p$-branes, referred to in this work as type I$^\prime_p$; the momentum states along the D9-branes are mapped to winding states in the directions transverse to the D$p$-branes, and the action on the radii and coupling constant is the standard T-duality relation $R\to{\ensuremath{l_{\rm I}}}^2/R$, ${\ensuremath{g_{\rm I}}}\to {\ensuremath{g_{\rm I}}}{\ensuremath{l_{\rm I}}}/R$. On the other hand, heterotic–type IIA duality only arises in 6 dimensions or lower, and identifies again two theories with inverse six-dimensional couplings ${\ensuremath{g_{\rm 6H}}}={\ensuremath{g_{\rm H}}}({\ensuremath{l_{\rm H}}}^4/V_4)^{1/2}$ and ${\ensuremath{g_{\rm 6IIA}}}={\ensuremath{g_{\rm II}}}({\ensuremath{l_{\rm II}}}^4/{\ensuremath{V_{K_3}}})^{1/2}$, with $V_4$ and ${\ensuremath{V_{K_3}}}$ the volumes of $T^4$ and ${\ensuremath{K_3}}$, respectively [@ht; @wi]: $${\ensuremath{l_{\rm II}}}={\ensuremath{g_{\rm 6IIA}}}{\ensuremath{l_{\rm H}}}\ \,\qquad {\ensuremath{g_{\rm 6IIA}}}=\frac{1}{{\ensuremath{g_{\rm 6H}}}}\ . \label{hetii0}$$ The identification of the scalar fields other than the dilaton can be obtained by decomposing the moduli space as $$\frac{SO(4,20)}{SO(4)\times SO(20)} \supset \left[ {\mathbb{R}}^+ \times \frac{SO(3,19)}{SO(3)\times SO(19)} \right] \ .$$ On the heterotic side, this involves choosing a preferred direction of radius $R_1$ in $T^4$, parametrising the ${\mathbb{R}}^+$ factor, whereas on the type IIA side, the two factors occur naturally as the overall volume and complex structure of ${\ensuremath{K_3}}$, respectively. The volume of ${\ensuremath{K_3}}$ in type IIA units is thus related to the radius $R_1$ in heterotic units as $$\label{rvk} \left( \frac{R_1}{{\ensuremath{l_{\rm H}}}} \right)^2 = \frac{{\ensuremath{V_{K_3}}}}{{\ensuremath{l_{\rm II}}}^4}\ .$$ Relating the other moduli fields requires a more precise understanding of the geometry of ${\ensuremath{K_3}}$ [@Aspinwall:1996mn], but in the following we will be able to obtain a partial identification from the M-theory point of view. Indeed, the above dualities can be understood in the M-theory description[^6], which subsumes the heterotic and type II descriptions at strong coupling. The type IIA string theory then appears as M-theory compactified on a vanishingly small circle of radius $R_s$ given by $$R_s = {\ensuremath{g_{\rm II}}}{\ensuremath{l_{\rm II}}}\ ,\qquad {\ensuremath{l_{\rm M}}}^3= {\ensuremath{g_{\rm II}}}{\ensuremath{l_{\rm II}}}^3$$ where ${\ensuremath{l_{\rm M}}}$ denotes the eleven-dimensional Planck length [@wi; @to]; the $E_8\times E_8$ heterotic string is obtained upon compactifying on a segment $I$ of length $R_I$ given by analogous formulae: $$\ \label{hetm} R_I = {\ensuremath{g_{\rm H}}}{\ensuremath{l_{\rm H}}}\ ,\qquad {\ensuremath{l_{\rm M}}}^3= {\ensuremath{g_{\rm H}}}{\ensuremath{l_{\rm H}}}^3\ .$$ The two nine-branes at the end of the segment support the non-abelian gauge-fields, whereas gravity propagates in the bulk [@hw]. In this framework, a double T-duality symmetry $(R_1,R_2)$ $\to$ $({\ensuremath{l_{\rm II}}}^2/R_2,{\ensuremath{l_{\rm II}}}^2/R_1)$ of type IIA acts as [@egkr; @Obers:1998fb] $$T_{ijk}~:~\tilde R_i=\frac{{\ensuremath{l_{\rm M}}}^3}{R_j R_k}\ ,\ \tilde R_j=\frac{{\ensuremath{l_{\rm M}}}^3}{R_i R_k}\ ,\ \tilde R_k=\frac{{\ensuremath{l_{\rm M}}}^3}{R_i R_j}\ ,\ \tilde l_M^3=\frac{{\ensuremath{l_{\rm M}}}^6}{R_i R_j R_k}\ ,\ \label{hmdf}$$ where one of the radii $i,j,k$ corresponds to the eleventh dimension; by eleven-dimensional general covariance, this symmetry still holds for any choice of the three radii. As for the heterotic T-duality $R_1\to {\ensuremath{l_{\rm H}}}^2/R_1$, it translates into a symmetry $$T_{Ii}~:~\tilde R_i=\frac{{\ensuremath{l_{\rm M}}}^3}{R_I R_i}\ ,\ \tilde R_I=\frac{R_I ^{1/2} {\ensuremath{l_{\rm M}}}^{3/2}}{R_i}\ ,\ \tilde l_M^3=\frac{{\ensuremath{l_{\rm M}}}^{9/2}}{R_I ^{1/2} R_1}\ ,\ \label{hmdf2}$$ where $R_i$ denotes the radius of any circular dimension and $R_I$ the length of the (single) segment direction. (300,300)(-150,-150) (0,0) (0,0) (0,0) (0,0) (20,20)[(1,1)[60]{}]{} (20,-20)[(1,-1)[60]{}]{} (-20,20)[(-1,1)[60]{}]{} (-20,-20)[(-1,-1)[60]{}]{} (0,9)[(0,0)[M-theory on]{}]{} (0,-9)[(0,0)[$S_1(R_1)\times I(R_I)\times T^3(R_2,R_3,R_4)$]{}]{} (105,114)[(0,0)[Het $SO(32)$ on]{}]{} (105,96)[(0,0)[$T^4(\tilde R_1,R_2,R_3,R_4)$ ]{}]{} (105,-96)[(0,0)[Het $E_8\times E_8$ on]{}]{} (105,-114)[(0,0)[$T^4(R_1,R_2,R_3,R_4)$]{}]{} (-105,114)[(0,0)[Type I $SO(32)$ on]{}]{} (-105,96)[(0,0)[ $T^4(\tilde R_I,R_2,R_3,R_4)$]{}]{} (-105,-96)[(0,0)[Type IIA on $K_3$=]{}]{} (-105,-114)[(0,0)[$I(R_I)\times T^3(\tilde R_2,\tilde R_3,\tilde R_4)$]{}]{} (50,50)[(0,0)[$T_{I1}$]{}]{} (-50,-50)[(0,0)[$T_{234}$]{}]{} (-50,50)[(0,0)[$T_{1I}$]{}]{} As depicted in Figure 1, we can now obtain the heterotic–type I–type II relationships by interpreting the compactification of M-theory on a manifold $S_1(R_1)\times I(R_I)\times T^3(R_2,R_3,R_4)$ in various ways. (i) Considering $I(R_I)$ as the eleventh dimension simply gives the $E_8\times E_8$ heterotic string on $T^4(R_1,R_2,R_3,R_4)$ with string length and coupling given by eq. (\[hetm\]). (ii) Considering $S_1(R_1)$ as the eleventh dimension gives type IIA on $I\times T^3$, or more properly type I$^\prime_8$ on $I\times T^3$; (iii) By T-duality $T_{I1}$ along the segment $I$, the theory (i) translates into heterotic $SO(32)$, whereas the theory (ii) turns into type I: it is straightforward to check that these two are related by the duality relations (\[heti\]) [@hw]. (iv) if we perform a $T_{234}$ duality on the torus $T^3$ before identifying $S_1(R_1)$ with the eleventh dimension, we obtain a type IIA theory compactified on a four-dimensional manifold $I(R_I) \times T^3(\tilde R_2,\tilde R_3,\tilde R_4)$, which by (i) is the same as the $E_8\times E_8$ heterotic string on $T^4(R_1,R_2,R_3,R_4)$. It is easy to check that these two theories are related by the heterotic–type II duality relations (\[hetii0\],\[rvk\]). It is therefore tempting to identify $${\ensuremath{K_3}}= I(R_I) \times T^3(\tilde R_2,\tilde R_3,\tilde R_4) \ , \label{K3}$$ where the type II parameters are related to the heterotic ones as \[hetii\] $$\begin{aligned} {\ensuremath{l_{\rm II}}}&=& \frac{{\ensuremath{g_{\rm H}}}{\ensuremath{l_{\rm H}}}^3}{\sqrt{R_1 R_2 R_3 R_4}}\ ,\qquad {\ensuremath{g_{\rm II}}}= \frac{\sqrt{R_1 ^3 R_2 R_3 R_4}}{{\ensuremath{g_{\rm H}}}{\ensuremath{l_{\rm H}}}^3}\ ,\qquad\label{hetii1}\\ R_I&=&{\ensuremath{g_{\rm H}}}{\ensuremath{l_{\rm H}}}\ ,\qquad \tilde R_i=\frac{{\ensuremath{g_{\rm H}}}{\ensuremath{l_{\rm H}}}^3}{R_j R_k}\ ,\ i,j,k=2,3,4\ .\label{hetii2}\end{aligned}$$ In units of the respective string length, this is[^7] $$\begin{aligned} \left(\frac{R_I}{{\ensuremath{l_{\rm II}}}}\right)^2 &=& \frac{R_1 R_2 R_3 R_4}{{\ensuremath{l_{\rm H}}}^4}\ ,\\ \left(\frac{\tilde R_2}{{\ensuremath{l_{\rm II}}}}\right)^2 &=& \frac{R_1 R_2}{R_3 R_4}\ ,\quad \left(\frac{\tilde R_3}{{\ensuremath{l_{\rm II}}}}\right)^2 = \frac{R_1 R_3}{R_2 R_4}\ ,\quad \left(\frac{\tilde R_4}{{\ensuremath{l_{\rm II}}}}\right)^2 = \frac{R_1 R_4}{R_2 R_3}\ ,\quad\end{aligned}$$ where we recognize a triality transformation in the $[SO(4)\times SO(4)] \backslash SO(4,4)$ subspace of the moduli space. Even though (\[K3\]) is not a proper ${\ensuremath{K_3}}$ surface (for one thing it is not simply connected), it still is na bona fide compactification manifold, albeit singular. Indeed, it has been argued that such a “squashed” shape arises in the decompactification limit of the heterotic torus $T^4$ at the $E_8 \times E_8$ enhanced symmetry point [@Aspinwall:1998eh]. This representation of ${\ensuremath{K_3}}$ will turn out to be very convenient in the discussion of large radius behaviour of heterotic and type II theories in the sequel. Large dimensions in heterotic string \[sechet\] =============================================== Here we consider the heterotic string compactified in four dimensions with a certain number $n$ of large internal dimensions. Keeping the four-dimensional gauge coupling ${\ensuremath{g_{\rm YM}}}$ of order unity, the heterotic theory is strongly coupled with a ten-dimensional string coupling and four-dimensional Planck length $${\ensuremath{g_{\rm H}}}={\ensuremath{g_{\rm YM}}}\left(\frac{R}{{\ensuremath{l_{\rm H}}}}\right)^{n/2}\gg 1\ ,\qquad {\ensuremath{l_{\rm P}}}={\ensuremath{g_{\rm YM}}}{\ensuremath{l_{\rm H}}}$$ where $R$ is the common radius of the large dimensions, while the remaining $6-n$ are assumed to be of the order of the string length ${\ensuremath{l_{\rm H}}}$. For $n<6$, the distinction between the $SO(32)$ and $E_8\times E_8$ heterotic theories is irrelevant, since a T-duality (\[hethet\]) along a heterotic-size direction converts one into another; we will therefore omit this distinction until we discuss the $n=6$ case, where such a dualisation is no longer innocuous. In order to obtain a perturbative description of this theory, we consider first its type I dual obtained through the relations (\[heti\]). The physical radii of the internal manifold are unaffected by this duality. In particular, there are still $6-n$ dimensions of size ${\ensuremath{l_{\rm H}}}$, and $n$ of size $R$. We therefore have the following type I string length and coupling $${\ensuremath{l_{\rm I}}}={\ensuremath{g_{\rm YM}}}^{1/2} R^\frac{n}{4} {\ensuremath{l_{\rm H}}}^{1-\frac{n}{4}}\ ,\qquad {\ensuremath{g_{\rm I}}}=\frac{1}{{\ensuremath{g_{\rm YM}}}}\left(\frac{{\ensuremath{l_{\rm H}}}}{ R}\right)^\frac{n}{2}\ ,$$ or more explicitely $$\mbox{Type I:}\quad \left\{ \begin{array}{lcccccc} n=1,2,3,4: & {\ensuremath{l_{\rm H}}}& < & {\ensuremath{l_{\rm I}}}& < &R \\ n=5,6: & {\ensuremath{l_{\rm H}}}& < & R & < &{\ensuremath{l_{\rm I}}}\end{array} \right.$$ In both cases, there are dimensions ($6-n$ or 6 respectively) with size smaller than the type I string length, which should be T-dualised in order to trade light winding modes for Kaluza-Klein (KK) field-theory states. In so doing, we move to a type I$^\prime$ description where the gauge interactions are localised on D-branes (extended in $3+n$ or 3 spatial directions, respectively). Using the standard ($\hat R= {\ensuremath{l_{\rm I}}}^2/R$, $\hat{\ensuremath{g_{\rm I}}}={\ensuremath{g_{\rm I}}}{\ensuremath{l_{\rm I}}}/R$) T-duality relations, we obtain the dual radii $$\hat {\ensuremath{l_{\rm H}}}={\ensuremath{g_{\rm YM}}}R^\frac{n}{2} {\ensuremath{l_{\rm H}}}^\frac{1-n}{2}\ ,\quad \hat R={\ensuremath{g_{\rm YM}}}R^\frac{n-2}{2} {\ensuremath{l_{\rm H}}}^\frac{2-n}{2}$$ and the hierarchies $$\mbox{Type I$^\prime$:}\quad \left\{ \begin{array}{lcccccc} n=1,2~:~ {\ensuremath{g_{\rm I'}}}={\ensuremath{g_{\rm YM}}}^\frac{4-n}{2} \left(\frac{R}{{\ensuremath{l_{\rm H}}}}\right)^\frac{n(4-n)}{4}\ ,& {\ensuremath{l_{\rm I}}}& < & \hat {\ensuremath{l_{\rm H}}}& < &R \\ n=3,4~:~ {\ensuremath{g_{\rm I'}}}={\ensuremath{g_{\rm YM}}}^\frac{4-n}{2} \left(\frac{R}{{\ensuremath{l_{\rm H}}}}\right)^\frac{n(4-n)}{4}\ ,& {\ensuremath{l_{\rm I}}}& < & R & < &\hat {\ensuremath{l_{\rm H}}}\\ n=5,6~:~ {\ensuremath{g_{\rm I'}}}={\ensuremath{g_{\rm YM}}}^2\ , & {\ensuremath{l_{\rm I}}}& < & \hat R & < &\hat {\ensuremath{l_{\rm H}}}\end{array} \right. \label{ip}$$ where the T-dualised hatted radii correspond to transverse dimensions. In the cases $n=1,2,3$, type I$^\prime$ theory is also strongly coupled, as seen from eqs. (\[ip\]), which is a consequence of the fact that the internal longitudinal directions of the D-branes (of size $R$) are larger than the type I$^\prime$ string length. In the cases $n=5,6$ however, the type I$^\prime$ theory does offer a perturbative description of the theory of interest, where the gauge interactions are confined on D3-branes with large transverse dimensions of size $\hat R$ ($n$ of them) and $\hat {\ensuremath{l_{\rm H}}}$ ($6-n$ of them). This is depicted in the following diagrams: [Het $n=5$]{}[I$^\prime_3, {\ensuremath{g_{\rm I'}}}={\ensuremath{g_{\rm YM}}}^2$]{}[scal0]{} [ (20,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][1]{}]{} (0,-10)[(0,0)\[t\][${\ensuremath{l_{\rm H}}},R_6$]{}]{} ]{}]{} [ (120,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\]]{} (0,-10)[(0,0)\[t\][R]{}]{} ]{}]{} [ (145,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][${\ensuremath{g_{\rm YM}}}^{1/2} R^{5/4}$]{}]{} (0,-10)[(0,0)\[t\][${\ensuremath{l_{\rm I}}}$]{}]{} ]{}]{} [ (170,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][    ${\ensuremath{g_{\rm YM}}}R^{3/2}$]{}]{} (0,-10)[(0,0)\[t\][    $\hat R_{1,2,3,4,5}$]{}]{} ]{}]{} [ (270,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][${\ensuremath{g_{\rm YM}}}R^{5/2}$]{}]{} (0,-10)[(0,0)\[t\][$\hat R_6$]{}]{} ]{}]{} [Het SO(32) $n=6$]{}[I$^\prime_3, {\ensuremath{g_{\rm I'}}}={\ensuremath{g_{\rm YM}}}^2$]{}[scal00]{} [ (20,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][1]{}]{} (0,-10)[(0,0)\[t\][${\ensuremath{l_{\rm H}}}$]{}]{} ]{}]{} [ (120,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][R]{}]{} (0,-10)[(0,0)\[t\]]{} ]{}]{} [ (170,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][${\ensuremath{g_{\rm YM}}}^{1/2} R^{3/2}$]{}]{} (0,-10)[(0,0)\[t\][${\ensuremath{l_{\rm I}}}$]{}]{} ]{}]{} [ (220,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][${\ensuremath{g_{\rm YM}}}R^2$]{}]{} (0,-10)[(0,0)\[t\][$\hat R_{1,2,3,4,5,6}$]{}]{} ]{}]{} In particular, the $SO(32)$ heterotic string with $n=6$ large dimensions, say at $10^8$ GeV, is dual to a type I$^\prime$ with string tension at the TeV and six transverse dimensions at 0.1 fermi. This is one of the examples that were treated recently in the context of TeV strings [@aadd]. The type I threshold appears in the strongly coupled heterotic theory below the KK scale of $10^8$ GeV [@ckm], which is then identified with the scale of the (superheavy) type I$^\prime$ winding states around the fermi-size transverse dimensions. In the $n=4$ case, two distinct type I$^\prime$ perturbative descriptions are possible, due to the proximity of the radius $R$ of the four large dimensions with the type I string scale ${\ensuremath{l_{\rm I}}}={\ensuremath{g_{\rm YM}}}^{1/2}R$. For ${\ensuremath{g_{\rm YM}}}<1$, it is sufficient to T-dualise the two directions of size ${\ensuremath{l_{\rm H}}}$, resulting in a D7-brane type I$^\prime$ description with string coupling unity, which can be lowered by increasing for instance the size of the two small dimensions slightly above the heterotic length. For ${\ensuremath{g_{\rm YM}}}>1$ on the other hand, one should T-dualise also the remaining four directions, resulting in a D3-brane type I$^\prime$ description as above. In both cases, the type I$^\prime$ scale is close to the size of the four heterotic large dimensions, say at the TeV scale. This provides another example of type I$^\prime$ TeV strings [@aadd]. The gauge interactions are confined on D-branes transverse to two large dimensions of (sub)millimeter-size. The type I threshold now appears at the same order as the KK scale (at the TeV) [@aq], while the heterotic scale – which is also the KK scale of the remaining two dimensions – is identified with the mass of the type I$^\prime$ winding modes around the two millimeter-size dimensions. This model is of particular interest, because it offers a possibility to keep the apparent unification of gauge couplings close to the heterotic scale, due to the logarithmic sensitivity of the gauge theory on the brane with respect to the size of the two-dimensional transverse space [@cb; @ab]. [Het $n=4$]{}[I$^\prime_7, {\ensuremath{g_{\rm I'}}}=1$]{}[scal01]{} [ (20,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][1]{}]{} (0,-10)[(0,0)\[t\][${\ensuremath{l_{\rm H}}}$]{}]{} ]{}]{} [ (100,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][${\ensuremath{g_{\rm YM}}}^{1/2} R$]{}]{} (0,-10)[(0,0)\[t\][${\ensuremath{l_{\rm I}}}$]{}]{} ]{}]{} [ (140,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][$R$]{}]{} (0,-10)[(0,0)\[t\][$R_{1,2,3,4}$]{}]{} ]{}]{} [ (260,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][${\ensuremath{g_{\rm YM}}}R^{2}$]{}]{} (0,-10)[(0,0)\[t\][$\hat R_{5,6}$]{}]{} ]{}]{} [Het $n=4$]{}[I$^\prime_3, {\ensuremath{g_{\rm I'}}}={\ensuremath{g_{\rm YM}}}^2>1$]{}[scal02]{} [ (20,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][1]{}]{} (0,-10)[(0,0)\[t\][${\ensuremath{l_{\rm H}}}$]{}]{} ]{}]{} [ (140,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][$R$]{}]{} (0,-10)[(0,0)\[t\]]{} ]{}]{} [ (170,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][${\ensuremath{g_{\rm YM}}}^{1/2} R$]{}]{} (0,-10)[(0,0)\[t\][${\ensuremath{l_{\rm I}}}$]{}]{} ]{}]{} [ (210,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][${\ensuremath{g_{\rm YM}}}R$]{}]{} (0,-10)[(0,0)\[t\][$\hat R_{1,2,3,4}$]{}]{} ]{}]{} [ (260,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][${\ensuremath{g_{\rm YM}}}R^{2}$]{}]{} (0,-10)[(0,0)\[t\][$\hat R_{5,6}$]{}]{} ]{}]{} In order to obtain a perturbative description for the cases $n=1,2,3$, we now consider the type IIA dual of the original heterotic theory. As described in Section \[trial\], the heterotic–type IIA duality selects four preferred dimensions of radii $R_{1,2,3,4}$ on the heterotic side, while at the $E_8\times E_8$ enhanced symmetry point the compactification manifold for the type IIA string takes the simplified form $${\ensuremath{K_3}}\times T^2 = \left[ I(R_I) \times T^3(\tilde R_2, \tilde R_3,\tilde R_4) \right] \times T^2( R_5,R_6)\ . \label{ktt}$$ The remaining two-torus of radii $R_5,R_6$ is common to both descriptions, which also have the same four-dimensional gauge coupling and Planck mass $$\begin{aligned} \frac{1}{{\ensuremath{g_{\rm YM}}}^2}&=&\frac{R_1 R_2 R_3 R_4 R_5 R_6} {{\ensuremath{g_{\rm H}}}^2 {\ensuremath{l_{\rm H}}}^6}=\frac{R_5 R_6}{{\ensuremath{l_{\rm II}}}^2}\ ,\\ \frac{1}{{\ensuremath{l_{\rm P}}}^2}&=&\frac{R_1 R_2 R_3 R_4 R_5 R_6}{{\ensuremath{g_{\rm H}}}^2 {\ensuremath{l_{\rm H}}}^8}= \frac{R_I \tilde R_2 \tilde R_3 \tilde R_4 R_5 R_6}{{\ensuremath{g_{\rm II}}}^2 {\ensuremath{l_{\rm II}}}^8}\ .\end{aligned}$$ In terms of these quantities, the duality map (\[hetii\]) takes the form $$\begin{aligned} {\ensuremath{l_{\rm II}}}&=&{\ensuremath{g_{\rm YM}}}\sqrt{R_5 R_6}\ ,\qquad {\ensuremath{g_{\rm II}}}=\frac{1}{{\ensuremath{g_{\rm YM}}}} \frac{R_1}{\sqrt{R_5 R_6}} \\ R_I&=&{\ensuremath{g_{\rm H}}}{\ensuremath{l_{\rm H}}}\ ,\qquad \tilde R_i=\frac{{\ensuremath{g_{\rm H}}}{\ensuremath{l_{\rm H}}}^3}{R_j R_k}\ ,\ i,j,k=2,3,4\end{aligned}$$ As we mentioned in the previous Sections, the four ${\ensuremath{K_3}}$ directions corresponding to $R_I$ and $\tilde R_i$ are transverse to the 5-brane where gauge interactions are localised. In order to obtain a weakly coupled type II description, we therefore need to carefully arrange the choice of the $n$ large dimensions on the heterotic side. For instance, in the $n=1$ case, choosing $R_1$ as the large radius results in a strongly coupled type II theory with ${\ensuremath{g_{\rm II}}}\sim R$ in units of the heterotic string length; choosing $R_2$ (or $R_3,R_4$) as the large dimensions gives a type II dual with moderate coupling ${\ensuremath{g_{\rm II}}}\sim 1/{\ensuremath{g_{\rm YM}}}$, but with radii $\tilde R_3,\tilde R_4 \sim 1/\sqrt{R}$ much smaller than the string length; after T-dualisation along these directions, the theory becomes strongly coupled. The last option is to take $R_5$ (or $R_6)$ as the large radius, which yields a weakly coupled type II dual string with ${\ensuremath{g_{\rm II}}}\sim 1/{\ensuremath{g_{\rm YM}}}\sqrt{R}$ and ${\ensuremath{l_{\rm II}}}\sim {\ensuremath{g_{\rm YM}}}\sqrt{R}$; after T-dualizing the heterotic-size direction $R_6$, we obtain a weakly coupled type IIB description with hierarchy [Het $n=1$]{}[IIB ${\ensuremath{g_{\rm II}}}=1$]{}[scal1]{} [ (20,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][1]{}]{} (0,-10)[(0,0)\[t\][${\ensuremath{l_{\rm H}}}$]{}]{} ]{}]{} [ (120,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][${\ensuremath{g_{\rm YM}}}R^{1/2}$]{}]{} (0,-10)[(0,0)\[t\][${\ensuremath{l_{\rm II}}},R_{I,\tilde 2,\tilde 3,\tilde 4}$]{}]{} ]{}]{} [ (220,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][${\ensuremath{g_{\rm YM}}}^2 R$]{}]{} (0,-10)[(0,0)\[t\][$\hat R_6$]{}]{} ]{}]{} [ (250,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][$R$]{}]{} (0,-10)[(0,0)\[t\][$R_5$]{}]{} ]{}]{} where we denoted by $\hat R_6$ the radius of the T-dual sixth dimension. This is one of the models discussed in Section \[secii\], with two radii at the TeV and a string scale at intermediate energies $10^{11}$ GeV. In the $n=2$ case, the same reasoning leads to choosing the radii $R_5,R_6\sim R$ as the large heterotic dimensions, and gives a weakly coupled type IIA description [Het $n=2$]{}[IIA ${\ensuremath{g_{\rm II}}}=\frac{1}{{\ensuremath{g_{\rm YM}}}R}$]{}[scal2]{} [ (20,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][1]{}]{} (0,-10)[(0,0)\[t\][${\ensuremath{l_{\rm H}}}$]{}]{} ]{}]{} [ (170,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][${\ensuremath{g_{\rm YM}}}R$]{}]{} (0,-10)[(0,0)\[t\][${\ensuremath{l_{\rm II}}},R_{I,\tilde 2,\tilde 3,\tilde 4}$]{}]{} ]{}]{} [ (220,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][$R$]{}]{} (0,-10)[(0,0)\[t\][$R_{5,6}$]{}]{} ]{}]{} without need of T-dualizing any direction. This is the other type II model discussed in Section \[secii\], with string scale and all internal dimensions at the TeV, and with an infinitesimal string coupling $10^{-14}$ accounting for the largeness of the four-dimensional Planck mass. In the cases $n=3,4,5,6$, we choose the directions of radii $R_1,R_5,R_6\sim R$ as three of the large heterotic dimensions, and for $n>3$ also switch on $n-3$ large dimensions in the heterotic $T^3(R_2,R_3,R_4)$ torus. The type II dual has string length ${\ensuremath{l_{\rm II}}}={\ensuremath{g_{\rm YM}}}R$ and coupling ${\ensuremath{g_{\rm II}}}=1/{\ensuremath{g_{\rm YM}}}$, while the ${\ensuremath{K_3}}$ manifold has size (in heterotic units): $$\begin{array}{|c|c|c|c|c|c|} \hline n & R_I & \tilde R_2 &\tilde R_3 &\tilde R_4 \\ \hline 3 & {\ensuremath{g_{\rm YM}}}R^{3/2} &{\ensuremath{g_{\rm YM}}}R^{3/2} &{\ensuremath{g_{\rm YM}}}R^{3/2} &{\ensuremath{g_{\rm YM}}}R^{3/2} \\ 4 & {\ensuremath{g_{\rm YM}}}R^2 &{\ensuremath{g_{\rm YM}}}R^2 &{\ensuremath{g_{\rm YM}}}R &{\ensuremath{g_{\rm YM}}}R \\ 5 & {\ensuremath{g_{\rm YM}}}R^{5/2} &{\ensuremath{g_{\rm YM}}}R^{3/2} &{\ensuremath{g_{\rm YM}}}R^{3/2} &{\ensuremath{g_{\rm YM}}}R^{1/2} \\ 6 & {\ensuremath{g_{\rm YM}}}R^3 &{\ensuremath{g_{\rm YM}}}R &{\ensuremath{g_{\rm YM}}}R &{\ensuremath{g_{\rm YM}}}R\\ \hline \end{array}$$ Except for the $n=5$, where the existence of the small radius $\tilde R_4$ implies strong coupling after T-duality, all these cases correspond to a weakly coupled type II dual. In the $n=3$ case, the type II dual provides a perturbative description of the heterotic theory that could not be reached on the type I side: [Het $n=3$]{}[IIA ${\ensuremath{g_{\rm II}}}=\frac{1}{{\ensuremath{g_{\rm YM}}}}$]{}[scal3]{} [ (20,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][1]{}]{} (0,-10)[(0,0)\[t\][${\ensuremath{l_{\rm H}}}$]{}]{} ]{}]{} [ (120,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][${\ensuremath{g_{\rm YM}}}R$]{}]{} (0,-10)[(0,0)\[t\][${\ensuremath{l_{\rm II}}}$]{}]{} ]{}]{} [ (150,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][$R$]{}]{} (0,-10)[(0,0)\[t\][$R_{5,6}$]{}]{} ]{}]{} [ (220,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][${\ensuremath{g_{\rm YM}}}R^{3/2}$]{}]{} (0,-10)[(0,0)\[t\][$R_{I,\tilde 2,\tilde 3,\tilde 4}$]{}]{} ]{}]{} This is the type II model discussed in Section \[secii\] with string scale and two longitudinal dimensions at the TeV, and an isotropic ${\ensuremath{K_3}}$ with 4 transverse directions at a fermi. In the $n=4$ case, the type II dual theory provides a perturbative description, alternative to the type I$^\prime$. The type II dual string has the same scale hierarchy as the type I, up to factors of ${\ensuremath{g_{\rm YM}}}$: [I$^\prime_7, {\ensuremath{g_{\rm I'}}}=1$]{}[Het $n=4$]{} [IIA ${\ensuremath{g_{\rm II}}}=\frac{1}{{\ensuremath{g_{\rm YM}}}}$]{}[scal4]{} [ (20,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][${\ensuremath{l_{\rm H}}}$]{}]{} (0,-35)[(0,1)[10]{}]{} (0,-20)[(0,0)\[b\][1]{}]{} (0,-40)[(0,0)\[t\][${\ensuremath{l_{\rm H}}}$]{}]{} ]{}]{} [ (110,0)[ (0,-35)[(0,1)[10]{}]{} (0,-20)[(0,0)\[b\][${\ensuremath{g_{\rm YM}}}R$]{}]{} (0,-40)[(0,0)\[t\][${\ensuremath{l_{\rm II}}},R_{\tilde 3,\tilde 4}$]{}]{} ]{}]{} [ (150,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][${\ensuremath{l_{\rm I}}}$]{}]{} (0,-10)[(0,0)\[t\][${\ensuremath{g_{\rm YM}}}^{1/2} R$]{}]{} ]{}]{} [ (180,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][$R_{1,2,3,4}$]{}]{} (0,-35)[(0,1)[10]{}]{} (0,-20)[(0,0)\[b\][$R$]{}]{} (0,-40)[(0,0)\[t\][$R_{5,6}$]{}]{} ]{}]{} [ (250,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][$\hat R_{5,6}$]{}]{} (0,-35)[(0,1)[10]{}]{} (0,-20)[(0,0)\[b\][${\ensuremath{g_{\rm YM}}}R^{2}$]{}]{} (0,-40)[(0,0)\[t\][$R_{I,\tilde 2}$]{}]{} ]{}]{} These two models should provide equivalent perturbative descriptions of the same theory. In the $n=6$ case, we now obtain a weakly coupled description of the $E_8\times E_8$ heterotic string with $n=6$ large radii as a type IIA string with string length ${\ensuremath{l_{\rm II}}}={\ensuremath{g_{\rm YM}}}R$: [Het $E_8\times E_8, n=6$]{}[IIA ${\ensuremath{g_{\rm II}}}=\frac{1}{{\ensuremath{g_{\rm YM}}}}$]{}[scal5]{} [ (20,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][1]{}]{} (0,-10)[(0,0)\[t\][${\ensuremath{l_{\rm H}}}$]{}]{} ]{}]{} [ (80,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][${\ensuremath{g_{\rm YM}}}R$]{}]{} (0,-10)[(0,0)\[t\][${\ensuremath{l_{\rm II}}},R_{\tilde 2,\tilde 3,\tilde 4}$]{}]{} ]{}]{} [ (120,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][$R$]{}]{} (0,-10)[(0,0)\[t\][$R_{5,6}$]{}]{} ]{}]{} [ (270,0)[ (0,-5)[(0,1)[10]{}]{} (0,10)[(0,0)\[b\][${\ensuremath{g_{\rm YM}}}R^3$]{}]{} (0,-10)[(0,0)\[t\][$R_I$]{}]{} ]{}]{} Due to the occurrence of gravitational KK states[^8] at the scale $R_I$, the type II string tension as well as the heterotic compactification scale cannot be lower than $10^8$ GeV, corresponding to the bound $R_I\simlt 1$ mm. This situation should be contrasted with the case of the $SO(32)$ heterotic string with $n=6$ large radii, which admits a perturbative description (\[scal00\]) as a type I string with length ${\ensuremath{l_{\rm I}}}={\ensuremath{g_{\rm YM}}}^{1/2} R^{3/2}$. The bound on $R$ still applies, corresponding now to a type I string scale at a TeV, and six transverse dimensions at 0.1 fermi. Note that the difference between the type I and type II string scales does not lead to any inconsistency, since the two perturbative descriptions are not simultaneously possible. Large dimensions in type II theories and their duals\[seciid\] ============================================================== Having discussed the large radius behaviour of the dual heterotic theory, we now reconsider the type IIA models we introduced in Section \[secii\], and discuss their dual descriptions. We therefore consider type IIA theory, compactified on the simplified model (\[ktt\]) of ${\ensuremath{K_3}}\times T^2$, with a weak string coupling ${\ensuremath{g_{\rm II}}}\ll 1$, two string-size directions $R_5,R_6\sim {\ensuremath{l_{\rm II}}}$ and possibly $\ell$ large transverse directions of size $R\gg {\ensuremath{l_{\rm II}}}$ within ${\ensuremath{K_3}}$. This theory is then identified to a strongly coupled heterotic string compactified on $T^6$, with parameters simply obtained by inverting eq. (\[hetii\]): \[iihet\] $$\begin{aligned} {\ensuremath{l_{\rm H}}}&=& \frac{{\ensuremath{g_{\rm II}}}{\ensuremath{l_{\rm II}}}^3}{\sqrt{R_I \tilde R_2 \tilde R_3 \tilde R_4}}\ ,\qquad {\ensuremath{g_{\rm H}}}= \frac{\sqrt{R_I ^3 \tilde R_2 \tilde R_3 \tilde R_4}}{{\ensuremath{g_{\rm II}}}{\ensuremath{l_{\rm II}}}^3}\ ,\qquad\label{iihet1}\\ R_1&=&{\ensuremath{g_{\rm II}}}{\ensuremath{l_{\rm II}}}\ ,\qquad R_i=\frac{{\ensuremath{g_{\rm II}}}{\ensuremath{l_{\rm II}}}^3}{\tilde R_j \tilde R_k}\ ,\ i,j,k=2,3,4\ . \label{iihet2}\end{aligned}$$ while the torus $T^2(R_5,R_6)$, common to both sides, is still at the type II string scale. In the $\ell=0$ case, the dual heterotic string has a scale ${\ensuremath{l_{\rm H}}}={\ensuremath{g_{\rm II}}}$ in type II units, of the same order as the radii $R_{1,2,3,4}$; $R_{5,6}$ on the other hand still have a type II string scale, and are much larger (at weak type II coupling) than the heterotic scale. This situation is therefore identical to the heterotic $n=2$ case in (\[scal2\]), which did not admit a perturbative type I dual. For $\ell\ge 1$, T-duality on the ${\ensuremath{K_3}}$ manifold allows us to choose one of the large directions as the interval of length $R_I=R\gg {\ensuremath{l_{\rm II}}}$. We therefore consider a regime where $$R_I = R\ ,\qquad \tilde R_2 \tilde R_3 \tilde R_4 = R^{\ell-1}\ ,\qquad R_{5}=R_6 = {\ensuremath{l_{\rm II}}}\ ,\qquad {\ensuremath{g_{\rm II}}}\ll 1$$ in units of the type IIA string length ${\ensuremath{l_{\rm II}}}$. The parameters for the dual heterotic string therefore scale as $${\ensuremath{l_{\rm H}}}= {\ensuremath{g_{\rm II}}}R^{-\ell/2}\ ,\qquad{\ensuremath{g_{\rm H}}}=\frac{R^\frac{2+\ell}{2}}{{\ensuremath{g_{\rm II}}}}\ ,\qquad R_1={\ensuremath{g_{\rm II}}}\ ,\qquad R_i = \frac{{\ensuremath{g_{\rm II}}}}{\tilde R_j \tilde R_k}\ .$$ A simple case by case study shows that the $\ell=1,2,4$ cases are identical to the $n=6,4,3$ heterotic cases, up to powers of ${\ensuremath{g_{\rm II}}}$ which we now consider of order 1. The $\ell=3$ case on the other hand is new, since it involves, after T-duality along the direction $R_4$, three large directions of size $R_{2,3,\hat 4}\sim{\ensuremath{l_{\rm H}}}(R/{\ensuremath{l_{\rm II}}})^{1/2}$, and three extra-large ones of size $R_{1,5,6}\sim {\ensuremath{l_{\rm H}}}(R/{\ensuremath{l_{\rm II}}})^{3/2}$. It does not yield, however, any perturbative description on the type I side. Again we see that the $n=5$ heterotic case does not appear, since it corresponds to a strongly coupled type II theory. We now turn to the type IIB theory, again compactified on the model (\[ktt\]) of ${\ensuremath{K_3}}\times T^2$ for simplicity. Since T-duality on one of the circles $R_{5,6}$ identifies the type IIA and IIB theories, it is sufficient to restrict our attention to the case where both circles are much larger than the type II string length, but still of comparable size in order to maintain a small gauge coupling ${\ensuremath{g_{\rm YM}}}^2=R_5/R_6$. The type IIB theory is then equivalent to a strongly coupled heterotic theory with parameters \[hetiib\] $$\begin{aligned} {\ensuremath{l_{\rm H}}}&=& \frac{{\ensuremath{g_{\rm IIB}}}{\ensuremath{l_{\rm II}}}^3}{R_6\sqrt{R_I \tilde R_2 \tilde R_3 \tilde R_4}}\ ,\qquad {\ensuremath{g_{\rm H}}}= \frac{R_6\sqrt{R_I ^3 \tilde R_2 \tilde R_3 \tilde R_4}} {{\ensuremath{g_{\rm IIB}}}{\ensuremath{l_{\rm II}}}^3}\ ,\qquad \label{hetiib1}\\ R_1&=&\frac{{\ensuremath{g_{\rm IIB}}}}{R_6} {\ensuremath{l_{\rm II}}}\ ,\qquad \hat R_6=\frac{{\ensuremath{l_{\rm II}}}^2}{R_6}\ ,\qquad \tilde R_i=\frac{{\ensuremath{g_{\rm IIB}}}{\ensuremath{l_{\rm II}}}^3}{\tilde R_j \tilde R_k}\ ,\ i,j,k=2,3,4\ , \label{hetiib2}\end{aligned}$$ where now the l.h.s. refers to type IIB variables. For $\ell=0$, the dual heterotic theory has one large dimension of radius $R_5=R$ and five heterotic-string–size dimensions, up to factors of ${\ensuremath{g_{\rm IIB}}}\sim 1$, which corresponds to the situation in (\[scal1\]). For $\ell\ge 1$, we obtain again a heterotic theory with more than two scales, and heterotic–type I duality does not yield any valuable perturbative description. Concluding remarks ================== In this paper, we studied new scenarios of TeV strings or large dimensions in weakly coupled type II theories and related them by duality to heterotic string compactifications with large dimensions. In particular, we described a type IIA theory with all compactification and string scales at the TeV, but with a tiny string coupling which explains the weakness of gravitational interactions. We also described a type IIB theory with two large non-transverse dimensions at the TeV and a fundamental string scale at $10^{11}$ GeV. The main features of our discussion are summarised in Figure \[sum\]. -------------------------------------------------------------------------------------------------------------------- $n$ $R^{-1}_{\rm H}$ Dual $l^{-1}_{\rm Dual}$ Radii QG Scale ----- ------------------ ---------- --------------------- ------------------------------------------ --------------- 1 TeV IIB $10^{11}$ GeV 2 at TeV$^{-1}$, 4 at $l_{\rm Dual}$ $10^{11}$ GeV 2 TeV IIA TeV all at TeV$^{-1}$ $10^{18}$ GeV 3 TeV IIA TeV 2 at TeV$^{-1}$, 4 transv. at fm TeV 4 TeV I or IIA TeV 4 at TeV$^{-1}$, 2 transv. at 0.1 mm TeV 5 $>10^{6}$ GeV I TeV 1 transv. at mm, 5 transv. at GeV$^{-1}$ TeV 6 $>10^{8}$ GeV I TeV 6 transv. at 0.1 fm TeV 6’ $>10^{8}$ GeV IIA $10^8$ GeV 1 transv. at mm, 5 at $l_{\rm $10^8$ GeV Dual}$ -------------------------------------------------------------------------------------------------------------------- As a result, the heterotic string with $n\le 4$ large dimensions at the TeV has a weakly coupled description in terms of type II or type I theory, as indicated in the table. When the number of large dimensions is $n=5$ or 6, there is an upper bound for the compactification scale because the string threshold of the weakly coupled dual theory appears in lower energies [@ckm]. The entries in the last three rows correspond to a saturation of this bound. Moreover, the case $n=5$ is generally forbidden since the dual type I theory has an anisotropic transverse space with one dimension very large compared to the others; this invalidates the decoupling of the gauge theory on the brane unless local tadpole cancellation is imposed [@ab]. In particular, we showed that the first two simple type II examples above describe the heterotic string with one or two large TeV dimensions. In fact, these are the only two cases that have been previously considered seriously in the context of the heterotic theory before knowing its strong coupling behavior [@ia; @abq]. Our analysis here showed that many of the properties and predictions of the heterotic string for these two cases remain valid, despite its strong ten-dimensional coupling. More precisely: (i) the existence of KK excitations for all Standard Model gauge bosons in $N=4$ supermultiplets, and their absence for quarks and leptons; (ii) the absence of visible quantum gravity effects at the TeV scale, above which there is a genuine six-dimensional gauge theory, regulated by the underlying type IIA or IIB theory; (iii) the possible relation of the TeV dimensions with the mechanism of supersymmetry breaking by the process of compactification. All soft breaking terms can then be studied reliably in the effective field theory due to the extreme softness of the breaking above the compactification scale [@soft]; (iv) the possibility that the unification of low energy gauge couplings remains at the experimentally inferred GUT scale, which is much higher than the fundamental string scale of the weakly coupled type II theory. Many questions and open problems remain of course to be done. Certainly, the possibilities discussed here give new “viable” directions of how string theory may be possibly connected with the description of our observed low energy world. We note however that the strong coupling regime of the heterotic string is traded for a strong (singular) curvature situation in the type II framework, which is only partially accounted for in the geometric engineering field theory approach. [*Acknowledgements*]{} : We would like to thank C. Bachas, S. Dimopoulos, P. Mayr and S. Shatashvili for useful discussions. [99]{} I. 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--- abstract: 'The parameter coclass has been used successfully in the study of nilpotent algebraic objects of different kinds. In this paper a definition of coclass for nilpotent semigroups is introduced and semigroups of coclass 0, 1, and 2 are classified. Presentations for all such semigroups and formulae for their numbers are obtained. The classification is provided up to isomorphism as well as up to isomorphism or anti-isomorphism. Commutative and self-dual semigroups are identified within the classification.' author: - Andreas Distler title: Finite nilpotent semigroups of small coclass --- Introduction ============ Nilpotency is an important concept in many areas of algebra. For semigroups there exist two common definitions; on the one hand the generalisation of nilpotency for groups introduced by Mal$'$cev [@Mal53], and on the other hand a natural adaptation of the notion for algebras. The latter is used in this paper: a semigroup $S$ is *nilpotent* if there exists $c \in {\mathbb{N}}_0$ such that $|S^{c+1}|=1$, the least such $c$ is the *(nilpotency) class* of $S$. A parameter used successfully in the studies of nilpotent groups and Lie algebras is the coclass of such objects [@LN80; @SZ97]. For a finite nilpotent semigroup $S$ of class $c$ we define the *(nilpotency) coclass* of $S$ to be $|S|-1-c$. It is immediate from the definition that every nilpotent semigroup $S$ contains a zero and that $|S|-1$ is an upper bound for both class and coclass. Throughout the paper attributing class or coclass to a semigroup $S$ shall imply that $S$ is nilpotent. The above definition of class and coclass is in parallel to that for other algebraic structures. It ensures for all $k\in {\mathbb{N}}$ that the class of a nilpotent semigroup $S$ equals the sum of the classes of the ideal $S^k$ and the quotient $S/S^k$. Furthermore class and coclass of $S$ equal the respective attributes of the nilpotent algebra naturally associated with $S$ over any given field, that is the contracted semigroup algebra of $S$ (compare [@DE12]). The main results obtained in this paper are complete classifications of nilpotent semigroups of coclass 1 and of coclass 2. These demonstrate that coclass is a useful parameter for the classification, seemingly better suited than the more directly defined nilpotency class. The forthcoming section contains technical background on basic properties of nilpotent semigroups. An immediate consequence is the description of semigroups of coclass $0$ in Lemma \[lem\_nil\_mono\]. In Sections \[sec\_cc1\] and \[sec\_cc2\] the main results of the paper are obtained. Classifications of semigroups of coclass $1$ respectively $2$ are given in Theorem \[thm\_cc1\] respectively Theorems \[thm\_cc2\_d2\] and \[thm\_cc2\_d3\]. Presentations of the semigroups are provided in all cases. Ideas and problems for an extension of the results to higher coclass are briefly discussed at the end of Section \[sec\_cc2\]. In the final section the main results are applied to the enumeration of semigroups of coclass $1$ and $2$. Formulae for the numbers of such semigroups are given along with a table containing the numbers for small orders. The lists of presentations from the main theorems have been implemented in <span style="font-variant:small-caps;">GAP</span> [@GAP4] by the author. The implementation is available in the package <span style="font-variant:small-caps;">Smallsemi</span> [@smallsemi] and can be accessed through the function `PresentationsOfNilpotentSemigroups`[^1]. Preliminaries and Coclass 0 {#sec_nil_rank} =========================== This section contains basic results about the structure of nilpotent semigroups and a characterisation of semigroups with coclass $0$. \[lem\_part\_nil\] Let $S$ be a nilpotent semigroup of class $c$. Then the following hold: 1. the sets $S^k\setminus S^{k+1}$ with $1 \leq k \leq c$ are non-empty and form a partition of $S\setminus S^{c+1}$; 2. if $s= s_1s_2\cdots s_k \in S^k\setminus S^{k+1}$ with $1\leq k\leq c$, then $s_i\cdots s_j \in S^{j-i+1}\setminus S^{j-i+2}$ for all $1\leq i \leq j \leq k$; 3. if $|S^l \setminus S^{l+1}|=1$ for $1 \leq l \leq c$ then $|S^k \setminus S^{k+1}|=1$ for all $l \leq k \leq c$. (i): For any three sets $A,B,C$ with $A\supseteq B\supseteq C$ the set $A\setminus C$ equals the disjoint union of $A\setminus B$ and $B\setminus C$. Hence it suffices to show that the sets $S^{k}\setminus S^{k+1}$ are non-empty for $1\leq k \leq c$. From $S^{k}= S^{k+1}$ it would follow that $S^{k}=S^{c+1}$ contains just one element and $c=k-1$, a contradiction to $k\leq c$. Thus $S^{k+1}$ is a proper subset of $S^{k}$ for $1\leq k \leq c$. (ii): The statement is shown by contradiction. Assume that $s_i\cdots s_j\in S^{j-i+2}$ for some $i,j$ with $1\leq i \leq j \leq k$. This means $s_i\cdots s_j$ can be expressed as a product $t_1\cdots t_{j-i+2} \in S^{j-i+2}$. Replacing $s_i\cdots s_j$ by $t_1\cdots t_{j-i+2}$ in $s$, that is $$s=s_1s_2\cdots s_k = s_1\cdots s_{i-1}t_1\cdots t_{j-i+2}s_{j+1}\cdots s_k,$$ yields $s \in S^{k+1}$, a contradiction. (iii): Let $s=s_1s_2\cdots s_k\in S^k\setminus S^{k+1}$ for some $l < k \leq c$. According to Part (ii) the product $s_1s_2\cdots s_l$ of length $l$ equals the unique element $t_1t_2\cdots t_l$ in $S^l \setminus S^{l+1}$. Hence $s_1s_2\cdots s_k = t_1t_2\cdots t_l s_{l+1}\cdots s_k$. The right hand side of this equation is still a product of length $k$ equalling $s$. Thus, again by Part (ii) $t_2t_3\cdots t_ls_{l+1}$ equals the unique element in $S^l \setminus S^{l+1}$ and can be replaced by $t_1t_2\cdots t_l$. Applying this argument repeatedly and always replacing the product of length $l$ with $t_1t_2\cdots t_l$ yields $s=t_1^{k-l+1}t_2\cdots t_l$. This implies $|S^k\setminus S^{k+1}|=1$, since $s \in S^k\setminus S^{k+1}$ was chosen arbitrarily. The previous lemma will be used repeatedly and it is worthwhile to gain some intuition for its content. First of all, for each element other than the zero, the length of a product equalling the element is restricted. An element in $S^k \setminus S^{k+1}$ can be written as product of length $k$, but not as product of length $k+1$. We obtain a partition of $S$ if we collect all elements with the same maximal length of a product equalling the element in a separate part. A semigroup of class $c$ is partitioned into $c+1$ sets: one for each maximal length between 1 and $c$ and the zero element in a set by itself. This yields precisely the partition in Lemma \[lem\_part\_nil\](i). Further, each part of a product of maximal length is clearly maximal itself, which is essentially what is stated in the second part of Lemma \[lem\_part\_nil\]. It follows in particular that every element in a product of maximal length lies in $S\setminus S^2$. That in addition each element in $S\setminus S^2$ clearly has to appear in every generating set, yields the following well-known result. \[lem\_gen\] Let $S$ be a nilpotent semigroup containing at least $2$ elements. Then $S\setminus S^2$ is the unique minimal generating set of $S$. We shall see that the previous lemma together with Lemma \[lem\_part\_nil\](i) yields that a semigroup with coclass $0$ can be generated by one element. A finite semigroup generated by a single element $u$ which satisfies $u^k=u^{k+l}$ for minimal $k,l\in{\mathbb{N}}$ is called *monogenic* of *index* $k$ and *period* $l$. \[lem\_nil\_mono\] Let $S$ be a semigroup of order $n\in{\mathbb{N}}$. Then the following statements are equivalent: 1. $S$ is nilpotent of coclass $0$; 2. $S$ is monogenic and nilpotent; 3. $S$ is monogenic with period $1$; 4. $\langle u\mid u^n=u^{n+1}\rangle$ is a presentation for $S$. We may assume $n\geq 2$ as the statement is obvious for $n=1$. \(i) $\Rightarrow$ (ii): According to Lemma \[lem\_part\_nil\](i) the sets $S^k \setminus S^{k+1}$ with $1\leq k \leq n-1$ are non-empty. Hence each set contains exactly one element. The set $S\setminus S^2$ is a generating set by Lemma \[lem\_gen\] and thus $S$ is monogenic. \(ii) $\Rightarrow$ (iii): If $u$ denotes the generator of $S$, then $u^n$ and $u^{n+1}$ both equal the zero element. Therefore the period of $S$ is $1$. \(iii) $\Rightarrow$ (iv): If $u$ denotes the generator of $S$ all of $u,u^2,\dots, u^n$ have to be pairwise different. And that the period of $S$ is $1$ implies that equality $u^n=u^{n+1}$ holds. \(iv) $\Rightarrow$ (i): Clearly $u^n$ is a zero and every product of $n$ elements equals $u^n$, making $S$ nilpotent. Moreover, $S^{k}\setminus S^{k+1} = \{u^{k}\}$ for $1\leq k \leq n-1$, showing that $S$ has nilpotency coclass $0$. In a nilpotent semigroup $S$ every element generates a nilpotent monogenic subsemigroup of coclass $0$. The class of each such subsemigroup is at most the class of $S$. Of special interest is the case when equality of the classes holds for some elements in $S$. \[lem\_struc\_nil\] Let $S$ be a nilpotent semigroup of class $c$. If $|S^{c-1}\setminus S^{c}|=1$ then there exists an element in $S$ that generates a nilpotent subsemigroup of class $c$. For $s \in S^{c}\setminus S^{c+1}$ take a product $s_1s_2\cdots s_{c}$ equal to $s$. Then both $s_1\cdots s_{c-1}$ and $s_2\cdots s_{c}$ are in $S^{c-1}\setminus S^{c}$ due to Lemma \[lem\_part\_nil\](ii) and hence $s_1\cdots s_{c-1}$ equals $s_2\cdots s_{c}$. This yields $s = s_1s_2s_3\cdots s_{c} = s_1s_1s_2\cdots s_{c-1}$. Repeating the process starting with $s=s_1s_1s_2\cdots s_{c-1}$ gives $s=s_1s_1s_1s_2\cdots s_{c-2}$ and leads after $c-1$ iterations to $s=s_1^{c}$. Conclude by Lemma \[lem\_part\_nil\](ii) that $s_i^k$ for $1\leq k\leq c$ are pairwise different and are also different from the zero $s_1^{c+1}$. Hence the semigroup generated by $s_1$ has size $c+1$ and class $c$. The condition in the previous lemma is rather technical. For a fixed coclass we can turn it into a restriction on the size of the semigroup. The restriction can be strengthened if also the size of the generating set is incorporated. \[coro\_bound\] Let $S$ be a finite, nilpotent semigroup of class $c$ and coclass $r$. If either $c \geq r+2$ or $c\geq r+4-|S\setminus S^2|$ then there exists an element in $S$ that generates a nilpotent subsemigroup of class $c$. By Lemma \[lem\_part\_nil\](iii) it is immediate that either of the two conditions in the statement implies that $S$ satisfies the prerequisite in Lemma \[lem\_struc\_nil\]. Note that for a nilpotent semigroup $S$ the only case in which $|S\setminus S^2|=1$ is, according to Lemma \[lem\_nil\_mono\], if $S$ has coclass $0$. It follows that the generating set for a semigroup of positive coclass contains at least two elements and that the second bound in the previous corollary is at least as good as the first bound, usually better. Classification for Coclass 1 {#sec_cc1} ============================ Some general remarks on the classification of semigroups are in place before stating the results of this section. Two semigroups are *anti-isomorphic* if one is isomorphic to the dual of the other, and a semigroup is *self-dual* if it is isomorphic to its dual. Semigroups are usually classified up to isomorphism or anti-isomorphism, and the results here are stated accordingly. For the sake of brevity we denote ‘isomorphism or anti-isomorphism’ by *(anti-)isomorphism* and correspondingly use *(anti-)isomorphic* to mean ‘isomorphic or anti-isomorphic’. For computational purposes it is often useful to work with a classification just up to isomorphism. In all results the self-dual semigroups are indicated thus allowing the reader to extract each classification up to isomorphism. We now use the structural information provided in the previous section to classify nilpotent semigroups of coclass 1. Though instead of doing so directly we prove a generalisation to arbitrary coclass which will also be useful for the classification of semigroups of coclass 2 in the forthcoming section. \[lem\_coclass\_d\] For $c,r\in{\mathbb{N}}$ with $c\geq 3$ the following is a complete list up to (anti-)isomorphism of representatives of nilpotent semigroups of class $c$ and coclass $r$, in which the unique minimal generating set is of size $r+1$ and that contain at least $r$ copies of the monogenic, nilpotent semigroup of class $c$: $$\begin{aligned} \mathcal{H}_k & \!\!\!\! = \langle u_1,u_2,\dots,u_r,v \mid &\!\!\!\! u_1^{c+1}=u_1^{c+2}; u_1^2=u_i^2=u_iu_j=u_ju_i, 1 \leq j < i \leq r;\\ &&\!\!\!\! u_iv=vu_i=u_1^k, 1 \leq i \leq r; v^2=u_1^{2k-2}\rangle, 2\leq k\leq c-1;\\ \mathcal{J}_k &\!\!\!\! = \langle u_1,u_2,\dots,u_r,v \mid &\!\!\!\! u_1^{c+1}=u_1^{c+2}; u_1^2=u_i^2=u_iu_j=u_ju_i, 1 \leq j < i \leq r;\\ &&\!\!\!\! u_iv=vu_i=u_1^k, 1 \leq i \leq r; v^2=u_1^{c}\rangle, \lfloor c/2\rfloor+2 \leq k\leq c-1;\\ \mathcal{X} &\!\!\!\! = \langle u_1,u_2,\dots,u_r,v \mid &\!\!\!\! u_1^{c+1}=u_1^{c+2}; u_1^2=u_i^2=u_iu_j=u_ju_i, 1 \leq j < i \leq r;\\ &&\!\!\!\! u_iv=vu_i=u_1^{(c+2)/2}, 1 \leq i \leq r; v^2=u_1^{c+1}\rangle, \mbox{ if } c\equiv 0\mod 2;\\ \mathcal{N}_{k,l,m}^e &\!\!\!\! =\langle u_1,u_2,\dots,u_r,v \mid & \!\!\!\! u_1^{c+1}=u_1^{c+2}; u_1^2=u_i^2=u_iu_j=u_ju_i, 1 \leq j < i \leq r;\\ &&\!\!\!\! vu_i=u_iv=u_1^{c+1}, 1\leq i \leq k;\\ &&\!\!\!\! vu_i=u_1^{c+1}, u_iv=u_1^{c}, k+1\leq i\leq l;\\ &&\!\!\!\! vu_i=u_1^{c}, u_iv=u_1^{c+1}, l+1\leq i\leq m;\\ &&\!\!\!\! vu_i=u_iv=u_1^{c}, m+1\leq i \leq r; v^2=u_1^{c+e}\rangle,\\ &&\!\!\!\! 0\leq k \leq m\leq r, k\leq l \leq \lfloor(k+m)/2\rfloor, e\in \{0,1\}.\end{aligned}$$ All semigroups in the list, except $\mathcal{N}^e_{k,l,m}$ with $l\neq (k+m)/2$, are self-dual. Let $S=\langle u_1,u_2,\dots,u_r,v \rangle$ be a nilpotent semigroup fulfilling the conditions described in the statement of the lemma where $u_1,u_2,\dots, u_r$ denote $r$ elements of $S$ that each generate a subsemigroup of class $c$. According to Lemma \[lem\_part\_nil\](i) the sets $S^k\setminus S^{k+1}$ for $2\leq k\leq c$ each contain exactly one element which consequently equals $u_i^k$ for all $1\leq i \leq r$. As the index $i$ does not influence the value of $u_i^k$ for $k\geq 2$ we simplify notation and write just $u^k$. For given $1\leq i,j \leq r$ the equality $u_iu_j=u^k$ holds for some $2\leq k \leq c+1$. It follows $u_i(u_iu_j) = u_iu^k= u^{k+1}$ and also $(u_iu_i)u_j = u^2u_j= u^3$. This proves $k=2$ using that $u^3$ is not the zero because $3 < c+1$ by assumption. As $i,j$ were arbitrary $u_iu_j$ equals $u^2$ for all $1\leq i,j\leq r$. We shall now conduct a case distinction depending on the value of $u_1v$. Note that the whole of $S$ is determined if we know the value of $v^2$ and the values of $u_jv$ and $vu_j$ for all $1\leq j\leq r$ because $u^kv$ and $vu^k$ can then be deduced for all $2\leq k \leq c+1$. The choices not contradicting associativity are determined below. Choices leading to (anti-)isomorphic semigroups and are identified; no two semigroups from different cases are (anti-)isomorphic. Note that any (anti-)isomorphism sends generators to generators and is as such induced by a permutation of $u_1,u_2,\dots,u_r,v$. [**Case 1:**]{} $u_1v = u^k$ with $2\leq k \leq \lceil c/2\rceil$. Let $l \in \{2,3,\dots,c+1\}$ such that $vu_1=u^l$. From $u^{k+1}= u_1vu_1 = u^{l+1}$ it follows that $k=l$ as $u^{k+1}$ is not the zero. Using the same type of argument considering $u_jvu_1 = u_ju^k=u^{k+1}$ it follows $u_jv=u^k$ and similarly $vu_j=u^k$ for all $2\leq j\leq r$. Also $v^2=u^m$ implies $u^{m+1}=v^2u=vu^k=u^{2k-1}$, and hence $m=2k-2$ as $u^{2k-1}$ is not the zero. It follows that the value of any proper product of generators is determined by its length and by how many times $v$ appears (A product of length $i+j$ that contains $j$ times $v$ equals $u^{i+j(k-1)}$.). This guarantees that the multiplication does not contradict associative. Hence $S$ is a homomorphic image of $\mathcal{H}_k$. Apart from the generators each presentation contains at most the elements $u^k$ for $2\leq k \leq c+1$ and has therefore size $c+2$ and class $c$. For all values of $k$ the semigroup is commutative and no two are (anti-)isomorphic since $v^2$ is different for different $k$. (Note that if $k=2$, than $\langle v\rangle$ is also of class $c$ and any permutation of the generators induces an automorphism of $S$.) [**Case 2:**]{} $u_1v = u^k$ with $\lceil c/2\rceil < k \leq c-1$. As in the previous case $vu_j=u_jv=u^k$ for all $1\leq j\leq r$. Now $vvu=u^{2k-1}$ equals the zero $u^{c+1}$. This leaves the two choices $u^{c}$ and $u^{c+1}$ for $v^2$. Again, the value of a product of generators only depends on its length and the number of times $v$ appears, making the multiplication associative. Similar to Case 1 it follows that $S$ is isomorphic to one of the presentations $\mathcal{H}_k$ or $\mathcal{J}_k$ respectively $\mathcal{X}$ if $k=(c+2)/2$ depending on the value of $v^2$. All these presentations define pairwise not (anti-)isomorphic, commutative semigroups. [**Case 3:**]{} $u_1v \in \{u^{c},u^{c+1}\}$. Let $l \in \{2,3,\dots,c+1\}$ such that $vu_1=u^l$. From $u_1(vu_1)=u^{l+1}$ and the fact that $(u_1v)u_1$ equals the zero, $u^{c+1}$, it follows that $l\in \{c,c+1\}$. Considering $u_jvu_1$ and $u_1vu_j$ one then shows with the same type of argument that $u_jv, vu_j \in \{u^{c},u^{c+1}\}$ for all $2\leq j\leq r$. In a similar way $v^2=u^m$ leads to $u^{m+1}=v(vu)=vu^l=u^{2l-1}=u^{c+1}$, and hence $m \in\{c,c+1\}$. Every choice gives an associative multiplication as all products of three elements involving $v$ equal $u^{c+1}$. Some multiplications lead to (anti-)isomorphic semigroups because every permutation of $u_1,u_2,\dots,u_r$ yields an isomorphism and an anti-isomorphism. To find representatives we partition the set $\{u_j\mid 1\leq j \leq r\}$ into four parts depending on $u_jv$ and $vu_j$ equalling $u^c$ or $u^{c+1}$. Two semigroups from this case are isomorphic if and only if they have the same number of generators of each of the four types and $v^2$ takes the same value; and they are anti-isomorphic if by interchanging the two types with $vu_j\neq u_jv$ in one of the semigroups we obtain two isomorphic semigroups. Hence a semigroup from this case is self-dual if there is the same number of generators of the two types with $vu_j\neq u_jv$. Presentations for every possible choice for the numbers of types and the value of $v^2$ up to (anti-)isomorphism are then given by $\mathcal{N}_{k,l,m}^e$ as in the statement of the lemma. For nilpotent semigroups of class $2$ the step in the proof of Lemma \[lem\_coclass\_d\] that restricts the possible results for products of two generators does not work. Indeed, for semigroups of class $2$ every combination can occur, a fact that is used in [@DM12] to count such semigroups of any order. \[thm\_cc1\] For $n\in{\mathbb{N}}$ with $n\geq 5$ the following is a complete list up to (anti-)isomorphism of representatives of nilpotent semigroups of order $n$ and coclass $1$: $$\begin{aligned} &H_k &\!\!\!\!= \langle u,v \mid u^{n-1}=u^n, uv=u^k, vu=u^k,v^2=u^{2k-2} \rangle, 2\leq k\leq n-1;\\ &J_k &\!\!\!\!= \langle u,v \mid u^{n-1}=u^n, uv=u^k, vu=u^k,v^2=u^{n-2} \rangle, n/2< k\leq n-1;\\ &X &\!\!\!\!= \langle u,v \mid u^{n-1}=u^n, uv=u^{n/2}, vu=u^{n/2},v^2=u^{n-1}\rangle, \mbox{ if } n\equiv 0 \mod 2;\\ &N_1 &\!\!\!\!= \langle u,v \mid u^{n-1}=u^n, uv=u^{n-1}, vu=u^{n-2},v^2=u^{n-2} \rangle;\\ &N_2 &\!\!\!\!= \langle u,v \mid u^{n-1}=u^n, uv=u^{n-1}, vu=u^{n-2},v^2=u^{n-1} \rangle. $$ All semigroups in the list, except $N_1$ and $N_2$, are self-dual. From Lemmas \[lem\_part\_nil\] and \[lem\_gen\] it follows that every semigroup of coclass $1$ is generated by $2$ elements. Together with Corollary \[coro\_bound\] this implies that Lemma \[lem\_coclass\_d\] covers all semigroups of coclass $1$ and order at least $5$. The presentations in the statement are obtained by choosing $r=1$ in Lemma \[lem\_coclass\_d\]. (There is a slight discrepancy in nomenclature as the four semigroups $H_{n-2}, H_{n-1}, J_{n-2},$ and $J_{n-1}$ correspond to $\mathcal{N}_{0,0,0}^1, \mathcal{N}_{1,1,1}^1,\mathcal{N}_{0,0,0}^0,$ and $\mathcal{N}_{1,1,1}^0$.) Note that there are no semigroups of coclass $1$ and order $1$ or $2$, and there is only the zero semigroup of order $3$. For $n=4$ the presentations given in Theorem \[thm\_cc1\] yield semigroups of order $4$ and coclass $1$, but $N_1$ becomes self-dual and the following two presentations of self-dual semigroups are missing from a complete list: $$\langle u,v \mid u^2=u^3, u^2=v^2, uv=vu, u^2=v^3 \rangle \mbox{ \ and \ }\langle u,v \mid u^2=u^3, u^2=v^2, u^2=uv, u^2=v^3 \rangle.$$ Classification for Coclass 2 {#sec_cc2} ============================ For some general remarks on the classification see the beginning of Section \[sec\_cc1\]. The natural next step is to consider nilpotent semigroups of coclass $2$. Due to Lemmas \[lem\_part\_nil\](i) and \[lem\_nil\_mono\] the minimal generating sets of such semigroups contain either $2$ or $3$ elements. These two possibilities shall be treated separately. \[thm\_cc2\_d2\] For $n\in{\mathbb{N}}$ with $n\geq 7$ the following is a complete list up to (anti-)isomorphism of representatives of nilpotent semigroups of order $n$ and coclass $2$ whose minimal generating set has size $2$: $$\begin{aligned} &T_{1,i} =\langle u,v \mid u^{n-2}=u^{n-1}, uv = vu, v^2 = uv, v^3 = u^{n-i}\rangle, i\in\{2,3\};\\ &T_{2,k} =\langle u,v \mid u^{n-2}=u^{n-1}, uv = vu, v^2 = u^{2k-4}, u^2v = u^k \rangle, 3 \leq k < n/2;\\ &T_{2,i,k} =\langle u,v \mid u^{n-2}=u^{n-1}, uv = vu, v^2 = u^{n-i}, u^2v = u^k \rangle, n/2 \leq k \leq n-2,i \in\{2,3,4\};\\ &T_{3} =\langle u,v \mid u^{n-2}=u^{n-1}, v^2 = uv, vu = u^2, uv^2 = u^3 \rangle;\\ &T_{3,i} =\langle u,v \mid u^{n-2}=u^{n-1}, v^2 = uv, vu = u^{n-i}, uv^2 = u^{n-2} \rangle,i \in\{2,3\};\\ &T_{4,k} =\langle u,v \mid u^{n-2}=u^{n-1}, uv = vu, uv = u^k, v^3 = u^{3k-3} \rangle, 2 \leq k < n/3;\\ &T_{4,i,k} =\langle u,v \mid u^{n-2}=u^{n-1}, uv = vu, uv = u^k, v^3 = u^{n-i}\rangle, n/3 \leq k \leq n-4, i\in\{2,3\};\\ &T_{4,i,j,k} =\langle u,v \mid u^{n-2}=u^{n-1}, uv = u^{n-i}, vu = u^{n-j}, v^3 = u^{n-k}\rangle, i,j,k\in\{2,3\};\\ &T_{5,k} =\langle u,v \mid u^{n-2}=u^{n-1}, uv = u^k, v^2 = u^{2k-2}, vu^2 = u^{k+1} \rangle, 2 \leq k < n/2;\\ &T_{5,i,k} =\langle u,v \mid u^{n-2}=u^{n-1}, uv = u^k, v^2 = u^{n-i}, vu^2 = u^{k+1} \rangle, n/2 \leq k < n-5, i\in\{2,3\};\\ &T_{5,i,j,k} =\langle u,v \mid u^{n-2}=u^{n-1}, uv = u^{n-i}, v^2 = u^{n-j}, vu^2 = u^{n-k} \rangle, i\in\{2,3,4\}, j,k\in\{2,3\}. $$ A semigroup from the list is self-dual if it is commutative or is $T_{5,2}$. The proof uses the same methods as those of Lemma \[lem\_coclass\_d\] and Theorem \[thm\_cc2\_d3\]. Let $S=\langle u,v\rangle$ be a semigroup of order $n$ and coclass $2$. By Corollary \[coro\_bound\] we may assume without loss of generality that $|\langle u\rangle|=n-2$. Further denote by $y$ the element in $S$, which is neither $v$ nor a power of $u$. As $y\in S^2\setminus S^3$ at least one of $uv,vu,v^2$ has to equal $y$. Note that the values of these products with two generators together with the values of $vu^2,vy,uy$ uniquely define $S$.[^2] We shall determine the choices of values not contradicting associativity in several cases depending on which products equal $y$. For all resulting multiplications the value of a product with three elements will only depend on the number of times $v$ and $y$ appear. Such multiplications are obviously associative. Each of the presentations from the list in the statement of the theorem leads to a semigroup with at most $n$ elements. Hence, whenever the considerations in the following imply that $S$ fulfils the relations $R$ of one of the presentations then $\langle u,v\mid R\rangle$ is a presentation for $S$. [**Case 1:**]{} $y=v^2=uv=vu$. Let $vy=u^k$. Since $u$ and $v$ commute, this yields $uy=uv^2=vuv=vy=u^k$ and $vu^2=uvu=uy=u^k$. Then $$u^{k+1}=uvy=uyv=u^kv=u^{k-1}uv=u^{k-1}y=u^{k-2}u^k=u^{2k-2},$$ which gives $k=3$ or $k\geq n-3$, and thus leads to 3 semigroups. For any $k\in\{3,n-3,n-2\}$ the respective semigroup fulfils the relations of $$\langle u,v \mid u^{n-2}=u^{n-1}, uv = vu, v^2 = uv, v^3 = u^k\rangle.$$ [**Case 2:**]{} $y=vu=uv$. Let $v^2=u^l$. It follows $vy=vvu=u^{l+1}$. Let $uy=u^k$ then $vu^2=uvu=uy=u^k$ and $$\label{eq_nil_n-2} u^{l+2}= v^2u^2=vuvu=vuy=vu^k=u^kv=u^{k-2}uy=u^{2k-2}.$$ If $2\leq l \leq n-5$ then gives $k=l/2+2$. That $k$ has to be an integer yields $\lceil n/2 \rceil - 3$ semigroups with presentations $T_{2,l}$. If $n-4\leq l \leq n-2$ then $k\geq n/2$ due to . Hence for each of the three choices for $l$ there are $3(\lfloor n/2 \rfloor -1)$ semigroups with presentation $T_{2,l,k}$. [**Case 3:**]{} $y=uv=v^2$. Let $vu=u^l$. Then $$u^{l+1}= uu^l=uvu=yu=vvu=vu^l=vuu^{l-1}=u^{2l-1}$$ yields either $l=2$ or $l\geq n-3$. Furthermore $vu^2=u^{l+1}$ and $$uy=uv^2=uvv=yv=vvv=vy=vuv=u^lv=u^{l-2}uy,$$ which shows that all three values for $l$ lead to valid choices for $vu^2,uy,$ and $vy$ defining an associative multiplication. Hence this case accounts for 3 semigroups with presentations $T_{3},T_{3,2}$ and $T_{3,3}$. [**Case 3’:**]{} $y=vu=v^2$. This case leads to the dual semigroups to those from Case 3. [**Case 4:**]{} $v^2=y$. Let $vu=u^k$ and $uv=u^l$. Then $u^{k+1}=uvu=u^{l+1}$ and hence $k=l$ or $k,l \in \{n-3,n-2\}$. Furthermore $vu^2=vuu=u^{k+1}$ and $uy = uvv=u^lv=u^{2l-1}$. For the value of $vy$ consider $uvy=u^lvv=u^{2l-1}v=u^{3l-2}$. If $2 \leq l < n/3$ then $vy=u^{3l-3}$ and $l < n-3$, which leads to $\lceil n/3\rceil -2$ semigroups with presentations $T_{4,l}$. If $n/3 \leq l \leq n-4$ then $vy \in \{u^{n-3},u^{n-2}\}$, leading in this case to $2(n-4-\lceil n/3\rceil +1)$ semigroups with presentations $T_{4,2,l}$ and $T_{4,3,l}$ . If $l\in\{n-3,n-2\}$ then again $vy \in \{u^{n-3},u^{n-2}\}$. Recall that here $k\in \{n-3,n-2\}$. The two semigroups in which one of $k$ and $l$ equals $n-3$ and the other one $n-2$ are anti-isomorphic. Hence this case leads to $6$ semigroups with presentations $T_{4,a,b,c}$ with $a,b,c \in \{2,3\}$ and $a\leq b$. [**Case 5:**]{} $vu=y$. Let $uv=u^k$ and $v^2=u^l$. It follows $vy=vvu=u^{l+1}$ and $uy=uvu=u^{k+1}$. For $vu^2$ consider $uvu^2=u^{k+2}$. If $2\leq k < (n-1)/2$ then $vu^2=u^{k+1}$. From $u^{l+1}=vvu=vu^k=u^{2k-1}$ it follows that $l = 2k-2$ which leads to $\lceil(n-1)/2\rceil -2$ semigroups with presentations $T_{5,k}$. If $(n-1)/2\leq k \leq n-5$ then $l \in\{n-3,n-2\}$ which gives $2(n-\lceil(n-1)/2\rceil-4)$ semigroups with presentations $T_{5,2,k}$ and $T_{5,3,k}$. If $n-4\leq k\leq n-2$ then $vu^2\in\{u^{n-3},u^{n-2}\}$ and $l \in\{n-3,n-2\}$, leading to $12$ semigroups with presentations $T_{5,k,a,b}$ with $a,b\in\{2,3\}$. [**Case 5’:**]{} $uv=y$. This case leads to the dual semigroups to those from Case 5. Only the transposition of the generators $u$ and $v$ might induce an isomorphism between two of the semigroups considered above. That no two semigroups from the same case are isomorphic follows immediately, but there are two semigroups from different cases which are isomorphic. These are the semigroups from Case 1 with $k=3$ and from Case 4 with $l=2$. For all other semigroups the transposition of $u$ and $v$ either induces an automorphism or an isomorphism to a semigroup in which $|\langle u\rangle|\neq n-2$. Numbers of semigroups of coclass $2$ with minimal generating set of size $2$ and order less than $7$ are contained in Table \[tab\_computed\] in Section \[sec\_enum\]. There are no such semigroups with less than $5$ elements. No further considerations will be undertaken for semigroups of orders $5$ or $6$ as semigroups of these orders have long been known [@MS55; @Ple67] and are available in the data library in  [@smallsemi]. From Lemma \[lem\_coclass\_d\] we know certain types of semigroups with coclass $2$ and minimal generating set of size $3$. The remaining types for this case are given in the next theorem. To simplify the statement of the theorem we introduce a total ordering $\prec$ on the semigroups of coclass $1$ from Theorem \[thm\_cc1\] given by their order of appearance therein. Moreover if a semigroup $S$ allows a presentation with relations $R$ we denote the inverted relations that naturally yield a presentation of the dual semigroup by $R^{\perp}$ and the dual semigroup itself by $S^{\perp}$. Note that a semigroup $S$ from Theorem \[thm\_cc1\] is commutative if and only if it is self-dual, that is $S\cong S^{\perp}$. \[thm\_cc2\_d3\] Given $n\in{\mathbb{N}}$ with $n\geq 6$ define for every two semigroups $V, W\not\cong H_2$ of order $n-1$ from Theorem \[thm\_cc1\] with $V \preceq W$ and with presentations $V=\langle u,v\mid Q\rangle$ respectively $W=\langle u,w\mid R\rangle$ the following presentation(s): 1. $\langle u,v,w \mid Q, R, vw= u^{k+l-1}, vw= wv\rangle$ if $k+l \leq n-2$; 2. $\langle u,v,w \mid Q, R, vw= u^{n-i}, wv= u^{n-j}\rangle, 2\leq i\leq j \leq 3$ if $k+l \geq n-1$ and $(W\cong W^{\perp}$ or $V\cong W)$; 3. $\langle u,v,w \mid Q, R, vw= u^{n-i}, wv= u^{n-j}\rangle, 2\leq i,j \leq 3$ if $V\prec W$ and $W\not\cong W^{\perp}$; 4. $\langle u,v,w \mid Q, R^{\perp}, vw= u^{n-i}, wv= u^{n-j}\rangle, 2\leq i,j \leq 3$ if $V\not\cong V^{\perp}$ and $W\not\cong W^{\perp}$, where $k$ and $l$ are given by the relations $uv=u^k$ and $uw=u^l$ in $Q$ respectively $R$. The presentations obtained in this way together with the presentations from Lemma \[lem\_coclass\_d\] for $c=n-3$ and $r=2$ form a complete list up to (anti-)isomorphism of representatives of nilpotent semigroups of order $n$ and coclass $2$ whose minimal generating set has size $3$. A semigroup from the above list is self-dual if and only if it is commutative or is from (iv) with $V\cong W$. Let $S=\langle u,v,w\rangle$ be a semigroup of order $n, n\geq 6$, coclass $2$. Due to Corollary \[coro\_bound\] we may assume without loss of generality that $|\langle u\rangle|=n-2$. If one of $v^2$ or $w^2$ equals $u^2$ then the conditions for Lemma \[lem\_coclass\_d\] are satisfied and the presentations follow from there. It remains to consider the case when neither $v^2$ nor $w^2$ equal $u^2$. The proof follows a similar approach as the one of Lemma \[lem\_coclass\_d\]. All eight products $uv,vu,v^2,uw,wu,w^2,vw,$ and $wv$ of two generators not both equal to $u$ are in the set $S^2 = \{u^k \mid 2\leq k \leq n-2\}$, and knowing them uniquely determines $S$. The different choices are discussed below. For each of the resulting multiplications the value of a product with three factors will depend only on the number of times $v$ and $w$ appear, making all multiplications associative. The only non-trivial permutation of generators possibly inducing an (anti-)isomorphism between different multiplications is the transposition of $v$ and $w$. Denote $V=\langle u,v\rangle$ and $W=\langle u,w\rangle$. Both $V$ and $W$ are semigroups with $n-1$ elements and coclass $1$. The possible choices for $V$ and $W$ up to (anti-)isomorphism are the semigroups listed in Theorem \[thm\_cc1\] except $H_2$ (taking into account that $v^2,w^2 \neq u^2$), and the equations $uv=u^k$ and $uw=u^l$ hold for some $3\leq k,l\leq n-2$. Without loss of generality we may assume that $V$ is isomorphic to a semigroup from Theorem \[thm\_cc1\] and that $V\preceq W$. The latter avoids considering isomorphic semigroups under the transposition of $v$ and $w$, except if $V$ and $W$ are of the same type. [**Case 1:**]{} $k+l\leq n-2$. From $uvw=uwv=u^{k+l-1}$ it follows that $vw=wv=u^{k+l-2}$. Using the latter equation as relation together with the relations of $V$ and $W$ yields the relations for a presentation of $S$. [**Case 2:**]{} $k+l\geq n-1$. From $uvw=uwv=u^{n-2}$ it follows that $vw,wv\in\{u^{n-3},u^{n-2}\}$. If $W$ and hence $V$ are commutative then the two choices with $vw\neq wv$ lead to a pair of anti-isomorphic semigroups; and if $V\cong W$ then these two choices lead to a pair of isomorphic semigroups (under the transposition of $v$ and $w$). Hence in both cases there are three choices up to (anti-)isomorphism with presentations as given in (ii). For all further considerations we have $V\prec W$. If in addition $W$ is isomorphic to a semigroup from Theorem \[thm\_cc1\], but not commutative then all four choices for $vw,wv\in\{u^{n-3},u^{n-2}\}$ yield not (anti-)isomorphic semigroups with presentations as given in (iii). Finally $W$ can be non-commutative and anti-isomorphic to a semigroup from Theorem \[thm\_cc1\], in other words isomorphic to $N_1^{\perp}$ or $N_2^{\perp}$. For commutative $V$ this yields semigroups that are anti-isomorphic to those in (iii). If $V$ is non-commutative then all four choices for $vw$ and $wv$ yield not (anti-)isomorphic semigroups with presentations as given in (iv). The semigroup $S$ is self-dual if and only if it is commutative or the transposition of $v$ and $w$ induces an isomorphism from $S$ to $S^{\perp}$. In the latter case $V\cong W^{\perp}$ is required. For commutative $V$ this yields $V\cong W$ and hence a presentation from (ii). Following the considerations from above these semigroups are not self-dual. This leaves semigroups from (iv) where it is easy to verify that the condition $V\cong W^{\perp}$ is sufficient for a semigroups to be self-dual. Numbers of semigroups of coclass $2$ with minimal generating set of size $3$ and order less than $6$ are contained in Table \[tab\_computed\] in Section \[sec\_enum\]. The only such semigroup with at most $4$ elements is the zero semigroup of order $4$. No further considerations will be undertaken for semigroups of order $5$ as semigroups of this order have long been known [@MS55] and are available in the data library  [@smallsemi]. The strategy in the proof of the previous theorem can be extended to inductively determine semigroups of coclass $r$ and generating set of size $r+1$, the maximal possible. Let $S$ be a semigroup of class $c$ and coclass $r$ with minimal generating set $\langle u_1,\dots,u_k,v_{k+1},\dots,v_{r+1}\rangle$ where each $u_i, 1\leq i\leq k$ generates a semigroup of class $c$ but none of the $v_i, k+1\leq i\leq r+1$ does. If we define $V=\langle u_1,\dots,u_k,v_{k+1},\dots,v_{r}\rangle$ and $W=\langle u_1,\dots,u_k,v_{r+1}\rangle$ then $V$ would be known by induction hypothesis and $W$ is one of the semigroups from Lemma \[lem\_coclass\_d\]. Hence every such $S$ can be constructed from a known semigroup with one fewer element. The possible values for the products $$\label{eq_products} v_iv_{r+1}\mbox{ \ and \ }v_{r+1}v_i\mbox{ \ for \ }k+1\leq i\leq r$$ in a semigroup constructed from $V$ and $W$ are either fixed by associativity or equal to one of $u_1^c$ and $u_1^{c+1}$. It remains to avoid (anti-)isomorphic copies of the same semigroup. A first step towards an orderly algorithm (see [@Rea78]) for the construction is to define an ordering $\prec$ on the semigroups listed in Lemma \[lem\_coclass\_d\] and require $$\langle u_1,\dots,u_k,v_{i}\rangle \preceq \langle u_1,\dots,u_k,v_{j}\rangle \mbox{ \ for all \ }k+1\leq i\leq j\leq r+1.$$ If $W$ is not (anti-)isomorphic to any of the subsemigroups of $V$ then the group of (anti-)automorphisms of $V$ determines which choices for the products in lead to (anti-)isomorphic semigroups. Depending on the size of the group the corresponding orbit calculations might be hard. The situation becomes even more difficult if $W$ is of the same type like $\langle u_1,\dots,u_k,v_{r}\rangle$ as permutations leading to isomorphisms to another constructed semigroup may move $v_{r+1}$. These problems prevent for the moment that an algorithm usable in practice can be derived from this inductive approach. To classify all semigroups with coclass $3$ the methods presented in Section \[sec\_cc1\] and in this section will need to be extended. For a semigroup $S$ of coclass $3$ it is not necessarily true that $S^3\setminus S^4$ contains only one element, while the proofs of Lemma \[lem\_coclass\_d\] and Theorems \[thm\_cc2\_d2\] and \[thm\_cc2\_d3\] rely on this as a key fact. Enumeration {#sec_enum} =========== We present formulae for the numbers of semigroups of coclasses $1$ or $2$ up to (anti-)isomorphism and up to isomorphism, and formulae for the numbers of commutative semigroups of coclasses $1$ or $2$ up to isomorphism. The results are obtained by counting the presentations in the respective theorems in the previous sections. For small orders the formulae have been verified computationally and the computed numbers are presented. The enumeration for coclass $1$ follows from Theorem \[thm\_cc1\]. \[coro\_cc1\] For $n\in{\mathbb{N}}$ with $n \geq 5$ the number of nilpotent semigroups of order $n$ and coclass $1$ … 1. …counting up to (anti-)isomorphism equals $n+\lfloor n/2 \rfloor$. 2. …counting up to isomorphism equals $n+\lfloor n/2 \rfloor+2$. 3. …counting commutative semigroups up to isomorphism equals $n+\lfloor n/2 \rfloor-2$. For coclass $2$ we shall determine the formulae depending on the size of the minimal generating set. If the set has size $2$ we need to count the presentations in Theorem \[thm\_cc2\_d2\]. For $n\in{\mathbb{N}}$ with $n \geq 7$ the number of nilpotent semigroups of order $n$, coclass $2$ and minimal generating set of size $2$ … 1. …counting up to (anti-)isomorphism equals $5n+\lfloor n/2\rfloor -\lceil n/3\rceil -1$. 2. …counting up to isomorphism equals $7n -\lceil n/3\rceil +5$. 3. …counting commutative semigroups up to isomorphism equals $3n+2\lfloor n/2\rfloor -\lceil n/3\rceil -8$. For each case considered in the proof of Theorem \[thm\_cc2\_d2\] also the number of semigroups has been given. It remains to note that the presentations that yield a commutative semigroup either have $uv=vu$ as relation or are $T_{4,i,j,k}$ with $i=j$. The enumeration of semigroups of coclass $2$ with minimal generating set of size $3$ follows from Lemma \[lem\_coclass\_d\] and Theorem \[thm\_cc2\_d3\]. For $n\in{\mathbb{N}}$ with $n \geq 6$ the number of nilpotent semigroups of order $n$, coclass $2$ and minimal generating set of size $3$ … 1. …counting up to (anti-)isomorphism equals $$\begin{aligned} &\frac{1}{8}(21n^2+22n-96) &\textrm{ if $n$ is even, and}\\ &\frac{1}{8}(21n^2+36n-81) &\textrm{ if $n$ is odd.}\end{aligned}$$ 2. …counting up to isomorphism equals $$\begin{aligned} &\frac{1}{8}(27n^2+94n-280) &\textrm{ if $n$ is even, and}\\ &\frac{1}{8}(27n^2+112n-243) &\textrm{ if $n$ is odd.}\end{aligned}$$ 3. …counting commutative semigroups up to isomorphism equals $$\begin{aligned} &\frac{1}{8}(15n^2-58n+24) &\textrm{ if $n$ is even, and}\\ &\frac{1}{8}(15n^2-48n+9) &\textrm{ if $n$ is odd.}\end{aligned}$$ We shall first count those semigroups that arise from the presentations in Lemma \[lem\_coclass\_d\]. We have $r=2$ and hence $c=n-3$ which leads to $n-5$ presentations of type $\mathcal{H}_k$, a joint $n-4-\lfloor n/2\rfloor$ presentations of type $\mathcal{J}_k$ or $\mathcal{X}$, and $14$ of type $\mathcal{N}_{k,l,m}^e$. To count the presentations listed in Theorem \[thm\_cc2\_d3\] we calculate that there is a total of $$\sum_{i=1}^{n-1+\lfloor \frac{n-1}{2} \rfloor-1}i = \frac{1}{2}\left( \left(n + \left\lfloor \frac{n-1}{2} \right\rfloor \right)^2 - 3 \left(n + \left\lfloor\frac{n-1}{2} \right\rfloor\right) \right)+ 1$$ choices for $V$ and $W$ with $V\preceq W$. Each pair fulfils only one of the conditions from (i), (ii) and (iii) in Theorem \[thm\_cc2\_d3\]. There are $$\sum_{k=3}^{\lceil \frac{n-1}{2} \rceil+1} \left(\sum_{l=k}^{n-k-2}1 + \sum_{l=\lceil\frac{n-1}{2}\rceil}^{n-k-2}1\right) = \sum_{k=3}^{\lceil \frac{n-1}{2} \rceil+1} \left(n-2k-1\right) + \left(n-\left\lceil\frac{n-1}{2}\right\rceil-k-1\right)$$ choices that lead to the $1$ presentation from (i) and $$\left(n+\lceil n/2\rceil -5 + n+\lceil n/2\rceil -4\right) =8n-8\lceil n/2 \rceil -36$$ that lead to the $4$ presentations from (iii). All other pairs fulfil the condition in (ii) and hence lead to those $3$ presentations. The pairs of non-commutative $V$ and $W$ fulfil the conditions in (iv) and each one leads to $3$ presentations. This yields and additional $4\cdot 3=12$ presentations from (iv). Simplifying the sum of all presentations separately for even and odd integer yields the stated formulae up to (anti-)isomorphism. The remaining formulae are obtained in a similar way and the proof is left out. For $n \geq 7$ the number of nilpotent semigroups of order $n$, coclass $2$ … 1. …counting up to (anti-)isomorphism equals $$\begin{aligned} &\frac{1}{8}(21n^2+66n-104) - \left\lceil \frac{n}{3} \right\rceil &\textrm{ if $n$ is even, and}\\ &\frac{1}{8}(21n^2+80n-93) - \left\lceil \frac{n}{3} \right\rceil &\textrm{ if $n$ is odd.}\end{aligned}$$ 2. …counting up to isomorphism equals $$\begin{aligned} &\frac{1}{8}(27n^2+150n-240) - \left\lceil \frac{n}{3} \right\rceil &\textrm{ if $n$ is even, and}\\ &\frac{1}{8}(27n^2+168n-203) - \left\lceil \frac{n}{3} \right\rceil &\textrm{ if $n$ is odd.}\end{aligned}$$ 3. …counting commutative semigroups up to isomorphism equals $$\begin{aligned} &\frac{1}{8}(15n^2-26n-40) - \left\lceil \frac{n}{3} \right\rceil &\textrm{ if $n$ is even, and}\\ &\frac{1}{8}(15n^2-16n-71) - \left\lceil \frac{n}{3} \right\rceil &\textrm{ if $n$ is odd.}\end{aligned}$$ For small orders the semigroups of coclass $1$ or $2$ where determined using the code from [@Dis10 Appendix C]. The numbers are given in Table \[tab\_computed\]. They coincide, where applicable, with the results obtained from the formulae given in this section. [lrrrrrrrrrrr]{} &[**3**]{} &[**4**]{} &[**5**]{} &[**6**]{} &[**7**]{} &[**8**]{} &[**9**]{} &[**10**]{} &[**11**]{} &[**12**]{} &[**13**]{}\ \ coclass $1$ &1 &8 &7 &9 &10 &12 &13 &15 &16 &18 &19\ coclass $2$ &0 &1 &84 &142 &184 &218 &288 &328 &412 &460 &557\ –, 2-generated &0 &0 &11 &43 &34 &40 &45 &50 &55 &61 &65\ –, 3-generated &0 &1 &73 &99 &150 &178 &243 &278 &357 &399 &492\ \ coclass $1$ &1 &9 &9 &11 &12 &14 &15 &17 &18 &20 &21\ coclass $2$ &0 &1 &118 &219 &284 &333 &434 &491 &610 &677 &813\ –, 2-generated &0 &0 &15 &62 &51 &58 &65 &71 &78 &85 &91\ –, 3-generated &0 &1 &103 &157 &233 &275 &369 &420 &532 &592 &722\ \ coclass $1$ &1 &5 &5 &7 &8 &10 &11 &13 &14 &16 &17\ coclass $2$ &0 &1 &23 &42 &67 &86 &123 &146 &193 &222 &278\ –, 2-generated &0 &0 &4 &15 &16 &21 &24 &28 &31 &36 &38\ –, 3-generated &0 &1 &19 &27 &51 &65 &99 &118 &162 &186 &240\ \[tab\_computed\] [10]{} Andreas Distler. . Shaker Verlag, Aachen, 2010. PhD thesis, University of St Andrews, 2010, <http://hdl.handle.net/10023/945>. Andreas Distler and Bettina Eick. Coclass theory for nilpotent semigroups via their associated algebras. , 2012. Andreas Distler and James D. Mitchel. The number of nilpotent semigroups of degree $3$. , 2012. Andreas Distler and James D. Mitchell. . <http://www-history.mcs.st-and.ac.uk/~jamesm/smallsemi/>, Oct 2011. A [GAP 4]{} package [@GAP4], Version 0.6.4. The GAP Group, `(http://www.gap-system.org)`. , 2008. V. L. Klee Jr. The [N]{}ovember meeting in [L]{}os [A]{}ngeles. , 62(1):13–23, 1956. C. R. Leedham-Green and M. F. Newman. Space groups and groups of prime-power order. [I]{}. , 35(3):193–202, 1980. A. I. Malcev. Nilpotent semigroups. , 4:107–111, 1953. T. S. Motzkin and J. L. Selfridge. Semigroups of order five. presented in [@Kle56], 1955. Robert J. Plemmons. There are [${\rm 15973}$]{} semigroups of order [${\rm 6}$]{}. , 2:2–17, 1967. Ronald C. Read. Every one a winner or how to avoid isomorphism search when cataloguing combinatorial configurations. , 2:107–120, 1978. Algorithmic aspects of combinatorics (Conf., Vancouver Island, B.C., 1976). Aner Shalev and Efim I. Zelmanov. Narrow [L]{}ie algebras: a coclass theory and a characterization of the [W]{}itt algebra. , 189(2):294–331, 1997. [^1]: Currently under development [^2]: The remaining products are deduced as follows: $vu^k=(vu^2)u^{k-2}$; $u^kv=u^{k-1}(uv)$; $$yu^k=\begin{cases} vuu^k=(vu^2)u^{k-1} & \textrm{ if } vu=y\\ v^2u^k=v(vu)u^{k-1}=(vu)u^{l+k+2}=u^{2l+k-2} & \textrm{ if } vu=u^l, v^2=y\\ uvu^k = u(vu)u^{k-1}=u^{l+k}& \textrm{ if } vu=u^l,uv=y; \end{cases}$$ $$yv=\begin{cases} v^2v=vv^2=vy & \textrm{ if } v^2=y\\ uvv = uu^l =u^{l+1} & \textrm{ if } v^2=u^l, uv=y\\ vuv = vu^k=(vu^2)u^{k-2} & \textrm{ if } uv=u^k, vu=y \end{cases}$$.
--- author: - | Gradeigh D. Clark and Janne Lindqvist\ \ bibliography: - 'gestures.bib' title: 'Engineering Gesture-Based Authentication Systems' ---
--- abstract: 'We present two integrable discretisations of a general differential-difference bicomponent Volterra system. The results are obtained by discretising directly the corresponding Hirota bilinear equations in two different ways. Multisoliton solutions are presented together with a new discrete form of Lotka-Volterra equation obtained by an alternative bilinearisation.' --- [**On various integrable discretisations of a general two component Volterra system** ]{} Corina N. Babalic$^{\dagger}$, A. S. Carstea$^*$[[^1]]{} $^\dagger$[*Dept. of Physics, University of Craiova, Romania\ $^*$National Institute of Physics and Nuclear Engineering, Dept. of Theoretical Physics, Atomistilor 407, 077125, Magurele, Bucharest, Romania\ *]{} Introduction ============ One of the main difficulty in the topic of integrable systems is obtaining an integrable discretisation of a given partial differential or differential-difference integrable system. Applying integrability criteria like complexity growth [@sebastien] or singularity confinement [@alfred] is not always quite easy because the discrete lattice equations have, in general, complicated forms. However, is much more convenient to start with some general lattice equation and impose some simpler (but more restrictive) integrability requirements (like, for instance, cube consistency [@suris] which leads immediately to the discrete variant of “zero curvature representation”). Among the methods of finding integrable discretisations, one of the powerful one is the Hirota bilinear method. The idea is quite simple. First, the integrable differential or differential-difference integrable system has to be correctly bilinearised (in the sense of allowing construction of general multisoliton solution). After that, in a first step, one has to replace differential Hirota operators with discrete ones preserving gauge invariance. Of course the resulting bilinear fully discrete system is not necesarily integrable, so in the second step, the multisoliton solution must be found [@side3]. If this exists then the discrete bilinear system is integrable and, in the final step which is rather complicated, the nonlinear form has to be recovered. Here, it is possible to introduce an auxiliary function, as Hirota have shown in [@hirota2000]. We have to clarify here the concept of integrability. Usually, the integrability for a given partial differential or partial discrete equation, is relying on the existence of an infinite number of independent integrals in involution that usually can be computed from the Lax pairs. Accordingly, if the equation can be written as a compatibility condition of two nontrivial linear operators (i.e. Lax pairs) then it is automatically completely integrable. However, the alternative formulation given by Hirota bilinear form, is also used in proving integrability by requiring the existence of general multisoliton solution. Here the word [*general*]{} means the solution describing multiple collisions of an [*arbitrary*]{} number of solitons having [*arbitrary*]{} parameters and phases and also for [*all*]{} branches of dispersion relations. Any constraint in the form of multisoliton solution breaks complete integrability. The integrals of motion are related to the soliton parameters which, because of elastic collisions, remain unchanged (and they are also related to the spectrum of Lax operator invariant as well since the equation itself is an isospectral deformation). Construction of the multisoliton solution is quite difficult, but it has been observed that once three general soliton solution is constructed then it can be proved by induction that the general N-soliton can be constructed as well. The existence of three-soliton solution has been used in classification of completely integrable bilinear equations [@hietarinta-mkdv], [@hietarinta-sg] as an integrability criterion. In this paper we are going to give two integrable discretisations of a general two component bidirectional Volterra system (in the sense that the dispersion relation has two branches). This system is rather old. It has been formulated for the first time by Hirota and Satsuma [@hirsat] but with constraints on parameters. Then, various solutions (rational, white and dark solitonic) have been obtained [@fane], [@narita], [@chow]. Curious enough, the fully discretisation has not been studied so far, although the initial system can be formulated as a coupled differential-difference focusing or defocusing mKdV system with non-zero boundary conditions. In this paper we present two fully discrete bilinear forms, show the structure of $N$-soliton solution and then recover the nonlinear form. For the second discretisation we extend the approach of Hirota presented in [@hirota2000] and use two auxiliary functions. An important fact is that the multisoliton solution in the case of second discretisation has practically the same phase factors and interaction terms as in the differential-difference one. The same fact has been observed by Hirota and Tsujimoto in [@hirota2000],[@hir2002], [@tsuji] for other cases. They have shown that for many examples including lattice mKdV, lattice NLS, lattice coupled mKdV, the structure of soliton solution remains the same as in the case of differential-difference analogs (although, it is true that all their examples support only unidirectional solitons). We think that this fact happens in many other cases and as an other example we construct a new completely integrable discretisation of Lotka-Volterra equation. Discretisation of general Volterra system ========================================= The differential-difference system under consideration is: $$\begin{aligned} \dot Q_n=(c_0+c_1Q_n+c_2Q_n^2)(R_{n+1}-R_{n-1})\\ \dot R_n=(c_0+c_1R_n+c_2R_n^2)(Q_{n+1}-Q_{n-1})\nonumber\end{aligned}$$ where $c_0, c_1, c_2$ are arbitrary constants and the dot means derivative with respect to time. In [@fane] we have shown that following the simple scalings and translations, $$u_n=\frac{2c_2}{\sqrt{4c_0c_2-c_1^2}}(Q_n+\frac{c_1}{2c_2}), \quad v_n=\frac{2c_2}{\sqrt{4c_0c_2-c_1^2}}(R_n+\frac{c_1}{2c_2})$$ we can cast the system in the following form: $$\begin{aligned} \dot u_n=(1+u_n^2)(v_{n+1}-v_{n-1})\\ \dot v_n=(1+v_n^2)(u_{n+1}-u_{n-1})\nonumber\end{aligned}$$ where $u_n, v_n\to \alpha$ as $n\to \pm\infty$ and $\alpha=c_1(4c_0c_2-c_1^2)^{-1/2}$. These transformations are valid only in the case of nonzero $c_2$. Of course, one must be careful about the square root appearing in the definition of $\alpha$. In case of negative argument the system is changed in a defocusing form: $$\begin{aligned} \dot u_n=(1-u_n^2)(v_{n+1}-v_{n-1})\\ \dot v_n=(1-v_n^2)(u_{n+1}-u_{n-1})\nonumber\end{aligned}$$ From the point of view of integrability this is not crucial since an overall imaginary unit factor for $u_n, v_n$ will change the system one into the other. But for the real soliton dynamics (which we are not discussing here)the solutions of the above systems are behaving completely different (only the unicomponent case has been analysed quite recently in [@chow]). For the system (2) the Hirota bilinear form is given by: $$\begin{aligned} \dot f_n g_n-f_n \dot g_n=(1+\alpha^2)(F_{n+1}G_{n-1}-G_{n+1}F_{n-1})\\ \dot F_n G_n-F_n \dot G_n=(1+\alpha^2)(f_{n+1} g_{n-1}-g_{n+1} f_{n-1})\\ (1+i\alpha)F_{n+1} G_{n-1}+(1-i\alpha)F_{n-1}G_{n+1}=2f_ng_n\\ (1+i\alpha)f_{n+1} g_{n-1}+(1-i\alpha)f_{n-1} g_{n+1}=2F_nG_n\end{aligned}$$ where the nonlinear substitutions are: $$u_n=\alpha-\frac{i}{2}\frac{\partial}{\partial t}\ln{\frac{f_n(t)}{g_n(t)}}, \quad v_n=\alpha-\frac{i}{2}\frac{\partial}{\partial t}\ln{\frac{F_n(t)}{G_n(t)}}$$ The bilinear system (4)-(7) is completely integrable [@fane] and has the following $N$-soliton solution: $$f_n=\sum_{\mu_1...\mu_N\in\{0,1\}}\left(\prod_{i=1}^{N}(\beta_i p_i^n e^{\omega_i t})^{\mu_i}\prod_{i<j}^{N}A_{ij}^{\mu_i\mu_j}\right)$$ $$F_n=\sum_{\mu_1...\mu_N\in\{0,1\}}\left(\prod_{i=1}^{N}(\beta_i' p_i^n e^{\omega_i t})^{\mu_i}\prod_{i<j}^{N}A_{ij}^{\mu_i\mu_j}\right)$$ $$g_n=\sum_{\mu_1...\mu_N\in\{0,1\}}\left(\prod_{i=1}^{N}(\gamma_i p_i^n e^{\omega_i t})^{\mu_i}\prod_{i<j}^{N}A_{ij}^{\mu_i\mu_j}\right)$$ $$G_n=\sum_{\mu_1...\mu_N\in\{0,1\}}\left(\prod_{i=1}^{N}(\gamma_i' p_i^n e^{\omega_i t})^{\mu_i}\prod_{i<j}^{N}A_{ij}^{\mu_i\mu_j}\right)$$ where the dispersion relation and phase factors are: $$\omega_i=\epsilon_i(1+\alpha^2)\frac{1-p_i^2}{p_i}$$ $$\beta_i=\frac{i\alpha(-1+\frac{\epsilon_i}{2}(p_i+p_i^{-1}))}{(p_i-p_i^{-1})}-\frac{\epsilon_i}{2},\quad \gamma_i=\frac{i\alpha(-1+\frac{\epsilon_i}{2}(p_i+p_i^{-1}))}{(p_i-p_i^{-1})}+\frac{\epsilon_i}{2}$$ $$\beta_i'=\frac{i\alpha\epsilon_i(1-\frac{\epsilon_i}{2}(p_i+p_i^{-1}))}{(p_i-p_i^{-1})}+\frac{1}{2},\quad \gamma_i'=\frac{i\alpha\epsilon_i(1-\frac{\epsilon_i}{2}(p_i+p_i^{-1}))}{(p_i-p_i^{-1})}-\frac{1}{2}$$ $$A_{ij}=\left(\frac{\epsilon_i p_i-\epsilon_j p_j}{1-\epsilon_i\epsilon_jp_ip_j}\right)^2$$ In order to construct an integrable discretisation we replace time derivatives in (4) and (5) with finite differences ($t\to m$) $$\dot f_n\to\frac{1}{\delta}(f(n,m+\delta)-f(n,m))$$ and impose the invariance of the resulting bilinear equation with respect to multiplication with $\exp(\mu n+\nu m)$ for any $\mu, \nu$ (bilinear gauge invariance). In this first discretisation we discretise [*also*]{} the equations (6) and (7) by assuming a gauge-invariant shift in the time variable. The fully discrete gauge invariant bilinear equations are given by: $$\begin{aligned} \tilde f_n g_n-f_n \tilde g_n=\delta(1+\alpha^2)(\tilde F_{n+1} G_{n-1}-\tilde G_{n+1} F_{n-1})\\ \tilde F_n G_n-F_n \tilde G_n=\delta(1+\alpha^2)(\tilde f_{n+1} g_{n-1}-\tilde g_{n+1} f_{n-1})\\ (1+i\alpha)\tilde F_{n+1} G_{n-1}+(1-i\alpha)F_{n-1}\tilde G_{n+1}=\tilde f_n g_n+f_n \tilde g_n\\ (1+i\alpha)\tilde f_{n+1} g_{n-1}+(1-i\alpha)f_{n-1}\tilde g_{n+1}=\tilde F_n G_n+F_n \tilde G_n\\end{aligned}$$ where $\tilde f_n=f(n,m+\delta)$ etc. In order to check the integrability one has to compute the general $N$-soliton solution of the above system. With the aid of a symbolic computation software like MATHEMATICA one can easily find 3-soliton solution. It can be extended to N-soliton solution in the form (the proof is given in the Appendix): $$\begin{aligned} f_n=\sum_{\mu_1...\mu_N\in\{0,1\}}\left(\prod_{i=1}^{N}(a_i p_i^n q_i^{m\delta})^{\mu_i}\prod_{i<j}^{N}M_{ij}^{\mu_i\mu_j}\right)\\ F_n=\sum_{\mu_1...\mu_N\in\{0,1\}}\left(\prod_{i=1}^{N}(A_i p_i^n q_i^{m\delta})^{\mu_i}\prod_{i<j}^{N}M_{ij}^{\mu_i\mu_j}\right)\\ g_n=\sum_{\mu_1...\mu_N\in\{0,1\}}\left(\prod_{i=1}^{N}(b_i p_i^n q_i^{m\delta})^{\mu_i}\prod_{i<j}^{N}M_{ij}^{\mu_i\mu_j}\right)\\ G_n=\sum_{\mu_1...\mu_N\in\{0,1\}}\left(\prod_{i=1}^{N}(B_i p_i^n q_i^{m\delta})^{\mu_i}\prod_{i<j}^{N}M_{ij}^{\mu_i\mu_j}\right)\end{aligned}$$ where the dispersion relation and phase factors are given by: $$q_i=\left(\frac{p_i+\delta\epsilon_i(1+\alpha^2)}{p_i+p_i^2 \epsilon_i\delta(1+\alpha^2}\right)^{1/\delta}$$ $$a_i=\frac{i\alpha(-1+\frac{\epsilon_i}{2}(p_i+p_i^{-1}))}{(p_i-p_i^{-1})(1+\delta+\delta\alpha^2)}-\frac{\epsilon_i}{2},\quad b_i=\frac{i\alpha(-1+\frac{\epsilon_i}{2}(p_i+p_i^{-1}))}{(p_i-p_i^{-1})(1+\delta+\delta\alpha^2)}+\frac{\epsilon_i}{2}$$ $$A_i=\frac{i\alpha\epsilon_i(1-\frac{\epsilon_i}{2}(p_i+p_i^{-1}))}{(p_i-p_i^{-1})(1+\delta+\delta\alpha^2)}+\frac{1}{2},\quad B_i=\frac{i\alpha\epsilon_i(1-\frac{\epsilon_i}{2}(p_i+p_i^{-1}))}{(p_i-p_i^{-1})(1+\delta+\delta\alpha^2)}-\frac{1}{2}$$ $$M_{ij}=\left(\frac{\epsilon_i p_i-\epsilon_j p_j}{1-\epsilon_i\epsilon_jp_ip_j}\right)^2$$ Accordingly, our bilinear system is an integrable one. Now we can proceed to recover the nonlinear form. Dividing (8) by (10) and (9) by (11) and taking into account that $\tan(\frac{i}{2}\log(G/F))=i(G-F)/(G+F)$ for any $G$ and $F$ we obtain the following system: $$\tan(\tilde{Q}_n-Q_n)=\frac{\delta(1+\alpha^2)\tan(\tilde{R}_{n+1}-R_{n-1})}{1+\alpha\tan(\tilde{R}_{n+1}-R_{n-1})}$$ $$\tan(\tilde{R}_n-R_n)=\frac{\delta(1+\alpha^2)\tan(\tilde{Q}_{n+1}-Q_{n-1})}{1+\alpha\tan(\tilde{Q}_{n+1}-Q_{n-1})}$$ where $Q_n=\frac{i}{2}\log(f_n/g_n), R_n=\frac{i}{2}\log(F_n/G_n)$. To our knowledge this system is a new one. We do not know how it is related to other integrable discretizations of Volterra systems. In the case of $\alpha\to 0$ the classical lattice self-dual network of Hirota is obtained. Since the phase factors are defined up to the multiplication with the same constant factor (which we take to be the imaginary unit) then $f_n$, $g_n$ (and $F_n$, $G_n$ as well) will be complex conjugated so the physical fields $Q_n$ and $R_n$ are real functions. Here we give the 1 and 2-soliton solution (see also the figure): $$Q_n=\frac{i}{2}\log\left(\frac{1+ia_1p_1^nq_1^{m\delta}}{1+ib_1p_1^nq_1^{m\delta}}\right),\quad R_n=\frac{i}{2}\log\left(\frac{1+iA_1p_1^nq_1^{m\delta}}{1+iB_1p_1^nq_1^{m\delta}}\right).$$ $$Q_n=\frac{i}{2}\log\left(\frac{1+ia_1p_1^nq_1^{m\delta}+ia_2p_2^nq_2^{m\delta}-a_1a_2M_{12}(p_1p_2)^n(q_1q_2)^{m\delta}}{1+ib_1p_1^nq_1^{m\delta}+ib_2p_2^nq_2^{m\delta}-b_1b_2M_{12}(p_1p_2)^n(q_1q_2)^{m\delta}}\right),$$ $$R_n=\frac{i}{2}\log\left(\frac{1+iA_1p_1^nq_1^{m\delta}+iA_2p_2^nq_2^{m\delta}-A_1A_2M_{12}(p_1p_2)^n(q_1q_2)^{m\delta}}{1+iB_1p_1^nq_1^{m\delta}+iB_2p_2^nq_2^{m\delta}-B_1B_2M_{12}(p_1p_2)^n(q_1q_2)^{m\delta}}\right).$$ ![One and two-soliton solution for $Q_n$ ($p_1=2, p_2=0.7, \alpha=1, \delta=2, \epsilon_1=-1, \epsilon_2=1$)](1SS.eps){width="6cm"} However one can obtain a different nonlinear form by taking $q_n=f_n/g_n$ and $r_n=F_n/G_n$, namely: $$\tilde{q}_n=q_n\frac{z_1 \tilde{r}_{n+1}+z_2 r_{n-1}}{z_2^* \tilde{r}_{n+1}+z_1^* r_{n-1}}, \quad \tilde{r}_n=r_n\frac{z_1 \tilde{q}_{n+1}+z_2 q_{n-1}}{z_2^* \tilde{r}_{n+1}+z_1^* r_{n-1}}$$ where $z_1=1+i\alpha+\delta(1+\alpha^2), z_2=1-i\alpha-\delta(1+\alpha^2).$ In the case of $q_n=r_n$ the system becomes the Nijhoff-Capel lattice mKdV equation [@frank] but with complex coefficients. Of course a complete description imposes finding the Lax pairs, hamiltonian structure using r-matrices and so on, [@suris2]. Second discretisation ===================== We may discretise the bilinear equations (4)-(7) in a simpler way. Namely we only discretise the dispersion equations (4) and (5) which depend explicitly on time and leave (6) and (7) unmodified. We obtain: $$\begin{aligned} \tilde f_n g_n-f_n \tilde g_n=\delta(1+\alpha^2)(\tilde F_{n+1} G_{n-1}-\tilde G_{n+1} F_{n-1})\\ \tilde F_n G_n-F_n \tilde G_n=\delta(1+\alpha^2)(\tilde f_{n+1} g_{n-1}-\tilde g_{n+1} f_{n-1})\\ (1+i\alpha) F_{n+1} G_{n-1}+(1-i\alpha)F_{n-1} G_{n+1}=2 f_n g_n\\ (1+i\alpha) f_{n+1} g_{n-1}+(1-i\alpha)f_{n-1} g_{n+1}=2 F_n G_n\end{aligned}$$ This bilinear system is again completely integrable. It has $N$-soliton solution of the same form, $$\begin{aligned} f_n=\sum_{\mu_1...\mu_N\in\{0,1\}}\left(\prod_{i=1}^{N}(\beta_i p_i^n q_i^{m\delta})^{\mu_i}\prod_{i<j}^{N}A_{ij}^{\mu_i\mu_j}\right)\\ F_n=\sum_{\mu_1...\mu_N\in\{0,1\}}\left(\prod_{i=1}^{N}(\beta_i' p_i^n q_i^{m\delta})^{\mu_i}\prod_{i<j}^{N}A_{ij}^{\mu_i\mu_j}\right)\\ g_n=\sum_{\mu_1...\mu_N\in\{0,1\}}\left(\prod_{i=1}^{N}(\gamma_i p_i^n q_i^{m\delta})^{\mu_i}\prod_{i<j}^{N}A_{ij}^{\mu_i\mu_j}\right)\\ G_n=\sum_{\mu_1...\mu_N\in\{0,1\}}\left(\prod_{i=1}^{N}(\gamma_i' p_i^n q_i^{m\delta})^{\mu_i}\prod_{i<j}^{N}A_{ij}^{\mu_i\mu_j}\right)\end{aligned}$$ The dispersion relation and the phase factors are: $$q_i=\left(\frac{p_i+\delta\epsilon_i(1+\alpha^2)}{p_i+p_i^2 \epsilon_i\delta(1+\alpha^2}\right)^{1/\delta}$$ $$\beta_i=\frac{i\alpha(-1+\frac{\epsilon_i}{2}(p_i+p_i^{-1}))}{(p_i-p_i^{-1})}-\frac{\epsilon_i}{2},\quad \gamma_i=\frac{i\alpha(-1+\frac{\epsilon_i}{2}(p_i+p_i^{-1}))}{(p_i-p_i^{-1})}+\frac{\epsilon_i}{2}$$ $$\beta_i'=\frac{i\alpha\epsilon_i(1-\frac{\epsilon_i}{2}(p_i+p_i^{-1}))}{(p_i-p_i^{-1})}+\frac{1}{2},\quad \gamma_i'=\frac{i\alpha\epsilon_i(1-\frac{\epsilon_i}{2}(p_i+p_i^{-1}))}{(p_i-p_i^{-1})}-\frac{1}{2}$$ $$A_{ij}=\left(\frac{\epsilon_i p_i-\epsilon_j p_j}{1-\epsilon_i\epsilon_jp_ip_j}\right)^2$$ The nonlinear form can be easily recovered although the system now is more complicated and it will involve two auxiliary functions $w_n$ and $v_n$. We divide (20) by $g_n\tilde{g}_n$ and (21) by $G_n\tilde{G}_n$. Calling $x_n=f_n/g_n, y_n=F_n/G_n, w_n=G_{n-1}\tilde{G}_{n+1}/g_n\tilde{g_n}, v_n=g_{n-1}\tilde{g}_{n+1}/G_n\tilde{G}_n$ we get: $$\tilde{x}_n-x_n=\delta(1+\alpha^2)(\tilde{y}_{n+1}-y_{n-1})w_n$$ $$\tilde{y}_n-y_n=\delta(1+\alpha^2)(\tilde{x}_{n+1}-x_{n-1})v_n$$ But one can see immediately that: $$w_{n+1}/v_n=\frac{G_n\tilde{G}_{n+2}G_n\tilde{G}_n}{g_{n+1}\tilde{g}_{n+1}g_{n-1}\tilde{g}_{n+1}}=\left(\frac{{G_{n}}^2}{g_{n+1}g_{n-1}}\right)\left(\frac{\tilde{G}_{n+2}\tilde{G}_n}{\tilde{g}_{n+1}^2}\right)$$ $$v_{n+1}/w_n=\frac{g_n\tilde{g}_{n+2}g_n\tilde{g}_n}{G_{n+1}\tilde{G}_{n+1}G_{n-1}\tilde{G}_{n+1}}=\left(\frac{{g_{n}}^2}{G_{n+1}G_{n-1}}\right)\left(\frac{\tilde{g}_{n+2}\tilde{g}_n}{\tilde{G}_{n+1}^2}\right)$$ The factors in the parantheses can be computed easily from (22) and (23) by dividing them to $G_{n-1}G_{n+1}$ and $g_{n-1}g_{n+1}$. Finally the nonlinear form of our sistem is: $$\tilde{x}_n-x_n=\delta(1+\alpha^2)(\tilde{y}_{n+1}-y_{n-1})w_n$$ $$\tilde{y}_n-y_n=\delta(1+\alpha^2)(\tilde{x}_{n+1}-x_{n-1})v_n$$ $$w_{n+1}=v_n\frac{x_{n+1}\tilde{x}_{n+1}(1+i\alpha)+x_{n-1}\tilde{x}_{n+1}(1-i\alpha)}{y_{n}\tilde{y}_{n+2}(1+i\alpha)+y_n\tilde{y}_{n}(1-i\alpha)}$$ $$v_{n+1}=w_n\frac{y_{n+1}\tilde{y}_{n+1}(1+i\alpha)+y_{n-1}\tilde{y}_{n+1}(1-i\alpha)}{x_{n}\tilde{x}_{n+2}(1+i\alpha)+x_n\tilde{x}_{n}(1-i\alpha)}$$ We can eliminate the auxiliary functions $w_n$ and $v_n$ and we get the following higher order system: $$\frac{\tilde{x}_{n+1}-x_{n+1}}{\tilde{y}_{n+2}-y_n}=\frac{\tilde{y}_n-y_n}{\tilde{x}_{n+1}-x_{n-1}}\frac{x_{n+1}\tilde{x}_{n+1}(1+i\alpha)+x_{n-1}\tilde{x}_{n+1}(1-i\alpha)}{y_{n}\tilde{y}_{n+2}(1+i\alpha)+y_n\tilde{y}_{n}(1-i\alpha)}$$ $$\frac{\tilde{y}_{n+1}-y_{n+1}}{\tilde{x}_{n+2}-x_n}=\frac{\tilde{x}_n-x_n}{\tilde{y}_{n+1}-y_{n-1}}\frac{y_{n+1}\tilde{y}_{n+1}(1+i\alpha)+y_{n-1}\tilde{y}_{n+1}(1-i\alpha)}{x_{n}\tilde{x}_{n+2}(1+i\alpha)+x_n\tilde{x}_{n}(1-i\alpha)}$$ An important remark is that we have, the [*same*]{} phase factors and interaction term as in the [*differential-difference*]{} case, namely the system (4)-(7)(and that is why we kept the same notation). Also in both equations (28), (29) the step of time discretisation $\delta$ dissapeared. So there is no trace of discretisation in the solutions, except the dispersion relation. This means that the structure of the soliton solution is the same at the level of tau functions. However, the nonlinear form is different. For instance the one and two-soliton solutions, $$x_n=\frac{1+\beta_1p_1^nq_1^{m\delta}}{1+\gamma_1p_1^nq_1^{m\delta}},\quad y_n=\frac{1+\beta_1'p_1^nq_1^{m\delta}}{1+\gamma_1'p_1^nq_1^{m\delta}},$$ $$x_n=\frac{1+\beta_1p_1^nq_1^{m\delta}+\beta_2p_2^nq_2^{m\delta}+\beta_1\beta_2A_{12}(p_1p_2)^n(q_1q_2)^{m\delta}}{1+\gamma_1p_1^nq_1^{m\delta}+\gamma_2p_2^nq_2^{m\delta}+\gamma_1\gamma_2A_{12}(p_1p_2)^n(q_1q_2)^{m\delta}},$$ $$y_n=\frac{1+\beta_1'p_1^nq_1^{m\delta}+\beta_2'p_2^nq_2^{m\delta}+\beta_1'\beta_2'A_{12}(p_1p_2)^n(q_1q_2)^{m\delta}}{1+\gamma_1'p_1^nq_1^{m\delta}+\gamma_2'p_2^nq_2^{m\delta}+\gamma_1'\gamma_2'A_{12}(p_1p_2)^n(q_1q_2)^{m\delta}}$$ are complex functions (since the phase factors are complex). But if we define new fields $\phi_n$ and $\psi_n$ by $x_n=\exp(i\phi_n)$ and $y_n=\exp(i\psi_n)$ then the soliton solutions expressed by $\phi_n$ and $\psi_n$ are real functions and have the same shape as the ones in the first discretisation. [**[Remark]{}**]{}: This type of discretization involving auxiliary functions has been done also by Hirota and Tsujimoto in [@hirota2000],[@hir2002], [@tsuji]. In their examples also the form and phase factors of soliton solutions remain the same. We believe that this fact happens in more general cases, although must be proved rigurously. Also we could have discretised the system (1) directly by taking $Q_n(t)=\Gamma_n(t)/\Phi_n(t), R_n(t)=H_n(t)/T_n(t)$ which give the bilinear system: $$D_t \Gamma_n\cdot \Phi_n=H_{n+1}T_{n-1}-H_{n-1}T_{n+1}$$ $$D_t H_n\cdot T_n=\Gamma_{n+1}\Phi_{n-1}-\Gamma_{n-1}\Phi_{n+1}$$ $$c_0\Phi_n^2+c_1\Gamma_n\Phi_n+c_2\Gamma_n^2=T_{n-1}T_{n+1}$$ $$c_0T_n^2+c_1H_nT_n+c_2H_n^2=\Phi_{n-1}\Phi_{n+1}$$ Discretising the first two equations and after making the same steps as in the previous case we would have obtained almost a similar form: $$\tilde{Q}_n-Q_n=\delta(R_{n+1}-R_{n-1})V_n$$ $$\tilde{R}_n-R_n=\delta(Q_{n+1}-Q_{n-1})W_n$$ $$W_{n+1}=V_n\frac{c_0+c_1\tilde{Q}_{n+1}+c_2\tilde{Q}_{n+1}^2}{c_0+c_1R_n+c_2R_n^2}$$ $$V_{n+1}=W_n\frac{c_0+c_1\tilde{R}_{n+1}+c_2\tilde{R}_{n+1}^2}{c_0+c_1Q_n+c_2Q_n^2}$$ The main drawback of this system is that the soliton solution has a very complicated form. A new form of the Lotka-Volterra equation ========================================= Let us give an nice and simple example related to the above construction, namely differential-difference Lotka-Volterra equation: $$\frac{du_n}{dt}=u_n(u_{n+1}-u_{n-1})$$ Of course this equation can be seen as a particular case of the system (1) for $Q_n=R_n\equiv u_n$ and $c_0=c_2=0, c_1=1$. However because the simplified form (2) has been obtained only for $c_2\neq 0$ this equation cannot be studied as a particular case of the bilinear system (4)-(7). We consider the substitution $u_n=G_n/F_n$ and we get the following: $$D_t G_n\cdot F_n=(G_{n+1}F_{n-1}-G_{n-1}F_{n+1})$$ $$G_nF_n=F_{n+1}F_{n-1}$$ Now we are going to discretise only the first bilinear equation (imposing gauge-invariance in the right hand side) and the second one will remain the same. We shall obtain ($t\to m\delta$): $$\tilde{G}_nF_n-G_n\tilde{F}_{n}=\delta(\tilde{G}_{n+1}F_{n-1}-G_{n-1}\tilde{F}_{n+1})$$ $$G_nF_n=F_{n+1}F_{n-1}$$ This bilinear system is integrable because it is equivalent (if we take $G_n=f_{n-1}f_{n+2}, F_n=f_nf_{n+1}$) with a quadrilinear equation reducible to: $$\tilde{f}_nf_{n+1}+\delta f_{n-1}\tilde{f}_{n+2}-(1+\delta)f_n\tilde{f}_{n+1}=0$$ which is the integrable bilinear form of the discrete Lotka-Volterra (with $u_n=f_{n-1}\tilde{f}_{n+2}/f_n\tilde{f}_{n+1}$) [@side3] $$\tilde{u}_n=u_{n}\frac{1-\delta+\delta u_{n-1}}{1-\delta+\delta\tilde{u}_{n+1}}$$ However the nonlinear form recovered from (30) and (31) is different and we proceed as in the case of the system (28)-(29). Calling $x_n=G_n/F_n$ we get: $$\tilde{x}_n-x_n=\delta(\tilde{x}_{n+1}-x_{n-1})w_n$$ $$w_{n+1}=w_n\frac{\tilde{x}_{n+1}}{x_n}$$ This is a different variant of lattice Lotka-Volterra equation. We can solve the second linear equation in $w_n$ and we get: $$\tilde{x}_n-x_n=\delta(\tilde{x}_{n+1}-x_{n-1})\prod_{k=-\infty}^n\frac{\tilde{x}_k}{x_{k-1}}$$ or we can eliminate $w_n$ from the first equation and we find the following nice form: $$\frac{\tilde{x}_{n+1}-x_{n+1}}{\tilde{x}_n-x_n}\frac{\tilde{x}_{n+1}-x_{n-1}}{\tilde{x}_{n+2}-x_n}\frac{x_n}{\tilde{x}_{n+1}}=1$$ The first two factors in the left hand side look like a discrete Schwartzian derivative [@frank] although the expression is not a cross ratio of four points but a cross ratio of diagonals of the two adjacent parallelograms formed by the six points. We think that this is a new equation although its symmetric form may be related in a way (unknown to us) to some well known one. Conclusions =========== In this paper we have presented two integrable discretisations of a general bicomponent differential-difference Volterra system. The main procedure was discretising differential Hirota bilinear operator and then recovery of nonlinear form with the aid of some auxiliary functions. This approach may lead to higher order nonlinear equations. We apply this procedure also to the well known Lotka-Volterra equation and we found a new discrete form However we started from a different bilinearization involving two tau functions. Relying on the fact that the structure of soliton solutions remains the same we believe that this procedure will be effective in discretising even nonintegrable equations of reaction-diffusion type (because it keeps the same structure of travelling waves). [**Aknowledgements:**]{} One of the authors (ASC) has been supported by the project PN-II-ID-PCE-2011-3-0137, Romanian Ministery of Education and Research Appendix: Proof of the N-soliton solution ========================================= As we said in the introduction checking the existence of three soliton solution for a bilinear system is enough to prove integrability. However because the structure of the soliton solution is rather complicated we are going to sketch the proof for the N-soliton solution in the case of the first discretisation (for the second one the phase factors and bilinear equations are simpler and everything goes in the same way). We follow the same procedure as in the old papers of Hirota [@h1], [@hs1], [@hirsat], [@hirkdv]. For simplicity lets call $\Delta=\delta(1+\alpha^2)$ and redefine $q_i\to q_i^{\delta}$. Plugging the N-soliton solution (12)-(15) into the first bilinear equation (8) we get $$\sum_{\mu=0,1}\sum_{\mu'=0,1}\left[\prod_{i=1}^N a_i^{\mu_{i}}b_i^{\mu'_i}\left(\prod_{i=1}^N q_i^{\mu_i}-\prod_{i=1}^N q_i^{\mu'_i}\right)- \Delta\prod_{i=1}^N A_i^{\mu_{i}}B_i^{\mu'_i}\left(\prod_{i=1}^N q_i^{\mu_i}p_i^{\mu_i-\mu'_i}- \prod_{i=1}^N q_i^{\mu'_i}p_i^{\mu'_i-\mu_i}\right)\right]\times$$ $$\times\prod_{i<j}M_{ij}^{\mu_i\mu_j+\mu'_i\mu'_j}\prod_{i=1}^N p_i^{(\mu_i+\mu'_i)n}q_i^{(\mu_i+\mu'_i)m}=0$$ Let the coefficient of the factor $\prod_{i=1}^{\nu_0}p_i^nq_i^m\prod_{i=\nu_0+1}^{\nu_1}p_i^{2n}q_i^{2m}$ be $F$. We have $$F=\sum_{\mu,\mu'=0,1}c_{\mu\mu'}\left[\prod_{i=1}^{\nu_1} a_i^{\mu_{i}}b_i^{\mu'_i}\left(\prod_{i=1}^{\nu_0} q_i^{\mu_i}-\prod_{i=1}^{\nu_0} q_i^{\mu'_i}\right)- \Delta\prod_{i=1}^{\nu_1} A_i^{\mu_{i}}B_i^{\mu'_i}\left(\prod_{i=1}^{\nu_0} q_i^{\mu_i}p_i^{\mu_i-\mu'_i}- \prod_{i=1}^{\nu_0} q_i^{\mu'_i}p_i^{\mu'_i-\mu_i}\right)\right]\times$$ $$\times\prod_{i<j}M_{ij}^{\mu_i\mu_j+\mu'_i\mu'_j}$$ where $c_{\mu\mu'}$ implies that the summation over families of indices $\mu$ and $\mu'$ should be done under the requirements: $$\mu_i+\mu'_i=1, i=1,...,\nu_0$$ $$\mu_i=\mu_i'=1, i=\nu_0+1,...,\nu_1$$ $$\mu_i=\mu_i'=0, i=\nu_1+1,...,N$$ Substituting the expressions of dispersion relation and phase factors (16)-(18) and introducing the multiindex $\sigma=\mu-\mu'$ we find that $F=const.\hat{F}$ where $$\hat{F}=\sum_{\sigma=\pm 1}\{\left(\prod_{i=1}^{\nu_0}(1+\epsilon_i\Delta p_i^{\sigma_i})-\prod_{i=1}^{\nu_0}(1+\epsilon_i\Delta p_i^{-\sigma_i})\right)\prod_{i=1}^{\nu_0}\left(i\alpha(-1+\cosh(\sigma_i k_i))-\epsilon_i\sinh(\sigma_i k_i)(1+\Delta)\right)-$$ $$-\Delta\left(\prod_{i=1}^{\nu_0}(1+\epsilon_i\Delta p_i^{\sigma_i})p_i^{\sigma_i}-\prod_{i=1}^{\nu_0}(1+\epsilon_i\Delta p_i^{-\sigma_i})p_i^{-\sigma_i}\right)\prod_{i=1}^{\nu_0}\left(i\alpha\epsilon_i(1-\epsilon_i\cosh(\sigma_i k_i))+\sinh(\sigma_i k_i)(1+\Delta)\right)\}$$ $$\times\prod_{i<j}\left(\epsilon_i\epsilon_j-\cosh(\sigma_i k_i-\sigma_j k_j)\right)^2$$ Making the notation $x_i=(\epsilon_i p_i)^{1/2}$, $z=(1+\Delta+i\alpha)/2, z^*=(1+\Delta-i\alpha)/2$ the above relation becomes proportional to the following expression: $$\hat{F}=\sum_{\sigma=\pm 1}\{\left(\prod_{j=1}^{\nu_0}(1+\Delta x_j^{2\sigma_j})-\prod_{j=1}^{\nu_0}(1+\Delta x_j^{-2\sigma_j})\right)\prod_{j=1}^{\nu_0}(-i\alpha-z^* x_j^{2\sigma_j}+zx_j^{-2\sigma_j})-$$ $$-\Delta\left(\prod_{j=1}^{\nu_0}(1+\Delta x_j^{2\sigma_j})x_j^{2\sigma_j}-\prod_{j=1}^{\nu_0}(1+\Delta x_j^{-2\sigma_j})x_j^{-2\sigma_j}\right)\prod_{j=1}^{\nu_0}(i\alpha+z^* x_j^{2\sigma_j}-zx_j^{-2\sigma_j})\}$$ $$\times\prod_{i<j}(x_i^{\sigma_i}x_j^{-\sigma_j}-x_i^{-\sigma_i}x_j^{\sigma_j})^2$$ Now in order to show that (35) holds we have to prove that $\hat{F}=0$ for any $n=1,...,N$. We shall prove this by induction. First, it is easily seen that this expression has the following properties: i)$\hat{F}(x_1,...,x_n)$ is a symmetric and even function of $x_1,...,x_n$ and invariant under the transformation $x_j\to1/x_j$ for any $j$. ii\) $\hat{F}(x_1,...,x_n)|_{x_1=1}=0$ iii)$\hat{F}(x_1,...,x_n)|_{x_1=x_2}=2(1+\Delta x_1^2)(1+\Delta x_1^{-2})(-i\alpha-z^*x_1^2+zx_1^{-2})(-i\alpha-z^*x_1^{-2}+zx_1^{2})\hat{F}(x_3,...,x_n).$ Now we assume that $\hat{F}=0$ for $n-1$ and $n-2$. From ii) and iii) $\hat{F}(x_1,...,x_n)|_{x_i=1}=0$ and $\hat{F}(x_1,...,x_n)|_{x_i=x_j}=0$ for any $i,j$. Then, based on the property i), Hirota proved [@h1], [@hs1], [@hirkdv] that the expression of $\hat{F}$ can be factored by a function: $$\frac{\prod_{i=1}^n(x_i^2-1)^2\prod_{i<j}^{(n)}(x_i^2-x_j^2)^2(x_i^2x_j^2-1)^2}{\prod_{i=1}^nx_i^2\prod_{i<j}^{(n)}x_i^4x_j^4}$$ which has $2n+2n(n-1)$ zeros of order $2$. Accordingly $\hat{F}$ would have at least $4n^2$ polynomials in the numerator. On the other hand the function: $$\hat{F}(x_1,...,x_n)\prod_{i=1}^nx_i^2\prod_{i<j}^{(n)}x_i^4x_j^4$$ are polynomials of degree $2n^2+6n$ at most. Hence $\hat{F}=0$ for any $n$. In the same way one can prove that the N-soliton solution obeys the other bilinear equations. For instance in the case of the third bilinear equation (10) introducing the expressing of N-soliton solution (12)-(15) we obtain $$\sum_{\mu,\mu'=0,1}\left[\prod_{j=1}^N a_j^{\mu_{j}}b_j^{\mu'_j}\left(\prod_{j=1}^N q_j^{\mu_i}+\prod_{j=1}^N q_j^{\mu'_j}\right)- \prod_{j=1}^N A_j^{\mu_{j}}B_j^{\mu'_j}\left(\prod_{j=1}^N(1+i\alpha) q_j^{\mu_j}p_j^{\mu_j-\mu'_j}+ \prod_{j=1}^N(1-i\alpha) q_j^{\mu'_j}p_j^{\mu'_j-\mu_j}\right)\right]\times$$ $$\times\prod_{i<j}M_{ij}^{\mu_i\mu_j+\mu'_i\mu'_j}\prod_{i=1}^N p_i^{(\mu_i+\mu'_i)n}q_i^{(\mu_i+\mu'_i)m}=0$$ From this expression we obtain by the same steps as above that the coefficient of the factor $\prod_{i=1}^{\nu_0}p_i^nq_i^m\prod_{i=\nu_0+1}^{\nu_1}p_i^{2n}q_i^{2m}$ is proportional to the following expression: $$\hat{K}=\sum_{\sigma=\pm 1}\{\left(\prod_{j=1}^{\nu_0}(1+\Delta x_j^{2\sigma_j})+\prod_{j=1}^{\nu_0}(1+\Delta x_j^{-2\sigma_j})\right)\prod_{j=1}^{\nu_0}(-i\alpha-z^* x_j^{2\sigma_j}+zx_j^{-2\sigma_j})-$$ $$-\left(\prod_{j=1}^{\nu_0}(1+i\alpha)(1+\Delta x_j^{2\sigma_j})x_j^{2\sigma_j}+\prod_{j=1}^{\nu_0}(1-i\alpha)(1+\Delta x_j^{-2\sigma_j})x_j^{-2\sigma_j}\right)\prod_{j=1}^{\nu_0}(i\alpha+z^* x_j^{2\sigma_j}-zx_j^{-2\sigma_j})\}$$ $$\times\prod_{i<j}(x_i^{\sigma_i}x_j^{-\sigma_j}-x_i^{-\sigma_i}x_j^{\sigma_j})^2$$ Immediately one can see that $\hat{K}$ is an invariant, symmetric and even function satisfying the same properties $\hat{K}(x_1,...,x_n)|_{x_1=1}=0$, $\hat{K}(x_1,...,x_n)|_{x_1=x_2}=2(1+\Delta x_1^2)(1+\Delta x_1^{-2})(-i\alpha-z^*x_1^2+zx_1^{-2})(-i\alpha-z^*x_1^{-2}+zx_1^{2})\hat{K}(x_3,...,x_n)$, so the induction method can be applied in the same way. [100]{} S. Tremblay, B. Grammaticos, A. Ramani, Phys. Lett. A 278, 319, (2001) B. Grammaticos, A. Ramani, V. Papageorgiou, Phys. Rev. Lett. 67, 1825, (1991) V. Adler, A. Bobenko, Yu. Suris, Comm. Math. Phys.233, 513, (2003) R.Hirota, [*Discretisation of soliton equations*]{} lecture presented at SIDE III; Symmetries and Integrability of Difference Equations, Sabaudia, Italy, 1998 R. Hirota, Chaos, Solitons and Fractals, 11, 77, (2000) K. Horiuchi,S. Oishi, R. Hirota, T. Tsujimoto, High precision simulation of soliton problems in [*Ultrafast and Ultraparallel Optoelectronics*]{}, Eds. T. Sueta, T. Okoshi, Wiley, New York, (1995) R. Hirota, J. Satsuma, Prog. Theor. Phys. 59, 64,(1976) A. S. Carstea, Phys. Lett. A, 233, 378, (1997) K. Narita, Chaos, Solitons and Fractals, 13, 1121, (2002) E. K. M. Shek, K. W. Chow, Chaos, Solitons and Fractals, 36, 296, (2008) J. Hietarinta, J. Math. Phys. 28, 2094, (1987) J. Hietarinta, J. Math. Phys. 28, 2586, (1987) F. Nijhoff, H. Capel, Acta Appl. Math. 39, 133, (1995) R. Hirota, in [*Bilinear Integrable systems: From classical to quantum and continuous to discrete*]{}, Editors L. Faddeev, P. Van Moerbeke, F. Lambert, NATO Science Series II-Mathematics Physics and Chemistry, vol. 201, pag.113-122, (2006) R. Hirota, J. Phys. Soc. Japan, 35, 289, (1973) R. Hirota, J. Satsuma, J. Phys. Soc. Japan, 40, 891, (1976) R. Hirota, J. Phys. Soc. Japan, 43, 1424, (1977) Yu. B. Suris, Rev. Math. Phys. 11, 727, (1999) [^1]: Corresponding author:carstea@gmail.com
--- abstract: 'Understanding of the parton distributions at small $x(g)$ is one of the most important issues for clarfiying of the QCD basics. In this paper potential of the LHeC for probing small $x(g)$ region via $c\bar{c}$ and $b\bar{b}$ production has been investigated. Comparison of the $ep$ and real $\gamma p$ options of the LHeC clearly show the advantage of $\gamma p$ collider option. Measurement of $x(g)$ down to $3\times10^{-6}$ with high statistics, especially at $\gamma p$ option, seems to be reachable which is two order smaller than HERA coverage.' author: - | U. Kaya$^{1}$, S. Sultansoy$^{1,2}$, G. Unel$^{3}$\ *$^{1}$TOBB University of Economics and Technology, Ankara, Turkey*\ *$^{2}$ANAS Institute of Physics, Baku, Azerbaijan*\ *$^{3}$UC Irvine,USA* date: '$\,$' title: 'Probing small $x(g)$ region with the LHeC based $\gamma p$ colliders' --- Introduction ============ The problem of precise measurement of parton distribution functions (PDF) is yet to be solved for the energy scales relevant to the LHC results. On the other hand, precison knowledge on the parton distribution of small $x_{B}$ and sufficiently large $Q^{2}$ is crucial for enlightening the QCD basics at all levels, from partons to nuclei. Besides, with the recent discovery of the 125 GeV scalar particle [@ATLAS; @Higgs], [@CMS; @Higgs] at the LHC, the basic components of the electroweak part of the Standard Model (SM) have been completed. Hovewer, the Higgs mechanism provides less than 2% of mass of the visible universe. Remaining 98% are provided by the QCD part of the SM. Therefore, clarifying of the basics of the QCD is important for a better understanding of our universe. That’s why the QCD explorer was proposed ten years ago (see review [@QCD; @2004] and references therein). One of the required measurements is the gluon PDF for low momentum fraction: small $x(g)$. The last machine that has probed $x(g)$ was HERA which had a reach of about $x(g)>$$10{}^{-4}$. Large Hadron-Electron Collider (LHeC) project [@J.; @Phys.; @G:; @Nucl.; @Part.; @Phys.; @39; @(2012); @075001] - the most powerful microscope ever designed - will provide a unique opportunity to probe extremely small $x(g)$ region. In this project, where proton-electron collisions are aimed the $e-beam$ can be obtained from a new circular or linear machine. Today, LR option is considered as the basic one for the LHeC [@arXiv:1211.4831v1; @[hep-ex]; @20; @Nov; @2012]. Actually this decision was almost obvious from the beginning due to the complications in constructing by-pass tunnels around the existing experimental caverns and installing the $e-ring$ in the already commissioned tunnel. Let us remind that the CDR stage of the LHC assumed also $ep$ collisions using the already existed LEP ring; but it turned out that LHC installation required dismantling of LEP from the tunnel. Within the linac-ring option of the LHeC, a proton beam from LHC can be hit with a high energy electron or photon beam. The photons may be virtual ones from the electron beam resulting in a typical DIS event or these can be real photons originating from the Compton Back Scattering process. In the latter case, the photon spectrum consists of the high energy photons peaking at about 80% of the electron beam energy on the continuum of Weizsacker-Williams photons. The present study aims to investigate the feasibility of a small $x(g)$ measurement with such a machine. Main parameters of $ep$ and $\gamma p$ options of the LHeC are presented in section 2. Section 3 is devoted to investigation of small $x(g)$ region using the processes $\gamma p$ $\rightarrow$ $c\bar{{c}}X$ and $\gamma p$ $\rightarrow$ $b\bar{{b}}X$. The generator level results are obtained using CompHEP [@CompHep] software package. Comparison with processes $ep$ $\rightarrow$ $ec\bar{{c}}X$ and $ep$ $\rightarrow$ $eb\bar{{b}}X$ shows an obvious advantage of the LHeC $\gamma p$ option, which will provide more than one order higher cross sections at small $x(g)$ region comparing to the $ep$ option. Finally, section 4, provides a summary of the conclusion together with some suggestions. Main parameters of $ep$ and $\gamma p$ options of the LHeC ========================================================== It should be emphasized that real $\gamma p$ collisions can be achieved only on the base of linac ring type $ep$ colliders (see review [@A.; @N.; @Akay] for history and status of linac-ring type collider proposals). Real $\gamma$ beam for $\gamma p$ collider [@S.; @I.; @Alekhin], [@A.; @K.; @Ciftci; @et; @al.], [@TESLA; @*; @HERA; @based], [@Conversion; @efficiency] will be produced using the Compton back scattering of laser beam off the high energy electron beam [@I.; @F.; @Ginzburg], [@Principles; @of; @photon; @colliders]. Possible application of this mechanism to the other LHeC option under consideration, namely to ring-ring type $ep$ colliders results in negligible $\gamma p$ luminosities, $L_{\gamma p}$$<10{}^{-7}$$L{}_{ep}$. Currently, two versions for the $ep$ option of the LHeC are under consideration: multi-pass energy recovery linac (ERL) yielding $L_{ep}=$$10^{33}$$cm^{-2}$$s{}^{-1}$ and pulsed single pass linac yielding $L{}_{ep}=$$10{}^{32}$$cm{}^{-2}$$s{}^{-1}$. In the first case, $E{}_{e}=60\, GeV$ has been chosen as a base electron energy, since higher energies are not available because of the synchrotron radiation loss in the arcs. In the second case, beam energies above $140\, GeV$ would be available [@J.; @Phys.; @G:; @Nucl.; @Part.; @Phys.; @39; @(2012); @075001]. These two options will be denoted as LHeC-1 and LHeC-2. Main parameters of the LHeC $ep$ collisions, in different options are presented in Table 1. $E_{e},\, GeV$ $E_{p},\, TeV$ $\sqrt{s},\, TeV$ $L,\, cm^{-2}$$s{}^{-1}$ -------- ---------------- ---------------- ------------------- -------------------------- ERL $60$ $7$ $1.30$ $10^{33}$ LHeC-1 $60$ $7$ $1.30$ $9\times$$10^{31}$ LHeC-2 $140$ $7$ $1.98$ $4\times10^{31}$ : Main parameters of ep collisions. \[TABLE1\] In the $\gamma p$ option the luminosity of $\gamma p$ collisions will be similar to the luminosity of $ep$ collisions for the pulsed single straight linac. In the ERL case, L$_{\gamma p}$ will be 10 times lower than L$_{ep}$ as the energy recovery does not work after Compton back scattering. Inclusive processes yielding $c\bar{c}$ and $b\bar{b}$ final states at LHeC =========================================================================== The final states that can be easily distinguished from the background events and that would give a good measure of the $x(g)$ are $eg\to eq\bar{q}$ and/or $\gamma g\to q\bar{q}$ where the gluon ($g$) is from the LHC protons, electrons and photons are from a new accelerator (namely, an electron linac providing beams tangential to the LHC) to be build and the letter $q$ stands for a heavy quark flavour, such as $b$ quark and possibly $c$ as well. The $b$ quark final states are easier to identify due to $b$-tagging possibility using currently available technologies: for example, ATLAS silicon detectors have about 70% $b$-tagging efficiency. In Table \[TABLE2\] we present the cross sections for heavy quark pair production via DIS, quasi real photons (WWA) and Compton Back Scattering (CBS) photons at the LHeC with $E{}_{e}=60\, GeV$ and $E{}_{e}=140\, GeV$. For comparison, we also give values for DIS and WWA processes at HERA. It is seen that WWA quasi real photons are advantageous comparing to DIS and CBS photons are advantageous comparing to WWA. All numerical calculations are performed using CompHep [@CompHep] with CTEQ6L1 [@CTEQ] PDF distributions. In Figure \[FIGURE1\], the differential cross section depending on the $x(g)$ has been shown for WWA photons at HERA and at LHeC. As expected, LHeC will give opportunity to investigate an order smaller $x(g)$ than HERA. ------------------------------ -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- Machine DIS WWA CBS DIS WWA CBS HERA $6.07\times10^{2}$ $4.57\times10^{3}$ - $4.66\times10^{4}$ $7.29\times10^{5}$ - LHeC-1($E{}_{e}=60$$\, GeV$) $4.26\times10^{3}$ $2.99\times10^{4}$ $2.41\times10^{5}$ $2.38\times10^{5}$ $3.44\times10^{6}$ $2.38\times10^{7}$ LHeC-2($E{}_{e}=140\, GeV$) $7.07\times10^{3}$ $4.91\times10^{4}$ $3.70\times10^{5}$ $3.72\times10^{5}$ $5.27\times10^{6}$ $3.46\times10^{7}$ ------------------------------ -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- : Heavy quark pair production cross sections via DIS, WWA, and CBS mechanisms. \[TABLE2\] The advantage of the CBS photons becomes even more obvious if one analyzes $x(g)$ distribution of differential cross sections for CBS, WWA and DIS. In Figure \[FIGURE2\], we show the $d\sigma/$$dx(g)$ at the LHeC-1 for $c\bar{c}$ production. It is seen that CBS at small $x(g)$ region provides more than one (two) order higher cross sections comparing to WWA (DIS). For example, differential cross section of $c\bar{c}$ pair production at the LHeC-1 achieves maximum value $94\,\mu b$ at $x(g)=1.44\times10^{-5}$ for CBS, whereas maximum value for WWA and DIS are $4\,\mu b$ at $x(g)=$$1.54\times10^{-5}$and $0.15\,\mu b$ at $x(g)=$$3.89\times10^{-5}$, respectively. Similar distributions for $b\bar{b}$ at LHeC-1, $c\bar{c}$ at LHeC 2 and $b\bar{b}$ at LHeC-2 are shown in Figures \[FIGURE3\], \[FIGURE4\] and \[FIGURE5\], respectively. Maximum values of differential cross sections and corresponding $x(g)$ values for DIS, WWA, and CBS at the LHeC-1 (2) are given in the Table \[TABLE 3\] (\[TABLE 4\]). The advantage of CBS due to large cross section is obvious from the comparison. ----- ---------------- --------------------- ---------------- --------------------- $d\sigma/$$dx$ $x$ $d\sigma/$$dx$ $x$ DIS $0.15\,\mu b$ $3.89\times10^{-5}$ $0.47\, nb$ $1.99\times10^{-4}$ WWA $4.0\,\mu b$ $1.54\times10^{-5}$ $5.02\, nb$ $1.25\times10^{-4}$ CBS $94\,\mu b$ $1.44\times10^{-5}$ $117\, nb$ $1.23\times10^{-4}$ ----- ---------------- --------------------- ---------------- --------------------- : Maximum values of differential cross sections and corresponding $x(g)$ values for DIS, WWA, and CBS at the LHeC-1. \[TABLE 3\] ----- ---------------- --------------------- ---------------- --------------------- $d\sigma/$$dx$ $x$ $d\sigma/$$dx$ $x$ DIS $0.44\,\mu b$ $1.54\times10^{-5}$ $1.73\, nb$ $9.12\times10^{-5}$ WWA $13.2\,\mu b$ $6.45\times10^{-6}$ $17\, nb$ $5.88\times10^{-5}$ CBS $312\,\mu b$ $6.02\times10^{-6}$ $408\, nb$ $5.01\times10^{-5}$ ----- ---------------- --------------------- ---------------- --------------------- : Maximum values of differential cross sections and corresponding $x(g)$ values for DIS, WWA, and CBS at the LHeC-2. \[TABLE 4\] ![The x(g) reach and differential cross sections in $c\bar{{c}}$ (left) and $b\bar{{b}}$ (right) final states for the HERA and the LHeC. []{data-label="FIGURE1"}](Figure1_Left "fig:"){width="0.4\paperwidth"}![The x(g) reach and differential cross sections in $c\bar{{c}}$ (left) and $b\bar{{b}}$ (right) final states for the HERA and the LHeC. []{data-label="FIGURE1"}](Figure1_Right "fig:"){width="0.4\paperwidth"} ![Differential cross sections for $c\bar{{c}}$ final states produced via CBS, WWA and DIS at the LHeC-1.[]{data-label="FIGURE2"}](Figure2_Left "fig:"){width="0.4\paperwidth"}![Differential cross sections for $c\bar{{c}}$ final states produced via CBS, WWA and DIS at the LHeC-1.[]{data-label="FIGURE2"}](Figure2_Right "fig:"){width="0.4\paperwidth"} ![Differential cross sections for $b\bar{{b}}$ final states produced via CBS, WWA and DIS at the LHeC-1.[]{data-label="FIGURE3"}](Figure3_Left "fig:"){width="0.4\paperwidth"}![Differential cross sections for $b\bar{{b}}$ final states produced via CBS, WWA and DIS at the LHeC-1.[]{data-label="FIGURE3"}](Figure3_Right "fig:"){width="0.4\paperwidth"} ![Differential cross sections for $c\bar{{c}}$ final states produced via CBS, WWA and DIS at the LHeC-2.[]{data-label="FIGURE4"}](Figure4_Left "fig:"){width="0.4\paperwidth"}![Differential cross sections for $c\bar{{c}}$ final states produced via CBS, WWA and DIS at the LHeC-2.[]{data-label="FIGURE4"}](Figure4_Right "fig:"){width="0.4\paperwidth"} ![Differential cross sections for $b\bar{{b}}$ final states produced via CBS, WWA and DIS at the LHeC-2 .[]{data-label="FIGURE5"}](Figure5_Left "fig:"){width="0.4\paperwidth"}![Differential cross sections for $b\bar{{b}}$ final states produced via CBS, WWA and DIS at the LHeC-2 .[]{data-label="FIGURE5"}](Figure5_Right "fig:"){width="0.4\paperwidth"} The angular dependency of the relevant processes is important to estimate the necessary $\eta$ coverage of the detector to be built and also to estimate the eventual electron machine selection. For illustration we consider $d\sigma/d\theta$ distribution where $\theta$ is the angle between $c$ $(b)$ quark and proton beam direction. These distributions for CBS at the LHeC-1 and LHeC-2 are presented in Figures \[FIGURE6\] and \[FIGURE7\], respectively. In Table \[TABLE 5\], we present reachable $x(g)$ for different $\theta$ coverage. One can notice that even for an angular loss of about 5 degrees, there is considerable drop in both the cross section and in the $x(g)$ reach. This effect can be understood by considering the $\eta$ dependence of the heavy quark pair production cross section in $\gamma p$ collisions which are shown in Figure \[Fig: eta\_dependency\_gp\] and \[FIGURE9\]. The vertical solid line is representative for a 1 degree, the dashed line for a 5 degree and the dot-dashed line is for 10 degree detector. Therefore, in order to have a good experimental reach the tracking should have an $\eta$ coverage up to 5. ---------- --------------------- --------------------- --------------------- --------------------- $\theta$ $c\bar{c}$ $b\bar{b}$ $c\bar{c}$ $b\bar{b}$ $0-180$ $7.94\times10^{-6}$ $6.91\times10^{-5}$ $3.16\times10^{-6}$ $3.02\times10^{-5}$ $1-179$ $8.31\times10^{-6}$ $6.91\times10^{-5}$ $3.36\times10^{-6}$ $4.36\times10^{-5}$ $5-175$ $1.44\times10^{-5}$ $7.94\times10^{-5}$ $1.20\times10^{-5}$ $4.78\times10^{-5}$ $10-170$ $2.39\times10^{-5}$ $1.00\times10^{-4}$ $2.28\times10^{-5}$ $7.58\times10^{-5}$ ---------- --------------------- --------------------- --------------------- --------------------- : Reachable $x(g)$ for different $\theta$ coverage. \[TABLE 5\] ![The effect of angular reach for $c\bar{c}$ (left) and $b\bar{b}$ (right) final states produced via CBS at the LHeC-1.[]{data-label="FIGURE6"}](Figure6_Left "fig:"){width="0.4\paperwidth"}![The effect of angular reach for $c\bar{c}$ (left) and $b\bar{b}$ (right) final states produced via CBS at the LHeC-1.[]{data-label="FIGURE6"}](Figure6_Right "fig:"){width="0.4\paperwidth"} ![The effect of angular reach for $c\bar{c}$ (left) and $b\bar{b}$ (right) final states produced via CBS at the LHeC-2.[]{data-label="FIGURE7"}](Figure7_Left "fig:"){width="0.4\paperwidth"}![The effect of angular reach for $c\bar{c}$ (left) and $b\bar{b}$ (right) final states produced via CBS at the LHeC-2.[]{data-label="FIGURE7"}](Figure7_Right "fig:"){width="0.4\paperwidth"} ![The $\eta$ dependency of the $c\bar{c}$ (left) and $b\bar{b}$ (right) production cross section via CBS at the LHeC-1. Vertical lines represent $1{}^{o}$ (solid line), $5{}^{o}$ (dashed line) and $10{}^{o}$ (dot-dashed line) detector cuts.[]{data-label="Fig: eta_dependency_gp"}](Figure8_Left "fig:"){width="0.4\paperwidth"}![The $\eta$ dependency of the $c\bar{c}$ (left) and $b\bar{b}$ (right) production cross section via CBS at the LHeC-1. Vertical lines represent $1{}^{o}$ (solid line), $5{}^{o}$ (dashed line) and $10{}^{o}$ (dot-dashed line) detector cuts.[]{data-label="Fig: eta_dependency_gp"}](Figure8_Right "fig:"){width="0.4\paperwidth"} ![The $\eta$ dependency of the $c\bar{c}$ (left) and $b\bar{b}$ (right) production cross section via CBS at the LHeC-2. Vertical lines are same as in Fig. 8.[]{data-label="FIGURE9"}](Figure9_Left "fig:"){width="0.4\paperwidth"}![The $\eta$ dependency of the $c\bar{c}$ (left) and $b\bar{b}$ (right) production cross section via CBS at the LHeC-2. Vertical lines are same as in Fig. 8.[]{data-label="FIGURE9"}](Figure9_Right "fig:"){width="0.4\paperwidth"} Conclusions =========== Measurements of $x(g)$ down to $3\times10^{-6}$ seem to be reachable in $\gamma p$ collisions which is two order smaller than the HERA coverage. These collisions provide higher cross section and better $x(g)$ reach with respect to the $ep$ collisions with the same electron beam energy. For the low $x(g)$ region, the enhancement factor compared to the DIS $ep$ collisions is about 700 for $c\bar{c}$ final states and about 230 for $b\bar{b}$ final states at the LHeC-2. Therefore, for the final states with heavy quarks, even if the $\gamma p$ luminosity is 10 times smaller than $ep$ luminosity (ERL option), the expected number of events in $\gamma p$ collisions would be 70 and 20 times higher than in $ep$ collisions for $c\bar{c}$ and $b\bar{b}$ final state respectively. The enhancement factor compared to WWA $ep$ collisions is about 24 for both final states. The angular sensitivity is very important for smallest $x(g)$ reach for either $e$ or $\gamma$ beams, therefore a detector with a pseudorapidity coverage up to $\eta=5$ is required. This coverage is already achieved at the ATLAS and CMS experiments using forward detector components. Finally, $ep$ option of LHeC will give an opportunity to shed light on the small $x(g)$ dynamics which is crucial for clarifying the QCD basics. On the other hand, the $\gamma p$ option of LHeC will essentially enlarge the LHeC capacity on the subject. Therefore, one pulse linac should be considered as a baseline for LHeC design. In this case, a higher center of mass energies can be achieved by lengthening of the electron linac which will provide an opportunity to investigate smaller $x(g)$ region. 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Aksakal et al., *Conversion efficiency and luminosity for* $\gamma-p$ *colliders based on the LHC-CLIC or LHC-ILC QCD explorer scheme*, Nucl. Instrum. Methods Phys. Res., Sect. A 576, 287 (2007). I. F. Ginzburg et al., *Colliding gamma e and gamma gamma Beams Based on the Single Pass e+ e- Accelerators. 2. Polarization Effects. Monochromatization Improvement*, Nucl. Instrum. Methods Phys. Res., Sect. A 219, 5 (1984). V. I. Telnov, *Principles of photon colliders*, Nucl. Instrum. Methods Phys. Res., Sect. A 355, 3 (1995). J. Pumplin et al., *New generation of parton distributions with uncertainties from global QCD analysis*, JHEP 0207 (2002) 012. V. Litvinenko, *LHeC with \~100% energy recovery linac*, 2nd CERN-ECFA-NuPECC workshop on LHeC, Divonne-les-Bains 1-3 Sep (2009).
--- abstract: 'Ensemble averages of a stochastic model show that, after a formation stage, the tips of active blood vessels in an angiogenic network form a moving two dimensional stable diffusive soliton, which advances toward sources of growth factor. Here we use methods of multiple scales to find the diffusive soliton as a solution of a deterministic equation for the mean density of active endothelial cells tips. We characterize the diffusive soliton shape in a general geometry, and find that its vector velocity and the trajectory of its center of mass along curvilinear coordinates solve appropriate collective coordinate equations. The vessel tip density predicted by the soliton compares well with that obtained by ensemble averages of simulations of the stochastic model.' author: - | L L Bonilla$^*$, M Carretero and F Terragni\ Gregorio Millán Institute,\ Fluid Dynamics, Nanoscience and Industrial Mathematics,\ and Department of Mathematics,\ Universidad Carlos III de Madrid, 28911 Leganés, Spain\ $^{*}$Corresponding author. E-mail: bonilla@ing.uc3m.es title: Two dimensional soliton in tumor induced angiogenesis --- [*Keywords*]{}: Two dimensional diffusive soliton, noise models, pattern formation, nonlinear dynamics, systems biology, tumor induced angiogenesis, chemotaxis Introduction {#sec:intro} ============ Angiogenesis is the growth of blood vessels out of a primary vessel, a complex multiscale process that determines organ growth and regeneration, tissue repair, wound healing and many other natural operations [@car05; @CT05; @GG05; @fru07; @fig08; @car11; @ton00]. Its imbalance contributes to numerous malignant, inflammatory, ischaemic, infectious, and immune diseases [@car05], such as cancer [@fol71; @fol74; @lio77; @fol06; @lia07; @zua18], rheumatoid arthritis [@mar06], neovascular age-related macular degeneration [@jag08; @niv14], endometriosis [@tay09], and diabetes [@mar03]. Normal angiogenesis proceeds as follows. During inflammation or under hypoxia, cells may activate signaling pathways that lead to secretion of pro-angiogenic proteins, such as Vessel Endothelial Growth Factor (VEGF). VEGF diffuses in the tissue, binds to extracellular matrix (ECM) components and forms a spatial concentration gradient in the direction of hypoxia. When VEGF molecules reach an existing blood vessel, they promote diminishing adhesion of its cells and the growth of new vessel sprouts. VEGF also activates the tip cell phenotype in endothelial cells (ECs) of the vessel, which then grow filopodia with many VEGF receptors. The tip cells pull the other ECs, open a pathway in the ECM, lead the new sprouts, and migrate in the direction of increasing VEGF concentration [@ger03]. [*Branching of new sprouts*]{} occurs as a result of signaling and mechanical cues between neighboring ECs [@hel07; @jol15; @veg20]. ECs in growing sprouts alter their shape to form a lumen connected to the initial vessel that is capable of carrying blood [@geb16]. Sprouts meet and merge in a process called [*anastomosis*]{} to improve blood circulation inside the new vessels. Poorly perfused vessels may become thinner and their ECs, in a process that inverts angiogenesis, may retract to neighboring vessels leading to a more robust blood circulation [@fra15]. Thus, the vascular plexus remodels into a highly organized and hierarchical network of larger vessels ramifying into smaller ones [@szy18]. In normal processes of wound healing or organ growth, the cells inhibit the production of growth factors when the process is finished. The previous picture changes in significant ways in pathological angiogenesis. Hypoxic cells of an incipient tumor experience lack of oxygen and nutrients. Then they produce VEGF that induces angiogenesis, and new vessel sprouts exit from a nearby primary vessel and move in the tumor direction [@fol74; @car05; @fol06]. Blood brings oxygen and nutrients that foster tumor growth. Instead of inhibiting production of growth factors, tumor cells continue secreting growth factors that attract more vessel sprouts and facilitate tumor expansion. Together with experiments, there are many models spanning from the cellular to macroscopic scales that try to understand angiogenesis [@lio77; @veg20; @sto91; @sto91b; @cha93; @cha95; @orm97; @and108; @ton01; @lev01; @pla03; @man04; @sun05a; @sun05; @cha06; @ste06; @bau07; @cap09; @owe09; @jac10; @das10; @swa11; @sci11; @sci13; @cot14; @dej14; @ben14; @bon14; @hec15; @per17; @pil17; @bon19]. Early models consider reaction-diffusion equations for densities of cells and chemicals (growth factors, fibronectin, etc.) [@lio77; @cha93; @cha95]. They cannot treat the growth and evolution of individual blood vessels. Tip cell stochastic models of tumor induced angiogenesis are among the simplest ones for this complex multiscale process. Their basic assumptions are that (i) the cells of a blood sprout tip do not proliferate and move towards the tumor producing growth factor, and (ii) proliferating stalk cells build the sprout along the trajectory of the sprout tip. Thus tip cell models are based on the motion of single particles representing the tip cells and their trajectories constitute the advancing blood vessels network [@sto91; @sto91b; @and108; @pla03; @man04; @cap09; @bon14; @hec15; @bon19; @ter16]. Tip cell models describe angiogenesis over distances that are large compared with a cell size, thereby renouncing to detailed descriptions of cellular processes. More complex models include tip and stalk cell dynamics, the motion of tip and stalk cells on the extracellular matrix outside blood vessels, signaling pathways and EC phenotype selection, blood circulation in newly formed vessels, and so on [@bau07; @jac10; @ben14; @hec15; @ber18; @veg20]. ![Angiogenic network generated by the stochastic process at (a) 24 hr and (b) 36 hr after sprouts issue from the primary blood vessel at $x = 0$. The level curves of the TAF density $C(t,\mathbf{x})$ are also depicted, showing a tumor located vertically at $x=L$ above the $x$-axis. \[fig1\]](figure1.pdf){width="14cm"} -6mm Previously, we have derived a deterministic description from a simple two dimensional (2D) tip cell model of tumor induced angiogenesis [@bon14]. This model considers tip cells subject to chemotactic, friction and white noise forces, and to random branching. When a moving sprout tip meets an existing sprout or a blood vessel, it fuses with it and stops moving, which is a simple model of anastomosis. A slightly more complicated earlier model by Capasso and Morale also includes haptotaxis [@cap09]. Fig. \[fig1\] shows two snapshots of a realization of the stochastic process. Our deterministic description [@bon14] consists of an integropartial differential equation for the density of active tip cells coupled to a reaction-diffusion equation for the tumor angiogenic factor (TAF), which is representative of VEGF and other relevant growth factors. Branching and anastomosis processes appear as source and sink terms in the equation for the density of active tip cells. The tip density and other mesoscopic quantities are ensemble averages over replicas of the stochastic process [@ter16]. A similar equation for the tip density on $\mathbb{R}^D$ ($D=2,3$) can be rigorously derived from the stochastic equations in the limit as the initial number of tips goes to infinity [@cap19]. However, when we consider the more realistic situation of angiogenesis issuing from a primary vessel and advancing in a bounded region, the number of tips is finite and limited by anastomosis. In this situation, the derivation of the deterministic equation from ensemble averages of the stochastic process makes more physical sense [@ter16], although a rigorous proof of its validity seems more difficult. Analysis and numerical solutions of both deterministic and stochastic descriptions of angiogenesis show that the tip density advances from primary blood vessel to tumor as a moving lump, which is a [*two dimensional diffusive soliton*]{} (2DDS). The 2DDS is a solution of a simplified version of the integropartial equation for the density of active tips moving on $\mathbb{R}^2$ and with constant TAF concentration. In simple one dimensional (1D) geometries, the longitudinal profile of the moving lump is a one dimensional diffusive soliton (1DDS) [@bon16]. Diffusive solitons are stable solitary waves of dissipative systems which, unlike true solitons, do not emerge unchanged from collisions [@rem99]. How does the soliton picture apply to angiogenesis starting from a blood vessel and advancing toward a tumor? The distance between the primary vessel and the tumor has to be sufficiently large, for otherwise the 2DDS does not have space and time to form. Provided the distance is large enough, there are three stages for the motion of active tips. First, the active tips proliferate through branching until the 2DDS forms. Secondly, the 2DDS advances far from primary vessel and tumor. This second stage can be approximated by the 2DDS solution obtained for the case of infinite space and constant TAF concentration if the latter changes slowly. The velocity and shape of the soliton vary slowly to accommodate the changes in TAF concentration and are determined by collective coordinate equations (CCEs) [@bon16; @bon16pre; @bon17]. The last stage describes how the angiogenic network reaches the tumor. The main result of this work is finding and validating an approximate description and CCEs for the 2DDS in the general case. Note that angiogenesis is a biological process very far from equilibrium. However, numerical simulations show that when the density of active vessel tips is far from the boundaries (primary vessel and tumor), it evolves rapidly to a particular pattern, the 2DDS or [*angiton*]{}. The latter is a stable uniformly moving solution for a Fokker-Planck equation with source terms and constant TAF concentration in the spatially unbounded case. It plays the same role as the thermal equilibrium solution for the Fokker-Planck equation describing a system with detailed balance. The 2DDS is characterized by a few parameters (velocity, height). For a finite geometry and slowly-varying TAF concentration, the 2DDS parameters change slowly to accommodate the angiton motion towards the boundaries and the varying TAF concentration. The description of the second stage of angiogenesis is far from obvious. The first step is to reduce the 2D equation for the marginal tip density to a 1D equation. For equations deriving from a variational principle, such as the Gross-Pitaevskii equation for a cigar shaped Bose condensate, an appropriate Ansatz is a Gaussian function of the transversal coordinate times a function of the axial coordinate [@sal02]. Inserting this Ansatz into the action of the variational principle produces an equation for the longitudinal function and an equation for the width of the Gaussian. When the latter is solved in terms of the longitudinal function, it yields an effective 1D nonlinear equation for it [@sal02]. The deterministic equation governing tumor induced angiogenesis does not derive from a variational principle, and, therefore, we cannot use the same ideas to reduce it to a 1D equation. For a general configuration, the 2DDS does not move on a straight line and, therefore, we have to use curvilinear coordinates to characterize both the 2DDS and the trajectory of its center of mass. The longitudinal coordinate is directed along the instantaneous velocity of the 2DDS and the transversal coordinate is measured perpendicularly to the velocity. By using the method of multiple scales with a fast transversal length scale that characterizes the 2DDS width, (i) we show that the marginal tip density has Gaussian shape (with a small variance), and (ii) derive an averaged 1D equation for the 2DDS longitudinal profile. The latter equation is the same as derived earlier for a simple 1D geometry [@bon16pre] except for a renormalized anastomosis coefficient and motion over the longitudinal curvilinear coordinate. The solution of the 1D is a 1DDS. While the soliton is a stable traveling wave moving rigidly on the infinite the real line, the slow evolution of the TAF concentration and the influence of boundary conditions change the 2DDS shape and velocity. We derive equations for collective coordinates of the 2DDS that include the magnitude and orientation of the soliton velocity, its shape parameter, and the location of its center of mass. The 2DDS rapidly adjusts its shape to the instantaneous values of the collective coordinates. From the numerical solutions of the CCEs, we can reconstruct the 2DDS, which then yields the marginal density of active tips. Comparison with the marginal density obtained from ensemble averages shows that the 2DDS provides a good approximation for the stochastic description of the tip cell model. Possible future applications of the present work to biology include investigating control of the 2DDS motion, e.g. by studying the effect of antiangiogenic drugs on its dynamics; cf Ref. [@lev01] for modifications of angiogenesis models due to angiostatin. The rest of the paper is as follows. We recall the reduced integropartial differential equation for the marginal density of active vessel tips [@ter16] in Section \[sec:model\]. To describe the lump of active tips moving toward the tumor, we assume that it is initially a Gaussian function of the transversal coordinate, check that it continues evolving as a Gaussian, derive an equation for its longitudinal part and find a one dimensional soliton as an approximate solution in Section \[sec:2DDS\]. The analytical formula for the soliton is analogous to that found in [@bon16; @bon16pre]. Section \[sec:cc\] contains a derivation of the differential equations for the collective coordinates of the 2DDS. The coefficients of the CCEs are spatial averages over the TAF density. The width of narrow 2DDSs is time independent and only three collective coordinates are needed to describe them. In Section \[sec:numerical\], we explain how to calculate the coefficients in the CCEs, solve them numerically, reconstruct the 2DDS and, through it, the marginal vessel tip density. We compare it with ensemble averages of the stochastic process. It turns out that the location of the 2DDS peak approximately gives the location of the maximum of the marginal density for any realization of the stochastic process. Thus, the 2DDS roughly describes the advancing vessel network for each realization although different realizations provide different looking vessel networks. The conclusions of this work appear in Section \[sec:conclusions\]. The Appendices are devoted to technical matters. Angiogenesis model {#sec:model} ================== Stochastic model ---------------- Early stages of tumor induced angiogenesis are described by a simple stochastic model of motion, creation and destruction of active tips [@bon14; @ter16]. The model comprises a system of Langevin-Ito equations for the extension of vessel tips, branching of tips (a birth process) and anastomosis (an annihilation process when the tips merge with existing vessels and cease to be active). In the model, the growing vessels are the trajectories of active vessel tips, whose motion obeys the following Langevin-Ito equations $$\begin{aligned} d\mathbf{X}^i(t)&=&\mathbf{v}^i(t)\, dt,\nonumber\\ d\mathbf{v}^i(t)&=& \!\left[- \mathbf{v}^i(t)+\mathbf{F}\!\left(C(t,\mathbf{X}^i(t))\right)\right]\!\beta dt + %\nonumber\\&+& \sqrt{\beta}\, d\mathbf{W}^i(t), \label{eq1}\end{aligned}$$ where the $\mathbf{X}^i(t)$ and $\mathbf{v}^i(t)$ are the position and velocity of the $i$th tip at time $t$, respectively, the $\mathbf{W}^i(t)$ are independent identically distributed (i.i.d.) standard Brownian motions, and $$\begin{aligned} \mathbf{F}(C)= \frac{\delta}{\beta}\,\frac{\nabla_x C(t,\mathbf{x})}{[1+\Gamma_1C(t,\mathbf{x})]^q}= \frac{\delta\nabla_x[1+\Gamma_1C(t,\mathbf{x})]^{1-q}}{(1-q)\Gamma_1\beta}. \quad \label{eq2}\end{aligned}$$ The equation for the TAF density $C(t,\mathbf{x})$ is $$\begin{aligned} \frac{\partial}{\partial t}C(t,\mathbf{x})\!&\!=\!& \! \kappa \Delta_x C(t,\mathbf{x}) -\chi C(t,\mathbf{x}) %\nonumber \\&\times& \sum_{i=1}^{N(t)} |\mathbf{v}^i(t)|\, \delta_{\sigma_x}(\mathbf{x}-\mathbf{X}^i(t)), \label{eq3}\end{aligned}$$ where $N(t)$ is the number of active tips at time $t$ and $\delta_{\sigma_x}$ are regularized delta functions: $$\begin{aligned} \delta_{\sigma_x}(\mathbf{x})= \frac{e^{-x^2/\sigma_x^2}\, e^{-y^2/\sigma_y^2}}{\pi\sigma_x \sigma_y}. \label{eq4}\end{aligned}$$ The $i$th active tip branches and fuses (anastomoses) with another tip or vessel at random times $T^i$ and $\Theta^i$, respectively. At time $T^i$ a new active tip is created, whereas at time $\Theta^i$ the $i$th tip merges with a vessel or reaches the tumor, thereby ceasing to be active and counting as such. We ignore branching from mature vessels or from existing vessel sprouts; see [@cap19]. The probability that a tip branches from one of the existing ones during an infinitesimal time interval $(t, t + dt]$ is proportional to $\sum_{i=1}^{N(t)}\alpha(C(t,\mathbf{X}^i(t)))dt$, where $\alpha(C)$ is given by $$\alpha(C)=\frac{A\, C}{1+C},\quad A>0. \label{eq5}$$ At time $T^i$, the velocity of the new tip that branches from tip $i$ is selected out of a normal distribution, $\delta_{\sigma_v}(\mathbf{v}-\mathbf{v}_0)$, with mean $\mathbf{v}_0$ and a narrow variance $\sigma_v^2$. The regularized delta function $\delta_{\sigma_v}(\mathbf{x})$ is given by Eq.  with $\sigma_x=\sigma_y=\sigma_v$. The change per unit time of the number of tips in boxes $d\mathbf{x}$ and $d\mathbf{v}$ about $\mathbf{x}$ and $\mathbf{v}$ is $$\begin{aligned} \nonumber &&\sum_{i=1}^{N(t)}\alpha(C(t,\mathbf{X}^i(t)))\, \delta_{\sigma_v}(\mathbf{v}^i(t)-\mathbf{v}_0)=\int_{d\mathbf{x}}\int_{d\mathbf{v}}\alpha(C(t,\mathbf{x)})\\ &&\times \delta_{\sigma_v}(\mathbf{v}-\mathbf{v}_0)\sum_{i=1}^{N(t)}\delta(\mathbf{x}-\mathbf{X}^i(t))\delta(\mathbf{v}-\mathbf{v}^i(t)) d\mathbf{x} d\mathbf{v}. \label{eq6}\end{aligned}$$ Representative values of all involved dimensionless parameters can be found in Table \[table2\] [@bon14; @ter16]. -------------------------------- -------------------------- ----------------------------------- -------------------------------- --------------- ----------------------------- ------------------ ------------ ------------ ------------ $\delta$ $\beta$ $A$ $\Gamma$ $\Gamma_1$ $\kappa$ $\chi$ $\sigma_v$ $\sigma_x$ $\sigma_y$ $\frac{d_1C_R}{\tilde{v}_0^2}$ $\frac{kL}{\tilde{v}_0}$ $\frac{\alpha_1L}{\tilde{v}_0^3}$ $\frac{\gamma}{\tilde{v}_0^2}$ $\gamma_1C_R$ $\frac{d_2}{\tilde{v}_0 L}$ $\frac{\eta}{L}$ - - - 1.5 5.88 $22.42$ 0.189 1 $0.0045$ 0.002 0.08 0.15 0.05 -------------------------------- -------------------------- ----------------------------------- -------------------------------- --------------- ----------------------------- ------------------ ------------ ------------ ------------ : Dimensionless parameters and representative values. []{data-label="table2"} Deterministic equations ----------------------- It is possible to derive deterministic equations for the density of [*active*]{} vessel tips and the vessel tip flux from the stochastic model. In all cases, the law of large numbers [@gar10] is involved. As the initial number of active tips $N(0)$ tends to infinity, the [*scaled tip density*]{}, defined as the number of active tips per unit phase volume divided by $N(0)$, is a self-averaging quantity obeying a deterministic integrodifferential equation [@cap19], similar to that derived earlier in [@bon14]. This can be proved rigorously as an initial value problem for tips moving on the infinite space [@cap19]. However, the situation is different for a slab geometry in two space dimensions, where tips are born from a primary vessel and advance toward a tumor placed at a finite distance. In this case, proliferation of active vessels due to branching is balanced by their inactivation due to anastomosis, which typically keeps the number of active tips below one hundred [@ter16]. Numerical simulations of the stochastic process also indicate that there are substantial velocity and density fluctuations. Thus, there is numerical evidence that the density of active tips is not self-averaging for bounded geometries. An average density satisfying a deterministic equation can be determined by a different usage of the law of large numbers. We consider a large number $\mathcal{N}$ of replicas (realizations) $\omega$ of the stochastic process and introduce the following empirical averages [@ter16]: $$\begin{aligned} p_{\mathcal{N}}\!(t,\mathbf{x},\mathbf{v})\!&=&\!\frac{1}{\mathcal{N}}\sum_{\omega=1}^\mathcal{N}\sum_{i=1}^{N(t,\omega)}\delta_{\sigma_x}(\mathbf{x}-\mathbf{X}^i(t,\omega))%\nonumber\\ &\times& \delta_{\sigma_v}(\mathbf{v}-\mathbf{v}^i(t,\omega)),\label{eq7}\\ \tilde{p}_{\mathcal N}(t,\mathbf{x})\!\!&=&\!\frac{1}{\mathcal{N}}\sum_{\omega=1}^\mathcal{N}\sum_{i=1}^{N(t,\omega)}\delta_{\sigma_x}(\mathbf{x}-\mathbf{X}^i(t,\omega)), \label{eq8}\\ j_{\mathcal N}(t,\mathbf{x})\!\!&=&\!\frac{1}{\mathcal{N}}\!\sum_{\omega=1}^\mathcal{N}\!\sum_{i=1}^{N(t,\omega)}\!\!|\mathbf{v}^i(t,\omega)|%\nonumber\\&\times&\! \delta_{\sigma_x}(\mathbf{x}-\mathbf{X}^i(t,\omega)).\label{eq9}\end{aligned}$$ Replicas of the stochastic process are independent by definition. By the law of large numbers as $\mathcal{N}\to\infty$, the limits of these empirical averages become the expected values of the tip density, the marginal tip density and the vessel tip flux according to the stochastic process [@gar10]. The averages over the set of replicas are the usual ensemble averages of statistical mechanics. The densities in Eqs.  and are not probability densities: note that the integral of the marginal density $\tilde{p}_{\mathcal N}(t,\mathbf{x})$ over space is not one. Instead, it is the average number of active tips at time $t$, $\langle\langle N(t)\rangle\rangle=\mathcal{N}^{-1}\sum_{\omega=1}^\mathcal{N}N(t,\omega)$, which is finite in the limit of infinitely many replicas and changes with time due to the branching and anastomosis processes. In Ref. [@ter16], we have shown that the ensemble averages, $p(t,\mathbf{x},\mathbf{v})$, $\tilde{p}(t,\mathbf{x})$, and $j(t,\mathbf{x})$, solve the following equations: $$\begin{aligned} &&\frac{\partial}{\partial t} p(t,\mathbf{x},\mathbf{v})=\alpha(C(t,\mathbf{x}))\, p(t,\mathbf{x},\mathbf{v})\delta_{\sigma_v}(\mathbf{v}-\mathbf{v}_0)%\nonumber\\&& - \Gamma\, p(t,\mathbf{x},\mathbf{v}) \!\int_0^t\! \tilde{p}(s,\mathbf{x})\, ds \nonumber\\ &&\!- \mathbf{v}\!\cdot\! \nabla_x p(t,\mathbf{x},\mathbf{v}) -\beta \nabla_v\! \cdot [(\mathbf{F}(C(t,\mathbf{x}))-\mathbf{v}) p(t,\mathbf{x},\mathbf{v})] %\nonumber\\ && + \frac{\beta}{2} \Delta_{v} p(t,\mathbf{x},\mathbf{v}), \label{eq10}\\ && \tilde{p}(t,\mathbf{x})=\int p(t,\mathbf{x},\mathbf{v}')\, d \mathbf{v'}. \label{eq11}\end{aligned}$$ Here, the first two terms on the right-hand side of Eq.  describe branching and anastomosis, respectively. The other terms are the usual ones appearing in the Fokker-Planck equation corresponding to the Langevin equations . The TAF equation becomes $$\begin{aligned} \frac{\partial}{\partial t}C(t,\mathbf{x})=\kappa \Delta_x C(t,\mathbf{x})- \chi\, C(t,\mathbf{x})\, j(t,\mathbf{x}),\label{eq12}\end{aligned}$$ where $$j(t,\mathbf{x})= \int |\mathbf{v}'|\, p(t,\mathbf{x},\mathbf{v}')\, d \mathbf{v'}. \label{eq13}$$ Unlike the case of the mean field limit $N(0)\to\infty$ in Ref. [@cap19], we lack a proof that the considered ensemble averages obey the deterministic equations -. However, we can compare the solution of the deterministic equations to the ensemble averages of the stochastic process and fit the anastomosis coefficient $\Gamma$ so that the average number of tips as a function of time is as close as possible in both descriptions [@ter16; @bon16pre]. The number of replicas $\mathcal{N}$ in our numerical simulations is selected so that the ensemble averages, $p_{\mathcal N}(t,\mathbf{x},\mathbf{v})$, $\tilde{p}_{\mathcal N}(t,\mathbf{x})$, and $j_{\mathcal N}(t,\mathbf{x})$, do not change by adding any more replicas to $\mathcal{N}$. Appendix \[app1\] contains appropriate initial and boundary conditions for solving the deterministic equations and . In principle, the ensemble average view of angiogenesis could be used to calculate higher moments, not only averaged quantities. This is largely unexplored. In Ref. [@ter16], we have used Ito’s formula with added branching and anastomosis to obtain an equation for the tip density. To this end, we need some closure assumptions of the type $\langle\langle f(x)\rangle\rangle= f(\langle\langle x\rangle\rangle)$. Justifying these assumptions would require taking into account and analyzing density fluctuations. This could be done by deriving a hierarchy of equations for $n$-particle densities and using closure assumptions as in kinetic theory [@bog46; @cer69; @akh81]. We could also include density fluctuations by keeping a Poisson noise (representing random branching) in Eq.  for the active tip density [@ter16]. This would give a formulation akin to the fluctuating lattice Boltzmann equation [@dun07]. In other cases, such as in the classical statistical mechanics of a crystal [@lew61] or in fluid turbulence [@lew62], it has been possible to derive and analyze functional equations. In recent years, there has been much progress in understanding rigorously the Kolmogorov-Hopf functional differential equation for fluid turbulence and the underlying invariant measure, [@bir13]. Marginal tip density -------------------- Assuming that the extension of the moving angiogenic sprouts is overdamped, Eqs.  and yield the following system of nondimensional equations for the marginal density of active vessel tips, $\tilde{p}(t,\mathbf{x})$, and the TAF density, $C(t,\mathbf{x})$, [@bon16pre] $$\begin{aligned} &&\frac{\partial\tilde{p}}{\partial t}+\nabla_x\!\cdot\!(\mathbf{F}\tilde{p})-\frac{1}{2\beta}\Delta_x\tilde{p}=\mu\,\tilde{p}%\nonumber\\ -\Gamma\tilde{p}\!\int_0^t\!\tilde{p}(s,\mathbf{x}) ds, \quad\label{eq14}\\ &&\frac{\partial}{\partial t}C(t,\mathbf{x})=\kappa \Delta_x C(t,\mathbf{x})- \tilde{\chi}\, C(t,\mathbf{x})\,\tilde{p}(t,\mathbf{x}).\label{eq15}\end{aligned}$$ Here $$\begin{aligned} &&\mu=\frac{\alpha}{\pi}\left[1+\frac{\alpha}{2\pi\beta(1+\sigma_v^2)}\ln\!\left(1+\frac{1}{\sigma_v^2}\right)\!\right]\!,\quad \tilde{\chi}=J\chi,\quad\nonumber\\ &&J=\int_0^\infty\! dV\frac{Ve^{-V^2}}{\pi}\!\int_{-\pi}^\pi\!d\phi\sqrt{1+V^2+2V\cos\phi}. \label{eq16}\end{aligned}$$ Note that the details of velocity selection during branching are lumped in the function $\mu(C)$ given by Eq. . In Ref. [@bon16pre], Eqs.  and are derived from Eqs.  and by a Chapman-Enskog perturbation method that shows the tip density to be $p\sim e^{-|\mathbf{V}|^2}\tilde{p}(t,\mathbf{x})/\pi$. Inserting this density in Eq.  and replacing $\mathbf{v}'=\mathbf{v}_0+\mathbf{V}$, we find $j(t,\mathbf{x})= J\tilde{p}(t,\mathbf{x})$ ($J\approx 1.28192$). -------------- --------------- ------------------------- ------------ ------------------------------ ------------------ $\mathbf{x}$ $\mathbf{v}$ $t$ $C$ $p$ $\tilde{p}$ $L$ $\tilde{v}_0$ $\frac{L}{\tilde{v}_0}$ $C_R$ $\frac{1}{\tilde{v}_0^2L^2}$ $\frac{1}{L^2}$ mm $\mu$m/hr hr mol/m$^2$ $10^{21}$s$^2$/m$^4$ $10^{5}$m$^{-2}$ $2$ 40 50 $10^{-16}$ 2.025 2.5 -------------- --------------- ------------------------- ------------ ------------------------------ ------------------ : Dimensional units with representative values. []{data-label="table1"} Typical values of the positive dimensionless parameters appearing in Eqs. - are given in Table \[table2\] whereas Table \[table1\] gives the units used to nondimensionalize them [@bon14]. Table \[table2\] shows that $1/\beta$, $\kappa$ and $\chi$ are small. This means that the evolution of the TAF density is slow compared to that of $\tilde{p}$ and that the diffusion in Eq.  is small. ![ Sketch of the unit vectors $\hat{\mathbf{V}}=(\cos\phi,\sin\phi)$ and $\hat{\mathbf{V}}^\perp=(-\sin\phi,\cos\phi)$. \[fig2\]](figure2.pdf){width="14cm"} -3mm Two dimensional diffusive soliton {#sec:2DDS} ================================= In this section, we derive the approximate 2DDS by using a method of multiple scales. Marginal tip density in curvilinear coordinates ----------------------------------------------- We now find an approximate lump-shaped solution of Eq. (\[eq14\]). Let $\mathbf{X}(t)$ be the center of mass of the lump at time $t$. Longitudinal and transverse coordinates based on the trajectory $\mathbf{X}(t)$ are $$\begin{aligned} &&\xi=(\mathbf{x}-\mathbf{X})\!\cdot\!\hat{\mathbf{V}}, \quad\eta=(\mathbf{x}-\mathbf{X})\!\cdot\!\hat{\mathbf{V}}^\perp,\label{eq17}\\ &&\mathbf{V}=\dot{\mathbf{X}}=c\hat{\mathbf{V}}=c(\cos\phi,\sin\phi), \quad%\nonumber\\&& \hat{\mathbf{V}}^\perp=(-\sin\phi,\cos\phi).\label{eq18}\end{aligned}$$ See Fig. \[fig2\]. Here and henceforth, $\dot{f}(t)=df/dt$ for any function of time, $c=|\mathbf{V}|$. Thus, $\mathbf{x}=(x,y)=(X,Y) + (\dot{X},\dot{Y})\,\xi/c+(-\dot{Y},\dot{X})\,\eta/c$. Eq.  can be written as $$\begin{aligned} &&\frac{\partial\tilde{p}}{\partial t}+\frac{\partial}{\partial\xi}\!\left((F_\xi-c+\eta\dot{\phi})\tilde{p}-\frac{1}{2\beta}\frac{\partial\tilde{p}}{\partial\xi}\!\right)\! %\nonumber\\&& +\frac{\partial}{\partial\eta}\!\left((F_\eta-\xi\dot{\phi})\tilde{p}-\frac{1}{2\beta}\frac{\partial\tilde{p}}{\partial\eta}\!\right)\!-\mu\,\tilde{p} \nonumber\\ &&=-\Gamma\tilde{p}\!\int_0^t\!\tilde{p}(s,\xi(s),\eta(s))\, ds, \label{eq19}\\ &&F_\xi=\hat{\mathbf{V}}\cdot\mathbf{F},\quad F_\eta=\hat{\mathbf{V}}^\perp\cdot\mathbf{F},\label{eq20}\end{aligned}$$ because $\dot{\xi}=-c+\eta\dot{\phi}$ and $\dot{\eta}=-\xi\dot{\phi}$. Method of multiple scales ------------------------- We shall now assume that the initial TAF concentration is a Gaussian with a small variance across the transversal direction $\eta$. We assume that $\tilde{p}$ depends on a fast variable $\eta/\sigma$ and a slowly varying $\eta$. We will use a method of multiple scales to find the slowly varying part of the marginal tip density [@kev96; @BT10]. The dominant terms in Eq.  are \[eq21\] $$\begin{aligned} \frac{\partial}{\partial\eta}(F_\eta\tilde{p})\sim\frac{1}{2\beta}\frac{\partial^2\tilde{p}}{\partial\eta^2}\Longrightarrow\tilde{p}\sim e^{2\beta\Omega[C]}\Psi(t,\xi),\label{eq21a}\\ \Omega[C]=\frac{\delta}{\beta}\frac{(1+\Gamma_1C)^{1-q}}{\Gamma_1(1-q)}\sim \Omega^0(\eta) - \frac{\eta^2}{2}\left.\!\left|\!\left(\frac{\partial^2 \Omega}{\partial\eta^2}\right)\!\right|_{\eta=0}\right|\!, \label{eq21b}\end{aligned}$$ in which $\partial\Omega/\partial\eta= F_\eta=0$ at $\eta=0$, where $C$ reaches its local maximum. Then the approximate density is a narrow Gaussian in $\eta$ that satisfies \[eq22\] $$\begin{aligned} &&\tilde{p}(t,\mathbf{x})\sim\frac{e^{-\eta^2/\sigma^2}}{\sqrt{\pi}\,\sigma} P(t,\xi,\eta)\sim\delta(\eta)P(t,\xi,0),\quad \label{eq22a}\\&& \sigma^2=\left.\frac{1}{\beta|\frac{\partial^2 \Omega}{\partial\eta^2}|}\right|_{\eta=0}=\left.\frac{[1+\Gamma_1C]^q}{\left|\frac{\partial^2C}{\partial\eta^2}\right|\delta}\right|_{\eta=0},\label{eq22b}\end{aligned}$$ as $\sigma\to 0$. We have identified $\sigma$ and, for the numerical values in Table \[table2\], $\sigma$ is indeed small; see Section \[sec:numerical\]. Then $P(t,\xi,\eta)$ varies slowly in $\eta$ compared to the Gaussian prefactor in Eq. . To obtain a reduced equation for $P$, we multiply Eq.  by $\sigma$ and integrate it with respect to the fast variable $\tilde{\eta}=\eta/\sigma$. After this, we use the second approximation in Eq.  and set the slow variable $\eta=0$. The resulting equation for $P(t,\xi,0)$ is $$\begin{aligned} &&\frac{\partial P}{\partial t}+\frac{\partial}{\partial\xi}\!\left((\overline{F_\xi}-c)P-\left.\frac{1}{2\beta}\frac{\partial P}{\partial\xi}\!\right)\right|_{\eta=0}\!+\frac{\partial}{\partial\eta}\!\left((\overline{F_\eta}-\xi\dot{\phi})P-\left.\frac{1}{2\beta}\frac{\partial P}{\partial\eta}\!\right)\right|_{\eta=0}\nonumber\\ &&\quad=\overline{\mu}\, P -\frac{\Gamma}{\sigma\sqrt{2\pi}}\, P \int_0^t\!P(s,\xi,0)\, ds,\label{eq23}\end{aligned}$$ \[eq24\] $$\begin{aligned} &&\overline{f(\tilde{\eta},\eta)}= \frac{1}{\sqrt{\pi}}\int_{-\infty}^\infty f(\tilde{\eta},0)\, e^{-\tilde{\eta}^2} d\tilde{\eta},\label{eq24a}\\ && \sigma\int_{-\infty}^\infty\frac{\partial}{\partial\eta}[f(\tilde{\eta},\eta)\tilde{p}]\,d\tilde{\eta} =\left. \frac{\partial}{\partial\eta}[P\,\overline{f(\tilde{\eta},\eta)}]\right|_{\eta=0},\label{eq24b}\\ &&\sigma\int_{-\infty}^\infty\tilde{p}\, d\tilde{\eta}=P(t,\xi,0). \quad\label{eq24c}\end{aligned}$$ One dimensional soliton ----------------------- Keeping Eq.  in mind, we will now look for a solitary wave profile of $P(t,\xi,0)$ as in Refs. [@bon16; @bon16pre], which, by abuse of language, we shall call [*soliton*]{}. The coefficients $\kappa$ and $\chi$ in Eq. (\[eq15\]) are very small [@bon14] and therefore the TAF concentration varies very slowly compared with the marginal tip density. When writing Eq. , we have used that $F_\xi$ and $F_\eta$ depend on $C$ and, therefore, vary slowly in time and space. Note that the slow variation of $C$ implies that an initial Gaussian TAF concentration with small variance produces an initial $\sigma\ll 1$, which does not change in time. Thus, Eq.  may hold initially due to a peaked initial $C$ and persist in time even if the coefficients in Eq.  are of order 1. For time and $\eta$ independent $P=P(\xi)$, with $dt=-d\xi/c$, we now define $$\rho(\xi)=-\frac{1}{c}\int_{\xi(0)}^\xi P(\xi')\, d\xi', \label{eq25}$$ insert it in Eq.  and integrate with respect to $\xi$. The result is $$\begin{aligned} \frac{\partial}{\partial\xi}\!\left(\!(\overline{F_\xi} - c)\rho-\frac{1}{2\beta}\frac{\partial\rho}{\partial\xi}\!\right)\! %\nonumber\\&& \!=\!\overline{\mu} \rho-\frac{g\rho^2}{2}+K,\quad g\!=\!\frac{\Gamma}{\sigma\sqrt{2\pi}},\, \label{eq26}\end{aligned}$$ in which $K$ is independent of $\xi$ and $g$ is a [*renormalized*]{} anastomosis coefficient. We will also assume that the initial TAF concentration varies on a larger spatial scale than the soliton size, which constitutes a good approximation [@bon14]. Then $\overline{F_\xi}$ and $\overline{\mu}$ are almost constant. Ignoring diffusion, we obtain $$\begin{aligned} \left(c-\overline{F_\xi}\right)\!\frac{2}{g}\frac{\partial\rho}{\partial\xi}=\rho^2-2\frac{\overline{\mu}}{g}\rho-\frac{2K}{g}. % = \!\left(\rho-\frac{\overline{\mu}}{g}\right)^2-\frac{2Kg+\overline{\mu}^2}{g^2}. \label{eq27}\end{aligned}$$ Setting $\rho=\frac{\overline{\mu}}{g}+\nu\tanh(\lambda\xi)$, we find $\nu^2=\frac{\overline{\mu}^2+2Kg}{g^2}$ and $2\nu\lambda(c-\overline{F_\xi})/g=-\nu^2$, thereby obtaining $$\begin{aligned} \rho=\frac{\overline{\mu}}{g}-\frac{\sqrt{2Kg+\overline{\mu}^2}}{g}\tanh\!\left[\frac{\sqrt{2Kg+\overline{\mu}^2}}{2(c-\overline{F_\xi})}\xi\right]\!. \label{eq28}\end{aligned}$$ Here a constant of integration has been absorbed in the definition of $\xi$. Thus $P=-c\partial\rho/\partial\xi$ yields [@bon16; @bon16pre] $$\begin{aligned} P_s(\xi)=\frac{(2Kg+\overline{\mu}^2)c}{2g(c-\overline{F_\xi})}\mbox{sech}^2\!\left[\frac{\sqrt{2Kg+\overline{\mu}^2}}{2(c-\overline{F_\xi})}\xi\right]\!. \label{eq29}\end{aligned}$$ As indicated in Refs. [@bon16; @bon16pre], Eq.  is similar to the usual soliton solution of the Korteweg-de Vries equation, except that the soliton velocity and shape now depend on three parameters, $c$, $K$, and (implicitly through $\xi$) $\phi$. Note that the existence of the 2DDS solution of Eqs.  and is a consequence of the quadratic anastomosis term in Eq.  first derived in Ref. [@bon14]. The function $\rho$ in Eq.  is the integral of the square hyperbolic secant given by Eq. , which has the same shape as the Korteweg-de Vries soliton [@lax68]. While the latter results from a balance of time derivative, nonlinear convection and dispersion [@lax68], the 1DDS soliton of Eq.  comes from a balance of time derivative, linear convection, branching and anastomosis (which contains a memory term). Center of mass -------------- Using Eqs.  and , we can calculate the center of mass of the 2DDS in curvilinear coordinates. As $\tilde{p}(t,\mathbf{x})=e^{-\eta^2/\sigma^2}P_s(\xi)/(\sqrt{\pi}\sigma)$ is even in both $\eta$ and $\xi$, the center of mass is the origin: $\int (\xi,\eta)\,\tilde{p}(t,\mathbf{x})d\xi\, d\eta/\int \tilde{p}(t,\mathbf{x}) d\xi\, d\eta=(0,0)$. Thus, the center of mass of the 2DDS coincides with the peak of the marginal tip density when the soliton is a good approximation for the latter. Collective coordinates {#sec:cc} ====================== Numerical simulations suggest that the 2DDS solution moving on unbounded space is stable. To obtain the soliton formula , we ignored the effects of small diffusion and a slowly varying TAF concentration. Without these terms, a proof that the traveling wavefront solution Eq.  is linearly stable (up to uniform spatial translations) follows along the same lines of Ref. [@bon91]. We expect the effects of diffusion and the slow TAF evolution to make the collective coordinates $c$, $K$ and $\phi$ slowly varying functions of time: due to its stability, the 2DDS adjusts its shape and velocity to the instantaneous values of the collective coordinates. For equations deriving from a variational principle, such as the Gross-Pitaevskii equation for a cigar shaped Bose condensate, a derivation of the CCEs first assumes that the soliton is a Gaussian function of the transversal coordinate times a function of the axial coordinate [@sal02]. Then this Ansatz is inserted into the variational principle and the corresponding Euler-Lagrange equations are the CCEs. In our case, we do not have a variational principle. Instead, the method of multiple scales has provided us with the splitting of the 2DDS in a Gaussian of the transversal coordinate times the longitudinal 1D diffusive soliton, cf Eq. . What is an Ansatz in Ref. [@sal02] is provided by our theory of the 2DDS. Slow variations of the collective coordinates --------------------------------------------- To obtain the 2DDS evolution without recourse to a variational principle (which does not exist for the present problem), we observe the following. As $F_\xi$ and $\mu$ are functions of $C(t,\mathbf{x})$, $P_s$ is primarily a function of $\xi$, but it is also a slowly varying function of $\xi$, $\eta$ and $t$ through the $\tilde{\eta}$-averages of the coefficients $\overline{\mu}$ and $\overline{F_\xi}$. Then $$P_s=P_s\!\left(\xi;K,c,\overline{\mu(C)},\overline{F_\xi\!\left(C,\frac{\partial C}{\partial\xi}\right)}\!\right)\!. \label{eq30}$$ The averages over the fast transversal coordinate $\tilde{\eta}$ still vary rapidly with the longitudinal coordinate $\xi$ of Eq.  and vary slowly on $\xi$ and $\eta$ through the the TAF concentration, which varies smoothly with distance. As indicated in Appendix \[app3\], we shall consider that $\mu(C)$ is approximately constant and set $\partial C/\partial t=0$ because the TAF concentration is varying slowly with time. Then we have $$\begin{aligned} &&P_s\!\!\left(\xi;K,c,\overline{\mu(C)},\overline{F_\xi\!\left(C,\frac{\partial C}{\partial\xi}\right)}\!\right)\!=P_s(\xi;K,c,\langle\overline{F_\xi}\rangle)\nonumber\\ &&\quad +\,\frac{\partial P_s}{\partial\overline{F_\xi}}(\xi;K,c,\langle\overline{F_\xi}\rangle)\,(\overline{F_\xi}-\langle\overline{F_\xi}\rangle)\!+\!\ldots, \label{eq31}\end{aligned}$$ in which we have dropped the dependence of $P_s$ on $\mu(C)$ and expanded the averages over the fast transversal coordinate, $\overline{f(C(t,\xi,\eta))}$, to first order in their differences with spatial averages $$\begin{aligned} \langle f(C(t,\psi,\eta))\rangle =\frac{1}{b-a}\int_a^bf(\psi)\, d\psi. \label{eq32}\end{aligned}$$ Here $f(\psi)\equiv \overline{f(C(t_0,\psi,0))}$. Eq.  is an average over the longitudinal coordinate, in which we have ignored time variation of the TAF concentration after some time $t=t_0$ and set $\eta=0$, cf Eq. . The time $t_0$ is selected after the formation stage of the 2DDS, and the interval $\mathcal{I}=(a,b)$ has to be appropriately chosen, as discussed in the next section. Using Eq. , we find $$\begin{aligned} \nabla_\xi P_s\!&=&\!(1,0)\,\frac{\partial P_s}{\partial\xi}(\xi;K,c,\langle\overline{F_\xi}\rangle)+ %\nonumber\\&+& \frac{\partial P_s}{\partial\overline{F_\xi}}(\xi;K,c,\langle\overline{F_\xi}\rangle)\nabla_\xi\overline{F_\xi}+\ldots,\label{eq33}\\ \Delta_\xi P_s\!&=&\!\frac{\partial^2P_s}{\partial\xi^2}(\xi;K,c,\langle\overline{F_\xi}\rangle)+ %\nonumber\\&+& \frac{\partial P_s}{\partial\overline{F_\xi}}(\xi;K,c,\langle\overline{F_\xi}\rangle)\Delta_\xi\overline{F_\xi} + \nonumber\\ &+& 2\frac{\partial^2P_s}{\partial\xi\partial\overline{F_\xi}}(\xi;K,c,\langle \overline{F_\xi}\rangle)\frac{\partial\overline{F_\xi}}{\partial\xi}+\ldots,\quad\label{eq34}\end{aligned}$$ where $\nabla_\xi=(\partial/\partial\xi,\partial/\partial\eta)$ and $\Delta_\xi=\nabla^2_\xi$. We now insert Eq. (\[eq30\]) into Eq. (\[eq23\]) and use Eqs. , and to simplify the result, thereby obtaining (see Appendix \[app3\]) $$\begin{aligned} &&\frac{\partial P_s}{\partial K} \dot{K}\!+\!\frac{\partial P_s}{\partial c}\dot{c}+\!\left( \frac{\partial P_s}{\partial\langle\overline{F_\xi}\rangle} \overline{F_\eta} - \xi\frac{\partial\overline{F_\xi}}{\partial\eta}\frac{\partial P_s}{\partial\langle\overline{F_\xi}\rangle} \right)\!\dot{\phi}=\mathcal{A},\qquad\, \label{eq35}\\ &&\mathcal{A}= \frac{1}{2\beta}\frac{\partial^2P_s}{\partial\xi^2} -\frac{\partial P_s}{\partial\langle\overline{F_\xi}\rangle}\!\left[\!\left((\overline{F_\xi}-c)\frac{\partial}{\partial\xi}+\overline{F_\eta}\frac{\partial}{\partial\eta}\right)\!\overline{F_\xi} %\right.\nonumber\\&&\quad\left. -\frac{\Delta_\xi\overline{F_\xi}}{2\beta}\right]\! \nonumber\\ &&\quad-P_s\!\left(\frac{\partial\overline{F_\xi}}{\partial\xi}+\frac{\partial\overline{F_\eta}}{\partial\eta}\right)\!+\frac{1}{\beta}\frac{\partial\overline{F_\xi}}{\partial \xi}\frac{\partial^2P_s}{\partial\xi\partial\langle\overline{F_\xi}\rangle} + \frac{1}{2\beta}\frac{\partial^2 P_s}{\partial\langle\overline{F_\xi}\rangle^2} \,|\nabla_{\xi}\overline{F_\xi}|^2. \quad\quad\label{eq36}\end{aligned}$$ In these equations, $P_s=P_s(\xi;K,c,\langle\overline{F_\xi}\rangle)$. We now find collective coordinate equations (CCEs) for $K$, $c$ and $\phi$. Of course, these equations hold only when the 2DDS is formed (far from primary vessel and tumor) after an initial stage that lasts a time $t_0>0$. Finding CCEs by using an approximate integral formula ----------------------------------------------------- We first multiply Eq. (\[eq35\]) by $\partial^2P_s/\partial\xi\partial\langle\overline{F_\xi}\rangle$ (which is odd in $\xi$) and integrate over $\xi$. As the soliton decays exponentially for $|\xi|\gg 1$, it is considered to be localized on some finite interval $(-\mathcal{L}/2,\mathcal{L}/2)$. The coefficients in the soliton formula and the coefficients in Eq.  depend on the slowly varying TAF concentration, therefore they are functions of $\xi$ and time and get integrated over $\xi$. The TAF varies slowly on the support of the soliton, hence we can approximate the integrals of functions $F(\xi;\psi,t)$ (varying rapidly on their first argument and slowly on their second argument) over $\xi$ by $$\begin{aligned} \!\int_a^b\! F(\xi;\psi,t)\, d\xi \approx\frac{1}{\mathcal{L}}\int_a^b\!\left(\int_{-\mathcal{L}/2}^{\mathcal{L}/2} F(\xi;\psi,t)\, d\xi\!\right)\! d\psi. \label{eq37}\end{aligned}$$ As in Eq. , the interval $\mathcal{I}=(a,b)$ over which we integrate should be large enough to contain most of the fully formed soliton of width $\mathcal{L}$. We have $b<L$ because the region near the tumor affects the soliton and should be excluded from the interval $\mathcal{I}$, to be specified in the next section. Similarly, $a>0$. The only odd terms in $\xi$ are the last term in the left-hand side of Eq.  and the second to last term in Eq. ; all other terms are even in $\xi$ and cancel out when multiplied by an odd function of $\xi$ and integrated over the interval $(-\mathcal{L}/2,\mathcal{L}/2)$. Then after integrating by parts the term proportional to $-\xi\dot{\phi}$ in Eq. , we obtain $$\begin{aligned} \dot{\phi}=\frac{2}{\beta}\frac{\int_{-\infty}^\infty\left\langle\frac{\partial\overline{F_\xi}}{\partial\xi}\left(\frac{\partial^2P_s}{\partial\xi\partial\langle\overline{F_\xi}\rangle}\right)^2\right\rangle d\xi }{\int_{-\infty}^\infty\!\left\langle\frac{\partial\overline{F_\xi}}{\partial\eta}\!\left(\frac{\partial P_s}{\partial\langle\overline{F_\xi}\rangle}\right)^2\!\right\rangle d\xi }. \label{eq38}\end{aligned}$$ Here, factors $1/\mathcal{L}$ in numerator and denominator cancel out and we have taken the limit as $\mathcal{L}\to\infty$ in the $\xi$-integrals with negligible error because the 2DDS decays exponentially to zero as $|\psi|\to\infty$. The brackets $\langle f(\psi)\rangle$ have been defined in Eq. . We now multiply Eq. (\[eq35\]) by $\partial P_s/\partial K$ (which is even in $\xi$) and integrate over $\xi$. Now all terms on the right hand side of Eq.  produce a nonzero contribution to the integral except for the second to last one. Acting similarly, we multiply Eq.  by $\partial P_s/\partial c$ (which is even in $\xi$) and integrate over $\xi$. From the two resulting formulas, we then find $\dot{K}$ and $\dot{c}$ as $$\begin{aligned} &&\dot{K}=\frac{\tilde{\mathcal{A}}_K I_{cc}-\tilde{\mathcal{A}}_c I_{Kc}}{I_{KK}I_{cc}-I_{Kc}^2},\label{eq39}\\ &&\dot{c}=\frac{\tilde{\mathcal{A}}_c I_{KK}- \tilde{\mathcal{A}}_K I_{Kc}}{I_{KK}I_{cc}-I_{Kc}^2}, \quad\label{eq40}\end{aligned}$$ in which we have used the following definitions: $$\begin{aligned} I_{ij}=\int_{-\infty}^\infty\left\langle\frac{\partial P_s}{\partial i}\frac{\partial P_s}{\partial j}\right\rangle d\xi, \, i,j=K,c,\quad \label{eq41}\\ \mathcal{A}_j=\int_{-\infty}^\infty\left\langle\frac{\partial P_s}{\partial j}\mathcal{A}\right\rangle d\xi,\quad j=K,c,\quad \label{eq42}\\ \tilde{\mathcal{A}}_j \!=\! \mathcal{A}_j \!-\! \dot{\phi} \!\int_{-\infty}^\infty\left\langle\frac{\partial P_s}{\partial j}\frac{\partial P_s}{\partial \langle\overline{F_\xi}\rangle}\,\overline{F_\eta} \right\rangle\! d\xi, \quad j=K,c. \label{eq43}\end{aligned}$$ Equations for collective coordinates ------------------------------------ The integrals of Eqs. - are calculated using Mathematica. As the coefficients $\chi$ and $\kappa$ are very small, the TAF concentration varies slowly, and terms containing them are ignored. We have also set $\mu$ to be a constant. Then Eqs. , - become $$\begin{aligned} \dot{K}\!&=&\! \frac{(2Kg\!+\!\langle\overline{\mu}\rangle^2)^2}{4g\beta(c\!-\!\langle\overline{F_\xi}\rangle)^2}\frac{\frac{4\pi^2}{75}\!+\!\frac{1}{5}\!+\!\!\left(\frac{2\langle\overline{F_\xi}\rangle}{5 c}\!-\!\frac{2\pi^2}{75}\!-\!\frac{9}{10}\right)\!\!\frac{\langle\overline{F_\xi}\rangle}{c}}{\left(1-\frac{4\pi^2}{15}\right)\!\left(1-\frac{\langle\overline{F_\xi}\rangle}{2c}\right)^2} \nonumber\\ \!&-&\! \frac{2Kg+\langle\overline{\mu}\rangle^2}{g\!\left(2c-\langle\overline{F_\xi}\rangle\right)}\!\left(\dot{\phi}\langle\overline{F_\eta}\rangle\!+\!c\left\langle\frac{\partial\overline{F_\eta}}{\partial\eta}\right\rangle+\!\left\langle\overline{\mathbf{F}}\cdot\nabla_\xi\overline{F_\xi}\right\rangle\!- \frac{\langle\Delta_\xi\overline{F_\xi}\rangle}{2\beta}\!\right)\! \nonumber\\ \!&+&\! \frac{2Kg+\langle\overline{\mu}\rangle^2}{2g\beta(c-\langle\overline{F_\xi}\rangle)^2}\langle |\nabla_\xi\overline{F_\xi}| \rangle^2% \nonumber \\ \!&\times&\! \frac{ 1-\frac{\pi^2}{30}\!-\!\frac{3 \langle\overline{F_\xi}\rangle}{2 c} \left(1-\frac{\pi^2}{90}\right) \!+\!\frac{\langle\overline{F_\xi}\rangle^2}{2c^2}}{\left(1-\frac{\langle\overline{F_\xi}\rangle}{2c}\right)^2} , \quad \label{eq44}\end{aligned}$$ $$\begin{aligned} \dot{c}&=&\!-\frac{7(2Kg+\langle\overline{\mu}\rangle^2)}{20\beta(c-\langle\overline{F_\xi}\rangle)}\frac{1-\frac{4\pi^2}{105}}{\left(1-\frac{4\pi^2}{15}\right)\!\left(1-\frac{\langle\overline{F_\xi}\rangle}{2c}\right)} %\nonumber\\&& -\frac{c}{2c-\langle\overline{F_\xi}\rangle}\!\left[c\left\langle\frac{\partial\overline{F_\xi}}{\partial\xi}\right\rangle\! \right. \nonumber\\ &+&(c\!-\!\langle\overline{F_\xi}\rangle)\!\langle\nabla_\xi\!\cdot\!\overline{{\mathbf F}}\rangle\!+\!\left.\frac{\langle\Delta_\xi\overline{F_\xi}\rangle}{2\beta}-\!\left\langle\overline{\mathbf{F}}\!\cdot\!\nabla_\xi\overline{F_\xi}\right\rangle-\dot{\phi}\langle\overline{F_\eta}\rangle\right]\! \nonumber\\ &-&\frac{ \langle |\nabla_\xi\overline{F_\xi} | \rangle^2\!\left( 1+\frac{\pi^2}{30} \right) }{\beta \left( c\!-\!\langle\overline{F_\xi}\rangle\right)\! \left( 2-\frac{\langle\overline{F_\xi}\rangle}{c} \right) }. \quad \label{eq45}\end{aligned}$$ If we set $\dot{\phi}=0$, these equations become Eqs. (C12)-(C13) of Ref. [@bon16pre] with $\mu_C=0$ and $\xi=x$. The coefficients entering Eqs.  and are spatial averages over $\psi$ (which is the slow variable $\xi$ that appears in the formulas through the TAF concentration) and have $\eta=0$ due to Eq. . The CCEs , and , describe the mean behavior of the 2DDS after its formation time, whenever it is far from primary vessel and tumor, as we will show in the next section. Numerical results {#sec:numerical} ================= In this paper, we obtain the vessel tip density by ensemble averages of stochastic simulations, as explained in Ref. [@ter16]. If we set up symmetric initial and boundary conditions so that the 2DDS moves on the $x$-axis, $\mathbf{X}=(X,0)$, $\hat{\mathbf{V}}=(1,0)$, $\hat{\mathbf{V}}^\perp= (0,1)$, $\xi=x-X$, $c=\dot{X}$, and $\eta=y$. Then the integrals in Eq.  are integrals over $y$ and the integrals in Eq.  are simply integrals over $x$ with $y=0$. From our simulations, we can obtain the evolution of the 2DDS collective coordinates thereby reconstructing the marginal tip density from Eqs. , , and - with $\dot{\phi}=0$. The 2DDS profile at $y=0$ is the 1D soliton studied in Refs. [@bon16; @bon16pre], which agrees with numerical simulations of the stochastic process and also with simulations of the corresponding deterministic equations. Initial and boundary conditions ------------------------------- The simplest asymmetric configuration consists of one initial tip moving toward a TAF source at $x=1$, above the $x$-axis. To the values of the dimensionless parameters in Table \[table2\], we have added the initial nondimensional TAF concentration $$\begin{aligned} C(0,x,y) = 1.1\, e^{-(x-1)^2/1.5^2 -(y-0.4)^2/0.6^2}, \label{eq46} \end{aligned}$$ and the nondimensional TAF flux boundary condition at $x = 1$ $$\begin{aligned} \frac{\partial C}{\partial x}(t,1,y) = 1.1 e^{-(y-0.4)^2/0.6^2}. \label{eq47} \end{aligned}$$ At $x=0$, the TAF flux is zero. These conditions correspond to having a TAF source at the border $x=1$, above the $x$-axis at $y=0.4$. ![Density plots of the marginal tip density $\tilde{p}(t,x,y)$ calculated from Eq.  with $N(0)=1$ and $\mathcal{N} = 400$ replicas, showing how tips are created at $x = 0$ and march toward the tumor at $x = L$. Snapshots at (a) 16 hr, (b) 24 hr, (c) 28 hr, (d) 32 hr.[]{data-label="fig9"}](figure9.pdf){width="14cm"} -3mm ![Profiles of the marginal tip density $\tilde{p}(t,\mathbf{x})$ for $\eta=0$ calculated as in Fig. \[fig9\], and for the same times. We have fit the 2DDS (the angiton) given by Eqs.  and with $\overline{F}_\xi$ calculated at $t=16$ hr and fixed for later times. For each time, $c$ and $K$ are calculated from the maximum of $\tilde{p}(t,\mathbf{x})$ and the trajectory of its center of mass.[]{data-label="fig10"}](figure10.pdf){width="14cm"} -3mm A single initial tip -------------------- Suppose that initially there is only one tip, $N(0)=1$, placed below the $x$-axis, say at $(0,-0.2)$. In a typical realization of the stochastic process, the initial active tip advances and undergoes repeated branching until the density of active tips approaches the 2DDS. For sufficiently large distance between the primary blood vessel and the tumor, the evolution of the active tip density comprises three stages: a soliton formation stage, evolution of the 2DDS far from the boundaries, and arrival at the TAF source. Here, we only describe the second stage of a 2DDS detached from the boundaries. A complete theory would require matching the detached soliton stage to reduced descriptions of the other stages, which we do not attempt in this paper. For shorter distances between primary blood vessel and TAF source, a 2DDS may not even form and our theory is then inapplicable. The evolution of the angiogenic network and the duration of the 2DDS formation period depend on the specific selection of the velocity in Eq. . For example, if $\mathbf{v}_0$ is parallel to the $x$ axis, it takes 18 hr to form the 2DDS, which finds it difficult to move upward to where the TAF source of Eq.  is. Furthermore, there are more than 30 realizations of the stochastic process for which anastomosis eliminates all active tips before they reach $x=1$. We need to discard these replicas when calculating the density of active tips by an ensemble average. A better choice of $\mathbf{v}_0$ decreases the number of replicas to be discarded and lifts the center of mass for the angiogenic network of active tips. Fig. \[fig9\] shows four snapshots of $\tilde{p}(t,\mathbf{x})$ after the 2DDS formation time for $\mathbf{v}_0=(1,0.4)$ and a 400-replica ensemble average. For this modified $\mathbf{v}_0$, only two replicas need to be discarded. From the trajectory of the maximum of $\tilde{p}$, which coincides with the 2DDS center of mass, we calculate $K$, $c$ and $\phi$. Fig. \[fig10\] displays a comparison of the snapshots of Fig. \[fig9\] for $\eta=0$ with the 2DDS obtained from Eqs.  and . The reasonably accurate fit shown in Fig. \[fig10\] confirms the validity of the 2DDS description after the formation stage and before the arrival at the tumor. Coefficients in the CCEs and initial conditions after the 2DDS formation stage ------------------------------------------------------------------------------ As explained before, when there is a single tip at $t=0$, the 2DDS formation stage takes longer, certain realizations of the stochastic process end up prematurely by anastomosis before the tips can reach $x=1$ and have to be discarded. In addition, the influence of the details of tip velocity selection at branching disappears in the overdamped limit. However, these different details still affect the 2DDS motion. Thus, we shall compare its motion to an initial configuration that has a faster formation stage and does not require discarding failed replicas of the stochastic process. Let us now consider an asymmetric configuration as in Fig. \[fig1\]. In each stochastic simulation (replica), $N(0)=20$ initial tips are placed at $x=0$ and uniformly distributed in the $y$-direction between -0.5 and 0.1 with $\mathbf{v}_0=(1,0)$. Stochastic simulations indicate that it takes a time $t_0=0.318$ (16 hours) after angiogenesis initiation to form the 2DDS. For $t>t_0$, the evolution of the soliton is given by Eqs. -. The variance in Eq.  is fixed as $\sigma=0.235$. As indicated before, we consider that the collective coordinates represent spatial averages over the spatial coordinate $x$ excluding regions affected by boundaries. The coefficients in Eqs. - are spatial averages involving $C(t_0,x,y)$. We calculate them by: (i) approximating all differentials by second order finite differences, (ii) approximating the integrals in Eq.  by Gaussian quadrature, and (iii) using Eq.  to average the coefficients by taking the arithmetic mean of their values at all grid points in the interval $x\in\mathcal{I}=(0.54,0.95]$ ($0.21<\xi\leq 0.63$). For $0<x\leq 0.54$ and for $0.95<x\leq 1$, the boundary conditions at $x=0$ and at $x=1$, respectively, influence the outcome and therefore we leave these values out of the averaging. ![ Evolution of the collective coordinates (a) $K(t)$, (b) $c(t)$, (c) $\phi(t)$, (d) $X(t)$, (e) $Y(t)$. \[fig3\]](figure3a.pdf "fig:"){width="6cm"} ![ Evolution of the collective coordinates (a) $K(t)$, (b) $c(t)$, (c) $\phi(t)$, (d) $X(t)$, (e) $Y(t)$. \[fig3\]](figure3b.pdf "fig:"){width="6cm"} ![ Evolution of the collective coordinates (a) $K(t)$, (b) $c(t)$, (c) $\phi(t)$, (d) $X(t)$, (e) $Y(t)$. \[fig3\]](figure3c.pdf "fig:"){width="6cm"} ![ Evolution of the collective coordinates (a) $K(t)$, (b) $c(t)$, (c) $\phi(t)$, (d) $X(t)$, (e) $Y(t)$. \[fig3\]](figure3d.pdf "fig:"){width="6cm"} ![ Evolution of the collective coordinates (a) $K(t)$, (b) $c(t)$, (c) $\phi(t)$, (d) $X(t)$, (e) $Y(t)$. \[fig3\]](figure3e.pdf "fig:"){width="6cm"} -3mm The initial conditions for the CCEs - are set as follows. We find the coordinates of the maximum of the marginal tip density $\tilde{p}(t_0,x,y)$ (calculated from ensemble average by Eq.  with $\mathcal{N}=400$) as $\mathbf{X}(t_0)=\mathbf{X}_0 = (0.34,0.08)$. Similarly, we set $K(t_0)= 5.9$, $c(t_0)=1.15$, $\phi(t_0)=0.245$, determined so that the maximum marginal tip density at $t=t_0$ coincides with the soliton peak. Solving the CCEs - with these initial conditions, we obtain the curves depicted in Fig. \[fig3\]. ![Evolution of (a) $x$-coordinate, (b) $y$-coordinate, and (c) value of the ensemble-averaged maximum marginal tip density (by $\mathcal{N} = 400$ replicas) as compared to the prediction by the collective coordinates of the 2DDS (the [*angiton*]{}). The maximum absolute error is reached at $t = 27$ hr, $28$ hr. It approximately equals (a) $\Delta x = 0.02$ and (b) $2\Delta x = 0.04$, where $\Delta x$ is the discretization space step. In (c) the relative error is around 4% at $t = 27$ hr, $28$ hr, while it is smaller than 2.5% for all other times. \[fig4\]](figure4a.pdf "fig:"){width="7cm"} ![Evolution of (a) $x$-coordinate, (b) $y$-coordinate, and (c) value of the ensemble-averaged maximum marginal tip density (by $\mathcal{N} = 400$ replicas) as compared to the prediction by the collective coordinates of the 2DDS (the [*angiton*]{}). The maximum absolute error is reached at $t = 27$ hr, $28$ hr. It approximately equals (a) $\Delta x = 0.02$ and (b) $2\Delta x = 0.04$, where $\Delta x$ is the discretization space step. In (c) the relative error is around 4% at $t = 27$ hr, $28$ hr, while it is smaller than 2.5% for all other times. \[fig4\]](figure4b.pdf "fig:"){width="7cm"} ![Evolution of (a) $x$-coordinate, (b) $y$-coordinate, and (c) value of the ensemble-averaged maximum marginal tip density (by $\mathcal{N} = 400$ replicas) as compared to the prediction by the collective coordinates of the 2DDS (the [*angiton*]{}). The maximum absolute error is reached at $t = 27$ hr, $28$ hr. It approximately equals (a) $\Delta x = 0.02$ and (b) $2\Delta x = 0.04$, where $\Delta x$ is the discretization space step. In (c) the relative error is around 4% at $t = 27$ hr, $28$ hr, while it is smaller than 2.5% for all other times. \[fig4\]](figure4c.pdf "fig:"){width="7cm"} -4mm ![Evolution of the position $(x,y)$ of the ensemble-averaged maximum marginal tip density (over $\mathcal{N}=400$ replicas) as compared to that of the 2DDS ([*angiton*]{}) predicted by collective coordinates. The absolute error is approximately equal to $2\Delta x = 0.04$ at $t = 27$ hr, $28$ hr, while it is $\Delta x = 0.02$ at most for all other times. Here $\Delta x$ is the space step in the discretization. \[fig5\]](figure5.pdf){width="14.0cm"} -4mm Comparison of CCE predictions with stochastic simulations --------------------------------------------------------- Using the 2DDS collective coordinates depicted in Fig. \[fig3\] and Eqs.  and (\[eq29\]), we reconstruct the marginal vessel tip density and find its maximum value and the location thereof for all times $t>t_0$. Fig. \[fig4\] shows that the position of the 2DDS as predicted from the CCEs and - compares very well with the location of the tip density maximum obtained by ensemble average of stochastic simulations (over $\mathcal{N}=400$ replicas). Fig. \[fig5\] shows that the overall trajectory of the 2DDS agrees well with the location of the tip density maximum, which coincides with the 2DDS center of mass. ![Density plots of the marginal tip density $\tilde{p}(t,x,y)$ calculated from Eq.  with $\mathcal{N} = 400$ replicas, showing how tips are created at $x = 0$ and march toward the tumor at $x = L$. Snapshots at (a) 16 hr, (b) 20 hr, (c) 24 hr, and (d) 28 hr. \[fig6\]](figure6.pdf "fig:"){width="14cm"}\ ![Same as in Fig. \[fig6\] but now the density plots of $\tilde{p}(t,x,y)$ are calculated from Eq.  and the CCEs , , for the 2DDS. Snapshots at (a) 16 hr, (b) 20 hr, (c) 24 hr, and (d) 28 hr. \[fig7\]](figure7.pdf){width="16cm"} Fig. \[fig6\] is the density plot of the ensemble-averaged marginal tip density in four snapshots taken at 16, 20, 24 and 28 hours after the initial time. Fig. \[fig7\] shows the density plot of the marginal tip density reconstructed from the 2DDS CCEs. The shape of the respective density plots is different but the sizes of their peaks are similar, which suggests correcting the leading order of the multiple scales theory, Eq. . However, Figs. \[fig4\] and \[fig5\] show that the 2DDS center of mass gives a good approximation for the motion of the marginal density peak. Thus, the 2DDS gives a good approximation of the advance of the marginal density and its order of magnitude. This agreement is all the more remarkable, as the parameters in Table \[table2\] used in our simulations are not particularly small. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![(a) $x$-coordinate and (b) $y$-coordinate of the 2DDS (angiton) peak compared to those of the maximum marginal tip density for different replicas of the stochastic process. \[fig8\]](figure8a.pdf "fig:"){width="12.0cm"} ![(a) $x$-coordinate and (b) $y$-coordinate of the 2DDS (angiton) peak compared to those of the maximum marginal tip density for different replicas of the stochastic process. \[fig8\]](figure8b.pdf "fig:"){width="12.0cm"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -4mm 2DDS center of mass and replicas of the stochastic process ---------------------------------------------------------- So far, our reconstructions have been based on ensemble averages, which produce mean values of the density of active tips and related functions. In past work, we have shown that fluctuations about the mean are large and, therefore, the stochastic process is not self-averaging [@ter16]. However, Fig. \[fig8\] indicates that the 2DDS center of mass is a good approximation to the location of the maximum marginal tip density for different replicas of the stochastic process. The $x$-coordinate of the maximum density location is approximated better than its $y$-coordinate. While vessel networks may widely differ from replica to replica, the position of the maximum marginal tip density is about the same for different replicas. As the maximum of the marginal tip density is a good measure of the advancing vessel network, the location of the 2DDS peak also characterizes it. Conclusions {#sec:conclusions} =========== On mesoscopic distances that are large compared to the size of one cell but small compared to the size of an organ, the early stage of tumor induced angiogenesis can be described by stochastic models that track the trajectories of active vessel tips. These models consider branching of blood capillaries as a stochastic process and renounce to describe cellular processes and scales. However, active tip models pose novel and interesting problems in nonequilibrium statistical mechanics. In previous works, we have shown that the ensemble-averaged density of active tips is described by an integrodifferential Fokker-Planck equation with source and sink terms [@bon14; @ter16]. Together with time derivative and linear convection, these terms make it possible for this equation to have an approximate soliton solution for simple one dimensional geometries [@bon16; @bon16pre; @bon17]. The soliton solution has the same shape as the well-known Korteweg-de Vries soliton, which results from a balance between time derivative, nonlinear convection and dispersion [@lax68]. In two dimensions, the marginal density of active tips acquires the form of a moving lump or 2DDS that advances towards the tumor in a curvilinear trajectory. Here we have used a method of multiple scales to show that the transversal section of the 2DDS is a narrow Gaussian and that its longitudinal section is a diffusive soliton. The slow variation of the tumor angiogenic factor changes slowly the shape and trajectory of the 2DDS. The latter can be reconstructed by solving collective coordinate equations for its speed, direction of velocity, shape parameter and coordinates of the center of mass. As the parameters used in numerical simulations are not particularly small, it is remarkable that the predictions from the 2DDS and its collective coordinates (based on the method of multiple scales and singular perturbation ideas) compare well with the predictions from numerical simulations of the stochastic model. The shape of the marginal density of active tips as obtained from ensemble averages of stochastic simulations is less symmetric than that reconstructed from the 2DDS. However, the size and position of its peak follow those given by Eqs.  and (the 2DDS) and the CCEs , and . In principle, the present model and the 2DDS construction can be extended to three spatial dimensions. We need to consider the angiogenic vessels as the tip trajectories plus a narrow region or tube around them and a criterion for anastomosis; cf Ref. [@cap19] for a possible way to do this. The derivation of CCEs could proceed along the lines explained in the present work. We would need to consider an additional curvilinear coordinate along the binormal and modify accordingly the CCEs. Extensions of the present work to more complete models based on tip cell motion are possible [@cap09; @hec15; @bon17], but we feel it is better to present these ideas in the simplest possible context. Future applications of the present work to biology include investigating possible control of the 2DDS motion, e.g. by studying the effect of antiangiogenic drugs; cf Ref. [@lev01]. In physics, our work could be useful to study dynamic phenomena that include stochastic branching and merging of advancing point defects. For example, propagation of cracks in brittle materials [@ala10] or dielectric breakdown [@bea88]. From the point of view of nonequilibrium statistical mechanics, at the present time there is no theory of the large fluctuations about the averaged tip density. Perhaps deriving functional equations for the moments and using ideas similar to those appearing in turbulence theory could be helpful [@bir13]. Acknowledgements {#acknowledgements .unnumbered} ================ This work has been supported by the FEDER/Ministerio de Ciencia, Innovación y Universidades – Agencia Estatal de Investigación grant MTM2017-84446-C2-2-R. Boundary and initial conditions for the deterministic equations {#app1} =============================================================== The governing equations and of the deterministic description have to be solved with appropriate initial and boundary conditions compatible with the stochastic description. The nondimensional initial and boundary conditions for the TAF are Eqs.  and , respectively, and we also have $\lim_{y\to\pm\infty} C= 0$ [@bon16pre]. We do not intend to follow the process of angiogenesis beyond the time when vessel tips have arrived at the tumor and therefore we do not give the latter a finite length. The boundary conditions for the tip density are [@bon14] $$\begin{aligned} p^+(t,0,y,v,w)=\frac{e^{-|\mathbf{v}-\mathbf{v}_0|^2}}{\int_0^{\infty}\!\int_{-\infty}^{\infty} |\mathbf{v}'|\, e^{-|\mathbf{v}'-\mathbf{v}_0|^2}dv'\,dw'} \nonumber \\ \!\times\! \!\left[j_0(t,y)\! -\!\! \int_{-\infty}^0\!\int_{-\infty}^{\infty}\!\! |\mathbf{v}'|\, p^-(t,0,y,v',w')d v' dw'\!\right]\!\!, \label{ap3} \\ p^-(t,1,y,v,w)=\frac{e^{-|\mathbf{v}-\mathbf{v}_0|^2}}{\int_{-\infty}^0\!\int_{-\infty}^{\infty} e^{-|\mathbf{v}'-\mathbf{v}_0|^2}dv'\,dw'} \nonumber\\ \!\times\! \!\left[\tilde{p}(t,1,y)\! -\!\!\int_0^{\infty}\!\!\int_{-\infty}^{\infty}\! p^+(t,1,y,v',w')dv' dw'\!\right]\!\!, \label{ap4}\\ p(t,\mathbf{x},\mathbf{v})\to 0 \mbox{ as } |\mathbf{v}|\to \infty,\quad\label{ap5}\end{aligned}$$ where $p^+=p$ for $v>0$ and $p^-=p$ for $v<0$, $\mathbf{v}=(v,w)$. At $x=0$, $j(t,\mathbf{x})$ given by Eq.  is $$j_0(t,y)= \alpha(C(t,0,y))\, p(t,0,y,v_0,w_0), \label{ap6}$$ for the vector velocity $\mathbf{v}_0=(v_0,w_0)$, with $|\mathbf{v}_0|=1$. The deterministic description including boundary conditions can be proved to have a solution [@car16; @CDN17]. A convergent numerical scheme to solve the initial boundary value problem corresponding to the deterministic description is studied in Ref. [@bon18]. Collective coordinates for a soliton far from primary vessel and tumor {#app3} ====================================================================== To obtain Eqs. -, we need to substitute the soliton Eq.  into Eq. . According to Eq. , the soliton is a function $$P_s=P_s\!\left(\xi;K,c,\overline{\mu(C)},\overline{F_\xi\!\left(C,\frac{\partial C}{\partial\xi}\right)}\!\right)\!, \label{a1}$$ in which we have distinguished the fast coordinate $\xi$ in Eq.  from the slowly varying coordinate $\xi=\psi$ resulting from averages of the TAF density. Assuming that $\mu(C)$ is approximately constant, we have the expressions in Eqs. , and $$\begin{aligned} \frac{\partial P_s}{\partial t}=\frac{\partial P_s}{\partial K}\dot{K}\!+\!\frac{\partial P_s}{\partial c}\dot{c}\!+\!\frac{\partial P_s}{\partial\overline{F_\xi}} \frac{\partial\overline{F_\xi}}{\partial t}, \label{a2}\end{aligned}$$ $$\begin{aligned} \overline{F_\xi}=\frac{\delta}{\beta}\overline{\frac{\hat{\mathbf{V}}\cdot\nabla_x C}{(1+\Gamma_1C)^q}}=\frac{\delta\,\overline{\hat{\mathbf{V}}\cdot\nabla_x(1+\Gamma_1C)^{1-q}}}{\beta(1-q)\Gamma_1} , \label{a3}\end{aligned}$$ $$\begin{aligned} &&\frac{\partial\overline{F_\xi}}{\partial t}=\dot{\phi}\overline{F_\eta}+ \frac{\delta/\beta}{(1-q)\Gamma_1} \frac{\partial}{\partial\xi}\overline{\frac{\partial}{\partial t}(1+\Gamma_1C)^{1-q}}\nonumber\\ &&\quad =\frac{\delta}{\beta}\frac{\partial}{\partial\xi}\!\!\left[\overline{\frac{\frac{\partial C}{\partial t}}{(1\!+\!\Gamma_1C)^q}}\right]\!+\dot{\phi}\overline{F_\eta},\label{a4}\end{aligned}$$ where we have used $\partial\hat{\mathbf{V}}/\partial t=\dot{\phi}\hat{\mathbf{V}}^\perp$. Setting $\partial C/\partial t=0$ on the right-hand side of Eq. , we obtain $$\begin{aligned} \frac{\partial\overline{F_\xi}}{\partial t}=\dot{\phi}\,\overline{F_\eta}. \label{a5}\end{aligned}$$ Note now that $P_s$ in Eq.  is a function of the ratios $\overline{F_\xi}/c$ and $\xi/c$. Then we have $$\begin{aligned} \frac{\partial P_s}{\partial c}= - \frac{\xi}{c} \frac{\partial P_s}{\partial\xi}- \frac{\overline{F_\xi}}{c}\frac{\partial P_s}{\partial\overline{F_\xi}}\Longrightarrow\frac{\partial P_s}{\partial\overline{F_\xi}}=- \frac{1}{\overline{F_\xi}} \!\left(\xi\frac{\partial P_s}{\partial\xi}+c\frac{\partial P_s}{\partial c}\right)\!.\label{a6}\end{aligned}$$ The soliton of Eq.  satisfies $$\begin{aligned} (\overline{F_\xi}-c)\frac{\partial P_s}{\partial\xi}=\overline{\mu} P_s-g P_s\int_0^t P_s(\xi(s))\, ds, \label{a7}\end{aligned}$$ in which $\overline{F_\xi}$ and $\overline{\mu}$ vary slowly with $\xi$ and $t$. Integration by parts shows that $$\begin{aligned} \int_{-\infty}^\infty \eta\,\frac{e^{-\frac{\eta^2}{\sigma^2}}}{\sqrt{\pi}\sigma}\Psi(\eta)d\eta=\frac{\sigma^2}{2}\int_{-\infty}^\infty\frac{e^{-\frac{\eta^2}{\sigma^2}}}{\sqrt{\pi}\sigma}\, \frac{\partial\Psi}{\partial\eta}\, d\eta. \label{a8}\end{aligned}$$ Using Eq. , Eq.  becomes $$\begin{aligned} \frac{\partial\tilde{p}}{\partial t}\!&+&\!\frac{\partial}{\partial\xi}\!\left((F_\xi-c)\tilde{p}+\frac{\sigma^2}{2}\dot{\phi}\frac{\partial\tilde{p}}{\partial\eta}-\frac{1}{2\beta}\frac{\partial\tilde{p}}{\partial\xi}\!\right)\!+\frac{\partial}{\partial\eta}\!\left((F_\eta-\xi\dot{\phi})\tilde{p}-\frac{1}{2\beta}\frac{\partial\tilde{p}}{\partial\eta}\!\right)\! \nonumber\\ \!&=&\!\mu\,\tilde{p} -\Gamma\tilde{p}\!\int_0^t\!\tilde{p}(s,\xi(s),\eta(s))\, ds. \label{a9}\end{aligned}$$ We now integrate this expression with respect to $\tilde{\eta}=\eta/\sigma$ using the Gaussian approximation Eq.  for $\tilde{p}$. 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--- abstract: 'We analyze the interaction of two conducting, charged polymer chains in solution using a minimal model for their electronic degrees of freedom. We show that a crossing of the two chains in which the polymers pass within Angstroms of each other leads to a decrease of the electronic energy of the combined system that is significantly larger than the thermal energy and thus promotes interchain aggregation. We consider the competition of this attractive interaction with the screened electrostatic repulsion and thereby propose a phase diagram for such polymers in solution; depending on the charge density and persistence length of the chains, the polymers may be unbound, bound in loose, braid-like structures, or tightly bound in a parallel configuration.' author: - 'Jeremy D. Schmit' - 'Alex J. Levine' title: Intermolecular adhesion in conducting polymers --- Introduction ============ Conjugated polymers have been the subject of intense research in the physics, chemistry, and materials science communities. These molecules provide both a laboratory to explore one-dimensional conductors [@Su:79; @Su:80; @Takayama:80] and the potential for a plethora of applications in such disparate devices as solar cells [@solar] and chemo/biosensors[@Heeger:98; @Heeger:01; @McQuade:00; @Gaylord:03]. The simplest conducting polymer system is polyacetylene. The underlying cause of its conductivity, however, is somewhat subtle owing to the fact that, due to the Peierls instability[@Peierls:55; @Yannoni:83] in one dimension, this molecule appears to be a semiconductor (based on an analysis of uncorrelated electrons). Here we consider the case of an electron-doped polymer in which the Fermi energy lies within the conduction band. The utility of conjugated polymers is generally realized in such complex chemical systems created by adding functional groups to the basic, conjugated backbone. Such modified polymers typically include side groups to modify the conductivity of the backbone by doping in either electrons or holes; polar or ionizable side groups can also dramatically increase the solubility of these molecules in water [@Shi:90]. Fluorescent, water-solubilized conjugated polymers such as polyparaphenylene vinylene (PPV) hold promise for use as biosensors since their fluorescence is readily quenched by electron acceptors such as methyl viologen (MV). Biomolecules tagged with MV can be detected optically at remarkably low concentration using these water soluble conjugated polymers. Fluorescence quenching can be quantified by the Stern-Volmer constant, $K_{\rm SV}$, which describes the loss of quantum efficiency per mole of quencher present. Since the $K_{SV}\simeq 10^7 M^{-1}$ for the PPV/MV system, MV tagged biomolecules can be detected via fluorescence quenching at 100 nanomolar concentrations[@Chen:99]. Before this analytic tool can be realized, a number of complications must be addressed. These include understanding and accounting for the dramatic effect of “bystander” molecules such as surfactants on the value of $K_{SV}$[@Chen:00], as well as addressing the role played by chain length of the PPV [@Gaylord:01; @Wang:01]. Additionally, the PPV/MV system is hindered as an analytical tool by the tendency of the PPV to aggregate in solution and to bind nonspecifically with other small molecules. Neutron and light scattering[@Wang:01a] demonstrate that PPV solutions form large aggregates under low-salt conditions. With added salt, the scattering data can be fit by rod-like structures having a persistence length of $80$nm [@Wang:01b] suggesting the bundling of the chains in solution. Understanding the tendency of metallic polymers to aggregate in solution is the central focus of this paper. We propose an interchain adhesion mechanism specific to conjugated polymers that should apply to doped, solubilized PPV and other conducting polymers. When two polymer chains approach each other to Angstrom-scale distances at some point along their backbones, the interchain tunnelling of the electrons (holes) in the conduction bands of the two molecules decreases the total electronic energy of the system. This decrease in the energy of the electronic degrees of freedom is primarily due to the creation of a low energy, localized state at the crossing point of the two chains. These states lead to the aggregation of the two chains since, for physically reasonable parameters the energy decrease per crossing point is on the order of a few $k_{\rm B} T$. Such a electron tunnelling mechanism has been previously discussed as a source of an [*intrachain*]{}, attractive interaction leading to polymer collapse in good solvent [@Hone:01]. The mechanism that we propose can be thought of as analogous to the creation of a molecular covalent bond with the distinction that in the present system, the bonding and anti-bonding states that are localized at the crossing point of the two chains are not created from normal atomic energy eigenstates, but rather are pulled out of the delocalized states of the charge carriers on the chains. In the remainder of the article, we first explore the consequences of this type of interchain bonding (in section \[adhesion\]) in three stages of increasing complexity. We consider the effect of a single chain-crossing point that leads to interchain tunnelling in section \[single\]. We then turn to the case of multiple crossing points by considering an ordered array of such points in section \[ordered\], before turning to the more physical case of a thermally disordered set of crossing points in \[disordered\]. Finally, we consider the attraction between two polymers aligned in a parallel configuration in \[railroad\]. In order to determine the adhesive properties of these conjugated polymers, we need to compare the decrease in the energy of the electronic degrees of freedom of the molecule, to the increase of chain conformational free energy associated with adhesion. We turn to the latter calculation in section \[polymer\], before combining these two parts of the analysis in section \[conclusions\] wherein we discuss the results and propose experimental tests of this work. Interchain Adhesion {#adhesion} =================== Since we suggest that the adhesive properties of the conducting polymers are a generic consequence of populating the conduction band of the these extended molecules, we develop our theory based on a minimalistic, tight-binding Hamiltonian for these conduction electrons on a chain of $N$ sites of the form $$\label{tight-binding} H_0=-t\sum_{\ell=-N/2}^{N/2} | \ell\rangle \langle \ell + 1 | + | \ell +1 \rangle \langle \ell |,$$ where $|\ell\rangle$ is a state vector for an electron on the $\ell^{\rm th}$ tight binding site. In the absence of strong electron–electron correlations[@Heeger:01] such a single particle approach is justified and many properties of such conjugated polymers have been predicted via such simple tight-binding models [@Fink:86]. The overlap integral $t$ is not precisely known for a number of these systems, but recent spectroscopic results on chemically related systems[@Fink:86] suggest that the $t$ is on the order of $1 - 4 eV$. To discuss the modification of the electronic states of this tight-binding model due to the proximity of another such polymer at one point or a set of points $\left\{ \ell_I \right\}$, we modify Eq. \[tight-binding\] by introducing another copy and including an interaction term between the chains parameterized by a second hopping matrix element $t'$: $$\begin{aligned} \label{tb-interchain} H_0 &=& -t \sum_{j=1,2} \sum_{\ell=1}^N \left( |\ell + 1,j\rangle \langle \ell,j | + |\ell,j\rangle \langle \ell +1,j | \right) \\ \label{tb-interaction} H_I &=& - t' \sum_{\bar{\ell} \in \left\{ \ell_I \right\} } \left( |\bar{\ell},1 \rangle \langle \bar{\ell},2| + |\bar{\ell},2\rangle \langle \bar{\ell},1| \right).\end{aligned}$$ In the above equations, the sum on $j$ is over the two chains, while the interaction Hamiltonian, $H_I$ allows for interchain hopping at a selected subset of sites along the polymer, $\left\{ \ell_I \right\}$. The interchain hopping matrix element has been shown numerically to be on the order of $0.1 eV$ for similar chemical systems[@Bredas:02]. It is essential to note that since the interchain tunnelling matrix element is exponentially sensitive to the interchain separation, it is acceptable to suppose that this matrix element will be finite at only isolated points. In fact, we will later show in section \[polymer\] that the positions of these “crossing-points” to be spatially correlated or anticorrelated along the arc length of the chain due to chain conformational free energy. A simple pictorial representation of the basic problem for the case of only one hopping site is shown in figure \[model-setup\]. ![A pictorial representation of the simplest crossed–chains configuration in which the inter-chain hopping is allowed at the central site, $\ell = 0$. The spheres represent the tight-binding sites for the conduction electrons along the chains. The chains need not be straight as shown here.[]{data-label="model-setup"}](crossedmodel.eps){width="8.0cm"} Because of the large separation of time scales between the electronic and conformational degrees of freedom of the polymer chain, we may ignore conformational changes in the polymer while discussing the modification of the electronic states due to the proximity of the two chains at a given point or set of points. To show that we may ignore the thermal motion of the point of closest approach in discussing the electronic structure of the molecules, we compare the time scale for monomer motion over distances of a Bohr radius (the distance over which one expects $t'$ to vary rapidly) $ T_{\rm chain} \sim a_{B} \sqrt{m/(k_{\rm B} T)}$ to characteristic times for electronic structure reorganization $T_{\rm electon} \sim \hbar/t$. We find that $T_{\rm electron}/T_{\rm chain} \sim 10^{-10}$ so the adiabatic approximation is eminently reasonable. Similarly, we note that the temperature of the system (typically $300K$ or less) is significantly less than the Fermi temperature of electronic system so in the remainder of the calculation we will determine the total free energy of the electronic degrees of the freedom in a zero temperature approximation whereas we will consider the role of temperature and consequently entropy while discussing the conformational and translational degrees of freedom of the polymer. We now develop our calculation for the binding energy of the two chains by computing separately the decrease in the energy of the electronic degrees of freedom due to interchain tunnelling and increase in chain free energy due to the loss of some translational and conformational entropy. From the sum of these two effects, we determine the effective binding energy. First we consider a single crossing point as shown in figure \[model-setup\]. A Single Crossing Point {#single} ----------------------- We take the single crossing point to lie in the middle of both chains $\{ \ell_I \} = \{ 0 \}$ as shown in figure \[model-setup\]. We return to crossing configurations of lower symmetry later. We note that the interaction Hamiltonian, Eq. \[tb-interaction\], can be diagonalized in the basis of symmetrized and antisymmetrized chain occupation states: $$\label{symm} | \ell, \pm \rangle = \frac{1}{\sqrt{2}} \left[ | \ell, 1 \rangle \pm | \ell, 2 \rangle \right].$$ so that in this basis Eq. \[tb-interaction\] takes the form $$\label{diag-interchain} H_I = -t' \sum_{\bar{\ell} \in \left\{ \ell_I \right\} } \left[ \, {|\bar{\ell},+\rangle} {\langle \bar{\ell},+|} - {|\bar{\ell},-\rangle} {\langle \bar{\ell},-|} \, \right]$$ It is immediately clear that the action of the Hamiltonian on the subspaces of the antisymmetrized and symmetrized states is identical except for the exchange of $t' \longrightarrow -t'$. We discuss the energies of states in both subspaces in parallel. The Hamiltonian is also symmetric under the parity operator, $P {|\ell,\pm\rangle} = {|-\ell, \pm\rangle}$; since only the states symmetric under $P$ will be affected by the crossing point, we focus on these even parity states in the following. We make an ansatz for the unnormalized, (anti-) symmetrized, even parity eigenstates of the Hamiltonian by writing $$\label{ansatz} | k, \pm \rangle = \sum_{\ell = -m}^{m} \cos \left( a k | \ell | + \phi^\pm_k \right) | \ell \rangle$$ where $\phi^\pm_k $ represents the wavevector-dependent break in the phase of the wavefunction due to the interchain interaction on chains of $N= 2m + 1$ tight binding sites. Boundary conditions require that the amplitude of the wavefunction at sites $\pm (m+1)$ vanishes, which with Eq. \[ansatz\] leads to $$\label{quantizationI} a k ( m+1) + \phi^\pm_k = \frac{\pi \left( 2 p + 1\right)}{2}, \hspace{0.2cm} p \in {\cal Z}.$$ One can show that the states $| k, \pm \rangle $ with arbitrary phase $\phi^\pm_k$ satisfy the time-independent Schrödinger equation all sites away from the crossing point since, via direct calculation, one finds $$\label{schro1} E^\pm_k \langle \ell \neq 0, \pm | k, \pm \rangle = \langle \ell \neq 0, \pm | H | k, \pm \rangle,$$ with $$\label{energy} E^\pm_k = - 2 t \cos ( a k ).$$ However, to simultaneously satisfy this eigenvalue equation at the crossing point as well as Eq. \[quantizationI\], we must choose the phase angle $\phi^\pm_k$ such that $$\label{quantizationII} \sin ( k a) = \mp \frac{t'}{2 t} \tan \left[ k a ( m+ 1 ) \right].$$ The solutions of this transcendental equation that determine the allowed wavevectors $k$ (and energies via Eq. \[energy\]) are shown graphically as the set of intersections in figure \[quant-fig\] where the left and right hand sides of Eq. \[quantizationII\] are plotted. ![Graphical solution of Eq. \[quantizationII\]. Allowed $k$ values lie at the intersections of the solid line with the long dashes (symmetric band) and with the short dashes (anti-symmetric band). The vertical lines represent the unperturbed $k$ values.[]{data-label="quant-fig"}](solution.eps){width="8.0cm"} The principal consequence of the crossing point is the creation of two localized bound states out of the symmetrized/anitsymmetrized states ${| k, \pm\rangle}$. We examine the symmetrized states first. The conditions on the tunnelling matrix element $t'$ for the appearance of this bound state can be inferred from Eq. \[quantizationII\]: If $$\label{condition} 1 < \frac{|t'|}{2 t} ( m+1)$$ then there is no solution for real wavevector $k a < \pi/[2 (m+1)]$, but a new solution appears along the imaginary axis in the complex $k$ plane at $k = i \kappa$, where $\kappa$ is given by $$\label{kappa} \sinh(a \kappa) = \frac{|t'|}{2 t} \tanh\left[a \kappa (m+1)\right].$$ From this relationship and Eq. \[energy\], we determine the energy of the resulting bound state to be $$\label{bound-state} E_b = - 2 t \sqrt{ 1 + \frac{t'^2}{4 t^2} \tanh^2[\kappa (m+1)] }$$ where $\kappa$ is the solution from Eq. \[kappa\]. From the magnitude of the imaginary wavevector, it is clear that this bound state is localized on the length scale of $a t/t' \sim a $. It is important to note that, due to the appearance of this bound state on any chain of reasonable length, the energy of the electronic degrees of freedom is decreased by a quantity on the order of the tunnelling matrix element $t'$. In the limit of an infinite chain where $m \longrightarrow \infty$, one sees from the above equations that $a \kappa \sim {\cal O}(1)$ and the $\tanh(\cdot)$ in Eq. \[bound-state\] becomes one in agreement with previous work [@economou:79]. This rearrangement of the electronic degrees of freedom has dramatic consequences for the polymer conformational dynamics as we show below. In addition to this lower energy bound state, a second localized state is pulled from the conduction band of antisymmetrized states. In this case $t' \longrightarrow -t'$ as discussed below Eq. \[diag-interchain\]. From Eqs. \[condition\],\[kappa\] we now find that the complex wavevector associated with the bound state becomes $k = i \kappa + \pi$; the bound state appears at the edge of the Brillouin zone and the energy of the state is $- E_b$. It is interesting to note that the appearance of these two localized states at energies $\pm E_b$ is directly analogous to the appearance of bonding/anitbonding states in covalently bonded atoms[@Pauling:55]. In the present case, however, we build these states out of spatially extended conduction-band states rather than the localized atomic orbitals. There are still other consequences of interchain tunnelling for the scattering states that remain in the conduction band. In other words, all the even symmetry, extended states of the unperturbed chains are shifted in energy due to their interaction with the crossing point at site zero. These scattering states remain extended, [*i.e.*]{} retain a purely real wavevector, but that wavevector shifts due to the interaction with the crossing point. For $t' = 0$ we trivially find these extended states at $k^{(0)}_p = \pi (2 p + 1)/(2 N)$ for integer $p$. Due to scattering at the crossing point, these wavevectors shift so that $k_p \longrightarrow k_p + \Delta k_p$. These shifts can be computed by expanding both sides of Eq. \[quantizationII\] near $k^{(0)}_p$. From this expansion we find to leading order in $t'/N$ that $$\label{shifts} \Delta k_p = \frac{-t'}{2t N \sin k_p} \left ( 1 + \frac{t'}{2t N} \frac{\cos k_p}{\sin^2 k_p } \right) + {\cal O} \left( \frac{1}{N^3} \right).$$ Using the above shifts in $k_p$, we can compute using Eq. \[energy\] the sum of the shifts in the energy levels of these scattering states in the symmetrized band. Recalling that we need to also include the analogous shifts in the chain antisymmetrized band, which eliminates terms odd in $t'$, we find that the total shift is $$\label{e-shifts} \Delta E = - \left(\frac{t'}{N}\right)^2 \, \sum_{p=1}^{(m_{\rm f})} \frac{\cos k_p}{2t \sin^2 k_p} ,$$ where the sum is over all filled energy levels from the bottom of the conduction band to $m_{\rm f}$ set by the Fermi level of the system. Since the sum of all such energy shifts of the extended states vanishes in the limit of large $N$ as $1/N$, the changes in the energy of the scattering states in the band is not significant for long chains. Hereafter we will ignore the finite length ($O(1/N)$) The principal result is that there is an attractive interchain interaction due to a reorganization of the electronic degrees of freedom of the system. The energetically dominant part of this reorganization is the appearance of the “bonding” bound state that lowers the electronic energy essentially by $t'^2/t$. Since the dominant contribution to the electronic energy shifts comes from a localized state at the crossing point, it is perhaps not surprising, that, when the above analysis is generalized to the case of a crossing point at an arbitrary point along the chain, the principal, qualitative result is unchanged. We will show below that this basic result is effectively insensitive to interactions between such crossing-points in chains that form a multicrossing-point “braids”. We do this in two steps by first considering an ordered array of crossing points and then by examining a spatially disordered set of crossing points-see figure \[braidmodel\]. An Ordered Array of Crossing Points {#ordered} ----------------------------------- One may wonder about the effect of a multiple set of interchain crossings. Specifically, if, as it appears above, one crossing point decreases the overall energy of the electronic states of the system, is it more profitable for the chains to form many such crossing points, and if so, at what density? To address this question, we first consider the most simple such structure – an ordered array of crossing points at every $M^{\rm th}$ site along the chains. The ordered structure forms, in effect, a superlattice in which the unit cell is constructed from one crossing point and a basis of $M-1$ tight binding sites between that consecutive crossing points. Qualitatively, one expects the localized states centered at each crossing point to merge into what may be termed an impurity band in the lattice. We explore this point below paying particular attention to the interchain binding energy per unit length of the polymers. ![Two polymers interacting at a series of localized sites. In the upper figure the two chains form an ordered structure in which a crossing point appears after exactly $M$ tight binding sites. The lower figure represents the disordered version of the braid. Below each braid is a pictorial representation of the appropriate tight-binding Hamiltonian in which the filled circles represent the interchain tunnelling sites. In the disordered two-chain braid, the number of sites between crossing points varies along the chains, $M \longrightarrow M_i$, but the same $M_i$ sites exist on both chains between the crossing points. []{data-label="braidmodel"}](orderedbraid.eps "fig:"){width="8.0cm"} ![Two polymers interacting at a series of localized sites. In the upper figure the two chains form an ordered structure in which a crossing point appears after exactly $M$ tight binding sites. The lower figure represents the disordered version of the braid. Below each braid is a pictorial representation of the appropriate tight-binding Hamiltonian in which the filled circles represent the interchain tunnelling sites. In the disordered two-chain braid, the number of sites between crossing points varies along the chains, $M \longrightarrow M_i$, but the same $M_i$ sites exist on both chains between the crossing points. []{data-label="braidmodel"}](randombraid.eps "fig:"){width="8.0cm"} To begin we note that within a single supercell of the superlattice, the tight-binding Hamiltonian becomes $$\begin{aligned} \label{supercell-H} H_n &=& -t \sum_{\ell = 1}^{M-1} \left\{ \, {|\ell,n\rangle} {\langle \ell+1,n|} + {|\ell+1,n\rangle} {\langle \ell,n|} \right\} \nonumber \\ & & + \lambda {|M,n\rangle} {\langle M,n|}\end{aligned}$$ where state ${|\ell,n\rangle}$ is a position eigenstate representing an electron at the $\ell^{\rm th}$ site in the $n^{\rm th}$ supercell. Due to the interchain tunnelling at the $M^{\rm th}$ site in each supercell, there is a localized potential for the electron to reside there. In the above equation and in what follows we find it convenient to suppress the $\pm$ notation describing the interchain symmetrized and antisymmetrized states. We have seen above in Eq. \[diag-interchain\] that the only difference in the form of the Hamiltonian acting on these two subspaces of opposite chain-exchange symmetry is the shift: $t' \longrightarrow -t'$. In the above equation, that $t'$ dependent term in the Hamiltonian is given in terms of $\lambda$ which takes on the value $t' \, (-t')$ for the antisymmetrized (symmetrized) states. To complete our superlattice Hamiltonian, we include a simple tunnelling term to couple the states in each supercell. Here, as before, we assume that the presence of the crossing chain does not affect the intrachain hopping matrix element, although such effects could be taken into account using the present tight-binding formalism. The coupling term is $$\label{supercell-coupling-H} H'_n = -t \left\{ {|M,n\rangle} {\langle 1, n+1|} + {|1,n+1\rangle} {\langle M,n|} \right\}.$$ The first term in the above equation couples the crossing-point site in the $n^{\rm th}$ supercell to the tight-binding site to its right (in the $(n+1)^{\rm th}$ supercell) with the usual intrachain hopping matrix element, while the second term allows for electron hopping from the first site in $(n+1)^{\rm th}$ supercell onto the crossing point site to its immediate left that is part of the $n^{\rm th}$ supercell. The full Hamiltonian of the system is simply the sum of these two terms summed over all $N$ supercells: $$\label{superlattice-full-H} H= \sum_{n=1}^N \left( H_n + H'_n \right),$$ where we have used periodic boundary conditions so that the $(N+1)^{\rm th}$ supercell is identified with the first supercell. Since principal effect of the interchain tunnelling on the energetics of the system has been shown to be the appearance of localized bound states for a single crossing point, the net effect of periodic boundary conditions used here is minimal. In order to diagonalize the Hamiltonian given by Eq. \[superlattice-full-H\], we introduce two wavevectors conjugate to the real-space degrees of freedom that index the supercell and the site within each supercell by writing unnormalized states $$\label{Bloch} {|q,k\rangle} = \sum_{n=1}^N \sum_{\ell = 1}^M \exp^{i q a M n} \left( e^{i a k \ell} + A_{q,k} e^{- i k a \ell} \right) {|\ell,n\rangle},$$ where $A_{q,k}$ will be determined below. The allowed values of $q$ are fixed by the periodic boundary conditions to be $ q = 2 \pi n/NMa$ where $n$ is an integer such that: $- N/2 < n < N/2$. We now insist that these states, which are eigenstates of the supercell translation operator are also energy eigenstates: $H {|q,k\rangle} = E(q,k) {|q,k\rangle}$. This relation can be translated into three related equations formed by projecting the above relation onto three carefully chosen states. We require that $$\begin{aligned} \label{req-one} {\langle \bar{\ell}, \bar{n}|} H {|q,k\rangle} &=& E(q,k) \langle \bar{\ell}, \bar{n} |q,k \rangle \\ \label{req-two} {\langle 1, \bar{n}|} H {|q,k\rangle} &=& E(q,k) \langle \bar{\ell}, \bar{n} |q,k \rangle \\ \label{req-three} {\langle M, \bar{n}|} H {|q,k\rangle} &=& E(q,k) \langle \bar{\ell}, \bar{n} |q,k \rangle,\end{aligned}$$ where $\bar{\ell} \neq 1,M$ and the index of the supercell $\bar{n}$ is arbitrary. The first condition determines the energy values to be $E(q,k) = -2 t \cos (k a)$. Note that since condition Eq. \[req-one\] applies to the interior of each supercell, the energy eigenvalues appear to be independent of $q$ and $\lambda$. This apparent lack of $q$- and $\lambda$-dependence is false, as we will soon be forced to require that the allowed values of $k$ depend on these parameters. The second and third conditions above acknowledge the special role of the crossing-point in the Hamiltonian. From the second condition, we determine the parameter $A_{q,k}$. In order to satisfy the eigenvector/eigenvalue equation at the left-most site in a supercell, we find $$\label{A-determined} A_{q,k} = \frac{e^{i (k a M - q)}- 1}{1 - e^{-i (k a M + q)}}.$$ In the above result we have used the energy eigenvalues obtained from enforcing the first condition. Finally, the combination of the third condition, Eq. \[req-three\] along with the energy eigenvalue used above leads to a relation between, $k$, $q$, and $\lambda$ of the form $$\label{k-q-condition} \frac{\lambda}{2 t} \tan \left( k M a \right) = - \left[ 1 - \frac{\cos (q M a)}{cos( k M a)} \right] \sin (k a).$$ The above equation may be read as an implicit form of the relation of $k$ upon $q$ and $\lambda$, [*i.e.*]{} $ k = k (q,\lambda)$. Note, that in the limit of a very sparse array of crossing-points, [*i.e.*]{} $M \longrightarrow \infty$ we recover the condition on imaginary $k$ for the isolated bound state given in Eq. \[kappa\]. However, as $M$ becomes smaller, we find a small modification in this condition and the introduction of a lattice momentum ($q$) dependence in $k$. We now turn to the calculation of the density of states $g(E) = \partial {\cal N}/\partial E$ where ${\cal N}$ is the total number of energy eigenstates with energy less than $E$. Using the density we compute the total energy of the electronic degrees of freedom of the braided polymers. The density of states can be written as $$\label{density-states} g(E) = \frac{\partial {\cal N} }{\partial k} \cdot \left( \frac{1}{\frac{\partial E}{\partial k}} \right).$$ The second derivative in the product above is computed trivially from the dispersion relation. Using the quantization of the wavevector $q$, the first derivative can be expressed as: $$\label{first-der} \frac{\partial {\cal N} }{\partial k} = \frac{\partial {\cal N}}{\partial n} \cdot \frac{\partial n}{\partial q} \cdot \frac{\partial q}{\partial k} = M \cdot \frac{N M a}{2 \pi} \cdot \frac{\partial q}{\partial k}$$ where the first term in the product reflects that fact that for each wavevector $q = q_n = 2 \pi n/(N a) $ there are $M$ allowed values of $k$, one for each of the $M$ bands in the system. The second term in the above product expresses the $q$-quantization condition; this leaves only the last term to be computed from Eq. \[k-q-condition\]. This derivative is given by $$\label{dqdk} \frac{\partial q}{\partial k} = \frac{ \frac{\lambda}{2 t} \left[ -M \cos ( k a M) + \sin(k a M) \frac{\cos(k a)}{ \sin^{2}} (k a) \right] + M \sin( k a M)}{M \sqrt{ 1 - \left( \frac{\lambda}{2 t} \sin (k a M) \csc ( k a ) + \cos ( k a M ) \right)^2 } }.$$ Now, the total energy of the electronic states of the chains can be computed from Eqs. \[density-states\], \[first-der\], and \[dqdk\]. One expects from the calculation of the isolated crossing-point that the change in electronic energy is almost entirely due to the creation of a narrow band of bonding states below the conduction band so it is reasonable to imagine that the total change in electronic energy is independent of the Fermi level. For convenience we choose the Fermi level to be $E_{\rm F} = 0$. Since there is at most one energy eigenstate for each (complex) value of $k$, we may write the total energy of the electronic system as an integral over $k$ of the density of states and the dispersion relation. All possible states below the Fermi level are explored by integrating along the following contour in the complex plane: ${\cal C} = (-i \infty, 0) \bigcup (0, \pi/2)$ which is the combination of the negative imaginary axis and the segment of the real axis spanning 0 to $\pi/2$. The density of states along this contour is shown in figure \[density-states-plot\] as a function of energy. Lastly, using Eq. \[k-q-condition\] we note that for there to be an energy eigenstate with a given $(q,k)$, it must be true that $| \lambda/(2t) \sin(M a k) \csc( a k ) + \cos (k a M) | < 1$ so that the denominator of Eq. \[dqdk\] must be real. To enforce this condition over the $k$ integration we take the real part of the integral to write $$\label{energy-final} E = \mbox{ Re } \int_{\cal C} \frac{\partial {\cal N}}{\partial k} E(k) dk .$$ The change in the total electronic energy as a function of $t'$ and $M$ can then be computed; these results are shown in figures \[threemodelst\] and \[binding-e-plot\]. The observed even-odd effect in figure \[threemodelst\] is an artifact of our choice of the Fermi level. For chains with even $M$, a band gap will open at $E_F=0$ resulting in an additional reduction in the energy. In practice, we expect that the Fermi level of doped polymers will not lie at a place of such high symmetry and the dominant contribution to the energy will be the creation of localized bound states. The principal result of this calculation is that, due to the highly localized nature of the bonding states at each crossing-point, the total interchain binding energy is essentially a linear function of the crossing point density. In other words the attractive interaction at each crossing point is highly independent of the local density of such crossing points at least until $M$ decreases to order unity. ![The density of states as a function of energy for an ordered array of crossing points (solid line) and a disordered array of crossing points (dashed line). In both cases the density of binding sites is $p = 1/M = 0.125$ and $\lambda/(2 t) = -0.25$.[]{data-label="density-states-plot"}](DOS-fixed.eps){width="8.0cm"} A Disordered Array of Crossing Points {#disordered} ------------------------------------- As a final point regarding the electronic contribution to the interchain adhesion, we now allow the set of crossing points to become disordered in a specific manner as shown in the lower part of figure \[braidmodel\]. We allow the number of tight-binding sites between two crossing points ($M$) now to vary along the chain in an uncorrelated manner so that the chains may be described by a sequences of such spacings $M$, $M_1, M_2, \ldots, M_p$. We restrict the form of the randomness to cases in which the [*same*]{} sequence can be used to describe both chains. The advantage of this restriction is that the interchain interaction part of the Hamiltonian is still diagonal in the basis of chain symmetrized/antisymmetrized wavefunctions. When considering the chain-symmetrized/antisymmetrized subspaces individually, the Hamiltonian given by Eqs. \[tb-interchain\],\[tb-interaction\] is identical to the two-component random lattice problem, which has been extensively studied [@Schmidt:57; @Soven:67; @Velicky:68; @Soven:69]. For completeness we recapitulate the discussion of the general formalism of the coherent potential approximation (CPA) developed therein while applying that formalism to the system at hand. We begin by expanding the propagator of the full, disordered system $G$ in terms of the propagator of a uniform system $G_0$ with a fixed, and as yet unspecified on-site potential. The propagator for the uniform system is given by $$\label{propagator-uniform} G_0 = \frac{1}{E - H_0}$$ where $$\label{uniform-H} H_0 = -t \sum_{\ell=1}^N \left( |\ell + 1\rangle \langle \ell | + |\ell \rangle \langle \ell +1 | \right) + v \sum_{\ell =1 }^N {|\ell\rangle} {\langle \ell|}$$ and $v$ is the spatially independent on-site energy. The full Hamiltonian of the disordered system is given by the sum of $H_0$ from Eq. \[uniform-H\] and scattering potentials at each site given by $$\label{random-int} H_I = \sum_{\ell =1 }^N (\varepsilon_\ell - v) {|\ell\rangle} {\langle \ell|}.$$ In the above equation, the random variable $\varepsilon_\ell$ introduces the quenched disorder by introducing uncorrelated tunnelling sites with density $p$ along the chain. This random variable is takes the value $\lambda$ with probability $p$ and is zero otherwise where, as before, $\lambda$ is $\pm t'$ for the chain-antisymmetrized/symmetrized Hamiltonian. One can show in the usual way that the propagator for the full, random system $G$ can be expressed perturbatively in terms of the tunnelling Hamiltonian $H_I$ and propagator of the uniform system so that $$\label{full-propagator} G = \frac{1}{E - H} = G_0 + G_0 H_I G_0 + \cdots \, .$$ Our goal, as before is to compute the density of states of the system, g(E), which is given in terms of the propagator as $$\label{DOS} g(E) = - \frac{1}{\pi} \mbox{Im} \sum_\ell {\langle \ell|} G {|\ell\rangle}$$ where the sum is over a complete set of states. To approximate this sum accurately, we follow the work of Soven [@Soven:69] and first reorganize the perturbation series given in Eq. \[full-propagator\] by introducing the $T$ operator representing the Born-series for scattering off a particular site, $\ell$, $$\begin{aligned} \label{T-operator} T_\ell &=& {|\ell\rangle} (\varepsilon_\ell - v) {\langle \ell|} + (\varepsilon_\ell - v)^2 {|\ell\rangle} {\langle \ell|} G_0 {|\ell\rangle} {\langle \ell|} + \cdots \\ \nonumber & =& \frac{1}{1 - {\langle \ell|} G_0 {|\ell\rangle} (\varepsilon_\ell - v)} {|\ell\rangle} {\langle \ell|},\end{aligned}$$ so that Eq. \[full-propagator\] implies $$\begin{aligned} \label{reorganized} {\langle \ell'|} G {|\ell\rangle} &=& {\langle \ell'|} G_0 {|\ell\rangle} + \sum_{\bar{\ell}} {\langle \ell'|} G_0 {|\bar{\ell}\rangle}T_{\bar{\ell}} {\langle \bar{\ell}|} G_0 {|\ell\rangle} + \\ \nonumber +& & \sum_{\bar{\ell} \neq \bar{\bar{\ell}}} {\langle \ell'|} G_0 {|\bar{\ell}\rangle}T_{\bar{\ell}} {\langle \bar{\ell}|} G_0 {|\bar{\bar{\ell}}\rangle} T_{\bar{\bar{\ell}}}{\langle \bar{\bar{\ell}}|} G_0 {|\ell\rangle} + \cdots \, .\end{aligned}$$ The successive terms in the above equation represent the propagator of the electron in the uniform system followed by a correction due to the interaction of that electron with the scattering potential at site $\bar{\ell}$, followed in turn by the interaction of that electron with sites $\bar{\ell}$ and $\bar{\bar{\ell}}$; the higher order terms (not shown above) have an analogous interpretation. It is important to note that, due to the reorganization the perturbation series in terms of the $T$ operators, the effect of the electron interacting with the [*same*]{} site successively has been already taken into account. Thus, in the sums representing the second and higher order terms it is necessary to restrict the summation to avoid revisiting the same site twice in a row, [*e.g.*]{} $\bar{\ell} \neq \bar{\bar{\ell}}$ in the second order term above. ![Plot of the change in electron energy as a function of linker density for cases of quenched, disordered positions (short dashes) and ordered positions (solid line). By assuming the independence of the binding sites so that each linker contributes $4 t^2-(4 t^2 +t'^2)^{1/2}$, one arrives at a reasonable linear approximation to the binding energy as a function of cross-link density (long dashes). Inset: Plot of the change in energy for an ordered lattice vs. 1/density. Both plots use $\lambda/2t =0.25$.[]{data-label="threemodelst"}](threemodels.eps){width="8.0cm"} We now choose the heretofore arbitrary on-site potential. One might imagine that it would be reasonable to choose that potential to simply be the mean of the on-site energies of the disordered system by taking $v = p \lambda $, however, we are not attempting to average the Hamiltonian over the disorder but rather we will be averaging the propagator. Thus, using the CPA approach, we choose that on-site potential to enhance the convergence of the perturbation series for $G$ given in Eq. \[reorganized\] by taking $v$ such that the average of the $T$ operator over the quenched disorder vanishes. Representing the disorder averages by $[\cdot ]$, we choose $v$ such that $$\label{v-picker} \left[ {\langle \ell|} T_{\ell} {|\ell\rangle} \right] = (1-p) \frac{-v }{1 + v {\langle \ell|} G_0 {|\ell\rangle} } + p \frac{\lambda - v}{1 - (\lambda - v) {\langle \ell|} G_0 {|\ell\rangle}} = 0.$$ Now, the reorganized perturbation series in Eq. \[reorganized\] is greatly simplified. Since $\left[ {\langle \ell|} T_{\ell} {|\ell\rangle} \right] = 0$ and $T$ operators at different sites are uncorrelated, $\left[ {\langle \ell|} T_{\ell} {|\ell\rangle} {\langle \ell'|} T_{\ell'} {|\ell'\rangle} \right] = \left[ {\langle \ell|} T_{\ell} {|\ell\rangle} \right] \cdot \left[{\langle \ell'|} T_{\ell'} {|\ell'\rangle} \right] = 0 $, we see that first correction to $G_0$ is fourth order in the $T$ operators. In the limit of weak scattering, $\left[G \right] \simeq G_0$ (with the advantageous choice for $v$ discussed above) makes an excellent approximation since the first correction is of order $(p \lambda/(2t))^4$. Finding the density of states in the disordered system requires that we evaluate the trace of the disorder–averaged propagator (Eq. \[DOS\]) as a function of $v$; we do this in the position representation by writing propagator in the uniform system as $$\label{prop} {\langle \ell |} G_0{|m\rangle} = G_0(\ell, m;E ) = \frac{a}{2 \pi} \int_{\pi/a}^{\pi/a} dk \frac{e^{ik a (\ell - n) }}{E - v + 2 t \cos (k a) }$$ and extracting the diagonal element $$\label{diagonal} G_0(\ell, \ell;E ) = \pm \left\{ (E-v)^2 - 4 t^2 \right)^{-1/2}$$ and the sign above is determined by whether the magnitude of $(E-v)/2t$ is greater or smaller than unity. In the uniform system the sum on position eigenstates is trivial as seen by the $\ell$-independence of the RHS of the above equation. However, determining $G_0(\ell, \ell;E ) $ and $v$ simultaneously requires that we solve two polynomial equations given by Eqs. \[v-picker\] and \[diagonal\]. The binding energy is determined by integrating Eq. \[DOS\] over the filled states. For convenience, we chose the Fermi level to again lie at $E_F=0$. The results are shown in Figs. \[threemodelst\], \[binding-e-plot\]. As with the ordered lattice of binding sites, we find that the binding energy of the disordered braided chains is very nearly a linear function of the density of sites. We can conclude that the primary effect of introducing binding points between the polymers is the creation of “impurity” bands centered at the energy given by Eq. \[bound-state\]. The states in the impurity bands are created at the expense of states near the band edges, $E=\pm 2t$. However, the density of states far from the band edges is essentially unchanged. This can be easily proved for a random arrangement of the impurities. We first note that the density of states is given by $$\begin{aligned} g(E)&=&\frac{1}{\pi}\frac{\partial \Phi}{\partial E} \\ &=&\frac{1}{\pi}\frac{\partial k}{\partial E}\left(N+Np\frac{\partial \overline{\theta}}{\partial k}\right),\end{aligned}$$ where $\Phi$ is the phase of the wavefunction at the last site on the chain, $p$ is the density of impurities, and $\overline{\theta}$ is the average phase-shift as the wavefunction passes through an impurity. For an impurity $\lambda$ at the origin $(\ell =0)$, the wavefunction takes the form $${|k\rangle}=\sum_{l\leq 0} \cos (kla +\phi){|l\rangle}+A\sum_{l> 0} \cos (kla +\phi+\theta){|l\rangle},$$ where $A$ is an amplitude to be determined. The phase-shift is given by $$\theta=\tan^{-1}\left(\tan(\phi)-\frac{\lambda}{t\sin(ka)}\right)-\phi.$$ Averaging over the incident phase, $\phi$, we find the average phase-shift $$\overline{\theta}=\frac{t \lambda \cos(ka)}{4t^2 \sin^2 (ka)+ \lambda^2}.$$ This shift is odd in the impurity strength, $\lambda$, therefore the change in the density of states from the symmetric and anti-symmetric bands will exactly cancel. Therefore, the reorganization of electronic energy must occur where this argument breaks down, namely when the separation between binding sites is comparable to the wavelength, [*i.e.*]{} when $\sin (ka)\simeq 0$. At this region near the band edges states are removed to form the imaginary wavenumber impurity states. The role of the impurity band as the source of the binding energy is further supported by the “single state" or independent binding site approximation (long dashes) line in figures \[threemodelst\] and \[binding-e-plot\]. This line is the binding energy that would be obtained if the sole effect of adding a binding site were to move a single electron from the bottom of the band $(E=-2t)$ to the energy of the isolated bound state $(E_b=-(4t^2+t'^2)^{1/2})$. This line is an excellent approximation for the binding energy of both the ordered lattice of binding sites (solid line) and the random array of binding sites (short dashes) which supports the suggestion that the creation of the impurity band is the dominant effect in the reorganization of electronic energy. ![Plot of the change in electron energy as a function of the interaction strength for random disorder (short dashes), perfectly ordered potential (solid line), and under the independent-linker assumption where each cross-link contributes $4t^2-(4t^2+t'^2)^{1/2}$ (long dashes). Here the cross-link density is $0.1$.[]{data-label="binding-e-plot"}](threemodels-t.eps){width="8.0cm"} Parallel Configurations {#railroad} ----------------------- If the chains adopt a parallel configuration, then there will be an overlap between each site and the corresponding site on the opposite chain. This interaction is easily diagonalized with the transformation given by Eq. \[symm\]. This yields the dispersion relation $$E_{\pm}(k)=-2t\cos{k}\pm t'.$$ We can solve for the binding energy of the polymers in this zipped configuration if the polymers are lightly doped (a general solution for the zipped configuration eigenstates may be found in Appendix A). If each polymer has $\delta/2$ electrons and $\delta \ll N$ then, prior to zipping, all electrons will have energy very near $-2t$. If the polymers bind over a length of $m$ monomers, then states are created with energy $E<-2t$ with a density of states $m(4t^2-(E+t')^2)^{-1/2}/\pi$. The electrons in these states have lowered their energy by $$\begin{aligned} \Delta E&=&-\frac{m}{\pi}\sqrt{4t^2-(-2t+t')^2} \nonumber \\ &&-\frac{m}{\pi}(2t-t')\left(\sin^{-1}(\frac{t'}{2t}-1)+\frac{\pi}{2}\right). \label{zippedE}\end{aligned}$$ This corresponds to a binding energy of $\simeq-2t'^{3/2}/3\pi \sqrt{t}$ per site. If the chains zip past a critical length $m_c=\delta\pi/\cos^{-1}(t'/2t-1)$ then all the electrons have energy $<-2t$. These electrons now have imaginary wavenumbers in the unzipped region of the chain, therefore the unzipped regions will no longer be metallic. The binding energy is now $$\Delta E=-\frac{2tm}{\pi}\sin (\frac{\delta \pi}{m})-\delta t'+2t\delta. \label{zipenergy}$$ Since the derivative of this binding energy with respect to the length of the binding region ($m$) is negative, we note that the electrons exert an effective pressure to increase the size of the zipped region [@Pincus:87]. When they form, we expect the length of the zipped regions to be limited only by the number of conduction electrons. The binding energy per binding site in a zipped region is $\sim t'^{3/2} t^{-1/2}$ whereas in the braided configurations the binding sites reduce the energy by $\sim t'^2 t^{-1}$. The zipped configuration exhibits stronger binding per site by $\sqrt{t/t'}$; since generically $t' < t$, the electronic degrees of freedom favor zipped configurations of the chains over braided ones. To determine whether a pair of chains will, however, form such zipped regions or the random braids, we must consider the energetics associated with the conformational degrees of freedom of these charged polymers in water. Conformational degrees of freedom of the chains and equilibrium structures {#polymer} ========================================================================== The equilibrium state of aggregating polymers depends on the combination of the binding energy due to the electronic degrees of freedom and the change in the translational/conformation free energy of the polymer. From the interplay of these two contributions to the total free energy, we explore the transition from free chains in solution to chain aggregates and discuss their equilibrium structure. We compare the free energies of (i) isolated chains in solution, (ii) braided pairs of chains consisting of unbound loops between isolated crossing points, and (iii) parallel or “zipped" configurations in which the chains have numerous consecutive tunnelling sites along their length. The aggregation of DNA in the presence of divalent ions has been approached in a similar manner[@Borukhov:01], however, in the present system there is no quantity analogous to the fixed concentration of linking divalent ions since the interchain binding sites are created spontaneously at chain intersections. We begin by considering the coexistence of isolated chains and paired ones in solution. The thermodynamic potential of a solution with a concentration of chains $n$ can be written up to a trivial constant as $$\frac{F(n_2)}{k_BT}=(n-2n_2)(\ln{(n-2n_2)}-1)+n_2(\ln{n_2}-1)+n_2\frac{{\cal E}_b}{k_BT}+n_2 \frac{S_0}{k_B}, \label{dimerenergy}$$ in terms of the concentration of bound-chain pairs $n_2$[@Landau:80]. In the above expression $S_0$ is the chain configurational entropy of one polymer: $S_0 \propto k_B L/l_p$. Its appearance in Eq. \[dimerenergy\] comes from the fact that in order to pair two chains, one of them must adopt the same configuration as the other and thereby lose its conformational entropy. The first two terms of Eq. \[dimerenergy\] represent only the translational entropy of the isolated chains and bound pairs; they do not incorporate this effect. Minimizing Eq. \[dimerenergy\], we find the concentration of dimers is $n^2 \exp ((-TS_0-{\cal E}_b)/k_BT)$. It remains now to calculate the interchain binding energy ${\cal E}_b$, which is a combination of electrostatic chain repulsion and chain binding due to the proposed tunnelling mechanism. We do not include van der Waals interactions between chains. This longer-ranged attractive interaction is generically stronger for the conducting polymers since the extended electronic states enhance their low-frequency polarizability. At higher frequencies one expects that the polarizability at higher frequencies is typical of small organic molecules since the dielectric spectrum at higher frequencies is primarily due to localized molecular states that are similar to all hydrocarbons. Such van der Waals interactions lead to a longer range attraction that, in high salt concentrations at least, are a subdominant correction to the interchain binding potential when the interchain distance is on the order of Angstroms. In addition the enhancement of the Van der Waals interaction is significant for only the metallic rather than the semiconducting state of the polymers. Thus we expect that the interchain tunnelling mechanism discussed in this article to dominate the binding energy of the observed bundles. We examine this point further in appendix B. We first consider the binding energy of the more loosely bound, braided configuration of the molecules, ${\cal E}_b^{braid}$. The binding energy is a combination of the electrostatic interaction of the chains, the bending of the molecules on scales short compared to their persistence length, and the electronic binding energy of the crossing point. At high salt the first of these terms is significant only near the crossing points. There the charged chains can be treated as crossed straight rods. If two charged rods cross at an angle $\theta$ with a distance of closest approach $d$ the electrostatic energy is $$E_{ES}=\frac{2\pi k_B T l_B e^{-\kappa d}}{\kappa a_c^2 \sin{\theta}},$$ where $l_B=e^2/\epsilon k_B T$ is the Bjerrum length, $a_c$ is the average separation between charges along the backbone, and $\kappa^{-1}$ is the Debye screening length. The bending energy of a polymer is $$E_{bend}=\frac{k_B T l_p}{2}\int \frac{dl}{R(l)^2},$$ where $l_p$ is the persistence length, and $R(l)$ is the local radius of curvature. If the separation between interaction points is comparable to the persistence length, then in the limit $\kappa^{-1}\ll l_p$ the crossing angle will be very close to $\pi/2$ and the arc the polymers follow will be very nearly circular. Then the energy per length of the polymers, $L$, is $$\label{braid-energy} E_{braid}(l)/L=(\frac{2\pi k_B T l_B e^{-\kappa d}}{\kappa a_c^2}-\frac{t'^2}{2t})/l+\frac{k_B T l_p\pi^2}{4l^2},$$ where $l$ is the arc length of the polymer between intersections. Minimizing with respect to $l$, the energy becomes $${\cal E}_b^{braid}=-\frac{(\frac{2\pi k_B T l_B e^{-\kappa d}}{\kappa a_c^2}-\frac{t'^2}{2t})^2}{k_B T l_p\pi^2}L. \label{Ebraid}$$ Here we have neglected the configurational entropy of the loops of chain making up the braid. As long as $l<l_p$ we expect this entropic contribution to be negligible. We now turn to the zipped configuration of the chains. To consider the electrostatic interaction between the two chains, we note that the potential, $\Phi$, around a thin charged rod in a salt solution is given by $$\Phi(r)=\frac{2l_B k_B T}{a_c}K_0(\kappa r), \label{potential}$$ so the energy of two polymers separated by a distance $d$ over a length $L$ is $${\cal E}_b^{zip}(L)= L \frac{2l_B k_B T}{a_c^2}K_0(\kappa d)-L\frac{2t}{\pi a_t}\sin \left(\frac{\delta \pi a_t}{L}\right)-t'\delta+2t\delta \label{Ezip}$$ where $K_0$ is the modified Bessel function, the distance between tight-binding sites is $a_t$, and we have assumed the binding energy is given by Eq. \[zipenergy\]. We observe that both Eqs. \[Ebraid\] and \[Ezip\] depend on the length over which the binding occurs. So, before we use either of these expressions in Eq. \[dimerenergy\] we must choose the suitable length, $L^\star$. For braided chains $L^\star$ is the length of the entire chain and for zipped chains $L^\star$ is determined from \[Ezip\] yielding $L^\star\simeq 2t\delta a_c^2/l_B k_B T K_0(\kappa d)$. Note that $S_0$ in Eq. \[dimerenergy\] is also proportional to $L^\star$. Based on these considerations we propose a schematic phase diagram for charged, conducting polymers in solution. This diagram (shown in figure \[phase-diagram\]), which represents the range of phase behavior of these polymers, is spanned by the persistence length and the (suitably scaled) inverse, linear charge density of the chains. This inverse charge density is measured in terms of the ratio of the distance $a_c$ between charged groups along the polymer backbone divided by the distance between potential tunnelling sites, $a_t$. In the upper right portion of the figure the chains are both stiff and have a high charge density relative to their density of potential binding sites. Thus, they are unbound in equilibrium. Generally, moving down and to the right in the diagram induces intermolecular binding due the mechanism presented in this work. The equilibrium bound configurations take the form of parallel, tightly bound pairs, which we denote as zipped chains or as more loosely bound braids. The onset of braiding is determined by setting that contour length between binding sites equal to the total contour length of the chain, $L$. This gives the free-chains/braid boundary as $$\label{free-braid-boundary} \frac{\ell_p}{L} = \frac{2}{k_{\rm B} T \pi^2} \left( \frac{t'^2}{2 t} - \frac{ 2 \pi k_{\rm B} T \ell_p e^{-\kappa d}}{\kappa a_c^2 } \right).$$ The term in parenthesis on the right-hand side of the above equation is collectively the effective binding of one interchain crossing. The first part of that term is the binding energy due to the reorganization of the electronic states of the polymers while the second term represents the reduction in binding energy associated with the electrostatic interaction of the two chains. For the cases of current interest in which a single interchain bond is favorable the right-hand side of Eq. \[free-braid-boundary\] is positive. ![Schematic phase diagram for two charged, conducting polymers. This diagram is spanned on the vertical axis by the persistence length scaled by the total contour length of the chain while the horizontal axis is the dimensionless ratio of the the distance between charges along the chain $a_c$ to the distance between potential tunnelling sites $a_t$. In the upper left region of the figure, chains of high charge density and long persistence length are unbound in solution (free); upon reducing the linear charge density along the chain and moving to the right in the diagram the chains generically become bound in a parallel or zipped configuration. For flexible enough chains of intermediate charge density, however, more loose, braided configurations are predicted.[]{data-label="phase-diagram"}](phasediagram.eps){width="8.0cm"} To explore the boundary between zipped and braided states of the polymers we compare the energy of the braided configuration given by Eq. \[Ebraid\] to the energy of the zipped configuration. This latter energy is a combination of the binding energy of a zipped region given by Eq. \[Ezip\]. For the case of lightly doped chains so that contour length of the zipped regions $L^\star$ is smaller than that of the whole polymer, one finds that the boundary between braided and zipped chain configurations to occur where $$\label{braid-zip} \frac{\ell_p}{L} = \frac{\left( \frac{t'^2}{2 t} - \frac{2 \pi k_{\rm B} T \ell_p e^{-\kappa d}}{\kappa a_c^2} \right)^2}{ L^\star k_{\rm B} T \pi^2 \left( \frac{2 t'^{3/2}}{2 \pi a_t t^{1/2}} - 2 \ell_p k_{\rm B} T K_0(\kappa d) a_c^{-2} \right) }.$$ This phase boundary continues to smaller $a_c/a_t$ until it intersects the free/braid boundary discussed above. There it terminates. Finally, there is a vertical line separating free and zipped chains that intercepts the intersection point of the free/braid and braid/zipped boundaries. This last separation is independent of chain persistence length since the energy of neither the zipped configurations nor the free chains depends on the persistence length. Discussion {#conclusions} ========== We find that conducting polymers in the metallic state, [*i.e.*]{} where the Fermi level lies within a band, will generically aggregate in solution due to the reorganization of the electronic degrees of freedom of these molecules upon the close (Angstrom-scale) approach to each other. This reorganization of the extended electronic states of the molecule results in the formation of localized bound states at these crossing-points and the consequent reduction of the electronic energy more that compensates for the reduction of chain configurational entropy due to the creation of the cross-links between the two polymers. It is interesting to note that these localized bound states are the direct analogs of the standard binding/anti-binding orbitals that are created out of atomic electronic states upon the close approach of their respective nuclei. In the case of conducting polymers, however, these localized bound states are created out of the extended states of the metallic molecules. The localized nature of these states has the consequence that the binding energy is very nearly linear in the density of binding sites when these sites are separated by at least a few monomers as they are in all braided configurations. However, when the polymers lie in parallel, zipped configurations, the resulting binding energy per site is stronger than the isolated binding sites by $O((t/t')^{1/2})$. The equilibrium structures formed by these polymers will depend not only on the interchain binding mechanism, but also the linear charge density of the polymers, solvent screening, and the persistence length of the polymer. The combination of these factors will determine whether the polymers form loosely braided structures, or tight bundles. Based on these considerations we have proposed a schematic phase diagram showing the expected equilibrium structures that may be observed in such systems. For large enough linear charge density, one finds that the lowest energy state of the aggregate is a braid consisting of loops of unbound polymer between cross-linking binding sites. For smaller linear charge density one finds that the energetically favorable configuration is two parallel chains where the interchain distance remains on the order of a few Angstroms all along their arc length. Clearly, the direct experimental observation of such structures is difficult. We note that SANS experiments [@Wang:01; @Wang:01b] are consistent with the formation of bundles in aqueous solutions, but the spatial resolution of these experiments is not fine enough to distinguish between the braid and parallel phases discussed above. Moreover, the current calculations are not directly applicable to these experiments since previous experimental studies focussed on semiconducting polymers. Nevertheless, our proposed binding mechanism mediated by the reorganization of the electronic degrees of freedom is also relevant to the semiconducting case[@Schmit:04]. Since it will likely remain difficult to quantitatively test the detailed structure of the aggregate one must directly probe the electronic states by studying the spectroscopic signature of the molecules as they aggregate. The authors would like to thank F. Pincus, G. Bazan, and A.J. Heeger for stimulating conversations. JDS would also like to thank D. Scalapino and L. Balents for helpful discussions. JDS acknowledges the hospitality of the University of Massachusetts, Amherst. This work was supported in part by the MRSEC Program of NSF DMR00-80034. Zipped configuration states =========================== Here we solve for the eigenstates for two chains that are zipped together over a portion of their length. The Hamiltonian is $$H_0 = -t \sum_{\ell=1}^{N-1} \left( |\ell + 1\rangle \langle \ell | + |\ell \rangle \langle \ell +1 | \right) + \sum_{\ell =1 }^{N-1} v_l{|\ell\rangle} {\langle \ell|}$$ where $v_l=0$ in the unzipped regions ($1\leq l \leq n$ and $n+m+1 \leq l \leq N$) and $v_l=\lambda$ in the zipped region ($n+1 \leq l \leq n+m$). Here $\lambda=\mp t'$ for the symmetric (anti-symmetric) states, $m$ is the number of sites in the zipped region, and $n(n')$ is the number of sites in the unzipped region to the left(right) of the zipped region. The chains have a total length of $N=n+m+n'$. The eigenfunctions are of the form $$\begin{aligned} {|k\rangle}=&&\sum_{l=1}^n \sin (lk) {|l\rangle}+A\sum_{l=n+1}^{n+m} \sin (lk'+\phi){|l\rangle} \nonumber \\ &&+B\sum_{l=n+m+1}^{N} \sin (lk+\theta){|l\rangle},\end{aligned}$$ where $A$ and $B$ are undetermined amplitudes, and $-2t\cos k=-2t\cos k' +\lambda$. The boundary conditions are $$\begin{aligned} \sin ((n+1)k)&=&A\sin ((n+1)k'+\phi) \nonumber \\ \sin (nk)&=&A\sin (nk'+\phi) \nonumber \\ A\sin((n+m+1)k'+\phi)&=&B\sin((n+m+1)k+\theta) \nonumber\\ A\sin((n+m)k'+\phi)&=&B\sin((n+m)k+\theta) \nonumber \\ (N+1)k+\theta&=&j\pi,\end{aligned}$$ where $j$ is an integer. After some algebra, we find two equations for $k$ and $\phi$ $$\begin{aligned} \frac{\sin (k n')}{\sin(k(n'+1))}&=&\frac{\sin(k'(n+m+1)+\phi)}{\sin(k'(n+m)+\phi)} \\ \cot (k'n+\phi)&=&\frac{\cot(kn)\sin (k)-\lambda}{\sin (k')}.\end{aligned}$$ From these equations we find the quantization condition for $k$ $$\xi \xi'+(\xi+\xi')\sin(k')\cot (k'(m+1))-\sin^2 (k')=0, \label{zippedquant}$$ where $$\begin{aligned} \xi&\equiv&\sin (k) \cot(nk)-\lambda \nonumber \\ \xi'&\equiv&\sin (k) \cot(n'k)-\lambda. \label{xi}\end{aligned}$$ From these conditions (Eqs. \[zippedquant\], \[xi\]) and the dispersion relation $$E_k=-2t\cos (k),$$ one obtains the energy eigenvalues. Intermolecular Van der Waals Attraction ======================================= The van der Waals interaction energy between two like molecules in a screened medium is given by $$E_{vdw}\simeq -I\frac{\alpha^2_0 e^{-2 \kappa r}}{\epsilon^2 r^6},$$ where $\alpha_0$ is the polarizability of the molecules, $\epsilon$ is the dielectric constant of the medium, $\kappa^{-1}$ is the Debye screening length, and $I$ is the ionization energy of the molecules[@Israelachvili:92]. At wide separations the polymers can be approximated as conducting spheres with a radius $R_g$, thus $\alpha_0 \simeq R_g^3$. In aqueous solutions $\epsilon \simeq 80$ and for an $I$ of order eV then $E_{vdw}\ll 1eV$ since $r\gg R_g$. If, on the other hand, $r\lesssim R_g$ it is more accurate to describe the polymers as narrow ellipsoids. In this situation we approximate the polymers as ellipsoids of length $l_p$ and thickness $a$ that is a monomeric dimension. The polarizability along the wire is found to be of order[@Landau:84]. $$\alpha_{\parallel}\simeq \frac{l_p^3}{\ln{\frac{l_p}{a}}},$$ and perpendicular to the wire it is $$\alpha_{\perp}\simeq a^2 l_p.$$ So, the interaction energy for parallel and perpendicular wires is $$\begin{aligned} E_{\parallel} & \simeq & I \frac{l_p^6 e^{-2\kappa r}}{\epsilon^2 r^6 (\ln{\frac{l_p}{a}})^2} \\ E_{\perp} & \simeq & I \frac{l_p^2 a^4 e^{-2\kappa r}}{\epsilon^2 r^6}.\end{aligned}$$ For $l_p \simeq 10$nm, $\epsilon \simeq 80$, $a \simeq r \simeq 1/2$nm, $I\simeq 1$eV, and $\kappa^{-1}\simeq 1$nm, we find that $E_{\parallel}\simeq 10^4 k_BT$ and $E_{\perp}\simeq 2 k_BT$. At high salt concentrations, when the electrostatic screening length $\kappa^{-1} \simeq 2 \AA$, these dipolar interactions are suppressed by a factor of $\simeq 10^4$ suggesting that the binding mechanism discussed in the article dominates at short interchain separation. 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--- author: - bibliography: - 'xu.2015.bib' title: 'Hierarchical image simplification and segmentation based on Mumford-Shah-salient level line selection' --- Introduction {#sec:introduction} ============ In natural images, meaningful contours are usually smooth and well-contrasted. Many authors ([*e.g.*]{}, [@caselles.99.ijcv; @cao.05.jmiv]) claim that significant contours of objects in images coincide with segments of the image level lines. The level lines are the boundaries of the connected components described by the [*tree of shapes*]{} proposed in [@monasse.00.itip], and also known as [*topographic map*]{} in [@caselles.99.ijcv]. Image simplification or segmentation can then be obtained by selecting meaningful level lines in that tree. This subject has been investigated in the past by [@pardo.02.icip; @cao.05.jmiv; @cardelino.06.icip]. In [@lu.07.itip], the authors have proposed a tree simplification method for image simplification purpose based on the binary partition tree. Classically, finding relevant contours is often tackled using an energy-based approach. It involves minimizing a two-term-based energy functional of the form $E_{\lambda_s} = \lambda_s C + D$, where $C$ is the regularization term controlling the regularity of contours, $D$ is a data fidelity term, and $\lambda_s$ is a parameter. A popular example is the seminal work of [@mumford.89.cpam]. Curve evolution methods are usually used to solve this minimization problem. They have solid theoretical foundations, yet they are often computational expensive. Current trends in image simplification and segmentation are to find a multiscale representation of the image rather than a unique partition. There exist many works about hierarchical segmentations such as the geodesic saliency of watershed contours proposed in [@najman.96.pami] and gpb-owt-ucm proposed by [@arbelaez.11.pami] and references therein. Some authors propose to minimize a two-term-based energy functional subordinated to a given input hierarchy of segmentations, in order to find an optimal hierarchical image segmentations in the sense of energy minimization. Examples are the works of [@guigues.06.ijcv; @kiran.14.pr]. Yet, the choice or the construction of the input hierarchy of segmentations for these methods is an interesting problem in itself. [@perret.15.ismm] compared different choices of morphological hierarchies for supervised segmentation. In this paper we propose a novel hierarchical image simplification and segmentation based on minimization of an energy functional ([*e.g.*]{}, the piecewise-constant Mumford-shah functional). The minimization is performed subordinated to the shape space given by the tree of shapes, a unique and equivalent image representation. The basis of our proposal was exposed in our previous study in [@xu.13.icip], in which we proposed an efficient greedy algorithm computing a locally optimal solution of the energy minimization problem. The basic idea is to take into account the meaningfulness of each level line [ which measures its “importance”. An example of meaningfulness function that we will use through the paper is the average of gradient’s magnitude along level lines. The order based on these meaningfulness values allows to get very quickly a locally optimal solution, which yields a well-simplified image while preserving the salient structures.]{} The current paper extends this idea to hierarchical simplification and segmentation. More precisely, following the same principle but without fixing the parameter $\lambda_s$ in the two-term-based energy, we compute an attribute function that characterizes the persistence of each shape under the energy minimization. Then we compute a saliency map, a single image representing the complete hierarchical simplifications or segmentations. To do so, we rely on the idea of hierarchy transformation via extinction value proposed by [@vachier.95.nlsp] and on the framework of tree-based shape space introduced in [@xu.14.filter]. This scheme of hierarchy transformation has been first used in [@xu.13.ismm] for a different input hierarchy and attribute function. Related algorithms were presented in [@xu.15.ismm]. The present paper extends on these ideas, focusing on the computation of an attribute function related to energy minimization. The main contribution of this current paper is [the proposition of a general framework of]{} hierarchical image simplification and segmentation method based on energy minimization subordinated to the tree of shapes, [contrary to the classical approaches that are subordinated to an initial hierarchy of segmentations. It is based on the introduction of a novel attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$ related to energy minimization. ]{} We have tested [the proposed framework]{} with a very simple segmentation model in this paper. Despite its simplicity, we obtain results that are competitive with the ones of some state-of-the-art methods on the classical segmentation dataset from [@alpert.12.pami]. In particular, they are on par with Gpb-owt-ucm proposed in [@arbelaez.11.pami] on this dataset. The rest of this paper is organized as follows: Some background information is provided in Section \[sec:background\]. Section \[sec:method\] is dedicated to depict the proposed method, followed by some illustrations and experimental results in Section \[sec:illustration\]. Section \[sec:relatedwork\] compares the proposed method with some similar works. We then conclude in Section \[sec:conclusion\]. Background {#sec:background} ========== The Tree of shapes {#subsec:tos} ------------------ For any $\lambda \in \mathbb{R} \textrm{ or } \mathbb{Z}$, the upper level sets ${\ensuremath{\mathcal{X}}\xspace}_\lambda$ and lower level sets ${\ensuremath{\mathcal{X}}\xspace}^\lambda$ of an image $f: \Omega \rightarrow \mathbb{R} \textrm{ or } \mathbb{Z}$ are respectively defined by ${\ensuremath{\mathcal{X}}\xspace}_\lambda(f) = \{ p \in \Omega \mid f(p) \ge \lambda \}$ and ${\ensuremath{\mathcal{X}}\xspace}^\lambda(f) = \{ p \in \Omega \mid f(p) \le \lambda \}.$ Both upper and lower level sets have a natural inclusion structure: $\forall \, \lambda_1 \leq \lambda_2, \; {\ensuremath{\mathcal{X}}\xspace}_{\lambda_1} \supseteq {\ensuremath{\mathcal{X}}\xspace}_{\lambda_2} \,\mbox{~and~}\, {\ensuremath{\mathcal{X}}\xspace}^{\lambda_1} \subseteq {\ensuremath{\mathcal{X}}\xspace}^{\lambda_2},$ which leads to two distinct and dual representations of an image, the max-tree and the min-tree. ![An image (left) and its tree of shapes (right).[]{data-label="fig:tree"}](tree_of_shapes){width="0.8\linewidth"} Another tree has been introduced in [@monasse.00.itip] via the notion of shapes. A *shape* is defined as a connected component of an upper or lower level set where its holes have been filled in. Thanks to the inclusion relationship of both kinds of level sets, the set of shapes gives a unique tree, called *tree of shapes*. This tree is a self-dual, non-redundant, and complete representation of an image. It is equivalent to the input image in the sense that the image can be reconstructed from the tree. And it is invariant to affine contrast changes. Such a tree also inherently embeds a morphological scale-space (the parent of a node/shape is a larger shape). An example on a synthetic image is depicted in Fig. \[fig:tree\]. Recently, an extension of the tree of shapes for color images has been proposed by [@carlinet.15.itip] through the inclusion relationship between the shapes of its three grayscale channels. Hierarchy of image segmentations or saliency maps {#subsec:hos} ------------------------------------------------- A hierarchy of image segmentation $H$ is a multiscale representation that consists of a set of nesting partitions from fine to coarse: $H = \{\mathcal{P}_i \, | \, 0 \leq i \leq n, \forall j, k, \, 0 \leq j \leq k \leq n \Rightarrow \mathcal{P}_j \sqsubseteq \mathcal{P}_k\},$ where $\mathcal{P}_n$ is the partition $\{\Omega\}$ of $\Omega$ into a single region, and $\mathcal{P}_0$ represents the finest partition of the image $f$. $\mathcal{P}_j \sqsubseteq \mathcal{P}_k$ implies that the partition $\mathcal{P}_j$ is finer than $\mathcal{P}_k$, which means $\forall \, R \in \mathcal{P}_j, \exists \, R' \in \mathcal{P}_k$ such that $R \subseteq R'$. As a multiscale representation, a hierarchy of segmentation satisfies the most fundamental principle for multiscale analysis: the causality principle presented by [@koenderink.84.bc]. From this principle, for any couple of scales $\lambda_{s_2} > \lambda_{s_1}$, the “structures” found at scale $\lambda_{s_2}$ should find a “cause” at scale $\lambda_{s_1}$. In the case of a hierarchy of segmentation, following the work of [@guigues.06.ijcv], the causality principle is applied to the edges associated to the set of partitions spanned by $H$: for any pair of scales $\lambda_{s_2} > \lambda_{s_1}$, the boundaries of partition $\mathcal{P}_{\lambda_{s_2}}$ are in a one-to-one mapping with a subset of the boundaries of $\mathcal{P}_{\lambda_{s_1}}$ (their “cause”). The pair $(H, \lambda_s)$ is called an indexed hierarchy. A useful representation of hierarchical image segmentations was originally introduced in [@najman.96.pami] under the name of [ *saliency map*]{}. A saliency map is obtained by stacking a family of hierarchical contours. This representation was then rediscovered independently by [@guigues.06.ijcv] through the notion of scale-set theory for visualization purposes, and it is then popularized by [@arbelaez.11.pami] under the name of [ *ultrametric contour map*]{} for boundary extraction and comparing hierarchies. It has been proved theoretically in [@najman.11.jmiv] that a hierarchy of segmentations is equivalent to a saliency map. Roughly speaking, for a given indexed hierarchy $(H, \lambda)$, the corresponding saliency map can be obtained by weighing each contour of the image domain with the highest value $\lambda_s$ such that it appears in the boundaries of some partition represented by the hierarchy $H$. The low level (resp. upper level) of a hierarchy corresponds to weak (resp. strong) contours, and thus an over-segmentation (resp. under-segmentation) can be obtained by thresholding the saliency map with low (resp. high) value. From shape-space filtering to hierarchy of segmentations {#subsec:ssf} -------------------------------------------------------- The three morphological trees reviewed in Section \[subsec:tos\] and the hierarchies of segmentations reviewed in Section \[subsec:hos\] have a tree structure. Each representation is composed of a set of connected components $\mathbb{C}$. Any two different elements $C_i \in \mathbb{C}, C_j \in \mathbb{C}$ are either disjoint or nested: $\forall \, C_i \in \mathbb{C}, \; C_j \in \mathbb{C}, C_i \, \cap \, C_j \neq \emptyset \, \Rightarrow \, C_i \subseteq C_j \textrm{ or } C_j \subseteq C_i$. This property leads to the definition of [ *tree-based shape space*]{} in [@xu.14.filter]: a graph representation $G_{\mathbb{C}} = (\mathbb{C}, E_{\mathbb{C}})$, where each node of the graph represents a connected component in the tree, and the edges $E_{\mathbb{C}}$ are given by the inclusion relationship between connected components in $\mathbb{C}$. In [@xu.14.filter], we have proposed to filter this shape space by applying some classical operators, notably connected operators on $G_{\mathbb{C}}$. We have shown that this shape-space filtering encompasses some classical connected operators, and introduces two families of novel connected operators: shape-based lower/upper leveling and shaping. Instead of filtering the shape space, another idea is to consider each region of the shape space as a candidate region of a final partition. For example, we weigh the shape space by a quantitative [attribute]{} ${\ensuremath{\mathcal{A}}\xspace}$. Then each local minimum of the node-weighted shape space is considered as a candidate region of a partition. The importance of each local minimum ([*i.e.*]{}, each region) can be measured quantitatively by the extinction value ${\ensuremath{\mathcal{E}}\xspace}$ proposed by [@vachier.95.nlsp]. Let $\prec$ be a strict total order on the set of minima $m_1\prec m_2 \prec\ldots$, such that $m_i\prec m_{i+1}$ whenever ${\ensuremath{\mathcal{A}}\xspace}(m_i) < {\ensuremath{\mathcal{A}}\xspace}(m_{i+1})$. Let $CC$ be the lowest lower level connected component (defined on the shape space) that contains both $m_{i+1}$ and a minimum $m_j$ with $j<(i+1)$. The extinction value for the minimum $m_{i+1}$ is defined as the difference of level of $CC$ and ${\ensuremath{\mathcal{A}}\xspace}(m_{i+1})$. An example of extinction values for three minima is depicted in Fig \[fig:extinction\]. We weigh the boundaries of the regions corresponding to the local minima with the extinction values. This yields a saliency map representing a hierarchical image simplification or segmentation. This scheme allows to transform any hierarchical representation into a hierarchical segmentation. It has been firstly used in [@xu.13.ismm], where the input hierarchy is a minimum spanning tree and the attribute is computed locally inspired from the work of [@felzenszwalb.04.ijcv]. For the sake of completeness, the algorithm [@xu.15.ismm] for the extinction-based hierarchy transformation is presented in Section \[subsec:algorithm\]. ![Illustration of the extinction values ${\ensuremath{\mathcal{E}}\xspace}$ of three minima. The order is $A \prec C \prec B$. $B$ merges with $C$, $C$ merges with $A$.[]{data-label="fig:extinction"}](extinct){width="0.6\linewidth"} Energy-based simplification and segmentation {#subsec:mumford} -------------------------------------------- There exist several works of hierarchical image segmentations based on energy minimization (see [@guigues.06.ijcv] and [@kiran.14.pr]). A general formulation of these methods involves minimizing a two term-based energy functional of the form $E_{\lambda_s} = \lambda_s C + D$. $C$ is a regularization term, $D$ is data fidelity term, and $\lambda_s$ is a parameter. Let $ \{R\} = R_1 \sqcup \dots \sqcup R_n$ be a partition of the image domain. If the energy functional can be written by $E_{\lambda_s} = \sum_{R_i \in \{R\}} \big(\lambda_s C(R_i) + D(R_i)\big)$, then $E_{\lambda_s}$ is called an affine separable energy functional. Furthermore, if either the regularization term $C$ decreases or the data fidelity term $D$ increases, the energy $E_{\lambda_s}$ is multiscale affine separable. A popular instance of such an energy functional that we will use as an example through this paper is the piecewise-constant Mumford-Shah functional proposed by [@mumford.89.cpam]. For an image $f$, it is given by $$E_{\lambda_s}(f, \partial \{R\}) \,= \iint_{\{R\}} \!\! (\tilde{f}_i - f)^2 \, dx dy \,+\, \lambda_s \, |\partial \{R\}|, \label{eq:simplemumford}$$ where $\tilde{f}_i = \frac{1}{|R_i|} \iint_{R_i} f \, dx dy$ inside each region $R_i \in \{R\}$, $\partial \{R\}$ is the set of contour, and $|\cdot|$ denotes the cardinality. Hierarchical image simplification and segmentation via level line selection {#sec:method} =========================================================================== Main idea {#subsec:mainidea} --------- The current proposal [is a general framework of hierarchical image simplification and segmentation based on energy minimization subordinated to the tree of shapes. It extends]{} a preliminary version of this study in [@xu.13.icip] [that selects]{} salient level lines based on Mumford-Shah energy functional minimization. We review this non-hierarchical version in Section \[subsec:llselection\], using a more general multiscale affine separable energy. [The hierarchical version proposed in the current paper is achieved thanks to:]{} - the introduction of a novel attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$ for each level line related to the energy regularization parameter $\lambda_s$, - and the idea of hierarchy transformation based on extinction values and on a tree-based shape space. This is detailed in Section \[subsec:hslls\]. Section \[subsec:algorithm\] provides algorithms for the whole process. Image simplification by salient level line selection {#subsec:llselection} ---------------------------------------------------- For a given tree of shapes ${\ensuremath{\mathcal{T}}\xspace}$ composed of a set of shapes $\{\tau_i\}$, any two successive shapes of ${\ensuremath{\mathcal{T}}\xspace}$ are related by an edge reflecting the inclusion relationship, also known as the parenthood between nodes of the tree. This tree structure ${\ensuremath{\mathcal{T}}\xspace}$ provides an associated partition of the image $\{R_{\ensuremath{\mathcal{T}}\xspace}\} = R_{\tau_1} \sqcup \dots \sqcup R_{\tau_n}$, where $R_\tau = \{p \, | \, p \in \tau, p \notin \mathit{Ch}(\tau) \}$ with $\mathit{Ch}(\tau)$ representing all the children of the shape $\tau$. We denote by $E_{\lambda_s}(f, {\ensuremath{\mathcal{T}}\xspace})$ the energy functional (see Section \[subsec:mumford\]) subordinated to the tree by considering its associated partition $\{R_{\ensuremath{\mathcal{T}}\xspace}\}$. This energy minimization is given by: $$\underset{{\ensuremath{\mathcal{T}}\xspace}'}{\min} \; E_{\lambda_s}(f, {\ensuremath{\mathcal{T}}\xspace}'), \label{eq:solution}$$ where ${\ensuremath{\mathcal{T}}\xspace}'$ is a simplified version of ${\ensuremath{\mathcal{T}}\xspace}$ by removing some shapes from ${\ensuremath{\mathcal{T}}\xspace}$ and by updating the parenthood relationship. The basic operation of the energy minimization problem given by Eq.  is to remove the contours of some shapes $\{\tau\}$ included in their corresponding parents $\{\tau_p\}$, which triggers the update of $R'_{\tau_p} \, = \, R_{\tau_p} \cup R_\tau$ for each shape $\tau$. The parent of its children $\tau_{c1}, \dots , \tau_{ck}$ should also be updated to the $\tau_p$. Fig. \[fig:merging\] shows an example of a such merging operation. ![Suppressing the node $\tau$ makes the “region” $R_\tau$ (covered with red oblique lines) merge with $R_{\tau_p}$; the result (depicted in the right image) is a simplified image.[]{data-label="fig:merging"}](remove_node){width="0.8\linewidth"} Observe that the minimization problem of Eq. (\[eq:solution\]) is a combinatorial optimization. The computation of the optimum has an exponential complexity. Hence a greedy algorithm is usually applied to compute a local optimum instead of a global optimum (see also [@ballester.07.jmiv]). It iteratively removes the shapes to decrease the energy functional. The greedy algorithm stops when no other shape can be removed that favors a decrease of the energy. The removability of a shape $\tau$ is decided by the sign of the energy variation $\Delta E_{\lambda_s}^\tau$ while $\tau$ is suppressed. For the multiscale affine separable energy described in Section \[subsec:mumford\], $\Delta E_{\lambda_s}^\tau$ is given by: $$\Delta E_{\lambda_s}^\tau = D(R'_{\tau_p}) - D(R_{\tau}) - D(R_{\tau_p}) - \lambda_s\big(C(R_{\tau}) + C(R_{\tau_p}) - C(R'_{\tau_p})\big). \label{eq:variation}$$ Taking the piecewise-constant Mumford-Shah functional given by Eq.  as an energy example, and let $S(f, R_i)$ be the sum of value of all the pixels inside $R_i$, Then the functional variation $\Delta E_{\lambda_s}^\tau$ is given by: $$\Delta E_{\lambda_s}^\tau = \frac{S^2(f,{R_\tau})}{|R_\tau|} + \frac{S^2(f, R_{\tau_p})}{|R_{\tau_p}|} - \frac{S^2(f, R'_{\tau_p})}{|R'_{\tau_p}|} - \lambda_s |\partial \tau|. \label{eq:variationmumford}$$ If $\Delta E_{\lambda_s}^\tau$ is negative, which means the suppression of $\tau$ decreases the functional, then we remove $\tau$. According to Eq. , the removability of a shape $\tau$ depends only on $R_\tau$ and $R_{\tau_p}$. As the suppression of the shape $\tau$ triggers the update of $R_{\tau_p}$, the removal of $\tau$ impacts also the removability of its parent, its children and siblings. So the order of level line removal is critical. In [@xu.13.icip], we proposed to fix the order by sorting the level lines in increasing order of a quantitative meaningfulness attribute ${\ensuremath{\mathcal{A}}\xspace}$ ([*e.g.*]{}, the average of gradient’s magnitude along the level line ${\ensuremath{\mathcal{A}}\xspace}_\nabla$). Meaningful contours in natural images are usually well-contrasted and smooth. Indeed, the minimization of energy functional in Eq (\[eq:simplemumford\]) favors the removal of level lines having small contrast (by data fidelity term) or being complex (by regularization term). So the shapes having small ([*resp.*]{}great) attribute ${\ensuremath{\mathcal{A}}\xspace}_\nabla$ are easier ([*resp.*]{}more difficult) to filter out under the energy minimization process of Eq. . Consequently, the level line sorting based on attribute ${\ensuremath{\mathcal{A}}\xspace}_\nabla$ provides a reasonable order to perform the level lines suppression that makes the energy functional decrease. Indeed, initially, each region $R_\tau$ has only several pixels. At the beginning, many “meaningless” regions are removed, which forms more proper regions in Eq.  for the “meaningful” regions. The removal decisions for these “meaningful” regions based on the sign of Eq.  are more robust. Hierarchical salient level line selection {#subsec:hslls} ----------------------------------------- The parameter $\lambda_s$, in the multiscale affine separable energy, controls the simplification/segmentation degree for the method described in Section \[subsec:llselection\], which is however not hierarchical. Because some level line $\tau$ may be removed with a parameter $\lambda_{s_1}$, but preserved for a bigger $\lambda_{s_2} > \lambda_{s_1}$. This contradicts the causality principle for hierarchical image simplification/segmentation described in Section \[subsec:hos\]. An example is given in Fig. \[fig:causality\]. Note that the simplification algorithm of [@ballester.07.jmiv] is not hierarchical either. [0.31]{} ![Illustration of causality principle violation. Left: input image; Middle and right: randomly colorized simplified image with $\lambda_{s_1} = 100$ and respectively $\lambda_{s_2} = 500$. The orange shape on top middle of the right image (surrounded by a black circle) is preserved for $\lambda_{s_2} = 500$, while it is removed for $\lambda_{s_1} = 100$ in the middle image.[]{data-label="fig:causality"}](plane "fig:"){width="1.0\linewidth"}   [0.31]{} ![Illustration of causality principle violation. Left: input image; Middle and right: randomly colorized simplified image with $\lambda_{s_1} = 100$ and respectively $\lambda_{s_2} = 500$. The orange shape on top middle of the right image (surrounded by a black circle) is preserved for $\lambda_{s_2} = 500$, while it is removed for $\lambda_{s_1} = 100$ in the middle image.[]{data-label="fig:causality"}](plane_100 "fig:"){width="1.0\linewidth"}   [0.31]{} ![Illustration of causality principle violation. Left: input image; Middle and right: randomly colorized simplified image with $\lambda_{s_1} = 100$ and respectively $\lambda_{s_2} = 500$. The orange shape on top middle of the right image (surrounded by a black circle) is preserved for $\lambda_{s_2} = 500$, while it is removed for $\lambda_{s_1} = 100$ in the middle image.[]{data-label="fig:causality"}](plane_500 "fig:"){width="1.0\linewidth"} Instead of fixing the parameter $\lambda_s$ in the energy functional ([*e.g.*]{}, $\lambda_s$ in Eq (\[eq:simplemumford\])), we propose to compute an individual $\lambda_s$ for each shape of the tree following the same principle of the energy minimization. For a given $\lambda_s$, the removability of a shape $\tau$ is based on the sign of energy variation $\Delta E_{\lambda_s}^\tau$ in Eq. , which is a linear decreasing function w.r.t. $\lambda_s$ ([*e.g.*]{}, the Eq (\[eq:variationmumford\]) for the piecewise-constant Mumford-Shah functional). When $\lambda_s$ is bigger than some value $\lambda_{s_{min}}$, $\Delta E_{\lambda_s}^\tau$ will be negative, which implies the removal of this shape decreases the energy functional. Thus $\lambda_{s_{min}}$ is a value of the transition for the removal decision of the underlying shape. Let us denote this value of transition as the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$, which is given by: $${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}(\tau) = \frac{D(R'_{\tau_p}) - D(R_{\tau}) - D(R_{\tau_p})}{C(R_{\tau}) + C(R_{\tau_p}) - C(R'_{\tau_p})}, \label{eq:lambda}$$ For the piecewise-constant energy functional in Eq. , it is given by: $${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}(\tau) = \big( \frac{S^2(f, R_\tau)}{|R_\tau|} + \frac{S^2(f, R_{\tau_p})}{|R_{\tau_p}|} -\frac{S^2(f, R'_{\tau_p})}{|R'_{\tau_p}|} \big) \, / \, |\partial \tau|. \label{eq:lambdamumford}$$ Note that for a given shape $\tau$, the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}(\tau)$ defined in Eq (\[eq:lambda\]) depends on $R_\tau$ and $R_{\tau_p}$, which means ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}(\tau)$ is decided by the shape $\tau$ itself, its parent, its siblings, and its children. Because the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$ is computed under the hypothesis that the shape $\tau$ under scrutiny is suppressed, we also need to update $R_{\tau_p}$, and update the parenthood relationship for its children to its parent. These update operations will also affect the computation of ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$ for the parent, children and siblings of $\tau$. So the computation order is again important. We follow the same principle as described in Section \[subsec:llselection\] to compute ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$, which is detailed as below: *step 1:* Compute ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$ for each shape $\tau \in {\ensuremath{\mathcal{T}}\xspace}$ supposing that only the shape under scrutiny is removed, and sort the set of shapes $\{\tau \, | \, \tau \in {\ensuremath{\mathcal{T}}\xspace}\}$ in increasing order of shape meaningfulness indicated by an attribute ${\ensuremath{\mathcal{A}}\xspace}$ ([*e.g.*]{}, ${\ensuremath{\mathcal{A}}\xspace}_\nabla$). *step 2:* Propagate the sorted shapes in increasing order, and remove the shape one by one. Compute the new value ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$ for the underlying shape $\tau$, and update it if the value is greater than the older one. Update also the parenthood relationship and the corresponding information for $R_{\tau_p}$. This attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$ is related to the minimization of the energy functional. It measures the persistence of a shape to be removed under the minimization problem of Eq (\[eq:solution\]). A bigger ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}(\tau)$ means that it is more difficult to remove the shape $\tau$. Thus the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$ is also some kind of quantitative meaningfulness deduced from the energy minimization. We use the inverse of the attribute ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$ described above as the final attribute function: ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow(\tau) \, = \, \underset{\tau' \in {\ensuremath{\mathcal{T}}\xspace}}{\textrm{max}}\big({\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}(\tau')\big) - {\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}(\tau)$. The local minima of the shape space weighted by this attribute function correspond to a set of candidate salient level lines. We make use of the scheme of hierarchy transformation described in Section \[subsec:ssf\] to compute a saliency map $\mathcal{M}_{\mathcal{E}}$. This saliency map $\mathcal{M}_{\mathcal{E}}$ represents hierarchical result of level line selections. Each thresholding of this map $\mathcal{M}_{\mathcal{E}}$ selects salient (of certain degree) level lines from which a simplified image can be reconstructed. An example of the proposed scheme on a synthetic image is illustrated in Fig. \[fig:example\]. The input image in Fig. \[fig:example\] (a) is both blurred and noisy. This blurring is also visible in Fig. \[fig:example\] (b) that illustrates the evolution of the average of gradient’s magnitude ${\ensuremath{\mathcal{A}}\xspace}_\nabla$ along the contours of shapes starting from regions inside the triangle, pentagon, and square regions to the root of the tree. The evolution of the initial values of the Attribute ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$ obtained at step 1 on the same branches of the tree are provided in Fig. \[fig:example\] (c). It is not surprising that those initial values ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$ are not effective to measure the [importance]{} of the shapes: indeed, this is due to the very small size of each region in $\{R_{\ensuremath{\mathcal{T}}\xspace}\}$. The evolution of the final values of the attribute ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$ (of step 2) is depicted in Fig. \[fig:example\] (d). We can see that the significant regions are highlighted by ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$. This experiment also demonstrates the relevance of the increasing order of average of gradient’s magnitude along the contour ${\ensuremath{\mathcal{A}}\xspace}_\nabla$ as a criterion to update the value of ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$. The saliency map ${\ensuremath{\mathcal{M}}\xspace}_{\ensuremath{\mathcal{E}}\xspace}$ using the attribute ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$ and one of the possible segmentations that can be obtaiend by thresholding ${\ensuremath{\mathcal{M}}\xspace}_{\ensuremath{\mathcal{E}}\xspace}$ are depicted in Fig. \[fig:example\]. (e) and (f). [0.45]{} ![An example of the proposed scheme on a synthetic image. (b-d): Evolution of attribute value starting from leaf regions (left end of each curve) inside the triangle (red), pentagon (green), and squares (blue) to the root of the tree (right end of each curve). Note that the length of these three branches is different (it explains that the root node appears at different abscissas.)[]{data-label="fig:example"}](example "fig:"){width="0.8\linewidth"}   [0.45]{} ![An example of the proposed scheme on a synthetic image. (b-d): Evolution of attribute value starting from leaf regions (left end of each curve) inside the triangle (red), pentagon (green), and squares (blue) to the root of the tree (right end of each curve). Note that the length of these three branches is different (it explains that the root node appears at different abscissas.)[]{data-label="fig:example"}](grad_mean "fig:"){width="0.9\linewidth"} [0.45]{} ![An example of the proposed scheme on a synthetic image. (b-d): Evolution of attribute value starting from leaf regions (left end of each curve) inside the triangle (red), pentagon (green), and squares (blue) to the root of the tree (right end of each curve). Note that the length of these three branches is different (it explains that the root node appears at different abscissas.)[]{data-label="fig:example"}](lambda_original "fig:"){width="0.9\linewidth"}   [0.45]{} ![An example of the proposed scheme on a synthetic image. (b-d): Evolution of attribute value starting from leaf regions (left end of each curve) inside the triangle (red), pentagon (green), and squares (blue) to the root of the tree (right end of each curve). Note that the length of these three branches is different (it explains that the root node appears at different abscissas.)[]{data-label="fig:example"}](lambda_final "fig:"){width="0.9\linewidth"} [0.45]{} ![An example of the proposed scheme on a synthetic image. (b-d): Evolution of attribute value starting from leaf regions (left end of each curve) inside the triangle (red), pentagon (green), and squares (blue) to the root of the tree (right end of each curve). Note that the length of these three branches is different (it explains that the root node appears at different abscissas.)[]{data-label="fig:example"}](example_saliency "fig:"){width="0.8\linewidth"}   [0.45]{} ![An example of the proposed scheme on a synthetic image. (b-d): Evolution of attribute value starting from leaf regions (left end of each curve) inside the triangle (red), pentagon (green), and squares (blue) to the root of the tree (right end of each curve). Note that the length of these three branches is different (it explains that the root node appears at different abscissas.)[]{data-label="fig:example"}](example_result "fig:"){width="0.8\linewidth"} Implementation {#subsec:algorithm} -------------- The proposed method is composed of [three]{} main steps: 1) Construction of the tree of shapes and computation of the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$; 2) Computation of the extinction values ${\ensuremath{\mathcal{E}}\xspace}$; 3) Computation of the saliency map ${\ensuremath{\mathcal{M}}\xspace}_{\ensuremath{\mathcal{E}}\xspace}$. Once we have the tree structure ${\ensuremath{\mathcal{T}}\xspace}$ represented by the parenthood image $parent$, and the corresponding information ${\ensuremath{\mathcal{A}}\xspace}$ for each node of the tree, we are able to compute the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$. [Note that the information ${\ensuremath{\mathcal{A}}\xspace}$ can be the area $A$, the sum of gray levels $S_f$, the contour length $L$, or the sum of gradient’s magnitude along the contour $S_\nabla$.]{} The computation of this attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$ is performed while computing the tree, and is detailed in Algorithm \[algo:attribute\]. We start by computing the initial values of attribute ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_S}$ according to Eq  by considering that only the underlying shape is removed (see line 11). Then we sort the shapes by increasing order of the average of gradient’s magnitude along the shape contour ${\ensuremath{\mathcal{A}}\xspace}_\nabla$. We process the shapes in this order. For each underlying shape $\tau$, we compute a new value according to Eq  (see line 15), and update the value ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}(\tau)$ if the new value is greater. Then we remove the shape $\tau$ from the tree and update the tree structure as well as the corresponding information. COMPUTE\_ATTRIBUTE($parent$, ${\ensuremath{\mathcal{T}}\xspace}$, ${\ensuremath{\mathcal{A}}\xspace}$)\ $\mathcal{R}_{\ensuremath{\mathcal{T}}\xspace}\leftarrow $ SORT\_NODES(${\ensuremath{\mathcal{T}}\xspace}$, ${\ensuremath{\mathcal{A}}\xspace}_\nabla$)\ The [second]{} step is to compute the extinction values ${\ensuremath{\mathcal{E}}\xspace}$ for the local minima of the tree-based shape space weighted by the attribute ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$. This is achieved thanks to a min-tree representation ${\ensuremath{\mathcal{T}}\xspace}{\ensuremath{\mathcal{T}}\xspace}$ constructed on the tree-based shape space. The algorithm is described in Algorithm \[algo:extinction\]. The image $original\_min$ tracks the smallest local minimum inside a lower level connected component $CC$ of the tree-based shape space. For each local minimum shape $\tau$, the lowest $CC$ that contains $\tau$ and a smaller minimum is the lowest ancestor node whose smallest local minimum shape is different from $\tau$. COMPUTE\_EXTINCTION\_VALUE(${\ensuremath{\mathcal{T}}\xspace}$, ${\ensuremath{\mathcal{A}}\xspace}$)\ $(parent_{\ensuremath{\mathcal{T}}\xspace}, \mathcal{R}_{\ensuremath{\mathcal{T}}\xspace}) \leftarrow$ COMPUTE\_TREE(${\ensuremath{\mathcal{A}}\xspace}$);\ \ To compute the final saliency map, we rely on the Khalimsky’s grid, [proposed by [@khalimsky.90.tia], and depicted in Fig. \[fig:interpolation\]]{}; it is composed of $0$-faces (points), $1$-faces (edges) and $2$-faces (pixels). The saliency map ${\ensuremath{\mathcal{M}}\xspace}_{{\ensuremath{\mathcal{E}}\xspace}}$ is based on the extinction values, where we weigh the boundaries ($0$-faces and $1$-faces) of each shape by the corresponding extinction value ${\ensuremath{\mathcal{E}}\xspace}$. More precisely, we weigh each $1$-face $e$ ([*resp.*]{}$0$-face $o$) by the maximal extinction value of the shapes whose boundaries contain $e$ ([*resp.*]{}$o$). The algorithm is given in Algorithm \[algo:saliency\]. It relies on two images $\mathit{appear}$ and $\mathit{vanish}$ defined on the 1-faces that are computed during the tree construction. The value $\mathit{appear}(e)$ encodes the smallest region $\mathcal{N}_a$ in the tree whose boundary contains the 1-face $e$, while $\mathit{vanish}(e)$ denotes the smallest region $\mathcal{N}_v$ that contains the 1-face $e$ inside it. COMPUTE\_SALIENCY\_MAP($f$)\ $(parent, {\ensuremath{\mathcal{T}}\xspace}, {\ensuremath{\mathcal{A}}\xspace}) \leftarrow$ COMPUTE\_TREE(f);\ ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s} \leftarrow$ COMPUTE\_ATTRIBUTE($parent$, ${\ensuremath{\mathcal{T}}\xspace}$, ${\ensuremath{\mathcal{A}}\xspace}$);\ $\lambda_s^M \leftarrow 0$;\ \ \ ${\ensuremath{\mathcal{E}}\xspace}\leftarrow$ COMPUTE\_EXTINCTION\_VALUE(${\ensuremath{\mathcal{T}}\xspace}, {\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$);\ \ ![Materialization of pixels with 0-faces (blue disks), 1-faces (green strips), and 2-faces (red squares). The original pixels are the 2-faces, the boundaries are materialized with 0-faces and 1-faces. The contour of the cyan region is composed of black 1-faces and 0-faces.[]{data-label="fig:interpolation"}](interpolation){width="0.25\linewidth"} We refer the reader to [@najman.06.itip; @berger.07.icip; @carlinet.14.itip; @geraud.13.ismm] for details about the tree construction, and to [@xu.15.ismm] for details about the efficient computation of some information ${\ensuremath{\mathcal{A}}\xspace}$ (namely $A$, $S_f$, $L$, and $S_\nabla$). Illustrations and experiments {#sec:illustration} ============================= In this section, we illustrate our proposed general framework with a simple segmentation model: piecewise-constant Mumford-Shah model. Using some more evolved energy functional will be one of our future work. For generic natural images, contours of significant objects usually coincide with segments of level lines. Our proposed method yields a hierarchical simplification rather than a hierarchical segmentation. So only qualitative illustrations are depicted in Section \[subsec:hierarchy\] for some images taken from the BSDS500 dataset introduced in [@arbelaez.11.pami]. For the Weizmann segmentation database proposed by [@alpert.12.pami], the objects’ contours coincide with almost full level lines. Our method provides a hierarchical segmentation. Quantitative results using the associated evaluation framework are depicted in Section \[subsec:segmentation\]. Hierarchical color image pre-segmentation {#subsec:hierarchy} ----------------------------------------- In Fig. \[fig:simpcolor\], we test our method on color images in the segmentation evaluation database proposed in [@alpert.12.pami]. Each image contains two objects to be segmented. We use the color tree of shapes proposed by [@carlinet.15.itip], where the input image $f$ in Eq  is a color image. A high parameter value $\lambda_s = 8000$ is used, and the grain filter proposed in [@monasse.00.itip] is applied to get rid of too tiny shapes ([*e.g.*]{}, smaller than 10 pixels). Less than 100 level lines are selected, which results in a ratio of level line selection around 1157. These selected level lines form less than 200 regions in each image. The simplified images illustrated in Fig. \[fig:simpcolor\] are obtained by taking the average color inside each region, where the boundaries between salient regions remain intact. Finding an actual segmentation becomes a lot easier with such a pre-segmentation. The extinction-based saliency maps ${\ensuremath{\mathcal{M}}\xspace}_{\ensuremath{\mathcal{E}}\xspace}$ using the attribute ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$ are depicted on the bottom of this figure. They represent hierarchical pre-segmentations. Some illustrations of the extinction-based saliency map ${\ensuremath{\mathcal{M}}\xspace}_{\ensuremath{\mathcal{E}}\xspace}$ applied on images in the dataset of BSDS500 [@arbelaez.11.pami] are also shown in Fig. \[fig:hierarchicalsimplify\]. Again, the input image $f$ is a color image, and the color tree of shapes is used. As shown in Fig. \[fig:hierarchicalsimplify\], [salient]{} level lines are highlighted in ${\ensuremath{\mathcal{M}}\xspace}_{\ensuremath{\mathcal{E}}\xspace}$ employing the attribute ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$. Hierarchical image simplification results can then be obtained by thresholding ${\ensuremath{\mathcal{M}}\xspace}_{\ensuremath{\mathcal{E}}\xspace}$. The saliency maps for all the images in the dataset of BSDS500 is available on <http://publications.lrde.epita.fr/xu.hierarchymsll>. [[0.31]{}]{} ![Some pre-segmentation results obtained with our proposed method on the segmentation evaluation database in [@alpert.12.pami]. Top: input images; Middle: pre-segmentations obtained with the simplification method; Bottom: inverted saliency maps for hierarchical simplifications. []{data-label="fig:simpcolor"}](pb134930 "fig:"){width="1.0\linewidth"}   [[0.31]{}]{} ![Some pre-segmentation results obtained with our proposed method on the segmentation evaluation database in [@alpert.12.pami]. Top: input images; Middle: pre-segmentations obtained with the simplification method; Bottom: inverted saliency maps for hierarchical simplifications. []{data-label="fig:simpcolor"}](b13vehicles "fig:"){width="1.0\linewidth"}   [[0.31]{}]{} ![Some pre-segmentation results obtained with our proposed method on the segmentation evaluation database in [@alpert.12.pami]. Top: input images; Middle: pre-segmentations obtained with the simplification method; Bottom: inverted saliency maps for hierarchical simplifications. []{data-label="fig:simpcolor"}](nopeeking "fig:"){width="1.0\linewidth"} [[0.31]{}]{} ![Some pre-segmentation results obtained with our proposed method on the segmentation evaluation database in [@alpert.12.pami]. Top: input images; Middle: pre-segmentations obtained with the simplification method; Bottom: inverted saliency maps for hierarchical simplifications. []{data-label="fig:simpcolor"}](pb134930_simp "fig:"){width="1.0\linewidth"}   [[0.31]{}]{} ![Some pre-segmentation results obtained with our proposed method on the segmentation evaluation database in [@alpert.12.pami]. Top: input images; Middle: pre-segmentations obtained with the simplification method; Bottom: inverted saliency maps for hierarchical simplifications. []{data-label="fig:simpcolor"}](b13vehicles_simp "fig:"){width="1.0\linewidth"}   [[0.31]{}]{} ![Some pre-segmentation results obtained with our proposed method on the segmentation evaluation database in [@alpert.12.pami]. Top: input images; Middle: pre-segmentations obtained with the simplification method; Bottom: inverted saliency maps for hierarchical simplifications. []{data-label="fig:simpcolor"}](nopeeking_simp "fig:"){width="1.0\linewidth"} [[0.31]{}]{} ![Some pre-segmentation results obtained with our proposed method on the segmentation evaluation database in [@alpert.12.pami]. Top: input images; Middle: pre-segmentations obtained with the simplification method; Bottom: inverted saliency maps for hierarchical simplifications. []{data-label="fig:simpcolor"}](pb134930_map "fig:"){width="1.0\linewidth"}   [[0.31]{}]{} ![Some pre-segmentation results obtained with our proposed method on the segmentation evaluation database in [@alpert.12.pami]. Top: input images; Middle: pre-segmentations obtained with the simplification method; Bottom: inverted saliency maps for hierarchical simplifications. []{data-label="fig:simpcolor"}](b13vehicles_map "fig:"){width="1.0\linewidth"}   [[0.31]{}]{} ![Some pre-segmentation results obtained with our proposed method on the segmentation evaluation database in [@alpert.12.pami]. Top: input images; Middle: pre-segmentations obtained with the simplification method; Bottom: inverted saliency maps for hierarchical simplifications. []{data-label="fig:simpcolor"}](nopeeking_map "fig:"){width="1.0\linewidth"} [[0.31]{}]{} ![Illustration of the hierarchical image simplification using the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$, applied on some images from the dataset of BSDS500 proposed by [@arbelaez.11.pami]. From top to bottom: input images; inverted saliency maps; slight simplification; moderate simplification; strong simplification.[]{data-label="fig:hierarchicalsimplify"}](145086_color "fig:"){width="1.0\linewidth"}   [[0.31]{}]{} ![Illustration of the hierarchical image simplification using the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$, applied on some images from the dataset of BSDS500 proposed by [@arbelaez.11.pami]. From top to bottom: input images; inverted saliency maps; slight simplification; moderate simplification; strong simplification.[]{data-label="fig:hierarchicalsimplify"}](5096_color "fig:"){width="1.0\linewidth"}   [[0.31]{}]{} ![Illustration of the hierarchical image simplification using the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$, applied on some images from the dataset of BSDS500 proposed by [@arbelaez.11.pami]. From top to bottom: input images; inverted saliency maps; slight simplification; moderate simplification; strong simplification.[]{data-label="fig:hierarchicalsimplify"}](86016_color "fig:"){width="1.0\linewidth"} [[0.31]{}]{} ![Illustration of the hierarchical image simplification using the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$, applied on some images from the dataset of BSDS500 proposed by [@arbelaez.11.pami]. From top to bottom: input images; inverted saliency maps; slight simplification; moderate simplification; strong simplification.[]{data-label="fig:hierarchicalsimplify"}](145086_color_map "fig:"){width="1.0\linewidth"}   [[0.31]{}]{} ![Illustration of the hierarchical image simplification using the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$, applied on some images from the dataset of BSDS500 proposed by [@arbelaez.11.pami]. From top to bottom: input images; inverted saliency maps; slight simplification; moderate simplification; strong simplification.[]{data-label="fig:hierarchicalsimplify"}](5096_color_map "fig:"){width="1.0\linewidth"}   [[0.31]{}]{} ![Illustration of the hierarchical image simplification using the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$, applied on some images from the dataset of BSDS500 proposed by [@arbelaez.11.pami]. From top to bottom: input images; inverted saliency maps; slight simplification; moderate simplification; strong simplification.[]{data-label="fig:hierarchicalsimplify"}](86016_color_map "fig:"){width="1.0\linewidth"} [[0.31]{}]{} ![Illustration of the hierarchical image simplification using the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$, applied on some images from the dataset of BSDS500 proposed by [@arbelaez.11.pami]. From top to bottom: input images; inverted saliency maps; slight simplification; moderate simplification; strong simplification.[]{data-label="fig:hierarchicalsimplify"}](145086_color_map_low "fig:"){width="1.0\linewidth"}   [[0.31]{}]{} ![Illustration of the hierarchical image simplification using the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$, applied on some images from the dataset of BSDS500 proposed by [@arbelaez.11.pami]. From top to bottom: input images; inverted saliency maps; slight simplification; moderate simplification; strong simplification.[]{data-label="fig:hierarchicalsimplify"}](5096_color_map_low "fig:"){width="1.0\linewidth"}   [[0.31]{}]{} ![Illustration of the hierarchical image simplification using the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$, applied on some images from the dataset of BSDS500 proposed by [@arbelaez.11.pami]. From top to bottom: input images; inverted saliency maps; slight simplification; moderate simplification; strong simplification.[]{data-label="fig:hierarchicalsimplify"}](86016_color_map_low "fig:"){width="1.0\linewidth"} [[0.31]{}]{} ![Illustration of the hierarchical image simplification using the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$, applied on some images from the dataset of BSDS500 proposed by [@arbelaez.11.pami]. From top to bottom: input images; inverted saliency maps; slight simplification; moderate simplification; strong simplification.[]{data-label="fig:hierarchicalsimplify"}](145086_color_map_mid "fig:"){width="1.0\linewidth"}   [[0.31]{}]{} ![Illustration of the hierarchical image simplification using the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$, applied on some images from the dataset of BSDS500 proposed by [@arbelaez.11.pami]. From top to bottom: input images; inverted saliency maps; slight simplification; moderate simplification; strong simplification.[]{data-label="fig:hierarchicalsimplify"}](5096_color_map_mid "fig:"){width="1.0\linewidth"}   [[0.31]{}]{} ![Illustration of the hierarchical image simplification using the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$, applied on some images from the dataset of BSDS500 proposed by [@arbelaez.11.pami]. From top to bottom: input images; inverted saliency maps; slight simplification; moderate simplification; strong simplification.[]{data-label="fig:hierarchicalsimplify"}](86016_color_map_mid "fig:"){width="1.0\linewidth"} [[0.31]{}]{} ![Illustration of the hierarchical image simplification using the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$, applied on some images from the dataset of BSDS500 proposed by [@arbelaez.11.pami]. From top to bottom: input images; inverted saliency maps; slight simplification; moderate simplification; strong simplification.[]{data-label="fig:hierarchicalsimplify"}](145086_color_map_high "fig:"){width="1.0\linewidth"}   [[0.31]{}]{} ![Illustration of the hierarchical image simplification using the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$, applied on some images from the dataset of BSDS500 proposed by [@arbelaez.11.pami]. From top to bottom: input images; inverted saliency maps; slight simplification; moderate simplification; strong simplification.[]{data-label="fig:hierarchicalsimplify"}](5096_color_map_high "fig:"){width="1.0\linewidth"}   [[0.31]{}]{} ![Illustration of the hierarchical image simplification using the attribute function ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}^\downarrow$, applied on some images from the dataset of BSDS500 proposed by [@arbelaez.11.pami]. From top to bottom: input images; inverted saliency maps; slight simplification; moderate simplification; strong simplification.[]{data-label="fig:hierarchicalsimplify"}](86016_color_map_high "fig:"){width="1.0\linewidth"} We have also tested our method on some cellular images, where the method is applied on the color input image $f$. As illustrated in Fig. \[fig:cellular\], the cellular image is strongly simplified, which almost leads to a uniform background. Finding an actual cellular segmentation result would become much easier. [[0.18]{}]{} ![image](cellular_input){width="1.0\linewidth"}   [[0.18]{}]{} ![image](cellular_map){width="1.0\linewidth"}   [[0.18]{}]{} ![image](cellular_less_simp){width="1.0\linewidth"}   [[0.18]{}]{} ![image](cellular_mid_simp){width="1.0\linewidth"}   [[0.18]{}]{} ![image](cellular_high_simp){width="1.0\linewidth"} Evaluation in context of segmentation {#subsec:segmentation} ------------------------------------- We have also evaluated our hierarchical image simplifications in context of segmentation on Weizmann segmentation evaluation database in [@alpert.12.pami]. For the 100 images containing 2 objects in this database (See Fig. \[fig:simpcolor\] for several examples), the saliency maps are thresholded with a fixed thresholding value to yield a partition result. And we filter out the regions whose area is less than 100 pixels. Note that, in order to perform a fair comparison with the state-of-the art, the saliency maps are constructed using grayscale tree of shapes computed by [@geraud.13.ismm] on grayscale versions of the input images $f$. We performed two tests as presented in [@alpert.12.pami] based on F-measure and number of fragments. For a segmentation *Seg* and a ground truth segmentation of the object *GT*, the F-measure is defined by $F = 2 \times precision \times recall/(precision + recall)$, where $precision = |\textit{Seg} \cap \textit{GT}|/|\textit{Seg}|$, $recall = |\textit{Seg} \cap \textit{GT}|/|\textit{GT}|$. The number of fragments is the number of regions selected from a partition to form the object segmentation result *Seg*. In the first test, for each foreground object, we select the segment that fits it the best based on F-measure score. The results of this single segment coverage test is depicted in Table \[tab:singlesegment\] (See  [@alpert.12.pami] for implementation details on the settings for the other methods). In this test, our method achieves F-measure score on par with the state-of-the-art methods, especially when replacing, in the attribute ${\ensuremath{\mathcal{A}}\xspace}_\nabla$, the classical grayscale gradient with the (learned) gradient computed (on the grayscale input image $f$) by [@dollar.15.pami] (see “Our2”). By using another gradient, we change the order in which the nodes of the tree are processed; thus this result highlights the importance of the sorting step in our algorithm. In Table \[tab:singlesegment\], note that [ *Gpb-owt-ucm without texture*]{} denotes the method of Gpb-owt-ucm computed without taking into account the texture information in the Gpb part. More precisely, in this case, the Gpb is computed using only brightness and color gradients. Note also that our method does not explicitely use any texture information either. In the second test, a combination of segments whose area overlaps considerably the foreground objects is utilized to assess the performance. For each union of segments, we measure the F-measure score and the number of segments composing it. This test is a compromise between good F-measure and low number of fragments. The results of this fragmentation test is given in Table \[tab:multisegment\]. In this test the averaged F-measure of different methods is fairly similar. However, our method, as a pre-segmentation method without using any texture information has a relatively high number of fragments. The saliency maps for these images containing two objects in the Weizmann dataset is available on <http://publications.lrde.epita.fr/xu.hierarchymsll>. -- -- -- -- -- -- -- -- : Results of single segment coverage test using F-measure on the two objects dataset in [@alpert.12.pami]. “Our2” stands for our method with the attribute ${\ensuremath{\mathcal{A}}\xspace}_\nabla$ using the gradient computed by [@dollar.15.pami].[]{data-label="tab:singlesegment"} -- -- -- -- -- -- -- -- -- -- -- -- -- -- Comparison with similar works {#sec:relatedwork} ============================= The tree of shapes has been widely used in connected operators, filtering tools that act by merging flat zones for image simplification and segmentation. The simplification and segmentation relies on relevant shapes extraction ([*i.e.*]{}, salient level lines), usually achieved by tree filtering based on some attribute function. A detailed review of tree filtering strategies can be found in [@salembier.09.spm]. In all these strategies, the attribute function ${\ensuremath{\mathcal{A}}\xspace}$ characterizing each node plays a very important role in connected filtering. The classical connected operators make filtering decisions based only on attribute function itself or the inclusion relationship of the tree ([*e.g.*]{}, [@xu.14.filter]). They are usually performed by removing the nodes whose attributes are lower than a given threshold. The method we propose in this paper combines this idea of classical connected operators with the energy minimization problem of Eq (\[eq:solution\]). It also makes use of the spatial information of the original image from which the tree is constructed. This might give more robust filtering decision. In this paper, we focus particularly on hierarchical relevant shapes selection by minimizing some multiscale affine separable energy functional ([*e.g.*]{}, piecewise-constant Mumford-Shah functional). The closely related work is the one in [@guigues.06.ijcv], where the authors proposed the scale-set theory, including an efficient greedy algorithm to minimize this kind of energy on a hierarchy of segmentations. More precisely, the authors use dynamic programming to efficiently compute two scale parameters $\lambda_s^+$ and $\lambda_s^-$ for each region $R$ of the input hierarchy $H$, where $\lambda_s^+$ ([*resp.*]{}, $\lambda_s^-$) corresponds to the smallest parameter $\lambda_s$ such that the region $R \in H$ belongs ([*resp.*]{}. does not belong) to the optimal solution of segmentation by minimizing $E_{\lambda_s}$, we have $\lambda_s^-(R) = \underset{R' \in H, R \subset R'}{\mathrm{min}}\lambda_s^+(R')$. There may exist some regions $R$ such that $\lambda_s^-(R) \leq \lambda_s^+(R)$, which implies that the region $R \in H$ does not belong to any optimal cut of $H$ by minimizing the energy $E_{\lambda_s}$. One removes these regions from the hierarchy $H$ and updates the parenthood relationship which yields a hierarchy $H'$, a hierarchy of global optimal segmentations on the input hierarchy. This work has been continued and extended by [@kiran.14.pr]. These methods work on an input hierarchy of segmentations, which is very different from the tree of shapes (a natural and equivalent image representation). [ Indeed, each cut of the tree of shapes is a subset of the image domain, while each cut of the hierarchy of segmentations forms a partition of the image domain; see [@ronse.14.jmiv]. This basic difference prohibits the direct use on the tree of shapes of the classical works which find optimal hierarchical segmentations by energy minimization.]{} In this sense, our approach can be seen as an extension of the scale-set theory proposed in [@guigues.06.ijcv] to the tree of shapes. Another related work is the one in [@ballester.07.jmiv]. It also selects meaningful level lines for image simplification and segmentation by minimizing the piecewise-constant Mumford-Shah functional. For this method, at each step the level line is selected which inflicts the largest decrease of functional. As a consequence, the iterative process of [@ballester.07.jmiv] requires not only computing a lot of information to be able to update the functional after each level line suppression, but also to find at each step, among all remaining level lines, the one candidate to the next removal. Consequently, the optimization process has a $O(N_{\ensuremath{\mathcal{T}}\xspace}^2)$ time complexity w.r.t. the number of nodes $N_{\ensuremath{\mathcal{T}}\xspace}$ of the tree. A heap-based implementation may improve the time complexity, but since at each removal, one has to update the corresponding energy variation for its children, parent, siblings, maintaining the heap structure is a costly process. In practice, the gain using heap-based implementation is relatively insignificant. Hence [@ballester.07.jmiv] is computationally expensive. We propose to fix that issue thanks to a reasonable ordering of level lines based on their quantitative meaningfulness measurement [([*e.g.*]{}, the average of gradient’s magnitude along the level line ${\ensuremath{\mathcal{A}}\xspace}_\nabla$)]{}. The time complexity of our optimization process is linear w.r.t. the number of nodes $N_{\ensuremath{\mathcal{T}}\xspace}$. We have implemented the method of [@ballester.07.jmiv] using the same tree construction algorithm and the same data structure based on heap. We have compared the running time on 7 classic images on a regular PC station. The comparison is detailed in Table. \[tab:comparison\]. Our proposal is significantly faster than that of [@ballester.07.jmiv]. Our approach is almost linear w.r.t. the number of nodes in the tree. Yet, the method of [@ballester.07.jmiv] seems to depend also on the depth of the tree. In [@ballester.07.jmiv], the authors proposed applying the simplification scheme successively with a set of augmenting parameters $\lambda_s$ so that to construct the input hierarchy. Then they employed the scheme of scale-set theory proposed by [@guigues.06.ijcv] on the obtained hierarchy to achieve a final hierarchy of optimal segmentations. In our case, rather than using a fixed parameter $\lambda_s$ or a set of fixed parameters, we propose to assign a measure related to $\lambda_s$ to each shape as an attribute function. Then we use the hierarchy transformation (reviewed in Section \[subsec:ssf\]) based on extinction values and on a tree-based shape space to compute a hierarchical salient level lines selection. -- -- -- -- -- -- -- -- -- -- -- -- : Comparison of computation times on seven classical images. The size for image “House” and “Camera” is $256\times256$, and $512\times512$ for the other images.[]{data-label="tab:comparison"} It is worth noticing that the minimization of Mumford-Shah-like functional has also been applied to shape analysis in [@tari.14.jmiv]. It consists of adding a non-local term, which is the squared average of the field in the energy functional. Its minimization tends to form negative field values on narrow or small parts as well as on protrusions, and positive field values on central part(s) of the input shape. The negative and positive regions inside the input shape yield some saddle points at which a crossing of a level curve occurs. This leads to a binary partition hierarchy $H_b$ of the shape. Then a probability measure based on the obtained field values inside the shape is assigned to each node of the partition hierarchy $H_b$. A set of hierarchical representations of the shape is obtained by removing some nodes from $H_b$ and update the parenthood relationship. Each candidate hierarchical representation is assigned with a saliency measure given by the products of the probability measure of the removed nodes. These hierarchical representations of the shape associated with the global saliency are used to analyze the shape. Our proposal is different from this framework in terms of the use of energy minimization. In [@tari.14.jmiv], the energy minimization is used to create an image with negative and positive regions for a given shape. Then one constructs a binary hierarchy of partitions of the created image via its saddle points, and weighs each node based on the obtained image values. In our case, the energy minimization is performed on an input image subordinated to its hierarchical representation by the tree of shapes. This yields a quantitative meaningfulness measure [ ${\ensuremath{\mathcal{A}}\xspace}_{\lambda_s}$]{} for each node of the tree of shapes. Conclusion {#sec:conclusion} ========== In this paper, we have presented an efficient approach of hierarchical image simplification and segmentation based on minimizing some multiscale separable energy functional on the tree of shapes, a unique and equivalent image representation. It relies on the idea of hierarchy transformation based on extinction values and on a tree-based shape space to compute a saliency map representing the final hierarchical image simplification and segmentation. The salient structures in images are highlighted in this saliency map. A simplified image with preservation of salient structures can be obtained by thresholding the saliency map. Some qualitative illustrations and quantitative evaluation in context of image segmentation on a public segmentation dataset demonstrate the efficiency of the proposed method. A binary executable of the proposed approach is available on <http://publications.lrde.epita.fr/xu.hierarchymsll>. In the future, we would like to explore some applications employing a strongly simplified image as pre-processing step. We believe that this could be useful for analyzing high-resolution satellite images and images with texts, where the contours of meaningful objects in images usually coincide with full level lines. Besides, as advocated in Table \[tab:singlesegment\] for Gpb-owt-ucm, the texture provides important information for image segmentation. An interesting perspective is to incorporate texture information in our proposed framework. Since the tree of shapes is a natural representation of the input image, a possible way to integrate texture information might consist in replacing the original image with a new grayscale image incorporating texture features. Although this is not directly appicable to our case, probability map incorporating region features, such as proposed in [@bai2009geodesic], are worth exploring. Computing the tree of shapes of such probability maps has already been proved valuable (see  [@dubrovina.14.icip]). In another direction, it would be interesting to investigate some other energy functionals for some specific tasks. Examples are the rate distortion optimization used in image or video compression coding system (see [@salembier.00.itip; @ballester.07.jmiv]) and the energy based on spectral unmixing used for hyperspectral image segmentation in [@veganzones.14.itip]. The energy functionals in these works are affine separable, which straightforwardly allows to use them in our proposed framework. Last, but not the least, given that using a learned gradient improves the results, a major research avenue is to combine our approach with learning techniques.
--- abstract: 'Bilayer graphene (BLG) offers a rich platform for broken symmetry states stabilized by interactions. In this work we study the phase diagram of BLG in the quantum Hall regime at filling factor $\nu=0$ within the Hartree-Fock approximation. In the simplest non-interacting situation this system has eight (nearly) degenerate Landau levels near the Fermi energy, characterized by spin, valley, and orbital quantum numbers. We incorporate in our study two effects not previously considered: ([*i*]{}) the nonperturbative effect of trigonal warping in the single-particle Hamiltonian, and ([*ii*]{}) short-range SU(4) symmetry-breaking interactions that distinguish the energetics of the orbitals. We find within this model a rich set of phases, including ferromagnetic, layer-polarized, canted antiferromagnetic, Kekulé, a “spin-valley entangled” state, and a “broken U(1) $\times$ U(1)” phase. This last state involves independent spontaneous symmetry breaking in the layer and valley degrees of freedom, and has not been previously identified. We present phase diagrams as a function of interlayer bias $D$ and perpendicular magnetic field $B_{\perp}$ for various interaction and Zeeman couplings, and discuss which are likely to be relevant to BLG in recent measurements. Experimental properties of the various phases and transitions among them are also discussed.' author: - Ganpathy Murthy - Efrat Shimshoni - 'H. A. Fertig' title: 'Spin-Valley Coherent Phases of the $\nu=0$ Quantum Hall State in Bilayer Graphene' --- Introduction {#section1} ============ Two-dimensional systems with discrete degrees of freedom in the quantum Hall regime support a variety of possible broken symmetry states, a phenomenon known as quantum Hall ferromagnetism (QHF)[@QHFM]. In this context graphene has presented itself as a particularly exciting system, both in its monolayer and bilayer forms. These systems differ from more conventional two dimensional electron gases in supporting a $\nu=0$ quantized Hall effect, a consequence of negative energy levels that are necessarily present in their non-interacting spectra [@Gusynin_2005; @McCann_2006]. Moreover, the presence of nearly-degenerate Landau levels (arising from internal degrees of freedom such as spin, valley, and layer) near the Fermi energy in undoped systems suggest that these systems offer a unique platform for QHF physics [@Yang2006]. In this work we study QHF in bilayer graphene (BLG) subject to magnetic and electric fields. In zero magnetic field, working in the tight-binding model with nearest-neighbor hoppings only, the system distinguishes itself from single layer graphene at the noninteracting level in supporting two quadratic band touching (QBT) points, at the $K$ and $K'$ points in the Brillouin zone, in contrast with monolayer graphene which supports Dirac points at these locations. When undoped, the Fermi energy passes through these QBT’s, opening the possibility of many-body instabilities when interactions are included in zero magnetic field [@Nandkishore-Levitov_2010; @Vafek-Yang_2010; @Zhang-MacDonald_2010; @Vafek_2010] . In the presence of a field, this system supports eight Landau levels near the Fermi energy, offering a particularly rich set of possibilities for groundstates with broken symmetries. These levels arise from spin and valley quantum numbers, as well as orbital states $n=0,1$ which are degenerate at any magnetic field in the simplest models, when no electric field $D_{\perp}$ is applied perpendicular to the system. Previous studies of this system have focused on models which differ in their choice of physical effects retained in the single-particle Hamiltonian, and in how interactions are modeled. Projection of the long-range Coulomb interaction into this 8-fold manifold yields an effective Hamiltonian with a layer-polarized state at large $D_{\perp}$ and a ferromagnetic state at small $D_{\perp}$, with a first order transition separating them [@Barlas_2008; @Gorbar_2011]. Distinguishing intra- and inter-layer Coulomb interactions, as well as inclusion of particle-hole symmetry-breaking terms, leads to the appearance of a state spontaneously breaking a U(1) symmetry [@Lambert_2013; @lukose_2016; @knothe_2016; @Jia_2017]. Interactions in general are, however, more complicated than the long-range Coulomb form, because at the microscopic scale they may have lower symmetry (e.g., on-site Hubbard interactions). Moreover, short-range interactions have greater effect than expected based on projection directly into the small set of Landau levels near the Fermi energy, because they impact the energetics of the Landau levels below them [@Herbut_2007; @Shizuya_2012; @Roy_2014_1; @Roy_2014; @Feshami_2016; @lukose_2016; @knothe_2016; @Jia_2017]. An effective method for dealing with this, introduced by Kharitonov [@kharitonov_bulk_monolayer; @kharitonov_bulk_bilayer], uses phenomenological short-range interactions consistent with the symmetries of the lattice, in principle incorporating renormalizations from the Landau levels deep within the Dirac sea. In this study, we adopt this general approach of effective interactions confined to the set of Landau levels near zero energy. Experimentally, evidence for phase transitions among states of different broken symmetries has been accumulating. Two-terminal conductance experiments reveal quantized Hall states at low and high $D_{\perp}$ at filling factor $\nu=0$, interrupted at intermediate $D_{\perp}$ scales by a region where the transport gap vanishes [@Weitz_2010; @velasco_2012; @maher_2013], indicating a phase transition between different quantized Hall states. The value of $D_{\perp}$ at which this transition occurs increases monotonically with increasing $B_{\perp}$, the magnetic field component perpendicular to the bilayer. The high $D_{\perp}$ phase is rather naturally identified with a layer polarized state, while the low $D_{\perp}$ phase is largely thought to represent a canted antiferromagnet (CAF) phase as was suggested in Ref. . More recent capacitance measurements [@Hunt_2016], however, show signatures of a separate intermediate gapped phase between the low and high $D_{\perp}$ limits, appearing above $B_{\perp} \sim$ 12T -13T. Finally, in some samples the region in $D_{\perp}$ separating the low and high $D_{\perp}$ states even at lower $B_{\perp}$ is not perfectly sharp, raising the possibility of other phases existing in the transition region[@maher_2013; @jun-pvt]. ![The theoretical phase diagram in the tuning parameters $B_\perp$ and the perpendicular electric field (labelled $D$ in the figure and proportional to $D_\perp$) of our model in a range of assumed couplings which exhibits the Broken-U(1)$\times$U(1) (BU(1)$^2$) state. Here and in all the figures following, $B_\perp$ is in Tesla, and $D$ is in arbitrary units. The boundaries of the BU(1)$^2$ state are the dashed red lines, while the boundaries of the partially orbitally polarized (POP) state are the solid black lines. The blue dash-dotted line is the upper boundary of the spin-valley entangled (SVE) phase, while the green dashed line with the + symbols is the upper boundary of the Kekule (KEK) state. The canted antiferromagnet (CAF) occupies the small $D$ part of the diagram at all values of $B_\perp$. For small values of $B_\perp\leqslant11$T, as one increases $D$ starting from zero, one successively encounters the CAF state, the BU(1)$^2$ state, the spin-valley-entangled (SVE) state, and finally, at large values of $D$, the fully layer polarized (FLP) state. At larger values of $B_\perp>11$T, again starting from $D=0$, one encounters the CAF, the BU(1)$^2$ state, the partially orbitally polarized (POP) state, the Kekule (KEK) state, and finally the FLP state. All solid lines indicate first-order phase transitions while the broken lines indicate second-order transitions.[]{data-label="PS1-Fig1-bperp-phase-dia-fig"}](fig1.eps){width="1.0\linewidth"} In this work, we explore the phase diagram of bilayer graphene at $\nu=0$ using a model of the form introduced in Ref. , within the Hartree-Fock approximation. Our model incorporates two ingredients which, to our knowledge, have not been considered before in the context of interacting BLG. The first is the nonperturbative inclusion of “trigonal warping” [@McCann_2006] (arising from a hopping amplitude $t_3$ between sites in different layers which are not above one another) in the single particle states comprising the low-energy manifold. Here and in the following, by “low-energy manifold” we will mean the states lying near the Fermi energy. The $t_3$ term is allowed by the spatial symmetries of the lattice, and generically arises in [*ab initio*]{} approaches to the band structure of BLG (see Ref. and references therein). This hopping term significantly distorts the QBT in zero field, replacing it with four Dirac points [@McCann_2006]. From a renormalization group (RG) perspective, recent work [@Pujari_2016] has shown that the $t_3$ term, being allowed by symmetry, is generated by short-range interactions, even if it is assumed to be zero in the bare theory. Once generated, it is relevant, and flows to large values at low energies. In large magnetic fields this term has a very small effect [@McCann_2006]. In consequence, this term has previously been either neglected [@kharitonov_bulk_bilayer; @Lambert_2013; @lukose_2016; @Jia_2017] or taken into account only perturbatively [@knothe_2016]. We find, however, that for experimentally relevant values of $B_{\perp}$ the nonperturbative effect of the $t_3$ term is crucial to stabilizing hitherto unknown broken symmetry states. The second crucial element in our theory is the inclusion of short range interactions not included in Ref. : a density-density coupling $g_0$, and an orbital anisotropy coupling $g_{nz}$, an Ising-like interaction energy for fluctuations in the density differences between the two spatial orbitals. Both these couplings are allowed by symmetry, and we find that including them yields a minimal model with a phase diagram qualitatively consistent with current experimental observations. The phases that we find to be stable in different parameter regimes include: (1) a fully layer polarized (FLP) state, (2) a fully spin polarized (ferromagnetic, FM) state; (3) a canted antiferromagnetic state (CAF), characterized by partial spin alignment along the direction of the total magnetic field and antiferromagnetic alignment between electrons in different valleys; (4) a Kekulé state (KEK), which may be regarded as an analog of the CAF in which the roles of spin and valley degrees of freedom have been interchanged; (5) a “spin-valley entangled” (SVE) state, in which the occupied single-particle states involve coherent superpositions of states of opposing spin and valley index, similar to the spin-layer coherent state of Refs. ; (6) a partial orbitally polarized (POP) state; and finally (7) a more exotic “Broken U(1)$\times$U(1)” state, which supports non-trivial coherence among different combinations of the single-particle states in the spin-valley manifold such that two different U(1) symmetries are spontaneously broken. This contrasts with the other coherent states that we find (which have been discussed in earlier literature as well [@kharitonov_bulk_bilayer; @Lambert_2013; @lukose_2016; @knothe_2016; @Jia_2017]) – the CAF, KEK, and SVE – which represent families of states with a single spontaneously broken U(1) symmetry. To our knowledge the Broken-U(1)$\times$U(1) (BU(1)$^2$) state has not been previously identified in the literature, though hints of it have been seen in the vanishing energy of collective modes even at $t_3=0$ [@denova_2017] at the CAF/FM to KEK/FLP phase boundary (we explain this connection in Sections \[sve\], \[instabilities\] and \[largebperp\]). Within our model, the BU(1)$^2$ phase requires a nonzero trigonal warping in the single particle Hamiltonian, as well as the $g_0$ and $g_{nz}$ couplings. We find that for physically reasonable sets of parameters it connects states with fewer broken symmetries, such as the CAF and KEK as the interlayer potential $D_{\perp}$ or the perpendicular field $B_{\perp}$ increases. Each of the two U(1) angles involved comes with a stiffness, one or the other of which vanishes continuously as the transition to another state is approached. This suggests the possibility of thermal or quantum disordering of the phase, and the possibility that the state does not manifest the quantized Hall effect at experimentally relevant temperatures. If so, this would introduce a broad transition region between, for example, CAF and FLP states as a function of $D_{\perp}$, rather than a sharp transition between them as would occur in a first-order transition. A typical phase diagram is illustrated in Fig. \[PS1-Fig1-bperp-phase-dia-fig\]. The rest of this article is organized as follows. In Section \[section2\] we introduce the noninteracting Hamiltonian for BLG and the low-energy basis states we will be using. These basis states include the effect of the trigonal warping nonperturbatively. In Section \[IntHam\] we will introduce the interacting Hamiltonian, and present the general formula for the energy of a Hartree-Fock (HF) state. In Section \[HF-and-Instabilities\] we describe the states that are encountered in our numerical calculation. We also present the linear instabilities of these states which helps us identify various second-order phase transitions. Most importantly, it helps us identify three different regimes of the coupling constants which result in different topologies of the phase diagram. In Section \[Results\], we present a brief analysis of the possible phase diagrams at small $B_\perp$ and large $B_\perp$. This distinction arises because the term in the Hamiltonian induced by the trigonal warping scales as $\sqrt{B_\perp}$, whereas other terms are proportional to $B_\perp$. Section \[Results\] also contains our main results. These include phase diagrams in $B_\perp - D$ space ($D$ is proportional to the perpendicular electric field applied on the sample) for three different regimes of coupling constants that produce different topologies for the phase diagrams. Section \[discussion\] includes a discussion of experimental consequences relevant to our phase diagrams, and notes a few limitations of our analysis. Section \[Conclusions\] concludes with a summary, open questions, and future directions. Noninteracting Hamiltonian and Low Energy States {#section2} ================================================ To set our notation from the start, we will use the index $n=0,1$ for the orbital degree of freedom, the Greek indices $\a=0,1$ for the valley (where $\a,\beta=0\equiv K$ and $\a,\beta=1\equiv K'$), and the indices $s,s'=0,1$ for spin ($s=0\equiv\uparrow$, and $s=1\equiv\downarrow$). As a starting point for analyzing the single-body part of the Hamiltonian we consider a Bernal stacked BLG, where the A site of one layer is directly on top of the B$^\prime$ site of the other. In the presence of a perpendicular electric field $D_\perp$ and a magnetic field ${\bf B}$ (introduced via a gauge choice where $A_y=B_\perp x$), the approximate effective Hamiltonian describing electron states on the remaining two sites of the BLG unit cell is given (for valley [**K**]{}, spin $s=0,1=\uparrow,\downarrow$ and wave-vector $k$ in the $\hat{y}$-direction) by [@McCann_2006; @Jung14] H\^[K s]{}\_[eff]{}&=&H\_0+H\_Z+H\_D \[eq:Heff\]\ H\_[0]{}&=&-\_c( [cc]{} -a\^a & (a\^)\^[2]{}+ a\ (a)\^[2]{}+a\^& -aa\^ ) ,\ H\_Z&=&-( [cc]{} E\_z(-1)\^s & 0\ 0 & E\_z(-1)\^s ) ,\ H\_D&=&-( [cc]{} D & 0\ 0 & -D ) .\[jungMacDH\]Here $E_z\propto |{\bf B}|$ is the Zeeman energy, $D\propto D_\perp$ is (half) the interlayer bias, and $a=\frac{\ell}{\sqrt{2}}[\partial_x+(x-X)/\ell^2]$ is the Landau level lowering operator (with $\ell=\sqrt{\hbar c/e B_\perp}$ the magnetic length and $X=k\ell^2$ the guiding center coordinate). The parameters of $H_0$ account for all the tight-binding parameters listed in Ref. , including the longer-range interlayer hopping coefficients $t_3$, $t_4$ and a particle-hole breaking onsite energy $\Delta$: \_c &= & \~B\_ ,\[omega\_c\]\ m && where $v_\perp=\sqrt{3}|t_\perp|a_0/2\hbar$, $v_4=\sqrt{3}|t_4|a_0/2\hbar$ with $t_\perp$ the inlayer hopping obeying $|t_\perp|\gg |t_4|,|t_1|\gg\Delta$. The dimensionless parameter = \[tea\_def\] determines the orbital anisotropy energy, and is independent of $B_\perp$, whereas ==\~ . \[lambda\_def\] Finally, $H^{K^\prime s}_{eff}$ (for the other valley [**K$^{\mathbf\prime}$**]{}) can be obtained from Eq. (\[eq:Heff\]) by trading $a^\dagger\leftrightarrow a$, $D\leftrightarrow -D$ and $\lambda\leftrightarrow -\lambda$. The spectrum and eigenstates of the above effective Hamiltonian are well-known for the case $\lambda=\tea=0$, i.e. when subleading hopping parameters are neglected. In particular, there is a two-fold orbitally degenerate manifold of zero energy eigenstates of $H_0$ (ignoring spin and the guiding center indices for the moment): |n,K=( [cc]{} |n\ 0 ) ,|n,K\^=( [cc]{} 0\ |n ), \[lambda0states\] where $|n\rangle$ with $n=0,1$ are Landau level (LL) wavefunctions. Their corresponding energies are $\epsilon_{n,\a,s}=-D(-1)^\a-E_z(-1)^s$. Note that the two-fold degeneracy of $n=0,1$ can be traced back to the quadratic band-touching (QBT) characteristic to BLG. Adding a finite $\tea$ to $H_0$ \[Eq. (\[eq:Heff\])\] maintains the eigenstates \[Eq. (\[lambda0states\])\], and merely lifts the degeneracy of the $n=0,1$ orbitals by a small asymmetry energy. However, the parameter $\lambda$ associated with the $t_3$-hopping term, which introduces trigonal warping of the QBT, fundamentally changes the structure of the electronic states. Moreover, using empirical estimates of the bare parameters [@Jung14; @t4t3exp] in Eq. (\[lambda\_def\]), one obtains $\lambda\equiv\lambda_1/\sqrt{B_\perp}$ where $B_\perp$ is in Tesla and $\lambda_1\sim 1$ is the value of $\lambda$ at $B_\perp=1$ T. This implies that its effect is not necessarily perturbative; its relative significance is tunable with $B_\perp$, and becomes especially pronounced for moderately low fields of the order of a Tesla. Indeed, as we show below, the resulting change in the structure of non-interacting electron states has dramatic consequences on the nature of broken-symmetry states when interactions are included. We therefore focus on the case where $\lambda\neq 0$ is arbitrary, and $\tea=0$ (corrections due to a finite $\tea$ will be accounted for later on as a perturbation). The eigenstates of $H^{K s}_{eff}$, $H^{K^\prime s}_{eff}$ can then be cast as (again ignoring spin and guiding center indices) |K=( [cc]{} |\_[\_K]{}\ 0 ) ,&&|K\^=( [cc]{} 0\ |\_[\_[K’]{}]{} )\ [where]{}(a\^2+(-1)\^a\^)|\_&=& 0 . \[psi\_general\] Using the operator identity $[a,f(a^\dagger)]=f^\prime(a^\dagger)$ (with $f(x)$ an analytic function), Eq. (\[psi\_general\]) can be cast as an operator version of the Airy equation $y^{\prime\prime}-xy=0$ whose solutions are the functions[@ASbook] $Ai(x)$, $Bi(x)$. Employing their integral form, we obtain the following basis for the states $|\psi_{_K}\rangle$ (i.e., for $\alpha=0$ and $\lambda>0$): \[psi\_AB\_integral\] |\_A,K&=&\_0\^dt|0,\ |\_B,K&=&\_0\^dt|0 .It is convenient to express these integral forms as power series in $\lambda$. This yields $|\psi_A\rangle$, $|\psi_B\rangle$ as linear combinations of the orthonormal orbital states (see Appendix \[Anm\]) \[psi\_01\_def\] |\_0,K&=&\_[m=0]{}\^A\_[0m]{}|3m ,A\_[0m]{}C\_0(-1)\^m ;\ |\_1,K&=&\_[m=0]{}\^A\_[1m]{}|3m+1 ,A\_[1m]{}C\_1(-1)\^m , where $|N\rangle=\frac{1}{\sqrt{N!}}(a^\dagger)^N|0\rangle$ are the LL states and the normalization factors $C_n$ guarantee $\sum_0^\infty A_{nm}^2=1$. Recalling Eq. (\[psi\_general\]), the solutions for the wavefunction $|\psi_{_{K'}}\rangle$ are directly obtained from Eq. (\[psi\_01\_def\]) by the substitution $\lambda\rightarrow -\lambda$. For convenience, we recall our label $\alpha$ for the valleys such that $\alpha=K=0$, $\alpha=K'=1$, and the corresponding orbital labels $n=0,1$, so that |n,\_[m=0]{}\^(-1)\^[m]{}A\_[nm]{}|3m+n . \[nalpha\_def\] The eigenstates of the effective Hamiltonian \[with $\tilde{\epsilon}_{\alpha}=0$ in Eq. (\[eq:Heff\])\] are then given by |0,K,s=( [cc]{} |0,0,s\ 0 ) ,&& |1,K,s=( [cc]{} |1,0,s\ 0 ) ,\ |0,K\^,s=( [cc]{} 0\ |0,1,s ) ,&& |1,K\^,s=( [cc]{} 0\ |1,1,s ) \[eigenstates\_final\] where the explicit dependence on the parameter $\lambda$ is given in Eqs. (\[psi\_01\_def\]) and (\[nalpha\_def\]) and the states $|n,\alpha,s\rangle \equiv |n,\alpha\rangle \otimes |s\rangle$ incorporate spin. Note that the wavevector $k$, or equivalently the guiding center $X=k\ell^2$, is also a quantum number of the states, but is suppressed in the above expressions. This basis of low-energy states, i.e., states close to the Fermi energy, has the full nonperturbative dependence on $t_3$ which will turn out to be important for the rest of our analysis. To evaluate the energy spectrum, we consider the full effective Hamiltonian where the anisotropy parameter $\tea$ in Eq. (\[eq:Heff\]) is finite but small \[see Eq. (\[tea\_def\])\], so that the corresponding terms can be treated perturbatively. Using the matrix elements 0,|a\^a|0,&=&\_[m=0]{}\^3m|A\_[0m]{}|\^2,\ 1,|a\^a|1,&=&\_[m=0]{}\^(3m+1)|A\_[1m]{}|\^2 ,and implementing the substitution $D\rightarrow -D$ for $K\rightarrow K^\prime$, we obtain the energy levels corresponding to the states Eq. (\[eigenstates\_final\]) to first order in $\tilde{\epsilon}_{\alpha}$: \[epsilon\_nalpha\] \_[0,[\_K]{}]{} &=& D+\_\_[m=0]{}\^3m|A\_[0m]{}|\^2,\ \_[1,[\_K]{}]{} &=& D+\_\_[m=0]{}\^(3m+1)|A\_[1m]{}|\^2,\ \_[0,[\_[K\^]{}]{}]{} &=& -D+\_\_[m=0]{}\^3m|A\_[0m]{}|\^2,\ \_[1,[\_[K\^]{}]{}]{} &=& -D+\_\_[m=0]{}\^(3m+1)|A\_[1m]{}|\^2 .For each valley, this introduces an orbital anisotropy \_a\_[1,]{}-\_[0,]{}=\_\_[m=0]{}\^\[epsilon\_a\_def\] which can be numerically evaluated for arbitrarily large $\lambda$ using the expressions for $A_{nm}$ \[Eq. (\[psi\_01\_def\])\]. The Interaction Hamiltonian and Hartree-Fock {#IntHam} ============================================ As explained above, there are three discrete quantum numbers for the non-interacting single particle states in BLG, representing spin, valley, and the $n=0,\ 1$ orbitals. To begin dealing with interactions we divide the basic Coulomb interaction into a long-range part that has the full SU(4) symmetry of spin and valley indices, and an effective short-range part. The short-range interactions (including those present at the bare level) should have SU(2) symmetry in the spin sector and a U(1) symmetry in the valley sector. There is no symmetry constraint in the orbital sector. Upon the application of a Zeeman field the symmetry of the spin-sector is also reduced to a U(1). Thus the symmetry of the full Hamiltonian is U(1)$_{spin}\times$U(1)$_{valley}$. Following previous work in single layer graphene[@kharitonov_bulk_monolayer], we will assume that the relevant interactions at low energy have no explicit spin-dependence. Translation invariance implies that at low energy there should be two kinds of interactions, those that transfer a momentum small compared to a reciprocal lattice vector, and those that transfer a momentum close to the intervalley momentum $\Delta{\bK}={\bK}-{\bK}'$. Taking all these conditions into account, we obtain a large set of possible interactions, each with its own coupling. Such a high-dimensional coupling constant space is very hard to analyze systematically. Hence, in this work, we will simplify the system by considering a “minimal” model which contains only four distinct couplings. Defining $c_{n\a s k}$ as the destruction operator for a particle in a $|n,\alpha,s,k\rangle$ state (here $k$ is the Landau guiding center label), our minimal interaction Hamiltonian takes the form H\_[int]{}=&\_[k\_1,k\_2,]{} e\^[-iq\_x(k\_1-k\_2-q\_y)\^2]{}\ &(v\_0()\_[n\_im\_i s\_1s\_2]{}\_[n\_1n\_2]{}\^()\_[m\_1m\_2]{}\^(-):[c]{}\^\_[n\_1s\_1,k\_1-q\_y ]{}[c]{}\_[n\_2s\_1,k\_1]{}[c]{}\^\_[m\_1s\_2,k\_2+q\_y ]{}[c]{}\_[m\_2s\_2,k\_2 ]{}:\ &+ v\_z()\_[n\_im\_i s\_1s\_2]{}\_[n\_1n\_2]{}\^()\_[m\_1m\_2]{}\^(-):[c]{}\^\_[n\_1s\_1,k\_1-q\_y ]{}\_z[c]{}\_[n\_2s\_1,k\_1 ]{}[c]{}\^\_[m\_1s\_2, k\_2+q\_y]{}\_z[c]{}\_[m\_2s\_2, k\_2]{}:\ &+ 2v\_[xy]{}()\_[n\_im\_i s\_1 s\_2]{}\_[n\_1n\_2]{}\^[KK’]{}()\_[m\_1m\_2]{}\^[K’K]{}(-):[c]{}\^\_[n\_1K s\_1, k\_1-q\_y]{}[c]{}\_[n\_2K’ s\_1, k\_1]{}[c]{}\^\_[m\_1K’ s\_2, k\_2+q\_y]{}[c]{}\_[m\_2K s\_2, k\_2]{}:\ &+v\_[nz]{}()\_[n\_1n\_2s\_1s\_2]{}(-1)\^[n\_1+n\_2]{}\_[n\_1n\_1]{}\^()\_[n\_2n\_2]{}\^(-):[c]{}\^\_[n\_1s\_1, k\_1-q\_y]{}[c]{}\_[n\_1s\_1, k\_1]{}[c]{}\^\_[n\_2s\_2, k\_2+q\_y]{}[c]{}\_[n\_2s\_2, k\_2]{}:). The matrix elements of the density $\tilde\rho_{n_1n_2}^{\alpha\beta}$ are defined using the states of Eq. (\[nalpha\_def\]) (with spin still suppressed but the guiding center indices now explicit) as n\_1k\_1|e\^[-i]{}|n\_2k\_2=\_[k\_1,k\_2-q\_y]{}e\^[-iq\_x(k\_1-q\_y/2)]{}\_[n\_1n\_2]{}\^(). \[eq:trho\] Some details about these matrix elements that are relevant to our study are provided in Appendix \[app:FF\]. The couplings $v_z,\ v_{xy}$ were originally introduced by Kharitonov for monolayer graphene[@kharitonov_bulk_monolayer], and have exactly the same meaning here as in the monolayer. In earlier work on the edge states of monolayer graphene [@us_2014; @us_2016], we introduced the coupling $v_0$, which treats all the discrete labels equally and endows the system with a spin stiffness for spatial variations of the order parameter. The new coupling we introduce is $v_{nz}$, which is analogous to $v_z$, but in the orbital sector. To proceed one must specify forms for $v_0(\bq),\ v_z(\bq),\ v_{xy}(\bq)$, and $v_{nz}(\bq)$. We make the simplest possible choices, that they are constants independent of $\bq$. This means the interactions are very short-ranged in space. We note that in the case of single-layer graphene $v_0$ does not alter the relative energies of the various possible bulk states. However, as we will see shortly, in bilayer graphene $v_0$ enters the energies of different states with different coefficients, and hence plays a role in picking the true ground state. The full effective Hamiltonian of our system truncated to the low-energy space is $H_0+H_{int}$ where H\_0=-\_[n s k]{}[c]{}\^\_[ns k]{}[c]{}\_[ns k]{}. Any Hartree-Fock (HF) state is fully determined by its one-body averages $\langle{c}^{\dagger}_i{c}_j\rangle$. We only consider states in the bulk that conserve the guiding center label $k$: Thus, the only possible translation symmetry breaking could arise via densities with momenta ${\bK}-{\bK}'$. We define the matrix $\Delta_{mn;ss'}^{\alpha\beta}$ via HF|[c]{}\^\_[ms k]{}[c]{}\_[ns’ k’]{}|HF\_[kk’]{}\_[mn;ss’]{}\^ where $|HF\rangle$ is a Hartree-Fock state. Note that $\Delta$ is independent of $k$. Now consider evaluating the average of $H_{int}$ in such a state. A generic term is a sum of direct and exchange contributions – i.e., HF| [c]{}\^\_[n\_1s\_1, k\_1-q\_y]{}[c]{}\^\_[m\_1s\_2, k\_2+q\_y]{}[c]{}\_[m\_2s\_2, k\_2]{}[c]{}\_[n\_2s\_1,k\_1 ]{}|HF=\_[q\_y,0]{}\_[n\_1n\_2;s\_1s\_1]{}\^\_[m\_1m\_2;s\_2s\_2]{}\^ -\_[k\_1,k\_2+q\_y]{}\_[n\_1m\_2;s\_1s\_2]{}\^\_[m\_1n\_2;s\_2s\_1]{}\^. The direct terms are easy to deal with because $\trho_{n_1n_2}^{\alpha\beta}(\bq=0)=\delta_{n_1n_2}\delta_{\alpha\beta}$. The exchange integrals are a bit more involved. In Appendix \[app:FF\] we show the following important result, which is relevant because of our assumption that all interactions $v_i(\bq)$ are constants $v_i$: \_[n\_1n\_2]{}\^()\_[m\_1m\_2]{}\^(-)=& r\_\^[(n\_1)]{}r\_\^[(n\_2)]{},\ r\_\^[(n)]{}=\_[j=0]{}\^ (-1)\^[j(+)]{} |A\_[nj]{}|\^2&= { [cc]{} 1&=\ r& }\ [where]{}r = \_[k=0]{}\^ (-1)\^[k]{} A\_[nk]{}\^2. \[eq:trho\_integral\] The number $r$ is independent of the orbital index $n$ (see App. B) but does depend on $B_{\perp}$ via the coefficient $\lambda$ \[Eqs. (\[lambda\_def\]) and (\[psi\_01\_def\])\] arising originally from the trigonal warping term $t_3$. The most important consequence of this relation is that only $\Delta$’s diagonal in the $n$-labels appear in the energy. Using the general reasoning of Ref. , since the inter-orbital exchange (zero here) is smaller than the intra-orbital exchange, this falls into the Ising anisotropy class: The system cannot lower its energy by superposing different orbitals in a single-particle state. Operationally, this leads to the enormous simplification that we need to consider only forms of $\Delta$ which are block-diagonal in $n$: \_[n\_1n\_2;s\_1s\_2]{}\^\_[n\_1n\_2]{}\_[n\_1;s\_1s\_2]{}\^ Let us now define the couplings $g_i=\frac{v_i}{2\pi\ell^2}$, and the number of flux quanta passing through the sample $N_\phi=\frac{L_xL_y}{2\pi\ell^2}$. Recalling the indexing of Section \[section2\] ($\alpha=0$ for the $K$ valley and 1 for the $K'$ valley, $s=0$ for spin up and 1 for spin down), the HF energy may then be written compactly as ({})=&-\_[n s]{}(\_a(-1)\^n+E\_z(-1)\^s+D(-1)\^)\_[n;ss]{}\^\ &+((\_[ns]{}\_[n;ss]{}\^)\^2-\_[s s’]{}r\_\^2(\_[n]{}\_[n;ss’]{}\^)(\_[n’]{}\_[n’;s’s]{}\^))\ &+((\_[ns]{}(-1)\^n\_[n;ss]{}\^)\^2-\_[s s’]{}r\_\^2(-1)\^[+]{}(\_[n]{}\_[n;ss’]{}\^)(\_[n’]{}\_[n’;s’s]{}\^))\ &+g\_[xy]{}(r\^2|\_[ns]{}\_[n;ss]{}\^[KK’]{}|\^2-\_[ss’]{}(\_[n]{}\_[n;ss’]{}\^[KK]{})(\_[n’]{}\_[n’,s’s]{}\^[K’K’]{}))\ &+((\_[ns]{}(-1)\^n\_[n:ss]{}\^)\^2-\_[nss’]{}r\_\^2\_[n:ss’]{}\^\_[n:s’s]{}\^). Hartree-Fock States and Linear Instabilities {#HF-and-Instabilities} ============================================ Before we present the numerical results, let us explore the nature of the states we will encounter, parametrize them analytically, and find critical values of $D$ at which one kind of state is unstable to another. At $\nu=0$ four single-particle states must be filled at each guiding center. All the states we consider are one of three types. (i) All four occupied states could be in the same ($n=0$) orbital, which would be a maximally orbitally anisotropic (MOA) state. (ii) Three of the occupied states could be in the $n=0$ orbital while one is in the $n=1$ orbital, a partially orbitally polarized (POP) state. (iii) Both the $n=0,\ 1$ orbitals support two occupied states. In this case the most natural choice is $\Delta_{0:ss'}^{\alpha\beta}=\Delta_{1:ss'}^{\alpha\beta}$, a state symmetric in the orbital label. We will analyze each of these possibilities in turn. In the following, when we represent $\Delta_0$ and $\Delta_1$ as $4\times4$ matrices, our ordering will be $K\ua,\ K\da,\ K'\ua,\ K'\da$. We will be guided by experiment in choosing our parameters; in particular, we will consider only $g_{xy}<0$, because of the evidence that a canted antiferromagnet (CAF) state is stable in BLG, determining the sign of $g_{xy}$. Maximally Orbitally Anisotropic State {#moa} ------------------------------------- This state is particularly simple. The $\Delta$ matrices are \_[0]{}=1\_[44]{},     \_1=0\_[44]{}. This state has orbital polarization, but no valley or spin polarization. The HF energy is \_[MOA]{}=-4\_a+6g\_0-2g\_z+6g\_[nz]{}. We find, for our choices of parameters, that this state is never the ground state. Partially Orbitally Polarized States {#pop} ------------------------------------ This state can be characterized by two different single-particle states in the spin-valley sector, which for the moment we generically label $|a\rangle$ and $|b\rangle$. The $\Delta$ matrices can be described as \_0=1\_[44]{}-|aa|,    \_1=|bb|. In principle, the states $|a\rangle,\ |b\rangle$ can be arbitrary, but at the HF minimum we find them to be parametrized by a single angle $\theta$ |a=&\[ [cccc]{} 0&0&-(/2)&(/2)\ \]\^T,\ |b=&\[ [cccc]{} (/2)&-(/2)&0&0\ \]\^T, where $\cos(\theta)=\frac{E_Z}{|g_{xy}|}$ for $E_Z<|g_{xy}|$ and $\cos(\theta)=1$ for $E_Z>|g_{xy}|$. The energy of this state is \_[POP]{}=&-2\_a-2D+5g\_0-g\_z+|g\_[xy]{}|-&\ &&E\_Z&lt;|g\_[xy]{}|,\ =&-2\_a-2D-2E\_Z+5g\_0-g\_z+2|g\_[xy]{}|&\ &&E\_Z&gt;|g\_[xy]{}|.Note that the POP states have an orbital polarization of 2, a valley polarization of 2, and variable spin polarization which can never exceed 2. They also spontaneously break the U(1) spin-rotation symmetry around the direction of $\bB$ for $E_Z<|g_{xy}|$. States Symmetric in Orbitals {#symmstates} ---------------------------- This class exhibits the richest set of HF states, and contains: (i) The canted antiferromagnet (CAF) which spontaneously breaks the U(1) spin-rotation symmetry around the direction of the total field $\bB$. The fully spin-polarized ferromagnet (FM) is a limit of the CAF. (ii) The Kekule state (KEK) which is a spin singlet but is canted in the valley sector and thus spontaneously breaks the valley U(1) symmetry. The fully layer polarized (FLP) state is a limit of the Kekule state. (iii) A spin-valley-entangled (SVE) state that entangles $K\da$ with $K'\ua$. (iv) A new state which is canted in both the spin and valley sectors, and thus has two distinct spontaneously broken U(1) symmetries. We will call this state the Broken-U(1)$\times$U(1), or BU(1)$^2$ state. It will prove convenient to look at the $4\times4$ matrix $\Delta_0=\Delta_1=\Delta$ rather than the occupied states themselves. In all the orbitally symmetric states $g_{nz}$ only appears via the combination $g_0+\half g_{nz}$. For future convenience we define &G\_0=g\_0+g\_[nz]{},\ =&(1-r\^2)G\_0+(1+r\^2)g\_z+|g\_[xy]{}|. \[G0Dtilde\] ### Canted Antiferromagnet (CAF) and Ferromagnet (FM) {#caf/fm} These states have a $\Delta$ matrix of the form =( [cccc]{} 1+&&0&0\ &1-&0&0\ 0&0&1+&-\ 0&0&-&1-\ ). The minimum occurs at $\cos\theta=\frac{E_Z}{2|g_{xy}|}$ for $E_z \le 2|g_{xy}|$ and $\cos\theta=1$ for $E_z > 2|g_{xy}|$. The energy is \_[CAF]{}=&8g\_0-4G\_0-4g\_z-&\[tcecaf\]\ &&E\_Z2|g\_[xy]{}|,\ \_[FM]{}=&8g\_0-4G\_0-4g\_z-4E\_Z+&4|g\_[xy]{}|\[tcefm\]\ &&E\_Z&gt;2|g\_[xy]{}|.The case $E_Z>2|g_{xy}|$ corresponds to the fully spin-polarized FM state. The CAF/FM state has only spin-polarization, and no orbital or valley polarization. The CAF state spontaneously breaks the U(1) spin-rotation symmetry around $\bB$. The FM state has no spontaneously broken symmetries. ### Kekule (KEK) and Fully Layer Polarized (FLP) States {#kek/flp} For this state, =( [cccc]{} 1+&0&&0\ 0&1+&0&\ &0&1-&0\ 0&&0&1-\ ). To specify the angle at the minimum, we need to define an energy $g_K$; g\_K=(3-r\^2)g\_z+(2r\^2-1)|g\_[xy]{}|-(1-r\^2)G\_0. \[gK\]In terms of $g_K$ the energy for arbitrary $\theta$ can be expressed as ()=&8g\_0-2(1+r\^2)G\_0-2(1-r\^2)g\_z\ &-2(2r\^2-1)|g\_[xy]{}|-4D+2g\_K\^2. It is clear that if $g_K<0$, $\theta=0$ is the minimum. For $g_K>0$ we find $\theta$ at the minimum to be =    D&lt;g\_K;    =1    D&gt;g\_K. The case $D>g_K$ corresponds to the fully layer polarized (FLP) state. The energy of the KEK/FLP state is \_[KEK]{}=&8g\_0-2(1+r\^2)G\_0-2(1-r\^2)g\_z\ &-2(2r\^2-1)|g\_[xy]{}|-    D&lt;g\_K,\[tcekek\]\ \_[FLP]{}=&8g\_0-4G\_0+4g\_z-4D    D&gt;g\_K.\[tceflp\] The KEK/FLP states have no orbital or spin polarization. They do have a valley polarization. The KEK state spontaneously breaks the valley U(1) symmetry. The FLP state does not spontaneously break any symmetry. ### Spin-Valley Entangled (SVE) State {#sve} This state has the $K\ua$ state occupied, but mixes the $K\da$ and $K'\ua$ states. In this case, =( [cccc]{} 2&0&0&0\ 0&1+&&0\ 0&&1-&0\ 0&0&0&0\ ). The energy of this state is evaluated to be ()=&4(2g\_0-G\_0-g\_z+|g\_[xy]{}|-E\_Z)-\ &4\^2(D+2|g\_[xy]{}|-E\_Z-)\ &+4(1-r\^2)(g\_z-G\_0)\^4. \[Esve\]The optimum value of $\cos^2\frac{\psi}{2}$ is easily found to be \^2=. Defining &D\^[SVE]{}\_[min]{}=+E\_Z-2|g\_[xy]{}|,\ &D\^[SVE]{}\_[max]{}=D\^[SVE]{}\_[min]{}+2(1-r\^2)(g\_z-G\_0), \[sveminmax\]the minimum energy of the SVE state for $D$ in the range $D^{SVE}_{min}< D < D^{SVE}_{max}$ is \_[SVE]{}=4(2g\_0-G\_0-g\_z-E\_Z+|g\_[xy]{}|)-. This state spontaneously breaks a single U(1), which is an entangled combination of valley and spin, and smoothly interpolates between the FLP and the FM states. Note that as $r^2\to1$, Eq. (\[sveminmax\]) implies that the range of $D$ over which the SVE state exists shrinks to zero. In fact, precisely at $r^2=1$ and $D=D^{SVE}_{min}=D^{SVE}_{max}$ the energy of Eq. (\[Esve\]) becomes independent of $\psi$. This means that there should be a zero energy $q=0$ collective mode at this value of $D$, which is indeed seen in a recent calculation [@denova_2017]. This is a hint of the potential existence of the SVE state [*even at $r^2=1$*]{}. ### Broken U(1)$\times$U(1) \[BU(1)$^2$\] State {#csvaf} This is an interesting state that spontaneously breaks the U(1) symmetries of both the spin and valley sectors. We will call this the BU(1)$^2$ state for short. The most general state for two filled levels, assuming real vectors, can be described by five real parameters. This can be seen as follows: The first filled state is an O(4) vector (real state) which can be specified by three angles. The second filled state also has three angles, but the constraint that it should be orthogonal to the first filled state reduces the total number of independent angles by one, to a total of five. We have numerically searched in this five-dimensional parameter space for the minimum energy HF state, and found that these minima can always be described by a state requiring only three real angles, which we call $\theta,\ \chi,\ \psi$. In addition to these there are two $U(1)$ angles upon which the energy does not depend, which we label $\phi$ and $\eta$. Defining $\gamma=\frac{\psi+\chi}{2}$ and $\zeta=\frac{\psi-\chi}{2}$ the resulting $\Delta$ matrix may be expressed as =( [cccc]{} 1+&e\^[-i]{}&e\^[i]{}&-e\^[i(-)]{}\ e\^[i]{}&1-&-e\^[i(+)]{}&e\^[i]{}\ e\^[-i]{}&-e\^[-i(+)]{}&1+&-e\^[-i]{}\ -e\^[-i(-)]{}&e\^[-i]{}&-e\^[i]{}&1-\ ). \[FullDelta\] The values of $\phi,\ \eta$ will be chosen in the true ground state by spontaneous symmetry breaking. In the limit $\psi=\chi=0$ this ansatz reduces to the CAF/FM where $\theta$ is the canting angle of the CAF/FM. Similarly, for $\chi=0,\ \psi=\pi$ it reduces to the KEK/FLP state, where $\theta$ now means the canting angle of the Kekule state. Thus Eq. (\[FullDelta\]) interpolates smoothly between the CAF/FM and the KEK/FLP states. Finally, $\theta=\chi=\pi$ and $\psi \ne 0,\pi$ corresponds to the SVE state. We will reserve the name “Broken-U(1)$\times$U(1)” for the state where all three angles $\theta,\ \chi,\ \psi$ are nontrivial, that is, different from $0$ or $\pi$. The energy for this ansatz is &=8g\_0-4G\_0-4g\_z-4(E\_Z+D) +2\^2(-2r\^2|g\_[xy]{}|)\ &+\^2(4|g\_[xy]{}|-4(1-r\^2)|g\_[xy]{}|\^2+2\^2+4\^2\^2(1-r\^2)\[g\_z-G\_0\]). \[energy3angle\] Unfortunately, we have not been able to analytically find the minima of $\tcE$ within its full three angle domain. Instabilities of CAF/FM, KEK/FLP, and SVE states {#instabilities} ------------------------------------------------ Since the three-angle ansatz can describe all the other states that only have a single broken U(1), we can use the three-angle ansatz to find the instabilities of the CAF/FM and the KEK/FLP. Motivated by experiment, we will analyze the situation where $B_{\perp}$ and $E_Z$ are fixed while $D$ is varied. We define the critical $D$ at which the CAF/FM becomes unstable to the three-angle ansatz as $D_{c1}$, while the $D$ at which the KEK/FLP or the SVE/FLP becomes unstable to the three-angle ansatz is defined as $D_{c2}$. Ignoring the POP state for the moment, a necessary (but not sufficient) condition for the Broken-U(1)$\times$U(1) state to exist as a HF state is $D_{c2}>D_{c1}$. To make the ideas concrete, Fig. \[Energies-PS1-bperp6p0-fig\] shows the energies of the various HF states as functions of $D$ for fixed $B_\perp=6$ T, and $E_Z=|g_{xy}|/3$. For this set of parameters, the FM, KEK, and POP states are always higher in energy than the others, and hence are not the ground state at any $D$. On the other hand, the CAF, the BU(1)$^2$, the SVE, and the FLP states are the lowest in energy, each in a corresponding range of $D$. As can be seen, the SVE state interpolates smoothly between the FM and the FLP states. Thus, the FLP and FM states must be linearly unstable to the SVE state at the appropriate values of $D$. Similarly, the BU(1)$^2$ state interpolates smoothly between the CAF and SVE states. Thus, the CAF and SVE states must be linearly unstable to the BU(1)$^2$ state at the appropriate values of $D$. In the following, we will analytically compute the values of $D$ corresponding to the various linear instabilities. ![The energies of the various HF states for a particular choice of parameters. We have fixed $B_\perp=6$ T, and $E_Z=|g_{xy}|/3$, and plotted the energies as functions of $D$. The dashed and the dot-dashed black lines are the FM and CAF energies respectively. They are independent of $D$. The solid blue line is the energy of the FLP state (with a slope of -4), while the solid black line is the energy of the POP state (with a slope of -2). The solid red line, where it exists, marks the energy of the BU(1)$^2$ state. The dot-double-dashed blue line is the energy of the SVE state, which interpolates smoothly between the FM and FLP states. The dot-double-dashed green line is the energy of the KEK state. For this particular choice of parameters, as $D$ increases, the system starts in the CAF state for small $D$, undergoes a second-order transition into the BU(1)$^2$ state, which then gives way to the SVE state, which in turn yields to the FLP state. All transitions are second-order.[]{data-label="Energies-PS1-bperp6p0-fig"}](fig2.eps){width="1.0\linewidth"} Let us consider $D_{c1}$ first. Since the CAF/FM state has $\chi=\psi=\gamma=\zeta=0$, we can consider $\gamma,\ \zeta\ll 1$ and expand the energy in powers of $\gamma,\ \zeta$. After doing so, we obtain a constant piece (the energy of the CAF/FM state) and a quadratic form in $\gamma$ and $\zeta$. The instability occurs when the quadratic form has a zero eigenvalue. For the CAF state with $E_Z<2|g_{xy}|$, after setting $\cos\theta=\frac{E_Z}{2|g_{xy}|}$, we find =4g\_0-4g\_z-2g\_[nz]{}-+M\_\^2+M\_\^2-,\ M\_=(|g\_[xy]{}|+(1-r\^2)G\_0+(1+r\^2)g\_z),\[mgg\]\ M\_=+2((1-r\^2)G\_0+(1+r\^2)g\_z-(2r\^2-1)|g\_[xy]{}|).\[mzz\] Recalling the definition of $\tD$ \[Eq. (\[G0Dtilde\])\] we can express Eqs. (\[mgg\]) and (\[mzz\]) as M\_=&,\ M\_=&2(-2r\^2|g\_[xy]{}|)+. The critical value $D_{c1}$ for the CAF case is then D\_[c1]{}\^[CAF]{}=&\ =&. \[Dc1CAF\]For the FM state ($E_Z>2|g_{xy}|$), setting $\theta=0$ we obtain &=4g\_0-4g\_z-2g\_[nz]{}-4E\_Z+4|g\_[xy]{}|\ &-4D+M\_\^2+M\_\^2,\ &M\_=M\_=2(2+E\_Z-2|g\_[xy]{}|). In this case the critical value is D\_[c1]{}\^[FM]{}=+E\_Z-2|g\_[xy]{}|. \[Dc1FM\] Now let us turn to $D_{c2}$, the critical value of $D$ where the KEK/FLP or the SVE/FLP state is unstable to the three-angle-ansatz. We start from large $D$ where the FLP state is clearly the HF ground state. In this case, since $\theta\approx0$, $\chi\approx0,\ \psi\approx\pi$ we can assume $\theta\ll1$, $\gamma=\frac{\pi}{2}-\xi,\ \zeta=\frac{\pi}{2}-\omega$, and expand the energy function for small $\theta,\ \xi,\ \omega$. Due to the fact that the energy function Eq. (\[energy3angle\]) depends on $\theta$ only via $\cos\theta$, we see that the quadratic fluctuations of $\theta$ decouple from those of $\gamma,\ \zeta$. The quadratic instability of the FLP state in the $\theta$ channel occurs at D\_[c]{}\^=D\_c\^[FLP/KEK]{}=g\_K and leads to the KEK state which we have already described. Ignoring the $\theta$ flucuations, the energy function near the FLP state can be expanded for small $\xi,\ \omega$ as =\_[FLP]{}+2(D-4g\_z+)(\^2+\^2)-4E\_Z. This leads to D\^[,]{}\_[c,FLP]{}=D\_c\^[FLP/SVE]{}=4g\_z-+E\_Z. \[Dc2FLP\]This instability leads to the SVE state which we have also described. Using Eqs. (\[G0Dtilde\]) and (\[gK\]), we note that D\_c\^[FLP/SVE]{}=D\_c\^[FLP/KEK]{}-2r\^2|g\_[xy]{}|+E\_Z \[Dc2SVE\_KEK\]implying that the SVE (KEK) is favored for $E_Z>2r^2|g_{xy}|$ ($E_Z<2r^2|g_{xy}|$). In either case, the linear instability of the FLP state leads to a state with a [*single*]{} broken U(1). Thus, in order to see where the BU(1)$^2$ state terminates as $D$ increases from $D_{c_1}$, we need to consider the linear instabilities of the KEK and SVE states. First consider the KEK state, which is stable when $D<g_K$. Once again the $\theta$ fluctuations decouple from those of the other two angles. The energy function to quadratic order in $\xi,\ \omega$ is =&+\^2-+M\_\^2,\ &M\_= 4r\^2|g\_[xy]{}|-2(1-). We infer the value of $D_{c2}$ from this equation to be D\_[c2]{}\^[KEK]{}=g\_K. \[Dc2KEK\]In order for the $KEK$ state to be stable we must impose $D_{c2}<g_K$, consistent with the requirement $E_Z<2r^2|g_{xy}|$. We thus identify a first parameter regime in which BU(1)$^2$ state is the groundstate for a non-vanishing range of $D$. For $E_Z>2r^2|g_{xy}|$ the KEK state has no linear instabilities. If its energy crosses that of the CAF/FM state it must do so as a first-order transition. Now we turn to the linear instabilities of the SVE state. The SVE state corresponds to $\theta=\xi=\pi$ while $\psi$ is nontrivial. The $\theta$ fluctuations once again decouple from the $\xi,\ \psi$ fluctuations. The $\xi,\ \psi$ fluctuations are innocuous, but the $\theta$ fluctuations do lead to an instability. A straightforward analysis shows that D\_[c2]{}\^[SVE]{}=D\_[min]{}\^[SVE]{}+(g\_z-G\_0) . Recalling the condition for the existence of the SVE state to be $D_{min}^{SVE}<D<D_{min}^{SVE}+2(1-r^2)(g_z-G_0)$, we indeed see that this is an actual instability only for $E_Z>2r^2|g_{xy}|$. We then arrive at a second scenario in which the BU(1)$^2$ state is stable, in this case connecting either the CAF state at $D=D_{c_1}^{CAF}$ (for $E_z < 2|g_{xy}|$) or the FM state at $D=D_{c_1}^{FM}$ (for $E_z > 2|g_{xy}|$) to the SVE state at $D=D_{c_2}^{SVE}$. Finally, if $E_Z<2r^2|g_{xy}|$ and the energy of the SVE state crosses that of the CAF/FM, it must do so as a first-order transition. Main Results and Phase Diagrams {#Results} =============================== As seen in the previous section, there are several different states that compete in different regimes of $B_{\perp},\ E_Z,\ D$. We will assume that all the couplings $g_i$ are proportional to $B_{\perp}$. It would then naively appear that one can scale out $B_{\perp}$ from the Hamiltonian. However, recall that the parameters $r$ and $\e_a$ depend on $B_{\perp}$ via their dependence on $\lambda$ \[see Eq. (4)\] arising from the trigonal warping coefficient $t_3$. ![$r^2$ vs. $B_\perp$ for $\lambda_1=3$ and $\lambda_1=4$. Note that $r^2$ tends vary rapidly to zero when $B_\perp$ falls below a characteristic scale set by $\lambda_1$. At large values of $B_\perp$, $r^2\to1$. The approach to $r^2=1$ is very slow. []{data-label="rsqvsBperp"}](fig3.eps){width="1.0\linewidth"} Introducing a field-independent parameter $\lambda_1=\lambda\sqrt{B_\perp}$ (which is the value of $\lambda$ at $B_\perp=1$T), in Fig. \[rsqvsBperp\] and Fig. \[anisovsBperp\] we show $r^2$ vs. $B_\perp$ and $\e_a$ vs. $B_\perp$ for $\lambda_1=3$ and 4. We see that both $r^2$ and $\e_a$ vanish very rapidly for $B_\perp$ smaller than a characteristic scale $B_\lambda$. For $B_{\perp}\gg B_\lambda$, we see that $r^2\to 1$ while $\e_a$ becomes linear in $B_\perp$. There are thus two regimes in which the analysis becomes simple. In the small $B_\perp$ regime we can essentially set $r^2\approx0$. In the large $B_\perp$ regime we can set $r^2\approx1$. With the parameters we use the small $B_\perp$ regime is far easier to realize at experimentally feasible values of $B_\perp$. Before presenting the numerical HF results we analyze the phase diagram for small and large $B_\perp$ analytically. This provides us with relations between the couplings $g_i$ that determine the topology of the phase diagram. ![Orbital anisotropy energy $\e_a$ vs. $B_\perp$ for $\lambda_1=3$ and $\lambda_1=4$. At very small values of $B_\perp$ below a characteristic scale set by $\lambda_1$, $\e_a$ vanishes rapidly as $B_\perp\to 0$. At large values of $B_\perp$, $\e_a$ becomes linear in $B_\perp$. []{data-label="anisovsBperp"}](fig4.eps){width="1.0\linewidth"} Possible Phase Diagrams at Small $B_\perp$ {#smallbperp} ------------------------------------------ The key idea is to analyse the ordering of the various special values of $D$ that we defined in Section \[instabilities\] in the limit $r^2\to0$. They are &g\_z+G\_0+|g\_[xy]{}|,\ &g\_K3g\_z-G\_0-|g\_[xy]{}|4g\_z-,\ &D\_[c1]{}\^[CAF]{},\ &D\_[c1]{}\^[FM]{}=+E\_Z-2|g\_[xy]{}|,\ &D\_[c2]{}\^[SVE]{}=+E\_Z-2|g\_[xy]{}|+(g\_z-G\_0). We have not included $D_{c2}^{KEK}$ because the condition for it to exist, $E_Z<2r^2|g_{xy}|$, cannot be satisfied when $r^2\to0$. The condition for the BU(1)$^2$ state to be the true HF ground state is $D_{c2}>D_{c1}$. For $E_Z<2|g_{xy}|$, this becomes g\_z&gt;G\_0+|g\_[xy]{}|. If $E_Z>2|g_{xy}|$ then $D_{c2}^{SVE}$ ceases to be physical (because it becomes less than $D_{min}^{SVE}$). In this case there is no BU(1)$^2$ state. Instead, as $D$ increases, the FM state gives way to the SVE state at $D_{c1}^{FM}$, which in turn continuously evolves to become the FLP state at $D^{FLP/SVE}_{c}=4g_z-\tD+E_Z$, as long as g\_z&gt;G\_0. Thus, we obtain the following three possibilities at small $B_\perp$: (i) If $g_z<G_0$ there will be a direct first-order transition of the CAF/FM into either of the SVE/FLP states at all values of $E_Z$. (ii) If $G_0<g_z<G_0+|g_{xy}|$ then there will be a direct first-order transition between the CAF and FLP/SVE states as $D$ increases as long as $E_Z<2|g_{xy}|$. However, for $E_Z>2|g_{xy}|$, the SVE state smoothly interpolates between the FM at small $D$ to the FLP state at large $D$. All transitions will now be continuous. (iii) If $g_z>G_0+|g_{xy}|$, then the BU(1)$^2$ state always intervenes between the CAF and the SVE states as $D$ is increased for $E_Z<2|g_{xy}|$. However, for $E_Z>2|g_{xy}|$, the BU(1)$^2$ state disappears, and instead the SVE smoothly connects the FM and FLP states. Now we consider the POP state, and the criteria for whether it is the true ground state for the small $B_\perp$ regime in which $r^2\to0$. Some insight can be obtained as follows. Consider the interlayer potential $D \equiv D_{FLP}^*$ at which the the CAF and FLP states are equal in energy \[Eqs. (\[tcecaf\]) and (\[tceflp\])\]. Recall that the slope of the POP state with respect to $D$ is $-2$, while that of the FLP state is $-4$. We evaluate the energy of the POP state at $D=D_{FLP}^*$. If $\tcE_{POP}(D_{FLP}^*)>\tcE_{CAF}$ then the POP state will not be the ground state for any $D$. For purely perpendicular field, assuming $E_Z\ll2|g_{xy}|$, we have $D_{FLP}^*\approx2g_z$. Since for $r^2\to0$ we have $\e_a\approx0$, this leads to the condition for the absence of the POP state, G\_0+|g\_[xy]{}|-g\_z+g\_[nz]{}&gt;0. Recall that in order to see the BU(1)$^2$ state at minimal $E_Z$, and assuming $E_Z\ll4|g_{xy}|$, we need $g_z>G_0+|g_{xy}|$. This means in order for the BU(1)$^2$ state to be the lowest in energy among the orbitally symmetric states, and for it to have a lower energy than the POP state, we need $g_{nz}$ greater than some critical value. This is easily understood, as a large, positive $g_{nz}$ penalizes orbital polarization. Let us now turn to the other extreme, very large values of $B_\perp$ such that $r^2\to1$ and $\e_a=\e_{a0}B_\perp$. Possible Phase Diagrams at large $B_\perp$ {#largebperp} ------------------------------------------ Setting $r^2\approx1$ we find &2g\_z+|g\_[xy]{}|,\ g\_K&2g\_z+|g\_[xy]{}|=,\ D\_[c1]{}\^[CAF]{}&,\ D\_[c1]{}\^[FM]{}&=+E\_Z-2|g\_[xy]{}|=2g\_z-|g\_[xy]{}|+E\_Z,\ D\_[c2]{}\^[KEK]{}&. For $E_Z<2|g_{xy}|$ we see that $D_c^{FLP/KEK}=g_K>D_{c}^{FLP/SVE}$, which means that one should consider $D_{c1}^{CAF}$ and $D_{C2}^{KEK}$. However, in the $r^2\to1$ limit, these are identical! This means the window for the BU(1)$^2$ state shrinks to zero as $r^2\to1$. The same is true for $E_Z>2|g_{xy}|$. At $r^2=1$ and $D=D_{c1}^{CAF}=D_{c2}^{KEK}$ the energy becomes independent of two of the three angles. This implies a $q=0$ collective mode whose energy vanishes, as has been found in a recent calculation [@denova_2017]. Thus, hints of the potential existence of the BU(1)$^2$ state can be seen in the collective mode spectrum even at $r^2=1$. We see then that the trigonal warping $t_3$, via the parameter $r^2<1$, is responsible for the existence of the BU(1)$^2$ state in a nonvanishing region of the parameter space. For this reason previous theoretical analyses, which in general have not included the effects of $t_3$, have not identified this state in the phase diagram. Hartree-Fock Phase Diagrams {#hfphasedias} --------------------------- Since the space of couplings is so large, we will take some guidance from experiments to narrow our choices. The POP state has been seen in experiments on BLG at $\nu=0$: at purely perpendicular fields, it makes its appearance for $B_\perp>$12 T [@Hunt_2016]. In some experiments a direct transition[@Hunt_2016] is seen between a putative CAF state at small $D$ and a putative FLP state at larger $D$, while in others there are intriguing hints that there may be an intermediate phase between the CAF and the FLP at small $B_\perp$ [@maher_2013; @jun-pvt]. Presumably, disorder, the screening environment, or perhaps microscopic features of how the samples are prepared, determine whether the intermediate phase is seen. A second result we will take from experiments is that when one tries to fit the observed sequence of transitions to a single-particle model, the anisotropy energy appears to be close to zero for $B_\perp<10$ T but turns on afterwards [@li_2017]. Looking at Fig. \[anisovsBperp\] we see that there is a similar behavior of $\e_a$ vs. $B_\perp$. This allows us to conjecture that the effective value of $\lambda$ is rather larger than conventionally assumed. To account for this diversity of observations, we will consider three sets of parameters embodying the three regimes of $g_z$ that we obtained in Section \[smallbperp\] for small $B_\perp$. [**Parameter Set 1**]{} (PS1) will have $g_z>G_0+|g_{xy}|$, so that there is an intervening Broken-U(1)$\times$U(1) phase as a function of $D$ between the CAF and the FLP phases for $E_Z<2|g_{xy}|$. [**Parameter Set 2**]{} (PS2) will have $G_0<g_z<G_0+|g_{xy}|$. This means that at the minimal $E_Z$ there is a direct first-order transition between the CAF and SVE phases, while for $E_Z>2|g_{xy}|$ the SVE phase smoothly connects the FM state to the FLP state. [**Parameter Set 3**]{} (PS3) will have $G_0>g_z$, so that there is always a direct first-order transition between the CAF and FLP phases. ### Parameter Set 1 {#PS1} The values we use (arbitrary units) are $g_0=0.5B_\perp$, $g_z=3.5B_\perp$, $g_{xy}=-1.65B_\perp$, and $g_{nz}=1.0B_\perp$. The dimensionless parameter $\lambda$ of Eq. (\[lambda\_def\]) is assumed to be $\lambda=5.0/\sqrt{B_\perp}$. In order to keep the POP state from appearing below about $12$T, we set the orbital anisotropy to $\tea=1.4$ Since we are using arbitrary units for the couplings $g_i$, our results for the values of $D$ at which transitions take place are also arbitrary. Therefore, in the phase diagrams that follow, we will not put units on the $D$ axis. Let us first consider the case of a perpendicular field only. From experimental measurements [@jun-pvt], the total field needed to spin-polarize a sample at $B_\perp=2$T is about $12$T. We combine this with the theoretical critical Zeeman coupling for full spin-polarization, $E_Z=2|g_{xy}|$, to obtain $E_Z=\frac{1}{3}|g_{xy}|$ for a purely perpendicular field. The phase diagram for this situation is shown in Fig. \[PS1-bperp-phase-dia-fig2\]. ![The $B_\perp-D$ phase diagram for PS1 for the case of only perpendicular field. Here and in the following, $D$ is in arbitrary units. This is identical to Fig. 1, reproduced here for convenience. At small $D$, the system is always in the CAF phase. For $B_\perp<11$T, as $D$ increases, the system undergoes a second-order phase transition (dashed red line) to the BU(1)$^2$ phase. Another second-order phase transition (dashed red line) takes the system at a slightly higher $D$ to the SVE state. Finally, at an even higher $D$ (dash-dotted blue line) the system goes into the FLP phase. For $B_\perp>11$T the POP state becomes lower in energy than the BU(1)$^2$ state for an intermediate range of $D$, and is the ground state between the two solid black lines. At higher $B_\perp$ the BU(1)$^2$ state gives way to the KEK state at the dashed red line, which in turn gives way to the FLP state at the green dashed line with the + symbols. []{data-label="PS1-bperp-phase-dia-fig2"}](fig5.eps){width="1.0\linewidth"} As can be seen, most of the phases discussed before appear in the phase diagram. Let us first focus on the small $B_\perp$ region, where we expect $r^2\ll1$. In accordance with the expectations of Section \[smallbperp\], we see that with increasing $D$, one encounters, in order, the CAF, BU(1)$^2$, SVE, and FLP states, all of which are identified from the numerically generated $\Delta$ matrix. Fig. \[Order-para-PS1-bperp-cut-6p0-fig\] illustrates the spin polarization $S_z$ and the valley polarization $\tau_z$ at fixed $B_\perp=6$T as a function of $D$. ![Order parameters at $B_\perp=6$T for PS1 with purely perpendicular field. Recall that $D$ is in arbitrary units. For $D<36$, the system is in the CAF phase. It makes a second-order transition to the BU(1)$^2$ phase at $D=36$ and remains in this phase till $D=46$, at which point it makes another second-order transition into the SVE phase. The SVE phase persists till about $D=50$, beyond which the system is fully layer polarized. []{data-label="Order-para-PS1-bperp-cut-6p0-fig"}](fig6.eps){width="1.0\linewidth"} At this field the CAF gives way to the BU(1)$^2$ state at around $D=36$. At $D\approx46$ a slight kink in the lines indicates that the system has made a transition to the SVE state. The SVE state is stable in the interval $46\leqslant D\leqslant 50$, and for $D>50$ the system is in the FLP state. At larger $B_\perp>11$T, the POP state makes its appearance by “eating-up” some of the regime that belongs to the BU(1)$^2$ state. An illustrative cut at $B_\perp=16$T is shown in Fig. \[Order-para-PS1-bperp-cut-16p0-fig\], which in addition to $S_z$ and $\tau_z$ illustrates $O_z$, the orbital polarization. 0.5cm ![Order parameters at $B_\perp=16$T for PS1 with purely perpendicular field. Once again, the system is in the CAF state for small $D$. At around $D=102$, there is a second-order transition into the BU(1)$^2$ state. This is followed by a first-order transition into the POP state at $D=107$. The POP state persists until $D=118$, at which point the system makes a first-order transition back into the BU(1)$^2$ state. At about $D=122$ there is another second-order transition, this time into the KEK state. Finally, at abour $D=130$, the KEK state gives way to the FLP state. []{data-label="Order-para-PS1-bperp-cut-16p0-fig"}](fig7.eps "fig:"){width="1.0\linewidth"} Now we see that the system undergoes a second-order transition from the CAF state to the BU(1)$^2$ state at $D\approx102$. This is followed by a first-order transition to the POP state at $D\approx107$, which then persists until $D\approx118$. The system now undergoes a first-order transition to a narrow sliver of the BU(1)$^2$ state, which gives way to the KEK state at $D\approx122$. The KEK state persists until $D\approx130$ beyond which the system is in the FLP state. For completeness, we present two other phase diagrams. In Fig. \[PS1-EZeqgxy-phase-dia-fig\], we consider an intermediate value of tilted field with $E_Z=|g_{xy}|$. The low $D$ phase is still the CAF state. ![Phase diagram for PS1 in a tilted field, such that $E_Z=|g_{xy}|$. The BU(1)$^2$ state appears between the dashed red lines, while the POP state appears between the solid black lines. The main qualitative difference between this figure and Fig. \[PS1-bperp-phase-dia-fig2\] is the absence of the KEK state at large $B_\perp$, where it has been supplanted by the SVE state. All transitions are second-order except for those into and out of the POP state. []{data-label="PS1-EZeqgxy-phase-dia-fig"}](fig8.eps){width="1.0\linewidth"} Note that the SVE state expands its domain compared to perperdicular field, and the BU(1)$^2$ state has a correspondingly smaller domain. The KEK state has disappeared altogether. This is because, unlike the SVE state, it has no spin polarization and thus cannot take advantage of the Zeeman field. The domain of the POP state has also expanded, and now it reaches down to $B_\perp=8$T. In Fig. \[PS1-EZ2p5gxy-phase-dia-fig\], we present the phase diagram for a very large tilted field of $E_Z=2.5|g_{xy}|$. The low $D$ phase is now the FM state. ![Phase diagram for PS1 at $E_Z=2.5|g_{xy}|$. The small $D$ region is now in the fully spin-polarized FM state, which makes a second-order transition (lower dashed blue line) to the SVE state, which in turn gives way to the FLP state via another second-order transition (upper dashed blue line). The BU(1)$^2$ state has disappeared and has been supplanted by the SVE state. The POP state intrudes into the SVE region via first-order transitions (solid black lines). []{data-label="PS1-EZ2p5gxy-phase-dia-fig"}](fig9.eps){width="1.0\linewidth"} We see that the BU(1)$^2$ state has disappeared. The SVE and POP states are better able to take advantage of the large $E_Z$ at intermediate values of $D$. ### Parameter Set 2 {#PS2} This set of parameters is identical to PS1, except $g_z=2.5B_\perp$. This change means that now $G_0<g_z<G_0+|g_{xy}|$. Furthermore, to keep the POP state from appearing below $\simeq 10$T, we need to increase the dimensionless orbital anisotropy to $\tea=1.77$. Fig. \[PS2-bperp-phase-dia-fig\] shows the phase diagram for PS2 with a purely perpendicular field ($E_Z=|g_{xy}|/3$). As can be seen, the BU(1)$^2$ phase has almost disappeared from the phase diagram. There is a tiny remnant of it for 8T$<B_\perp<$10T. ![Phase diagram for PS2 at $E_Z=|g_{xy}|/3$ (purely perpendicular field.) The small $D$ region is in the CAF phase. For small $B_\perp<8$T the CAF makes a direct first-order transition into the SVE phase (solid blue line), which then gives way to the FLP phase via a second-order transition (dashed blue line). Between 8T and 10T, the situation is very complicated at intermediate $D$, where many phases are almost identical in energy. At 10T, as one increases $D$, there is a direct first-order transition from the CAF phase into the POP state (lower solid black line). The system exits the POP state via another first-order transition into a narrow sliver of the BU(1)$^2$ state, which exists between the upper solid black line and the red line with circles. The BU(1)$^2$ state enters the KEK state via a second-order phase transition. Finally, the KEK state gives way to the FLP state. At larger $B_\perp$ the situation simplifies: The CAF makes a first-order transition into the POP, which makes another first-order transition into the KEK, which finally makes a second-order transition to th FLP state (dashed green line with + symbols).[]{data-label="PS2-bperp-phase-dia-fig"}](fig10.eps){width="1.0\linewidth"} There are several differences in the phase diagrams between PS1 and PS2. Focusing first on small $B_\perp$, the CAF goes into the SVE phase via a first-order transition, without going through the BU(1)$^2$ phase. The SVE phase gives way to the FLP phase at larger $D$ via a second-order transition. Fig. \[Order-para-PS2-bperp-cut-2p0-fig\] shows the evolution of the order parameters with $D$ for fixed $B_\perp=2$T. ![Order parameters as a function of $D$ at $B_\perp=2$T in PS2 for purely perpendicular field. The first-order nature of the transition between the CAF and the SVE states is clear. The SVE order parameters smoothly go over to those of the FLP. []{data-label="Order-para-PS2-bperp-cut-2p0-fig"}](fig11.eps){width="1.0\linewidth"} In Fig. \[Order-para-PS2-bperp-cut-10p0-fig\] we show the evolution of the order parameters at $B_\perp=10$T, which includes a sliver of the BU(1)$^2$ state. ![Order parameters as a function of $D$ at $B_\perp=10$T in PS2 for purely perpendicular field. At small $D$ the system is in the CAF phase. It makes a first-order transition into the POP state at $D=50.1$. The POP state gives way to the BU(1)$^2$ state via a first-order transition at $D=50.75$. The BU(1)$^2$ state persists until $D=51.25$, at which point the system makes a second-order transition to the KEK state. Finally, at $D=52$, the KEK state gives way to the FLP state via a second-order transition. []{data-label="Order-para-PS2-bperp-cut-10p0-fig"}](fig12.eps){width="1.0\linewidth"} The evolution of the order parameters at $B_\perp=16$T is presented in Fig. \[Order-para-PS2-bperp-cut-16p0-fig\]. ![Order parameters as a function of $D$ at $B_\perp=16$T in PS2 for purely perpendicular field. As $D$ increases, the first two transitions, from the CAF into the POP, and from the POP into the KEK state, are first-order. The final transition from the KEK to the FLP state is second-order. []{data-label="Order-para-PS2-bperp-cut-16p0-fig"}](fig13.eps){width="1.0\linewidth"} For completeness we examine PS2 for larger Zeeman values. In Fig. \[PS2-EZeqgxy-phase-dia-fig\] we present the phase diagram for PS2 at $E_Z=|g_{xy}|$. For $B_\perp<8$T, there are only two transitions as $D$ increases. First the CAF goes into the SVE state via a first-order phase transition, and then the SVE state gives way to the FLP state via a second-order transition. For larger $B_\perp>8$T, the CAF goes directly into the POP state via a first-order transition. The system then makes another first-order transition into the SVE state, which finally undergoes a second-order transition into the FLP state. Note also that the POP state, being able to take advantage of the larger Zeeman coupling, now appears at smaller values of $B_\perp$ as compared to the case of perpendicular field only. ![Phase diagram for PS2 in a tilted field, such that $E_Z=|g_{xy}|$. Only the CAF, the SVE, FLP and the POP appear. The transitions between the CAF, SVE and FLP states (dashed blue lines) are second-order, while those from the POP state (solid black lines) are first-order. []{data-label="PS2-EZeqgxy-phase-dia-fig"}](fig14.eps){width="1.0\linewidth"} In Fig. \[PS2-EZ2p5gxy-phase-dia-fig\] we present the phase diagram for PS2 at large Zeeman coupling, $E_Z=2.5|g_{xy}|$. ![The phase diagram for PS2 in a large Zeeman field $E_Z=2.5|g_{xy}|$. Only the FM phase at small $D$, the SVE, the FLP and the POP phases appear. []{data-label="PS2-EZ2p5gxy-phase-dia-fig"}](fig15.eps){width="1.0\linewidth"} The low $D$ phase is now the FM state. This implies that the transition from the FM to the SVE state should be second-order, since the SVE smoothly interpolates between the FM and the FLP. Indeed, in Fig. \[Order-para-PS2-EZ2p5gxy-cut-2p0-fig\], a cut at $B_\perp=2$T showing the evolution of the order parameters as a function of $D$ exhibits the second-order nature. ![Order parameters for $B_\perp=2$T in PS2 at large Zeeman coupling, such that $E_Z=2.5|g_{xy}|$. The small $D$ phase is the fully spin-polarized FM. This makes a second-order phase transition into the SVE, which smoothly interpolates to the FLP state via another second-order phase transition. []{data-label="Order-para-PS2-EZ2p5gxy-cut-2p0-fig"}](fig16.eps){width="1.0\linewidth"} At larger values of $B_\perp$, the POP state intervenes and two additional first-order phase transitions, into and out of the POP state, appear, as seen in Fig. \[Order-para-PS2-EZ2p5gxy-cut-12p0-fig\]. ![Order parameters at $B_\perp=12$T in PS2 at a large Zeeman coupling $E_Z=2.5|g_{xy}|$. The small $D$ phase is the fully spin-polarized FM. This makes a second-order phase transition into the SVE. The POP state intrudes via a first-order transition into the SVE. Another first-order transition takes the system back into the SVE, which smoothly interpolates to the FLP state via another second-order phase transition. []{data-label="Order-para-PS2-EZ2p5gxy-cut-12p0-fig"}](fig17.eps){width="1.0\linewidth"} ### Parameter Set 3 {#PS3} For PS3, we need to have $g_z<G_0$. So we choose the following values: $g_0=1.5B_\perp,\ g_z=1.75B_\perp,\ g_{xy}=-1.65B_\perp,\ g_{nz}=B_\perp$, and keep $\lambda_1=5$. In order to have the POP state not appear below $B_\perp=12$T at purely perpendicular field, we have to increase the value of the dimensionless orbital anisotropy to $\tea=3.8$. In Fig. \[PS3-bperp-phase-dia-fig\] we show the phase diagram for PS3 at purely perpendicular field. This is the simplest topology of the phase diagram, and only the CAF, FLP and POP states appear. All the transitions are first-order. ![Phase diagram for PS3 at purely perpendicular field, $E_Z=|g_{xy}|/3$. All the transitions are first-order. The CAF gives way directly to the FLP at small $B_\perp$, whereas the POP state intrudes for larger $B_\perp$. []{data-label="PS3-bperp-phase-dia-fig"}](fig18.eps){width="1.0\linewidth"} In Fig. \[PS3-EZeqgxy-phase-dia-fig\] we show the phase diagram at an intermediate value of the Zeeman coupling, $E_Z=|g_{xy}|$. Apart from the POP state appearing at lower $B_\perp$, and extending to larger $D$, there are no qualitative differences from the case of purely perpendicular field. ![Phase diagram for PS3 at an intermediate value of Zeeman coupling, $E_Z=|g_{xy}|$. This is very similar to the phase diagram of PS3 at perpendicular field. All transitions are first order. []{data-label="PS3-EZeqgxy-phase-dia-fig"}](fig19.eps){width="1.0\linewidth"} ![Phase diagram for PS3 at a large value of Zeeman coupling, $E_Z=|g_{xy}|$. The small $D$ phase is the FM, otherwise the phase diagram is very similar to those for PS3 at smaller $E_Z$. []{data-label="PS3-EZ2p5gxy-phase-dia-fig"}](fig20.eps){width="1.0\linewidth"} Discussion ========== Experimental Signatures of the Phase Transitions {#expt-sig-PT} ------------------------------------------------ We begin this section by discussing possible experimental signatures of the phases and transitions discussed above. To our knowledge three types of measurements have been performed on BLG in the quantum Hall regime: transport, compressibility, and layer polarizability. With respect to transport, all the bulk states we have analyzed are insulators with a charge gap. Deep within a phase, transport occurs only at the edges. In BLG, all quantum numbers except spin are broken by the edge potential; because of this, the FM state is expected to be a quantum spin Hall state [@Abanin_2006; @Fertig2006; @SFP; @pasha] whereas the others are trivial non-conducting states [@Kharitonov_edge; @us_2014; @us_2016; @kharitonov_2016]. At a transition between two bulk phases, there can be conduction by two distinct mechanisms. Firstly, if the transition is second-order and has at least one broken U(1) symmetry on at least one side of the transition (all our second-order transitions have this property), we may expect the stiffness of the broken U(1) angle to vanish at the transition. This leads to gapless charged [*edge*]{} excitations, as the present authors have established in monolayer graphene [@us_2014; @us_2016]. Secondly, if the transition is first-order, one may expect the formation of domains due to disorder. Presumably charged excitations are attracted to the domain walls, and if they percolate, there may be [*bulk*]{} conduction [@jungwirth_2001; @dhochak_2015]. Thus, both first- and second-order transitions are expected to be visible in transport. Bulk excitations can also provide information about the nature of the ground state. For example, gapless modes associated with broken U(1) symmetries should have clear signatures in heat transport[@pientka_2017]. Bulk excitations can also be probed via the compressibility. Several of the transitions we have described involve a U(1) symmetry breaking as the transition is crossed. In the broken symmetry phase, near the transition where there is a soft stiffness one expects very low energy, charged merons [@QHFM]. Nevertheless, we expect the system to remain incompressible at zero temperature: in order to inject an electron, one has to combine this low-energy meron (which is expected to support a small charge) with a high-energy antimeron (carrying the remaining charge of the electron). The resulting bimeron, the form in which electrons can be injected into the system, will have non-vanishing energy in spite of the low energy of one of its components. By contrast, at a first-order transition, if the domain walls percolate we expect that electrons can be injected at arbitrarily low energy, and the system becomes compressible. At $T>0$, the key criterion is whether the phase with spontaneously broken U(1) is below its Kosterlitz-Thouless transition temperature $T_{KT}$. In particular, the appearance of unbound (charged) vortices above $T_{KT}$ may lead to singular behavior in the compressibility as a function of temperature. Finally, layer polarizability measurements have recently become feasible for this system [@Hunt_2016]. The level of charge in each layer continuously varies in any state for which there is a broken U(1) symmetry involving the valley degree of freedom. Thus, the FM, CAF, POP and FLP states have a vanishing linear layer polarizability, while the BU(1)$^2$, SVE, and KEK states are layer polarizable. Such experiments thus allow one to probe when the U(1)$_{valley}$ symmetry is spontaneously broken in the bulk. Current experiments on BLG suggest that the CAF, FM, POP, and FLP states can be stable in BLG. In a subset of samples, at small $B_\perp$, an intermediate state[@maher_2013; @jun-pvt] [*may*]{} have been seen between the CAF and the FLP phases, suggesting that such samples are in parameter regimes consistent with PS1 or PS2. In some samples, an intermediate phase is also seen at small $B_\perp$, albeit at large tilted field, between the FM and the FLP phases[@jun-pvt]. Again, this is consistent with both PS1 and PS2. In other experiments, however, no intermediate phases are seen between the CAF and the FLP at small $B_\perp$, suggesting that those samples are consistent with PS3. What precisely determines in which parameter regime a particular sample might be remains unclear at this time, and is a subject for further investigation. Detailed observations at small $B_{\perp}$ in extremely clean and cold samples would greatly clarify the parameter regime to which pure BLG belongs. It is interesting to carry out a thought experiment in which we assume that the bulk spin susceptibility $\frac{\partial S_z}{\partial E_Z}$ can be measured, in addition to the layer polarizability $\frac{\partial \tau_z}{\partial D}$ and the cross-susceptibilities $\frac{\partial S_z}{\partial D}\equiv \frac{\partial \tau_z}{\partial E_Z}$ (the last identity is a Maxwell relation arising from $S_z=-\frac{\partial \tcE}{\partial E_z}$ and $\tau_z=-\frac{\partial \tcE}{\partial D}$). Such measurements may indeed be accessible, e.g. using the technique of Reznikov [*et al*]{} [@Reznikov]. The combined measurement allows one to distinguish between the different possible states. The FM, CAF, POP, and FLP have a vanishing layer polarizability. The FM, KEK, and FLP have a vanishing spin susceptibility. The SVE state has both layer polarizability and spin susceptibility nonvanishing, but satisfies $S_z+\tau_z=4$, which implies +=0=+ \[SVE-sus-constraint\]Finally, the BU(1)$^2$ state also has all susceptibilities nonvanishing, but is not subject to the condition of Eq. (\[SVE-sus-constraint\]). This allows us, in principle at least, to distinguish the BU(1)$^2$ state from other possibilities. Caveats and Omissions {#Caveats} --------------------- We next briefly review some of the underlying assumptions that lead to the model analyzed in this work. We first separated the Coulomb and other lattice scale interactions into an SU(4) symmetric part (which plays no role in choosing the ground state) and a part that does not respect SU(4) symmetry,. We assumed that the part that does not respect SU(4) symmetry can be represented as short-range interactions. These short-range interactions respect the spin-SU(2) but have only a U(1) symmetry in the valley indices. Finally we assumed that all interaction parameters $g_i$ are proportional to $B_\perp$, corresponding to ultra-short-range interactions. Each of these assumptions can be challenged. Consider first our assumption that $g_i\propto B_\perp$. This seems reasonable from the renormalization group (RG) standpoint, as can be seen from the following argument. At high energies, the dispersion is Dirac-like, and short-range interactions are irrelevant as one scales down in energy: \_i()=\_i(0)e\^[-]{}, where $\tg_i$ are the dimensionless couplings (the ratio of the dimensionful couplings to the kinetic energy scale), $\ell$ is the RG flow parameter defined by $e^{-\ell}=\Lambda(\ell)/\Lambda(0)$, and $\Lambda(0)$ is the bandwidth. At a scale proportional to the interlayer hopping $t_\perp$ (corresponding to RG scale $\ell_\perp$, say) the quadratic band touching manifests itself, and the one-loop RG flow of $\tg_i$, if one neglects $t_3$, becomes marginal [@Vafek_2010]. In general the RG flows may be written in the form =C\_[ijk]{}\_j\_k, and should be stopped at a kinetic energy scale $\sim B_{\perp}$ which is of relevance to the system we are studying. Since they are marginal, the values of $g_i$ will follow the kinetic energy scale, thus becoming proportional to $B_\perp$. Complications arise when $t_3$ enters the picture. At the quadratic band touching $t_3$ is a relevant coupling and will grow. Further, we know that $t_3$ is generated by the interactions[@Pujari_2016], and will in turn affect the flow of the $g_i$. Thus, it is likely that the couplings $g_i$ do have some $B_\perp$ dependence in the presence of trigonal warping. Since we have not worked out the RG flow equations in the presence of $t_3$, we have not taken this into account, and have made the naïve assumption that $g_i\propto B_\perp$, which follows from directly computing the interaction matrix elements for our model in the Landau levels of interest, without including any renormalization effects. Secondly, we assume that all our interactions are ultra-short-range. Here we are on somewhat firmer footing. Introducing a $\bq$-dependence of the form $e^{-|\bq|^2\xi^2}$ into the interactions $v_i(\bq)$ will leave the Hartree terms unchanged, but reduce the exchange terms by a factor close to unity. This does change some of the inequalities which we use to define the different parameter sets (PS1, PS2, and PS3), but does not change the qualitative nature of the phases or the topologies of the phase diagrams. As an aside, introducing such a $\bq$-dependence into the Kharitonov model [@kharitonov_bulk_bilayer] will lead to a BU(1)$^2$ phase in the phase diagram. Thirdly, we reiterate that the four couplings retained in our interaction model are only a subset of many such couplings which are allowed by the symmetry of the system. This was largely to keep a tractable parameter space size for our study; however, we believe that other couplings will not qualitatively alter the topologies of the phase diagrams or the nature of the phases we encounter. Finally, our analysis has been carried out within the Hartree-Fock approximation. Quantum fluctuations could play an important role near second-order phase transitions, particularly for states with broken U(1) symmetries. These are generically accompanied by soft stiffnesses when they are first entered, so that low-energy excitations around the HF state will necessarily exist. Conclusions and Open Questions {#Conclusions} ============================== In this work we have studied the possible zero-temperature ground states of bilayer graphene (BLG) at charge neutrality in a quantizing perpendicular magnetic field $B_\perp$. This $\nu=0$ system is very rich, possessing three sets of discrete labels: spin, valley, and orbital, leading to eight nearly degenerate Landau levels in the low-energy manifold. (Recall that by “low-energy manifold” we mean the manifold of states near the Fermi energy.) Experimentally, the system can be probed by applying a tilted magnetic field (to increase the Zeeman coupling $E_Z$) and/or by applying a perpendicular electric field $D$ which induces layer polarization. In the presence of these external fields, the symmetry of the problem is reduced to U(1)$_{spin}\times$U(1)$_{valley}$. Our philosophy is to ignore the SU(4) symmetric, long-range part of the Coulomb interaction completely, because it plays no role in ground state selection at $\nu=0$. Our model is based on an effective Hamiltonian, containing only short-range interactions, in the truncated Hilbert space of the low-energy manifold. Effects of the filled Dirac sea [@Herbut_2007; @Shizuya_2012; @Roy_2014_1; @Roy_2014] are assumed to be absorbed into renormalizations of the couplings of the effective Hamiltonian [@kharitonov_bulk_monolayer; @kharitonov_bulk_bilayer]. We incorporate two aspects distinct from previous work [@kharitonov_bulk_bilayer; @lukose_2016; @knothe_2016; @Jia_2017]: (i) We include the effect of the trigonal warping $t_3$ (an interlayer hopping term allowed by the lattice symmetries) nonperturbatively in the one-body states of the low-energy manifold that form our basis. (ii) In addition to interactions introduced in previous work [@kharitonov_bulk_monolayer; @kharitonov_bulk_bilayer] ($g_z$ and $g_{xy}$ which correspond to $U(1)_{valley}$ symmetric interactions), we introduce two new interactions into our effective Hamiltonian, one ($g_0$) which treats all discrete labels equally, and another ($g_{nz}$) which is an Ising-like interaction in the orbital sector. The dependence of the dimensionless coupling constant associated with $t_3$ on $B_\perp$, together with suitable values of the interaction strengths, leads to the stabilizition of a hitherto unknown phase. This phase, which we dub the Broken-U(1)$\times$U(1) or BU(1)$^2$ phase, spontaneously breaks two distinct U(1) symmetries, and is one of the central findings in this work. Hints of its existence can be gleaned from unexpected zero modes in the collective spectrum [@denova_2017] even at $t_3=0$. In contrast, all phases known previously at $\nu=0$ are either symmetric under U(1)$_{spin}\times$U(1)$_{valley}$ or spontaneously break a single U(1). The spin-polarized ferromagnet (FM) and the fully layer polarized (FLP) phases are symmetric, while the canted antiferromagnet (CAF), the Kekule (KEK), and the spin-valley entangled (SVE) phases break a single U(1) symmetry. We explored three parameter sets of couplings characterized by inequalities among them. For parameter set 1 (PS1), $g_z>g_0+\half g_{nz}+|g_{xy}|$, and the BU(1)$^2$ phase invariably appears in the $B_\perp-D$ phase diagram at small $B_\perp$ and small $D$ when the $\bB$-field is not tilted. In this regime, transitions between the CAF, BU(1)$^2$, SVE, and FLP phases are driven by increasing $D$ and are all second-order. At large $B_\perp$ a partially orbitally polarized (POP) phase, and the Kekulé (KEK) phase intervene between the CAF and the FLP phases for intermediate values of $D$. Transitions between the POP and other states are always first-order, while the transition from the KEK state to the FLP state is second-order. As the field is tilted and the Zeeman energy increased, the BU(1)$^2$ phase shrinks and disappears from the $B_\perp-D$ phase diagram. Parameter set 2 (PS2) satisfies the inequalities $g_0+\half g_{nz}+|g_{xy}|>g_z>g_0+\half g_{nz}$. In this case the BU(1)$^2$ phase, if it appears at all, is confined to a small sliver of $D$ and $B_\perp$ near the onset of the POP state when the $\bB$-field is untilted. At small $B_\perp$ the CAF state transitions directly to the SVE state via a first-order transition as $D$ is increased, which in turn smoothly goes over into the FLP state via a second-order transition at even higher $D$. As above, at larger $B_\perp$, the POP state intervenes at intermediate $D$, and a KEK state may appear at higher $D$ which ultimately gives way to the FLP state. Parameter set 3 (PS3) satisfies $g_z <g_0+\half g_{nz}$, and has the simplest phase diagram of all. The CAF/FM state at small $D$ undergoes a first-order transition to either the FLP or the POP state, depending on the value of $B_\perp$. All transitions in PS3 are first-order. The BU(1)$^2$ phase, if it exists, always appears in a narrow window of $D$. Since it undergoes second-order phase transitions to states with a single broken U(1) at its $D$-boundaries, one (pseudo)spin-stiffness must always vanish at each transition. In previous work we have shown that in such cases the gap to edge transport vanishes at the transition. Depending on the details of the stiffnesses, and the temperature at which measurements are made, the BU(1)$^2$ phase may appear to be metallic. An alternative possibility is that quantum fluctuations disorder at least one of the broken U(1)’s to form a symmetric phase with vanishing gap at either $D$-boundary. Our results also raise a host of interesting questions. Foremost among them is the issue of edge conduction in the various states. The BLG edge is expected to break all lattice symmetries, but preserve spin-rotation symmetry, because spin-orbit coupling is tiny. For the CAF state in monolayer graphene the present authors showed that edge conduction occurs via topological vortex excitations of the CAF order parameter bound to an image antivortex near the edge [@us_2014; @us_2016]. In a quantum Hall state such topological objects carry charge due to the spin-charge relation[@QHFM]. In BLG, the SVE and KEK states are valley analogues of the CAF, and it remains to be seen whether this edge physics carries over to the two latter phases. Perhaps the most interesting is the edge BU(1)$^2$ phase, because the bulk supports several flavors of topological excitations (vortices can be formed from either of the two broken U(1)’s). The effects of thermal and/or quantum disordering of the BU(1)$^2$ state should also be explored. Another set of interesting questions concerns fillings close to $\nu=0$, particularly in the range $-4\le \nu\le 4$. All these fillings nominally involve only the nearly degenerate set of Landau levels around the Fermi energy for undoped BLG. Trigonal warping likely impacts the phase diagram at such fillings, and a detailed investigation could help identify the appropriate interaction regime for BLG. Lastly, on the theoretical side, a full renormalization-group analysis for the short-range couplings in the presence of $t_3$ and a quantizing magnetic field, while challenging, would in principle indicate the scale of couplings that apply to models such as we have analyzed, in which the degrees of freedom are projected to a small number of Landau levels. There are also intriguing connections between the phase transitions in BLG at $\nu=0$ and recent ideas of critical deconfinement [@senthil], which is the phenomenon whereby the emergent degrees of freedom at a phase transition are fractionalized in terms of the order parameter fields on either side of the transition. The canonical example of critical deconfinement is the Neel to Valence Bond Solid transition in a class of two-dimensional quantum antiferromagnets. Recall that in the absence of Zeeman coupling, the CAF state would become an antiferromagnet (AF). Recently, it was argued[@Lee-Sachdev_2015] that the transition between the AF and the KEK phase would be critically deconfined. Adding the Zeeman coupling will convert the deconfined transition into a region where the two order parameters coexist[@senthil]. The BU(1)$^2$ phase does have both CAF and KEK order parameters but, in our model, exists even at zero Zeeman coupling. Last, but not least, it has been proposed[@xu-etal] that the fully polarized FM state in BLG (achieved at large Zeeman coupling) could be a realization of a bosonic symmetry-protected topological insulator[@lu-vishwanath]. Precisely what set of interaction parameters would realize such a state remains an open question. We are grateful to Jun Zhu, Jing Li, Andrea Young, Mike Zaletel, and Juan Ramon de Nova for illuminating conversations, and to the Aspen Center for Physics (NSF Grant 1066293), where this work was begun and completed. GM thanks the NSF (DMR-1306897) and the Gordon and Betty Moore Foundation for support. HAF acknowledges the support of the NSF through grant Nos. DMR-1506263 and DMR-1506460. ES thanks support of the Israel Science Foundation (ISF) via grant no. 231/14, of the Simons Foundation, and thanks the hospitaliy of the Kavli Institute for Theoretical Physics (NSF PHY-11-25915). Finally we would like to acknowledge support for all the present authors by the US-Israel Binational Science Foundation (BSF-2012120). Derivation of the coefficients $A_{nm}$ {#Anm} ======================================= In this Appendix we derive a power-series expansion in $\lambda$ for the states $|\psi_A\rangle$, $|\psi_B\rangle$ \[Eq. (\[psi\_AB\_integral\])\], and consequently the expressions for the coefficients $A_{nm}$ in Eq. (\[psi\_01\_def\]). We start by considering the integral \[t\_to\_xi\] & &\_0\^dte\^[-ita\^]{} =\ & &()\^[1/3]{}e\^\_0\^d\^[-2/3]{}e\^[-]{}{e\^[-]{}(3)\^[1/3]{}a\^}where we have used the change of variables $t^3=i3\lambda\xi$. Implementing a power-series expansion of the last exponential factor in Eq. (\[t\_to\_xi\]), and performing the integration over $\xi$, we obtain \[int\_to\_series\] \_0\^dte\^[-ita\^]{} = ()\^[1/3]{}e\^\_[n=0]{}\^ e\^[-]{}(a\^)\^n . Employing Eq. (\[psi\_AB\_integral\]), we thus find \[psi\_A\_series\] |\_A= ()\^[1/3]{}\_[n=0]{}\^{(2n-1)} (a\^)\^n|0 . To get a similar expansion for $|\psi_B\rangle$, we repeat the same steps for the purely real integral \[realint\_to\_series\] \_0\^dte\^[--ta\^]{} = ()\^[1/3]{}\_[n=0]{}\^ (-1)\^n(a\^)\^n ; substituting in Eq. (\[psi\_AB\_integral\]), this yields \[psi\_B\_series\] |\_B= ()\^[1/3]{}\_[n=0]{}\^ (a\^)\^n|0 . We next examine the oscillating factors in Eqs. (\[psi\_A\_series\]) and (\[psi\_B\_series\]), which exhibit a 3-fold periodicity in $n$: for any integer $m$, n &=& 3m-1{(2n-1)}=(-1)\^n-{(2n-1)}=0\ n &=& 3m{(2n-1)}=(-1)\^m ,(-1)\^n-{(2n-1)}=(-1)\^m\ n &=& 3m+1{(2n-1)}=(-1)\^m ,(-1)\^n-{(2n-1)}=(-1)\^[m+1]{} . \[osc\_factors\] Inserting Eq. (\[osc\_factors\]) in (\[psi\_A\_series\]), (\[psi\_B\_series\]) and using $|N\rangle=\frac{1}{\sqrt{N!}}(a^\dagger)^N|0\rangle$, we obtain |\_A&=& (|\_0+|\_1)\ |\_B&=& |\_0-|\_1where |\_0&=& \_[m=0]{}\^(-1)\^m(m+) |3m\ |\_1&=& \_[m=0]{}\^(-1)\^m(m+) |3m+1 . \[psi\_01\_unnorm\] By definition, $|\tilde{\psi}_n\rangle$ are orthogonal ($\langle\tilde{\psi}_0 |\tilde{\psi}_1\rangle$=0) for arbitrary prefactors of each. Hence, introducing the normalization factors $C_0$, $C_1$, we arrive at the orthonormal basis states Eq. (\[psi\_01\_def\]). Once this form has been obtained, it is straightforward to verify that these states satisfy $(a^2+\lambda a^{\dagger})|\tilde{\psi}_0\rangle=0$. Form Factors {#app:FF} ============ In this Appendix we discuss some details relevant to the calculation of the density matrix elements, Eq. (\[eq:trho\]), and in particular how their form leads to Eq. (\[eq:trho\_integral\]). We begin with the basis states $|n,\alpha,k\rangle$ in Eq. (\[nalpha\_def\]), |n,,k\_[m=0]{}\^(-1)\^[m]{}A\_[nm]{}|3m+n,k , for which the coefficients $A_{nm}$ are defined in Eq. (\[psi\_01\_def\]). Direct substitution yields the explicit form $$\tilde\rho^{\alpha\beta}_{n_1n_2}({\bf q})= \sum_{k_1=0}^{\infty}\sum_{k_2=0}^{\infty}(-1)^{k_1\alpha+k_2\beta} A_{n_1k_1}A_{n_2k_2}\rho_{3k_1+n_1,3k_2+n_2}(\bf{q})$$ where the usual Landau level matrix elements are defined as $$\rho_{n_1n_2}({\bf q})=(-1)^{n_<+n_2}e^{-q^2\ell^2/4} \sqrt{\frac{n_{<}!}{n_{>}!}} \,e^{i(n_1-n_2)(\theta_q-\pi/2)} \left(\frac{q\ell}{\sqrt{2}}\right)^{n_{>}-n_{<}} L_{n_<}^{|n_{1}-n_{2}|}\left(\frac{q^2\ell^2}{2}\right).$$ In this equation, $n_{<}$ ($n_{>}$) is the smaller (larger) of $n_1$ and $n_2$, $L_m^n$ is an associated Laguerre polynomial, and $\theta_q$ is the angle formed by ${\bf q}$ with the $\hat{x}$-axis. Now consider the exchange integral $$\begin{aligned} &\int\frac{d^2q}{(2\pi)^2}&\!\!\!\!\!\!\!\!\!\! v({\bf q}) \tilde\rho_{n_1n_2}^{\alpha\beta}({\bf q})\tilde\rho_{m_1m_2}^{\gamma\delta}(-{\bf q}) \nonumber \\ &=\sum_{k_1k_2k_3k_4}&(-1)^{k_1\alpha+k_2\beta+k_3\gamma+k_4\delta} A_{n_1k_1}A_{n_2k_2}A_{m_1k_3}A_{m_2k_4} \int \frac{d^2q}{(2\pi)^2} v({\bf q}) \rho_{3k_1+n_1,3k_2+n_2}({\bf q})\rho_{3k_3+m_1,3k_4+m_2}(-{\bf q}).\nonumber\\ \label{rhorhoint}\end{aligned}$$ Writing $N_1\equiv 3k_1+n_1$, $N_2\equiv 3k_2+n_2$, $M_1\equiv 3k_3+m_1,$ and $M_2\equiv 3k_4+m_2,$ Eq. (\[rhorhoint\]) can be reexpressed as $$\begin{aligned} \int\frac{d^2q}{(2\pi)^2}v({\bf q}) \tilde\rho_{n_1n_2}^{\alpha\beta}({\bf q})\tilde\rho_{m_1m_2}^{\gamma\delta}(-{\bf q}) =\sum_{k_1k_2k_3k_4}(-1)^{k_1\alpha+k_2\beta+k_3\gamma+k_4\delta} A_{n_1k_1}A_{n_2k_2}A_{m_1k_3}A_{m_2k_4} \nonumber\\ \times \, \int \frac{d^2q}{(2\pi)^2} v({\bf q}) e^{-q^2\ell^2/2} (-1)^{N_{<}+N_2+M_<+M_2+M_1+M_2}\sqrt{\frac{N_<!M_<!}{N_>!M_>!}} \,e^{i(\theta_1-\frac{\pi}{2})(N_1-N_2+M_1-M_2)} \nonumber\\ \times\,\left(\frac{q\ell}{\sqrt{2}}\right)^{|N_1-N_2|+|M_1-M_2|} L_{N_<}^{|N_1-N_2|}\left(\frac{q^2\ell^2}{2}\right) L_{M_<}^{|M_1-M_2|}\left(\frac{q^2\ell^2}{2}\right), \label{B5}\end{aligned}$$ where we used the property $$\rho_{n_1n_2}(-{\bf q})=(-1)^{n_1+n_2}\rho_{n_1n_2}({\bf q}).$$ The integration over $\theta_q$ forces the integral to vanish unless $N_1+M_1=N_2+M_2$. Moreover, specializing to the case where $v({\bf q})$ has no ${\bf q}$ dependence, the orthogonality relation $$\int_0^\infty dx \, e^{-x}x^{\alpha} L_m^{\alpha}(x) L_n^{\alpha}(x) = \frac{\Gamma(n+\alpha+1)}{n!}\delta_{mn} \label{ortho_relation}$$ guarantees that the integral in Eq. (\[B5\]) vanishes unless $N_<=M_<$. Writing $v({\bf q}) \rightarrow \tilde v$, we arrive at the relation $$\int\frac{d^2q}{(2\pi)^2}v({\bf q}) \tilde\rho_{n_1n_2}^{\alpha\beta}({\bf q})\tilde\rho_{m_1m_2}^{\gamma\delta}(-{\bf q}) = \frac{\tilde v}{2\pi \ell^2} \delta_{n_1m_2} \delta_{m_1n_2} r_{\alpha\delta}^{(n_1)} r^{(n_2)}_{\beta\gamma}$$ where $$r_{\alpha\beta}^{(n)} = \sum_{k=0}^{\infty} (-1)^{k(\alpha+\beta)} A_{nk}^2 \equiv\delta_{\alpha\beta}+r(1-\delta_{\alpha\beta}). \label{rabn}$$ $r_{\alpha\beta}^{(n)}$ turns out to be unity if $\alpha = \beta$ because of the normalization condition that the wavefunctions coefficients $A_{nk}$ must obey. 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--- abstract: 'The current generation of D-Wave quantum annealing processor is designed to minimize the energy of an Ising spin configuration whose pairwise interactions lie on the edges of a [*Chimera*]{} graph ${\mathcal{C}}_{M,N,L}$. In order to solve an Ising spin problem with arbitrary pairwise interaction structure, the corresponding graph must be minor-embedded into a Chimera graph. We define a combinatorial class of [*native clique minors*]{} in Chimera graphs with vertex images of uniform, near minimal size, and provide a polynomial-time algorithm that finds a maximum native clique minor in a given induced subgraph of a Chimera graph. These minors allow improvement over recent work and have immediate practical applications in the field of quantum annealing.' author: - Tomas Boothby - 'Andrew D. King' - Aidan Roy bibliography: - 'bibtex.bib' title: 'Fast clique minor generation in Chimera qubit connectivity graphs[^1]' --- Introduction and motivation =========================== D-Wave quantum annealing processors are designed to sample low-energy spin configurations in the Ising model using open-system quantum annealing [@Albash2015; @Boixo2014; @Dickson2013; @Johnson2011]. Input to the processor consists of an [*Ising Hamiltonian*]{} $(h,J)$, where $h \in \mathbb R^n$ is a vector of [*local fields*]{} and $J\in \mathbb R^{n\times n}$ is a matrix of [*couplings*]{}, which we assume here to be symmetric. The [*energy*]{} of a spin configuration $s\in \{-1,1 \}^n$ is given as $$E(s) = E(h,J,s) = s^TJ s + s^Th.$$ The output of an [*anneal*]{} (i.e. a run) of the processor is a low-energy state $s$, which consists of an Ising spin (either $-1$ or $1$) for each [*qubit*]{}. Nonzero terms of $J$ are physically realized using programmable [*couplers*]{}. These couplers only exist between certain pairs of physically proximate qubits. The input $(h,J)$ is therefore restricted such that if $J_{i,j}\neq 0$, there must be a coupler between qubit $i$ and qubit $j$. We rephrase this in graph-theoretic terms: The [*connectivity graph*]{} of $(h,J)$, which we denote by $G_J$, is the undirected graph on $n$ vertices whose adjacency matrix has the same nonzero entries as $J$. Likewise, each processor has a [*hardware graph*]{} $G$ representing the available qubits and couplers in the processor. For $(h,J)$ to be input directly to the processor, $G_J$ must be a subgraph of $G$. If this is not the case, we can indirectly input $(h,J)$ to the hardware by embedding $G_J$ as a graph minor of $G$. Implementing graph minors in the Ising model involves putting a strong ferromagnetic coupling $J_{i,j}\ll 0$ between any two adjacent vertices $i,j$ of $G$ in the same vertex image. This coupling compels multiple qubits to take the same spin, thus acting like a single logical qubit. The method is studied in greater detail elsewhere [@Choi2008; @Venturelli2014; @Perdomo-Ortiz2015; @King2014; @Cai2014]. In this paper we consider the problem of finding large clique minors in the hardware graph. This is sufficient for minor-embedding any problem of appropriate size in a given hardware graph, and allows the study of random, fully-connected spin glass problems, as in recent work [@Venturelli2014]. Klymko et al. first provided a polynomial-time algorithm for generating large clique minors in subgraphs of hardware graphs [@Klymko2014]. In Section \[sec:previouswork\] we provide evidence that our algorithm uses fewer physical qubits and allows the embedding of larger minors. The Chimera graph and triangle embeddings ----------------------------------------- D-Wave processors currently operate using a [*Chimera*]{} hardware graph ${\mathcal{C}}_{M,N,L}$, which is an $M\times N$ grid of $K_{L,L}$ complete bipartite graphs (unit cells). [ Two]{}  processors use a 512-qubit ${\mathcal{C}}_{8,8,4}$ hardware graph, and the most recent processors use a 1152-qubit ${\mathcal{C}}_{12,12,4}$ graph. ![(Color online) “Triangle” clique embedding in ${\mathcal{C}}_{4,4,4}$, which motivated the design of the Chimera graph. Every chain in this embedding has 5 qubits.[]{data-label="fig:tridia"}](tridia.pdf) The Chimera graph was chosen, in part, because it contains a particularly nice clique minor [@Choi2011] (see Figure \[fig:tridia\]). This [*triangle embedding*]{} is uniform in the sense that each vertex image (or [*chain*]{}) has the same number of vertices, and is near-optimal in the sense that it gives a $K_{LM}$ minor in ${\mathcal{C}}_{M,M,L}$, whereas ${\mathcal{C}}_{M,M,L}$ has treewidth $LM$ and therefore contains no $K_{L M +2}$ minor[^2]. A degree argument also shows that any uniform $K_{LM}$ minor requires chains of size at least $M$, while the triangle embedding has chains of size $M+1$. In practice, a given processor will have a number of inoperable qubits. If there are $t$ inoperable qubits, up to $t$ chains in a particular clique embedding can be rendered useless. These inoperable qubits force us to find a clique minor in an induced subgraph of ${\mathcal{C}}_{M,N,L}$. In the face of this, note that there are at least four triangle embeddings, which we can find by simply rotating the embedding shown in Figure \[fig:tridia\]. So a first attempt at minimizing the impact of inoperable qubits is to choose the triangle embedding for which the greatest number of chains survive. Triangle embeddings can be generalized further. A triangle embedding consists of overlapping ell-shaped (L-shaped) “bundles” of chains, and each chain in a bundle joins a horizontal “wire” with a vertical “wire” via a matching at the corner of the bundle. First, the structure of overlapping ell-shaped bundles can be generalized from the triangle (we can avoid all four corner unit cells, for example). Second, the corner matchings can be chosen arbitrarily to minimize the impact of inoperable qubits (if there are three intact vertical wires and three intact horizontal wires, we can ensure that they are matched together to make three intact chains). These generalizations result in exponential expansion of the number of clique embeddings available, but we can optimize over them in polynomial time using a dynamic programming approach. Defining and efficiently optimizing over these clique embeddings are the main goals of this work. In the next section we formalize the definition of [*native clique embeddings*]{} that generalize triangle embeddings, and give a combinatorial characterization of the same. In Section \[sec:3\] we give a dynamic programming technique that, given an induced subgraph of ${\mathcal{C}}_{M,N,L}$, finds a maximum-sized native clique embedding in polynomial time. One desirable feature of native clique embeddings is uniform chain length, which results in uniform, predictable chain dynamics throughout the anneal [@Venturelli2014]. Clique embeddings found through heuristic methods such as the algorithm described by Cai et al. [@Cai2014] generally lack this property. In Section \[sec:previouswork\] we compare the results of our approach to those of the somewhat similar approach by Klymko et al. [@Klymko2014]. The shorter chains and larger cliques generated by our approach lead to improved tunneling dynamics and error insensitivity [@Dziarmaga2005; @Venturelli2014; @Young2013]. Native Clique Embeddings ======================== ![(Color online) (*l*) A maximum native clique embedding of $K_{24}$ in an induced subgraph of ${\mathcal{C}}_{8,8,4}$ with 26 randomly selected vertices deleted. (*r*) The corresponding block clique embedding, with dots indicating corners of the ell blocks. []{data-label="fig:nativeembedding"}](nativeembedding_a.pdf "fig:") ![(Color online) (*l*) A maximum native clique embedding of $K_{24}$ in an induced subgraph of ${\mathcal{C}}_{8,8,4}$ with 26 randomly selected vertices deleted. (*r*) The corresponding block clique embedding, with dots indicating corners of the ell blocks. []{data-label="fig:nativeembedding"}](nativeembedding_b.pdf "fig:") We now formally define the structures required to construct and analyze native clique embeddings. Recall that Chimera ${\mathcal{C}}_{M,N,L}$ is an $M\times N$ grid of $K_{L,L}$ unit cells. Specifically, ${\mathcal{C}}_{M,N,L}$ has vertices $V = \{1,\cdots,M\} \times \{1,\cdots,N\} \times \{0,1\} \times \{1,\cdots,L\}$ and edges: -------------------------------- -------------------------------------- $(x,y,0,k) \sim (x+1,y,0,k)$ (horizontal inter-cell couplings), $(x,y,1,k) \sim (x,y+1,1,k)$ (vertical inter-cell couplings), and $(x,y,0,k_1) \sim (x,y,1,k_2)$ (intra-cell couplings). -------------------------------- -------------------------------------- We construct native clique embeddings using [*wires*]{}. For $t\geq 1$, a [*horizontal wire*]{} of length $t$ is a contiguous set of vertices $\{(x+i,y,0,k): i\in[0,t-1]\}$, whose induced subgraph is a path on $t$ vertices. Likewise, a [*vertical wire*]{} is a set $\{(x,y+i,1,k): i\in[0,t-1]\}$. An [*ell*]{} is the union of a horizontal wire and a vertical wire where there is an edge between one end of the horizontal wire and one end of the vertical wire. Note that these ends are necessarily in the same unit cell, which we call the [*corner*]{} of the ell; for an ell $\ell$ we denote the corner by $c(\ell)$. In the embeddings we study, each chain is an ell. Our aim now is to specify an orderly way of arranging them into a clique minor. Looking at Figure \[fig:nativeembedding\], one may notice that chains appear in sets that intersect the same unit cells. With this in mind, for an ell $\ell$ we define its [*ell block*]{} $(X(\ell),c(\ell))$: $X(\ell)$ is the set of unit cells intersecting $\ell$, and again $c(\ell)$ is the corner of $\ell$, which we must specify for the ell block in order to avoid ambiguity in the case of horizontal or vertical wires of length $1$. We define an [*ell bundle*]{} $B$ as a (possibly empty) set of vertex-disjoint ells $\ell_1,\dots,\ell_p$ with the same ell blocks, i.e. such that $|\{(X(\ell),c(\ell)) \mid \ell \in B\}| \leq 1$. A *block clique embedding* is a set ${\mathcal{X}}$ of $n$ ell blocks $\{(X_1,c_1),\dots,(X_n,c_n)\}$ such that - each $X_i$ contains $n$ unit cells (so ells have length $n+1$), and - every distinct pair $X_i$, $X_j$ in ${\mathcal{X}}$ intersects at exactly one unit cell, which is in the horizontal component of one ell block and the vertical component of the other. A [*native clique embedding*]{} respecting a block clique embedding ${\mathcal{X}}$ is a collection ${\mathcal{B}}$ of ell bundles $\{B_i,\dots,B_n\}$ such that for each $i$ and for each $\ell\in B_i$, $(X_i,c_i) = (X(\ell),c(\ell))$, i.e.  $(X_i,c_i)$ is the ell block for each ell in $B_i$. From this definition and the above, we infer that - any two ells $\ell$ and $\ell'$ in the same bundle have exactly two edges between them, both in the unit cell $c(\ell) = c(\ell')$, and - any two ells $\ell$ and $\ell'$ in different bundles have exactly one edge between them, and it is in the unit cell $X(\ell)\cap X(\ell')$. Hence a native clique embedding is a clique embedding. It turns out that in a block clique embedding ${\mathcal{X}}=\{(X_1,c_1),\dots,(X_n,c_n)\}$, the corners $c_1,\dots,c_n$ form a permutation in the $n\times n$ matrix representing the unit cells of the graph $\mathcal C_{n,n,L}$. These permutations have a specific structure that is in direct correspondence with a class of permutations representable by circular point sets studied recently by Vatter and Waton [@circlepoints]. The following theorem provides a constructive classification of block clique embeddings and shows that, in contrast with triangle embeddings, native clique embeddings exist in abundance. \[thm:blockclique\] In a ${\mathcal{C}}_{n,n,L}$ Chimera graph for $n\geq 2$, there are $4^{n-1}$ block clique embeddings that contain $n$ ell blocks. In particular, they are in natural bijection with the set $\{\mathsf{E,W}\}\times \{\mathsf{NE,NW,SE,SW}\}^{n-2}\times\{\mathsf{N,S}\}$. To prove Theorem \[thm:blockclique\], we first show that each ell block in a block clique embedding has a distinct shape. Define the [*width*]{} of an ell as the number of vertices in its horizontal wire, and its [*height*]{} as the number of vertices in its vertical wire. All ells in an ell bundle have the same width and height, so define the width and height of an ell bundle or an ell block as the width and height of its constituent ells. \[lem:height\] Let ${\mathcal{X}}=\{(X_1,c_1),\dots,(X_n,c_n)\}$ be a block clique embedding in ${\mathcal{C}}_{n,n,L}$. Then the ell blocks of ${\mathcal{X}}$ have distinct heights. We will establish that ${\mathcal{X}}$ has a unique ell block of height $i$ for each $1 \leq i \leq n$. Clearly two ells of height 1 or two ells of height $n$ cannot intersect properly (i.e. in exactly one unit which is horizontal for one ell block and vertical for the other). Assume for contradiction that ${\mathcal{X}}$ contains two ells $(X,c)$ and $(X',c')$ of height $1 < i < n$. Their horizontal and vertical components must occupy different rows and columns respectively. Up to symmetry, there are two cases to consider where ells of the same height intersect properly, shown in Figure \[fig:lemmacases\]. In both cases, we name the upper ell block $(X,c)$. -------------------------------------------------------------------------------------------------- -- -------------------------------------------------------------------------------------------------- ![Two cases of intersecting ell blocks with the same shape.\[fig:lemmacases\]](lem1a.pdf "fig:") ![Two cases of intersecting ell blocks with the same shape.\[fig:lemmacases\]](lem1b.pdf "fig:") [Case 1]{} [Case 2]{} -------------------------------------------------------------------------------------------------- -- -------------------------------------------------------------------------------------------------- Since there are $n$ ell blocks and $n$ cells per block, every non-corner cell of every ell block must intersect another ell block. In Case 1, we have shaded cells for which the horizontal or vertical coordinate is unique among the two ells. Consider an ell block $(Y,d)$ which properly intersects $(X,c)$ in a shaded cell. To properly intersect $(X',c')$, $(Y,c)$ cannot intersect $(X,c)$ again, so must intersect $(X',c')$ in a shaded cell also. Therefore, $Y$ must size greater than $n$, a contradiction. In Case 2, we have shaded the cell which lies directly north of a corner. In this case, it is clear that no ell block can properly intersect $(X,c)$ at the gray cell and also intersect $(X',c')$ properly. Without loss of generality, we will assign labels to each ell block $(X_i,c_i) \in {\mathcal{X}}$ so that $X_i$ has height $i$ and width $n-i+1$. We first give a mapping from the set of $4^{n-1}$ words to the set of block clique embeddings and then provide the inverse mapping. Let $W=s_1,\dots,s_n$ be a word for which $s_1\in\{\mathsf{E,W}\}$, each of $s_2,\dots,s_{n-1}$ is in $\{\mathsf{NE,NW,SE,SW}\}$, and $s_n\in \{\mathsf{N,S}\}$. We construct a block clique embedding ${\mathcal{X}}=\{(X_1,c_1),\dots,(X_n,c_n)\}$ from $W$ in such a way that each ell block $(X_i,c_i)$ corresponds to the subword $s_i$. We denote the location of a corner $c_i$ by its Cartesian coordinates $(x_i,y_i) \in \{1,\dots,n\}^2$ and coordinates increase to the east (for $x$) and north (for $y$). If $s_1=\mathsf{W}$ we place $c_1$ so that it is west of the remaining corners, i.e. $x_1=1$. If $s_1=\mathsf{E}$, we place $c_1$ so that it is east of the remaining corners, i.e. $x_1=n$. We select $y_1$ so that there are $y_1-1$ $\mathsf{S}$s following the subword $s_1$ in the word $W$. In Figure \[fig:thm1first\], we show two initial ell block placements for words where $*$ denotes a (possibly empty) subword containing zero $\mathsf{S}$s. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![Two examples of the first ell block’s orientation and location. $*$ denotes a (possibly empty) subword containing zero $\mathsf{S}$s. \[fig:thm1first\]](thm1a.pdf "fig:") ![Two examples of the first ell block’s orientation and location. $*$ denotes a (possibly empty) subword containing zero $\mathsf{S}$s. \[fig:thm1first\]](thm1b.pdf "fig:") [$\mathsf{E*S*S}\,*$]{} [$\mathsf{W}\,*$]{} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ The placement of the first ell block defines a [*working rectangle*]{} $R_1$ of cells intersecting horizontal wires but not vertical wires in the partially-constructed block clique embedding. Every time we place an ell block, the working rectangle becomes one unit taller and narrower, and the vertical component of the recently-placed ell covers either the left-most or right-most column of the previous working rectangle. Suppose the working rectangle after the selection of $i$ ell blocks is $R_i=[x, x'] \times [y, y']$ in Cartesian coordinates, where $x\leq x'$ and $y \leq y'$. We choose corner $c_i$ based on subword $s_i$ as follows, noting that $x'=x+n-i-1$ and $y'=y+i-1$: - If $s_i=\mathsf{NE}$ or $\mathsf N$, $c_i = (x',y'+1)$. - If $s_i=\mathsf{NW}$ or $\mathsf N$, $c_i = (x,y'+1)$. - If $s_i=\mathsf{SE}$ or $\mathsf S$, $c_i = (x',y-1)$. - If $s_i=\mathsf{SW}$ or $\mathsf S$, $c_i = (x,y-1)$. Letting $c_i = (x_i,y_i)$, we then choose $ X_i = (\{x_i\}\times [y,y']) \cup ( [x,x']\times \{y_i\}) $ and update the working rectangle $ R_{i+1} = ([x, x']\backslash \{x_i\}) \times ([y, y'] \cup \{y_i\}). $ Observe that every time that we find a $\mathsf{W}$, $x$ increases by 1 and every time we find a $\mathsf{E}$, $x'$ decreases by 1; every time we find an $\mathsf{N}$, $y$ increases by 1 and every time an $\mathsf{S}$, $y'$ decrease by 1. Figure \[fig:thm1second\] shows an example of how the construction might proceed. -------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------- -- -- -- -- ![An example of the iterative construction of a block clique embedding.\[fig:thm1second\]](thm1c.pdf "fig:") ![An example of the iterative construction of a block clique embedding.\[fig:thm1second\]](thm1d.pdf "fig:") ![An example of the iterative construction of a block clique embedding.\[fig:thm1second\]](thm1e.pdf "fig:") ![An example of the iterative construction of a block clique embedding.\[fig:thm1second\]](thm1f.pdf "fig:") ![An example of the iterative construction of a block clique embedding.\[fig:thm1second\]](thm1g.pdf "fig:") [$\mathsf{E}\cdots$]{} [$\mathsf{ENE}\cdots$]{} [$\mathsf{ENENW}\cdots$]{} [$\mathsf{ENENWSE}\cdots$]{} [$\mathsf{ENENWSES}$]{} -------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------- -- -- -- -- Note that $R_i$ never shares a column with a corner in $\{c_j \mid j\leq i\}$, and the set of rows intersected by $R_i$ is the same set of rows intersected by $\{c_j \mid j\leq i\}$. It therefore follows from the construction that $\{c_1,\dots,c_n\}$ is a permutation. Furthermore, by the construction, for $i<j$ there is always a unit cell $X_i\cap X_j$ in the working rectangle $R_i$ – specifically, it is $(x_j,y_i)$. So ${\mathcal{X}}=\{(X_1,c_1),\dots,(X_n,c_n)\}$ is indeed a block clique embedding. Now, we invert our construction to show that it is in fact bijective. Let ${\mathcal{X}}=\{(X_1,c_1),\dots,(X_n,c_n)\}$ be a block clique embedding in which $X_i$ has height $i$; we will reconstruct our word $W$. We call $(x_n,y_1) = X_1 \cap X_n$ the center of ${\mathcal{X}}$, and say that a point $(x,y)$ lies to the east or west of the center if $x > x_n$ or $x < x_n$ respectively, and likewise north and south. - Let $s_1 = \mathsf{E}$ if the corner $c_1$ lies to the east of the center, and $\mathsf{W}$ otherwise. - For $1 < i < n$, let $s_i = a_ib_i$, where $a_i = \mathsf{N}$ if $c_i$ lies to the north of the center, and $\mathsf{S}$ otherwise; and $b_i = \mathsf{E}$ if $c_i$ lies to the east of the center, and $\mathsf{W}$ otherwise. - Let $s_n = \mathsf{N}$ if the corner $c_n$ lies to the north of the center, and $\mathsf{S}$ otherwise. Therefore, $W = s_1s_2\ldots s_n$ is in $\{\mathsf{E,W}\}\times \{\mathsf{NE,NW,SE,SW}\}^{n-2}\times\{\mathsf{N,S}\}$. To see that this construction is in fact the inverse, we observe that each ell block $(X_i,c_i)$ is associated to the direction $s_i$ in which the corner $c_i$ lies from the center, so we obtain the same word we began with. Finding optimal native clique embeddings in induced subgraphs\[sec:3\] ====================================================================== In this section we describe an algorithm to find a largest native clique embedding ${\mathcal{B}}=\{B_i,\dots,B_n\}$ in an induced subgraph $G$ of ${\mathcal{C}}_{M,N,L}$ with $n\leq M,N$. Our algorithm necessarily takes as input a parameter $n$, which determines the size of the chains, i.e. $n+1$. Our algorithm is informed by the proof of Theorem \[thm:blockclique\]. We use dynamic programming to maximize the block clique embeddings with each working rectangle $R$, and do so in an orderly way which results in a polynomial-time algorithm. As a preprocessing step, for each ell block $(X,c)$ we compute a maximum bundle and store it as ${\mathit{maxBundle}}(X,c)$ (it is straightforward to do this in $O(nL)$ time per ell block). We then use this information to construct block clique embeddings as in the proof of Theorem \[thm:blockclique\]: adding one ell block at a time, with working rectangles of increasing height. Let $(X,c)$ be an ell block of height $i$. If $1 \leq i \leq n-1$, there is a unique working rectangle ${\mathit{R_{from}}}(X,c)$ that can be in effect immediately after $(X,c)$ is placed. If $2\leq i \leq n$, there is a unique working rectangle ${\mathit{R_{to}}}(X,c)$ that can be in effect immediately before $(X,c)$ is placed. Note that for each working rectangle $R$, the sets $${\mathit{X_{from}}}(R) := \{ (X,c) \mid R = {\mathit{R_{to}}}(X,c) \}$$ and $${\mathit{X_{to}}}(R):= \{ (X,c) \mid R = {\mathit{R_{from}}}(X,c) \}$$ each have size at most four (see Figure \[fig:extensions\]). ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- $R$ $R_1$ $R_2$ $R_3$ $R_4$ ![The four rectangles $R_i = {\mathit{R_{to}}}(X_i,c_i)$ with $(X_i,c_i) \in {\mathit{X_{to}}}(R)$. Gray ell blocks represent precomputed ${\mathit{maxPartialEmbedding}}(R_i)$ as in Lemma \[lem:mpemax\]. []{data-label="fig:extensions"}](alg1a.pdf "fig:") ![The four rectangles $R_i = {\mathit{R_{to}}}(X_i,c_i)$ with $(X_i,c_i) \in {\mathit{X_{to}}}(R)$. Gray ell blocks represent precomputed ${\mathit{maxPartialEmbedding}}(R_i)$ as in Lemma \[lem:mpemax\]. []{data-label="fig:extensions"}](alg1b.pdf "fig:") ![The four rectangles $R_i = {\mathit{R_{to}}}(X_i,c_i)$ with $(X_i,c_i) \in {\mathit{X_{to}}}(R)$. Gray ell blocks represent precomputed ${\mathit{maxPartialEmbedding}}(R_i)$ as in Lemma \[lem:mpemax\]. []{data-label="fig:extensions"}](alg1c.pdf "fig:") ![The four rectangles $R_i = {\mathit{R_{to}}}(X_i,c_i)$ with $(X_i,c_i) \in {\mathit{X_{to}}}(R)$. Gray ell blocks represent precomputed ${\mathit{maxPartialEmbedding}}(R_i)$ as in Lemma \[lem:mpemax\]. []{data-label="fig:extensions"}](alg1d.pdf "fig:") ![The four rectangles $R_i = {\mathit{R_{to}}}(X_i,c_i)$ with $(X_i,c_i) \in {\mathit{X_{to}}}(R)$. Gray ell blocks represent precomputed ${\mathit{maxPartialEmbedding}}(R_i)$ as in Lemma \[lem:mpemax\]. []{data-label="fig:extensions"}](alg1e.pdf "fig:") ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- For any a set of ell bundles ${\mathcal{B}}= \{B_1, \cdots, B_n\}$ where each $B_i$ is contained in the ell block $(X_i, c_i)$ and ${\mathcal{X}}= {\left\{{(X_1, c_1), \cdots, (X_n, c_n)}\right\}}$, $$\|{\mathcal{X}}\| := {\left|{\bigcup_{i=1}^n {\mathit{maxBundle}}(X_i,c_i)}\right|} \geq {\left|{\bigcup_{i=1}^n B_i}\right|}$$ provided that the ell blocks are all distinct. This enables us to construct maximum native clique embeddings while only considering the ell blocks involved. For each working rectangle $R$ of height $i$ (and width $n-i$), our algorithm computes and stores a *partial block clique embedding* with that working rectangle: a set of ell blocks ${\mathcal{X}}_i = \{(X_1,c_1), \cdots, (X_i,c_i)\}$ such that - $R = {\mathit{R_{from}}}(X_i,c_i)$, - $X_j$ has height $j$ for all $1 \leq j \leq i$ - ${\mathit{R_{to}}}(X_j,c_j) = {\mathit{R_{from}}}(X_{j-1},c_{j-1})$ for all $2 \leq j \leq i$. In particular, we compute *maximum* partial block clique embeddings with working rectangle $R$; those which maximize $\|{\mathcal{X}}_i\|$ over all partial block clique embeddings with a given working rectangle. We denote a particular maximum partial block clique embedding for a given working rectangle $R$ by ${\mathit{maxPartialEmbedding}}(R)$, though many may exist. Our algorithm will operate by extending maximum partial block clique embeddings by ells, so we define ${\mathit{maxExtension}}(X,c) = {\mathit{maxPartialEmbedding}}({\mathit{R_{to}}}(X,c)) \cup \{(X,c)\}$ when $X$ has height $h > 1$, and ${\mathit{maxExtension}}(X,c) = {\left\{{(X,c)}\right\}}$ otherwise. The following lemma encodes the key step of the algorithm: to find a maximum partial block clique embedding with working rectangle $R$, we need only consider partial block clique embeddings that include an ell block $(X,c)$ such that $R={\mathit{R_{from}}}(X,c)$. \[lem:mpemax\] Given a rectangle $R$ of height $i \geq 1$ and width $n-i$, $$\|{\mathit{maxPartialEmbedding}}(R)\| = \max_{(X,c) \in {\mathit{X_{to}}}(R)} {\left|{{\mathit{maxExtension}}(X,c)}\right|}.$$ We proceed by induction on $i$. When $i=1$, our claim follows immediately from definitions. Assume that $i>1$ and that our claim holds for all working rectangles of height $i-1$. We consider the complete set of partial block clique embeddings with working rectangle $R$, $ S = \{{\mathcal{X}}_i \mid {\mathit{R_{from}}}(X_i,c_i) = R\} = \{{\mathcal{X}}_i \mid (X_i,c_i) \in {\mathit{X_{to}}}(R)\} $ where $(X_i,c_i)$ is the ell block with height $i$ in ${\mathcal{X}}_i$. By definition, $ \|{\mathit{maxPartialEmbedding}}(R)\| = \max_{{\mathcal{X}}_i \in S} \|{\mathcal{X}}_i\|. $ For contradiction, pick some maximum ${\mathcal{X}}_i \in S$ and suppose that $$\|{\mathcal{X}}_i\| > \|{\mathit{maxPartialEmbedding}}({\mathit{R_{to}}}(X,c)) \cup \{(X,c)\}\|$$ for all $(X,c) \in {\mathit{X_{to}}}(R)$. In particular, letting ${\mathcal{X}}_{i-1} = {\mathcal{X}}_i \setminus {\left\{{(X_i,c_i)}\right\}}$, $$\begin{aligned} \|{\mathcal{X}}_i\|&=& {\left\|{{\mathcal{X}}_{i-1}}\right\|} + {\left\|{{\left\{{(X_i,c_i)}\right\}}}\right\|}\\ &>& \|{\mathit{maxPartialEmbedding}}({\mathit{R_{to}}}(X_i,c_i))\| + |{\mathit{maxBundle}}(X_i,c_i)|, \end{aligned}$$ a contradiction since ${\mathcal{X}}_{i-1}$ has working rectangle ${\mathit{R_{to}}}(X_i,c_i)$. We now present the algorithm. The idea follows from Lemma \[lem:mpemax\], to compute ${\mathit{maxPartialEmbedding}}(R)$ for rectangles of increasing height. To do so we treat the set of possible working rectangles as a digraph, where $R\rightarrow R'$ if and only if there is an ell block $(X,c)$ for which $R\in {\mathit{R_{from}}}(X,c)$ and $R'\in{\mathit{R_{to}}}(X,c)$. This means that if $R\rightarrow R'$, the height of $R'$ is one more than the height of $R$. The number of edges in this digraph is equal to the number of ell blocks. To compute ${\mathit{maxPartialEmbedding}}(R')$, assuming that we have computed ${\mathit{maxPartialEmbedding}}(R)$ for all rectangles of lesser height, we simply set ${\mathit{maxPartialEmbedding}}(R')$ to be a maximum partial block clique embedding in the set $$\left\{ {\mathit{maxExtension}}(X,c) \mid (X,c)\in {\mathit{X_{to}}}(R') \right\}.$$ Once we have computed ${\mathit{maxPartialEmbedding}}(R)$ for all rectangles $R$ of height $n-1$ and width $1$, we pick a maximum-sized clique embedding from the set $$\left\{ {\mathit{maxExtension}}(X,c) \mid (X,c)\textrm{ has height }n-1\textrm{ and width }1 \right\}.$$ Pseudocode is given in Algorithm \[alg:cliqueembed\]. ${\mathit{maxPartialEmbedding}}(R)\gets \emptyset$ ${\mathcal{B}}\gets{\mathit{maxPartialEmbedding}}({\mathit{R_{to}}}(X,c))\cup\{(X,c)\}$ ${\mathit{maxPartialEmbedding}}({\mathit{R_{from}}}(X,c)) \gets {\mathcal{B}}$ ${\mathcal{B}}_{\mathit{max}} \gets \emptyset$ \[line:finalpass\] ${\mathcal{B}}\gets {\mathit{maxPartialEmbedding}}({\mathit{R_{to}}}(X,c)) \cup{\left\{{(X,c)}\right\}}$ \[line:lastextension\] ${\mathcal{B}}_{\mathit{max}} \gets {\mathcal{B}}$ \[line:return\] The ${\mathit{NativeCliqueEmbed}}$ algorithm finds a maximum-sized native clique embedding with chain length $n+1$ in polynomial time. We first prove correctness, then the bound on running time. Note that the loop beginning on line \[line:forblock\] of Algorithm \[alg:cliqueembed\] iterates over all ell blocks of height $i$ and width $n-i$. Given a rectangle $R$, there are up to four ell blocks $(X,c)$ for which $R = {\mathit{X_{to}}}(X,c)$. If we ignore all ell blocks except those incident to a particular rectangle $R$, this loop implements Lemma \[lem:mpemax\] directly. Therefore, when we reach line \[line:finalpass\], we have computed ${\mathit{maxPartialEmbedding}}(R)$ for all $R$ with height $n-1$ and width 1. Let ${\mathcal{B}}= {\left\{{B_1,\cdots,B_n}\right\}}$ be a maxmimum native clique embedding where $B_i$ is an ell bundle in the ell block $(X_i,c_i)$ with height $i$. By Theorem \[thm:blockclique\], ${\mathcal{X}}_{n-1} = {\left\{{(X_1,c_1),\cdots,(X_{n-1},c_{n-1})}\right\}}$ is a partial block clique embedding with working rectangle ${\mathit{R_{to}}}(X_n,c_n)$. By Lemma \[lem:mpemax\], $${\left\|{{\mathit{maxPartialEmbedding}}({\mathit{R_{to}}}(X_n,c_n))}\right\|} \geq {\left\|{{\mathcal{X}}_{n-1}}\right\|},$$ so we see a clique embedding of size at least ${\left\|{{\mathcal{X}}_{n-1}}\right\|} + |{\mathit{maxBundle}}(X_n,c_n)|$ when line \[line:lastextension\] is reached with $(X,c) = (X_n,c_n)$. Therefore, when line \[line:return\] is reached, ${\left\|{{\mathcal{B}}_{max}}\right\|} \geq {\left\|{{\mathcal{B}}}\right\|}$ and a native clique embedding of maximum size has been found. We can compute ${\mathit{maxBundle}}(X,c,G)$ in $O(nL)$ time, and there are polynomially many ell blocks and rectangles: There are at most $MN$ possible locations of a rectangle’s lower-left corner, and $n$ possible shapes, thus at most $nMN$ rectangles. Likewise there are at most $MN$ possible locations for an ell block’s corner, and at most $4(n-1)$ ell blocks containing $n$ unit cells with a given corner, thus at most $4nMN$ ell blocks. It follows that each line in Algorithm \[alg:cliqueembed\] is evaluated $O(nMN)$ times, and the preprocessing step of computing ${\mathit{maxBundle}}(X,c,G)$ for each ell block $(X,c)$ naively takes $O(n^2MNL)$ time. Consequently, with the rough bound that each line in Algorithm \[alg:cliqueembed\] takes $O(nL)$ time for a single evaluation, we can bound the total running time of our algorithm by $O(n^2MNL)$. #### Remark. The $O(n^2MNL)$-time bound on Algorithm \[alg:cliqueembed\] is quadratic in the number of vertices in ${\mathcal{C}}_{N,M,L}$, i.e. $O(n^2MNL)\subseteq O((MNL)^2)$. Assume $M \leq N$ and $L$ is constant. With a little more care, we can modify Algorithm \[alg:cliqueembed\] to achieve a bound of $O(N^3)$ instead of $O(N^4)$. Doing this involves (a) precomputing all maximum horizontal and vertical line bundles in time $O(LNM^2 + LMN^2)$ with a dynamic programming approach, which allows us to compute $|{\mathit{maxBundle}}(X,c,G)|$ in $O(1)$ time, and (b) exploiting the fact that throughout the algorithm, we need only keep track of the size of maximum partial embeddings and the route used to reach it (replacing ${\mathit{maxPartialEmbedding}}(R)$ with a mapping $R \mapsto (X,c) \in {\mathit{X_{to}}}(R)$), rather than the embeddings themselves. NativeCliqueEmbed gives a maximum native clique embedding for a fixed chain length. To find a maximum native clique embedding over all chain lengths for a given graph, we simply repeat the process for each choice of $n\in \{2,\dots,\min\{M,N\}\}$. For $M \leq N$ and constant $L$, this gives an overall running time on a subgraph of ${\mathcal{C}}_{M,N,L}$ of $O(N^5)$ with the naive implementation and $O(N^4)$ with the refinement discussed above. Induced and general subgraphs ----------------------------- We now discuss the motivation of using induced subgraphs and how to approach more general subgraphs. Recall that we have restricted our attention to induced subgraphs rather than more general subgraphs because failed couplers adjoining working qubits are relatively rare. In an induced subgraph, we will still focus on horizontal and vertical wires, and it is easy to find a maximum set of such wires in a line of unit cells. A [*maximum ell bundle*]{} is a maximum-sized set of vertex-disjoint ells that occupy an ell block, and the [*size*]{} of an ell bundle is the number of ells contained therein. It is simple to find a maximum ell bundle in a given ell block: finding maximum sets $S_H$ and $S_V$ of horizontal and vertical wires spanning a line of cells is trivial, and these lines may be paired off arbitrarily since the corner is a complete bipartite graph. Here there may be difficulty in generalizing even this relatively simple optimization problem in the face of arbitrary edge deletion. Finding a maximum ell bundle in an arbitrary Chimera subgraph is polynomially equivalent to finding a maximum clique in $D_2(\mathcal{L}(B))$ where $B$ is a bipartite graph, $\mathcal{L}(B)$ is the line graph of $B$, and $D_2(G)$ is the distance-2 graph with vertex set $V(G)$ and edge set $\{uv : d_G(u,v) = 2\}$. We are unsure of the complexity of this problem, but expect that it is NP-complete. Note that we can easily relax the requirement of an induced subgraph when couplers between unit cells are defective – in any case, one just computes the number of wires in a line of cells where all qubits and couplers are contained in the subgraph. In short, failed inter-cell couplers don’t increase the difficulty of the problem. If we restrict our attention to Chimera graphs ${\mathcal{C}}_{N,N,L}$ where $L = O(\log N)$, or assume $L$ to be a constant, then this difficulty at the corners can be swept under the rug. At present, this appears to be a reasonable consideration, as it is much easier from a manufacturing and design standpoint to increase $N$ than it is to increase $L$. However, there is a further challenge introduced by intra-cell couplers. Our algorithm intrinsically relies upon the assumption that couplers exist between any two ells whose unit cells intersect. Missing intra-cell couplers void that assumption. The easiest remedy for this obstruction is to consider vertex covers of the failed intra-cell couplers: Given a graph $G$ that is the subgraph of $ {\mathcal{C}}_{M,N,L}$ induced by vertex set $W$, i.e. $G = {\mathcal{C}}_{M,N,L}[W]$, let $U_1, \dots, U_s$ be the list of minimal vertex covers of the failed intra-cell couplers. Then, for each $1 \leq i \leq s$, compute a largest clique minor considering $G[W]$ for the purpose of constructing ells, and $G[W \setminus U_i]$ for the purpose of growing cliques. This approach gives a fixed-parameter tractable algorithm for finding the maximum native clique embedding in an arbitrary subgraph of Chimera in terms of the number of missing edges with both endpoints intact – for the D-Wave Two processors installed at NASA Ames [@Venturelli2014] and ISI [@Albash2015], this parameter was zero. Comparison with previous work\[sec:previouswork\] ================================================= Klymko, Sullivan and Humble gave a greedy embedding algorithm that quickly produces similar embeddings to those in this paper [@Klymko2014]. Their algorithm produces plus-shaped chains (having nearly twice as many qubits), and restricts its search to embeddings where the vertical and horizontal components of pluses are only allowed to meet at the diagonal of a fixed square. While our algorithm is slower, taking $O(N^4)$ time compared to their $O(N^3)$ for ${\mathcal{C}}_{N,N,4}$, it is exhaustive, empirically embeds larger cliques, and produces chains of roughly half the size. ![(Color online) Comparison of clique yields for ell-shaped (red) and plus-shaped (blue) chains. The solid lines denote the median, and the shaded regions encompass the middle two quartiles.[]{data-label="fig:ellvplus"}](yield.pdf) Given a family of clique minors, the [*clique yield*]{} of a graph $G$ over that family is the size of the largest clique minor in the family, in $G$. In Figure \[fig:ellvplus\], we compare ell- and plus-shaped chains as the grid size grows for several fixed percentages of operational qubits. In both families, a similar asymptotic behavior becomes clear: for a fixed qubit failure rate, increasing the grid size gives diminishing returns in terms of clique yield. However, the difference between these curves is significant, with ell-shaped chains producing much larger clique minors. [^1]: This research was partially supported by the Mitacs Accelerate program. [^2]: Taking the triangle embedding and making an image of all the unused qubits gives a $K_{LM +1}$ minor.
--- abstract: 'We discuss two-parameter deformations of an universal enveloping algebra $U(g[u])$ of a polynomial loop algebra $g[u]$, where $g$ is a finite-dimensional complex simple Lie algebra (or superalgebra). These deformations are Hopf algebras. One deformation called Drinfeldian is a quantization of $U(g[u])$ in the direction of a classical r-matrix which is a sum of the simplest rational and trigonometric r-matrices. Another deformation (discussed only for the case $g=sl_{2}$) is a twisting of the usual Yangian $Y_{\eta}(sl_{2})$.' --- 5.8in -7pt \[section\] \[section\] \[section\] \[section\] \[section\] [**TWO-PARAMETER DEFORMATIONS OF LOOP ALGEBRAS AND SUPERALGEBRAS**]{} [Valeriy N. Tolstoy]{} Introduction ============ As it is well known, an universal enveloping algebra $U(g[u])$ of a polynomial loop (current) Lie algebra $g[u]$ , where $g$ is a finite-dimensional complex simple Lie algebra, admits two type deformations: a trigonometric deformation $U_q(g[u])$ and a rational deformation or Yangian $Y_{\eta}(g)$ [@D]. (In the case $g=sl_n$ there also exists an elliptic quantum deformation of $U(sl_n[u])$, which is not discussed in this report). The algebras $U_q(g[u])$, and $Y_{\eta}(g)$ are quantizations of $U(g[u])$ in the direction of the simplest trigonometric and rational solutions of the classical Yang-Baxter equation over $g$, respectively. These deformations are one-parameter. It turns out that $U(g[u])$ also admits two-parameter deformations. Here we discuss two type of such deformations which are Hopf algebras. A Hopf algebra of the first type called the rational-trigonometric quantum algebra or the Drinfeldian $D_{q\eta}(g)$ [@T] is a quantization of $U(g[u])$ in the direction of a classical r-matrix which is a sum of the simplest rational and trigonometric r-matrices. The Drinfeldian $D_{q\eta}(g)$ contains $U_{q}(g)$ as a Hopf subalgebra, and $U_{q}(g[u])$ and $Y_{\eta}(g)$ are its limit quantum algebras when the deformation parameters of $D_{q\eta}(g)$ $\eta$ goes to $0$ and $q$ goes to $1$, respectively. These results are easy generalized to a supercase, i.e. when $g$ is a finite-dimensional contragredient simple superalgebra. A Hopf algebra of the second type discussed only for the case $g=sl_{2}$ is obtained by twisting of the usual Yangian $Y_{\eta}(sl_{2})$. The twisted Yangian $Y_{\eta\zeta}(sl_{2})$ is a quantization $U(sl_{2}[u])$ in the direction of a classical r-matrix $r(u,v)=\eta{\bf c}_{2}/(u-v)+\zeta h_{\alpha}\wedge e_{-\alpha}$, where ${\bf c}_{2}$ is the $sl_{2}$ Casimir element. Detailed describtion of $Y_{\eta\zeta}(sl_{2})$ is given in [@KST]. Drinfeldian $D_{q\eta}(g)$ ========================== Let $g$ be a finite-dimensional complex simple Lie algebra of a rank $r$ with a standard Cartan matrix $A=(a_{ij})_{i,j=1}^r$, with a system of simple roots $\Pi:= \{\alpha_1,\ldots, a_r\}$, and with a maximal positive root $\theta$. Let $U_{q}(g)$ be a standard q-deformation of the universal enveloping algebra $U(g)$ with Chevalley generators $k_{\alpha_i}^{\pm 1}$, $e_{\pm\alpha_i}$ $(i=1,2,\ldots, r)$ and with the defining relations $$[k_{\alpha_i},k_{\alpha_j}]=0~,\qquad k_{\alpha_i}e_{\pm\alpha_j}k^{-1}_{\alpha_i}= q^{\pm(\alpha_i,\alpha_j)}e_{\pm\alpha_j}~, \label{D1}$$ $$[e_{\alpha_i},e_{-\alpha_i}]= \frac{k_{\alpha_i}-k_{\alpha_i}^{-1}}{q-q^{-1}}~,\qquad ({\rm ad}_{q}e_{\pm\alpha_{i}})^{1-a_{ij}} e_{\pm\alpha_{j}}=0 \quad\;{\rm for}\,\, i\neq j~, \label{D2}$$ The Drinfeldian $D_{q\eta}(g)$ is generated as an associative algebra over $C\!\!\!\!I\,[[\eta]]$ by the algebra $U_{q}(g)$ and the elements $\xi_{\delta-\theta}$, $k_{\delta}^{\pm 1}$ with the relations: $$[k_{\delta}^{\pm 1},{\rm everything}]=0~\qquad k_{\alpha_i}\xi_{\delta-\theta}k^{-1}_{\alpha_i}= q^{-(\alpha_i,\theta)}\xi_{\delta-\theta}~, \label{D3}$$ $$[e_{-\alpha_i},\xi_{\delta-\theta}]= a\,[e_{-\alpha_i},\tilde{e}_{-\theta}],\qquad ({\rm ad}_{q}e_{\alpha_i})^{n_{i0}}\xi_{\delta-\theta}= a\,({\rm ad}_{q}e_{\alpha_i})^{n_{i0}}\tilde{e}_{-\theta} \label{D4}$$ for $n_{i0}=1+2(\alpha_i,\theta)/(\alpha_i,\alpha_i)$ , and $$\begin{aligned} [[e_{\alpha_i},\xi_{\delta-\theta}]_{q},\xi_{\delta-\theta}]_{q} &\!\!\!\!\!=&\!\!\!\!\! -a^{2}[[e_{\alpha_i},\tilde{e}_{-\theta}]_{q},\tilde{e}_{-\theta}]_{q} \nonumber \\ &&\!\!\!\! +a\,[[e_{\alpha_i},\tilde{e}_{-\theta}]_{q},\xi_{\delta-\theta}]_{q}+ a\,[[e_{\alpha_i},\xi_{\delta-\theta}]_{q},\tilde{e}_{-\theta}]_{q}\end{aligned}$$ \[D5\] for $g\ne sl_2$ and $(\alpha_i,\theta)\ne 0$, $$\begin{aligned} &&\!\!\!\!\!\!\!\!\!\!\!\!\! [[[e_{\alpha},\xi_{\delta-\alpha}]_{q}, \xi_{\delta-\alpha}]_{q},\xi_{\delta-\alpha}]_{q}=a^{3} [[[e_{\alpha},\tilde{e}_{-\alpha}]_{q},\tilde{e}_{-\alpha}]_{q}, \tilde{e}_{-\alpha}]_{q}\qquad \label{D6} \nonumber \\ &&-a^{2}[[[e_{\alpha},\tilde{e}_{-\alpha}]_{q},\tilde{e}_{-\alpha}]_{q}, \xi_{\delta-\alpha}]_{q}-a^{2}[[[e_{\alpha},\tilde{e}_{-\alpha}]_{q}, \xi_{\delta-\alpha}]_{q},\tilde{e}_{-\alpha}]_{q} \nonumber\\ &&-a^{2}[[[e_{\alpha},\xi_{\delta-\alpha}]_{q},\tilde{e}_{-\alpha}]_{q}, \tilde{e}_{-\alpha}]_{q}+a\,[[[e_{\alpha},\tilde{e}_{-\alpha}]_{q}, \xi_{\delta-\alpha}]_{q},\xi_{\delta-\alpha}]_{q} \nonumber \\ &&+a\,[[[e_{\alpha},\xi_{\delta-\alpha}]_{q}, \tilde{e}_{-\alpha}]_{q},\xi_{\delta-\alpha}]_{q}+a\,[[[e_{\alpha}, \xi_{\delta-\alpha}]_{q},\xi_{\delta-\alpha}]_{q}, \tilde{e}_{-\alpha}]_{q}\end{aligned}$$ for $g=sl_2$. The Hopf structure of $D_{q\eta}(g)$ is defined by the formulas $\Delta_{q\eta}(x)=\Delta_{q}(x)$, $S_{q\eta}(x)=S_{q}(x)$ ($x\in U_{q}(g)$) and it is the same for the elements $k_{\delta}^{\pm}$ and $k_{\alpha_i}$. The comultiplication and the antipode of $\xi_{\delta-\alpha}$ are given by $$\begin{aligned} \Delta_{q\eta}(\xi_{\delta-\theta})\!\!\!&=&\!\!\! \xi_{\delta-\theta}\otimes 1+k_{\delta-\theta}^{-1} \otimes \xi_{\delta-\theta} \nonumber \\ &&\!\!\!+ a\left(\Delta_{q}(\tilde{e}_{-\theta}) -\tilde{e}_{-\theta}\otimes 1- k_{\delta-\theta}^{-1}\otimes\tilde{e}_{-\theta}\right), \label{D7}\end{aligned}$$ $$S_{q\eta}(\xi_{\delta-\theta})= -k_{\delta-\theta}\xi_{\delta-\theta} +a\left(S_{q}(\tilde{e}_{-\theta})+ k_{\delta-\theta}\tilde{e}_{-\theta}\right)~, \label{D8}$$ where $a:=\eta/(q-q^{-1})$, $({\rm ad}_{q}e_{\beta})e_{\gamma}=[e_{\beta},e_{\gamma}]_{q}$, and the vector $\tilde{e}_{-\theta}$ is any $U_{q}(g)$ element of the weight $-\theta$, such that $g\ni\lim_{q\to1}\tilde{e}_{-\theta}\neq 0$. The right-hand sides of the relations (\[D4\])-(\[D8\]) are nonsingular at $q=1$. \(i) The Drinfeldian $D_{q\eta}(g)$ is a two-parameter quantization of $U(\overline{g[u]})$, where $\overline{g[u]}$ is a central extension of $g[u]$, in the direction of a classical r-matrix which is a sum of the simplest rational and trigonometric r-matrices.\ (ii) The Hopf algebra $D_{q=1,\eta}(g)$ is isomorphic to the Yangian $Y_{\eta}'(g)$ (with a central element). Moreover, $D_{q\eta=0}(g)=U_q(\overline{g[u]})$. \[DT1\] In the supercase, i.e. when $g$ is a simple finite-dimensional contragredient Lie superalgebra all the commutators and the q-commutators are replaced by the supercommutators and the q-supercommutators. Moreover we have to add some additional Serre relations if they exist. Twisted Yangian $Y_{\eta\zeta}(sl_2)$ ===================================== In the case $sl_{2}$ from (\[D3\])-(\[D8\]) at $q=1$ we can obtain that the Yangian $Y_{\eta}(sl_2)$ is generated by the $sl_2$ elements $h_{\alpha},\;e_{\pm\alpha}$ and the element $\xi_{\delta-\alpha}$ with the relations: $$[h_{\alpha},\xi_{\delta-\alpha}]=-2\xi_{\delta-\alpha}~,\quad [e_{-\alpha},\xi_{\delta-\alpha}]=\eta e_{-\alpha}^2~, \label{TY2}$$ $$[e_{\alpha},[e_{\alpha},[e_{\alpha},\xi_{\delta-\alpha}]]]= 6\eta e_{\alpha}^{2}~,\quad [[[e_{\alpha},\xi_{\delta-\alpha}],\xi_{\delta-\alpha}],\xi_{\delta-\alpha}]= 6\eta \xi_{\delta-\alpha}^{2}~. \label{TY3}$$ $$\Delta(\xi_{\delta-\alpha})=\xi_{\delta-\alpha}\otimes 1 + 1\otimes \xi_{\delta-\alpha}+ \eta e_{-\alpha}\otimes h_{\alpha}~,\;\; S(\xi_{\delta-\alpha})=-\xi_{\delta-\alpha}+\eta e_{-\alpha}h_{\alpha}~, \label{TY5}$$ where we put $(\alpha,\alpha)=2$. Using the twisting element $F=\sum_{k\geq 0}(\zeta^{k}/k!) \big(\prod_{i=0}^{k-1}(h_{\alpha}+2i)\big)\otimes e_{-\alpha}^k~$ one can calculate the new coproduct $\Delta^{(F)}(x):=F\Delta(x)F^{-1}$ and antipode $S^{(F)}:=uS(x)u^{-1}$ ($x\in Y_{\eta}(sl_2)$) where $u:=\sum_{k\geq 0}((-\zeta)^k/k!) \big(\prod_{i=0}^{k-1}(h_{\alpha}+2i)\big)e_{-\alpha}^k$ . The result is the twisted Yangian $Y_{\eta\zeta}(sl_2)$. It is not difficult to show that $Y_{\eta\zeta}(sl_2)$ is a quantization of $U(sl_{2}[u])$ with the classical r-matrix $r(u,v)=\eta{\bf c}_{2}/(u-v)+\zeta h_{\alpha}\wedge e_{-\alpha}$, where ${\bf c}_{2}$ is the $sl_{2}$ Casimir element. See [@KST] for details. Acknowledgments {#acknowledgments .unnumbered} =============== The author is grateful to the Organizing Committee for a partial support of his participation at WigSym5. The work was supported by the Russian Foundation for Fundamental Research, grant No. 96-01-01421. [\*\*]{} V.G. Drinfeld, [*Proc. ICM-86 (Berkeley USA) vol.1*]{}, 798-820. Amer. Math. Soc. Providence, RI (1987), 798-820. V.N. Tolstoy, [*Proc. of the Max Born Symp., Wroclav 1996, (eds. J. Lukierski, M.Mozrzymas) PWN - Polish Sci. Publ.- Warsaw*]{} (1997), 99-117. S.M. Khoroshkin, A.A. Stolin, and V.N. Tolstoy, [*From Field Theory to Quantum Groups. (Vol. dedict. to J. Lukierski) World Sci. Publ., Singapore*]{} (1996), 53-75.
--- abstract: 'From recent analysis of the $\pi\pi$ scattering amplitude, it has been claimed that there exists a broad and light $\sigma$ meson. However, if this meson really exists, it must also appear in other observables such as the pion scalar form factor. With the use of unitarity and dispersion relations together with chiral perturbation theory, this form factor is analyzed in the complex energy plane. The result agrees well with the empirical information in the elastic region and reveals a resonance pole at $\sqrt{s}=445-i235$ MeV. This gives further strong evidence for the existence of the $\sigma$ meson.' address: 'Nordita, Blegdamsvej 17, DK-2100 Copenhagen [Ø]{}, Denmark' author: - 'Torben Hannah[@THannah]' title: Pion Scalar Form Factor and the Sigma Meson --- [2]{} It is still controversial whether a broad and light isosinglet scalar meson really exists. After having been omitted from the Particle Data Group for more than twenty years, this scalar meson has reappeared in the last two editions of the Particle Data Group under the entry $f_0$(400-1200) or $\sigma$ [@ref:PDG98]. This reintroduction is based upon several new theoretical analyses [@ref:TR96; @ref:Ishida97; @ref:KLL97], which have given support to the existence of a broad and light scalar meson. In addition, there has also recently been some experimental indications of $\sigma$ production from central $pp$ collisions [@ref:Alde97]. The theoretical evidence for the existence of the $\sigma$ meson comes mainly from model-dependent analysis of $\pi\pi$ scattering [@ref:TR96; @ref:Ishida97; @ref:KLL97]. However, if this meson in fact exist, it must also appear in other processes containing $\pi\pi$ in the final state. Therefore, in order to further investigate whether the $\sigma$ meson really exists, other processes should also be analyzed. The pion scalar form factor is such a process which can give further important information on the $\sigma$ meson. This is somewhat similar to the well-known case of the $\rho$(770) meson, where the information on this resonance can be obtained both from $\pi\pi$ scattering and from the pion vector form factor. The pion scalar form factor has previously been calculated using the inverse amplitude method (IAM) [@ref:Han97; @ref:Tru88]. This method is based upon the combination of unitarity and dispersion relations together with chiral perturbation theory (ChPT) [@ref:We79; @ref:GL84] and has been used in order to extend the range of applicability of ChPT. In particular, the IAM has been applied in order to account for possible resonances since this method, contrary to ChPT, can produce resonance poles in the complex energy plane. In this brief report, the scalar form factor is investigated further using the IAM and the result is analyzed in the complex energy plane in order to find possible resonance poles corresponding to the $\sigma$ meson. The pion scalar form factor $F$ is given by the matrix element of the quark density $$\label{eq:defF} \langle\pi^i(p_2)|\bar{u}u+\bar{d}d|\pi^j(p_1)\rangle = \delta^{ij}F(s) ,$$ where $s=(p_2-p_1)^2$. This form factor is analytical in the complex $s$ plane with a unitarity cut starting at the $\pi\pi$ threshold. In the elastic region the unitarity relation is given by $$\label{eq:uni} {\rm Im}F(s) = \sigma (s)F^{\ast}(s)t^0_0(s) ,$$ where $\sigma (s)$ is the phase-space factor and $t^0_0$ is the isosinglet scalar $\pi\pi$ partial wave. This relation implies that the phase of $F$ will coincide with the $\pi\pi$ phase shift $\delta^0_0$ in accordance with Watson’s final-state theorem [@ref:Wat54]. The first important inelastic effect starts at around 1 GeV and is due to the $K\bar{K}$ intermediate state. Since the main interest will be in energies well below this inelastic effect, in the following the form factor will only be calculated using the elastic approximation. The scalar form factor has been calculated to two loops in ChPT both by a dispersive analysis [@ref:GM91] and more recently by a full field theory calculation [@ref:BCT98]. The result can be written as $$\label{eq:ChPTF} F(s) = F^{(0)}(s)+F^{(1)}(s)+F^{(2)}(s) ,$$ where $F^{(0)}$ is the leading order result, $F^{(1)}$ the one-loop correction, and $F^{(2)}$ the additional two-loop correction. Since the form factor will be normalized to $F(0)=1$ in the following, one has for the leading order term $F^{(0)}=1$. The one-loop correction is given in terms of the single one-loop low-energy constant $l^r_4$ [@ref:GL84], whereas the two-loop correction contains the additional one-loop low-energy constants $l^r_1$, $l^r_2$, and $l^r_3$ together with the two-loop low-energy constants $r^r_{S2}$ and $r^r_{S3}$ [@ref:BCT98]. The superscript $r$ indicates that these low-energy constants depend on the renormalization scale $\mu$, whereas the full form factor is scale-independent. Since ChPT is a perturbative expansion, the unitarity relation (\[eq:uni\]) will only be satisfied perturbatively $$\begin{aligned} \label{eq:puni} {\rm Im}F^{(0)}(s) & = & 0 , \nonumber \\ {\rm Im}F^{(1)}(s) & = & \sigma (s)t^{0(0)}_0(s) , \nonumber \\ {\rm Im}F^{(2)}(s) & = & \sigma (s) \left[ {\rm Re}F^{(1)}(s) t^{0(0)}_0(s)+{\rm Re}t^{0(1)}_0(s) \right] .\end{aligned}$$ This perturbative unitarity will restrict the applicability of ChPT to the very low-energy region. However, with the use of the IAM, the range of applicability of ChPT can be substantially extended to also include resonance regions. The starting point for this method is to write down a dispersion relation for the inverse of the form factor $\Gamma =1/F$ [@ref:Han97; @ref:Tru88]. In this dispersion relation the unitarity relation (\[eq:uni\]) gives ${\rm Im}\Gamma =-{\rm Im}F/|F|^2=-\sigma t/F$. Expanding this quantity to two loops in ChPT gives ${\rm Im}\Gamma =-\sigma [t^{(0)}(1-{\rm Re}F^{(1)})+{\rm Re}t^{(1)}]$ which can be used on the unitarity cut in the dispersion relation for $\Gamma$. The subtraction constants may also be evaluated by expanding the function $\Gamma$ to two-loop order as $\Gamma^{(2)}=1-F^{(1)}+{F^{(1)}}^2-F^{(2)}$. Thus, neglecting possible zeros in the form factor, one has the following dispersion relation $$\begin{aligned} \label{eq:disp1} \frac{1}{F(s)} & = & 1+a_1s+a_2s^2-\frac{s^3}{\pi} \int^{\infty}_{4M^2_{\pi}}ds' \sigma (s')\times \nonumber \\ && \frac{t^{0(0)}_0(s')\left[ 1-{\rm Re}F^{(1)}(s')\right] +{\rm Re}t^{0(1)}_0(s')}{s'^3(s'-s-i\epsilon )} ,\end{aligned}$$ where three subtractions are used in order to make the dispersion integral convergent. This relation can be simplified by writing a dispersion relation for the function $\Gamma^{(2)}$. Using perturbative unitarity (\[eq:puni\]) one finds the this dispersion relation will be exactly similar to the one given in Eq. (\[eq:disp1\]). Thus, the IAM to two loops in the chiral expansion gives the form factor as [@ref:Han97] $$\label{eq:IAM2} F(s) = \frac{1}{1-F^{(1)}(s)+{F^{(1)}}^2(s)-F^{(2)}(s)} .$$ This expression for the form factor is formally equivalent to the \[0,2\] Padé approximant applied on ChPT and will therefore coincide with the chiral expansion up to two loops. However, with the IAM the range of applicability of ChPT is substantially extended. This is based upon the fact that the expansion of $t/F$ used in the IAM works well over a much larger region than the corresponding expansion of $F^{\ast}t$ used in ChPT. In fact, the former expansion works well throughout the elastic region, even when the form factor has a resonant character [@ref:Tru88]. However, the IAM may generate poles on the physical sheet which violate the analyticity requirement. These poles are caused by the high-energy part of the dispersion integral in Eq. (\[eq:disp1\]). Since this part is not expected to be well approximated, it may cause the right-hand side of Eq. (\[eq:disp1\]) to vanish and thereby generate spurious poles in the form factor. These poles should in principle be removed without any significant influence in the region of applicability of the IAM. A rather general method to remove possible poles and thereby restore analyticity is to put the imaginary part of the IAM back into a dispersion relation. With three subtractions the result can be written as $$\label{eq:polIAM2} F(s) = 1+\mbox{$\frac{1}{6}$}\langle r^2\rangle s+cs^2 +\frac{s^3}{\pi}\int^{\infty}_{4M^2_{\pi}}\frac{{\rm Im}F(s')ds'} {s'^3(s'-s-i\epsilon )} ,$$ where both the subtraction constants and ${\rm Im}F$ are calculated from the IAM (\[eq:IAM2\]). Here, it is assumed that ${\rm Im}F$ does not contain any poles on the unitarity cut and three subtractions are used in order to suppress the high-energy part of the dispersion integral. Without any poles Eq. (\[eq:polIAM2\]) is just an identity, but with poles the output will in general be different from the input. However, in the region where the IAM is applicable, the difference between the form (\[eq:polIAM2\]) and the form (\[eq:IAM2\]) should be small. In fact, this method to remove possible poles is equivalent to the subtraction of the poles on the physical sheet from the original IAM (\[eq:IAM2\]). This will be discussed in more detail elsewhere [@ref:HT98]. The IAM to two loops depends on a number of low-energy constants which have to be determined phenomenologically. Unfortunately, the scalar form factor is not directly accessible to experiment. However, in the elastic region the phase of $F$ is given by the $\pi\pi$ $I=0$ $S$ phase shift $\delta^0_0$, which is known experimentally. Fitting these phase shifts up to 0.9 GeV [@ref:Ros77; @ref:Pro73; @ref:Hyams73; @ref:EM74] and using the value of the pion scalar radius $\langle r^2\rangle =0.60$ ${\rm fm}^2$ [@ref:GM91; @ref:DGL90], some of the low-energy constants in the IAM to two loops have previously been determined without taking possible poles into account [@ref:Han97]. Here, in order to remove spurious poles, this fit is repeated with the form factor given by Eq. (\[eq:polIAM2\]). The result is shown in Fig. \[Fig1\] together with the experimental $\pi\pi$ phase shifts, from where it is observed that the IAM agrees rather well with the main bulk of the data all the way up to 0.9 GeV. Thus, the IAM satisfies Watson’s final-state theorem [@ref:Wat54] quite well in the whole elastic region. This fit gives the following values for the low-energy constants $$\begin{aligned} \label{eq:lec} l^r_4 & = & 1.53\times 10^{-3} ,\nonumber \\ r^r_{S2} & = & 2.25\times 10^{-3} ,\nonumber \\ r^r_{S3} & = & 7.60\times 10^{-5}\end{aligned}$$ at the renormalization scale $\mu =M_{\rho}=770$ MeV. Since the experimental data are not very consistent with each other, there has not been assigned any error bars on these low-energy constants. However, the obtained values of $l^r_4$ and $r^r_{S3}$ agree rather well with the recent determination of these low-energy constants using two-loop ChPT [@ref:BCT98]. As for $r^r_{S2}$, this low-energy constant has so far only been estimated on the basis of the resonance saturation hypothesis [@ref:Eck89] with a result [@ref:BCT98] that is somewhat smaller than the value obtained above. In the future, it might be possible to determine this two-loop low-energy constant from independent observables [@ref:Han96] and thereby check the value obtained here. The scalar form factor can be defined in the whole complex $s$ plane. Since it contains cuts starting at the $\pi\pi$ threshold, this will involve different Riemann sheets. In the elastic approximation there are two Riemann sheets, which are defined according to the sign of the center of mass momenta $q=\sqrt{s-4M^2_{\pi}}/2$. The first or physical sheet has positive values of ${\rm Im}q$, whereas the second or unphysical sheet has negative values of ${\rm Im}q$. The form factor given by either Eq. (\[eq:IAM2\]) or Eq. (\[eq:polIAM2\]) can indeed be extended analytically to the whole complex $s$ plane. This analytic continuation will involve infinitely many Riemann sheets since the cut in the IAM comes from logarithmic functions. However, only two of these sheets correspond to the first and second Riemann sheet that the form factor should reproduce. In Fig. \[Fig2\] the absolute square of the form factor (\[eq:polIAM2\]) is shown in the complex energy plane on the first Riemann sheet. On the real axis the result agrees very well with the result of a dispersive analysis [@ref:GM91; @ref:DGL90], where the scalar form factor has been determined from the experimental $\pi\pi /K\bar{K}$ phase shifts. Furthermore, the form factor (\[eq:polIAM2\]) is analytic in the whole complex energy plane with the correct cut structure starting at the $\pi\pi$ threshold. This is different from the original IAM (\[eq:IAM2\]) which generates a pole on the negative $s$ axis. However, this pole is removed by using the form (\[eq:polIAM2\]) without any significant influence on the result in the region shown in Fig. \[Fig2\]. From this figure it is also observed that around $0.4-0.5$ GeV the form of $|F|^2$ is somewhat reminiscent of a resonant structure. However, in order to investigate whether this form is really associated with a resonance, one has to consider the second Riemann sheet. On this sheet resonances are characterized by poles in the complex energy plane, where the mass ($M_R$) and width ($\Gamma_R$) of the resonance can be related to the position of the pole by $$\label{eq:resonance} \sqrt{s_{pole}} = M_R-i\frac{\Gamma_R}{2} .$$ In Fig. \[Fig3\] the absolute square of the form factor (\[eq:polIAM2\]) is shown in the complex energy plane on the second Riemann sheet. One finds that $|F|^2$ indeed generates two complex conjugated poles corresponding to a broad and light resonance. In fact, the position of these poles is the same for the two expressions of the form factor given by Eq. (\[eq:IAM2\]) and Eq. (\[eq:polIAM2\]), respectively. From the position of the pole, the mass and width of this $\sigma$ meson is given by $$\label{eq:mwsigma} M_{\sigma} = 445\;{\rm MeV}\;\; ,\;\;\Gamma_{\sigma} = 470\;{\rm MeV} .$$ This compares rather well with the values $M_{\sigma}=470$ MeV and $\Gamma_{\sigma}=500$ MeV obtained in Ref. [@ref:TR96]. However, rather different values for $M_{\sigma}$ and $\Gamma_{\sigma}$ have also been obtained with other theoretical models [@ref:Ishida97; @ref:KLL97]. Therefore, it is important to reduce the model-dependence when the mass and width of the $\sigma$ meson are determined. The IAM is based solely on the use of unitarity and dispersion relations together with ChPT. Therefore, within this approach, any model-dependence in the mass and width of the $\sigma$ meson is due to higher order terms in the chiral expansion together with the present uncertainties in the values of the low-energy constants. The IAM is in fact a systematic approach which can be applied to any given order in the chiral expansion. Originally, this method was applied to the scalar form factor in the one-loop approximation [@ref:Tru88] with a result that is formally equivalent to the \[0,1\] Padé approximant applied on ChPT. In this case the IAM depends on the single one-loop low-energy constant $l^r_4$ which can be determined from the ratio $F_K/F_{\pi}$ [@ref:GL84; @ref:Han96]. With this low-energy constants fixed, the IAM to one loop agrees rather well with the empirical information up to about 0.5 GeV. The one-loop approximation also contains a pole on the negative $s$ axis. However, this pole can be removed by using the method discussed previously without any significant influence on the result in the elastic region. Extending this result to the whole complex energy plane, one finds that the IAM to one loop also generates a resonance pole, where the corresponding mass and width of this resonance are given by $$\label{eq:mwsigma1} M_{\sigma} = 463\;{\rm MeV}\;\; ,\;\;\Gamma_{\sigma} = 393\;{\rm MeV} .$$ Comparing these values with the values obtained from the IAM to two loops (\[eq:mwsigma\]), it is observed that the masses are very similar, whereas the difference in the widths is somewhat larger. However, this difference is not significant compared to the large uncertainty in the width of the $\sigma$ meson given by the Particle Data Group [@ref:PDG98]. In view of this the convergence of the IAM is satisfactory for both the mass and width of the $\sigma$ meson. Hence, the corrections to the values given in (\[eq:mwsigma\]) due to even higher orders in the chiral expansion are expected to be of little importance. There is also an uncertainty in the obtained values for $M_{\sigma}$ and $\Gamma_{\sigma}$ coming from the uncertainties in the values of the low-energy constants. However, this effect should also be rather small and could be estimated when the values of the low-energy constants in the IAM are determined more accurately. The IAM is not restricted to the scalar form factor but this method is in fact quite general and has been applied to other processes as well. In particular, the IAM has been applied to $\pi\pi$ scattering where it also generates a resonance pole corresponding to the $\sigma$ meson [@ref:DP97]. In fact, the mass and width of the $\sigma$ resonance obtained from $\pi\pi$ scattering, $M_{\sigma}=440$ MeV and $\Gamma_{\sigma}=490$ MeV, agree very well with the values obtained in (\[eq:mwsigma\]). This strongly supports the consistency of the IAM and gives additional evidence for the existence of the $\sigma$ meson. To summarize, the pion scalar form factor has been calculated by the use of unitarity and dispersion relations together with the chiral expansion. In order to satisfy the analyticity requirement, possible poles on the physical sheet are removed from this IAM. The result agrees well with both the experimental $\pi\pi$ phase shifts and a dispersive analysis in the whole elastic region. 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--- abstract: 'We report on the production of a rubidium Bose-Einstein condensate in a simplified vacuum apparatus. Magneto-optical traps with large numbers and ultra-high vacuum for moderately long conservative trap lifetimes of 16 seconds are sequentially obtained with light induced rapid atomic vapor pressure modulation. Subsequent evaporative cooling is carried out in two stages in a hybrid magnetic quadrupole plus optical dipole trap. High evaporation efficiencies are observed in both stages and $^{87}$Rb BECs with more than $10^{5}$ atoms can be reliably produced with total evaporation time of only 9.5 seconds.' author: - 'Dezhi Xiong, Fudong Wang, Xiaoke Li, Ting-Fai Lam, Dajun Wang' title: 'Production of a rubidium Bose-Einstein condensate in a hybrid trap with light induced atom desorption' --- Introduction ============ Bose-Einstein condensates (BEC) of dilute atomic gases were first produced in 1995[@Anderson95; @Davis95; @Bradley95] using laser cooling followed by evaporative cooling. To date, these two cooling methods are still the standard and indispensable steps for ultracold quantum gas studies. Evaporative cooling prefers ultra-high vacuum (UHV) for efficient thermalization, whereas collecting large number of atoms by laser cooling needs enough particles in the background. These two contradictory requirements are the main reasons behind the complexity of typical BEC UHV setups. For instance, the Zeeman slower[@Phillips82] and the double magneto-optical trap (MOT) system [@Myatt96] are two of the most common implementations to overcome this issue. Here we describe a setup consists of a simple single glass cell. With the ultraviolet (UV) light induced atomic desorption (LIAD), we meet the vacuum conditions at both ends and are able to produce Rb BECs with more than 10$^5$ atoms repeatably. LIAD has been used to produce Rb BECs [@Du04] and $^{40}$K degenerate Fermi gases [@Aubin06] on atomic chips in single glass cells previously. The strong confinement provided by the chip makes evaporation happen very rapidly. However, the presence of a surface only hundreds of microns from the atoms makes the application of an optical dipole trap (ODT) very challenging. The ODT is especially important for our future plan of studying Feshbach resonances[@Kohler06; @Chin10] and ultracold polar molecules[@Ni08]. In a stainless chamber, Mimoun et al.[@Mimoun10] achieved a sodium BEC with LIAD. Evaporation was carried out in pure optical traps. To maintain efficient evaporation to reach degeneracy, besides a crossed-beam ODT, an additional tightly focused “dimple” beam had to be added. In the current work, we have adapted the hybrid trap developed by Lin et al. [@Lin09] where a spherical quadrupole trap is used for the first stage of evaporative cooling. The pre-cooled atoms are then transfered to crossed or single beam ODTs with foci displaced from the magnetic field zero for further evaporation to BEC. Compared with the ODT, a magnetic trap typically has deeper trap depth and larger trap volume. It can be mode-matched well with laser cooled atom clouds for efficient conservative trap loading. The simplest magnetic trap is of the spherical quadrupole configuration, which can be easily generated from a pair of anti-Helmholtz coils. Due to its superior confinement[@Ketterle99], evaporation in this linear trap is more efficient compared with harmonic Ioffe-Pritchard traps before Majorana loss worsens at low temperatures. The Majorana hole can be plugged with a blue detuned laser beam focused to the quadrupole trap center [@Davis95; @Dubessy12; @Heo11]. Lin et al. instead used a far red detuned laser focused below the quadrupole center to displace the overall trap potential minimum away from zero B field [@Lin09]. This method is easier to implement in terms of optical alignment. It also results in a BEC in ODT without other intermediate steps. With full strength quadrupole trap, this ODT does not prevent Majorana loss completely. Subsequent quadrupole trap decompression transfers a large portion of the pre-cooled atoms to the new trap center defined by the ODT. Majorana loss is eliminated this way with the added huge gain of phase-space density because of the trap shape deformation and evaporation during the transfer. In this work, we explore evaporative cooling of Rb to BEC in this hybrid trap in a very simple vacuum setup. We believe that this compact system can be useful for new groups who would like to setup a BEC apparatus quickly with relatively low cost. Experimental setup ================== Vacuum system and LIAD source ----------------------------- At the center of our BEC setup is a single rectangular glass cell without anti-reflection coating. The cell has outer dimensions of 100 mm$\times$40 mm$\times$40 mm and is connected to a standard CF35 cube. Rubidium (and sodium for future experiments) dispensers (Alvatec GmbH) are directly inserted into the glass cell from the opposite side of the cube. The distance between the dispensers and the cell center is $\sim$ 12 cm, ensuring alkali atoms released from the dispensers have the most direct line-of-sight with cell walls for fast atom absorption. The vacuum is maintained by an ion pump(Gamma Vacuum 45S). A titanium sublimation pump (Varian Vacuum TSP Cartridge) is also installed, but it has never been fired after the initial vacuum preparation stage. Standard procedures are followed to obtain UHV. At the ion pump position, the pressure reads $1.8\times10^{-11}$ torr. The pressure at the center of the glass cell is estimated to be 5 to 6 times higher limited by conductance. This is consistent with the measured magnetic trap lifetime as discussed in later subsections. Initially, to coat the glass cell walls with Rb atoms, 2.5 A current is applied to one dispenser for a day. Later, firing the dispenser for 15 minutes once a week is enough to replenish atoms. It is then left off during further experiment cycles. The LIAD light is provided by a 365 nm UV LED (Thorlabs M365L2) with $\sim$ 200 mW uncollimated power output at its maximum current. It is mounted close to the cell to reach enough localized intensity for efficient desorption. We do observe saturation behavior of the Rb MOT number with increasing UV light power. But it is suspected that with multiple LEDs to cover different parts of the cell walls, atom numbers could be further increased. The LED current driver can be quickly modulated by a TTL signal synchronized with other events. Similar to other groups[@Telles10; @Klempt06] using LIAD, we find that the vacuum pressure can recover within a short time scale after switching off the UV light. To allow vacuum recovery, the LED is typically switched off one second before magnetic trap loading. The MOT number loss during this interval is less than 10$\%$. Laser cooling ------------- Our MOT is in the real six-beam configuration with total laser power of 70 mW and $1/e^2$ beam diameters of 25 mm. The more convenient three beam retro-reflection configuration does not produce as good initial phase-spaced densities (PSD) in the MOT and molasses cooling stages. We use two home-built external cavity diode lasers (ECDL) [@Arnold98] to provide the trapping and repumping beams. The trap laser frequency is empirically detuned -19 MHz from the $F=2$ to $F'=3$ cycling transition. The repump beam has a power of 10 mW and is on resonance with the $F=1$ to $F'=2$ transition. The trap power is boosted up with a 150 mW laser diode injection locked by the trap ECDL and then delivered to the UHV cell by polarization maintaining fibers. With LIAD, the MOT loading time constant is typically 10 seconds. More than $2\times10^8$ Rb atoms can be collected after 20 seconds loading. The atoms then undergo a 24 ms compressed MOT stage by reducing the repump power to $\sim$ 150 $\mu$W and increasing the trap laser detuning to -32 MHz. A 6 ms polarization gradient cooling stage is then applied by abruptly turning off the 10 G/cm gradient and further detuning the trap laser to -78 MHz. The repump laser beams are then turned off first, followed by the trap beams 1 ms later. Following these steps, $>$ 95$\%$ atoms in the $F$ = 1 manifold with temperatures around 15 $\mu$K and densities $>$ 10$^{11}$/cm$^3$ are routinely achieved. After optically pumping the atoms to the $|F = 1, m_F = -1 \rangle$ state, $> $ 65$\%$ of them can be loaded into the magnetic quadrupole trap. The hybrid trap --------------- The magnetic trap is generated from the same anti-Helmholtz coil pair used for the MOT field. They are wound with standard 4 mm outer diameter refrigerator copper tubes, in-house insulated by doubly-wrapped Kapton tapes. Pressurized water is running continuously inside to remove the resistive heating. The current applied to the coils is actively stabilized with the help of a current transducer (LEM IT200-S) and a simple electronic feedback servo. For this experiment, we use an axial field gradient of 160 G/cm. To capture the laser cooled atoms with minimum PSD loss, we first abruptly turn on the magnetic field gradient to 60 G/cm. After a holding time of 70 ms, it is ramped up to 160 G/cm in 130 ms. This adiabatic compression heats the cold atoms up to 90 $\mu$K, but the peak collision rate increases by a factor of 3.7. The PSD of 2.1$\times$10$^{-6}$ is an excellent starting point for evaporation. ![\[fig:lifetime\] (color online). Lifetime of Rb atoms in a magnetic quadrupole trap loaded from a MOT filled with the help of LIAD. Black dots are atom fluorescences measured using MOT recapture after different holding times in the magnetic trap (see text for detail). The red line is an exponential fitting to the measurement, which gives a $1/e$ lifetime of $\sim$ 16 s. The measurement is carried out one second after turning the UV light off. ](lifetime){width="0.85\linewidth"} We have measured the magnetic trap lifetime by MOT recapturing. After different holding times, the atoms are released from the quadrupole trap by suddenly reducing the gradient from 160 G/cm to the value used for normal MOT operation. The MOT beams are then turned back on and the remaining atoms reveal themselves by fluorescences. As shown in Fig. \[fig:lifetime\], the 16 s trap lifetime coupled with the large atom number is a clear evidence of the LIAD technique’s effectiveness. Direct forced radio (RF) or microwave (MW) frequency evaporation in the quadrupole trap stops at a PSD of 10$^{-4}$ due to Majorana loss. To partially mitigate this loss, we superimpose a crossed ODT to the atoms together with the quadrupole field. The crossed ODT is produced by a 1070 nm, multi-frequency, linear polarized fiber laser(IPG Photonics). A 110 MHz acousto-optical modulator (Crystal Technology) is used for intensity stabilization and rapid trap switching off in less than 1 $\mu$s. We divide the laser power into two arms using a $\lambda$/2 waveplate and a polarizing beamsplitter cube. They intersect each other with an angle of $62^{\circ}$. The beam waists are 90 $\mu$m, while the vertical offset between the foci and the magnetic field zero is 150 $\mu$m. Only 4.5 W power is used in each beam, which produces a trap depth of 110 $\mu$K for $^{87}$Rb. They are ramped up in 200 ms after the quadrupole trap has reached its full strength and remain there during the MW evaporation. ![\[fig:potential\] (color online). Variation of the hybrid trap potential along $y$ (gravity) direction during the magnetic quadrupole gradient ramping down. The black dashed line is the magnetic trap potential at 160 G/cm without the dipole trap. The dotted lines show the overall potential with different magnetic trap strengths ranging from gradients of 160 G/cm to 26 G/cm. The optical trap and magnetic trap centers are displaced by 150 $\mu$m from each other.](potential){width="0.85\linewidth"} The effective potential for atoms in this hybrid trap is $$\begin{aligned} U(r)=&&\frac{1}{2}\mu_B B^{'}\sqrt{\dfrac{x^2}{4}+y^2+\dfrac{z^2}{4}}+mgy\nonumber\\ &&-U_0 e^{-2(x^2+(y-y_0)^2)/w_0^2} + E_0 \label{eq:four}.\end{aligned}$$ where $B'$ is the quadrupole field gradient along the vertical direction $\hat{y}$. $U_0$ is the optical trap depth at $y = y_0$ and $w_0$ is the beam waist. $E_0$ is the potential difference between the quadrupole trap center and the final trap minimum. $\mu$$_B$, $m$ and $g$ are the Bohr magneton, $^{87}$Rb atomic mass and the acceleration of gravity, respectively. As illustrated in Fig. \[fig:potential\], at 160 G/cm, there are two potential minima as a result of the displaced optical and magnetic trap centers. The potential difference between these two minima is $\sim$ 40 $\mu$K. This is comparable to the 20 $\mu$K cloud temperature where Majorana loss becomes severe. Thus this loss cannot be suppressed completely. Majorana loss ------------- In an effort to quantify the Majorana loss in this setup, we evaporate the atoms to different temperatures by controlling the final evaporation cut frequency, and then measure the lifetimes to obtain loss rates. The measured data points are shown in Fig. \[fig:majorana\]. Following ref.[@Petrich95], the Majorana induced loss rate can be estimated as: $$\Gamma_m=\chi\dfrac{\hbar}{m}(\dfrac{0.5 \mu_B B'}{k_B T})^2. \label{eq:AtomLoss}$$ Here $\chi$ is a proportional constant, $T$ is the temperature, $\hbar$ is the Planck constant over 2$\pi$ and $k_B$ is the Boltzmann constant. ![\[fig:majorana\] (color online). Majorana loss in the bare quadrupole trap. Black dots are measured atom loss rates at different temperatures obtained by MW evaporative cooling to different final frequencies. The red solid curve is a fit of the data with a loss model.(see text for detail).](majorana){width="0.85\linewidth"} To account for the background loss, we modify this equation as $\Gamma_L=aT^{-2}+\Gamma_b$ [@Heo11], where $\Gamma_L$, and $\Gamma_b$ are the measured total loss rate and the background loss rate, respectively. The coefficient $a =\chi\dfrac{\hbar}{m}(\dfrac{0.5 \mu_B B'}{k_B})^2$, as a result of equation (\[eq:AtomLoss\]). The fitting yields $a = 8.3(9)\mu K^2/s$ and $\Gamma_b = 0.071(4)$. Further analysis is not pursued due to the comparably large background loss rate, but the $T^{-2}$ dependence is already evident. For the full strength quadrupole trap, we have observed that Majorana loss is still severe and evaporative cooling below 20 $\mu$K is inefficient even with the displaced ODT beams. This is consistent with our understanding of the potential following Fig. \[fig:potential\]. The situation is marked improved after the magnetic gradient is reduced to 26 G/cm. With the ODT beam still at full power, the trap lifetime is measured to be 13(1) s for a 8 $\mu$K cloud. This matches with the background limited value well and suggests that the Majorana loss is fully suppressed. BEC production ============== Evaporation in the magnetic trap -------------------------------- Evaporative cooling in magnetic trap starts right after the loading is finished. Condensates of similar numbers can be obtained with either RF or MW evaporation. Here only the current MW setup will be described. The MW signal driving the transition between $\left|1,-1\right\rangle$ and $\left|2,-2\right\rangle$ hyperfine levels is generated by doubling the output of a signal generator (Anapico ASPIN6000-HC). After a 3 W amplifier (Minicircuit ZVE-3W-183), it is broadcast to the atoms by a microwave horn antenna. Thanks to the excellent initial PSD, this evaporation step can be completed within 6 seconds during which the frequency is swept from 6774 MHz to 6822 MHz. The sweep is divided into several segments with different slopes and powers. This is done empirically by optimizing the PSD obtained after each segment. During the whole procedure, a truncation factor of $\eta$ $\approx$ 6 is roughly maintained. After the MW evaporation, we typically end up with $2.5\times10^7$ atoms at temperatures of $\sim$ 29 $\mu$K and calculated densities of 10$^{12}$/cm$^3$. This corresponds to a PSD of $7\times10^{-5}$. Transfer to the optical dipole trap ----------------------------------- After the MW evaporation in magnetic trap, the atoms are transfered into the ODT by ramping down the $B'$ linearly from 160 G/cm to 26 G/cm in 500 ms. This final $B'$ value is chosen to below 30.5 G/cm, which is the minimum gradient required for levitating Rb $\left|1,-1\right\rangle$ atoms against gravity. During this process, the confinement provided by the magnetic trap gradually decreases. The atoms sag down under the influence of gravity until they are trapped by the ODT. The trap potential deformation during the quadrupole trap decompression is shown in Fig.\[fig:potential\]. At the final $B$ field gradient, the atoms occupy mainly the near bottom part of the potential and Majorana loss is suppressed by the high potential wall. ![\[fig:BEC\] (color online). Absorption images after 30 ms time of flight showing evidence of the BEC phase transition following evaporation in the ODT. Bottom panels: (a) thermal cloud at just above the transition temperature; (b) bimodal distribution; (c) a quasi-pure condensate with $10^5$ atoms. Field of view: 900 $\mu$m by 900 $\mu$m. Top panels: the integrated optical densities of corresponding images. Blue dots are experimental data, red solid lines are fittings to Gaussian (thermal atoms) or/and parabola (condensed atoms) functions. The red dashed line is the Gaussian fitting of thermal atoms in the bimodal phase. ](BEC){width="0.85\linewidth"} The magnetic field gradient ramping speed is selected experimentally for best final condensate numbers. This rate is rather fast compared with the Lin et al.’s experiment [@Lin09] (2 seconds for the same gradient range). We suspect that this is due to the compromise between the larger background loss under our vacuum condition and the adiabaticity requirement. We also observe that no MW sweep is necessary in this step, while a factor of 40 increase in PSD is still observed because of continuous evaporation during the potential deformation. When the gradient reaches the final value, the atoms are loaded into the crossed ODT with temperatures of 14.6 $\mu$K, which is about one-eighth of the trap depth. Correspondingly, the PSD increases to $3\times10^{-3}$. The overall transfer efficiency from the quadrupole trap to the ODT is about $15\%$. ![\[fig:trajectory\](color online). Evaporation trajectory showing the temperature(red circles) and peak phase-space density(blue dots) vs. atom number during the cooling process. Dashed vertical lines delimit three regions labeled(i),(ii)and (iii), which correspond to three evaporation steps. (i) forced MW evaporation in hybrid trap; (ii) loading into the dipole trap; (iii) evaporation in the optical dipole trap.](trajectory){width="0.85\linewidth"} Evaporation in the optical trap ------------------------------- With $4\times10^6$ atoms of PSD $3\times10^{-3}$ in the crossed ODT, condensate production is straightforward. Forced evaporation is carried out by lowering the ODT power and thus the trap depth. We control the trap power following the scaling law $P(t) = P_i (1+t/\tau)^{-\beta}$, where $P_i$ is the initial power. Both $\tau$ and $\beta$ are determined experimentally for best final condensate number. According to the reference[@OHara01], power ramping like this should result in optimal evaporation efficiency with fixed truncation factors. However, our case is complicated by thermal effect induced foci position shifts, mainly coming from the 3 mm thick Pyrex cell wall. These shifts will further loosen the trap in addition to the power reduction. Indeed, phase transition is already observed after lowering the power by a factor of 12. While following the scaling law, a reduction factor of $\sim$100 is needed. Judging from the final BEC number repeatability, we conclude that the thermal shift is reproducible to a great extent and thus further effort to minimize the shifts is not pursued. As shown in Fig. \[fig:trajectory\], the ODT evaporation efficiency defined as $\alpha = -\frac{log(PSD/PSD_0)}{log(N/N_0)}$ is 2.7, which is still quite high. Here PSD$_0$ and $N_0$ are the initial phase-space density and atom number, respectively. The ODT evaporation lasts for 3.5 s, which is also on the fast side. The phase transition happens at $T = 240$ nK with $2.7\times10^5$ atoms, signified by a bimodal distribution after 30 ms time-of-flight, as shown in Fig. \[fig:BEC\]. Further evaporation leads to a quasi-pure condensate with $10^5$ atoms. The final trap frequencies are measured to be 2$\pi \times$($98,112, 61$)Hz, along $x$, $y$ and $z$ directions, respectively. We emphasize that the excellent initial PSD resulted short evaporation time is vital for producing condensates with relatively large numbers in our vacuum condition. As shown in part (i) of Fig. \[fig:trajectory\], the PSD and temperature scale with the number as $N^{-2.17}$ and $N^{0.75}$ respectively in the magnetic trap evaporation. The evaporative is obviously less efficient compared with other experiments which typically have ten times better vacuum. In those cases, evaporation efficiencies close to $\alpha = 3$ have been achieved routinely [@Heo11; @Lin09; @McCarron11]. While the current evaporation efficiency is still high enough for reliable condensate production, with lower initial PSD, the evaporation has to be slowed down to allow enough thermalization. One-body number loss will limit the density gain in each evaporation steps and the evaporation efficiency could be further deteriorated. In the worst case, condensate production might even become impossible. Conclusion ========== We have described a compact experimental setup capable of producing $^{87}$Rb Bose-Einstein condensates with more than $10^5$ atoms. Each experiment cycle takes about 30 seconds, contributed largely by the 20 s MOT loading time. The short evaporation time is made possible by the excellent initial PSD, and the high evaporation efficiency of the hybrid trap. Our success in condensate production with vacuum limited lifetime of 16 seconds further demonstrated the merit of the hybrid trap technique. Its combination with LIAD makes a nice shortcut to quantum gas productions. The current setup is limited by the MOT laser power. Using a high power tapered amplifier and larger beam size, higher initial number and shorter MOT loading times can be readily achieved. Larger condensates with even less evaporation time in similar setups are thus conceivable. We thank Jie Ma for his contribution at the early stage of this experiment. We are grateful to Ruquan Wang and Shuai Chen for useful discussions and the loan of some essential optics. Our work was supported by Hong Kong RGC CUHK Grant No. 403111 and CUHK Direct Grant for Research No. 2060416. 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--- abstract: 'The long–standing issues of determination of the mass distribution and nature of the center of our Galaxy could be probed by a lensing experiment capable of testing the spatial and velocity distributions of stars nearby and beyond it. We propose a lensing toy–model which could be a further evidence that a massive consensation (a neutrino condensation) is a good candidate to explain the data ruling out the presence of a supermassive black hole.' author: - | S. Capozziello$^{1}$[^1] and G. Iovane$^{1}$[^2]\ [*$^{1}$Dipartimento di Scienze Fisiche “E.R. Caianiello”,*]{}\ [*INFN Sez. di Napoli, Gruppo collegato di Salerno*]{}\ [*Università di Salerno, I-84081 Baronissi (SA) Italy.*]{} title: Probing the nature of compact dark object at the Galactic Center by gravitational lensing --- 8.5in 6.25in -.25in \#1[$^{\ref{#1}}$]{} ====== å[[*Astron. Astrophys.*]{} ]{} ø[ò ]{} ù V() ¶[****]{} ł[L]{} Keywords: supermassive neutrino star, dark matter, gravitational lensing. The puzzle to explain the nature of matter condensation at the center of our Galaxy is more than twenty years old problem [@oort]. Various observational campaigns [@genzel1] have identified such a center with the supermassive compact dark object Sagittarius A$^*$ (Sgr A$^*$) which is an extremely loud radio source. Detailed information comes from dynamics of stars moving in the gravitational field of such a central object. The statistical properties of spatial and kinematical distributions are of particular interest [@sellgreen]: Using them, it is possible to establish the mass and the size of the object which are $(2.61\pm 0.76)\times 10^6 M_{\odot}$ concentrated within a radius of 0.016 $pc$ (about 30 $lds$)[@ghez],[@genzel96]. More precisely, Ghez [*et al.*]{} [@ghez] have made a campaign of observations where velocity measurements in the central arcsec$^2$ are extremely accurate. From this bulk of data, it is possible to state that a supermassive compact dark object is present at the center of Galaxy and, furthermore, it is revealed by the motion of stars moving within a projected distance of less than 0.01 $pc$ from the radio source Sgr A$^*$ at projected velocities in excess of 1000 $km/s$. In other words, a high increase of velocity dispersion of the stars toward the dynamic center is revealed. Furthermore, a large and coherent counter–rotation, expecially of the early–type stars, is revealed, supporting their origin in a well–defined epoch of star formation. Besides, observations of stellar winds nearby Sgr A$^*$ give a mass accretion rate of ${\displaystyle \frac{dM}{dt}=6\times 10^{-6}M_{\odot}yr^{-1}}$ [@genzel96]. Hence, the dark mass must have a density $\sim 10^9 M_{\odot}pc^{-3}$ or greater and a mass–to–luminosity ratio of at least $100M_{\odot}/L_{\odot}$. The conclusion is that the central dark mass is statistically very significalt $(\sim 6-8\sigma)$ and cannot be removed even if a highly anisotropic stellar velocity dispersion is assumed. Given that the majority of stars in a cluster are of solar mass, such a large density contrast excludes that the dark mass could be a cluster of almost $2\times 10^6$ neutron stars or white dwarfs. As a first conclusion, several authors state that in the Galactic center there is either a single supermassive black hole or a very compact cluster of stellar size black holes [@genzel96]. The first hypothesis is supported by several authors since similar supermassive black holes have been inferred to explain the central dynamics of several galaxies as M87 [@ford],[@macchetto], or NGC4258 [@greenhill]. However, due to the above mentioned mass accretion rate, if Sgr A$^*$ is a supermassive black hole, its luminosity should be more than $10^{40}erg\,s^{-1}$. On the contrary, observations give a bolometric luminosity of $10^{37}erg\,s^{-1}$. This discrepancy is the so–called “blackness problem” which has led to the notion of a “black hole on starvation” at the center of Galaxy. Besides, the most recent observations probe the gravitational potential at a radius larger than $4\times 10^{4}$ Schwarzschild radii of a black hole of mass $2.6\times 10^{6}M_{\odot}$ [@ghez] so that the supermassive black hole hypothesis at the center of Galaxy is far from being conclusive. On the other hand, stability criteria rule out the hypothesis of a very compact stellar cluster in Sgr A$^{*}$ [@sanders]. In fact, detailed calculations of evaporation and colision mechanisms give maximal lifetimes of the order of $10^8$ years which are much shorter than the estimed age of the Galaxy [@maoz]. Another viable and, in some sense more attractive alternative model for the supermassive compact object in the center of our Galaxy (and in the center of several other galaxies) has been recently proposed by Viollier [*et al.*]{} [@viollier]. The main ingredient of the proposal is that the dark matter at the center of galaxies is made by nonbaryonic matter (massive neutrinos or gravitinos) which interacts gravitationally forming supermassive balls in which the degeneracy pressure of fermions balances their self–gravity. Such neutrino balls could have formed in the early epochs during a first–order gravitational phase transition and their dynamics could be reconciled with some adjustments to the Standard Model of Cosmology (for an exhaustive discussion of the problem, see [@viollier]). Furthermore, several experiments are today running to search for neutrino oscillations. LSND [@lsnd] finds evidence for oscillations in the $\nu_{e}-\nu_{\mu} $ channel for pion decay at rest and in flight. On the contrary KARMEN [@karmen] seems to be in contradiction with LSND evidence. CHORUS and NOMAD at CERN are just finished the phase of (’94–’95) data analysis. In any case, it is very likely that exact preditions for and $\nu_{\mu}-\nu_{\tau}$ oscillations will be available at the end of millennium or in first years of the next. From all this bulk of data, and thanks to the fact that it is possible to give correct values for the masses to the quarks [*up*]{}, [*charm*]{}, and [*top*]{}, it is possible to infer reasonable values of mass for $\nu_{e}$, $\nu_{\mu}$, and $\nu_{\tau}$. For our purposes, we are particularly interested in fermions which masses range between 10 and 25 keV$/c^{2}$ which cosmologically fall into the category of [*warm*]{} dark matter[^3]. Choosing fermions like neutrinos or gravitinos in this mass range allows the formation of supermassive degenerate objects (from $10^6 M_{\odot}$ to $10^9 M_{\odot}$). As we said, the existence of such objects avoids to invoke the supermassive black hole hypothesis in the center of galaxies and quasars and it is able to justify the large amount of radio emission coming from such unseen objects. The theory of heavy neutrino condensates, bound by gravity, can be easily sketched [@leimgruber]. Let us consider the Thomas–Fermi model for fermions. We can set the Fermi energy $E_{F}$ equal to the gravitational potential which binds the system, that is $$\label{n1} \frac{\hbar^2 k_{F}^{2}(r)}{2 m_{\nu}}-m_{\nu}\Phi(r)=E_{F} =-m_{\nu}\Phi(r_{0})\,,$$ where $\Phi(r)$ is the gravitational potential, $k_{F}$ is the Fermi wave number and $\Phi(r_0)$ is a constant chosen to cancel the gravitational potential for vanishing neutrino density. The length $r_{0}$ is the estimed size of the halo. If we take into account a degenerate Fermi gas, we get $k_{F}(r)=\left(6\pi^{2}n_{\nu}(r)/g_{\nu}\right)^{1/3},$ where $n_{\nu}(r)$ is the neutrino number density and we are assuming that it is the same for neutrinos and antineutrinos within the halo. The number $g_{\nu}$ is the spin degeneracy factor. Immediately we see that the number density is a function of the gravitational potential, [*i.e.* ]{} $n_{\nu}=f(\Phi),$ and the model is specified by it. If in the center of the neutrino condensate there is a baryonic star (which we approximate as a point source), the gravitational potential will obey a Poisson equation where neutrinos (and antineutrinos) are the source term, [*i.e.* ]{} $$\label{n4} \triangle\Phi=-4\pi Gm_{\nu}n_{\nu}\,.$$ Such an equation is valid everywhere except at the origin. We can assume, for the sake of simplicity, the spherical symmetry and define the variable $% u=r[\Phi(r)-\Phi(r_{0})]$ then the Poisson equation reduces to the radial Lané–Emden differential equation $$\label{n5} \frac{d^2 u}{dr^2}= -\left(\frac{4\sqrt{2}m_{\nu}^{4}G g_{\nu}}{3\pi\hbar^{3}% }\right) \frac{u^{3/2}}{\sqrt{r}}\,,$$ with polytropic index $n=3/2$. This equation is equivalent to the Thomas–Fermi differential equation of atomic physics, except for the minus sign that is due to the gravitational attraction of the neutrinos as opposed to the electrostatic repulsion between the electrons. If $M_{B}$ is the mass of the baryonic star internal to the condensation, the natural boundary conditions are $$u(0)=GM_{B}\,,\;\;\;\;\;\;\;u(r_{0})=0\,.$$ Recasting the problem in a dimensionless form, we have $$\label{n6} \frac{d^{2}v}{dx^2}=-\frac{v^{3/2}}{\sqrt{x}}\,,$$ and the boundary conditions $$\label{n7} v(0)=\frac{M_{B}}{M_{\odot}}\,,\;\;\;\;\;\;\;v(x_{0})=0\,,$$ with the positions $$\label{n8} v=\frac{u}{GM_{\odot}}\,,\;\;\;\;\;\;\;\;x=\frac{r}{a}\,,$$ and $$\label{n9} a=\left(\frac{3\pi \hbar^3}{4\sqrt{2}m_{\nu}^4g_{\nu} G^{3/2}M_{\odot}^{1/2}}% \right)^{2/3}= 2.1377\left(\frac{17.2\mbox{keV}}{m_{\nu}c^2}% \right)^{8/3}g_{\nu}^{-2/3} \mbox{lyr}\,.$$ We have to note that for $m_{\nu}=17.2\mbox{keV}/c^2$ the characteristic scale $a$, the corresponding of the Bohr radius for an electron bound to a nucleus, is, for a neutrino halo bound by a baryonic star, of the order of the average distance between stars. It strongly depends on neutrino mass. All the quantities characterizing the condensate can be written in terms of $v$ and $x$ (or $u$ and $r$): $$\label{n10} \Phi(r)=\Phi(r_{0})+\frac{u}{r}\,,\;\; n_{\nu}(r)=\frac{m_{\nu}^3 g_{\nu}}{6\pi^2\hbar^3} \left(\frac{2u}{r}% \right)^{3/2}\,,\;\; P_{\nu}(r)=\left(\frac{6}{g_{\nu}}\right)^{2/3} \frac{\pi^{4/3}\hbar^{2}}{% 5m_{\nu}}n_{\nu}(r)^{5/3}\,,$$ which are, respectively, the gravitational potential, the number density, and the degeneracy pressure. As shown in [@viollier], the general solution of (\[n5\]), or equivalently (\[n6\]), has scaling properties and it is able to reproduce the observations. In particular, it could well fit the observations toward the center of our Galaxy which estimate, considering the proper motion ($\leq 20$ km sec$^{-1}$) of the source Sgr A$^*$, a massive object of $M=(2.6\pm 0.7)\times 10^{6}M_{\odot}$ which dominates the gravitational potential in the inner ($\leq 0.5$pc) region of the bulge [@sagitta]. In summary a degenerate neutrino star of mass $M=2.6\times 10^{6}M_{\odot}$, consisting of neutrino with mass $% m\geq 12.0$ keV$/c^{2}$ for $g_{\nu}=4$, or $m\geq 14.3$ keV$/c^{2}$ for $% g_{\nu}=2$, does not contradict the observations. Considering a standard accretion disk, the data are in agreement with the model if Sgr A$^*$ is a neutrino star with radius $R=30.3$ ld ($\sim 10^5$ Schwarzschild radii) and mass $M=2.6\times 10^{6}M_{\odot}$ with a luminosity $L\sim 10^{37}$erg sec$% ^{-1}$. Similar results hold also for the dark object ($M\sim 3\times 10^{9}M_{\odot} $) inside the center of M87. Now the problem is: How much is the model consistent? Could it be improved at boundaries? Actually, due to the Thomas–Fermi theory, the model fails at the origin and, in any case, we have to consider the effect of the surrounding baryonic matter which, in some sense, have to give stability to the neutrino condensate. In fact, an exact solution of Eq.(\[n5\]) or Eq.(\[n6\]) is $u(r)\sim r^{-3}$ from which $\Phi(r)\sim r^{-4}$ which is clearly unbounded from below. Let us now assume a thermodynamical phase where a constant neutrino number density can be taken into consideration. This is quite natural for a Fermi gas at temperature $T=0$. The Poisson equation is $$\label{n13} \triangle\Phi=-4\pi Gm_{\nu}n_{\nu}=\mbox{cost}\,.$$ which, for spherical symmetry, can be recast in a Lané–Emden form $$\label{n14} \frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{d\Phi}{dr}\right)=\mbox{cost}\,,$$ with polytropic index $n=0$. The solution of such an equation is $\Phi(r)\sim r^{2},$ which is clearly bounded. For $T\neq 0$ the above theory holds so that we get a solution of the form $\Phi(r)\sim r^{-4}$. Matching the two results it is possible to confine the neutrino condensate. On the other hand, a similar result is recovered using the Newton Theorem for a spherically symmetric distribution of matter of radius $R$ [@binney]. In that case, the potential goes quadratically inside the sphere while it goes as $\Phi(r)\sim r^{-1}$ matching on the boundary. In our case, the situation is similar assuming the matching with a steeper potential. Assuming the existence of such a neutrino condensate in the center of Galaxy, it could act as a spherical lens for the stars behind so that their apparent velocities will be larger than in reality. Comparing this effects with the proper motion of the stars of the cluster near Sgr A$^*$, exact determinations of the physical parameters of the neutrino ball could be possible. In this case, gravitational lensing, always used to investigate baryonic objects, could result useful in order to detect a nonbaryonic compact object. Furthermore, since the astrophysical features of the object in Sgr A$^*$ are quite well known [@genzel96], accurate observations by lensing could contribute to the exact determination of particle constituents which could be, for example, neutrinos or gravitinos. Besides, microlensing by cold dark matter particles and noncompact objects has been widely considered in literature [@gurevich], being gravitational lensing independent of the nature and the physical state of deflecting mass. In fact, any gravitationally condensed structure can act, in principle, as a gravitational lens. Our heavy neutrino ball, being massive, extended and transparent, can be actually considered as a magnifying glass for stars moving behind it. If an observer is on Earth and he is looking at the center of our Galaxy (which is at a distance of 8.5 $Kpc$), he should appreciate a difference in the motion of stars since lensed stars and non-lensed stars should have different projected velocity distributions. In other words, depending on the line of sight (toward the ball or outside the ball) it should be possible to correct or not the projected velocities by a gravitational lensing contribution and try to explain the bimodal distribution actually observed [@ghez],[@genzel96]. Let us discuss the physical reasons why a heavy neutrino ball can be treated as a thick lens. For a static gravitational field, the refraction index is connected to the Newtonian potential by the equation $$\label{pot} n({\bf r})=1-2\frac{\Phi ({\bf r})}{c^{2}}\,$$ easily derived by $$g_{00}\simeq 1+2\frac{\Phi ({\bf r})}{c^{2}}\;;\;\;\;\;\;g_{ik}\simeq -\delta _{ik}\left( 1-2\frac{\Phi ({\bf r})}{c^{2}}\right) \;;$$ assuming the weak field $\Phi /c^{2}<<1$ and the slow motion approximation $% \left| v\right| <<c$ [@straumann]. In this situation, almost all the usual geometric optics works. Our neutrino ball gives rise to a static gravitational field, it is an extended object and, due to neutrinos, it can be reasonably approximated by a “transparent” medium. In this case, it is essentially a thick spherical lens which can be replaced by two thin lenses at a distance “$d$” [@fleury]. It is easy to show, by elementary optics arguments, that this double dioptric system can be described by the equation $$\frac{1}{f}=\frac{1}{f_{1}}+\frac{1}{f_{2}}-\frac{d}{f_{1}f_{2}}\,,$$ where $f_{1,2}$ are focal lengths. The relations with the gravitational field and the size of the neutrino ball are given by $$\frac{1}{f_{i}}=(n-1)\frac{1}{r_{i}}\,,\;\;\;\;\;\;\;\;\;i=1,2$$ and $r_{i}=R\simeq d/2$, where $n\left( r\right) $ is the refraction index induced by the Newtonian potential $R$ is the neutrino star radius. Since we are assuming a spherically symmetric distribution of matter inside a radius $R$, the Newton theorem holds [@binney] so that $$\Phi(r)=\frac{1}{2}\omega ^{2}r^{2}\,,$$ with $\omega ^{2}=4\pi G\rho /c^{2}$. The focal length is then given by [@waveguide] $$L_{foc}=\sqrt{\frac{\pi c^{2}}{4G\rho }}\,,$$ which has the same order of magnitude of $f$ and $f_{1,2}$. If $M=2.6\times 10^6M_{\odot }$ and $R\simeq 30.3\, ld$, $L_{f_{oc}}\simeq 70ly$ and we have to expect lensing effects approximately in this range for the stars behind the ball. Depending on the radius, Eq.(\[pot\]) gives the refraction index so that the model is completely determined. Further considerations give also the range of validity of paraxial approximation. In fact, given a ray of light entering in the neutrino ball, it is possible to calculate, the angle of incidence $\alpha $ of the ray on the back surface, the angle of deflection $\delta $ and the entering angle $\eta $, (with respect to the normal) which produces the minimal deflection. See the Fig.1. By knowing these parameters, we could say if an intervening star undergoes this magnifying glass effect. Snell’s law gives $n\sin \alpha =\sin \eta$ so that the relation $$\sin \alpha =\frac{1}{n}\sin \eta <\frac{1}{n}\,,$$ holds. The angle of incidence $\alpha $ has to be smaller than the critical angle ${\displaystyle \arcsin \frac{1}{n}}$ so that the incident light is reflected partially at the back spherical surface. The net effect is that we should lose a part of the luminosity of the star population behind the ball. As $\alpha =(\eta -\alpha )+\chi$, we have $$\delta =\pi -2\chi =\pi -4\alpha +2\eta ;$$ which is the deflection angle. The minimal deflection is deduced in a straightforward way. We require $$\frac{d\delta }{d\eta }=-4\frac{d\alpha }{d\eta }+2=0,$$ which is ${\displaystyle \frac{d\alpha }{d\eta }=\frac{1}{2}}$. Being ${\displaystyle \alpha =\arcsin (\frac{1}{n}\sin \eta)}$, immediately we get ${\displaystyle \frac{d\alpha }{d\eta } =\frac{\cos \eta }{n\cos \alpha }}$ and then $$1-\frac{1}{n}\sin ^{2}\eta =\frac{4}{n^{2}}\cos ^{2}\eta\,.$$ Finally $$\label{glass} \cos ^{2}\eta =\frac{n^{2}-1}{3},$$ which relates the entering incidence angle with the refraction index. Summing up all these information, we should say if a given star undergoes or not a significant lensing effect behind the neutrino ball. Let us now take into account the projected positions $(x,y)$ in the sky of the observed early- and late-type stars as reported in [@genzel96]. The gravitational refraction index as given by Eq.(\[pot\]) for a supermassive neutrino of mass $M=2.6\times 10^{6}M_{\odot }$ and radius $R=30.3\,ld$ is $n=1+5\times 10^{-6}$. From the above considerations, the angles $\eta $ and $\alpha $ are given by $$\eta =\tan ^{-1}\frac{y}{x}\,,\quad \sin \alpha =\frac{\sin \eta }{n}$$ and then we can calculate the deflection angle $\delta$ considering the refraction index given by the magnifying glass model (Eq.(\[glass\])) or given by the gravitational potential (\[pot\]). The mean values, taking into account the Genzel [*et al*]{} data [@genzel96] are $$\stackrel{-}{\delta }_{opt}=3.25\pm 0.89\;\mbox{arcsec}\,,\quad \stackrel{-}{\delta }_{grav}=3.40\pm 1.91\;\mbox{arcsec},$$ for the sample of late–type stars, and $$\stackrel{-}{\delta }_{opt}=3.48\pm 0.58\;\mbox{arcsec}\,,\quad \stackrel{-}{\delta }_{grav}=3.49\pm 1.71\;\mbox{arcsec},$$ for the sample of early-type stars. In both cases the “optical" and “gravitational" results are in a good agreement. In other words, a magnifying glass model seems to reproduce the spatial distribution of stars behind Sgr A$^*$. Fig.2 shows such a spatial distribution for the late and early-type samples. The histograms for the deflection angle $\delta $ are given in Fig.3. It is clear that a correlation exists between the deflection angle evaluated in both approaches. From gravitational lensing point of view, there is no relevant difference between early and late type samples (see Fig.4). A more striking result concerns kinematics. The lensing effect of a possible neutrino ball at the center of our Galaxy will magnify stars up to 70$lyrs$ behind, as we showed above, and their apparent velocities will be larger than in reality, as well (the effect is similar to that of looking at red fishes moving in a spherical water–jug of glass!). Taking into account the projected velocities, they will be corrected by lensing effects, $$v_{\perp }^{observed}(t)=v_{\perp }^{not\ lensed}(t)+R_{E}/T,$$ where $$R_{E}=\vartheta _{E}D_{ol}=\sqrt{\frac{4GM}{c^{2}}\frac{D_{ls}}{D_{ol}D_{os}}}, \quad T=\frac{R_{B}}{v_{\perp }^{not\ lensed}}$$ are respectively the Einstein radius and the time of crossing. $D_{ls}$ is the distance between lens and source; $D_{ol}$ the distance between observer and lens; $D_{os}$ the distance between observer and source; $R_{B}$ is the radius of the neutrinos ball and $M$ its mass. We can assume $$D_{ls}\approx D_{os}\approx D_{os}/2\approx L_{foc}=f\,.$$ We get $R_{E}\propto f^{1/2}$ and then $$v_{\perp }^{observed}(t)=v_{\perp }^{not\ lensed}(t)\left( 1+\frac{\sqrt{% \frac{2GMf}{c^{2}}}}{R_{B}}\right) \simeq \left( 1+1.9\right) \cdot v_{\perp }^{not\ lensed}(t)$$ Fig.5 shows $v_{\perp }^{observed}(t)$ as measured by Genzel [*et al.*]{} [@genzel96], while the result of our evaluation is in Fig.6. It is clear that if we take into account a lensing effect, the velocity dispersion (the sigma into the plots) of early–type stars becomes smaller and comparable with the late-type ones, considering that the standard deviation measured and reported in [@genzel96] is about $30\,Km/s$. In conclusion, using such a magnifying neutrino ball, late and early–type stars could not have different spatial and kinematical distributions, the only difference should be if they are “behind” or “nearby” the neutrino ball from the observer point of view. In fact, as we have shown, early type–stars undergo a major lensing effect. If these considerations work, the central compact object could be investigated in such an alternative way and accurate kinematical and photometric data could give final answers on its size and nature. [**Acknowledgments**]{}The authors are grateful to G. Scarpetta and R.D. Viollier for fruitful suggestions and discussions which allow to improve the paper. [99]{} J. 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[^1]: E-mail:capozziello@vaxsa.csied.unisa.it [^2]: E-mail:geriov@vaxsa.csied.unisa.it [^3]: A good estimated value for the mass of $\tau$-neutrino is $$m_{{\nu_{\tau}}}=m_{{\nu_{\mu}}}\left(\frac{m_{t}}{m_{c}}\right)^{2} \simeq 14.4 \mbox{keV}/c^{2}\,,$$ which well falls into the above range.
--- abstract: 'A switchable hyperbolic material (SHM) is investigated, with which one can turn on or off the hyperbolic dispersion of the material via magnetic control. The SHM has simple structure, with a one-dimensional periodic stacking of dielectric layer and gyromagnetic layer. The hyperbolic dispersion of SHM is due to the negative effective permeability of gyromagnetic layers, and it can be transformed into a regular circular dispersion when the d.c. magnetic field is switched off. This switchable dispersion transition is reversible, which may have great potential applications in many fields.' author: - Wei Li - Zheng Liu - Xiaogang Zhang - Xunya Jiang title: Switchable Hyperbolic Metamaterials With Magnetic Control --- Metamaterials (MMs)[@PendryPRL2000; @PendrySmithScience2006; @LeonhardtScience2006; @SoukoulisWegenerScience2010; @ChenShengNM2010] have shown many new striking physics that can be used to control the electromagnetic properties of materials and go beyond the limit that is attainable with naturally existing substances. Unlike photonic crystals, MMs can be viewed as homogeneous media described by effective permittivity $\varepsilon_e$ and effective magnetic permeability $\mu_e$, since their period is much smaller than the working wavelength. For anisotropic MMs, their $\varepsilon_e$ and/or $\mu_e$ become tensors. Recently, an important type of the anisotropic MMs, so called “hyperbolic material", in which one of the diagonal effective permittivity tensor is negative and results in an anomalous hyperbolic dispersion, has attracted growing attention[@JacobOE06; @SalandrinoPRB06; @YaoScience08; @LiuScience07; @ZhangPRL11; @YangOL10; @LiAPL08; @LiuAPL10]. Many interesting phenomena which are difficult to be realized by natural materials, such as hyperlensing[@JacobOE06; @SalandrinoPRB06; @YaoScience08; @LiuScience07; @ZhangPRL11], all-angle nonreflection[@LiAPL08; @YangOL10], and all-direction pulse compression[@LiuAPL10], etc., can be achieved by these hyperbolic materials. The realization of a hyperbolic material has recently advanced by adopting the MMs with a periodic metal-dielectric layered structure[@SalandrinoPRB06], or periodic metallic lines[@YaoScience08]. For the performance of the hyperbolic material realized by such structure, one of the most important challenges is the high dissipative losses of MMs, due to the metallic nature of their constituent meta-molecules. To overcome the losses of MMs, one can use gain medium for loss-compensation[@WuestnerPRL10]. However, this solution need more complex structures, which may bring new challenges in realization. Besides, this loss-compensation is frequency-sensitive, which is a drawback for broadband capacity. On the other hand, the ability to tune and switch the properties of MMs has greatly broaden the applications of MMs in many fields[@ZheludevScience10], which has been widely studied both experimentally and theoretically in recent years. Very recently, a “big flash" event (a large number of photons emit instantaneously, which has some similarities with the cosmological “big bang"), is predicated to occur during the metric signature transition in hyperbolic material[@SmolyaninovPRL10]. The metric signature transition in hyperbolic material is a transition, in which the dispersion relation of the material is changed between hyperbolic and elliptic[@SmolyaninovPRL10]. Therefore, it is significant to design a switchable hyperbolic material (SHM), with which one can switch on or off the hyperbolic dispersion. Reviewing the existing efforts, we think the SHM should include at least three characteristics: (I) broadband working frequency; (II) low loss; and (III) switchable hyperbolic dispersion. In this Letter, our design will be presented to serve this purpose. ![\[\] The model of our SHM, which is a one-dimensional periodic stacking of dielectric layer and gyromagnetic layer, with the thickness $d_1=5.5\mu m$ and $d_2=4.5\mu m$, the relative permittivity $\varepsilon_1=2.25$ and $\varepsilon_2=15$, the relative permeability $\mu_1=1$, $\mu_2=\stackrel{\leftrightarrow}{\mu}_2$, respectively. An external uniform d.c. magnetic field is applied along $z$ direction.](fig1.eps){width="0.6\columnwidth"} The model of our SHM is schematically shown in Fig.1, in which the SHM is a periodic stacking of dielectric layer and gyromagnetic layer such as yttrium-iron-garnet (YIG) with an external d.c. magnetic field $H_0$ along $z$ direction. In our model, the thickness of dielectric layer $d_1=5.5\mu m$ and gyromagnetic layers $d_2=4.5\mu m$ are both much smaller than the working wavelengths (about $10^{-2}m$ in this work). Therefore, our material can be viewed as the effective anisotropic MMs, which can be described by $\varepsilon_e$ tensor for TM mode or $\mu_e$ tensor for TE mode. In this work, *only* TE mode is considered[@ZhangSup10]. An obvious difference between our structure and the traditional ones[@SalandrinoPRB06; @LiAPL08] is that we use the low-loss gyromagnetic material YIG[@DasPattonAM2009; @lossOfYIG] instead of the high-loss metallic material.That being said, the dissipative losses of our SHM is naturally very low, which can be neglected in this work because both of the dielectric material and the YIG material in the SHM are nearly lossless. Unlike the traditional hyperbolic material, the dispersion of our SHM is due to the $\mu_e$ tensor, which can be controlled by the external d.c. magnetic field. Therefore, with magnetic control, one can switch (or tune) the hyperbolic dispersion of the SHM reversibly. According to real materials, the dielectric constants of dielectric layers and gyromagnetic layers in our model for the SHM are $\varepsilon_1$=$2.25$, and $\varepsilon_2$=$15$, respectively. With respect to the relative permeability, the dielectric layers is nonmagnetic with $\mu_1$=1, and the gyromagnetic layers has a gyromagnetic form[@LiuPRB08]: $$\label{permeability} \stackrel{\leftrightarrow}{\mu}_2=\left[\begin{array}{ccc} \mu_a&\pm j\mu_b&0\\ \mp j\mu_b&\mu_a&0\\ 0&0&1 \end{array} \right]$$ where $\mu _a=1+\frac{\omega_m(\omega _0-i\alpha\omega)}{(\omega _0-i\alpha\omega)^2-\omega^2}$, $\mu _b=\frac{\omega \omega _m}{(\omega _0-i\alpha\omega)^2-\omega ^2}$, $\pm$ and $\mp$ describe the direction of the external d.c. magnetic field, respectively. $\omega_0=\gamma H_0$ is the resonance frequency with $\gamma$ as the gyromagnetic ratio. $\alpha$ is the damping coefficient. The characteristic circular frequency is $\omega_m=5.32GHz$, corresponding to a wave vector $k_m=\omega_m/c$. $c$ is the speed of light in vacuum. When $H_0$ is 0.16T, the tensor element in YIG [@WangPRL08] is at 4.28 GHz with $\mu_a=14$ and $\mu_b=12.4$. When $H_0=0$, we have $\mu_a=1$ and $\mu_b=0$. Using the transfer-matrix method and imposing the Bloch theorem, the dispersion of our SHM associated with an effective permeability tensor $\mu_e$=$diag(\mu_x,\mu_y,\mu_z)$ can be obtained as $$\label{dispersion} \frac{k_x^2}{\mu_{y}}+\frac{k_y^2}{\mu_{x}}=\overline{\varepsilon}\frac{\omega^2}{c^2},$$ where $k_x$ and $k_y$ is the Bloch’s wave vector along $x$ axis and the wave vector along $y$ axis, respectively, with $\overline{\varepsilon}=f_1\varepsilon_1+f_2\varepsilon_2$, $\mu_{y}=\mu_{z}=f_1\mu_1+f_2\mu_2$, and $$\label{mux} \mu_{x}=\left[\frac{f_1}{\mu_1}+\frac{f_2}{\mu_{2e}}-\frac{f_1f_2}{(\mu_1/\mu_{2e})(\mu_b^2/\mu_a^2)}\right]^{-1}.$$ Here $f_1=d_1/(d_1+d_2)=0.55$, $f_2=d_2/(d_1+d_2)=0.45$, and $\mu_{2e}=(\mu_a^2-\mu_b^2)/\mu_a$ is the effective permeability of the gyromagnetic layers. From Eq.(\[dispersion\]), one can find that the dispersion of the SHM becomes hyperbolic when $\mu_x\mu_y<0$. The sign of $\mu_x\mu_y$ at different frequency with different external d.c. magnetic field is shown in Fig.2(a), where the white and the red regions indicate positive and negative $\mu_x\mu_y$, corresponding to the elliptic dispersion and the hyperbolic dispersion, respectively. From this figure, we can see the hyperbolic dispersion has two frequency ranges (red regions) for each $H_0$. The width of each range is about $0.28\omega_m$, which is nearly independent on $H_0$. Between two red regions, there is a very narrow white region with the width of frequency range about $0.03\omega_m$ for each $H_0$, corresponding to the elliptic dispersion. Therefore, in the SHM, the total width of frequency ranges that correspond to hyperbolic dispersion is about $0.56\omega_m$ ($\simeq 3.0GHz)$ for each $H_0$, which is quite a considerable broad range for microwave signal processing. As an typical example, when $\omega=2.02\omega_m$ and $H_0=0.3T$, we have $\mu_x=2.3$ and $\mu_y=-2.3$, and the corresponding equifrequency curve is plotted in Fig.2(b) (the red one). Obviously, the dispersion in this case is hyperbolic. Why the dispersion of our SHM can exhibit hyperbolic with some $H_0$? Physically, it can be understood as follows. First, the effective permeability of the gyromagnetic medium $\mu_{2e}$, as a function of $\omega$ and $H_0$, can be negative at some $H_0$ [@ZhangSup10; @ZhangJiangAPL2012]. Second, with the negative $\mu_{2e}$ of gyromagnetic layers in our material, its contribution on the hyperbolic dispersion is similar with that of negative permittivity of metallic layers in the traditional hyperbolic material. As a result, the expression of $\mu_x$ for the SHM has many similarities with its counterpart for the traditional hyperbolic materials[@SalandrinoPRB06; @LiAPL08], except the last term in brackets of Eq.(\[mux\]). This term is due to gyromagnetism, which provides a modification for $\mu_x$ and it disappears in the nongyromagnetic material. ![image](fig3.eps){width="1.0\columnwidth"} Our material is tunable. With magnetic control, the dispersion of our SHM can be transformed, due to the change of $\mu_e$. As shown in Fig.2(a), for a given frequency, the dispersion of the SHM is able to be changed between hyperbolicity and ellipticity with the control of $H_0$. To show it more clearly, the dispersion of the SHM at frequency $\omega=2.02\omega_m$ with $H_0=0$, $H_0=0.2T$, $H_0=0.3T$ and $H_0=0.4T$ are calculated, and the corresponding equifrequency curves are respectively plotted in Fig.2(b). In this figure, we can find the dispersion of the SHM can be hyperbolic or elliptic, by changing $H_0$. When $H_0$ is switched off, i.e., $H_0=0$, the dispersion turns into a regular circular one, because our material becomes nonmagnetic in this case. Our SHM can be used to realize a reversibly tunable metric signature transition by switching $H_0$ on and off. This tunable transition may have a great many potential applications in various fields, i.g., it can be used to control the light on the interface of the SHM. In our previous work, we have demonstrated that the all-angle zero-reflection [@LiAPL08] and all-direction pulse compression[@LiuAPL10] can occur on the interface between the anisotropic medium with hyperbolic dispersion and the isotropic dielectric. Here we would like to show that these striking phenomena can be controlled by $H_0$ with our SHM. For this, we show the control of electromagnetic wave on the interface between the SHM and air. As shown in Fig.3(a), the surface of the SHM is terminated with a slanted angle $\theta$, and the electromagnetic wave is incident from the SHM to the air. For the sake of simplicity, we choose a new Cartesian coordinate system $x'$-$y'$ with the same origin, and the angle between $x'$ and $x$ axis is ($\pi/2-\theta$). After some algebra, the dispersion of the SHM in the new coordinate can be obtained as: $$\label{dispersion2} \frac{k_{x'}^2\mu_{x'x'}+k_{x'}k_{y'}(\mu_{x'y'}+\mu_{y'x'})+k_{y'}^2\mu_{y'y'}}{\mu_{x'x'}\mu_{y'y'}-\mu_{x'y'}\mu_{y'x'}}=\overline{\varepsilon}\frac{\omega^2}{c^2},$$ where $$\label{} \begin{split} \left[\begin{array}{ccc} \mu_{x'x'},&\mu_{x'y'}\\ \mu_{y'x'},&\mu_{y'y'} \end{array} \right]=\left[\begin{array}{ccc} \mu_xs_{\theta}^2+\mu_yc_{\theta}^2,&(\mu_x-\mu_y)s_{\theta}c_{\theta} \\ (\mu_x-\mu_y)s_{\theta}c_{\theta},&\mu_xc_{\theta}^2+\mu_ys_{\theta}^2 \end{array} \right] \end{split}$$ with $c_{\theta}=\cos\theta$ and $s_{\theta}=\sin\theta$. The dispersion of our SHM at frequency $\omega=2.02\omega_m$ with $H_0=0$, $H_0=0.3T$ can be obtained from Eq.(\[dispersion2\]) and the corresponding equifrequency curve are shown in Fig.3(b) and Fig.3(c), respectively. In this work, the slanted angle is chosen as $\theta=\pi/4$. For the slanted angle $\theta$, when a beam from our material with hyperbolic dispersion is incident to the air, there is a critical condition $\theta=\theta_c=\arctan(|\mu_y/\mu_x|)$ which means the interface ($x'$ axis) perpendicular to one of the hyperbola-dispersion asymptotes. When $H_0=0.3T$, the critical condition is $\theta=\theta_c=\pi/4$. At this condition, the beam has zero reflection and zero transmission for all incident angles, which can be seen from the equifrequency curve analysis of reflection and transmission on the interface shown in Fig.3(b). From this figure, we can find that for a incident beam from the SHM to the air with any angle, the transmitted wave is always evanescent wave, so the transmitted energy flux is zero, i.e., zero transmission occurs. Meanwhile, the zero reflection also occurs in this critical condition, for the reflected beam is absent, as shown in Fig.3(b). With this zero-reflection and zero-transmission effect, the incident pulses from the SHM with hyperbolic dispersion can be totally compressed or stopped on the interface. On the other hand, when $H_0$ is switched off, our material become regular circular dispersion, as shown in Fig.3(c). In this case, the effects of zero reflection, zero transmission, and pulse compression on the interface are also “switched off". In Fig.3(c), the “zero reflection" becomes “total reflection" with the same incident beam. Therefore, one can control the electromagnetic wave on the interface of our material, just by switching $H_0$ on or off. In reality, here we emphasis that, control of the electromagnetic wave on the interface of our material is still feasible. Different from the ideal model as discussed above in which the ideal hyperbolic dispersion is still valid when $k_{x'(y')}\rightarrow\infty$, the strict “zero reflection" and “zero transmission" as well as the completely compressed or stopped pulse on the interface are impossible in reality, since the hyperbolic dispersion described by Eq.(\[dispersion2\]) is available only when $k_{x'(y')}<k_{max}=2\pi/d\simeq 2.8\times 10^{3} k_m$. However, the “near zero reflection", “near zero transmission", the strongly compressed and slow pulse[@suppl1] could be achieved easily on the interface of our SHM, because $k_{max}$ in our SHM is a considerable large value comparing with $k_m$. To show more about it, a numerical experiment based on the finite-differential-time-domain (FDTD)[@eastfdtd] method is presented, with the calculated model shown in Fig.3(a). An incident Gaussian pulse at the center frequency $\omega=2.02\omega_m$ from the SHM with $H_0=0.3T$ is just arriving at the interface, with the $E_z$ field distribution shown in Fig.4(a). When $H_0$ is kept turned on, as shown in Fig.4(b), the pulse is strongly compressed, and the compressed pulse propagates along the interface very slowly. Furthermore, the “near zero reflection" and “near zero transmission" can also be observed in this dynamical process. On the contrast, when $H_0$ is switched off before the pulse arriving at the interface, the pulse is almost reflected, as shown in Fig.4(c). From this figure, we can see that “near total reflection" occurs on the interface, rather than “near zero reflection", since the dispersion of the SHM in the case of $H_0=0$ is changed into a circular one with the radium larger than that of air, which can also be seen in Fig.3(c). ![\[\](a) A Gaussian pulse at the center frequency $\omega=2.02\omega_m$ from our material is incident to the air, with $H_0=0.3T$. (b) When $H_0$ is kept turned on, the Gaussian pulse is strongly compressed on the interface, almost without any reflection and transmission. (c) When $H_0$ is switched off before the Gaussian pulse arriving at the interface, the pulse is nearly totally reflected on the interface. The white arrows indicate the direction of energy flow of the pulse. The dashed black line indicates the interface.](fig4.eps){width="0.7\columnwidth"} Beyond the example presented in this Letter, there can be much more applications of the SHM, for instance, it may be used for the observation on “big flash"[@SmolyaninovPRL10], since the SHM is very easy to realize the metric signature transition. The related study of “big flash" in the SHM will be published in our another paper elsewhere. In conclusion, in this Letter we have presented our design for the SHM, in which one can turn on or off the hyperbolic dispersion via magnetic control. The hyperbolic dispersion of the SHM is due to the negative effective permeability of the gyromagnetic material, and it can be transformed into a regular circular one when the external d.c. magnetic field is switched off. This tunable dispersion transition is reversible, which may have great potential applications in many fields, such as the control of electromagnetic wave on the interface of our SHM. In addition, the SHM may be used to observe “big flash” since the SHM is very easy to realize the metric signature transition. *Acknowledgement*. This work was supported by the NSFC (Grant Nos. 11004212, 11174309, and 60938004), and the STCSM (Grant Nos. 11ZR1443800 and 11JC1414500). [s1]{} J. Pendry, Phys. Rev. Lett. 85, 3966 (2000). J. Pendry, D. Schurig, and D. Smith, Science 312, 1780 (2006). U. Leonhardt, Science 312, 1777 (2006). C. Soukoulis and M. Wegener, Science 330, 1633 (2010). H. Chen, C. Chan and P. Sheng, Nat. Mat. 9, 387 (2010). Z. Jacob, L. Alekseyev, and E. Narimanov, Opt. Express 14, 8427 (2006). J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, Science 321, 930 (2008). Z.W. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007). S. Zhang, Y. Xiong, G. Bartal, X. Yin, and X. Zhang, Phys. Rev. Lett. 106, 243901, (2011). A. Salandrino and N. Engheta, Phys. Rev. B 74, 075103 (2006). X. Li, Z. Liang, X. Liu, X. Jiang, and J. Zi, Appl. Phys. Lett. 93, 171111 (2008). J. Yang, X. Hu, X. Li, Z. Liu, X. Jiang, and J. Zi, Opt. Lett. 35, 16 (2010). Z. Liu, Z. Liang, X. Jiang, X. Hu, X. Li, and J. Zi, Appl. Phys. Lett. 96, 113507 (2010). S. Wuestner, A. Pusch, K. Tsakmakidis, J. Hamm, and O. Hess, Phys. Rev. Lett. 105, 127401 (2010). N. Zheludev, Science 328, 582 (2010). I. Smolyaninov and E. Narimanov, Phys. Rev. Lett. 105, 067402 (2010). See supplementary material at \[URL will be inserted by AIP\] for the dispersion relation and the effective wave vector in an infinite gyromagnetic medium. J. Das, Y. Song, N. Mo, P. Krivosik and C. Patton, Adv. Mater. 21, 2045 (2009). The imaginary part of the relative permittivity and the relative permeability of a typical YIG material are both as low as $10^{-3}\sim 10^{-4}$ in microwave region, which are at least $10^{8}$ times smaller than that of a metal such as gold. S. Liu, J. Du, Z. Lin, R. X. Wu, and S. T. Chui, Phys. Rev. B 78, 155101 (2008). Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljačić, Phys. Rev. Lett. 100, 013905 (2008). X. Zhang, W. Li and X. Jiang, Appl. Phys. Lett. 100, 041108 (2012). The group velocity of the compressed pulse on the interface of our SHM with hyperbolic dispersion is very slow, with the slow limit $v_{gx'}$ (perpendicular to the interface) and $v_{gy'}$ (parallel to the interface) satisfying: $v_{gx'}=\partial \omega/\partial k_{x'} \propto 1/\gamma_s$ and $v_{gy'}=\partial \omega/\partial k_{y'}\propto 1/\gamma_s^2$, where $\gamma_s=k_{max}/k_0=\frac{1}{d_1+d_2}\frac{2\pi}{k_0}$ is the slowing coefficient. See also Ref.[@LiuAPL10]. EastFDTD V3.0, DONGJUN Information Technology Co., Ltd., China.
--- abstract: 'We describe a completely algebraic axiom system for intertwining operators of vertex algebra modules, using algebraic flat connections, thus formulating the concept of a [*tree algebra*]{}. Using the Riemann-Hilbert correspondence, we further prove that a vertex tensor category in the sense of Huang and Lepowsky gives rise to a tree algebra over $\C$. We also show that the chiral WZW model of a simply connected simple compact Lie group gives rise to a tree algebra over $\Q$.' author: - Igor Kriz and Yang Xiu title: 'Tree algebras: An algebraic axiomatization of intertwining vertex operators' --- Introduction ============ The aim of the present paper is to describe a set of purely algebraic axioms designed to capture the structure of genus $0$ (or “tree level”) amplitudes in conformal field theory. To explain this very briefly, in mathematical physics, a chiral quantum conformal field theory consists of a system of state spaces indexed by a set of [*labels*]{} $\Lambda$. A Riemann surface with parametrized boundary components whose boundary components are labelled by elements of $\Lambda$ should specify, roughly, a linear operator from the tensor product of the state spaces corresponding to the inbound boundary components to the tensor product of the state spaces corresponding to the outbound boundary components. In fact, the operator is not specified uniquely. Rather, there should be a finite vector space of such operators, and these vector spaces should form a holomorphic bundle on the moduli space of Riemann surfaces, called a [*modular functor*]{}. Certain axioms should be satisfied, in particular, when cutting a Riemann surface $\Sigma$ along an analytically parametrized simple curve, the space of operators corresponding to $\Sigma$ should be isomorphic, via a prescribed isomorphism, to the direct sum of the vector spaces corresponding to all possible labels on the cutting curve (the spaces should be $0$ for all but finitely many choices of labels on the cutting curve). The isomorphisms should be subject to certain canonical coherence diagrams. The operators on state spaces corresponding to the uncut and cut Riemann surfaces should be related by trace. There should also be a special label called the [*zero label*]{} such that the operator corresponding to annuli approaching the unit circle “converge” to the identity. The main issue with such an approach is inherited from the usual issues of quantum mechanics: in interesting examples, the state spaces are infinite-dimensional, and to make the approach mathematically rigorous, we must select which category of vector spaces we will be working in. Quantum mechanics suggests Hilbert spaces (cf. [@scft]), in which case we are firmly in the realm of analysis. Involving analysis certainly seems necessary if we want to make rigorous the entire structure outlined in the last paragraph. In the 80’s, however, it has been noticed by Borcherds [@bor] and Frenkel, Lepowsky and Meurman [@flm] that the structure present on the state space corresponding to the $0$ label and genus $0$ Riemann surfaces can be axiomatized purely algebraically. In this approach, the state space is not a Hilbert space, but simply a graded vector space over a field of characteristic $0$. Operators corresponding to round domains (the unit disk with a finite set of disks removed, all boundary components parametrized linearly) are encoded in a structure called [*(graded) vertex algebra*]{}. If one wishes to look beyond round domains, one does not consider arbitrary Riemann surfaces of genus $0$, but rather infinitesimal variations of boundary parametrization, the effect of which is described by what is known as the [*conformal element*]{} or [*energy-momentum tensor*]{}. The goal of the present paper is to propose an extension of the vertex algebra axiomatization to [*intertwining operators*]{}, i.e. operators on state spaces involving the entire set of labels $\Lambda$ of a chiral conformal field theory, purely in the language of algebra. In particular, a test of the success of such an approach is that it should be valid over any field of characteristic $0$. (Note: in this paper, we only deal with round domains, we do not consider conformal elements). This advances the program of making conformal field theory a purely algebraic object, which is a well known desideratum. It is related, for example, to the Grothendieck-Teichmüller program [@esquisse; @ideal], and, viewed from a different perspective, the geometric Langlands program (cf. [@beidr]). There is, however, a major difficulty. It is well known that in interesting examples (e.g. parafermions or the chiral WZW models), the intertwining operators corresponding to non-zero labels (even for round domains) fail to be algebraic. Typically, rather, one gets hypergeometric functions (which are transcendental) and their generalizations (cf. [@gh; @hl1]). Considering this, our task may seem impossible. Because of this, axiomatic systems have been developed which combine algebra and analysis, in particular the concept of [*vertex tensor category*]{} by Huang and Lepowsky [@hl]. Yet, the same examples point toward a possible solution: although the correlation functions for non-zero labels are not algebraic, they satisfy [*differential equations*]{} which are algebraic. An example is given by the Knizhnik-Zamolodchikov equations in the case of the WZW model (cf. [@hl1]). In fact, similar equations have been formulated by Y.Z. Huang for a large range of examples in [@h7]. What we do in the present paper is formulate purely algebraic axioms the Huang-Knizhnik-Zamolodchikov equations should satisfy in an algebraic model of chiral conformal field theory in genus $0$. The main obstacle to doing so is the sheer complexity of the structure involved. Even the algebraic axiomatization of a vertex algebra [@bor; @flm] (i.e. the case of the $0$ label) involves the remarkably complicated Jacobi identity. To make things worse, a straightforward generalization of the Jacobi identity to intertwining operators is false. Although Huang [@hjac] has obtained a more complicated generalization of the Jacobi identity for intertwining operators, devicing a workable system of axioms from this approach seems daunting. To get a handle on the structure in the present paper, we make substantial use a simple interpretation of the vertex algebra axioms obtained by Hortsch, Kriz and Pultr [@hkp]: a graded vertex algebra is essentially the same thing as an algebra over a certain co-operad (the “correlation function co-operad”), which can be easily described explicitly. The present paper is organized as follows: We write down the general axioms in Section \[sdef\] below; we call the resulting algebraic structure a [*tree algebra*]{}. In Section \[sreg\], we discuss a version of the Riemann-Hilbert correspondence (cf. [@d]) in the present setting, and prove that a vertex tensor category in the sense of Huang and Lepowsky [@hl] can be realized as a tree algebra over $\C$ “with regular singularities”, which is a completely algebraic object. At the end of Section \[sreg\], we also give an example showing that the chiral WZW model for a simply connected simple compact Lie group can be realized as a tree algebra over $\Q$. The basic definitions {#sdef} ===================== Let $F$ be any field of characteristic $0$. We recall from [@hkp] the ordered configuration space $C(n)=C(z_1,...,z_n)$ of $n$ distinct points $z_1,...,z_n$ in $\A^1$. Denote the coefficient ring of the affine variety $C(n)$ by $\mc(n)$. It is advantageous to consider $\mc(n)=\mc(z_1,...,z_n)$ a $\Z$-graded ring with grading by total degree of a homogeneous rational function. In [@hkp], it is shown that the system $(\mc(n))$ is a graded co-operad: the co-multiplication [$$\label{egeneral+}\begin{array}{l}\mc(z_{11},...,z_{1n_1},...,z_{k1},...,z_{kn_k})_\ell\\ \r\mc(z_1,...,z_k)_{\ell_0}\otimes\mc(t_{11},...,t_{1n_1})_{\ell_1} \otimes ...\otimes \mc(t_{k1},...,t_{kn_k})_{\ell_k}\end{array}$$]{} with $\ell_0+...+\ell_k=\ell$ is given by setting $$z_{ij}=t_{ij}+z_i$$ and expanding [$$\label{egeneral++}(t_{ij}+z_i-t_{i^\prime j^\prime}-z_{i^\prime})^{-1}, \; i\neq i^\prime$$]{} by rewriting [(\[egeneral++\])]{} as $$(t_{ij}+(z_i-z_{i\prime}) -t_{i^\prime j^\prime})^{-1},$$ and expanding in increasing powers of $t_{ij}$ and $t_{i^\prime j^\prime}$. Moreover, it is shown in [@hkp] that vertex algebras in a suitable sense can be characterized as graded algebras over the graded co-operad $\mc$. Define, for $\Z$-graded $F$-vector spaces $A$, $B$, a $\Z$-graded vector space $$(A\widehat{\otimes}B)_n={\begin{array}{c} {\scriptstyle }\\ colim\\ {\scriptstyle k_0}\end{array}}{\begin{array}{c} {\scriptstyle }\\ \prod\\ {\scriptstyle k\geq k_0}\end{array}} A_k\otimes B_{n-k}$$ If $A$, $B$ are (commutative) rings, so is (in a natural way) $A\widehat{\otimes} B$. One checks that the structure map [(\[egeneral+\])]{} takes the form [$$\label{egen+1}\mc(m_1+...+m_n)\r (\mc(m_1)\otimes ...\otimes \mc(m_n)) \widehat{\otimes} \mc(n),$$]{} and is a homomorphism of graded rings. Note that $\mc(0)=F$, so selecting $i$ among $n$ coordinates gives us a “co-augmentation” [$$\label{eaug}\mc(n-i)\r \mc(n).$$]{} The purpose of this paper is to extend this definition to a fully algebraic treatment of tree-level amplitudes of chiral conformal field theories. This makes it possible to define, at least on the level of chiral tree-level amplitudes, “rational conformal field theory” as “conformal field theory over $\Q$”. This is possible despite of the fact that these amplitudes are usually transcendental: typical examples are hypergeometric functions [@gh]. The reason that an algebraic treatment is possible is because we can describe [*algebraic flat connections with regular singularities*]{} whose solutions are the desired amplitudes, and characterize the precise algebraic structure these connections are required to obey. To this end, we must first develop the theory of such connections: the reason this is non-trivial is that we need a suitable treatment of the grading, which in particular should capture a concept of “total degree” of the solution, which is to be an element of the field $F$. We consider the ring of Kähler differentials $\Omega^{1}_{\mc(n)/F}$, and equip it with a $\Z$-grading so that the universal differentiation $$d:\mc(n)\r \Omega^{1}_{\mc(n)/F}$$ is a graded homomorphism of $F$-modules (of degree $0$). Then $\Omega^{1}_{\mc(n)/F}$ is a free $\mc(n)$-module on the basis $dz_1,...,dz_n$. Let $M$ be a projective graded $\mc(n)$-module. Then a [*homogeneous connection*]{} is a map of graded $F$-modules $$\nabla:M\r M\otimes_{\mc(n)}\Omega^{1}_{\mc(n)/F}$$ which satisfies, for $f\in \mc(n)$, $m\in M$, $$\nabla(fm)=mdf+f\nabla(m).$$ We say that the connection $\nabla$ is [*flat*]{} when it satisfies the Maurer-Cartan equation [$$\label{emc*}(1\otimes d)\nabla=-(\nabla\otimes 1)\nabla$$]{} where both sides are considered as maps into $M\otimes_{\mc(n)}\Omega^{2}_{\mc(n)/F}$. Next, our aim is to define the [*degree*]{} of a flat homogeneous connection. We first define the degree of the [*difference*]{} of two flat homogeneous connections. In effect, such difference $$E=\nabla_1-\nabla_2,$$ written in matrix form as [$$\label{emc1}E=E_1dz_1+...+E_ndz_n,\; E_i\in Hom_{\mc(n)}(M,M)$$]{} is said to [*have a degree*]{} when [$$\label{emc2}E_1z_1+...+E_nz_n = k(z_1,...,z_n)Id_{M}$$]{} for some function $$k(z_1,...,z_n)\in \mc(n).$$ Clearly, this property is invariant under linear change of the variables $z_1,...,z_n$. \[lmc1\] When a difference $E$ of two flat connections has a degree, then $deg(E)=k(z_1,...,z_n)$ is a constant function of $z_1,...,z_n$. In coordinates, [(\[emc\*\])]{} reads [$$\label{emc3}\frac{\partial{E_i}}{\partial z_j}-\frac{\partial{E_j}}{\partial z_i} =-\frac{1}{2}[E_i,E_j].$$]{} Now compute [$$\label{emc4}\begin{array}{l} {\frac{\displaystyle \partial(E_1z_1+...+E_nz_n)}{\displaystyle \partial z_1}}=E_1+ z_1{\frac{\displaystyle \partial E_1}{\displaystyle \partial z_1}} +...+ z_n {\frac{\displaystyle \partial E_n}{\displaystyle \partial z_1}}=\\[3ex] =E_1+z_1{\frac{\displaystyle \partial E_1}{\displaystyle \partial z_1}}+...+z_n{\frac{\displaystyle \partial E_1}{\displaystyle \partial z_n}}- {\frac{\displaystyle 1}{\displaystyle 2}}[E_1,z_1E_1+...+z_nE_n]. \end{array}$$]{} The Lie bracket [(\[emc4\])]{} vanishes because of our assumption [(\[emc2\])]{}. The first term vanishes by the following Lemma and the fact that $E$ is a homogeneous connection. Thus, [(\[emc4\])]{} vanishes. An analogous argument holds with $z_1$ replaced by any $z_i$, which proves the statement of the Lemma. \[lmc2\] A function $f$ of $n$ variables $z_1,...,z_n$ is homogeneous of degree $k$ if and only if [$$\label{ehomog}z_1\frac{\partial f}{z_1}+...+z_n\frac{\partial f}{\partial z_n}= kf.$$]{} Suppose $f$ is homogeneous of degree $k$. By the chain rule, $$\begin{array}{l} ka^{k-1}f(z_1,...,z_n)={\frac{\displaystyle da^k}{\displaystyle da}}f(z_1,...,z_n)={\frac{\displaystyle df(az_1,...,az_n)}{\displaystyle da}}=\\[3ex] =\left.{\frac{\displaystyle \partial f(t_1,...,t_n)}{\displaystyle \partial t_1}}\right|_{(az_1,...,az_n)}\cdot z_1 +...+ \left.{\frac{\displaystyle \partial f(t_1,...,t_n)}{\displaystyle \partial t_n}}\right|_{(az_1,...,az_n)}\cdot z_n= \\[4ex] =a^{k-1}\left(z_1{\frac{\displaystyle \partial f}{\displaystyle \partial z_1}}+...+z_n{\frac{\displaystyle \partial f}{\displaystyle \partial z_n}}\right). \end{array}$$ Conversely, if [(\[ehomog\])]{} holds, $$\frac{df(\lambda z_1,...,\lambda z_n)}{d\lambda}=\left.\sum z_i\frac{\partial f(t_1,...,t_n)}{ \partial t_i}\right|_{t=\lambda z_i}=$$ $$=\frac{kf(\lambda z_1,...,\lambda z_n)}{\lambda}.$$ Studying the solutions of the ODE $y^\prime(\lambda)=\frac{ky}{\lambda}$ gives the result. \[lmon\] Suppose $M$ is a graded $\mc(n)$-module and $\nabla$ is a flat homogeneous connection. Suppose $g:M\r M$ be an isomorphism of $\mc(n)$-modules which is homogeneous of degree $\ell$. Then [$$\label{eEg} \nabla+g^{-1}dg$$]{} is a flat homogeneous connection, and the difference $g^{-1}dg$ has degree $\ell$. A direct consequence of Lemma \[lmc2\]. The connections $\nabla$, [(\[eEg\])]{} of Lemma \[lmon\] will be said to [*have the same monodromy*]{}. Now we define the degree of a homogeneous flat connection $\nabla$ on $M$. Let us first assume that $M$ is a free graded $\mc(n)$-module. Let $\phi=(\phi_1,...,\phi_k)$ be a $\mc(n)$-basis of $M$ consisting of elements of degree $0$. Then we say that the flat homogeneous connection $$\nabla_\phi:a_1\phi_1+...+a_k\phi_k \mapsto (da_1)\phi_1+...+(da_k)\phi_k$$ has degree $0$. An arbitrary flat homogeneous connection $\nabla$ is said to have degree $0$ when the difference $\nabla-\nabla_\phi$ has degree $0$. This definition is consistent since by Lemma \[lmon\], for two degree $0$ bases $\phi$, $\psi$, $\nabla_\phi-\nabla_\psi$ has degree $0$. When $M$ is a projective $\mc(n)$-module, consider a direct summand embedding $M\subseteq N$ where $N$ is a free $\mc(n)$-module. We say that a flat connection $\nabla$ on $M$ has degree $0$ when there exists a commutative diagram [$$\label{ehomog+}\diagram M\rto^(.3){\nabla}\dto_{\iota}^{\subseteq} & M\otimes_{\mc(n)}\Omega^{1}_{\mc(n)/k} \dto^{\iota\otimes Id}_{\subseteq}\\ N\rto^(.3){\nabla^\prime} & N\otimes_{\mc(n)}\Omega^{1}_{\mc(n)/k} \enddiagram$$]{} where $N$ is a free $\mc(n)$-module and $\nabla^\prime$ is a flat homogeneous connection of degree $0$, and $\iota$ is an inclusion of a graded direct summand. \[lhomoga\] This notion of degree $0$ flat homogeneous connection does not depend on the choice of the graded direct summand inclusion $\iota$. Consider two such direct summand embeddings $\iota:M\r N$, $\iota^\prime:M\r N^\prime$. Clearly, we may assume [$$\label{elhomogai}N=N^\prime.$$]{} Let the direct complements of $\iota$, $\iota^\prime$ be $K,K^\prime$, respectively. We may assume [$$\label{elhomogaii}K\cong K^\prime$$]{} (by replacing, if necessary, $N$ with $N\oplus M\oplus K$). Now assuming [(\[elhomogai\])]{}, [(\[elhomogaii\])]{}, we can produce a diagram of $\mc(n)$-modules $$\diagram M\dto_=\rto^\iota & N\dto_{\cong}^{f}\\ M\rto_{\iota^\prime} & N \enddiagram$$ for some graded isomorphism $f:N\r N$ (of degree $0$) of graded $\mc(n)$-modules. The statement then follows from Lemma \[lmon\]. [**Comment:**]{} We do not know whether every projective $\mc(n)$-module $M$, or even one endowed with a flat homogeneous connection of degree $0$, is free. Let $M_i$ be modules over $\Z$-graded commutative rings $R_i$, and suppose we have homogeneous connections $E_i$ on $M_i$. Then there is a natural homogeneous connection $E$ on the $\bigotimes R_i$-module $\bigotimes M_i$ given by [$$\label{emcprod}E(v_1\otimes ...\otimes v_n)={\begin{array}{c} {\scriptstyle n}\\ \sum\\ {\scriptstyle i=1}\end{array}} v_1\otimes...\otimes v_{i-1} \otimes E_i(v_i)\otimes v_{i+1}\otimes...\otimes v_n.$$]{} The same formula also defines a natural homogeneous connection on $M_1\widehat{\otimes} M_2$. If $R\r S$ is a map of $\Z$-graded commutative rings, $M$ is a graded $R$-module and we have a homogeneous flat connection $E$ on $M$, then there is a natural “pushforward" homogeneous flat connection $E\otimes_R S$ on $M\otimes_R S$, since we have a natural map $$\Omega^{1}_{R/F}\otimes_R S\r \Omega^{1}_{S/F}.$$ Tree functors and tree algebras {#stree} =============================== Let $\Lambda$ be a set (called [*set of labels*]{}). A [*pre-tree functor*]{} $(\Lambda,M)$ is a system of graded $\mc(n)$-modules $M_{\lambda_1,...,\lambda_n,\lambda_\infty}$ and graded module isomorphisms [$$\label{emcsum}\diagram\protect M_{\lambda_{11},...,\lambda_{nm_n},\lambda_{\infty}}\otimes_{\mc(m_1+...+m_n)} (\mc(m_1)\otimes...\otimes\mc(m_n)\widehat{\otimes}\mc(n))\dto^(.4)\cong \\ \protect{\begin{array}{c} {\scriptstyle }\\ \bigoplus\\ {\scriptstyle \lambda_1,...,\lambda_n}\end{array}} \left({\begin{array}{c} {\scriptstyle n}\\ \bigotimes\\ {\scriptstyle i=1}\end{array}} M_{\lambda_{i1},...., \lambda_{im_i},\lambda_i}\right) \protect \widehat{\otimes} M_{\lambda_1,....,\lambda_n,\lambda_\infty} \enddiagram$$]{} which must make $M_{\lambda_1,...,\lambda_n,\lambda_\infty}$ a $\Lambda$-sorted $\Z$-graded $\mc$-module co-operad. We will also assume that there be a distinguished element $1\in \Lambda$ such that [$$\label{eunit1}\begin{array}{lll}M_{\lambda_\infty}= & 0 &\text{if $\lambda_\infty\neq 1$}\\ & F=\mc(0) &\text{if $\lambda_\infty=1$}. \end{array}$$]{} It follows that the co-augmentation [(\[eaug\])]{} induce an isomorphism [$$\label{eaug1}\diagram\protect M_{\lambda_{i+1},...,\lambda_{n},\lambda_\infty}\otimes_{\mc(n-i)}\mc(n) \rto^\cong & \protect M_{1,...,1,\lambda_{i+1},...,\lambda_{n},\lambda_\infty} \enddiagram$$]{} A pre-tree functor is called [*finite*]{} if all the projective $\mc(n)$-modules $M_{\lambda_1,...,\lambda_n,\lambda_\infty}$ are finite rank. A [*pre-tree algebra*]{} $(\Lambda,M,V,\alpha,\phi)$ consists of a pre-tree functor $M$, a system of $\Z+\alpha_\lambda$-graded vector spaces $V_\lambda$ such that $\alpha_1=0$, and homomorphisms of $\mc(n)$-modules [$$\label{epta}\phi_{\lambda_1,...,\lambda_n,\lambda_\infty}:M_{\lambda_1,...,\lambda_n,\lambda_\infty}\r Hom((V_{\lambda_1})\otimes...(\otimes V_{\lambda_n}), (V_{\lambda_\infty})\otimes_F \mc(n))$$]{} homogeneous of degree [$$\label{epdeg}\alpha_\infty-\alpha_1-...-\alpha_n,$$]{} such that if we choose labels $\lambda_{11},...,\lambda_{nm_n},\lambda_\infty$, and denote, for a graded $\mc(m_1+...+m_n)$-module $X$, put $$X^\prime :=X\otimes_{\mc(m_1+...m_n)}(\mc(m_1)\otimes...\otimes \mc(m_n)\widehat{\otimes}\mc(n)),$$ and put $$\mathcal{M}=\protect{\begin{array}{c} {\scriptstyle }\\ \bigoplus\\ {\scriptstyle \lambda_i}\end{array}} \left({\begin{array}{c} {\scriptstyle n}\\ \bigotimes\\ {\scriptstyle i=1}\end{array}} M_{\lambda_{i1},...\lambda_{in_i},\lambda_{i}}\right) \protect \widehat{\otimes}M_{\lambda_1,...,\lambda_n,\lambda_\infty},$$ $$\mathcal{V}=\protect{\begin{array}{c} {\scriptstyle }\\ \bigotimes\\ {\scriptstyle \lambda_i}\end{array}} \left( {\begin{array}{c} {\scriptstyle n}\\ \bigotimes\\ {\scriptstyle i=1}\end{array}}Hom({\begin{array}{c} {\scriptstyle }\\ \bigotimes\\ {\scriptstyle j}\end{array}} V_{\lambda_{ij}}, V_{\lambda_i}\otimes \mc(m_i))\right)\widehat{\otimes} \protect Hom({\begin{array}{c} {\scriptstyle }\\ \bigotimes\\ {\scriptstyle i}\end{array}}V_{\lambda_i},V_{\lambda_\infty}\otimes \mc(n)),$$ we have a commutative diagram [$$\label{epta1}\diagram (M_{\lambda_{11},...,\lambda_{nm_n},\lambda_\infty})^\prime \rto & \left(Hom(V_{\lambda_{11}}\otimes...\otimes V_{\lambda{nm_n}},\mc( {\begin{array}{c} {\scriptstyle n}\\ \sum\\ {\scriptstyle i=1}\end{array}} m_{i})) \right)^\prime\\ \mathcal{M}\uto^\cong\rto_{\phi_*\hat{\otimes}\phi_*} & \mathcal{V},\uto \enddiagram$$]{} where the top and bottom row are induced by the appropriate cases of [(\[epta\])]{}, the left column is induced by an appropriate case of [(\[emcsum\])]{}, and the right hand column is induced by composition. There is also the obvious equivariance axiom and a unitality axiom which asserts that $1\in M_{\lambda,\lambda}$ maps in $$Hom(V_\lambda,V_{\lambda}\otimes \mc(1))$$ to $Id\otimes 1$. A pre-tree algebra is called [*finite*]{} if its pre-tree functor is finite. A [*tree functor*]{} $(\Lambda,M,E)$ consists of a pre-tree functor $(\Lambda,M)$, and a system of homogeneous flat connections [$$\label{efcon}E_{\lambda_1,...,\lambda_n,\lambda_\infty}$$]{} of degree [(\[epdeg\])]{} on $M_{\lambda_1,...,\lambda_n,\lambda_\infty}$ such that 1. The connections [(\[efcon\])]{} are equivariant with respect to the obvious action of $\Sigma_n$ 2. The connection [(\[efcon\])]{} for $\lambda_1=...=\lambda_i=1$ has the same monodromy as the pushforward of the connection $E_{\lambda_{i+1},...,\lambda_n,\lambda_\infty}$ along the natural map $\mc(n-i)\r\mc(n)$ induced by co-inserting $\mc(0)$ to the first $i$ coordinates. 3. \[efcon3\] The pushforward of the connection $$E_{\lambda_{11},...,\lambda_{nm_n}}$$ via the structure map [$$\label{efcon31}\mc(m_1+...+m_n)\r \mc(m_1)\otimes...\mc(m_n)\widehat{\otimes} \mc(n)$$]{} is equal to $${\begin{array}{c} {\scriptstyle }\\ \bigoplus\\ {\scriptstyle \lambda_i,...,\lambda_n}\end{array}}\left({\begin{array}{c} {\scriptstyle n}\\ \bigotimes\\ {\scriptstyle i=1}\end{array}} E_{\lambda_{i1},...,\lambda{im_i},\lambda_i}\right) \widehat{\otimes} E_{\lambda_1,...,\lambda_n}.$$ 4. The connection $E_{\lambda,\lambda}$ is the pushforward of the connection $dz$ via the map $\mc(1)\r M_{\lambda,\lambda}$. A [*tree algebra*]{} $(\Lambda,M,V,\alpha,\phi,E)$ consists of a pre-tree algebra $(\Lambda,M,V,\alpha,\phi)$, and a structure $(\Lambda,M,E)$ of a tree functor on its pre-tree functor $(\Lambda,M)$. This data are somewhat redundant. There is no natural choice of the numbers $\alpha_\lambda$, they are only determined modulo $1$. Because of that, it is important to define isomorphism of tree algebras. To this end, there is an obvious notion of isomorphism of tree functors, and an obvious notion of isomorphism involving graded isomorphisms of the spaces $V_\lambda$, so all we need to discuss is isomorphism of tree algebra on the same tree functor and the same data $V,\phi$. An isomorphism of tree algebras $(\Lambda,M,V,\alpha,\phi,E)$, $(\Lambda,M,V,\beta,\phi,F)$ consists of homogeneous isomorphisms [$$\label{eiso1}\diagram g_{\lambda_1,...,\lambda_n,\lambda_\infty}:M_{\lambda_1,...,\lambda_n,\lambda_\infty} \rto^(.6)\cong & M_{\lambda_1,...,\lambda_n,\lambda_\infty} \enddiagram$$]{} homogeneous of degree $$(\beta_{\lambda_\infty}-\alpha_{\lambda_\infty}) -(\beta_{\lambda_1}-\alpha_{\lambda_1})-...-(\beta_{\lambda_n}-\alpha_{\lambda_n})$$ such that $$F_{\lambda_1,...,\lambda_n,\lambda_\infty}-E_{\lambda_1,...,\lambda_n,\lambda_\infty}= g^{-1}_{\lambda_1,...,\lambda_n,\lambda_\infty}dg_{\lambda_1,...,\lambda_n,\lambda_\infty}$$ and the pushforward of $g_{\lambda_{11},...,\lambda_{nm_n},\lambda_\infty}$ via [(\[emcsum\])]{} is equal to $$\protect{\begin{array}{c} {\scriptstyle }\\ \bigoplus\\ {\scriptstyle \lambda_1,...,\lambda_n}\end{array}} \bigotimes g_{\lambda_{i1},...., \lambda_{im_i},\lambda_i}\protect\widehat{\otimes} g_{\lambda_1,....,\lambda_n,\lambda_\infty}.$$ Regularity {#sreg} ========== Our treatment of regular connections follows Deligne [@d]. Let $C$ be a smooth algebraic curve over $F$, and let $\overline{C}$ be a smooth projective curve containing $C$. Let $M$ be a (finite-dimensional) algebraic vector bundle on $C$, and let [$$\label{er1}E: M\r M\otimes_{\mathcal{O}_C} \Omega^{1}_{C/Spec F}$$]{} be an algebraic connection. We say that $E$ has [*regular singularities*]{} if for every smooth projective curve $\overline{C}$ and every embedding $C\subset \overline{C}$, and every point $x\in \overline{C}-C$, $$M\otimes_{\mathcal{O}_C}\mathcal{O}_x$$ has a basis in which the pushforward of the connection $E$ has simple poles. A connection $E$ on an $n$-dimensional smooth separated algebraic variety $X$ is said to have [*regular singularities*]{} if for every smooth algebraic curve $C$ in $X$, the restriction of $E$ to $C$ has regular singularities. We will call a tree functor $(\Lambda,M,E)$ [*regular*]{} when all the connections $E_{\lambda_1,...,\lambda_n,\lambda_\infty}$ have regular singularities. A tree algebra is called regular when its tree functor is regular. This notion is clearly invariant under isomorphism of tree algebras. Our main goal is to establish a version of the Riemann-Hilbert correspondence for tree functors and tree algebras over $\C$. To this end, we must define the analytic versions of our concepts. In effect, we may define $\mc(n)_{an}$ to be the $\Z$-graded ring of all holomorphic functions $f$ on the configuration space $C(n)$ of $n$ ordered distinct points in $\C$, homogeneous in the sense that $$f(\lambda z_1,...,\lambda z_n) =\lambda^k f(z_1,...,z_n);$$ In our grading, the degree of $f$ is $-k$. We would like to have a $\Z$-graded co-operad structure [$$\label{ecoopan0} \mc(m_1+...+m_n)_{an}\r (\mc(m_1)_{an}\otimes...\otimes \mc(m_n)_{an})\widehat{\otimes} \mc(n)_{an},$$]{} but the difficulty is that expanding an analytic function by a Laurent series may present elements of arbitrarily low degrees. Replacing $\widehat{\otimes}$ by the product of its bigraded summands of the same total degree, we do get a $\Z$-graded co-operad, but the resulting objects $\mathcal{P}(m_1,...,m_n,n)$ aren’t rings, so we cannot study vector bundles in the sense of finite rank projective modules. Our solution is to replace [(\[ecoopan0\])]{} by [$$\label{ecoopan} \mc(m_1+...+m_n)_{an}\r \chi(m_1,...,m_n,n)_{an},$$]{} where the right hand side denotes the ring of all partially defined holomorphic functions $f$ on [$$\label{ecooprod}C(m_1)\times...\times C(m_n)\times C(n)$$]{} where, if we denote the coordinates of [(\[ecooprod\])]{} by [$$\label{ecooprod1}t_{11},...,t_{1m_1},....,t_{n1},...,t_{nm_n}, z_1,...,z_n,$$]{} then for each choice of $z_1,...,z_n$ there exists a locally uniform $\epsilon>0$ such that $f$ is defined on [(\[ecooprod1\])]{} when [$$\label{ecooprod2}||t_{ij}||<\epsilon.$$]{} Using [(\[ecoopan\])]{} and the natural inclusion [$$\label{ecooprod3}\mc(m_1)_{an}\otimes...\otimes \mc(m_n)_{an}\otimes \mc(n)_{an}\subset \chi(m_1,...,m_n,n)_{an},$$]{} we may define analytic (pre)-tree functors and (pre)-tree algebras in precise analogy with the algebraic definitions. We start with $\C^\times$-equivariant holomorphic vector bundles $\Xi$ on $C(n)$. By this we mean a holomorphic bundle with a holomorphic action of $\C^\times$ on the total space which is compatible with the $\C^\times$-action on $C(n)$ by $$\lambda(z_1,...,z_n)=(\lambda z_1,...,\lambda z_n).$$ Then the space $M$ of global sections of $\Xi$ is then naturally a graded $\mc(n)_{an}$-module. We will called a projective module [*of finite rank*]{} if it is a direct summand of a free module on a finite set of generators. \[lstein\] Let $X$ be a Stein manifold. The global sections functor defines an equivalence of categories from the category of finite-dimensional holomorphic vector bundles over $X$ (and holomorphic maps over $Id_X$) and the category of finite rank projective modules over the ring $Hol(X)$ of holomorphic functions on $X$. First we will prove that a holomorphic vector bundle $\xi$ of finite dimension $n$ on $X$ is always a holomorphic direct summand of a finite dimensional trivial vector bundle. First, note that this is true topologically: $X$ is of the homotopy type of a $dim(X)$-dimensional CW complex, so by Whitehead’s theorem, the topological classifying map $\phi$ of $\xi$ factors through a map $\phi^\prime$ into the $dim(X)$-skeleton of $BU(n)$: $$\diagram X\rto^\phi\drdotted|>\tip^{\phi^\prime}& BU(n)\\ & BU(n)_{dim(X)}\uto_{\subseteq} \enddiagram$$ But $BU(n)_{dim(X)}$ is compact, so the restriction $\gamma^{\prime}_{n}$ of the universal $n$-bundle $\gamma_n$ on $BU(n)$ to $BU(n)_{dim(X)}$ is a direct summand of a finite-dimensional trivial vector bundle. Hence, the same is true for $\xi$ topologically, which we indicate by the subscript $(?)_{top}$: [$$\label{elstein1}\xi_{top}\oplus\eta_{top}\cong N_{top}.$$]{} But now the data [(\[elstein1\])]{} may be represented topologically by a $GL_n(\C)\times GL_{N-n}(\C)$-principal bundle, so by Grauert’s principle [@grauert], Satz 2, the data [(\[elstein1\])]{} can be represented in the holomorphic category, which we indicate by the subscript $(?)_{an}$: [$$\label{elstein2}\xi_{an}\oplus\eta_{an}\cong N_{an}.$$]{} Applying Grauert’s principle again for the groups $GL_n(\C)$, $GL_N(\C)$, however, we see that $$\xi_{an}\cong \xi,\; N_{an}\cong N.$$ This shows that the global section functor in the statement of the Lemma lands in the category indicated. To show that the functor is onto on isomorphism classes of objects, recall that a finite rank projective $Hol(X)$-module can be constructed from a free module by applying an idempotent matrix; since a free module always arises from a trivial bundle, applying the same matrix on the bundle gives the bundle corresponding to the finite rank projective module. We now need to prove that the global section functor is fiathfully full. Since, however, every object in the source is a holomorphic direct summand of a trivial finite-dimensional vector bundle, it suffices to prove that the functor is faithfully full on the subcategory of finite-dimensional trivial holomorphic vector bundles, which in turn reduces to showing that the functor induces bijection of the set of holomorphic self-maps of the $1$-dimensional trivial vector bundle to the set of holomorphic self-maps of the $1$-dimensional free $Hol(X)$-modules. Obviously, however, both sets are (compatibly) bijective to $Hol(X)$. \[cstein\] The category of finite-dimensional $\C^\times$-equivariant holomorphic vector bundles over $C(n)$ and holomorphic $\C^\times$-equivariant homomorphisms (over the identity on $C(n)$) is equivalent, via the global sections functor, to the category of finite rank projective graded $\mc(n)_{an}$-modules and (degree $0$) homomorphisms of graded modules. In the case $n=1$, the isomorphism class of a $\C^\times$-equivariant bundle is determined by the representation on the $0$ fiber, which shows that the functor is a bijection on isomorphism classes of objects. Additionally, non-equivariantly, the bundles are trivial by Grauert’s theorem [@grauert], so in the rank $1$ case, maps are simply holomorphic functions on $\C$, and therefore homogeneous functions are simply $z^k$, $k\geq 0$. This is a graded morphism if and only if the $\C^\times$-action on $0$-fiber of the target is $z^{-k}$ tensored with the action of the $0$-fiber of the source. This gives the required statement. In the case $n>1$, the category of $\C^\times$-equivariant holomorphic bundles and $\C^\times$-equivariant holomorphic maps is equivalent to the category of holomorphic vector bundles on $C(n)_0$. Since $C(n)_0$ is a Stein manifold, this is in turn equivalent to the category of finite rank projective $\mc(n)_0$-modules, which is in turn equivalent to the category of graded finite type projective $\mc(n)$-modules. Also, a holomorphic connection on $\Xi$ gives rise to a connection in the $\mc(n)_{an}$-module sense [$$\label{econnal}E:M\r M\{dz_1,...,dz_n\};$$]{} note that we have a canonical differentiation $$\mc(n)_{an}\r \mc(n)_{an}\{dz_1,...,dz_n\}.$$ We may then define flat, homogeneous connections and connections with a given degree in terms of the algebraic connection [(\[econnal\])]{}, and we can mimic the definitions of the previous section using the space $\chi(m_1,...,m_n,n)_{an}$. One complication to the compatibility of our notions however is that $(\mc(m_1)\otimes...\otimes \mc(m_n))\widehat{\otimes}\mc(n)$ is not a subspace of $\chi(m_1,...,m_n,n)_{an}$. Nevertheless, if we denote by $\chi(m_1,...,m_n,n)$ the intersection of $(\mc(m_1)\otimes...\otimes \mc(m_n))\widehat{\otimes}\mc(n)$ and $\chi(m_1,...,m_n,n)_{an}$ in $P(m_1,...,m_n,n)$, then the right hand side of [(\[efcon31\])]{} can be replaced by $\chi(m_1,...,m_n,n)$, which allows a comparison. We will need to define a regular flat homogeneous connection $$E:M^\prime\r M^\prime\{dt_{ij},dz_i\}$$ on a finitely generated projective $\chi(m_1,...,m_n,n)$-module $M$ where $$M^\prime =M\otimes_{\chi(m_1,...,m_n,n)}\chi(m_1,...,m_n,n)_{an}.$$ We will say that a flat connection $E$ on $M^\prime$ is regular if each of its solutions has moderate growth. We say that a solution $g$ has moderate growth if for every locally closed smooth algebraic curve $C$ in [(\[ecooprod\])]{}, every holomorphic embedding $D\subset \overline{C}$ with non-zero derivative at $0\in D$ (where $C$ is a smooth compactification of $C$) and every continuous function $\epsilon$ on $D$ such that for every $x=(t_{ij},z_i)\in D-\{0\}$, $g$ is defined for $x^\prime=(\delta t_{ij}, z_i)$ whenever $\delta_i\leq \epsilon(x)$, there exists an $N>0$ such that [$$\label{eregcond}g(x^\prime)<(\prod\delta_i)^{-N} ||u||^{-N}$$]{} where $u$ denotes the standard holomorphic coordinate on $D$ and in [(\[eregcond\])]{}, $g$ denotes a branch of $g$ on a sector $S$ of $D-\{0\}$ ([@d]); we may define a branch of a solution of $E$ on $S$ as a holomorphic function on $S$ which satisfies $\nabla g=0$ where $\nabla$ is the pullback of $E$ to $S$. The multi-valued section $g$ will also be referred to as [*regular*]{}. Our first version of Riemann-Hilbert correspondence is the following \[lrh1\] There is an equivalence of categories (canonical up to canonical isomorphism) between the following categories: [$$\label{ecat1} \parbox{3.5in}{Finite-dimensional $\mc$-equivariant holomorphic vector bundles on $C(n)$ with a homogeneous flat connection of degree $k$}$$]{} [$$\label{ecat2}\parbox{3.5in}{Finite rank projective $\Z$-graded $\mc(n)$-modules with a homogeneous regular flat connection of degree $k$.}$$]{} When $n=1$, then every then every local system on $C(1)$ is trivial. Treating the local system as a free $\mc(1)_{an}$-module $M$ with a flat connection $E$, a priori the free generators of $E$ may not be graded. However, obviously we have a decreasing filtration $F^k M$ consisting of all elements of degree $\geq k$. Then, we can consider the associated graded object [$$\label{easgr}F^k M/F^{k+1}M.$$]{} Looking at the lowest $k$ for which [(\[easgr\])]{} is non-trivial, the fact that $M$ clearly implies that [(\[easgr\])]{} is generated by homogeneous elements of degree $k$. Further, the connection must be $0$ on these generators by homogeneity. Consider the free graded submodule generated by these elements by $M_0$. Then $M/M_0$ is also a free module (the category of finite dimensional free $\mc(1)_{an}$-modules with flat connection is equivalent to the category of finite dimensional $\C$-vector spaces), so we may repeat the argument with $M$ replaced by $M/M_0$ to show by induction that $M$ is free as a graded $\mc(1)_{an}$-module with generators annihilated by the connection. Since clearly an analogous argument applies to the algebraic category, the statement follows. Assume now $n>1$. Then any $z_i$ defines an isomorphism of vector spaces from $\mc(n)_k$ to $\mc(n)_{k-1}$. The category of graded $\mc(n)$-modules and morphisms of degree $0$ is therefore equivalent to the category of $\mc(n)_0$-modules. The statement for flat connections is that flat connections o Similar statements are also valid for the corresponding analytic categories. But now $\mc(n)_0$ is in fact the coefficient ring of a smooth affine variety $C(n)_0$, which comes with an embedding into $\P^{n-1}$ with homogeneous coordinates $z_1,...,z_n$ (in fact, the embedding factors through the copy of $\A^{n-1}$ which is the complement of the locus of $z_1-z_2$). The statement for connections is that flat connections (algebraic or analytic) on $C(n)_0$ correspond precisely to homogeneous connections on $\mc(n)$ resp. $\mc(n)_{an}$ of degree $0$: This is because $$\Omega^{1}_{\mc(n)_0/F}=\{a_1 \frac{dz_1}{z_1}+...+a_n\frac{dz_n}{z_n}\in (\Omega^{1}_{\mc(n)/F})_0 | a_1+...+a_n=0\}.$$ Therefore, for homogeneous flat connections of degree $0$, the result follows from Theorem 5.9 of [@d], applied to the variety $C(n)_0=Spec(\mc(n)_0)$. The case of general degree $k$ can be reduced to the case of degree $0$ by subtracting an algebraic connection of degree $k$. Our main interest is in the following statement: \[trh\] There is an equivalence of categories (canonical up to canonical isomorphism) between the category of finite analytical tree functors (resp. algebras) and finite regular tree functors (resp. algebras). Given Lemma \[lrh1\], and the fact that the two connections whose isomorphism we are seeking are obviously regular, the statement amounts to asserting that an isomorphism in [$$\label{ehatan}\chi(m_1,...,m_n,n)_{an}$$]{} between two graded regular connections on [$$\label{ehat1}\chi(m_1,...,m_n,n)$$]{} of the same degree is algebraic. However, this amounts to saying that every regular function in [(\[ehatan\])]{} is in [(\[ehat1\])]{}. This can be shown as follows: consider a regular function in [(\[ehatan\])]{}. Then considering $g$ as a function of the $t_{ij}$’s for fixed $z_i$, expand in the total degree of all the $t_{ij}$’s for a fixed $i$. Then the coefficients of fixed total degree $d_i$ in the $t_{ij}$’s for each $i$ are obviously elements of $$\mc(m_1)_{d_1}\otimes...\otimes\mc(m_n)_{d_n}$$ and further each of the numbers $d_i$ where the coefficient is non-zero is bounded below by a bound $N(z_1,...,z_n)$. Since this bound, however, is locally constant, it must be in effect constant because the function involved are analytic. Now we claim that for each given assortment of degrees $d_1,...,d_n$, the corresponding component [$$\label{egcomp1}g_{d_1,...,d_n}$$]{} of $g$ is in effect an element of $$\mc(m_1)\otimes...\otimes\mc(m_n)\otimes\mc(n).$$ In fact, select the lowest $(d_1,...,d_n)$ (say, lexicographically) such that [(\[egcomp1\])]{} is non-trivial. Then by [(\[eregcond\])]{}, all summands of higher degree can be neglected, and [(\[egcomp1\])]{} must be regular. Subtracting this component, we may show by induction that all of the components are regular. \[dvt\] Recall the definition of vertex tensor category of Huang and Lepowsky [@hl], Definition 4.1. We call a vertex tensor category [*semisimple*]{} if there are finitely many objects (called [*irreducible objects*]{}) whose endomorphism groups are $\mc$ and such that there are no nonzero morphisms between non-isomorphic irreducible objects, and every object is isomorphic to a direct sum of irreducible objects, and the unit object is irreducible. \[thuang\] There is a ‘realization’ functor (canonical up to natural equivalence) from the category of vertex tensor categories and isomorphisms, to the category of analytic tree algebras and isomorphisms, and consequently, by Theorem \[trh\], to the category of tree algebras and isomorphisms. [**Comment:**]{} By the work of Lepowsky and Huang, vertex algebras which satisfy certain ‘rationality’ conditions supply examples of vertex tensor categories in the sense of [@hl]. A good survey is Huang [@hrep]. It should also be noted that the present result overlaps with Huang’s construction [@h7] of genus $0$ correlation functions for modules of vertex algebras. The major point of interest of our result is that it extends to the algebraic category of tree algebras over $\C$. It should be noted that we model only a part of the structure of vertex tensor category in our axioms (e.g. we do not treat the conformal element = energy-momentum tensor), which is one of the reasons why we do not get an equivalence of categories in our statement. [**Proof of Theorem \[thuang\]:**]{} We will study the definition of vertex tensor category [@hl], Definition 4.1. First of all, because we do not treat conformal element data, we restrict attention to the subspaces $K(n)_0$ of the moduli spaces $K(n)$ (see Huang [@huangbook], p.65) where the tube functions are the identity, and the scaling constant is $1$. In this setting, there is a canonical splitting of the determinant bundle, so we get a canonical map [$$\label{ekn0}\psi:K(n)_0\r \widetilde{K}^c(n).$$]{} Next, for a semisimple vertex tensor category $\mathcal{V}$, the set of labels $\Lambda$ is the set of representatives of isomorphism classes of irreducible objects (we shall also write $V_\lambda=\lambda$), and $1$ corresponds to the unit object [@hl], Definition 4.1. (3). Now for and $Q\in K(2)_0$, and irreducible objects $V_\lambda$, $V_\mu$, we consider their tensor product [@hl], Definition 4.1 (1) [$$\label{ehltensor}W:=V_\lambda \boxtimes_{\psi(Q)}V_\mu.$$]{} Then by our definition of semisimple vertex tensor category, we have a unique decomposition up to isomorphism [$$\label{eqd1}W={\begin{array}{c} {\scriptstyle }\\ \bigoplus\\ {\scriptstyle \nu\in \Lambda}\end{array}}W_\nu$$]{} where, non-canonically, [$$\label{egd2}W_\nu\cong N\otimes V_\nu$$]{} for a finite-dimensional complex vector space $N$. Further, by our assumptions, non-canonically, we have [$$\label{egd3}Aut(W_\nu)\cong GL(N).$$]{} Property (6), together with axioms (1) and (2) of Definition 4.1 of [@hl] imply that [(\[egd3\])]{} define a principal smooth $GL(N)$-bundle with flat connection on the space $C(2)$. We let $M_{\lambda,\mu,\nu}$ be the dual of the associated vector bundle (which the flat connection automatically makes analytic). The correlation function $\phi_{\lambda,\mu,\nu}$ (the analytic version of [(\[epta\])]{}) then follows from the universal intertwining operator from $V_\lambda\otimes V_\mu$ to [(\[ehltensor\])]{}. General correlations functions with an arbitrary number of arguments are then produced in an analogous way by iterating the tensor product [(\[ehltensor\])]{}. Co-operad associativity resp. unitality resp. equivariance follow from property (4) resp. (7) resp. (5) of [@hl], Definition 4.1. Vertex algebra unitality follows from property (8). [**Example:**]{} The chiral WZW model. Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\C$. In [@hl1], Huang and Lepowsky construct a vertex tensor category of finite sums of irreducible lowest weight-modules $L(k,\lambda)$ over the quotient $L(k,0)$ of the affine vertex algebra $M(k,0)$ by its maximal ideal (=maximal proper graded submodule) for $k=0,1,2,...$. By Theorem \[thuang\], this gives rise to a tree algebra $T$ over $\C$. In effect, we have the following refinement: \[thuang1\] Let $\mathfrak{g}$ be defined over $\C\supseteq k\supseteq \Q$. Then the tree algebra corresponding to the Huang-Lepowsky construction can be defined over $k$. The key point is to study the so called Knizhnik-Zamolodchikov equations [@hl1], (2.14), which define the desired homogeneous flat connection on the tree functor. The connection is defined on the trivial $\mc(n)$-module [$$\label{ekz*}N_{\lambda_1,...,\lambda_n,\lambda_\infty}= L(\lambda_1)\otimes...\otimes L(\lambda_n)$$]{} where $L(\lambda)$ is the summand of lowest degree of $L(k,\lambda)$. The flat connection defined by the KZ-equations maps $$f:L(\lambda_1)\otimes...\otimes L(\lambda_n)$$ to [$$\label{ekz+}df-\frac{1}{k+h^{\vee}}{\begin{array}{c} {\scriptstyle }\\ \sum\\ {\scriptstyle p\neq\ell}\end{array}} \frac{1}{z_\ell-z_p}{\begin{array}{c} {\scriptstyle }\\ \sum\\ {\scriptstyle i}\end{array}} f(Id\otimes...\otimes g^i\otimes...\otimes g_i\otimes...\otimes Id)dz_\phi$$]{} where on the right hand side, $(g^i)$ and $(g_i)$ are dual bases of $\mathfrak{g}$ with respect to the Killing form, and are inserted at the $\ell$’th and $p$’th coordinate, respectively. Manifestly, the connection [(\[ekz+\])]{} is defined over $\Q$. The tree functor is actually a direct summand of [(\[ekz\*\])]{}. It corresponds to $\mathfrak{g}$-equivariant maps from the lowest weight summand of $$(L(k,\lambda_1)\boxtimes...\boxtimes L(k,\lambda_n))_Q$$ to $L(k,\lambda_\infty)$ where $Q$ encodes the moduli data [@hl], which, in our case, is just the $n$-tuple of points $(z_1,...,z_n)$. By definition, all this is defined over $k$. [99]{} A. Beilinson and V. Drinfeld, [*Chiral algebras*]{}, American Mathematical Society Colloquium Publications, 51. AMS, Providence, RI, 2004 R.E.Borcherds: Monstrous moonshine and monstrous Lie superalgebras, [*Invent. Math.*]{} 109 (1992) 405-444 P. Deligne: Equations Differentielles a Points Singuliers Reguliers, [*Lecture Notes in Math.*]{} 163, Springer Verlag T.Fiore, P.Hu, I.Kriz: Laplaza sets, or how to select coherence diagrams for pseudo algebras, [*Adv. Math.*]{} 218 (2008), 1705-1722 I.Frenkel, I.Lepowsky, A.Meurman: [*Vertex operator algebras and the Monster*]{}, Academic Press, Boston, MA, 1988 H.Grauert: Analytische Faserungen über holomorph-vollständigen Räumen, [*Math. Ann.*]{} 135 (1958) 263-273 P.A.Griffin, O.F.Hernandez: Structure of irreducible $SU(2)$ parafermion modules derived vie the Feigin-Fuchs construction, [*Internat. Jour. Modern Phys.*]{} A 7 (1992) 1233-1265 A. Grothendieck: Esquisse d’un programme, in [*Geometric Galois Actions*]{}, L.Schneps and P. Lochak eds., London Math. Soc. Lecture Notes 242, Cambridge University Press, 1997 R.Hortsch, I.Kriz, A.Pultr: A universal approach to vertex algebras, [*J. of Algebra*]{} 324 (2010) 7, 1731-1753 P.Hu, I.Kriz: On modular functors and the ideal Teichmüller tower, [*Pure Appl. Math. Q.*]{} 1 (2005) no.3, part 2, 665-682 Y.Z.Huang: Differential equations, duality and intertwining operators, [*Comm. Contemp. Math.*]{} 7 (2005), 649-706 Y.Z.Huang: Generalized rationality and a “Jacobi identity" for intertwining operator algebras, [*Selecta Math.*]{} 6 (2000) 3, 225-267 Y.Z. Huang: Representations of vertex operator algebras and braided finite tensor categories, in: [*Vertex algebras and related areas*]{}, Contemp. Math. 497, Amer. Math. Soc., Providence, RI, 2009, pp. 97-111 Y.Z. Huang: [*Two-dimensional conformal geometry and vertex operator algebras*]{}, Prog. Math. 148, Birkhäuser, Boston, 1997 Y.Z.Huang, J.Lepowsky: Tensor products of modules for a vertex operator algebra and vertex tensor categories, in: [*Lie theory and geometry*]{}, Prog. Math. 123, Birkhäuser, Boston, 19914, pp. 349-383 Y.Z.Huang, J.Lepowsky: Intertwining operator algebras and vertex tensor categories for affine Lie algebras, [*Duke J. Math.*]{} 99, 1 (1999) 113-134 G. 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--- abstract: 'The Schwinger mechanism, the production of charged particle-antiparticle pairs in a macroscopic external electric field, is derived for 2+1 dimensional theories. The rate of pair production per unit area for four species of massless fermions, with charge $q$, in a constant electric field $E$ is given by $ \pi^{-2} \, \hbar^{-3/2} \,\tilde{c}^{-1/2}\, ( q E)^{3/2} $ where $\tilde{c}$ is the speed of light for the two-dimensional system. To the extent undoped graphene behaves like the quantum field-theoretic vacuum for massless fermions in 2+1 dimensions, the Schwinger mechanism should be testable experimentally. A possible experimental configuration for this is proposed. Effects due to deviations from this idealized picture of graphene are briefly considered. It is argued that with present day samples of graphene, tests of the Schwinger formula may be possible.' author: - Danielle Allor - 'Thomas D. Cohen' - 'David A. McGady' title: The Schwinger mechanism and graphene --- Introduction ============ The Schwinger mechanism refers to the production of charged particle-antiparticle pairs out of the vacuum by a classical electric field which is homogenous over a large volume in space. More than half a century ago, Schwinger computed[@Schwinger] the rate associated with vacuum breakdown via pair production per unit volume for the case of a constant electric field. Over the years, the Schwinger mechanism has spawned a vast literature. Invoked to gain insights on topics as diverse as the string breaking rate in QCD[@CNN; @Neuberger] and on black hole physics[@GR], this mechanism has become a textbook topic in quantum field theory[@IZ]. Topics such as back reaction[@BR] and finite size effects[@Wang_Wong] have been addressed. A key issue is the rate of pair production from a static electric field. Schwinger originally addressed this question by noting that the probability that a system with zero electrons or positrons remains in the fermionic vacuum state decays exponentially in time[@Schwinger]. Treating the electric field classically—*i.e.* the formal limit of $q\rightarrow 0$, $E \rightarrow \infty$ with $q E$ fixed—Schwinger calculated the vacuum persistence probability, $P_{\rm vac}(t)$, as a function of time: $$\begin{aligned} P_{\rm vac}(t)& \equiv& |\langle {\rm vac}| U(t) |{\rm vac} \rangle |^2 = \exp(-w V t) \label{SF1}\\ {\rm with} \; \; w &=& \frac{(q E)^2}{4 \pi^3 \hbar^2 c} \, \sum_{n=1}^\infty \frac {1}{n^2} \, \exp \left( -\frac{n \pi m^2 c^3}{q E \hbar}\right ), \label{SF2}\end{aligned}$$ where $V$ is the spatial volume of the system and $w$ is the rate of vacuum decay per unit volume. Schwinger’s interpretation[@Schwinger] of Eqs. (\[SF1\]) and (\[SF2\]) was straightforward: $w$ was taken to be the local rate of production per unit volume of fermion-antifermion pairs by the electric field. This interpretation has been widely accepted in much of the literature on the Schwinger mechanism. However, while the Schwinger formula of Eq. (\[SF2\]) is very well known, it has been argued that its interpretation as the pair production rate is not correct[@Niki]. Despite the very natural interpretation of $w$ in Eq. (\[SF2\]) as the rate of production of pairs per unit volume, an explicit calculation gives the rate of pair production per unit volume, which we denote $\Gamma$, as $$\Gamma = \frac{(q E)^2}{4 \pi^3 \hbar^2 c} \, \exp \left( -\frac{\pi m^2 c^3}{q E \hbar} \right ) \; . \label{SF3}$$ This rate does [*not*]{} agree with $w$: the entire rate is given by the first term in the series for $w$. For a recent, pedagogical, discussion highlighting the theoretical distinction between these two rates, see Ref. [@Cohen_McGady]. Despite its theoretical significance, there has been no direct experimental signature of the Schwinger mechanism, of charged pair creation in electric fields. This is particularly unfortunate given the common confusion between the rate of pair creation $\Gamma$ and $w$ the rate of vacuum decay. It would be [*very*]{} useful to concoct an experimental test to distinguish between the two directly. Moreover, apart from the distinction between $w$ and $\Gamma$ the derivation of the rate of pair production via the Schwinger mechanism raises a number of subtle issues associated with the implementation of appropriate boundary conditions[@GR; @GG; @Wang_Wong; @Neuberger]; it is important to test whether these are handled correctly. Ultimately the most compelling test would be experimental. The reason that the Schwinger mechanism has never been tested experimentally is very easy to understand: the exponential factor in the pair production rate is [*very*]{} small for static macroscopic $E$ fields realizable in the lab. It only becomes of order unity when $E$ is large enough so that $q E$ times the electron’s Compton wavelength is greater than $m c^2$: this requires an electric field of order $10^{16} {\rm V/cm}$; for $E = 10^6 \frac{\rm V}{\rm cm}$, $\exp \left(-\frac{ \pi m^2 c^3}{q E \hbar} \right ) \approx \exp\left(-4 \times 10^{10} \right)$. One might hope to test the Schwinger formula experimentally in a condensed matter system which simulates light or massless, electrically charged, relativistic fermions. In this context, the condensed matter system acts as an analog computer to test the underlying result from relativistic field theory. Fortunately, it has been known for more than two decades that charged quasi-particle excitations in a potential with a two-dimensional hexagonal symmetry have a region of momenta over which their dispersion relation is linear-$(\epsilon - \epsilon_0)^2 = \tilde{c}^2 (p_x^2 + p_y^2)$[@Semenov]. This is precisely the dispersion relation of a massless relativistic particle with energy measured relative to $\epsilon_0$ and has $\tilde{c}$ playing the role of $c$. Graphene ([*i.e.*]{}, a single sheet of graphite) has such a symmetry. Moreover, in undoped graphene, the Fermi level is at $\epsilon_0$. Thus, to the extent that a single particle description holds in graphene, the quantum ground state of a filled Fermi sea is the precise analog of a filled Dirac sea—[*i.e.*]{} the vacuum of a two-dimensional non-interacting field theory for fermions. The recent development of techniques to produce samples of graphene and measure its properties[@GrExp; @GrExp2] has focused significant attention to its analogy with massless Dirac particles: graphene has been proposed as a testing ground for the standard relativistic quantum mechanical effects of zwitterbewegung and Klein/Landau-Zener tunneling [@KP; @GPNJ]. This letter explores the possibility of using graphene to test experimentally the more subtle dynamics of the Schwinger mechanism. The Schwinger pair creation rate in graphene ============================================ One can adapt Schwinger’s calculation[@Schwinger] for smaller dimensions[@1plus1; @2plus1; @GG]; in (2+1) dimensions the probability that the system has remained in the (fermionic) vacuum after time $t$, $P_{\rm vac} (t)$, is $$\begin{aligned} P_{\rm vac}^{2+1}(t)& =& \exp(-w^{2+1} A t) \\ {\rm with} \; \; w^{2+1} &=& \frac{f \, (q E)^{3/2}}{4 \pi^2 \, \hbar^{3/2} \, \tilde{c}^{1/2}} \, \sum_{n=1}^\infty \frac {1}{n^{3/2}}\, \exp \left( -\frac{n \pi m^2 \, \tilde{c}^3}{q E \, \hbar} \right )\nonumber \\ & =& \frac{f \, \zeta \left ( \frac{3}{2} \right )\,( q E)^{3/2}}{4 \pi^2 \, \hbar^{3/2} \, \tilde{c}^{1/2} } \; \; \; \; {\rm for} \; \; m=0 \label{2plus1w}\end{aligned}$$ where $A$ is the spatial area, $\tilde{c}$ is the speed of light for the 2-d system, $\zeta$ is the Riemann zeta function with $\zeta (3/2) \approx 2.612$, and $f$ is the number of species of fermion ([*i.e.*]{}, four for graphene). Similarly, the local rate of pair creation, $\Gamma^{2+1}$, is given by the first term of this series[@Cohen_McGady; @Niki; @GG], $$\Gamma^{2+1} = \frac{f (q E)^{3/2}}{4 \pi^2 \, \hbar^{3/2} \, \tilde{c}^{1/2}} \exp \left( -\frac{n \pi m^2 \, \tilde{c}^3}{q E \, \hbar} \right ). \label{2plus1G}$$ These derivations hold for classical, constant, and externally fixed electric fields. Consequently these relations are valid only when it is legitimate to neglect: i) macroscopic back reaction to the applied field due to this same charged particle production rate[@BR]; ii) the production of real photons as the charged particles accelerate in the electric field; and iii) interactions between the fermions mediated by the exchange of virtual photons. Alternative derivations[@CNN; @Cohen_McGady] to Schwinger’s original calculation[@Schwinger], focus on the nature of a single particle level for the Dirac equation in the presence of an electric field, switched on in the distant past. These derivations have the twin virtues of clarifying both the role of boundary conditions, and the distinction between the rate associated with vacuum breakdown, $w$, and the rate of pair creation, $\Gamma$. Here we will discuss the derivation done in the time independent gauge, $E_i = -\partial_i A_0(x_j)$[@CNN; @Wang_Wong]. In this gauge, we see that the electric field alters the single particle energy levels. The shifts in energy due to the potential have opposite signs on either side of the field region, allowing filled levels on one side to become degenerate with empty ones on the other. This yields an effective potential which filled levels in the elevated Dirac sea tunnel through, leaving a hole on one side and yielding a particle type state on the other. This tunneling problem is conceptually simple for an electric field of limited spatial extent[@Wang_Wong]. Consider an infinite plane with an electric field independent of $y$, with magnitude $E$, oriented in the $-x$ direction and confined to the region between $-L/2 \le x\le L/2$. A useful basis is the in-state wave functions which correspond to solutions of the Dirac equation with unit flux moving towards the region of the electric field from either the left or right. The states have amplitude $T$ (times a flux-normalizing kinematic factor) to be found on the far side of the field region. $T$ may be computed directly as a tunneling problem in an effective energy-dependent Schrödinger equation derivable from the underlying Dirac equation. ![The dispersion relations are shifted on either side of the electric field. The image on the left is a cartoon picture of the in-state vacuum, where the Schwinger mechanism applies. After $\tau_{fill}$ the level occupations change, and the system falls into a p-n type system, depicted on the right.[]{data-label="electronic_structure"}](45thEDBand.eps){width="3in"} Suppose that the system is in the “in-state vacuum” (the left image in Fig. \[electronic\_structure\]). All of the the in-states below the Dirac sea on the left, $\psi^{L \, \rm in}_{\epsilon,p_T}$, are occupied for $\epsilon < - q E L/2$ and all of the states below the Dirac sea on the right, $\psi^{R \, \rm in}_{\epsilon,p_T}$, are occupied for $\epsilon < q E L/2$. Turning on the electric field in the distant past has merely shifted the energies of the states relative to the local vacuum, not their occupation number. The pair production rate for a particle with energy $\epsilon$ and transverse momentum $p_T$ is proportional to the transmission probability for a filled in-state from the left $-q E L/2 \le \epsilon \le q E L/2$. In essence a filled level moving in the elevated Dirac sea towards the region of the electric field propagates through and emerges on the other side, with probability $|T|^2$, where it appears as a particle. The rate of pair production per unit width per unit $\epsilon$ per unit $p_T$ is computed analogously to the 3+1 dimensional case[@CNN]. Integration over $\epsilon$ and $p_T$ gives the rate of pairs per unit width: $$\begin{aligned} ~ \frac{d^2 N}{ d t\, d W } & = & \int_{-\epsilon^{\rm max}}^{\epsilon^{\rm max}} d \epsilon \int_{-p_T^{\rm max}}^{p_T^{\rm max}} d p_T \frac{d^4 N}{d \epsilon \, d p_T \, d W \, d t} \nonumber \\ \epsilon^{\rm max} = \frac{q E L}{2}&-&m \tilde{c}^2 \; \; , \; \; p_T^{\rm max} = \frac {\sqrt{(2 \epsilon -q E L)^2 - 4 m^2 \tilde{c}^4}}{2 \tilde{c}} \nonumber \\ \frac{d^4 N}{d \epsilon \, d p_T \, d W \, d t} & = & f \frac{|T(\epsilon,p_T)|^2 }{4 \pi^2 \, \hbar^2} \; .\label{rate}\end{aligned}$$ In the large $L$ limit, the WKB result is increasingly valid, and $|T|^2= \exp (-\frac{\pi (m^2+p_T^2) \, \tilde{c}^3}{q E \, \hbar})$ [@CNN]. Evaluating $T$ within the WKB approximation for arbitrary $L$, and equating $ \frac{d^2 N}{ d t\, d W } $ with $\Gamma_{2+1}$, one arrives at the 2+1 dimensional Schwinger formula of Eq. (\[2plus1G\]) for $L^2 \gg \hbar \tilde{c}/q E .$ If $L$ is not large one must compute $T$ from the Dirac equation and numerically integrate over $\epsilon$ and $p_T$. The ratio of the rate at finite $L$ to the Schwinger rate depends only on the ratio $L / \sqrt{\hbar \tilde{c} /q E }$. The numerically calculated rate smoothly converges to the Schwinger rate; see Fig. \[finiteL\]. It should be noted that while the preceding derivation is modeled on the derivation (in 3+1 dimensions) of Ref. [@Wang_Wong] there is one crucial distinction: Ref. [@Wang_Wong] computed $\Gamma$ (the vaccum decay rate) which was implicitly assumed to be the pair production rate $w$. This difference only appears in the last line of Eq. (\[rate\]): to compute $w_{2+1}$ rather than $\Gamma_{2+1}$ one replaces $-\log ( 1- |T(\epsilon,p_T)|^2 )$ (used in Ref. [@Wang_Wong]) with $|T(\epsilon,p_T)|^2$. A proposed experiment ===================== The previous derivation depends on the system being in the in-state vacuum. Intuitively, after a certain transient time during which the system equilibrates, the rate should be dominated by incoming levels from far away. Since the time scale for such transient behavior is finite at infinite $L$[@Neuberger], the natural time scale associated with transients does not depend on $L$. This transient time scale can be obtained via simple dimensional analysis: $\tau_{\rm trans} = \sqrt{{\hbar}/{(\tilde{c} \, q E)}}$. ![The ratio of the pair production rate at finite $L$ to the (infinite $L$) Schwinger formula rate for a 2+1 dimensional system. $L$ is measured in units of $L_0 \equiv {\sqrt {\hbar \tilde{c} / (q E)}}$. []{data-label="finiteL"}](LLLfiniteLLL.eps){width="3in"} The analysis crucially requires the fermionic “vacuum” to be placed in an external electric field which remains constant over time. To experimentally realize this with graphene, one must fix an external voltage by placing a region of the graphene sheet between two conducting plates, held at [*constant*]{} voltage. Pair production is driven by differences in level occupation on either side of the field. Particles and holes created in the field region are carried into the regions of the graphene sample on either side of the field regions which serve as [*reservoirs*]{} for particles and holes. After a finite time, the accessible states fill substantially and the system leaves the regime of validity of the Schwinger formula; the reservoirs develop an excess of holes or particles and the system more closely resembles p-n junctions, under extensive study in the context of Landau-Zener tunneling[@GPNJ]; see Fig. \[electronic\_structure\]. The natural timescale needed to deplete a substantial fraction of the reservoir and depart from the Schwinger regime is $\tau_{\rm fill} \equiv \sqrt{ q E/(\hbar \tilde{c}^3)}\, L L_{\rm res}$. An experiment to test the Schwinger formula is thus conceptually straightforward: a sheet of graphene of width $W$, and length $2 L_{\rm res} + L$ is placed in an apparatus whose cross-section is given schematically in Fig. \[experiment\]; $L$ is the length the field region and $L_{\rm res}$ is the length of the reservoirs to either side. An electric field of magnitude $V_0/L$ is turned at $t=0$; the Schwinger formula (\[2plus1G\]) should be accurate for $\tau_{\rm trans } \ll t \ll\tau_{\rm fill}$, bringing about a two-dimensional current density ${\cal J}$ from the Schwinger pairs. For $\tau_{\rm trans } \ll t \ll \tau_{\rm fill}$, the current density just beyond the field regions to good approximation is $${\cal J}_{\rm Sch}\equiv q \, \Gamma^{2+1} \ L = \, \frac{ q \,( q V_0)^{3/2}}{ \pi^2 \, \hbar^{3/2} \, \tilde{c}^{1/2} \, L^{1/2} } \; . \label{dens}$$ As charge flows into the reservoirs, an external current $I={\cal J}_{\rm Sch} W$ must flow to the conductors to maintain them at fixed voltages of $\pm V_0/2$. This current $I$ can be monitored to determine ${\cal J}$. Note that the graphene sheet is fully insulated electrically and is [*not*]{} part of a closed circuit. Thus, the current is necessarily transient and the experiment does [*not*]{} measure the usual conductance. This proposed method of testing Schwinger mechanism in graphene has a key feature in common with conductance in graphene p-n junctions (GPNJ) in the ballistic regime: both systems depend on quantum tunnelling of massless Dirac particles[@GPNJ]. Effects due to non-zero temperature, non-zero lattice spacing, impurities, finite size effects and temporal transients present in real systems can affect the results of the measurements. One expects to be in the regime of validity for the Schwinger formula if the parameters satisfy: $$\begin{aligned} \sqrt{\frac{\, q E_0 }{\hbar \tilde{c}^3}} \, L \, L_{\rm res}&\gg& t \gg \sqrt{\frac{\hbar }{q E_0\tilde{c}}} \label{tcond}\\ L , \, W , \, l_{\rm mfp} & \gg &\sqrt{\frac{\hbar \tilde{c}}{q E_0}} \label{Lcond}\\ V_0 & \ll & \frac{\hbar \tilde{c} }{ q a} \approx 2.5 V \label{Vcond}\\ T & \ll & \sqrt[4]{\hbar \tilde{c} ( q E_0)^3 L^2/k_B^4} \; , \label{Tcond}\end{aligned}$$ where $E_0 \equiv V_0/L$, $a$ is the lattice spacing and $l_{\rm mfp}$ is an effective mean-free path. Conditions (\[tcond\]) and (\[Lcond\]) for $L$ relate to time transient effects and finite length issues associated with the WKB formalism, respectively; they are discussed above. The analogous condition for $W$ follows from the requirement that the discrete mode sum in $p_T$ is approximated by the Gaussian integral in Eq. (\[rate\]). Since the distance scale for the creation of pairs in the Schwinger mechanism is $L_0 \equiv \sqrt{\hbar \tilde{c}/ (q E)}$, the Schwinger formula only applies when the dynamics are well described by the simple Dirac description over that scale, requiring $l_{mfp} \gg L_0$. Condition (\[Vcond\]) encodes the requirement of a linear dispersion relation; this fails for momenta comparable to $a$. Finally, Condition (\[Tcond\]) applies to thermal fluctuations, which imply a density of particles and holes in the reservoirs. These can randomly wander into, and get transported across, the field region, creating a current. The condition ensures that the Schwinger current dominates. ![Schematic depiction of the cross-sectional view of a possible experiment measuring the rate of production of Schwinger pairs. []{data-label="experiment"}](experiment.eps){width="3.3in"} The experimental configuration does not directly measure conductivity since current flow is necessarily transient. Still one might worry that the dynamics associated with the usual conductivity could mask the Schwinger effect. However, Eqs. (\[2plus1G\]) and (\[dens\]) imply that ${\cal J}_{\rm Sch} = \left( \frac{4 q^2 E_0}{h} \right ) \left [ \frac{ 1}{2 \pi}\right ] \sqrt{ \frac{L^2 q E_0}{\hbar \tilde{c}}}$. The term in parenthesis is of the scale expected from standard conductivity mechanisms: in graphene it is $\sigma \sim 4 q^2/h$[@GrExp2]. The term in square brackets is of order unity; Condition (\[Lcond\]) implies that the square root factor is large. Thus, the transient current density induced by the Schwinger mechanism dominates over what is expected from the usual conductivity. Whether Conditions (\[tcond\])-(\[Tcond\]) can be satisfied in practice depends critically on both the size of the graphene sample and its purity. In practice, with samples with sizes restricted to $\sim 100 \, \mu {\rm m}$ and impurity concentrations reported in [@GrExpMFP; @rev], the conditions appear to require extremely fast measurements but do not seem to be beyond current technology. For the purpose of making estimates we will assume that the size of the sample is $\sim 100 \, \mu {\rm m}$; for concreteness we will take $W =100 \, \mu {\rm m}$, $L =1 \, \mu {\rm m}$ and $L_{\rm res}= 49 \, \mu {\rm m}$. We also take $V_0 =1$V which effectively satisfies Condition (\[Vcond\]) [@rev]. With the values above,$\sqrt{\hbar \tilde{c}/q E_0} = 25 {\rm nm}$, and Condition (\[Lcond\]) is satisfied by a factor of 40 for $L$ and a factor of 4000 for $W$. Condition (\[Tcond\]) is also well satisfied provided $T \ll 1800^\circ$K. Condition (\[Lcond\]) for $l_{mfp}$ depends on a quasi-particle’s energy, $\epsilon$. It has been argued with current samples the dominant contribution is from Coulomb impurities[@GrExpMFP]. For the purposes of estimating $l_{mfp}$, this will be assumed to be correct. Coulomb scattering has an infinite cross-section and the mean-free path is not well defined. However, an effective mean free path in the sense of the characteristic distance a quasiparticle travels before it is substantially affected by the impurities can be estimated. It is the distance a quasi-particle travels before entering a region in which the Coulomb energy is comparable to the kinetic energy, yielding $l_{mfp} \sim \kappa \epsilon (\hbar \, c \, \alpha \, n_{imp})^{-1}$ where $\kappa$ the dielectric constant of the insulator and $n_{imp}$ is the density of impurities. For Schwinger pairs, $\epsilon$ is of order (but less than) $q V_0$; Condition (\[Lcond\]) becomes $ \frac{ \kappa \epsilon}{\hbar c \alpha n_{imp}} \gg \sqrt{\frac{\hbar \tilde{c} L}{q V_0}}$. Samples with $n_{imp}$ as small as $\sim 2 \times 10^{11} cm^{-2}$ have been reported. Using this value, the left-hand side of the inequality is 350 $\kappa$ nm. The right-hand side is 25 nm. Thus, the condition appears to be satisfied at least moderately well. Using the results in Fig. \[finiteL\] as a guide in estimating errors suggests that at the least, a semi-quantitative test of the Schwinger mechanism should be possible. If a substrate with relatively large $\kappa$ proves viable, the condition may be well-satisfied. It appears possible that the mean-free paths are long enough to allow for a meaningful test of the Schwinger mechanism with currently available samples. Precision tests will probably require cleaner samples which one hopes may become available in the future. With the parameters given above, $\tau_{fill} \equiv \sqrt{\frac{\, q V_0 L}{\hbar \tilde{c}^3}} \, L_{\rm res} \approx 1.9 \times 10^{-9} {\rm s}$ while $\tau_{trans}\equiv \sqrt{\frac{\hbar L }{q V_0\tilde{c}}} \approx 2.6 \times 10^{-14} {\rm s}$. It is easy to ensure $t \gg \tau_{trans}$. The restriction $t \ll \tau_{fill}$, however, requires taking data at a very high rate—considerably faster than one GHz. Fortunately, it [*is*]{} possible to take data at rates much faster than one GHz. If future sample sizes increase significantly, one could increase the size of $L_{res}$ and thereby $\tau_{fill}$ and thus reduce the technical challenges associated with very rapid measurements. Conclusion ========== To summarize, even with presently available samples, there is good reason to believe that the regime of validity of the Schwinger formula can be realized experimentally, at least at a semi-quantitative level. As larger and higher quality samples become available in the future, practical tests of the Schwinger formula with increasing accuracy ought to become possible. It is reasonable to expect that such experimental probes should become sufficient to test quantitatively the pair production of the Schwinger mechanism; it is is important that such measurements are accurate enough to distinguish between the predicted rate of pair of production and the rate of vacuum decay. [*Acknowledgments.*]{} The authors benefitted greatly from discussions with S. Adam, S. Das Sarma, M. Fogler, M. Furher, V. Galitsky, S.P. Gavrilov, T. Jacobson, K.G. Klimenko, K.S. Novoselov and I.A. Shovkovy. D.A. and D.A.M were supported by the University of Maryland through its Senior Summer Scholars program. T.D.C. was supported by the United States D.O.E. through grant number DE-FGO2-93ER-40762. [99]{} J. Schwinger, Phys. Rev.  [**82**]{}, 664 (1951). C. Itzykson and J.-B. Zuber, [*Quantum Field Theory*]{}, McGraw-Hill (1980). A. Casher, H Neuberger and S. Nussinov, Phys. Rev. D [**20**]{}, 179 (1979). H. Neuberger , Phys. Rev. D [**20**]{} 2936 (1979). 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--- abstract: 'For a large class of possibly singular complete intersections we prove a formula for their Chern-Schwartz-MacPherson classes in terms of a single blowup along a scheme supported on the singular loci of such varieties. In the hypersurface case our formula recovers a formula of Aluffi proven in 1996. As our formula is in no way tailored to the complete intersection hypothesis, we conjecture that it holds for all closed subschemes of a smooth variety. If in fact true, such a formula would provide a simple characterization of the Chern-Schwartz-MacPherson class which does not depend on a resolution of singularities. We also show that our formula may be suitably interpreted as the Chern-Fulton class of a scheme-like object which we refer to as an ‘$\mathfrak{f}$-scheme’.' address: - | Department of Mathematics\ University of Hong Kong\ Pokfulam Road, Hong Kong. - | Department of Mathematics\ Dongbei University of Finance and Economics\ 217 Jianshan St, Shahekou District, Dalian, Liaoning, China. author: - James Fullwood and Dongxu Wang bibliography: - 'CSMC2.bib' title: 'Towards a simple characterization of the Chern-Schwartz-MacPherson class' --- Introduction ============ Influenced by ideas of Grothendieck, in the 1960s Deligne conjectured the existence of a natural transformation $$c_*:\mathscr{C}\to H_*,$$ where $\mathscr{C}$ is the covariant constructible function functor and $H_*$ is the integral homology functor, such that for a smooth complex algebraic variety $X$ $$c_*(\mathbbm{1}_X)=c(TX)\cap [X] \in H_*X,$$ where $\mathbbm{1}_X$ denotes the indicator function of $X$ and $c(TX)\cap [X]$ denotes the total homological Chern class of $X$. Such a natural transformation is necessarily unique, and the existence of such would deem the class $c_*(\mathbbm{1}_X)$ for singular $X$ a natural extension of Chern class to the realm of singular varieties. Moreover, functoriality would imply $$\int_X c_*(\mathbbm{1}_X)=\chi(X),$$ where the integral sign is notation for taking the degree-zero component of a homology class and $\chi(X)$ denotes topological Euler characteristic with compact support, thus generalizing the Gau[ß]{}-Bonnet-Chern theorem to the singular setting. In 1974 MacPherson explicitly constructed such a $c_*$, thus proving Deligne’s conjecture [@RMCC]. Then in 1981, Brasselet and Schwartz showed that the class $c_*(\mathbbm{1}_X)$ was the Alexander-dual in relative cohomology of a class constructed by Schwartz in the 1960s using radial vector fields [@BSCC], thus MacPherson’s Chern class $c_*(\mathbbm{1}_X)$ eventually became known as the *Chern-Schwartz-MacPherson class* – or *CSM class* for short – which from here on will be denoted $c_{\text{SM}}(X)$. Kennedy then generalized MacPherson’s construction to hold over an arbitrary algebraically closed field of characteristic zero, for which the integral homology functor is promoted to the Chow group functor $A_*$ [@KCC], and this is the context in which we will work throughout. While the functorial CSM classes occupy a central role in the study of characteristic classes of singular varieties, MacPherson’s construction of $c_*$ involves such machinery as the graph construction, local Euler obstructions and Nash blowups, thus rendering them quite difficult to define, let alone compute. As such, simple characterizations of the CSM class are quite desirable for anyone interested in their study. Aluffi has given two characterizations for a general variety over an algebraically closed field of characteristic zero in terms of a resolution of singularities [@ADF][@ALCG], and also a characterization for a subvariety of projective space in terms of its general linear sections [@AE]. Our aim here is then to suggest a characterization of the CSM class which does not depend on resolution of singularities, not only for the sake of simplicity, but in the hopes that such a characterization may be used to generalize the CSM class to fields of positive characteristic. We take as our starting point a formula proven by Aluffi in the hypersurface case, which depends on the notion of the *singular scheme* of a variety. For $X$ a subvariety of a smooth variety $M$ (over an algebraically closed field of characteristic zero), let $\mathcal{J}_X$ be the subscheme of $X$ whose ideal sheaf is locally generated by the $m\times m$ minors of the matrix valued function $$(a_{ij})=\frac{\partial F_i}{\partial x_j},$$ where $F_i=0$ are the defining equations for $X$ and $m$ is the codimension of $X$ in $M$. We refer to $\mathcal{J}_X$ as the singular scheme of $X$, as it is an intrinsic subscheme of $X$ supported on its singular locus. In [@ASCC], Aluffi proves that if $X$ is hypersurface then $$\label{mf} c_{\text{SM}}(X)=c(TM)\cap p_*\left(\frac{\widetilde{X}-E}{1+\widetilde{X}-E}\right),$$ where $p:\widetilde{M}\to M$ is the blowup of $M$ along $\mathcal{J}_X$, $E$ denotes the class of the exceptional divisor of the blowup, $\widetilde{X}$ denotes the class of $p^{-1}(X)$, $(1+\widetilde{X}-E)^{-1}$ is notation for the inverse Chern class of $\mathscr{O}(\widetilde{X}-E)$ and $p_*$ denotes the proper pushforward of algebraic cycles associated with $p$ as defined in §1.4 of [@IT][^1]. As none of the ingredients of the RHS of equation (\[mf\]) depend on $X$ being a hypersurface, it is natural to surmise equation (\[mf\]) still holds in higher codimension, but this is not so. Moreover, while many formulas for CSM classes have appeared in the literature in the hypersurface case [@PPCC][@MM][@ASCC], little progress has been made in generalizing such formulas to higher codimension. In particular, in [@ASCC] Aluffi states of his formula “We do not know whether our result is an essential feature of hypersurfaces, or whether a formula similar to (\[mf\]) may compute (Chern-)Schwartz-MacPherson’s class of arbitrary varieties. While this is a natural question, the approach of this paper does not seem well suited to address it....”. We conjecture that the key to generalizing formula (\[mf\]) to higher codimension lies in the simple observation that in the hypersurface case, the blowup of $M$ along $\mathcal{J}_X$ coincides with the blowup of $M$ along the *scheme-theoretic union* $X\cup \mathcal{J}_X$ (i.e., the scheme whose ideal sheaf is the product of the ideal sheaves of $X$ and $\mathcal{J}_X$), and that it is precisely the blowup along $X\cup \mathcal{J}_X$ one should use for the generalization of (\[mf\]) to higher codimension. We provide evidence for this conjecture by proving it for complete intersections in arbitrary codimension which we refer to as *almost smooth*. If $X$ is a complete intersection in $M$ of codimension $m$ we refer to it as almost smooth if there exist $m$ hypersurfaces $X_1,\ldots, X_m$ in $M$ such that $X=X_1\cap \cdots \cap X_m$ with $X_1\cap \cdots \cap X_{m-1}$ being smooth. All hypersurfaces are vacuously almost smooth. The main result of this note is then given by the following \[mt\] Let $X$ be an almost smooth complete intersection in a smooth variety $M$ and let $p:\widetilde{M}\to M$ be the blowup of $M$ along $X\cup \mathcal{J}_X$. Then $$\label{mf1} c_{\emph{SM}}(X)=c(TM)\cap p_*\left(\frac{\widetilde{X}-E}{1+\widetilde{X}-E}\right),$$ where $\widetilde{X}$ and $E$ denote the classes of $p^{-1}(X)$ and the exceptional divisor of the blowup respectively. The proof of Theorem \[mt\] is given in §\[proof\]. We note that the moniker almost smooth is not to imply that the singularities of an almost smooth complete intersection are necessarily mild. For example, for a hypersurface in projective space with arbitrary singularities all of its general linear sections are almost smooth complete intersections. The almost smooth assumption on $X$ implies that the ideal sheaf $\mathscr{I}=\mathscr{I}_X\cdot \mathscr{I}_{\mathcal{J}_X}$ of the closed subscheme $X\cup \mathcal{J}_X\hookrightarrow M$ is of *linear type*, which means the canonical surjection $$\label{ilt} \text{Sym}^d(\mathscr{I})\to \mathscr{I}^d$$ is an isomorphism for all $d$, and this is crucial for our proof of Theorem \[mt\]. An algorithm to compute CSM classes of almost smooth complete intersections was developed in [@MHP], for which many explicit examples appeared. In any case, it would be surprising if Theorem \[mt\] didn’t hold sans the almost smoothness assumption, as formula (\[mf1\]) is in no way tailored to it. We conclude with $\S\ref{fs}$, where – in the spirit of unification of Chern classes for singular varieties – we show that formula $(\ref{mf1})$ may be interpreted as the *Chern-Fulton class* of an object we refer to as an $\mathfrak{f}$-*scheme*, which we think of as a scheme with ‘negatively thickened’ components. *A word of caution.* For the sake of aesthetics, we will often make an abuse of notation by making no notational distinction between a class and its pushforward and/or pull back by an inclusion map, and the same goes for bundles and their restrictions. Proof of main theorem {#proof} ===================== Before proving Theorem \[mt\] we introduce some notation which will streamline our computations. So let $S$ be an algebraic scheme over a field, denote its Chow group by $A_*S$ and denote by $d$ the dimension of the largest component of $S$. For $\alpha=\sum_i\alpha^i\in A_*S$ with $\alpha^i\in A_{(d-i)}S$ we let $$\label{dn} \alpha^{\vee}=\sum_i(-1)^i\alpha^i,$$ and refer to it as the *dual* of $\alpha$. We remark that by replacing $-1$ by a positive integer $n$ in formula (\[dn\]) yields the $n$-th *Adams* of $\alpha$, usually denoted $\alpha^{(n)}$. Thus we may think of $a^{\vee}$ as the ‘$-1$th Adams’ of $\alpha$. For a line bundle $\mathscr{L}\to S$ we let $$\label{tn} \alpha \otimes \mathscr{L}=\sum_i\frac{\alpha^i}{c(\mathscr{L})^i},$$ and refer to it as $\alpha$ *tensor* $\mathscr{L}$. After identifying $\mathscr{L}$ with its class in the Picard group $\text{Pic}(S)$ it is then straightforward to show the map $\alpha\mapsto \alpha \otimes \mathscr{L}$ defines an action of $\text{Pic}(S)$ on $A_*S$, so that for any other line bundle $\mathscr{M}\to S$ we have $$\label{tf2} (\alpha \otimes \mathscr{L})\otimes \mathscr{M}=\alpha \otimes (\mathscr{L}\otimes \mathscr{M}).$$ This fact is proven in [@MFCH], along with the fact that if $\mathscr{E}$ is a class in the Grothendieck group of vector bundles on $S$ then $$\label{df} \left(c(\mathscr{E})\cap \alpha\right)^{\vee}=c(\mathscr{E}^{\vee})\cap \alpha^{\vee},$$ and $$\label{tf} \left(c(\mathscr{E})\cap \alpha\right) \otimes \mathscr{L}=\frac{c(\mathscr{E}\otimes \mathscr{L})}{c(\mathscr{L})^r}\cap \left(\alpha \otimes \mathscr{L}\right),$$ where $r\in \mathbb{Z}$ denotes the rank of $\mathscr{E}$. We will also need the following Let $Y$ be a closed subscheme of $S$ .The *Segre class* of $Y$ (relative to $S$) is denoted $s(Y,S)$, and is defined as $$\label{scd} s(Y,S)=\begin{cases} c(N_YS)^{-1}\cap [Y] \quad \quad \quad \text{for $Y$ regularly embedded in $S$} \\ f_*\left(c(N_E\widetilde{S})^{-1}\cap [E]\right) \quad \hspace{1.05cm} \text{otherwise,}\end{cases}$$ where $c(N_YS)$ denotes the Chern class of the normal bundle to $Y$ in $S$ (in the case that $Y$ is regularly embedded), $f:\widetilde{S}\to S$ is the blowup of $S$ along $Y$ with exceptional divisor $E$ and $N_E\widetilde{S}$ denotes the normal bundle to $E$ in $\widetilde{S}$. We now recall the assumptions of Theorem \[mt\]. So let $M$ be a smooth variety over an algebraically closed field of characteristic zero, and let $X=X_1\cap \cdots \cap X_m$ be a complete intersection of $m$ hypersurfaces in $M$ such that $Z=X_1\cap \cdots \cap X_{m-1}$ is smooth. We denote the blowup of $M$ along $X\cup \mathcal{J}_X$ by $p:\widetilde{M}\to M$, where $\mathcal{J}_X$ denotes the singular scheme of $X$ and $X\cup \mathcal{J}_X$ is the subscheme of $M$ corresponding to the ideal sheaf $\mathscr{I}_X\cdot \mathscr{I}_{\mathcal{J}_X}$, i.e., the product of the ideal sheaves of $X$ and $\mathcal{J}_X$. We first prove the following \[l1\] Let $k$ be a positive integer and let $X\cup \mathcal{J}_X^k$ be the closed subscheme of $M$ corresponding to the ideal sheaf $\mathscr{I}_X\cdot \mathscr{I}^k_{\mathcal{J}_X}$. Then $$s(X\cup \mathcal{J}_X^k,Z)=s(X,Z)+c(\mathscr{O}(X))^{-1}\cap \left(s(\mathcal{J}_X,Z)^{(k)}\otimes_Z \mathscr{O}(X)\right),$$ where $\mathscr{O}(X)$ is the line bundle on $Z$ corresponding to the divisor $X$, and $s(\mathcal{J}_X,Z)^{(k)}$ denotes the $k$th-Adams of $s(\mathcal{J}_X,Z)$. The proof of this fact follows along the lines of the proof of Proposition 3 in [@MFCH]. Indeed, let $q:\widetilde{Z}\to Z$ be the blowup of $Z=X_1\cap \cdots \cap X_{m-1}$ along $\mathcal{J}_X$, viewing $\mathcal{J}_X$ as a subscheme of $Z$, denote the exceptional divisor of the blowup by $E$ and denote the class of $q^{-1}(X)$ by $\widetilde{X}$. By Proposition 4.2 (a) of [@IT], we have $$\begin{aligned} s(X\cup \mathcal{J}^k_X,Z)&=&q_*\left(s(q^{-1}(X\cup \mathcal{J}^k_X),\widetilde{Z})\right) \\ &=&q_*\left(\frac{\widetilde{X}+kE}{1+\widetilde{X}+kE}\right) \\ &=&q_*\left((\widetilde{X}+kE)\otimes \mathscr{O}(\widetilde{X}+kE)\right),\end{aligned}$$ where the last equality comes from formula (\[tn\]). Now we compute: $$\begin{aligned} q_*\left((\widetilde{X}+kE)\otimes \mathscr{O}(\widetilde{X}+kE)\right)&=&q_*\left((\widetilde{X}+kE)\otimes (\mathscr{O}(kE)\otimes \mathscr{O}(\widetilde{X}))\right) \\ &=&q_*\left(\left(\widetilde{X}\otimes \mathscr{O}(kE)+kE\otimes \mathscr{O}(kE)\right)\otimes \mathscr{O}(\widetilde{X})\right) \\ &=&q_*\left(\left(\frac{\widetilde{X}}{1+kE}+\frac{kE}{1+kE}\right)\otimes \mathscr{O}(\widetilde{X})\right) \\ &=&q_*\left(\left(\widetilde{X}-\frac{\widetilde{X}\cdot kE}{1+kE}+\frac{kE}{1+kE}\right)\otimes \mathscr{O}(\widetilde{X})\right) \\ &=&q_*\left(\left(\widetilde{X}+c\left(\mathscr{O}(-\widetilde{X})\right)\cap \frac{kE}{1+kE}\right)\otimes \mathscr{O}(\widetilde{X})\right) \\ &=&q_*\left(s(\widetilde{X},\widetilde{Z})+c(\mathscr{O}(\widetilde{X}))^{-1}\cap \left(s(E,\widetilde{Z})^{(k)}\otimes \mathscr{O}(\widetilde{X})\right)\right) \\ &=&s(X,Z)+c(\mathscr{O}(X))^{-1}\cap \left(s(\mathcal{J}_X,Z)^{(k)}\otimes \mathscr{O}(X)\right),\end{aligned}$$ where $s(\mathcal{J}_X,Z)^{(k)}$ denotes the $k$th Adams of $s(\mathcal{J}_X,Z)$. The second to last equality follows from the definition of Segre class and formula \[tf\], while the last equality follows from the projection formula. This proves the lemma. We now proceed with the For every positive integer $k$ we may factor the embedding $X\cup \mathcal{J}^k_X\hookrightarrow M$ as $X\cup \mathcal{J}^k_X\hookrightarrow Z\hookrightarrow M$, and since $Z$ is smooth, this implies the ideal sheaf of $X\cup \mathcal{J}^k_X$ is of linear type (as defined via equation (\[ilt\])), thus by Theorem 2 of [@KLI] we have $$s(X\cup \mathcal{J}^k_X,M)=c(N_ZM)^{-1}\cap s(X\cup \mathcal{J}^k_X,Z).$$ Moreover, by the birational invariance of Segre classes (Proposition 4.2 (a) of [@IT]), we have (recall $p$ denotes the blowup of $M$ along $X\cup \mathcal{J}_X$) $$s(X\cup \mathcal{J}^k_X,M)=p_*\left(\frac{\widetilde{X}+kE}{1+\widetilde{X}+kE}\right),$$ where $E$ and $\widetilde{X}$ denote the class of the exceptional divisor and the pullback of $X$ respectively. Thus by Lemma \[l1\] we have $$\label{e1} p_*\left(\frac{\widetilde{X}+kE}{1+\widetilde{X}+kE}\right)=c(N_ZM)^{-1}\cap \left(s(X,Z)+c(\mathscr{O}(X))^{-1}\cap \left(s(\mathcal{J}_X,Z)^{(k)}\otimes_Z \mathscr{O}(X)\right)\right),$$ where we use a subscript $Z$ on the tensor notation to emphasize the fact that we are tensoring the class $s(\mathcal{J}_X,Z)^{(k)}$ with respect to codimension in $Z$. Now for $k=-1$, the geometric meaning of the equation (\[e1\]) as the Segre class of a closed subscheme of $M$ is lost, but equation (\[e1\]) still holds nonetheless. And since the dual of a class may be suitably interpreted as its ‘$-1$th Adams’, we have $$\label{e2} p_*\left(\frac{\widetilde{X}-E}{1+\widetilde{X}-E}\right)=c(N_ZM)^{-1}\cap \left(s(X,Z)+c(\mathscr{O}(X))^{-1}\cap \left(s(\mathcal{J}_X,Z)^{\vee}\otimes_Z \mathscr{O}(X)\right)\right),$$ where we recall $s(\mathcal{J}_X,Z)^{\vee}$ denotes the dual of $s(\mathcal{J}_X,Z)$ (\[dn\]). Now let $$\mathscr{E}=\mathscr{O}(X_1) \oplus \cdots \oplus \mathscr{O}(X_m),$$ and note that the restriction to $Z$ of the bundles $\mathscr{O}(X_1)\oplus \cdots \oplus \mathscr{O}(X_{m-1})$ and $\mathscr{O}(X_m)$ coincide with its normal bundle $N_ZM$ and $\mathscr{O}(X)$ respectively, so that $c(N_ZM)c(\mathscr{O}(X))=c(\mathscr{E})$. Thus $$\begin{aligned} p_*\left(\frac{\widetilde{X}-E}{1+\widetilde{X}-E}\right)&=&c(N_ZM)^{-1}\cap \left(s(X,Z)+c(\mathscr{O}(X))^{-1}\cap \left(s(\mathcal{J}_X,Z)^{\vee}\otimes_Z \mathscr{O}(X)\right)\right) \\ &=&s(X,M)+c(\mathscr{E})^{-1}\cap \left(s(\mathcal{J}_X,Z)^{\vee}\otimes_Z \mathscr{O}(X)\right) \\ &=&s(X,M)+c(\mathscr{E})^{-1}\cap \left(\left(c(N_ZM)\cap s(\mathcal{J}_X,M)\right)^{\vee}\otimes_Z \mathscr{O}(X)\right) \\ &=&s(X,M)+c(\mathscr{E})^{-1}\cap \left(\left(c(N_ZM)^{\vee}\cap s(\mathcal{J}_X,M)^{\vee}\right)\otimes_Z \mathscr{O}(X)\right), \\\end{aligned}$$ where in the second and third equalities we used the fact that $$c(N_ZM)^{-1}\cap s(X,Z)=s(X,M) \quad \text{and} \quad s(\mathcal{J}_X,Z)=c(N_ZM)\cap s(\mathcal{J}_X,M)$$ (both of which follow by Theorem 2 of [@KLI]), and in the fourth equality we use formula (\[df\]). Our theorem is then proved once we show $$c(TM)\cap \left(s(X,M)+c(\mathscr{E})^{-1}\cap \left(\left(c(N_ZM)^{\vee}\cap s(\mathcal{J}_X,M)^{\vee}\right)\otimes_Z \mathscr{O}(X)\right)\right)=c_{\text{SM}}(X).$$ For this, let $$M(X)=c(\mathscr{E})^{-1}\cap \left(\left(c(N_ZM)^{\vee}\cap s(\mathcal{J}_X,M)^{\vee}\right)\otimes_Z \mathscr{O}(X)\right).$$ As Theorem 1.1 of [@FMC] is equivalent to the statement that $$c_{\text{SM}}(X)-c(TM)\cap s(X,M)=c(TM)\cap \left((-1)^{m-1}\frac{c(\mathscr{E}^{\vee}\otimes \mathscr{O}(X_m))}{c(\mathscr{E})}\cap \left(s(\mathcal{J}_X,M)^{\vee}\otimes \mathscr{O}(X_m)\right)\right),$$ showing $$M(X)=(-1)^{m-1}\frac{c(\mathscr{E}^{\vee}\otimes \mathscr{O}(X_m))}{c(\mathscr{E})}\cap \left(s(\mathcal{J}_X,M)^{\vee}\otimes \mathscr{O}(X_m)\right)$$ then finishes the proof of the theorem. Indeed, $$\begin{aligned} M(X)&=&c(\mathscr{E})^{-1}\cap \left(\left(c(N_ZM^{\vee})\cap s(\mathcal{J}_X,M)^{\vee}\right)\otimes_Z \mathscr{O}(X)\right) \\ &=&(-1)^{m-1}\frac{c(\mathscr{O}(X_m))^{m-1}}{c(\mathscr{E})}\cap \left((c(N_ZM^{\vee})\cap s(\mathcal{J}_X,M)^{\vee})\otimes_M \mathscr{O}(X_m)\right) \\ &\overset{(\ref{tf})}=&(-1)^{m-1}\frac{c(\mathscr{O}(X_m))^{m-1}}{c(\mathscr{E})}\cap \left(\frac{c(N_ZM^{\vee}\otimes \mathscr{O}(X_m))}{c(\mathscr{O}(X_m))^{m-1}}\cap (s(\mathcal{J}_X,M)^{\vee}\otimes_M \mathscr{O}(X_m))\right)\\ &=&(-1)^{m-1}\frac{c(\mathscr{E}^{\vee}\otimes \mathscr{O}(X_m))}{c(\mathscr{E})}\cap(s(\mathcal{J}_X,M)^{\vee}\otimes_M \mathscr{O}(X_m)),\end{aligned}$$ where in the second equality we used the fact that for all classes $\alpha$ we have (see Lemma 2.1 of [@FMC]) $$\alpha\otimes_{Z} \mathscr{O}(X) =c(\mathscr{O}(X_m))^{m-1} \cap \left(\alpha\otimes_M \mathscr{O}(X_m)\right),$$ and the factor of $(-1)^{m-1}$ appears due to the fact that we switch from taking duals in $Z$ to duals in $M$. In the last equality we used that since $\mathscr{E}=N_ZM\oplus \mathscr{O}(X_m)$, $\mathscr{E}^{\vee}\otimes \mathscr{O}(X_m)=(N_ZM^{\vee}\otimes \mathscr{O}(X_m))\oplus \mathscr{O}$, thus $$c(\mathscr{E}^{\vee}\otimes \mathscr{O}(X_m))=c(N_ZM^{\vee}\otimes \mathscr{O}(X_m)),$$ which concludes the proof. CSM classes via Chern-Fulton classes of $\mathfrak{f}$-schemes {#fs} ============================================================== There are various notions of Chern class for singular varieties and schemes which coincide with the usual Chern class in the smooth case. One such class is the *Chern-Fulton class*, which is defined for all closed subschemes of a smooth variety $M$ over an arbitrary field. In contrast to the CSM classes, they are easy to define. In particular, for a closed subscheme $V\hookrightarrow M$ its Chern-Fulton class $c_{\text{F}}(V)$ is given by $$c_{\text{F}}(V)=c(TM)\cap s(V,M).$$ In Example 4.2.6 of [@IT], Fulton proves his classes are intrinsic to the scheme $V$, and are thus independent of any embedding of $V$ in a smooth variety. While CSM classes generalize the Gau[ß]{}-Bonnet-Chern theorem to singular varieties, Chern-Fulton classes provide a deformation-invariant extension of the Gau[ß]{}-Bonnet-Chern theorem, since if $\mathcal{Z}\to \Delta$ is a family over a disk $\Delta\subset \mathbb{C}$ whose fibers are all smooth except for possibly the central fiber, then for all $t\neq 0$ in $\Delta$ we have $$\int_{Z_0} c_{\text{F}}(Z_0)=\chi(Z_t),$$ where $Z_0$ denotes the central fiber of the family and $Z_t$ denotes the fiber over $t\neq 0$. Furthermore, Chern-Fulton classes are sensitive to possible non-reduced scheme structure, while it is known that the CSM class of a non-reduced scheme coincides with the CSM class of its reduced support. In spite of these differences however, the Chern-Schwartz-MacPherson class and the Chern-Fulton class are closely related. In particular, if $X$ is a complex hypersurface with isolated singularities then $$\int_X c_{\text{SM}}(X)-c_{\text{F}}(X)=\sum_{x_i\in \text{Sing}(X)} \mu(x_i),$$ where $\mu(x_i)$ denotes the *Milnor number* of the singular point $x_i\in \text{Sing}(X)$. For arbitrary $X$ the class $$\mathcal{M}(X)=c_{\text{SM}}(X)-c_{\text{F}}(X)$$ is then an invariant of the singularities of $X$ which generalizes the notion of global Milnor number to all varieties and schemes, and is referred to as the *Milnor class* of $X$. In this section we take this relationship between the two classes a step further, and show how the main formula (\[mf1\]) of Theorem \[mt\] may be interpreted as the Chern-Fulton class of a formal object we refer to as an $\mathfrak{f}$-*scheme*. So let $X$ be an almost smooth complete intersection in a smooth variety $M$ (over an algebraically closed field of characteristic zero), and let $p:\widetilde{M}\to M$ denote the blowup of $M$ along the singular scheme $\mathcal{J}_X$ of $X$. We recall that for all $k>0$ Proposition 4.2 (a) of [@IT] implies $$p_*\left(\frac{\widetilde{X}+kE}{1+\widetilde{X}+kE}\right)=s(X\cup \mathcal{J}^k_X,M),$$ where $X\cup \mathcal{J}^k_X$ denotes the closed subscheme of $M$ corresponding to the ideal sheaf $\mathscr{I}_X\cdot \mathscr{I}^k_{\mathcal{J}_X}$. It immediately follows that $$c_{\text{F}}(X\cup \mathcal{J}^k_X)=c(TM)\cap p_*\left(\frac{\widetilde{X}+kE}{1+\widetilde{X}+kE}\right).$$ Now view $k$ as a parameter. For $k=0$ we have $$c_{\text{F}}(X)=\left.c(TM)\cap p_*\left(\frac{\widetilde{X}+kE}{1+\widetilde{X}+kE}\right)\right|_{k=0},$$ while Theorem \[mt\] amounts to the assertion $$\label{ae1} c_{\text{SM}}(X)=\left.c(TM)\cap p_*\left(\frac{\widetilde{X}+kE}{1+\widetilde{X}+kE}\right)\right|_{k=-1},$$ so that $c_{\text{SM}}(X)$, $c_{\text{F}}(X)$ and $c_{\text{F}}(X\cup \mathcal{J}^k_X)$ all correspond to evaluating a single expression at different values of $k$. Moreover, since we think of the scheme $X\cup \mathcal{J}^k_X$ as a $k$th thickening of $X$ along $\mathcal{J}_X$, the CSM class of $X$ may be interpreted formally as the Chern-Fulton class of a scheme-like object which is a ‘negative thickening’ of $X$ along $\mathcal{J}_X$, or rather, as a geometric object associated with the ‘fraction’ $\mathscr{I}_X\cdot \mathscr{I}^{-1}_{\mathcal{J}_X}$. We make this qualitative interpretation more precise as follows. We first need the following Let $\mathfrak{I}$ denote the monoid generated by quasicoherent ideal sheaves over $M$, with binary operation induced by the usual product of ideal sheaves. The group of *$\mathfrak{f}$-schemes* of $M$, denoted $\mathfrak{F}(M)$, is then defined as the Grothendieck group of the monoid $\mathfrak{I}$. *Some terminology*. An $\mathfrak{f}$-scheme of the form $[\mathscr{I}_S]^{-1}$ with $S$ non-empty will be referred to as an *antischeme*. An $\mathfrak{f}$-scheme $[\mathscr{I}_A]$ different from the identity is said to be a *factor* of $[\mathscr{I}_S]$ if there exists a non-empty closed subscheme $B\hookrightarrow M$ such that $[\mathscr{I}_S]=[\mathscr{I}_A]\cdot [\mathscr{I}_B]$. An $\mathfrak{f}$-scheme $U\in \mathfrak{F}(M)$ is said to have *support* $S\cup T$ if there exists closed subschemes $S\hookrightarrow M$ and $T\hookrightarrow M$ such that $U=[\mathscr{I}_S]\cdot [\mathscr{I}_{T}]^{-1}$ with $[\mathscr{I}_S]$ and $[\mathscr{I}_T]$ having no common factors. In such a case we will often make an abuse of notation and denote $[\mathscr{I}_S]\cdot [\mathscr{I}_{T}]^{-1}\in \mathfrak{F}(M)$ with support $S\cup T$ simply by $S\cdot T^{-1}$.\ We now extend the domain of Segre classes and Chern-Fulton classes to $\mathfrak{f}$-schemes via the following \[d2\] Let $U=[\mathscr{I}_{S_1}]\cdot [\mathscr{I}_{S_2}]^{-1}\in \mathfrak{F}(M)$ be an $\mathfrak{f}$-scheme with support $S_1\cup S_2$, and let $p:\widetilde{M}\to M$ be the blowup of $M$ along $S_1\cup S_2$. Then the *Segre class* of $U$ is defined via the formula $$s(U,M)=p_*\left(\frac{\widetilde{S_1}-\widetilde{S_2}}{1+\widetilde{S_1}-\widetilde{S_2}}\right),$$ where $\widetilde{S_i}$ denotes the class of $p^{-1}(S_i)$. The Chern-Fulton class of $U$ is then given by $$c_{\text{F}}(U)=c(TM)\cap s(U,M).$$ We note that if $S_2$ in Definition \[d2\] is the empty subscheme of $M$ then $s(U,M)$ coincides with the usual Segre class $s(S_1,M)$, and if $S_1$ is empty then $s(U,M)=s(S_2,M)^{\vee}$. Now let $X$ be an almost smooth complete intersection in $M$ with singular scheme $\mathcal{J}_X$. Then the RHS of equation (\[ae1\]) coincides with the Chern-Fulton class of the $\mathfrak{f}$-scheme $X\cdot \mathcal{J}_X^{-1}$, thus Theorem \[mt\] may be reformulated in the language of $\mathfrak{f}$-schemes via the formula $$\label{mf2} c_{\text{SM}}(X)=c_{\text{F}}(X\cdot \mathcal{J}_X^{-1}),$$ where we recall $X\cdot \mathcal{J}_X^{-1}$ is notation for the $\mathfrak{f}$-scheme $[\mathscr{I}_X]\cdot [\mathscr{I}_{\mathcal{J}_X}]^{-1}$. As for the Milnor class, we then have $$\mathcal{M}(X)=c_{\text{F}}(X\cdot \mathcal{J}_X^{-1})-c_{\text{F}}(X),$$ so that $c_{\text{SM}}(X)$, $c_{\text{F}}(X)$ and $\mathcal{M}(X)$ may all be formulated in terms of Chern-Fulton classes of $\mathfrak{f}$-schemes. Moreover, we conjecture formula (\[mf2\]) holds for $X$ *any* closed subscheme of $M$. The language of $\mathfrak{f}$-schemes along with formula (\[mf2\]) yields a simple proof in the hypersurface case that CSM classes are not sensitive to non-reduced scheme structure. Indeed, let $X$ be a reduced hypersurface given by the equation $F=0$ and denote its singular scheme by $\mathcal{J}_X$. Then the scheme $X^k$ corresponding to the ideal sheaf $\mathscr{I}_X^k$ is given by $F^k=0$, and $d(F^k)=kF^{k-1}dF$. Now since $\mathcal{J}_X$ corresponds to the equation $dF=0$, the singular scheme of $X^k$ corresponds to the ideal sheaf $\mathscr{I}_X^{k-1}\cdot \mathscr{I}_{\mathcal{J}_X}$. Thus $$c_{\text{SM}}(X^k)=c_{\text{F}}(X^k\cdot (X^{(k-1)}\cdot \mathcal{J}_X)^{-1})=c_{\text{F}}(X^k\cdot (X^{(1-k)}\cdot \mathcal{J}_X^{-1}))=c_{\text{F}}(X\cdot \mathcal{J}_X^{-1})=c_{\text{SM}}(X),$$ where the first and last equalities follow from formula (\[mf2\]). We conclude with a quote from 18th century mathematician Fancis Maceres in regards to negative numbers:\ “Quantities marked with a minus sign darken the very whole doctrines of the equations, and make dark of the things which are in their nature excessively obvious and simple.” [^1]: The LHS of equation (\[mf\]) actually denotes $\iota_*c_{\text{SM}}(X)$, where $\iota:X\hookrightarrow M$ denotes the natural inclusion.
--- abstract: 'In this article, a new reproducing kernel approach is developed for obtaining numerical solution of nonlinear three-point boundary value problems with fractional order. This approach is based on reproducing kernel which is constructed by shifted Legendre polynomials. In considered problem, fractional derivatives with respect to $\alpha$ and $\beta$ are defined in Caputo sense. This method has been applied to some examples which have exact solutions. In order to shows the robustness of the proposed method, some numerical results are given in tabulated forms.' author: - Mehmet Giyas Sakar - 'Onur Sald[i]{}r' date: 'Received: date / Accepted: date' title: 'A new reproducing kernel approach for nonlinear fractional three-point boundary value problems' --- [example.eps]{} gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore Introduction ============ In this paper, a new iterative reproducing kernel approach will be constructed for obtaining the numerical solution of nonlinear fractional three-point boundary value problem, $$\begin{aligned} \label{eq1} a_{2}(\xi){\ }^{c}D^{\alpha}z(\xi)+a_{1}(\xi){\ }^{c}D^{\beta}z(\xi)+a_{0}(% \xi)z(\xi)=g(\xi,z(\xi),z^{\prime }(\xi)),\quad \xi\in\lbrack 0,1]\end{aligned}$$ with following boundary conditions, $$\begin{aligned} \label{eq2} z(0)=\gamma_{0},\,z(\theta)=\gamma_{1},\ z(1)=\gamma_{2}, \,\,\ 0<\theta<1,\ 1<\alpha\leq2,\,\ 0<\beta\leq1.\end{aligned}$$ Here, $a_{0}(\xi),$ $a_{1}(\xi),$ $a_{2}(\xi)$ $\in$ $C^{2}(0,1)$ and $% g(\xi,z)\in$ $L_{\rho}^{2}[0,1]$ are sufficiently smooth functions and fractional derivatives are taken in Caputo sense. Without loss of generality, we pay regard to $z(0)=0$, $z(\theta)=0$ and $z(1)=0$. Because, $z(0)=\gamma_{0}$, $z(\theta)=\gamma_{1}$ and $z(1)=\gamma_{2}$ boundary conditions can be easily reduced to $z(0)=0$, $z(\theta)=0$ and $% z(1)=0$. Nonlinear fractional multi-point boundary value problems appear in a different area of applied mathematics and physics [@1; @2; @3; @4; @5; @6; @7] and references therein. Many important studies have been concerned in engineering and applied science such as dynamical systems, fluid mechanics, control theory, oil industries, heat conduction can be well-turned by fractional differential equations [@8; @9; @10]. Some applications, qualitative behaviors of solution and numerical methods to find approximate solution have been investigated for differential equation with fractional order [@11; @12; @13; @14]. More particularly, it is not easy to directly get exact solutions to most differential equations with fractional order. Hence, numerical techniques are utilised largely. Actually, in recent times many efficient and convenient methods have been developed such as the finite difference method [@15], finite element method [@16], homotopy perturbation method [@17], Haar wavelet methods [@18], Adomian decomposition method [19]{}, collocation methods [@20], homotopy analysis method [@21], differential transform method [@22], variational iteration method [23]{}, reproducing kernel space method [@24; @25] and so on [@26; @27; @28]. In 1908, Zaremba firstly introduced reproducing kernel concept [@29]. His resarches with regard to boundary value problems which includes Dirichlet condition. Reproducing kernel method (RKM) produces a solution in convergent series form for many differential, partial and integro-differential equations. For more information, we refer to [30,31]{}. Recently, this RKM is applied for different type of problem. For example, fractional order nonlocal boundary value problems [@32], Riccati differential equations [@33], forced Duffing equations with nonlocal boundary conditions [@34], Bratu equations with fractional order Caputo derivative [@35], time-fractional Kawahara equation [36]{}, two-point boundary value problem [@37], nonlinear fractional Volterra integro-differential equations [@38]. Recently, Legendre reproducing kernel method is proposed for fractional two-point boundary value problem of Bratu Type Equations [@39]. The main motivation of this paper is to extend the Legendre reproducing kernel approach for solving nonlinear three-point boundary value problem with Caputo derivative. The remainder part of the paper is prepared as follows: some fundamental definitions of fractional calculus and the theory of reproducing kernel with Legendre basis functions are given in Section 2. The structure of solution with Legendre reproducing kernel is demonstrated in Section 3. In order to show the effectiveness of the proposed method, some numerical findings are reported in Section 4. Finally, the last section contains some conclusions. Preliminaries ============= In this section, several significant concepts, definitions, theorems, and properties are provided which will be used in this research. **Definition 2.1** Let $z(\xi)\in C[0,1]$ and $\xi\in[0,1]$. Then, the $% \alpha$ order left Riemann-Liouville fractional integral operator is given as [@8; @12; @13]: $$\begin{aligned} J_{0+}^{\alpha}z(\xi)=\frac{1}{\Gamma(\alpha)}\int\limits_{0}^{\xi} {% (\xi-s)^{\alpha -1}z(s)ds},\end{aligned}$$ here $\Gamma(.)$ is Gamma function, $\alpha\geq0$ and $\xi>0$. **Definition 2.2** Let $z(\xi)\in AC[0,1]$ and $\xi\in[0,1]$. Then, the $\alpha$ order left Caputo differential operator is given as [@8; @12; @13]: $$\begin{aligned} {\ }^{c}D_{0+}^{\alpha}z(\xi)=\frac{1}{\Gamma(m-\alpha)}\int_{0}^{\xi} \frac{ \partial^{m}}{\partial \xi^{m}}\frac{z(s)}{(\xi-s)^{m-\alpha-1}}ds, \,\ m-1<\alpha< m, m\in\mathbb{N} \,\ \hbox{and} \,\ \xi>0.\end{aligned}$$**Definition 2.3** In order to construct polynomial type reproducing kernel, the first kind shifted Legendre polynomials are defined over the interval $[0,1]$. For obtaining these polynomials the following iterative formula can be given: $$\begin{aligned} P_{0}(\xi) &=& 1, \\ P_{1}(\xi) &=& 2\xi-1, \\ &\vdots& \\ (n+1)P_{n+1}(\xi) &=&(2n+1)(2\xi-1)P_{n}(\xi)-nP_{n-1}(\xi), \,\ n=1,2,...\end{aligned}$$ The orthogonality requirement is $$\begin{aligned} \langle P_{n},P_{m} \rangle=\int_{0}^{1}\rho_{[0,1]}(\xi)P_{n}(\xi)P_{m} (\xi)d\xi=\left\{ \begin{array}{ll} 0, & n\neq m, \\ 1, & n=m=0, \\ \frac{1}{2n+1}, & n=m\neq0,% \end{array} \right.\end{aligned}$$ here, weighted function is taken as, $$\begin{aligned} \label{eq4} \rho_{[0,1]}(\xi)=1.\end{aligned}$$ Legendre basis functions can be established so that this basis function system satisfy the homogeneous boundary conditions as: $$\begin{aligned} \label{eq5} z(0)=0 \,\,\hbox{and}\,\ z(1)=0.\end{aligned}$$ Eq. (\[eq5\]) has a advantageous feature for solving boundary value problems. Therefore, these basis functions for $j\geq2$ can be defined as; $$\begin{aligned} \label{eq6} \phi_{j}(\xi)= \left\{ \begin{array}{ll} P_{j}(\xi)-P_{0}(\xi), & \hbox{$j$ is even,} \\ P_{j}(\xi)-P_{1}(\xi), & \hbox{$j$ is odd.}% \end{array} \right.\end{aligned}$$ such that this system satisfy the conditions $$\begin{aligned} \label{eq7} \phi_{j}(0)=\phi_{j}(1)=0.\end{aligned}$$ It is worth noting that the basis functions given in Eq. (\[eq6\]) are complete system. For more information about orthogonal polynomials, please see [@41; @42; @43]. **Definition 2.4** Let $\Omega \neq \emptyset$, and $\mathbb{H} $ with its inner product $\langle\cdot,\cdot\rangle_\mathbb{H}$ be a Hilbert space of real-valued functions on $\Omega$. Then, the reproducing kernel of $% \mathbb{H}$ is $R:\Omega\times \Omega\rightarrow \mathbb{R}$ iff 1. $R(\cdot,\xi) \in \mathbb{H}, \forall \xi \in \Omega$ 2. $\langle\phi,R(\cdot,\xi) \rangle_\mathbb{H} = \phi(\xi), \forall\phi\in \mathbb{H}, \forall \xi \in \Omega$. The last condition is known as reproducing property. Especially, for any $x$, $\xi$ $\in$ $\Omega$, $$\begin{aligned} R(x,\xi)=\langle R(\cdot,x),R(\cdot,\xi) \rangle_\mathbb{H}. \notag\end{aligned}$$ If a Hilbert space satisfies the above two conditions then is called reproducing kernel Hilbert space. Uniqueness of the reproducing kernel can be shown by use of Riesz representation theorem [@40]. **Theorem 2.1** Let $\{e_{j}\}_{j=1}^{n}$ be an orthonormal basis of $n$-dimensional Hilbert space $\mathbb{H}$, then $$\begin{aligned} \label{eq8} R(x,\xi )=R_{x}(\xi )=\sum_{j=1}^{n}\bar{e}_{j}(x)e_{j}(\xi )\end{aligned}$$ is reproducing kernel of $\mathbb{H}$ [@30; @31].**Definition 2.5** Let $W_{\rho }^{m}[0,1]$ polynomials space be pre-Hilbert space over $[0,1]$ with real coefficients and its degree $\leq m$ and inner product as: $$\label{eq9} \langle z,v\rangle _{W_{\rho }^{m}}=\int_{0}^{1}\rho _{\lbrack 0,1]}(\xi )z(\xi )v(\xi )d\xi ,\,\,\ \forall z,v\in W_{\rho }^{m}[0,1]$$ with $\rho _{\lbrack 0,1]}(\xi )$ described by Eq. (\[eq4\]), and the norm $$\label{eq10} \Vert z\Vert _{W_{\rho }^{m}}=\sqrt{\langle z,z\rangle }_{W_{\rho }^{m}},\,\,\ \forall z\in W_{\rho }^{m}[0,1].$$With the aid of definiton of $L^{2}$ Hilbert space, $L_{\rho }^{2}[0,1]=\{g|\int_{0}^{1}\rho _{\lbrack 0,1]}(\xi )|g(\xi )|^{2}d\xi <\infty \}$ for any fixed $m$, $W_{\rho }^{m}[0,1]$ is a subspace of $% L_{\rho }^{2}[0,1]$ and $\forall z,v\in W_{\rho }^{m}[0,1]$, $\langle z,v\rangle _{W_{\rho }^{m}}=\langle z,v\rangle _{L_{\rho }^{2}}$ **Theorem 2.2** $W_{\rho }^{m}[0,1]$ Hilbert space is a reproducing kernel space. **Proof.** From Definition 2.5, it is quite apparent that $W_{\rho }^{m}[0,1]$ functions space is a finite-dimensional. It is well known that all finite-dimensional pre-Hilbert space is a Hilbert space. Herewith, using this consequence and Theorem 2.1, $W_{\rho }^{m}[0,1]$ is a reproducing kernel space.For solving problem (\[eq1\])-(\[eq2\]), it is required to describe a closed subspace of $W_{\rho }^{m}[0,1]$ so that satisfy homogeneous boundary conditions. **Definition 2.6** Let $${\ }^{0}W_{\rho }^{m}[0,1]=\{z\text{ }|\text{ }z\in W_{\rho }^{m}[0,1],\text{ }z(0)=z(1)=0\}.$$One can easily demonstrate that ${\ }^{0}W_{\rho }^{m}[0,1]$ is a reproducing kernel space using Eq. (\[eq6\]). From Theorem 2.1, the kernel function $R_{x}^{m}(\xi )$ of ${\ \ }^{0}W_{\rho }^{m}[0,1]$ can be written as $$\label{eq11} R_{x}^{m}(\xi )=\sum_{j=2}^{m}h_{j}(\xi ){h_{j}}(x).$$Here, $h_{j}(\xi )$ is complete system which is easily obtained from basis functions in Eq. (\[eq6\]) with the help of Gram-Schmidt orthonormalization process. Eq. (\[eq11\]) is very useful for implementation. In other words, $R_{x}^{m}(\xi )$ and $W_{\rho }^{m}[0,1]$ can readily re-calculated by increasing $m$. Main Results ============ In this section, some important results related to reproducing kernel method with shifted Legendre polynomials are presented. In the first subsection, generation of reproducing kernel which is satify three-point boundary value problems is presented. In the second subsection, representation of solution is given ${\ }^{\theta}W_{\rho}^{m}[0,1]$. Then, we will construct an iterative process for nonlinear problem in third subsection. Generation of reproducing kernel for three-point boundary value problems ------------------------------------------------------------------------ In this subsection, we shall generate a reproducing kernel Hilbert space ${\ }^{\theta }W_{\rho }^{m}[0,1]$ in which every functions satisfies $z(0)=0$, $% z(\theta )=0$ and $z(1)=0$. ${\ }^{\theta }W_{\rho }^{m}[0,1]$ is defined as ${\ }^{\theta }W_{\rho }^{m}[0,1]=\{z|z\in W_{\rho }^{m}[0,1],z(0)=z(\theta )=z(1)=0\}$. Obviously, ${\ }^{\theta }W_{\rho }^{m}[0,1]$ reproducing kernel space is a closed subspace of ${\ }^{0}W_{\rho }^{m}[0,1]$. The reproducing kernel of ${% \ }^{\theta }W_{\rho }^{m}[0,1]$ can be given with the following theorem. **Theorem 3.1** The reproducing kernel ${\ }^{\theta }R_{x}^{m}(\xi )$ of ${\ }^{\theta }W_{\rho }^{m}[0,1]$, $$\label{eq12} {\ }^{\theta }R_{x}^{m}(\xi )=R_{x}^{m}(\xi )-\frac{R_{x}^{m}(\theta )R_{\theta }^{m}(\xi )}{R_{\theta }^{m}(\theta )}.$$ **Proof.** Frankly, not all elements of ${\ }^{0}W_{\rho }^{m}[0,1]$ vanish at $\theta $. This shows that $R_{\theta }^{m}(\theta )\neq $ 0. Hence, it can be easily seen that ${\ }^{\theta }R_{x}^{m}(\theta )={\ }% ^{\theta }R_{\theta }^{m}(\xi )=0$ and therefore ${\ }^{\theta }R_{x}^{m}(\xi )\in {\ }^{\theta }W_{\rho }^{m}[0,1]$. For $\forall z\left( x\right) \in $ $^{\theta }W_{\rho }^{m}[0,1]$, clearly, $z\left( \theta \right) =0$, it follows that $${<z(x),}^{\theta }R_{x}^{m}(\xi )>_{^{\theta }W_{\rho }^{m}[0,1]}={<z(x),}% \text{ }R_{x}^{m}(\xi )>-\frac{R_{x}^{m}(\alpha )z(\theta )}{R_{\theta }^{m}(\theta )}=z(\xi ).$$ Namely, ${\ }^{\theta }R_{x}^{m}(\xi )$ is of reproducing kernel of $% ^{\theta }W_{\rho }^{m}[0,1],$. This completes the proof. Representation of solution in ${\ }^{\protect\theta}W_{\protect% \rho }^{m}[0,1]$ Hilbert space ---------------------------------------------------------------- In this subsection, reproducing kernel method with Legendre polyomials is established for obtaining numerical solution of three-point boundary value problem. For Eqs. (\[eq1\])-(\[eq2\]), the approximate solution shall be constructed in ${\ }^{\theta }W_{\rho }^{m}[0,1]$. Firstly, we will define linear operator $L$ as follow, $$L:{\ }^{\theta }W_{\rho }^{m}[0,1]\rightarrow L_{\rho }^{2}[0,1]$$such that $$Lz(\xi ):=a_{2}(\xi ){\ }^{c}D^{\alpha }z(\xi )+a_{1}(\xi ){\ }^{c}D^{\beta }z(\xi )+a_{0}(\xi )z(\xi ).$$The Eqs.(\[eq1\])-(\[eq2\]) can be stated as follows $$\left\{ \begin{array}{ll} Lz=g(\xi ,z(\xi ),z^{\prime }(\xi )) & \\ z(0)=z(\theta )=z(1)=0. & \end{array}% \right. \label{eq13}$$Easily can be shown that linear operator $L$ is bounded. We will obtain the representation solution of Eq. (\[eq13\]) in the ${\ }^{\theta }W_{\rho }^{m}[0,1]$ space. Let $^{\theta }R_{x}^{m}(\xi )$ be the polynomial form of reproducing kernel in ${\ }^{\theta }W_{\rho }^{m}[0,1]$ space.**Theorem 3.2** Let $\{\xi _{j}\}_{j=0}^{m-2}$ be any $(m-1)$ distinct points in open interval $(0,1)$ for Eqs. (\[eq1\])-(\[eq2\]), then $\psi _{j}^{m}(\xi )=L^{\ast }$ $^{\theta }R_{\xi _{j}}^{m}(\xi )=L_{x}$ $^{\theta }R_{x}^{m}(\xi )|_{x=\xi _{j}}.$**Proof.** For any fixed $\xi _{j}\in (0,1)$, put $$\begin{aligned} \psi _{j}^{m}(\xi ) &=&L^{\ast \text{ }\theta }R_{\xi _{j}}^{m}(\xi )=\langle L^{\ast \text{ }\theta }R_{\xi _{j}}^{m}(\xi ),^{\theta }R_{\xi }^{m}(x)\rangle _{{\ }^{\theta }W_{\rho }^{m}} \notag \label{eq14} \\ &=&\langle ^{\theta }R_{\xi _{j}}^{m}(\xi ),L_{x}\text{ }^{\theta }R_{\xi }^{m}(x)\rangle _{L_{\rho }^{2}}=L_{x}\text{ }^{\theta }R_{\xi }^{m}(x)|_{x=\xi _{j}}.\end{aligned}$$It is quite obvious that $^{\theta }R_{\xi }^{m}(x)=$ $^{\theta }R_{x}^{m}(\xi )$. Therefore $\psi _{j}^{m}(\xi )=L^{\ast }$ $^{\theta }R_{\xi _{j}}^{m}(\xi )=L_{x}$ $^{\theta }R_{x}^{m}(\xi )|_{x=\xi _{j}}$. Here, $L^{\ast }$ shows the adjoint operator of $L$. For any fixed $m$ and $% \xi _{j}\in (0,1)$, $\psi _{j}^{m}\in {\ }^{\theta }W_{\rho }^{m}[0,1]$.**Theorem 3.3** Let $\{\xi _{j}\}_{j=0}^{m-2}$ be any $(m-1)$ distinct points in open interval $(0,1)$ for $m\geq 2$, then $\{\psi _{j}^{m}\}_{j=0}^{m-2}$ is complete in ${\ }^{\theta }W_{\rho }^{m}[0,1]$.**Proof.** For every fixed $z\in {\ }^{\theta }W_{\rho }^{m}[0,1]$, let $$\langle z(\xi ),\psi _{j}^{m}(\xi )\rangle _{{\ }^{\theta }W_{\rho }^{m}}=0,$$this result shows, for $j=0,1,...,m-2$, $$\begin{aligned} \langle z(\xi ),\psi _{j}^{m}(\xi )\rangle _{{\ }^{\theta }W_{\rho }^{m}} &=&\langle z(\xi ),L^{\ast \text{ }\theta }R_{\xi _{j}}^{m}(\xi )\rangle _{{% \ }^{\theta }W_{\rho }^{m}} \notag \\ &=&\langle Lz(\xi ),^{\theta }R_{\xi _{j}}^{m}(\xi )\rangle _{L_{\rho }^{2}} \notag \\ &=&Lz(\xi _{j})=0.\end{aligned}$$In Eq. (15), by use of inverse operator, it is decided that $z\equiv 0$. Thus, $\{\psi _{j}^{m}\}_{j=0}^{m-2}$ is complete in ${\ }^{\theta }W_{\rho }^{m}[0,1]$. This completes the proof. Theorem 3.3 indicates that in Legendre reproducing kernel approach, using a finite distinct points are enough. But, in traditional reproducing kernel method need to dense sqeuence on the interval. Namely, this new approach is vary from traditional method in [@28; @32; @33; @34; @35; @38].The orthonormal system $\{\bar{\psi}_{j}^{m}\}_{j=0}^{m-2}$ of ${\ }^{\theta }W_{\rho }^{m}[0,1]$ can be derived with the help of the Gram-Schmidt orthogonalization process using $\{\psi _{j}^{m}\}_{j=0}^{m-2}$, $$\bar{\psi}_{j}^{m}(\xi )=\sum_{k=0}^{j}\beta _{jk}^{m}\psi _{k}^{m}(\xi ), \label{eq16}$$here $\beta _{jk}^{m}$ show the coefficients of orthogonalization. **Theorem 3.4** Suppose that $z_{m}$ is the exact solution of Eqs. ([eq1]{})-(\[eq2\]) and $\{\xi _{j}\}_{j=0}^{m-2}$ shows any $(m-1)$ distinct points in open interval $(0,1)$, in that case $$z_{m}(\xi )=\sum_{j=0}^{m-2}\sum_{k=0}^{j}\beta _{jk}^{m}g(\xi _{k},z_{m}(\xi _{k}), z_{m}^\prime(\xi _{k}))\bar{\psi}_{j}^{m}(\xi ). \label{eq17}$$**Proof.** Since $z_{m}\in {\ }^{\theta }W_{\rho }^{m}[0,1]$ from Theorem 3.3 can be written $$z_{m}(\xi )=\sum_{i=0}^{m-2}\langle z_{m}(\xi ),\bar{\psi}_{j}^{m}(\xi )\rangle _{{\ }^{\theta }W_{\rho }^{m}}\bar{\psi}_{j}^{m}(\xi ).$$On the other part, using Eq. (\[eq14\]) and Eq. (\[eq16\]), we obtain $% z_{m}(\xi )$ which is the precise solution of Eq. (\[eq10\]) in ${\ }% ^{\theta }W_{\rho }^{m}[0,1]$ as, $$\begin{aligned} z_{m}(\xi ) &=&\sum_{j=0}^{m-2}\langle z_{m}(\xi ),\bar{\psi}_{j}^{m}(\xi )\rangle _{{\ }^{\theta }W_{\rho }^{m}}\bar{\psi}_{j}^{m}(\xi ) \\ &=&\sum_{j=0}^{m-2}\langle z_{m}(\xi ),\sum_{k=0}^{j}\beta _{jk}^{m}\psi _{k}^{m}(\xi )\rangle _{{\ }^{\theta }W_{\rho }^{m}}\bar{\psi}_{j}^{m}(\xi ) \\ &=&\sum_{j=0}^{m-2}\sum_{k=0}^{j}\beta _{jk}^{m}\langle z_{m}(\xi ),\psi _{k}^{m}(\xi )\rangle _{{\ }^{\theta }W_{\rho }^{m}}\bar{\psi}_{j}^{m}(\xi ) \\ &=&\sum_{j=0}^{m-2}\sum_{k=0}^{j}\beta _{jk}^{m}\langle z_{m}(\xi ),L^{\ast }{}^{\theta }R_{\xi _{k}}^{m}(\xi )\rangle _{{\ }^{\theta }W_{\rho }^{m}}% \bar{\psi}_{j}^{m}(\xi ) \\ &=&\sum_{j=0}^{m-2}\sum_{k=0}^{j}\beta _{jk}^{m}\langle Lz_{m}(\xi ),^{\theta }R_{\xi _{k}}^{m}(\xi )\rangle _{L_{\rho }^{2}}\bar{\psi}% _{j}^{m}(\xi ) \\ &=&\sum_{j=0}^{m-2}\sum_{k=0}^{j}\beta _{jk}^{m}\langle g(\xi ,z_{m}(\xi ),z^\prime_{m}(\xi)),^{\theta }R_{\xi _{k}}^{m}(\xi )\rangle _{L_{\rho }^{2}}\bar{\psi}% _{j}^{m}(\xi ) \\ &=&\sum_{j=0}^{m-2}\sum_{k=0}^{j}\beta _{jk}^{m}g(\xi _{k},z_{m}(\xi _{k}),z^\prime_{m}(\xi _{k}))\bar{\psi}_{j}^{m}(\xi ).\end{aligned}$$ The proof is completed. **Theorem 3.5** If $z_{m}(\xi )\in {\ }^{\theta }W_{\rho }^{m}[0,1]$, then $|z_{m}^{(s)}(\xi )|\leq F\Vert z_{m}\Vert _{{\ }^{\theta }W_{\rho }^{m}}$ for $s=0,\ldots ,m-1$, where $F$ is a constant. **Proof.** We have $z_{m}^{(s)}\left( \xi \right) =\langle z_{m}\left( x\right) ,\partial _{\xi }^{s}$ $^{\theta }R_{\xi }^{m}\left( x\right) \rangle _{{\ }^{\theta }W_{\rho }^{m}}$ for any $\xi ,\,x\in \left[ {0,1}% \right] $, $s=0,\ldots ,m-1.$ From the expression of $^{\theta }R_{\xi }^{m}\left( x\right) $, it pursue that $\left\Vert {\partial _{\xi }^{s}}% ^{\theta }R_{\xi }^{m}\left( x\right) \right\Vert _{{\ }^{\theta }W_{\rho }^{m}}\leq F_{s},\,s=0,\ldots ,m-1.$So, $$\begin{aligned} |z_{m}^{(s)}(\xi )| &=&|{\langle z_{m}(\xi ),\partial _{\xi }^{s}}\text{{\ }}% ^{\theta }R_{\xi }^{m}\left( x\right) {\rangle _{{\ }^{\theta }W_{\rho }^{m}}% }| \\ &\leq &\Vert {z_{m}(\xi )}\Vert _{{\ }^{\theta }W_{\rho }^{m}[0,1]}\Vert {% \partial _{\xi }^{s}}\text{{\ }}^{\theta }R_{\xi }^{m}\left( \xi \right) \Vert _{{\ }^{\theta }W_{\rho }^{m}} \\ &\leq &F_{s}\Vert {z_{m}(\xi )}\Vert _{{\ }^{\theta }W_{\rho }^{m}},s=0,\ldots ,m-1.\end{aligned}$$Therefore, $|z_{m}^{(s)}(\xi )|\leq \max \{F_{0},\ldots ,F_{m-1}\}\left\Vert {z_{m}\left( \xi \right) }\right\Vert _{{\ }^{\theta }W_{\rho }^{m}},\,s=0,\ldots ,m-1$. **Theorem 3.6** $z_{m}(\xi )$ and its derivatives $z_{m}^{(s)}(\xi )$ are respectively uniformly converge to $z(\xi )$ and $z^{(s)}(\xi )$ ($% s=0,\ldots ,m-1$). **Proof** By using Theorem 3.5 for any $\xi \in \lbrack 0,1]$ we get $$\begin{aligned} |z_{m}^{(s)}(\xi )-z^{(s)}(\xi )| &=&|\langle z_{m}(\xi )-z(\xi ),\partial _{\xi }^{s}\text{ }^{\theta }R_{\xi }^{m}\left( \xi \right) \rangle |_{{\ }% ^{\theta }W_{\rho }^{m}} \\ &\leq &\Vert \partial _{\xi }^{s}\text{ }^{\theta }R_{\xi }^{m}\left( \xi \right) \Vert _{{\ }^{\theta }W_{\rho }^{m}}\Vert z_{m}(\xi )-z(\xi )\Vert _{% {\ }^{\theta }W_{\rho }^{m}} \\ &\leq &F_{s}\Vert z_{m}(\xi )-z(\xi )\Vert _{{\ }^{\theta }W_{\rho }^{m}},\,\ s=0,\ldots ,m-1. \end{aligned}$$where $F_{0},\ldots ,F_{m-1}$ are positive constants. Therefore, if $% z_{m}(\xi )\rightarrow z(\xi )$ in the meaning of the norm of ${\ }^{\theta }W_{\rho }^{m}[0,1]$ as $m\rightarrow \infty $, $z_{m}(\xi )$ and its derivatives $z_{m}^{^{\prime }}(\xi ),\ldots ,z_{m}^{(m-1)}(\xi )$ are respectively uniformly converge to $z(\xi )$ and its derivatives $% z^{^{\prime }}(\xi ),\ldots ,z^{(m-1)}(\xi )$. This completes the proof.\ If considered problem is linear, numerical solution can be directly get from (\[eq17\]). But, for nonlinear problem the following iterative procedure can be construct. Construction of iterative procedure ----------------------------------- In this subsection, we will use the following iterative sequence to overcome the nonlinearity of the problem, $y_{m}(\xi )$, inserting, $$\label{eq18} \left\{ {{\begin{array}{*{20}c} {Ly_{m,n}\left( \xi \right) = g\left( {\xi,z_{m,n-1}(\xi),z^\prime_{m,n-1}(\xi)} \right)} \hfill \\ {z_{m,n}\left( \xi \right) = P_{m-1} y_{m,n} (\xi)} \hfill \\ \end{array}}}\right.$$ here, orthogonal projection operator is defined as $P_{m-1}:{\ }^{\theta }W_{\rho }^{m}[0,1]\rightarrow span\{\bar{\psi}_{0}^{m},\bar{\psi}% _{1}^{m},\ldots ,\bar{\psi}_{m-2}^{m}\}$ and $y_{m,n}(\xi )\in {\ }^{\theta }W_{\rho }^{m}[0,1]$ shows the $n$-th iterative numerical solution of ([eq18]{}). Then, the following important theorem will be given for iterative procedure. **Theorem 3.7** If $\{\xi _{j}\}_{j=0}^{m-2}$ is distinct points in open interval $(0,1)$, then $$\label{eq19} y_{m,n}(\xi )=\sum_{j=0}^{m-2}\sum_{k=0}^{j}\beta _{jk}^{m}g(\xi _{k},z_{m,n-1}(\xi _{k}),z^\prime_{m,n-1}(\xi _{k}))\bar{\psi}_{j}^{m}(\xi )$$ **Proof.** Since $y_{m,n}(\xi )\in {\ }^{\theta }W_{\rho }^{m}[0,1]$, $% \{\bar{\psi}_{j}^{m}(\xi )\}_{j=0}^{m-2}$ is the complete orthonormal system in ${\ }^{\theta }W_{\rho }^{m}[0,1]$, $$\begin{aligned} y_{m,n}(\xi ) &=&\sum_{j=0}^{m-2}\langle y_{m,n}(\xi ),\bar{\psi}% _{j}^{m}(\xi )\rangle _{{\ }^{\theta }W_{\rho }^{m}}\bar{\psi}_{j}^{m}(\xi ) \\ &=&\sum_{j=0}^{m-2}\langle y_{m,n}(\xi ),\sum_{k=0}^{j}\beta _{jk}^{m}\psi _{k}^{m}(\xi )\rangle _{{\ }^{\theta }W_{\rho }^{m}}\bar{\psi}_{j}^{m}(\xi ) \\ &=&\sum_{j=0}^{m-2}\sum_{k=0}^{j}\beta _{jk}^{m}\langle y_{m,n}(\xi ),\psi _{k}^{m}(\xi )\rangle _{{\ }^{\theta }W_{\rho }^{m}}\bar{\psi}_{j}^{m}(\xi ) \\ &=&\sum_{j=0}^{m-2}\sum_{k=0}^{j}\beta _{jk}^{m}\langle y_{m,n}(\xi ),L^{\ast \text{ }\theta }R_{\xi _{k}}^{m}(\xi )\rangle _{{\ }^{\theta }W_{\rho }^{m}}\bar{\psi}_{j}^{m}(\xi ) \\ &=&\sum_{j=0}^{m-2}\sum_{k=0}^{j}\beta _{jk}^{m}\langle Ly_{m,n}(\xi ),^{\theta }R_{\xi _{k}}^{m}(\xi )\rangle _{L_{\rho }^{2}}\bar{\psi}% _{j}^{m}(\xi ) \\ &=&\sum_{j=0}^{m-2}\sum_{k=0}^{j}\beta _{jk}^{m}\langle g(\xi ,z_{m,n-1}(\xi ),z^\prime_{m,n-1}(\xi )),^{\theta }R_{\xi _{k}}^{m}(\xi )\rangle _{L_{\rho }^{2}}\bar{\psi}% _{j}^{m}(\xi ) \\ &=&\sum_{j=0}^{m-2}\sum_{k=0}^{j}\beta _{jk}^{m}g(\xi _{k},z_{m,n-1}(\xi _{k}),z^\prime_{m,n-1}(\xi _{k}))\bar{\psi}_{j}^{m}(\xi )\end{aligned}$$This completes the proof.Taking $z_{m,0}(\xi )=0$ and define the iterative sequence $$\label{eq20} z_{m,n}(\xi )=P_{m-1}y_{m,n}(\xi )=\sum_{j=0}^{m-2}\sum_{k=0}^{j}\beta _{jk}^{m}g(\xi _{k},z_{m,n-1}(\xi _{k}),z^\prime_{m,n-1}(\xi _{k}))\bar{\psi}_{j}^{m}(\xi ),\,\ n=1,2,\ldots$$ Numerical applications ====================== In this section, some nonlinear three-point boundary value problems are considered to exemplify the accuracy and efficiency of proposed approach. Numerical results which is achieved by L-RKM are shown with tables.**Example 4.1** We consider the following fractional order nonlinear three-point boundary value problem with Caputo derivative: $$\label{eq21} {\ }^{c}D^{\alpha }z(\xi ) + (\xi +1) {\ }^{c}D^{\beta }z(\xi ) + \xi z(\xi)-z^{2}(\xi )=f(\xi),\quad 1<\alpha \leq 2.\quad 0<\beta \leq 1.$$ $$\label{eq22} z(0)=z(\frac{1}{2}) =z(1)=0.$$Here, $f(\xi)$ a known function such that the exact solution of this problem is $z(\xi )=\xi(\xi-\frac{1}{2})(\xi-1)$. By using proposed approach for Eqs. (\[eq21\])-(\[eq22\]), and choosing nodal points as $\xi _{j}=\frac{j+0.3}{m},\,j=0,1,\,2,...,m-2$, the approximate solution $z_{m,n}\left( \xi \right)$ is computed by Eq. ([eq20]{}). For (\[eq21\])-(\[eq22\]), comparison of absolute errors for different $\alpha$, $\beta$ values are demonstrated in Table 1 and Table 2 and comparison of exact solution and numerical solution for $\alpha=1.75$ and $\beta=0.75$ is given in Table 3. **Example 4.2** We take care of the following nonlinear three-point boundary value problem with Caputo derivative $$\label{eq23} \xi^2{\ }^{c}D^{\alpha }z(\xi ) + (\xi^2-1) {\ }^{c}D^{\beta }z(\xi ) + \xi^3 z(\xi)-z(\xi)z^\prime(\xi)-z^3(\xi)=f(\xi),\quad 1<\alpha \leq 2.\quad 0<\beta \leq 1.$$ $$\label{eq24} z(0)=z(\frac{3}{5}) =z(1)=0.$$ Here, $f(\xi)$ a known function such that the exact solution of this problem is $z(\xi )=\xi(\xi-\frac{3}{5})(\xi-1)$. By using proposed approach for Eqs. (\[eq23\])-(\[eq24\]), and choosing nodal points as $\xi _{j}=\frac{j+0.3}{m},\,j=0,1,\,2,...,m-2$, the approximate solution $z_{m,n}\left( \xi \right)$ is computed by Eq. ([eq20]{}). For (\[eq23\])-(\[eq24\]), comparison of absolute errors for different $\alpha$, $\beta$ values are demonstrated in Table 4 and Table 5 and comparison of exact solution and numerical solution for $\alpha=1.75$ and $\beta=0.75$ is given in Table 6. Conclusion ========== In this research, a novel numerical approach which is called L-RKM has been proposed and successfully implemented to find the approximate solution of nonlinear three-point boundary value problems with Caputo derivative. For nonlinear problem, a new iterative process is proposed. Numerical findings show that the present approach is efficient and convenient for solving three-point boundary value problems with fractional order. [99]{} Y. Lin, J. Niu, M. Cui. A numerical solution to nonlinear second order three-point boundary value problems in the reproducing kernel space. Applied Mathematics and Computation, 218(14) (2012) 7362-7368. M. Rehman, R. A. Khan, N. A. Asif. 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Tables {#tables .unnumbered} ====== --------------------------------------------------------------------------------------------------------------------------------- $x$ $\alpha=2, \beta=1 $ $\alpha=1.9, \beta=0.9$ $\alpha=1.8, \beta $\alpha=1.7, \beta= 0.7$ $\alpha=1.6, \beta= 0.6$ =0.8$ ----- ---------------------- ------------------------- -------------------- -------------------------- -------------------------- 0.1 3.25E-11 6.08E-11 3.70E-12 4.45E-10 3.14E-9 0.2 5.45E-11 8.75E-11 4.93E-12 8.70E-10 5.51E-9 0.3 6.97E-11 9.00E-11 4.34E-12 1.32E-9 7.36E-9 0.4 8.17E-11 7.81E-11 2.55E-12 1.85E-9 8.96E-9 0.5 0 0 0 0 0 0.6 1.10E-10 5.14E-11 2.03E-12 3.36E-9 1.24E-8 0.7 1.34E-10 5.65E-11 3.56E-12 4.44E-9 1.49E-8 0.8 1.70E-10 8.70E-11 3.73E-12 5.80E-9 1.81E-8 0.9 2.21E-10 1.53E-11 1.89E-12 7.49E-9 2.25E-8 --------------------------------------------------------------------------------------------------------------------------------- : Comparison absolute error of Example 4.1 for various $\protect\alpha% , \protect\beta$ ($m=3$, $n=3$) --------------------------------------------------------------------------------------------------------------------------------- $x$ $\alpha=2, \beta=1 $ $\alpha=1.9, \beta=0.9$ $\alpha=1.8, \beta $\alpha=1.7, \beta= 0.7$ $\alpha=1.6, \beta= 0.6$ =0.8$ ----- ---------------------- ------------------------- -------------------- -------------------------- -------------------------- 0.1 3.78E-17 1.33E-16 2.49E-19 4.86E-15 7.48E-14 0.2 5.27E-17 1.94E-16 3.36E-19 1.13E-14 1.36E-13 0.3 5.10E-17 2.03E-16 3.80E-19 1.99E-14 1.91E-13 0.4 3.91E-17 1.81E-16 5.20E-19 3.09E-14 2.43E-13 0.5 0 0 0 0 0 0.6 1.05E-17 1.36E-16 1.56E-18 6.18E-14 3.60E-13 0.7 6.65E-18 1.57E-16 2.74E-18 8.25E-14 4.36E-13 0.8 1.81E-17 2.34E-16 4.50E-18 1.07E-14 5.30E-13 0.9 5.14E-17 3.91E-16 7.02E-18 1.36E-14 6.48E-13 --------------------------------------------------------------------------------------------------------------------------------- : Comparison absolute error of Example 4.1 for various $\protect\alpha% , \protect\beta$ ($m=3$, $n=5$) $x$ Exact Sol. Approximate Sol. Absolute Error ----- -------------------------- -------------------------- ---------------- 0.0 0.000000000000000000000 0.000000000000000000000 0 0.1 0.036000000000000000000 0.036000000000000000018 1.80E-20 0.2 0.048000000000000000000 0.048000000000000000044 4.40E-20 0.3 0.042000000000000000000 0.042000000000000000061 6.10E-20 0.4 0.024000000000000000000 0.024000000000000000071 7.10E-20 0.5 0.000000000000000000000 0.000000000000000000000 0 0.6 -0.024000000000000000000 -0.023999999999999999899 1.01E-19 0.7 -0.042000000000000000000 -0.041999999999999999819 1.81E-19 0.8 -0.048000000000000000000 -0.047999999999999999694 3.06E-19 0.9 -0.036000000000000000000 -0.035999999999999999491 5.09E-19 1.0 0.000000000000000000000 0.000000000000000000000 0 : Numerical results of Example 4.1 for $m=5$, $n=9$ values ($\protect% \alpha=1.75$, $\protect\beta=0.75$) --------------------------------------------------------------------------------------------------------------------------------- $x$ $\alpha=2, \beta=1 $ $\alpha=1.9, \beta=0.9$ $\alpha=1.8, \beta $\alpha=1.7, \beta= 0.7$ $\alpha=1.6, \beta= 0.6$ =0.8$ ----- ---------------------- ------------------------- -------------------- -------------------------- -------------------------- 0.1 5.00E-11 4.11E-15 2.78E-13 6.10E-12 1.40E-12 0.2 8.39E-11 3.45E-15 1.87E-13 1.17E-11 2.42E-11 0.3 1.06E-10 1.24E-15 1.87E-13 1.45E-11 7.08E-11 0.4 1.23E-10 9.21E-15 7.60E-13 1.20E-11 1.32E-10 0.5 1.39E-10 1.97E-14 1.44E-12 1.75E-12 2.02E-10 0.6 0 0 0 0 0 0.7 1.88E-10 4.52E-14 2.81E-12 5.17E-11 3.45E-10 0.8 2.31E-10 5.88E-14 3.32E-12 9.98E-11 4.05E-10 0.9 2.93E-10 7.18E-14 3.60E-12 1.65E-10 4.49E-10 --------------------------------------------------------------------------------------------------------------------------------- : Comparison absolute error of Example 4.2 for various $\protect\alpha% , \protect\beta$ ($m=3$, $n=8$) --------------------------------------------------------------------------------------------------------------------------------- $x$ $\alpha=2, \beta=1 $ $\alpha=1.9, \beta=0.9$ $\alpha=1.8, \beta $\alpha=1.7, \beta= 0.7$ $\alpha=1.6, \beta= 0.6$ =0.8$ ----- ---------------------- ------------------------- -------------------- -------------------------- -------------------------- 0.1 3.49E-13 1.93E-18 4.48E-16 2.68E-14 4.75E-15 0.2 5.87E-13 8.32E-19 2.99E-16 5.58E-14 7.68E-14 0.3 7.47E-13 2.90E-18 3.06E-16 7.18E-14 2.25E-13 0.4 8.65E-13 8.81E-18 1.23E-15 6.00E-14 4.21E-13 0.5 9.76E-13 1.65E-17 2.34E-15 5.54E-15 6.45E-13 0.6 0 0 0 0 0 0.7 1.31E-12 3.55E-17 4.55E-15 2.91E-13 1.09E-12 0.8 1.61E-12 4.60E-17 5.37E-15 5.64E-13 1.29E-12 0.9 2.05E-12 5.65E-17 5.83E-15 9.39E-13 1.43E-12 --------------------------------------------------------------------------------------------------------------------------------- : Comparison absolute error of Example 4.2 for various $\protect\alpha% , \protect\beta$ ($m=3$, $n=10$) $x$ Exact Sol. Approximate Sol. Absolute Error ----- -------------------------- -------------------------- ---------------- 0.0 0.000000000000000000000 0.000000000000000000000 0 0.1 0.045000000000000000000 0.045000000000480782793 4.80E-13 0.2 0.064000000000000000000 0.064000000000580045412 5.80E-13 0.3 0.063000000000000000000 0.063000000000488602930 4.88E-13 0.4 0.048000000000000000000 0.048000000000398645450 3.98E-13 0.5 0.025000000000000000000 0.025000000000468872800 4.68E-13 0.6 0.000000000000000000000 0.000000000000000000000 0 0.7 -0.021000000000000000000 -0.020999999998651962040 1.34E-12 0.8 -0.032000000000000000000 -0.031999999998006864020 1.99E-12 0.9 -0.027000000000000000000 -0.026999999997598991320 2.40E-12 1.0 0.000000000000000000000 0.000000000000000000000 0 : Numerical results of Example 4.2 for $m=5$, $n=9$ values ($\protect% \alpha=1.75$ , $\protect\beta=0.75$)
--- abstract: 'In the last decade, Hawkes processes have received a lot of attention as good models for functional connectivity in neural spiking networks. In this paper we consider a variant of this process; the Age Dependent Hawkes process, which incorporates individual post-jump behaviour into the framework of the usual Hawkes model. This allows to model recovery properties such as refractory periods, where the effects of the network are momentarily being suppressed or altered. We show how classical stability results for Hawkes processes can be improved by introducing age into the system. In particular, we neither need to a priori bound the intensities nor to impose any conditions on the Lipschitz constants. When the interactions between neurons are of mean field type, we study large network limits and establish the propagation of chaos property of the system.' author: - 'Mads Bonde Raad [^1]\' - Susanne Ditlevsen - 'Eva L[ö]{}cherbach[^2]' title: - - Age Dependent Hawkes Process --- [*Key words*]{}: Multivariate nonlinear Hawkes processes, Multivariate point processes, Mean-field approximations, Age dependency, Stability, Coupling, Piecewise deterministic Markov processes. [*AMS Classification*]{} : 60G55; 60G57; 60K05 Introduction ============ In nature, macroscopic events caused by many microscopic events in an interacting network of units often exhibit a cascading structure, so that they come in waves, for example caused by some events inhibiting or exciting the occurrence of other events. Technological development permitting high-frequency data sampling in recent decades has made it possible to perform detailed analysis of the dependencies between units interacting in a cascade structure. It is therefore relevant to develop models which can quantify temporal interactions between many units on a microscopic level. There are many examples of phenomena of interest which occur in cascades, and which have been analyzed by Hawkes processes. Examples are bankruptcies in finance that propagate through a market, giving rise to volatility clustering observations [@Hawkes-In-Finance], interactions on social media [@Social-Media], and pattern dependencies in DNA [@pat]. Hawkes processes are a class of multivariate point processes, where the intensity function is stochastic and allowed to depend on the past history, introducing memory in the temporal evolution of the stochastic process. Hawkes processes have commonly been used to model neurophysiological processes [@chevallier; @ccdr; @SusEva; @GerhardDegerTruccolo2017; @patvin], and the application we have in mind is to model functional connectivity between neurons in a network. When neurons send an electric signal, the so-called [*action potential*]{} or [*spike*]{}, they excite or inhibit recipient neurons in the network (the [*post-synaptic*]{} neurons). Jumps of the $i$th unit of the Hawkes process are then identified with the spike times of the $i$th neuron. Moreover, the biological process imposes a strong self-inhibition on a neuron that has just emitted a spike. This period of about $2 ms$ is called the *absolute refractory period*, and in this phase it is virtually impossible for a neuron to spike again. The neuron then gradually regains its ability to spike in the longer *relative refractory period*. It was proposed in [@chevallier] to model absolute and relative refractory periods in neuronal spike trains by [*Age Dependent Hawkes processes*]{}, where the age of a unit is defined as the time passed since last time it jumped, and thus, it resets to zero at each jump time. The present article is devoted to a thorough study of its stability properties and associated mean field limits. It turns out that the classical stability results for Hawkes processes can be improved by introducing age into the system. In particular, we neither need to a priori bound the intensities nor to impose any conditions on the Lipschitz constants. This is interesting not only from a mathematical but also from a biological point of view. If a network of neurons is transmitting some information over time, it is not operational nor realistic that intensities should be bounded, a given signal should be transmitted without any necessary delay. However, if the activity explodes, the entire system breaks down. Introducing a refractory period on the individual neuron stabilizes the system, while at the same time the information is still transferred effectively by the network. More precisely, we consider counting processes $(Z_t^i)_{t \geq 0}, i = 1, \ldots , N$, where $N$ is the number of units in the network. The intensity $\lambda_t^i$ of the $i$th process $Z_{t}^i$ can informally be described as the instantaneous jump rate $$\lambda_t^i dt \approx P ( Z^{i} \mbox{ has a jump in } \left( t , t + dt \right]).$$ The basic idea of the Hawkes process is that $\lambda$ denotes a *conditional* intensity which is stochastically depending on the history ${\mathcal F}_{t}$ of the entire network of neurons, that is $$\lambda_t^i dt \approx P ( Z^{i}\mbox{ has a jump in } \left( t , t + dt \right]\vert {\mathcal F}_{t}).$$ This allows $\lambda$ to be updated each time a unit jumps, and individual intensities may depend on individual age processes. To be more precise, we consider an $N-$dimensional point process ${\left( }Z^{i}{\right) }_{i\leq N}$, where each coordinate $Z^{i}$ counts the jump events of the $i$th unit. The intensity of this process is a predictable process depending on the history ${\mathcal F}_{t}$ before and up to time $t.$ It is assumed to have the form $$\lambda^{i}_{t}=\psi^{i}{\left( }X^{i}_{t}({\mathcal F}_{t}),A^{i}_{t}{\right) },$$ where $X^{i}_{t}$ is the *memory process*, $\psi^{i}$ the *rate function,* and $A_t^{i} $ the [*age process*]{} of $Z^{i}$. Here follows a brief introduction of the involved objects. **The Rate Function ${\left( }x,a{\right) }\mapsto \psi^{i}{\left( }x,a{\right) }$** describes how the memory and the age influence the intensity of the $i$th unit. The existence of a non-exploding Hawkes process is generally ensured by assuming that $\psi^{i}$ is sub-linear in $x$. Often, stronger assumptions such as a uniform bound on $\psi^{i}$ is also imposed to prove basic properties. In this article we will work under standard Lipschitz- and linear-growth-conditions, but we shall not need to bound the rate function. **The Age Process $A^{i}_t$** associated to the $i$th process $Z^{i}$ is the time elapsed since the last jump time of $Z^{i}$ before time $t$, that is, $$\begin{aligned} A^{i}_{t} =\begin{cases} A^{i}_{0}+t, & \mbox{if } Z^{i} \text{ has not jumped between time 0 and time }t,\\ t-\sup{\left\lbrace}s<t : \Delta Z^{i}_{s}>0{\right\rbrace}, & \text{otherwise} \end{cases}\end{aligned}$$ where $\Delta Z^{i}_{s} = Z^{i}_{s}-Z^{i}_{s-}=Z^{i}_{s}-\lim_{\varepsilon \rightarrow 0+}Z^{i}_{s-\varepsilon}$ are the jumps. **The Memory Process** $X^{i}_t$ integrates the effects of previous jumps in the network, where the influence from the past is a weighted average of all previous jumps of all units that directly affect unit $i$ (the [ *pre-synaptic*]{} neurons). Each unit has its own memory process, even if they all depend on the same common history of all units, but they are affected in individual ways. More precisely, the $i$th memory process is assumed to have the structure $$\begin{aligned} X^{i}_{t} = \sum_{j=1}^{N}\sum_{\tau < t : \Delta Z^{j}_\tau = 1 } h_{ij}{\left( }t-\tau{\right) }+R^{i}_{t}=\sum_{j=1}^{N}\int_{0}^{t-}h_{ij}{\left( }t-s{\right) }Z^{j}(ds) + R^{i}_{t}.\end{aligned}$$ In the definition of the memory process we have introduced two new objects. **The Weight Function** $h_{ij}{\left( }t{\right) }$ determines how much a jump of unit $j$ that occurred $t$ time units ago, contributes to the present memory of unit $i$. Positive $h_{ij}(t)$ means excitation of unit $i $ when a jump of unit $ j $ occurred $t$ time units ago, while negative $h_{ij}(t)$ means inhibition. **The Initial Signal** $R^{i}_t$ is a process assumed to be known at time $t=0.$ It should be thought of as a memory process which the process inherits from past time. When $R^{i}\equiv 0$ and $\psi^{i}{\left( }x,a{\right) }=f^{i}{\left( }x{\right) }$ for a suitable function $f^{i},$ we obtain the usual non-linear Hawkes process which has been studied in detail, e.g., in [@bm]. In Section \[sec:stability\], we discuss stability of the Age Dependent Hawkes process. The main assumption is a post-jump bound on the intensity, corresponding to a strong self-inhibition for a short time interval after a spike. This models the refractory period. We do not impose any a priori bounds on the intensities. Within this sub-model, we are able to prove stability properties for the $N-$dimensional Hawkes process. The results are similar to what has been shown for ordinary nonlinear Hawkes processes in [@bm], but it turns out that the natural self-inhibition by the age processes eliminates the need of controlling the Lipschitz constant of $\psi$. We also discuss which starting conditions (that is, which form of an initial process) will ensure coupling to the invariant process. These results are collected within our first main theorem, Theorem \[Stab\]. During the proof of the stability properties, other interesting properties of the model are discussed, such as a nice-behaving domination of the intensities (Lemma \[Xbound\]). In Section \[sec:mf\], we study a mean-field setup and associated mean-field limits. More precisely, we consider $N$ interacting units which are organized within ${{\cal K}}$ classes of populations. Each unit belongs to one of these classes, and any two units within the same class $k$ is assumed to be similar, $k=1,\dots ,{{\cal K}}$. This means that they have the same rate function $\psi^{k}$, memory process $X^k$ and initial signal $R^{k}$, and the weight function describing the influence of any unit belonging to another class $l$ is given by $N^{-1}h_{kl}$. Note that each unit still has its own age process. In this setup we establish a limiting distribution for a large scale network, $N{\rightarrow}\infty$. Let $N_{k}$ be the number of units in class $k$ with proportion $N_{k}/N$, and assume $\lim_{N\rightarrow \infty} N_k/N = p_k > 0$. We index the $j$th unit within class $k$ by $Z^{kj}$, $k=1, \ldots, {{\cal K}}, j=1, \ldots , N_{k}$. It is sensible to assume that small contributions from unit $Z^{lj}$ to the memory process $X^{k}$ of the $k$th class disappear in the large-scale dynamics, meaning that $N^{-1}\sum_{j=1}^{N_{l}} \int_0^{t-} h_{kl} (t-s) Z^{lj} (ds ) \approx p_{l} \int_0^t h_{kl} (t-s) {\mathbb E} Z^{l1} (ds) $ for large $N$, for any $ 1 \le l \le {{\cal K}}.$ Therefore, if a limiting point process ${\left( }Z^{kj}{\right) }_{j\in {\mathbb{N}}}$ exists, we expect that $Z^{kj}$ should have intensity $$\begin{aligned} \lambda^{kj}_{t} = \psi^k{\left( }x^{k}_{t},A^{kj}_{t}{\right) },\end{aligned}$$ where $A^{kj}$ is the age process of $Z^{kj}$, and where the process $x^{k}_{t}$ is deterministic, given by $$\begin{aligned} x^{k}_{t} = \sum_{l=1}^{{{\cal K}}}p_{l}\int_{0}^{t} h_{kl}{\left( }t-s{\right) }{\mathbb E} Z^{l1} (ds) +r^{k}_{t},\end{aligned}$$ where $r_t^k$ is a suitable limit of the initial processes in population $k$. In Theorem \[koroet\], we discuss criteria under which such a system exists. Our second main theorem, Theorem \[theo:prop\], shows that this system will indeed be a limit process for the generalized Hawkes processes for $N{\rightarrow}\infty$. We also discuss in Lemma \[weightapproximation\] how robust the system is to adjusting the weight functions. Not only is this robustness a good model feature in itself, but it also allows approximation of an arbitrary Age Dependent Hawkes process, using weight functions with better features. For example, weight functions such as Erlang densities or exponential polynomials induce Markovian systems, see [@SusEva]. We close our article with an Appendix where we collect some proofs and useful results about counting processes. Notation, Definitions and Core Assumptions {#notation-definitions-and-core-assumptions .unnumbered} ------------------------------------------ Throughout this article, we will be working on a background probability space ${\left( }\Omega, {\mathcal F},P{\right) }$ and all random variables are assumed to be defined on this space. If $v= (v_1, \ldots , v_d ) $ is a $d-$dimensional Euclidian vector, then ${\left\vert}v{\right\vert}= \sum_{i=1}^d {\left\vert}v_{i}{\right\vert}$ denotes the $1$-distance. Moreover, for a $d-$dimensional process $X$ we define the running-supremum of the $1$-distance as $\| X \|_{t}=\sup_{s\leq t}{\left\vert}X_{s} {\right\vert}$. We shall also need the basic Stieltjes integration notation. A function of finite variation $f$ induces a Stieltjes signed measure $\mu_{f}$ satisfying $\mu_{f}{\left( }{\left( }a,b{\right]}{\right) }= f{\left( }b{\right) }-f{\left( }a{\right) }$. We use the notation ${\left\vert}df {\right\vert}$ for the corresponding variation measure. The *Lebesgue-Stieltjes integral* is defined as $$\begin{aligned} \int_{a}^{b} f{\left( }x{\right) }dg{\left( }x{\right) }= \int {\mathbbm{1}}{\left\lbrace}{\left( }a,b{\right]}{\right\rbrace}(x) f{\left( }x{\right) }d\mu_{g}{\left( }x{\right) }, \end{aligned}$$ see e.g. [@Halmos]. If $\nu$ is a measure on ${\mathbb{R}}^{2}$ we shall also use the following notation for the integral over semi-closed boxes $$\begin{aligned} \int_{a}^{b}\int_{c}^{d} f{\left( }x,y{\right) }d\nu{\left( }x,y{\right) }= \int {\mathbbm{1}}{\left\lbrace}{\left( }a,b{\right]}{\right\rbrace}{\left( }x{\right) }{\mathbbm{1}}{\left\lbrace}{\left( }c,d{\right]}{\right\rbrace}{\left( }y{\right) }f{\left( }x,y{\right) }d\nu{\left( }x,y{\right) }. \end{aligned}$$ In the following we introduce the core mathematical objects and assumptions needed to discuss the Age Dependent Hawkes process. - $\pi $ and $\pi^{i}, i \in {\mathbb{N}}, $ are i.i.d. Poisson Random Measures (PRMs) on ${\mathbb{R}}_+ \times {\mathbb{R}}$ with Lebesgue intensity measure. For any $t \in {\mathbb{R}},$ we define the $\sigma$-algebra $\tilde {\mathcal F}_{t}$ induced by the projections $$\begin{aligned} \pi{\left( }A\cap {\left( }{\left( }-\infty,t {\right]}\times {\mathbb{R}}_{+} {\right) }{\right) }, \; \pi^{i} {\left( }A\cap {\left( }{\left( }-\infty,t {\right]}\times {\mathbb{R}}_{+} {\right) }{\right) }\mbox{ for } A\in {\mathcal B}{\left( }{\mathbb{R}}\times {\mathbb{R}}_{+}{\right) }, \, i \in {\mathbb{N}}. \end{aligned}$$ We equip the space ${\left( }\Omega, {\mathcal F},P{\right) }$ with the filtration $( {\mathcal F}_t)_{t \in {\mathbb{R}}} $ which is the completion of ${\left( }\tilde {\mathcal F}_{t} {\right) }_{t\in {\mathbb{R}}}.$ - weight functions: For all $ 1 \le i , j \le N, $ $h_{ij} :{\mathbb{R}}_{+}\rightarrow {\mathbb{R}}$ is a locally integrable function. - initial signals: For all $ 1 \le i \le N, $ $(R^i_{t})_{t \geq 0}$ is an $\mathcal{F}_{0 }\otimes \mathcal{B} $ measurable process on $t\in {\mathbb{R}}_{+}$ such that $t \mapsto {\mathbb E} R^i_{t}$ is locally bounded. - rate functions: For all $1 \le i \le N, $ $\psi^i : {\mathbb{R}}\times {\mathbb{R}}_{+}{\rightarrow}{\mathbb{R}}_{+}$ is a measurable function which is $L$-Lipschitz in $x$ when the age variables agree, and otherwise sub-linear in $x,$ i.e., $$\begin{aligned} \label{psias1} {\left\vert}\psi^i{\left( }x,a{\right) }-\psi^i{\left( }x',a'{\right) }{\right\vert}\leq L {\left\vert}x-x' {\right\vert}{\mathbbm{1}}{\left\lbrace}a=a'{\right\rbrace}+L{\left( }\max{\left( }{\left\vert}x'{\right\vert},{\left\vert}x{\right\vert}{\right) }+1 {\right) }{\mathbbm{1}}{\left\lbrace}a\neq a' {\right\rbrace}, \end{aligned}$$ for some $L \geq 1. $ Note in particular that $\psi^i $ is sub-linear in $x,$ i.e., $$\begin{aligned} \label{psias2} \psi^i{\left( }x,a{\right) }\leq L{\left( }1+|x| {\right) }\quad \forall x,a\in {\mathbb{R}}\times {\mathbb{R}}_{+}. \end{aligned}$$ - initial ages ${\left( }A^{i}_{0} {\right) }_{i\in {\mathbb{N}}}$ are $ {\mathcal F}_0-$measurable random variables with support in ${\mathbb{R}}_{+}$. With these definitions, we introduce the Age Dependent Hawkes process. \[defiHawkes\] Let $N\in {\mathbb{N}}$ and let ${\left( }Z,X,A{\right) }= ( (Z^i)_{1 \le i \le N}, (X^i )_{1 \le i \le N }, (A^i )_{1 \le i \le N } ) $ be a triple consisting of an $N-$dimensional counting process $Z$, an $N-$dimensional predictable process $X$, and an $N-$dimensional adapted càglàd process $A$. The triple is an $N-$dimensional Age Dependent Hawkes process with weight functions ${\left( }h_{ij} {\right) }_{i,j\leq N}$, spiking rates ${\left( }\psi^{i} {\right) }_{i\leq N}$, initial ages ${\left( }A^{i}_{0}{\right) }_{i\leq N}$, and initial signals ${\left( }R^{i} {\right) }_{i\leq N}$ if almost surely all sample paths solve the system $$\begin{aligned} Z_t^{i} &= &\int_{0}^{t}\int_{0}^{\infty}{\mathbbm{1}}{\left\lbrace}z\leq \psi^{i}{\left( }X^{i}_{s},A^{i}_{s}{\right) }{\right\rbrace}\pi^{i}{\left( }dz,ds{\right) }, \nonumber \\ X^{i}_{t}&=&\sum_{j=1}^{N}\int_{0}^{t-} h_{ij}{\left( }t-s{\right) }Z^{j}(ds)+R^{i}_{t} , \label{system}\\ A^{i}_{t}-A^{i}_{0} &=& t-\int_{0}^{t-} A^{i}_{s} \, Z^i (ds) ,\nonumber \end{aligned}$$ for all $ t \geq 0.$ Let $ z^i (ds) , 1 \le i \le N, $ be ${\cal F}_0-$measurable point measures on $ [ - \infty , 0 ] $ which we interpret as the initial condition of the Age Dependent Hawkes process. We then typically think of initial processes $R_t^i$ of the form $$R_t^i = \sum_{j=1}^N \int_{- \infty }^0 h_{ij} (t-s) z (ds) ,$$ provided the above expression is well-defined. Well-posedness of the system follows from \[prop:13\] Almost surely, there is a unique sample path ${\left( }Z,X,A{\right) }$ solving . Moreover, $Z$ is non-exploding. Let $h=\sum_{i,j=1}^{N}{\left\vert}h_{ij}{\right\vert}$ and [ $R = \sum_{i = 1 }^{N }{\left\vert}R^{i }{\right\vert}$. Assume first that $R $ is bounded by some constant $M>0 .$]{} Consider the linear Hawkes processes $$\begin{aligned} \label{eq:ztilde} \tilde{Z}^{i}_{t} &=& \int_{0}^{t}\int_{0}^{\infty}{\mathbbm{1}}{\left\lbrace}z\leq L{\left( }1+Y_{s}{\right) }{\right\rbrace}\pi^{i}{\left( }dz,ds{\right) }, 1 \le i \le N, \nonumber \\ Y_{t}&=& \sum_{j=1}^{N}\int_{0}^{t-} h{\left( }t-s{\right) }\tilde{Z}^{j} (ds) +M. \end{aligned}$$ Notice that $\tilde{Z}$ is driven by the same PRMs as $Z$. It is well known that the system almost surely has a path-wise unique solution, which is defined for all $t \in {\left[}0 ,\infty{\right) }$ (see for example [@dfh], Theorem 6). By induction over jump times of $\sum_{j=1}^{N}\tilde{Z}^{j}$ it follows that has a unique solution satisfying $Z_t^{i}\leq \tilde{Z}_t^{i}$ for all $t\in {\mathbb{R}}_+,$ implying that $ Z$ does not explode.\ \ [In the general case define for each $m \in {\mathbb{N}}$ the $ N-$dimensional Age Dependent Hawkes processes $ ( Z^m , X^m , A^m ) $ with the same starting conditions and parameters as ${\left( }Z,X,A {\right) }$ except for the initial processes, which are instead defined as $R^{mi } = -m\vee R^i \wedge m. $ Define also $ \lambda_t^{ m , i} = \psi^i ( X^{m, i }_t, A_t^{m, i } ) $ for the associated intensity and $ \lambda_t^{m \wedge m+1, i } := \lambda^{m, i}_t \wedge \lambda^{m+1, i }_t .$ We have that $$\begin{aligned} P &( Z_{t }^m \neq Z_{t }^{m+1},\;t \in [0,T] ) \\ &\le \, P \left( \exists i :\int_0^\infty \int_{0}^{T } {\mathbbm{1}}\{ z \in ( \lambda_s^{m \wedge m+1, i } , \lambda_s^{m \wedge m+1, i } + L | R^{m, i}_s - R^{m+1 , i}_s | \} \pi^i (dz, ds) \geq 1 \right) \\ &\le \, L \sum_{i=1}^N {\mathbb E}\int_0^T | R^{m, i}_s - R^{m+1 , i}_s | \, ds .\end{aligned}$$ Since $ {\mathbb E}\int_0^T |R^i_s | ds < \infty , $ we conclude that $$\sum_m P ( Z^m_{t } \neq Z_{t }^{m+1},\;t \in [0,T] ) < \infty ,$$ implying that almost surely, the limit $Z = \lim_{m\to \infty }Z^{m } $ exists. It is straightforward to show that $Z $ solves .]{} Stability {#sec:stability} ========= We start by discussing stability of the Age Dependent Hawkes Process within a sub-model where age acts as an inhibitor. For nonlinear Hawkes processes with no age dependence, a thorough investigation of invariant distributions and couplings was done in [@bm]. However, for results where boundedness is not forced upon the system (Theorem 1 of [@bm]), stability depends on the Lipschitz constant $L$ of $\psi$ in . As we show in below, such restrictions are not necessary when age is incorporated as an inhibitor. The fact that the model has desirable stability properties is indicated by the result of Lemma \[Xbound\], where we state a strong control of the intensity. Throughout this section, the random measures are defined on the entire real line ${\mathbb{R}}$ unless otherwise mentioned. We do this for the following reason. When studying stability and thus the existence of stationary versions of infinite memory processes such as (Age Dependent) Hawkes Processes, a widely used approach is to construct the process starting from $t = - \infty .$ If such a construction is feasible, this implicitly implies that the state of the process at time $t=0$ must be in a stationary regime. Therefore, throughout this section we will work with random measures $ Z $ defined on the entire real line, with the usual identification of processes and random measures given by $Z_t = Z ( (0, t ] ) ,$ for all $ t \geq 0,$ and $Z_t = - Z ( (t, 0]), $ for all $ t < 0 .$ We shall also use the shift operator $ \theta^r $ which is defined for any $ r \in {\mathbb{R}}$ by $$\label{eq:shiftoperator} \theta^r Z ( C ) := Z ( r + C ) := Z ( \{ r + x : x \in C \} ) ,$$ for any $ C \in {\cal B} ( {\mathbb{R}}).$ #### Setup in this section: We consider a system with a fixed number of units $N$. In addition to the fundamental assumptions, we strengthen Assumption such that the spiking rate functions satisfy $$\begin{aligned} \label{gassump} \exists K,\delta >0 : \psi^i {\left( }x,a{\right) }&\leq K \text{ for all } 1 \le i \le N, a\in [0,\delta],x\in {\mathbb{R}}. \end{aligned}$$ This assumption excludes instantaneous bursting by imposing a bound on the immediate post-jump intensity. In order to ensure some regeneration properties in the system (no units will eventually stop spiking), we also suppose that there exist $x^* , a^* > 0 $ such that for all $ |x| \le x^*, a \geq a^* $ and for all $ 1 \le i \le N, $ $$\label{eq:doeblin} \psi^i ( x, a ) \geq c > 0 .$$ Finally, we introduce $$\overline{h}_{ij}( t) :=\sup_{s \geq t} | h_{ij}| ( s), h{\left( }t {\right) }=\sum_{i,j = 1 }^{N }{\left\vert}h_{ij }{\right\vert}$$ and we assume that $$\label{eq:hbar} [0, \infty [ \ni t\mapsto \overline{h}_{ij}( t) \in {\mathcal L}^{1} \cap {\cal L}^2 \, \mbox{ and } \, \; [0, \infty [ \ni t\mapsto t h_{ij}( t) \in {\mathcal L}^1.$$ As a consequence, $$\label{eq:hbar2} \overline{h}:=\sum_{i,j=1}^{N} \overline{h}_{ij} \in {\mathcal L}^{1} \cap {\cal L}^2$$ is a decreasing function that dominates $h_{ij}$ for all $i,j\leq N.$ Throughout this section we use the following notation. For $K > 0 $ as in above, we denote the PRMs $$\label{eq:pik} \pi_K ( ds ) := \pi ( [0, K] , ds ), \quad \pi^i_K ( ds) := \pi^i ( [0, K], ds) \, \mbox{ and } \, \pi_{N K} := \sum_{i=1}^N \pi^i_K .$$ The assumption $ \overline{h} \in {\mathcal L}^1 $ is natural, at least in the context of modeling interacting neurons. To obtain stability, it is usually assumed that the weight functions are integrable. Here we impose the slightly stronger assumption that $ \bar h_{ij} \in {\mathcal L}^1 ;$ that is, there exists a decreasing integrable function dominating $ h_{ij } .$ Possible choices of rate functions are $ \psi^i ( x, a ) = f^i (x) g^i (a) $ for all $ a > \delta,$ and bounded otherwise, where $f^i$ is Lipschitz and $ g^i$ bounded. In particular, it is possible to model an absolute refractory period of length $\delta$ by putting $ \psi^i ( x, a) = f^i (x) {\mathbbm{1}}\{ a > \delta\} .$ \[ex:erlang\] For fixed constants $ c_{ij} \in {\mathbb{R}}, \nu_{ij} > 0 $ and $n_{ij} \in {\mathbb{N}}\cup \{0\} ,$ we take $$h_{ij } (t) = c_{ij} t^{ n_{ij }} e^{ - \nu_{ij} t } .$$ Such kernels allow to model delay in the information transmission. The order of the delay is given by $n_{ij } .$ The delay of the influence of particle $j$ on particle $i$ is distributed and taking its maximum absolute value at $n_{ij}/\nu_{ij}$ time units back in time. The sign of $c_{ij }$ indicates if the influence is inhibitory or excitatory, and the absolute value of $c_{ij }$ scales how strong the influence is. All $h_{ij}$ clearly satisfy . The main result of this chapter shows existence of a unique stationary $N-$dimensional Age Dependent Hawkes process following the dynamics of . In order to state the result, we first introduce the notion of *compatibility* (see e.g. [@bm]). Let $M_{{\mathbb{R}}_{-}} $ be the set of all bounded measures defined on $ {\mathbb{R}}_{-} $ equipped with the weak-hat metric and the associated Borel $\sigma-$algebra $ {\cal M}_{{\mathbb{R}}_-} $ (see Appendix for details). We shall say that $Z$ is *compatible* (to $\pi^{1},\dots,\pi^{N})$ if there is a measurable map $H: M^{N}_{{\mathbb{R}}_{-}}{\rightarrow}M_{{\mathbb{R}}_{-}}$ such that $${\left( }\theta^{t} Z{\right) }_{\vert{\mathbb{R}}_{-}}=H{\left( }{\left( }\theta^{t}\pi^1,\dots,\theta^{t}\pi^{N}{\right) }_{\vert{\mathbb{R}}_{-}}{\right) }$$ Likewise, we say that a stochastic process $X$ is compatible, if $X_{t}=H{\left( }{\left( }\theta^{t}\pi^1,\dots,\theta^{t}\pi^{N}{\right) }_{\vert{\mathbb{R}}_{-}}{\right) }$ for an appropriate measurable mapping $H$. Note that if $Z^{1},\dots Z^{n}$ are compatible random measures, then ${\left( }Z^{1},\dots Z^{n}{\right) }$ is a stationary and ergodic n-tuple of random measures. Let $Z={\left( }Z^{i} {\right) },$ $1 \le i\leq N, $ be compatible random measures on $\mathbb{R}$. Let $X={\left( }X^{i } {\right) }_{i\leq N }$, $A={\left( }A^{i } {\right) }_{i\leq N }$ be compatible processes defined on $t\in {\mathbb{R}}$ such that $A_{t }^{i }$ is adapted and càglàd and $X_{t }^{i }$ is predictable for all $1 \le i\leq N$. We say that $Z $ is an N-dimensional Age Dependent Hawkes process on $t\in {\mathbb{R}}$, if almost surely $$\begin{aligned} Z^{i} (t_1, t_2 ] &= &\int_{t_1}^{t_2 }\int_{0}^{\infty}{\mathbbm{1}}{\left\lbrace}z\leq \psi^{i}{\left( }X^{i}_{s},A^{i}_{s}{\right) }{\right\rbrace}\pi^{i}{\left( }dz,ds{\right) }, \nonumber \\ X^{i}_{t}&=&\sum_{j=1}^{N}\int_{-\infty }^{t-} h_{ij}{\left( }t-s{\right) }Z^{j}(ds) \quad t\in \mathbb{R},\label{systembis}\\ A^{i}_{t_2} - A^i_{t_1} &=& t_2- t_1 -\int_{t_1}^{t_2- } A^{i}_{t} \, Z^i (dt), \nonumber \end{aligned}$$ for all $-\infty< t_1 \le t_2. $ \[Stab\] 1. There exists an $N$-dimensional Age Dependent Hawkes process $ Z $ on ${\mathbb{R}},$ compatible to ${\left( }\pi^{1},\dots , \pi^{N}{\right) }$. 2. Let $\check{Z}$ be another $N-$dimensional Age Dependent Hawkes process with the same weight functions $(h_{ij})_{i, j \le N}$ and driven by the same PRMs $ {\left( }\pi^{1},\dots , \pi^{N}{\right) },$ following the dynamics , that is; starting at time $0$ with arbitrary initial ages $(\check A^i_{0})_{i \le N} $ and initial signals ${\left( }R^{i} {\right) }_{ i \le N} $ such that $$\begin{aligned} \label{eq:rint} {\mathbb E} \int_{0}^{\infty}{\left\vert}R^i_{s}{\right\vert}ds<\infty\end{aligned}$$ for all $ 1 \le i \le N. $ Then almost surely, $\check{Z}$ and $Z$ couple eventually, i.e., $$\begin{aligned} \exists \, t_{0}\in {\mathbb{R}}_+ : \;\; \check{Z}_{| [t_0, \infty) } = Z_{| [t_0, \infty ) }.\end{aligned}$$ 3. If $Z'$ is another $N$-dimensional Age Dependent Hawkes process on ${\mathbb{R}}, $ compatible to ${\left( }\pi^{1},\dots , \pi^{N}{\right) },$ then $Z = Z'$ almost surely. An immediate corollary of the above theorem is an ergodic theorem for additive functionals of Age Dependent Hawkes processes depending only on a finite time horizon. More precisely, let $ T > 0$ be a fixed time horizon and let $M_T $ be the set of all bounded measures defined on $ ]- T, 0 ], $ equipped with its Borel $\sigma-$algebra $ {\cal M}_T $ (see Appendix). Grant the assumptions of Theorem \[Stab\], let $ (Z, X, A) $ be the stationary Age Dependent Hawkes process and let $\check{Z}$ be as in item 2. of Theorem \[Stab\]. Let moreover $ f : M_T \to {\mathbb{R}}$ be any measurable function such that $$\label{eq:intz} \pi ( f) := {\mathbb E} f (( Z_{| (-T, 0 ]} ) ) < \infty .$$ Then $$\label{eq:erg} \frac1t \int_0^t f (( \theta^s \check Z_{| (-T, 0 ]} ) )ds = \frac1t \int_0^t f (( \check Z_{| (s-T, s ]} ) )ds \to \pi ( f)$$ almost surely, as $t \to \infty .$ By ergodicity of $ (Z, X, A) $, it holds that $$\frac1t \int_0^t f (( \theta^s Z_{| (-T, 0 ]} ) )ds = \frac1t \int_0^t f (( Z_{| (s-T, s ]} ) )ds \to \pi ( f).$$ Since $\check{Z}_{| [t_0, \infty) } = Z_{| [t_0, \infty) } ,$ we have that $$f (( \theta^s \check Z_{| (-T, 0 ]} ) ) = f (( \theta^s Z_{| (-T, 0 ]} ) ) \mbox{ for all } s \geq t_0 + T,$$ which implies . Proof of Theorem \[Stab\] ------------------------- This section is devoted to the proof of Theorem \[Stab\]. We start with the following useful result which provides bounds on the intensities. \[Xbound\] Let $K,\delta$ be the constants from . Let $X$ be a progressive stochastic process and assume that $(Z,A)$ solves the system $$\begin{aligned} Z{\left( }t_{1},t_{2}{\right]}=\int_{t_{1}}^{t_{2}}\int_{0}^{\infty} {\mathbbm{1}}{\left\lbrace}z\leq \psi {\left( }X_{s},A_{s}{\right) }{\right\rbrace}\pi {\left( }dz,ds{\right) },\quad t_{1}\leq t_{2}\in {\mathbb{R}},\end{aligned}$$ where $A$ is the age process of $Z$ and where $\psi$ satisfies and . Suppose moreover that is satisfied. Then almost surely, for any $1 \le i,j\leq N$, $t_{1}\leq t_{2}, $ $$Y_{ij}{\left( }t_{1},t_{2}{\right) }=\int_{-\infty}^{t_{1}-} h_{ij}{\left( }t_{2}-s {\right) }Z(ds)$$ is well-defined and $$\begin{aligned} \label{eq:yineq} {\left\vert}Y_{ij}{\left( }t_{1},t_{2}{\right) }{\right\vert}\leq \sum^{\infty}_{k=0}\overline{h}_{ij}{\left( }t_{2}-t_{1} + A_{t_1} + k \delta{\right) }+\int_{-\infty}^{t_{1}- A_{t_1} } \overline{h}_{ij}{\left( }t_{2}-s{\right) }\pi_{K}{\left( }d s{\right) }.\end{aligned}$$ Moreover, $${\mathbb E}\int_{-\infty}^{t} \overline{h}_{ij}{\left( }t-s{\right) }\pi_{K}{\left( }d s{\right) }< \infty$$ for all $t .$ \[cor:hbounded\] If we suppose in addition that $ \bar h (0) < \infty ,$ then $${\mathbb E} Y _{ij}{\left( }t,t {\right) }\le K \int_0^\infty \bar h ( u ) du + \sum_{k\geq 0 } \bar h ( k \delta ) < \infty .$$ For any $i,j\leq N$, $t\leq t_{2}$ we have $$\begin{aligned} {\left\vert}Y_{ij}{\left( }t,t_{2}{\right) }{\right\vert}&\leq \int_{-\infty}^{t-} \overline{h}_{ij}{\left( }t_{2}-s {\right) }{\mathbbm{1}}{\left\lbrace}A_{s}\geq \delta{\right\rbrace}Z (ds) +\int_{-\infty}^{t-} \overline{h}_{ij}{\left( }t_{2}-s {\right) }{\mathbbm{1}}{\left\lbrace}A_{s}< \delta{\right\rbrace}Z (ds)\\ &= \int_{-\infty}^{t-A_t} \overline{h}_{ij}{\left( }t_{2}-s {\right) }{\mathbbm{1}}{\left\lbrace}A_{s}\geq \delta{\right\rbrace}Z (ds) +\int_{-\infty}^{t-A_t} \overline{h}_{ij}{\left( }t_{2}-s {\right) }{\mathbbm{1}}{\left\lbrace}A_{s}< \delta{\right\rbrace}Z (ds)\\ &\leq \int_{-\infty}^{t-A_t} \overline{h}_{ij}{\left( }t_{2}-s {\right) }G (ds)+\int_{-\infty}^{t-A_t} \overline{h}_{ij}{\left( }t_{2}-s {\right) }d\pi_{K}{\left( }s {\right) }\\ &:=\hat{Y}_{ij}{\left( }t,t_{2}{\right) }+\tilde{Y}_{ij}{\left( }t,t_{2}{\right) },\end{aligned}$$ where $G (dt) ={\mathbbm{1}}{\left\lbrace}A_{t}\geq \delta {\right\rbrace}Z (dt) $. Define now for fixed $t\in {\mathbb{R}}$ and for all $ l \in \mathbb{N}, $ $\tau_{l}(t) := \sup \{ s < \tau_{l-1}(t) : \Delta G_s = 1 \} , $ where we have put $ \tau_0 (t) := t-A_t .$ Thus, $ \tau_l (t) $ is the $l$th jump-time of $G$ before $t - A_t $ – which is itself the last jump-time of $ Z$ strictly before time $t.$ We may upper bound $\hat{Y}_{ij}$ by $$\begin{aligned} \hat{Y}_{ij}{\left( }t,t_{2}{\right) }\le \sum_{l=0}^{G (- \infty , t ) } \overline{h}_{ij}{\left( }t_{2}-\tau_{l} {\left( }t {\right) }{\right) }.\end{aligned}$$ Since $\tau_{l}{\left( }t {\right) }-\tau_{l-1}{\left( }t {\right) }\geq \delta$ by construction of $G$ and since $\overline{h}_{ij}$ is decreasing, we get the bound $$\begin{aligned} \hat{Y}_{ij}{\left( }t,t_{2}{\right) }\leq \sum_{l=0}^{\infty}\overline{h}_{ij}{\left( }t_2 - t + A_t+ l\delta {\right) }.\end{aligned}$$ Note that almost surely, $A_t$ never attains the value 0 for any $t\in {\mathbb{R}}$, and in that event, each term in the above sum is finite for all $t\leq t_2 \in {\mathbb{R}}$. Moreover, since $\overline{h}_{ij}$ is ${\mathcal L}^{1}$ and decreasing the sum is finite as well. The expectation $t\mapsto {\mathbb E}\tilde{Y}{\left( }t,t {\right) }$ is given by $$\begin{aligned} {\mathbb E}\tilde{Y}_{ij}{\left( }t,t{\right) }&=& \lim_{T{\rightarrow}\infty} {\mathbb E} \int_{-T}^{t- A_t} \overline{h}_{ij}{\left( }t-s{\right) }\pi_{K}{\left( }d s {\right) }\\ &\le& \lim_{T{\rightarrow}\infty} {\mathbb E} \int_{-T}^{t} \overline{h}_{ij}{\left( }t-s{\right) }\pi_{K}{\left( }d s {\right) }=K\int_{0}^{\infty} \overline{h}_{ij}{\left( }u{\right) }du<\infty .\end{aligned}$$ In the sequel, we shall use the upper bound appearing in several times. In particular, we will rely on the following result. Recall that $ L \geq 1 $ is the Lipschitz constant appearing in and let $K, \delta $ be the constants from . \[prop:zhat\] Let $ C \geq \max \left\{ 1+ \sum_{k\geq 1 } \bar h ( k \delta ) , K \right\} .$ There exists a compatible process $( \hat Z , \hat A , \hat \lambda) $ which is defined for any $t \in {\mathbb{R}}$ by $$\label{eq:hatlambda} \hat \lambda_t = L \left( C + \int_{-\infty}^{t-} \bar h ( t- s) \pi_{NK} (ds) + \bar h ( \hat A_t)\right) ,$$ where $$\label{eq:hatz} \hat Z ( t_1, t_2 ) = \sum_{i=1}^N \int_{t_1}^{t_2} \int_0^\infty {\mathbbm{1}}\{ z \le \hat \lambda_s \} \pi^i ( dz, ds )$$ for all $ t_1 \le t_2, $ together with its age process $ \hat A_t.$ Moreover, we have that $${\mathbb E}( \hat \lambda_t ) < \infty .$$ By construction, $\hat \lambda_t \geq K $ for all $t$, and therefore, any jump time $\tau $ of $\pi^i_K $ is also a jump of $\hat Z.$ Hence, at $\tau ,$ the age process $ \hat A_t $ is reset to $0.$ It is therefore possible to construct a unique solution to on $t \in {\left( }\tau,\infty{\right) }$. This solution is non-exploding since the process is stochastically dominated by a classical linear Hawkes process $Z' $ having intensity $ L \left( C + \int_{-\infty}^{\tau - } \bar h ( t- s) \pi_{NK} (ds) + 2 \int_\tau^{t-} \bar h (t - s ) Z' (d s ) \right) $ which is non-exploding by Proposition \[prop:13\] since $ \bar h \in {\cal L}^1.$ A solution on the entire real line may be constructed by pasting together the solutions constructed in between the successive jump times of $ \pi_{NK}.$ It is unique and compatible by construction. It remains to prove that $ {\mathbb E}( \hat \lambda_t) < \infty .$ Due to stationarity, it is sufficient to prove that $ {\mathbb E}\bar h ( \hat A_0) < \infty .$ Also from stationarity, writing $ T_1 $ for the first jump time of $\hat Z $ after time $0, $ it follows that $ {\cal L} (T_1 ) = {\cal L} ( \hat A_0 ) .$ It follows from in the Appendix that $$P ( T_1 > t ) = P{\left( }Z[0,t] = 0 {\right) }= {\mathbb E}\left( \exp \left( - \int_0^t ( L C + L \xi_s + L \bar h ( \hat A_0 + s ) )ds \right) \right) ,$$ where $ \xi_s := \int_{ - \infty }^{0-} \overline h (s-u) \pi_{NK} ( du ) ,$ implying that, since $ \hat A_0 \geq 0 $ and $ \bar h $ is decreasing, $$\begin{aligned} {\mathbb E}( \bar h ( \hat A_0 ) ) &= \int_0^\infty {\mathbb E}\left( \bar h ( t) e^{ - \int_0^t ( L C + L \xi_s + L \bar h ( \hat A_0 + s ) ) ds } ( L C + L \xi_t + L \bar h ( \hat A_0 + t ) ) \right) dt \\ &\le L \int_0^\infty \left( \bar h ( t)^2 + \bar h (t) {\mathbb E}( \xi_0) + \bar h ( t) C \right) dt < \infty ,\end{aligned}$$ since $ \bar h \in {\cal L}^1 \cap {\cal L}^2 .$ Assumption will enable us to construct common jumps for any two point processes $ Z^1 , Z^2 $ having intensity $\psi ( \tilde X^1_t, A^1_t) $ and $ \psi ( \tilde X_t^2, A_t^2 ),$ where $ A^i_t $ is the age process of $ Z^i $, and $\tilde X^i_t $ is a predictable process such that $ \psi ( X^i_t, A^i_t ) \le \hat \lambda_t ,$ for $ i = 1, 2 . $ We do this by introducing the following sequence of events. Fix some $p > a^* $ such that $$\begin{aligned} \label{eq:varrho} \sum_{k \geq 1 } {L} \bar{h } ( p + k \delta ) < \frac{x^*}{3N},\end{aligned}$$ where $a^* $ and $x^* $ are given in , and fix some $M > LC $ where $L$ and $C$ are as in . Then necessarily $M \geq K.$ Introduce for all $t \in {\mathbb{R}}$ the events $$\begin{aligned} && E^1_t := \nonumber \\ && \quad \{ \pi^1 ( [0, c ] , ds ) \mbox{ has a unique jump $\tau^1$ in } ( t - 2N p + p , t- 2N p + 2 p ] )\} \nonumber \\ &&\quad \cap \bigcap_{j=1}^N \left\{ \int_{ t- 2N p }^{ \tau^1 - } \int_{{\mathbb{R}}_+} {\mathbbm{1}}\{ z \le M \} \pi^j (dz, ds) = 0 \right\} \nonumber \\ && \quad \cap \bigcap_{j =1}^N \left\{ \int_{\tau^1}^{t - 2N p + 2 p} \int_{{\mathbb{R}}_+} {\mathbbm{1}}\{ z \le M + {2L}\bar h (s - \tau^1) \} \pi^j (dz, ds ) = 0 \right\} ,\end{aligned}$$ and for all $ i = 2, \ldots , N ,$ $$\begin{aligned} \label{eq:ai} && E^i_t := \nonumber \\ && \quad \{ \pi^i ( [0, c ] , ds ) \mbox{ has a unique jump $\tau^i$ in } ( t - 2N p + 2 ( i- 1) p + p , t- 2N p + 2 i p ] )\} \nonumber \\ &&\quad \cap \bigcap_{j=1}^N \left\{ \int_{ t- 2N p + 2 (i-1) p }^{ \tau^i - } \int_{{\mathbb{R}}_+} {\mathbbm{1}}\{ z \le M + {2L} \bar h (s- (t- 2N p + 2 (i-1) p) ) \} \pi^j (dz, ds) = 0 \right\} \nonumber \\ && \quad \cap \bigcap_{j =1}^N \left\{ \int_{\tau^i}^{t - 2N p + 2 i p} \int_{{\mathbb{R}}_+} {\mathbbm{1}}\{ z \le M + {2L}\bar h (s - \tau^i) \} \pi^j (dz, ds ) = 0 \right\} ,\end{aligned}$$ where the constant $c$ is given in . This event splits the interval $(t-2Np,t)$ up in intervals of length $2p$, where the $i$th truncated PRM has exactly one jump in the second part, and no other events (of truncated PRMs) occur. To control the past up to time $t-2Np$ we also introduce the event $$E^0_t := \{ \hat \lambda_{ t- 2N p } +x^{* } \le {M} \} \cap \{ \int_{- \infty}^{ t- 2N p } \bar h ( t - 2N p - s) \pi_{NK } (ds) \le \frac{x^*}{3N} \}$$ and put $$\label{eq:et} E_t := \bigcap_{i=0}^N E_t^i .$$ Using induction and the strong Markov property, it follows from integrability of $ \bar h$ that $P{\left( }\bigcap_{i = 0}^{j }E^{i }_{t } {\right) }>0 $ for all $t\in {\mathbb{R}},j\leq N$. In particular $P{\left( }E_{t } {\right) }>0 $. The event $E_{2Np}$ is illustrated in Figure \[fig:Et\], for $N=2$ and $ \overline{h}{\left( }t {\right) }\approx t^{-0.4}$. The grey area is the relevant part for the truncated PRMs. ![\[fig:Et\] [An illustration of the event $ E_{2Np }$ with $N=2$ and $ \overline{h}{\left( }t {\right) }\approx t^{-0.4}$. The figure shows a superposition of $\pi^{1 }{\left( }\bullet {\right) },\pi^{2 }{\left( }\blacktriangle {\right) }$, including the jump times $\tau^1$ and $\tau^2$, and three curves:\ **the dotted curve** is the constant $c $\ **the dashed curve** is the intensity process $\hat{\lambda } $\ **the solid curve** is enclosing the area (in grey) of the plane that is relevant to the event $ E_{2Np }$, and it is given by $M \mbox{ for } 0<s \le \tau^1, M + 2L\overline{h} (s-\tau^1) \mbox{ for } \tau^1 < s \le 2p ,\, M + 2L\overline{h} (s-2p)\mbox{ for } 2p<s \le \tau^2$ and $ M + 2L\overline{h} (s-\tau^2) \mbox{ for } \tau^2 < s \le 4p $. ]{}](Esetpicture.jpg) Let us define $$\label{eq:yt} Y_t := \int_{-\infty }^{ t - \hat A_t } \overline{h} (t-s) \pi_{NK }(ds) + \sum_{ k=0}^{ \infty } \bar h ( \hat A_t + k \delta ) .$$ We summarize the most important features of the event $E_t$ in the next lemma. \[lem:ok\] On $E_t, $ each measure $ \pi^i ( [0, c ], ds ) $ has a jump at time $\tau^i \in (t- 2N p, t) $ such that $$\label{eq:1} \int_{ t- 2N p}^t \int_{{\mathbb{R}}_+} {\mathbbm{1}}\{ s \neq \tau^i , z \le \hat \lambda_s\} \pi^i (ds, dz ) = 0 ,$$ $$\label{eq:kl} \hat \lambda_{\tau^i } \le \hat \lambda_{ t - 2N p } + \frac{2i }{3N} x^* ,$$ $$\label{eq:21} Y_{\tau^i } \le \frac{2i }{3N} x^* .$$ Moreover, $ | \tau^i - \tau^{i-1} | \geq a^* , $ where we put $\tau^0 = t - 2N p.$ Let ${\left( }\tau^{i } {\right) }_{i \leq N } $ be the jump times as given in the definition of $E_t$. By construction, the inter-distances are at least equal to $p $. We shall prove by induction over $j $ that $$\int_{ t- 2N p}^{t- 2N p + 2jp} \int_{{\mathbb{R}}_+} {\mathbbm{1}}\{ s \neq \tau^i , z \le \hat \lambda_s\} \pi^i (ds, dz ) = 0\;\;\; \forall i \leq N$$ as well as and hold for $i \in \{ 0,\dots,j \} $ in the event $E_{t }$. The induction start is trivial, so assume that the assertion is true up to $j - 1 $. Notice that by the induction assumption $$\hat{\lambda}_{s } \leq \hat \lambda_{ t - 2N p } + \frac{2(j-1) }{3N} x^* +2 L\overline{h}{\left( }s - \tau^{j - 1 } {\right) },$$ for $s \geq \tau^{j - 1 } $ and until the next jump of $\hat{Z} $. It follows from the construction of $E^{j }\cap E^{j - 1 } $ that $\hat{Z }{\left( }\tau^{j- 1 },\tau^{j } {\right) }= 0 $. This proves the first claim. It also shows that $\hat A_{\tau^{j } } >p $ so the properties of $p $ gives $2L\overline{h }{\left( }\tau^{j } - \tau^{j - 1 } {\right) }\leq \frac{2x^{* } }{3N } $ which implies the remaining claims.\ The proof of relies on a Picard iteration of that alternately updates $ {\left( }X^{i } {\right) }_{i \leq N } $ and $ {\left( }Z^{i },A^{i } {\right) }_{i \leq N } .$ The next theorem ensures that for a well-behaving processes $X^{i }$ there exist couples ${\left( }Z^{i },A^{i } {\right) }$ such that $Z^{i }$ has intensity $\psi^{i }{\left( }X^{i },A^{i } {\right) }$ and $A^{i }$ is the age of $ Z^{i}$. \[NoJumpLemma\] Let $( \hat Z , \hat A , \hat \lambda) $ be as in Proposition \[prop:zhat\] and let ${\left( }X^i_{t}{\right) }_{t\in {\mathbb{R}}}, 1 \le i \le N , $ be compatible and predictable stochastic processes satisfying that almost surely, $$\label{eq:upper bound} |X^i_t| \le Y_{t } ,$$ for all $ 1 \le i \le N, $ $ t \in {\mathbb{R}}.$ Then there exist random counting measures $Z^i , 1 \le i \le N, $ on ${\mathbb{R}}$ which are compatible, and compatible càglàd processes $A^i , 1 \le i \le N, $ which almost surely satisfy $$\label{eqtime} Z^i {\left( }B {\right) }= \int_{B}\int_{0}^{\infty} {\mathbbm{1}}{\left\lbrace}z\leq \psi^i {\left( }X^i_{s},A^i_{s} {\right) }{\right\rbrace}\pi^i {\left( }dz,ds {\right) },\quad \forall B\in {\mathcal B}{\left( }{\mathbb{R}}{\right) },$$ for all $ 1 \le i \le N, $ where $A^i$ is the age process of $Z^i .$ The proof follows the ideas of the proof of Theorem 4 in [@bm] and uses Picard iteration. For that sake, define recursively for all $ n \geq 1, $ $$\begin{aligned} Z^{ni}{\left( }t_{1},t_{2}{\right]}=\int_{t_{1}}^{t_{2}}\int_{0}^{\infty} {\mathbbm{1}}{\left\lbrace}z\leq \psi^i {\left( }X^i_{s},A_{s}^{n- 1, i }{\right) }{\right\rbrace}\pi^i {\left( }dz,ds{\right) },\quad t_{1}<t_{2}\in {\mathbb{R}},\end{aligned}$$ where $A^{n- 1, i}$ is the age process corresponding to $Z^{n- 1 , i}.$ We initialize the iteration with $Z^{0, i}\equiv \pi^i_K .$ We start by proving inductively over $n$ that the Picard iteration is well-posed, and $Z^{n}$ is non-exploding and compatible. The induction start is trivial. We assume that the hypothesis holds for $n-1. $ Clearly $Z^{n}$ is compatible. Moreover, $Z^{ni} $ has intensity $\psi^i ( X^i_t , A^{n- 1, i }_t ) \le L ( 1 + Y_t), $ and $Y_t$ is almost surely locally integrable, implying that $ Z^{ni} $ does not explode.\ \ We will now prove the convergence of the above scheme. To do so, define measures $ \underline Z^i $ and $ \overline Z^i $ by $$\underline Z^i [t] = \liminf_{n} Z^{ni}[t] , \; \overline Z^i [t] = \limsup_{n} Z^{ni } [t]$$ for any $ t \in {\mathbb{R}},$ and $$\tilde {Z} = \sum_{i=1}^N ( \overline Z^i -\underline Z^i ) .$$ That is, $\tilde {Z}$ counts the sum of the differences of the superior and inferior limit processes. We claim that $\tilde {Z}$ is almost surely the trivial measure. It will follow that $Z^{ni}$, and thus also $A^{ni}$ converge. To prove this claim, consider the event $$G_{t}= \left\{ \tilde {Z}{\left( }t,\infty {\right) }=0 \right\} .$$ Notice that $\{ \tilde {Z}{\left( }t,\infty{\right) }=0\} = \{ \theta^{t} (\pi^i)_{i=1}^N\in V\}$ for some $V\in {\mathcal M}$ and thus $\{ \tilde {Z}{\left( }t,\infty{\right) }= 0 \; \text{infinitely often} \}$ is an invariant set, and thus also a $0 /1$ event. It follows by standard arguments that $P{\left( }\tilde {Z}{\left( }{\mathbb{R}}{\right) }= 0{\right) }= 1$ if $P{\left( }G_{0}{\right) }>0$. We now prove that $P ( G_0 ) > 0 $ by showing that $ E_0 \subset G_0 , $ where $E_0$ was defined in above (that is, we choose $t = 0$). The assumption $ |X_t^i | \le Y_t $ implies that $\lambda_t^{n, i } \le \hat \lambda_t $ for all $i, n $ and $t.$ Lemma \[lem:ok\] implies that on $E_{t } $ we have $ \hat A_{\tau^i } \geq a^* , $ and therefore also $ A^{n, i }_{\tau^i } \geq a^*.$ Moreover, implies that $ |X^{i }_{\tau^i}| \le x^*.$ Therefore, implies $$\label{eq:lblambda} \lambda_{\tau^i }^{n, i} \geq c$$ for all $ n, i .$ As a consequence, at time $ \tau^i ,$ all $Z^{ni } $ have a common jump. From it follows that $ Z^{n i } ( \tau^i , 0) = 0 ,$ and therefore, $ A^{ni }_{0 } = t - \tau^i . $ In particular, they are all equal. We may now conclude that on $E_0, $ $ Z^{ni }_{| {\mathbb{R}}_+} $ is a constant sequence over $n, $ for all $i.$ In particular, we have $ \tilde Z ( 0, \infty ) = 0.$ To resume, we have proven that $ E_0 \subset G_0, $ and thus $$P( G_0 \cap E_0 ) = P( E_0 ) > 0 ,$$ implying the result. We are now able to give the Proof of Theorem \[Stab\].\ [*Proof of Theorem \[Stab\].*]{} The proof follows the ideas of the proof of Theorem 4 in [@bm]. First we construct a stationary solution to . For this sake we consider the Picard iteration $$\begin{aligned} X^{ni}_{t}&=\sum_{j=1}^{N}\int_{-\infty}^{t-}h_{ij}{\left( }t-s{\right) }Z^{n-1,j} {(ds)} , \\ Z^{ni}{\left( }t_{1},t_{2}{\right]}&=\int_{t_{1}}^{t_{2}}\int_{0}^{\infty} {\mathbbm{1}}{\left\lbrace}z\leq \psi^{i}{\left( }X^{ni}_{s},A_{s}^{ni}{\right) }{\right\rbrace}\pi^{i}{\left( }dz,ds{\right) },\quad t_{1}<t_{2}\in {\mathbb{R}},\end{aligned}$$ where $A^{nj}$ is the age process of $Z^{nj}.$ We initialize the iteration with $Z^{0i} \equiv \pi^i_K ,X^{0}\equiv 0.$ We start by proving inductively over $n$ that the Picard iteration is well-posed, $Z^{ni}$ is non-exploding and compatible, and almost surely $$\label{eq:bound} L( 1 + |X_t^{ni }|) \leq \hat{\lambda}_{t}\quad \forall t\in {\mathbb{R}},$$ for all $n,i$, where $\hat \lambda$ is defined in above. The induction start is trivial. Suppose now that the assertion holds for $n-1.$ We apply Lemma \[NoJumpLemma\] with $X^i = X^{n, i },$ and show that the conditions of this Lemma are met, then well-posedness, ergodicity and stationarity of $Z^{ni},A^{ni}$ follow. So we prove the upper bound on $ X^{n, i} .$ By construction, $$X_t^{n , i } = \sum_{j=1}^N \int_{-\infty}^{t-} h_{ij} (t-s) Z^{n-1, j } (ds) = \sum_{j=1}^N \int_{-\infty }^{ t - A^{n-1, j }_t } h_{ij} ( t-s) Z^{n-1, j } (ds ) .$$ We apply Lemma \[Xbound\] to each of the $N$ terms within the above sum and obtain $$\label{eq:xni} \int_{-\infty }^{ t - A^{n-1, j }_t } h_{ij} ( t-s) Z^{n-1, j } (ds ) \le \sum_{k \geq 0} \bar h_{ij} ( A^{n-1, j }_t + k \delta ) + \int_{-\infty}^{t- A_t^{n-1, j } }\bar h_{ij} (t-s) \pi^j_K ( ds ) .$$ Since $ \hat \lambda_t \geq \psi^i ( X_t^{n-1, i }, A_t^{n-1, i } ) $ for all $i, $ it follows that $ \hat A_t \le A_t^{n-1, i } $ for all $ i, $ implying that $$| X_t^{n, i } | \le \int_{-\infty}^{t - \hat A_t} \bar h ( t- s ) \pi_{NK } (ds) + \sum_{k \geq 0} \bar h ( \hat A_t + k \delta),$$ which is . Finally, since $A^{n-1},Z^{n-1}$ are compatible, it is straight-forward to show that $Z^{n}$ is compatible as well. Define now $$\begin{aligned} \underline{\lambda}^{i}_{t}= \liminf_{n{\rightarrow}\infty} \psi^{i}{\left( }X^{ni}_{t},A_{t}^{ni}{\right) },\quad \overline{\lambda}^{i}_{t}= \limsup_{n{\rightarrow}\infty} \psi^{i}{\left( }X^{ni}_{t},A_{t}^{ni}{\right) }, 1 \le i \le N .\end{aligned}$$ Note that by and , $ \underline{\lambda}^{i}_{t} \le \overline{\lambda}^{i}_{t} \le \limsup_{n \to \infty} L( 1 + |X_t^{ni }|) \leq \hat{\lambda}_{t} .$ So almost surely, $\underline{\lambda}^{i},\overline{\lambda}^{i}$ have finite sample paths. Note also that they are limits of predictable processes (see in Appendix), and thus they are predictable as well. Define also $$\begin{aligned} \tilde {Z}^{i} [t] &= \limsup_{n}Z^{ni}[t] -\liminf_{n} Z^{ni}[t] = \pi^i \left ( ] \underline{\lambda}^{i}_{t},\overline{\lambda}_{t}^{i} ] \times \{ t \} \right) ,\end{aligned}$$ for $i\leq N,t \in {\mathbb{R}}$. That is, $\tilde {Z}^{i}$ counts the difference of the superior and inferior limit process. We claim that $$\label{eq:tildeZ} \tilde{Z}=\sum_{j=1}^{N} \tilde {Z}^{j}$$ is almost surely the trivial measure. It will follow that $Z^{nj}$, and thus also $X^{ni},A^{ni}$ converge. Moreover, it is straight forward to check that the limit variables solve . To prove this claim, note that we may also find measurable $H^{i}: M_{{\mathbb{R}}\times{\mathbb{R}}_{+}}\rightarrow {\mathbb{R}}^{2} $ such that almost surely $$\begin{aligned} H^{i}{\left( }\theta^{- t}(\pi^i)_{i=1}^N {\right) }= {\left( }\underline{\lambda}^{i}_{t},\overline{\lambda}^{i}_{t}{\right) },\quad \forall t\in {\mathbb{R}}.\end{aligned}$$ Consider the events $E_t$ defined in above as well as $$G_{t}= {\left( }\tilde {Z}{\left( }t,\infty {\right) }=0 {\right) }.$$ Using the functionals obtained previously, it follows that $\{ \tilde {Z}{\left( }t,\infty{\right) }=0\} = \{ \theta^{t}(\pi^i)_{i=1}^N \in V\}$ for some $V\in {\mathcal M}$ and thus $\{ \tilde {Z}{\left( }t,\infty{\right) }= 0 \; \text{infinitely often} \}$ is an invariant set, and thus also a $0 /1$ event. As before this implies that $P{\left( }\tilde {Z}{\left( }{\mathbb{R}}{\right) }= 0{\right) }= 1$ if $P{\left( }G_{0} \cap E_0 {\right) }>0$. To prove that $P ( G_0 \cap E_0 ) > 0, $ note that we have $\lambda^{n, i }_t = \psi ( X_t^{n, i }, A_t^{n , i } ) \le \hat \lambda_t $ and $ |X_t^{n, i } |\le Y_t .$ Therefore, the same arguments as those exposed in the proof of Lemma \[NoJumpLemma\], show that on $E_{0}$, we have $A^{n i }_{0} = A^{m , i }_0 ; $ for all $ n , m $ and $ i, $ that is, the age variables are all equal at time $0.$ Moreover, on $ G_0,$ either no jumps happen any more, or they happen conjointly, and so the Lipschitz criterion ensures the bound $$\begin{aligned} {\left\vert}\overline{\lambda}^{i}_{t}-\underline{\lambda}^{i}_{t}{\right\vert}\leq L \lim_{n{\rightarrow}\infty}\sup_{m,k\geq n}{\left\vert}X^{m}_{t}-X^{k}_{t}{\right\vert}\leq L\int_{-\infty }^{0-} h{\left( }t-s{\right) }\tilde {Z}(ds):=\bar{X}_{t},\end{aligned}$$ for all $ t \geq 0,$ which holds on $G_0 \cap E_0.$ Therefore we may write $$\begin{aligned} P{\left( }G_{0} \cap E_0 {\right) }\geq P{\left( }E _{0} \cap \left\{ \sum_{j=1}^{N}\int_{0}^{\infty}\int_{0}^{\infty}{\mathbbm{1}}{\left\lbrace}z\in {\left( }\underline{\lambda}^{j}_{s},\underline{\lambda}^{j}_{s}+\bar{X}_{s}{\right]}{\right\rbrace}\pi^{j}{\left( }dz,ds{\right) }=0\right\} {\right) }.\end{aligned}$$ Note that in Appendix reveals that the compensator of the integral-sum above is $$\begin{aligned} t\mapsto N\int_{0}^{t} \bar{X}_{s} ds=NL \int_{0}^{t} \int_{-\infty}^{0-} h{\left( }s-u{\right) }\tilde {Z}(du) ds.\end{aligned}$$ The same lemma gives an expression for $P ( \bar Z ( 0, \infty) = 0 | \mathcal F_0 ) , $ and so implies the lower bound $$\begin{aligned} P{\left( }G_{0}\cap E_0 {\right) }&\geq {\mathbb E} {\mathbbm{1}}{\left\lbrace}E_{0} {\right\rbrace}\exp{\left( }-NL \int_{0}^{\infty}\int_{-\infty}^{0-}h {\left( }s-u{\right) }\tilde {Z}(du)ds {\right) }.\end{aligned}$$ To prove that the right hand side is positive, it suffices to show that the double integral inside the exponential is almost surely finite. Notice that by construction, $ \tilde Z_t \le \hat Z_t,$ and recall from that $ \hat\lambda$ is stationary with $\mathbb{E}\hat\lambda_{0 }<\infty$. After taking expectation, $$\begin{aligned} {\mathbb E}\int_{0}^{\infty}\int_{-\infty}^{0-} h{\left( }s-u{\right) }\tilde {Z}(du)ds & \le&\label{lastintegrals} {\mathbb E}\int_{0}^{\infty}\int_{-\infty}^{0-} h{\left( }s-u{\right) }\hat {Z}(du)ds \nonumber\\ &=& {\mathbb E}\int_{0}^{\infty}\int_{-\infty}^{0-} h{\left( }s-u{\right) }\hat \lambda_u du ds\nonumber \\ &=& {\mathbb E}( \hat \lambda_0 ) \int_{0}^{\infty}\int_{-\infty}^{0-} h{\left( }s-u{\right) }du ds\nonumber\\ & =&{\mathbb E}( \hat \lambda_0 ) \int_0^\infty t h (t ) dt < \infty \nonumber.\end{aligned}$$ This proves the desired result. We now prove the coupling part. This will be done in two steps. First suppose that $ |R_t | $ is bounded by a constant $C_R.$ We suppose w.l.o.g. that $\hat \lambda $ defined in is such that also $ C \geq C_R.$ Let ${\left( }{\left( }\check Z^{i}{\right) }_{i\leq N},{\left( }\check X^{ i}{\right) },{\left( }\check A^{ i}{\right) }_{i\leq N}{\right) }$ be the $N$-dimensional Age Dependent Hawkes process with initial conditions ${\left( }\check A^{i}_{0}{\right) },{\left( }R^{i}{\right) }$ and denote $\check \lambda^i_t := \psi ( \check X^i_t , \check A_t^i ) . $ Then we clearly have that $$\check \lambda^i _0 \le \hat \lambda_0$$ for all $ i ,$ and it can be shown inductively over the successive jumps of $ \hat Z, $ using Lemma \[Xbound\], that this inequality is preserved over time, that is, $$\label{eq:good} \check \lambda^i_t \le \hat \lambda_t$$ for all $ t \geq 0.$ Now introduce $$\check E^0_t := \left\{\sum_{i=1}^N \int_{t - 2N p }^t {\mathbbm{1}}\left \{ z \le \frac{3 N c}{ x^* } | R _s| \right \} \pi^i (dz, ds) = 0 \right\}$$ and put $$\label{eq:etprime} E'_t := E_t \cap \check E^0_t .$$ Let ${\left( }\tau^i {\right) }_{i \le N} $ be the jump times from . We necessarily have that on $E'_t, $ $$| R_{ \tau^i }| \le \frac{x^*}{ 3 N} ,$$ for all $ 1 \le i \le N.$ As a consequence, implies that $$| \check X_{\tau^i }| \le Y_{\tau^i } + |R_{\tau^i }| \le x^* .$$ Due to , we have $ \check A^i_{\tau^i } \geq \hat A_{\tau^i } \geq a^* . $ Therefore, using , we conclude that $$\check \lambda^i_{\tau^i } \geq c ,$$ implying that $ \tau^i $ is also a jump of $ \check Z^i.$ Thus, on $ E_t', $ at time $ t, $ all $ ( \check A^i , A^i ) , 1 \le i \le N, $ are coupled. Therefore, we have a Lipschitz bound under the event $E'_{t}; $ so with $Z=\sum_{j=1}^{N}Z^{j},$ $\check {Z}=\sum_{j=1}^{N}\check{Z}^{j},$ we may write $$\begin{gathered} \sum_{i=1}^N {\left\vert}\psi^{i}{\left( }X^{i}_{t},A^{i}_{t}{\right) }-\psi^{i}{\left( }\check X^{i}_{t}, \check A^{i}_{t}{\right) }{\right\vert}\leq L \sum_{i=1}^N {\left\vert}X^{i}_{t}-\check X ^{i}_{t} {\right\vert}\\ \leq L{\left( }\int_{-\infty}^{0-} h{\left( }t-s{\right) }Z(ds) + {\left\vert}R_{t}{\right\vert}+ \int_{0}^{t-}h{\left( }t-s{\right) }\tilde Z(ds){\right) },\end{gathered}$$ which holds on $E_t' . $ As before $ \tilde Z := | Z - \check Z | $ and $ G_t := \{ \tilde Z (t, \infty ) = 0 \} .$ Equivalent considerations as in the first part of the proof yield $$\begin{aligned} P&{\left( }G_{t} \cap E'_t \; \vert \; {\mathcal F}_{t}{\right) }\nonumber \\ &\geq{\mathbbm{1}}{\left\lbrace}E'_{t}{\right\rbrace}\exp{\left( }-L\int_{t}^{\infty} \left[ \int_{-\infty}^{0} h{\left( }s-u{\right) }Z(du) + {\left\vert}R_{s}{\right\vert}+ \int_{0}^{t-} h{\left( }s-u {\right) }\tilde{Z}(du)\right] ds {\right) }\nonumber \\ &\geq {\mathbbm{1}}{\left\lbrace}E'_{t}{\right\rbrace}\exp{\left( }-L\int_{t}^{\infty} \left[ \int_{-\infty}^{t- } h{\left( }s-u{\right) }Z(du) + {\left\vert}R_{s}{\right\vert}+ \int_{0}^{t-} h{\left( }s-u {\right) }\check {Z}(du)\right] ds {\right) }.\end{aligned}$$ Since $ \check \lambda_t \le \hat \lambda_t $ for all $ t \geq 0, $ $$\int_{0 }^{t- } h{\left( }s-u{\right) }\check Z(du) \le \int_{-\infty}^{t- } h{\left( }s-u{\right) }\hat Z(du) .$$ As a consequence, $$\begin{gathered} \int_{t}^{\infty} \left[ \int_{-\infty}^{t- } h{\left( }s-u{\right) }Z(du) + {\left\vert}R_{s}{\right\vert}+ \int_{0}^{t-} h{\left( }s-u {\right) }\check {Z}(du)\right] ds \\ \le 2 \int_{t}^{\infty} \left[ \int_{-\infty}^{t- } h{\left( }s-u{\right) }\hat Z(du) \right] ds + \int_t^\infty |R_s| ds = C_t +D_t ,\end{gathered}$$ where $$C_t := 2 \int_{t}^{\infty} \left[ \int_{-\infty}^{t- } h{\left( }s-u{\right) }\hat Z(du) \right] ds$$ is stationary and ergodic, and where $$D_t := \int_t^\infty |R_s| ds.$$ Clearly, $ D_t \to 0 $ as $t \to \infty $ almost surely. Now apply in Appendix with $ U_t := {\mathbbm{1}}{E_t} e^{-L C_t}, $ $ r_t := {\mathbbm{1}}{E_t} e^{-LC_t} - {\mathbbm{1}}{E_t'} e^{- L C_t - L D_t }.$ Clearly, $U_t $ is ergodic and satisfies $ P ( U_t > 0 ) > 0.$ To see that $r_t \to 0 $ almost surely as $t \to \infty , $ it suffices to prove that $ {\mathbbm{1}}E'_t - {\mathbbm{1}}E_t \to 0 $ almost surely, as $t \to \infty , $ which is equivalent to proving that $ {\mathbbm{1}}\check E^0_t \to 1 $ almost surely. But this follows from $$\sum_{i=1}^N {\mathbb E}\int_0^\infty \int_{{\mathbb{R}}_+} {\mathbbm{1}}\left \{ s \le \frac{3N }{x^* } |R _s| \right \} \pi^i (dz, ds) < \infty ,$$ which follows from . This finishes the first part of the proof of the coupling result. Finally, suppose that the initial process only satisfies . Take then a sequence $ (\check Z^m , \check X^m , \check A^m ) $ of $ N-$dimensional Age Dependent Hawkes processes with starting condition $ ( \check A^i_0)_{ i \le n } $ and initial processes $ (-m\vee R^i \wedge m)_{i \le N } .$ Write $ \check \lambda_t^{ m , i} = \psi^i ( \check X^{m, i }_t, \check A_t^{m, i } ) $ for the associated intensity and $ \check \lambda_t^{m \wedge m+1, i } := \check \lambda^{m, i}_t \wedge \check \lambda^{m+1, i }_t .$ Denote $R_t^{m,i} := -m \vee R_t^i \wedge m.$ As in the proof of Proposition \[prop:13\], we have that $$P ( \check Z^m \neq \check Z^{m+1} ) \le L \sum_{i=1}^N {\mathbb E}\int_0^\infty | R^{m, i}_t - R^{m+1 , i}_t | dt .$$ Since $ {\mathbb E}\int_0^\infty |R^i_t | dt < \infty , $ we conclude that $$\sum_m P ( \check Z^m \neq \check Z^{m+1} ) < \infty ,$$ implying that almost surely, $ \check Z^m = \check Z $ on $ {\mathbb{R}}_+ $ for sufficiently large $m. $ Since $ \check Z^m $ and $ Z$ couple eventually almost surely, this proves the coupling part. It remains to prove uniqueness of the stationary solution. Let ${\left( }Z',X',A' {\right) }$ be another Age Dependent Hawkes Process on $t\in {\mathbb{R}}$ compatible to $\pi^{1},\dots , \pi^{N}$. gives the inequality $${\left\vert}X^{'i }_{t }{\right\vert}\leq \sum_{j=1 }^{N }{\left( }\sum_{k \geq 0} \bar h_{ij} ( A^{'j }_t + k \delta ) + \int_{-\infty}^{t- A^{'j }_t }\bar h_{ij} (t-s) \pi^j_K ( ds ) {\right) }.$$ Let $\tau $ be a jump of $\hat{Z}$. Using the above inequality, it is shown inductively over future jumps of $\hat{Z}$ that $\hat{\lambda}_{t }\geq {\left\vert}\lambda^{'i }{\right\vert}$ for all $t\in {\left( }\tau,\infty {\right) }$. Thus it follows that $\hat{\lambda}_{t }\geq {\left\vert}\lambda^{'i }{\right\vert}$ for all $t\in {\mathbb{R}}$. Note that the $ {\left( }Z',X',A' {\right) }$ system may be written in terms of with initial signals $R^{'i }_{t }:= \sum_{j=1 }^{N }\int_{\infty }^{0 }h_{ij }{\left( }t-s {\right) }dZ^{'j }_{s }$. The same arguments as in give $$\mathbb{E}\int_{0 }^{\infty }{\left\vert}R^{'i}_{s }{\right\vert}ds \leq \mathbb{E}\int_{0 }^{\infty }\int_{\infty }^{0- } h{\left( }s-u {\right) }\hat{Z}{\left( }du {\right) }ds<\infty.$$ Therefore it follows from the 2nd point of this theorem that $$P{\left( }\exists \, t_{0}\in {\mathbb{R}}: \;\; Z'_{| [t_0, \infty) } = Z_{| [t_0, \infty ) }{\right) }= P\bigcup_{n=-\infty}^{\infty}{\left( }Z'_{| [n, \infty) } = Z_{| [n, \infty ) }{\right) }= 1.\label{uniqueeq}$$ Since $Z $ and $Z' $ are both compatible, it follows that ${\left( }Z,Z'{\right) }$ is compatible and therefore also stationary. Thus, the events ${\left\lbrace}Z'_{| [n, \infty) } = Z_{| [n, \infty ) }{\right\rbrace}$ have the same probability for all $n\in \mathbb{Z}$, and from it follows that the probability is equal to $1$. This proves that $Z=Z'$ almost surely. $\qed$ Age Dependent Hawkes processes with Erlang weight functions ----------------------------------------------------------- Here we show how the above results can be applied for weight functions given by Erlang kernels as in Example \[ex:erlang\], and consider a one-dimensional ($N=1$) Age Dependent Hawkes process $(Z, X, A), $ solution of $$\begin{aligned} Z_t &= &\int_{0}^{t}\int_{0}^{\infty}{\mathbbm{1}}{\left\lbrace}z\leq \psi {\left( }X_{s},A_{s}{\right) }{\right\rbrace}\pi {\left( }dz,ds{\right) },\nonumber\\ X_{t}&=&\int_{0}^{t-} h_{}{\left( }t-s{\right) }Z(ds) +R_t, \label{systemerlang}\\ A_{t}-A_{0} &=& t-\int_{0}^{t-} A_{s} Z(ds) ,\nonumber \end{aligned}$$ where $$\label{eq:erlangkernel2} h(t) = b \frac{t^n }{n!} e^{- \nu t } ,$$ for some $b \in {\mathbb{R}}, \nu > 0 $ and $ n \geq 0 ,$ and where $$R_t = \int_{- \infty}^0 h(t-s) z (ds) ,$$ for some fixed discrete point measure $z$ defined on $ ( - \infty , 0 ) $ such that $ \int_{- \infty}^0 h(t- s) z (ds) $ is well defined. It is well-known (see e.g. [@SusEva]) that the above system can be completed to a Markovian system $ ( X_t^{(0)} := X_t, X_t^{(1) }, \ldots , X_t^{(n)}, A_t) ,$ by introducing $$X_t^{(k) } := \int_{0}^{t-} b \frac{ (t-s)^{n-k}}{(n-k)!} e^{ - \nu ( t-s) } Z(ds) + \int_{- \infty}^0b \frac{ (t-s)^{n-k}}{(n-k)!} e^{ - \nu ( t-s) } z (ds) , \mbox{ for all } 0 \le k \le n.$$ By [@SusEva], these satisfy the system of coupled differential equations, driven by the PRM $\pi, $ given by $$d X_{t+}^{(k)} = - \nu X_{t+}^{(k)}dt + X_{t+}^{(k+1)} dt , \quad 0 \le k < n , \qquad d X_{t+}^{(n)} = - \nu X_{t+}^{(n)}dt + b Z(dt) ,$$ and $$A_{t}-A_{0} = t-\int_{0}^{t-} A_{s} Z(ds),$$ for $ t \geq 0. $ Evidently, $h$ satisfies . We suppose that $\psi ( x, a) $ satisfies , and we strengthen to $$\label{eq:psi} \psi (x, a ) \mbox{ is continuous in $x$ and $a; $ and $\psi ( x, a ) \geq c > 0 $ for all $ x, a$ with $ a \geq a^* .$}$$ By Lemma \[Xbound\] and Corollary \[cor:hbounded\], $ t \mapsto {\mathbb E} ( \lambda_t) = {\mathbb E}( \psi ( X_t, A_t) )$ and $ t \mapsto {\mathbb E}( |X_t|) = {\mathbb E}(|X_t^{(0)}|) $ are bounded on $ {\mathbb{R}}.$ By the same argument, also $ t \mapsto {\mathbb E}(|X_t^{(k)}|) $ is bounded for $ 1 \le k \le n.$ Therefore, $ ( X_t^{(0)} , X_t^{(1) }, \ldots , X_t^{(n)}) $ is a $1-$ultimately bounded Feller process (the Feller property follows from the continuity of $ \psi $), see e.g.[@miha]. Finally, for any $ x = ( x^0, \ldots , x^n ) \in {\mathbb{R}}^{n+1} $ write $ P_t ( (x, a ) , \cdot ) $ for the transition semigroup of $ (X^{(0)}_t, \ldots , X_t^{(n)} , A_t ) $ and let $ B_k = \{ (x, a ) : | x| + |a| \le k \} .$ Then $$P_t ( (x_0, a_0) , B_k^c ) \le P_t((x_0, a_0), |X_t| \geq \frac{k}{2} ) + P_t((x_0, a_0), A_t \geq \frac{k}{2} ) ,$$ where $$P_t((x_0, a_0), | X_t | \geq \frac{k}{2} ) \le \frac{ 2 \sup_t {\mathbb E}_{(x_0, a_0 ) } ( |X_t | )}{k} ,$$ and $$P_t((x_0, a_0), A_t \geq \frac{k}{2} ) \le e^{ - c ( \frac{k}{2}-a^*)_+} ,$$ implying inequality (6) of [@miha]. Thus, by Theorem 1 of [@miha], $ (X^{(0)}_t, \ldots , X_t^{(n)} , A_t)_{t \geq 0} $ possesses invariant measures (not necessarily unique ones), providing a different approach to prove Theorem \[Stab\]. Yet, Theorem \[Stab\] implies a much stronger result, since it proves the coupling property between the stationary random measure $ Z $ and $ \tilde Z .$ More precisely, in what follows we write $ (Z, X, A ) $ for the stationary version of , which exists according to Theorem \[Stab\]. Moreover, we write $ (\tilde Z, \tilde X, \tilde A ) $ for a version of starting at time $t= 0 $ from an arbitrary initial age $ a_0$ and an initial configuration $x_0= (x_0^{(0)}, \ldots , x_0^{(n)} ) $ with $ x_0^{(k)} = b\int_{- \infty }^0 \frac{ (-s)^{n-k}}{(n-k)!} e^{-\nu s } z (ds ) .$ Write $$\tau_c := \inf \{ t > 0 : Z \mbox{ and } \tilde Z \mbox{ couple at time } t \}\vee 1 .$$ Note that $$| h(s+u )| \le C |h (s)| | h(u)|$$ for all $ s,u \geq 1, $ where $C$ is an appropriate constant. It follows that almost surely, for all $ t \geq \tau_{c } + 1$ $$|X_t - \tilde X_t| \le \overline{h} ( t- \tau_c) (Z ( [0, \tau_c]) + \tilde Z ( [0, \tau_c] ))\tilde Z ( [0, \tau_c] )) + C |h(t- \tau_{c})| ( |X_{\tau_{c }} | + | \tilde X_{\tau_{c }} |) ,$$ showing that $$\lim_{t \to \infty } |X_t - \tilde X_t| = 0$$ almost surely, since $| h ( t - \tau_{c } )| \to 0 $ as $ t \to \infty.$ In the same way one proves that also $$\lim_{t \to \infty } |X^{(k)}_t - \tilde X^{(k)}_t| = 0$$ almost surely, for all $ 1 \le k \le n.$ Moreover, we obviously have that $ \tilde A = A $ on $ [ T_1 \circ \theta_{\tau_c}, \infty [ , $ where $ T_1 \circ \theta_{\tau_c} = \inf \{ t > \tau_c : Z ( [t]) = \tilde Z ( [t] ) = 1 \}.$ Notice that since $ \psi( x, a ) \geq c > 0 $ for all $ a \geq a^*, $ $ T_1 \circ \theta_{\tau_c} < \infty $ almost surely. This implies the uniqueness of the invariant measure. It also implies the Harris recurrence of the process $ ( X^{(0)}_t,\ldots , X_t^{(n)} , A_t) $ which can be seen as follows. An adaptation of the proof of Theorem 3 in [@aline] shows the following local Doeblin lower bound. For all $ (x^{**}, a^{**} ) \in {\mathbb{R}}^{n+1}\times {\mathbb{R}}_+ , $ there exist $R > 0 , $ an open set $ I \subset {\mathbb{R}}^{n+1}\times {\mathbb{R}}_+ $ and a constant $\beta \in ] 0, 1 [, $ such that for any $T > (n+2) a^* , $ $$\label{doblinminorization} P_{T} ( (x_0, a_0) , \cdot ) \geq \beta {\mathbbm{1}}_C ( x_0, a_0) U ( \cdot) ,$$ where $ C = B_R ( (x^{**}, a^{**} )) $ is the (open) ball of radius $R$ centered at $(x^{**}, a^{**} ) ,$ and where $ U $ is the uniform measure on $ I.$ We may apply the above result with $ (x^{**}, a^{**} ) \in supp ( \pi ) $ where $ \pi $ is the (unique) invariant measure of the process. Then for the stationary version of the process, $(X_t^{(0)}, \ldots, X_t^{(n)} , A_t) \in B_{R/2} ( (x^{**}, a^{**} ) $ infinitely often. Thus, also $(\tilde X^{(0)}_t, \ldots , \tilde X_t^{(n)} , \tilde A_t) \in B_{R} ( x^{**}, a^{**} ) = C $ infinitely often, almost surely, and then the classical regeneration technique (Nummelin splitting) allows to show that indeed the process is positively recurrent in the sense of Harris. Mean-field limit and propagation of chaos {#sec:mf} ========================================= In this section we focus on a multi-class mean-field setup of the Age Dependent Hawkes process. We propose a limit system, and show how the high dimensional system couples with the limit system. This can be seen as a generalization of the work of Chevallier [@chevallier] where a single class is considered under the assumption that the spiking rate function $\psi$ is uniformly bounded. The multi-class setup is similar to the one in [@SusEva] for ordinary Hawkes Processes, and to the data transmission model in [@carl2]. We also discuss how to approximate a Hawkes process induced by one weight function, by another Hawkes process, induced by different weight functions. **Setup in this section:** In addition to the fundamental assumptions, we introduce the following specifications for the mean-field setup, which will be used throughout this section. We partition the indices of individual units into ${{\cal K}}\in {\mathbb{N}}$ different populations, where ${{\cal K}}> 0$. More precisely, for each fixed total population size $N \in {\mathbb{N}},$ $$N_{k}:=N_{k}{\left( }N{\right) }:=\#{\left\lbrace}i\leq N : i\text{ in population } k {\right\rbrace}$$ will denote the number of units belonging to population $k, 1 \le k \le {{\cal K}},$ and $$N = N_1 + \ldots + N_{{{\cal K}}} .$$ We assume that each population represents an asymptotic part of all units, i.e., there exists $p_{k}>0$ such that $$\frac{N_{k}}{N} {\stackrel{N{\rightarrow}\infty}{\rightarrow}} p_{k}.$$ For a fixed $N\in {\mathbb{N}}, $ we re-index the $N$-dimensional Age Dependent Hawkes process of as $${\left( }Z^{kj}{\right) }_{k\leq {{\cal K}},j\leq N_{k}},$$ where the superscript $ kj $ denotes the $j$th unit in population $k.$ The weight function from the $i$th unit of population $l$ to the $j$th unit of population $k$ is given by $N^{-1}h_{kl}.$ Moreover, all units within the same population have the same spiking rate $\psi^{k}.$ Finally, we assume that all units in population $k$ have the same initial signal $R^{k},$ and that the initial ages are interchangeable in groups and mutually independent in and between groups. With this set of parameters, the Age Dependent Hawkes process ${\left( }Z,X,A{\right) }$ from becomes $$\begin{aligned} Z_t^{kj} &= \int_{0}^{t}\int_{0}^{\infty} {\mathbbm{1}}{\left\lbrace}z\leq \psi^{k}{\left( }X^{k}_{s},A^{kj}_{s}{\right) }{\right\rbrace}d\pi^{kj}{\left( }dz,ds{\right) },\quad j\leq N_{k},k\leq {{\cal K}},\label{meanfieldsystem} \\ X^{k}_{t}&=N^{-1}\sum_{l=1}^{{{\cal K}}}\sum_{j=1}^{N_{l}}\int_{0}^{t-}h_{kl}{\left( }t-s{\right) }Z^{lj}(ds)+R_t^{k},\quad \quad \quad \quad \quad \quad \;\;\; k\leq {{\cal K}}, \nonumber\end{aligned}$$ where $ A^{kj}$ is the age process of $Z^{kj}$ starting at $A^{kj}_{0}$. Sometimes, to explicitly indicate the dependency on $N, $ we add $N$ to the superscript and write $ Z_t^{Nkj } .$ ### Model observations {#model-observations .unnumbered} 1. Suppose that the initial ages $(A^i_0 )_{ i \in {\mathbb{N}}} $ are exchangeable. Then the symmetry of the system gives interchangeability between units in same population, i.e., $Z^{kj}\stackrel{\mathcal L}{=}Z^{ki}$ for $i,j\leq N_{k},k\leq {{\cal K}}$. 2. In the mean-field setup, all units within a population $k$ share the same memory process $X^{k}$. The Limit System ---------------- We propose a limit system for $N{\rightarrow}\infty$. To pursue this goal, take finite variation functions $t\mapsto \alpha^{k}_{t}$, locally bounded functions $t\mapsto \beta^{k}_{t}$, and PRMs $\pi^{k}$ for $k\leq {{\cal K}},$ and consider the stochastic convolution equation $$\begin{aligned} \phi^{k}_{t}&=&\int_{0}^{t} {\mathbb E}\psi^{k}{\left( }x^{k}_{s},A^{k}_{s} {\right) }ds , \nonumber\\ x^{k}_{t}&=&\sum_{l=1}^{{{\cal K}}}p_{l}\int_{0}^{t} h_{kl}{\left( }t-s {\right) }d\alpha^{l}_{s}+\beta^{k}_{t} , \label{diff}\\ A^{k}_{t}-A^{k}_{0}&=&t-\int_{0}^{t-}\int_{0}^{\infty} A^{k}_{s}{\mathbbm{1}}{\left\lbrace}z\leq \psi^{k}{\left( }x^{k}_{s},A^{k}_{s}{\right) }{\right\rbrace}\pi^{k}{\left( }dz,ds{\right) }, \nonumber \end{aligned}$$ with unknown ${\left( }\phi,x,A{\right) }= {\left( }\phi^{k}, x^{k}, A^{k}{\right) }_{k\leq {{\cal K}}}$. Notice that only $A$ is stochastic and that $\phi$ depends on the law of $A$. We are motivated by what for the moment is a heuristic: $N^{-1}\sum_{j=1}^{N_{k}}Z^{N kj}\approx p_{k}{\mathbb E} Z^{k}$ for large $N,$ where $ (Z^1, \ldots, Z^{{{\cal K}}} ) $ denotes the limit process such that each $Z^k $ describes the jump activity of a typical unit belonging to population $k.$ This relation invites the idea that the memory process for $N{\rightarrow}\infty$, $t\mapsto x_{t}$ would satisfy the integral system with $\phi^{k}_{t}=\alpha^{k}_{t}={\mathbb E} Z^k _{t}$ and $\beta^{k}_{t} = {\mathbb E} R^{k1}_{t}$. This motivates the following result. \[odelm\] Let $\beta_{t}={\left( }\beta^{k}_{t}{\right) }_{k\leq {{\cal K}}}$ be measurable and locally bounded. There is a unique function $\alpha$ such that $\alpha = \phi$, where ${\left( }\phi,x,A {\right) }$ is the solution to . Moreover, $\phi$ is continuous and $x$ is bounded on ${\left[}0,T {\right]}$ by a constant $C$ which depends on $h:= \sum_{k, j } | h_{kj } |,$ ${\left\Vert}\beta{\right\Vert}_T$, $T$ and $L$. The proof is given in the Appendix. Once this lemma is established, we can ensure existence of the limit process. \[koroet\] Let $\beta_{t}={\left( }\beta^{k}_{t}{\right) }_{k\leq K}$ be measurable and locally bounded. There is a unique solution ${\left( }Z,A {\right) }$ to the integral equation $$Z^{k}_{t} = \int_{0}^{t}\int_{0}^{\infty}{\mathbbm{1}}{\left\lbrace}z\leq \psi^{k} {\left( }\sum_{l=1}^{{{\cal K}}}p_{l}\int_{0}^{s} h_{kl}{\left( }s-u {\right) }d{\mathbb E} Z^{l}_{u}+\beta^{k}_{s},A^{k}_{s}{\right) }{\right\rbrace}\pi^{k}{\left( }dz,ds{\right) }, 1 \le k \le {{\cal K}},$$ where $A^{k}$ is the age process corresponding to $Z^{k}$, initialized at $A^{k}_{0}$. Let ${\left( }\phi,x,A {\right) }_{k\leq {{\cal K}}}$ be the tuple given in . Define the counting process $$Z^{k}_{t}:=\int_{0}^{t}\int_{0}^{\infty}{\mathbbm{1}}{\left\lbrace}z\leq \psi^{k}{\left( }x^{k}_{s},A^{k}_{s}{\right) }{\right\rbrace}\pi^{k}{\left( }dz,ds{\right) }.$$ It is clear that $A^{k}$ is the age process of $Z^{k}$, and since $d{\mathbb E} Z^{k}_{t}={\mathbb E}\psi^{k}{\left( }x^{k}_{t},A^{k}_{t} {\right) }dt $, $Z^{k}$ will satisfy the desired identity. For uniqueness, consider another solution ${\left( }\tilde{Z}^{k},\tilde{A}^{k} {\right) }_{k\leq {{\cal K}}}$, which satisfies the same identity : $$\tilde{Z}^{k}_{t}= \int_{0}^{t}\int_{0}^{\infty}{\mathbbm{1}}{\left\lbrace}z\leq \psi^{k} {\left( }\sum_{l=1}^{{{\cal K}}}p_{l}\int_{0}^{s} h_{kl}{\left( }s-u {\right) }d{\mathbb E} \tilde{Z}^{l}_{u}+\beta^{k}_{s},\tilde{A}^{k}_{s}{\right) }{\right\rbrace}\pi^{k}{\left( }dz,ds{\right) }.$$ Defining $\tilde{x}^{k}_{t}=\sum_{l=1}^{{{\cal K}}}p_{l}\int_{0}^{t} h_{kl}{\left( }t-s {\right) }d{\mathbb E} \tilde{Z}^{l}_{u}+\beta^{k}_{t}$ and $\tilde{\phi}^{k}_{t}=\int_{0}^{t} {\mathbb E} \psi^{k}{\left( }\tilde{x}^{k}_{s},\tilde{A}^{k}_{s} {\right) }ds$, we note that ${\mathbb E} \tilde{Z}^{k}_{t}=\tilde{\phi}^{k}_{t}$. Thus, if we insert $\alpha = \tilde{\phi}$ in , ${\left( }\tilde{\phi}, \tilde{x},\tilde{A} {\right) }$ is a solution, and hence the uniqueness part of gives that ${\left( }\tilde{\phi}, \tilde{x},\tilde{A} {\right) }= {\left( }\phi, x , A {\right) }$ and thus also $Z = \tilde{Z}$. Large network asymptotics and weight approximations --------------------------------------------------- In this section we couple the limit system proposed in the previous section with the $N$-dimensional Hawkes process. We now proceed to show how the finite-dimensional systems converge to the limit system. The result is traditionally named [*Propagation of Chaos*]{}, a typical result within mean-field theory. Specifically for Hawkes processes, there are several variants of this result. Some of the recent results may be found in [@chevallier], [@dfh] and in [@SusEva]. ### Framework for Propagation of Chaos {#framework-for-propagation-of-chaos .unnumbered} - Define for each $k\leq {{\cal K}}$ a sequence ${\left( }R^{Nk}{\right) }_{N\in {\mathbb{N}}}$ of initial signals with $\sup_{k\leq {{\cal K}},N\in {\mathbb{N}}} {\left\Vert}{\mathbb E} R^{Nk} {\right\Vert}_{t}<\infty$, and assume that there is a locally bounded function $t\mapsto r^{k}_{t}$ such that $$\begin{aligned} \int_{0}^{t}{\mathbb E} {\left\vert}R^{Nk}_{s}-r^{k}_{s}{\right\vert}ds \to 0 \, \mbox{ as } \, N \to \infty, \label{Rconv} \end{aligned}$$ for all $t\geq 0$. - Assume that the initial ages $ A_0^{k i }, 1 \le k \le {{\cal K}}, 1 \le i < \infty $ are i.i.d. - Define for $N\in {\mathbb{N}}, $ $1 \le k,l\leq {{\cal K}},$ weight functions $h^{N}_{kl} : {\mathbb{R}}_{+}{\rightarrow}{\mathbb{R}}.$ Assume that $h^{N}_{kl}{\stackrel{}{\rightarrow}} h_{kl}$ as $N \to \infty $ locally in ${\mathcal L}^{1}$ and that $h_{kl}\in {\mathcal L}^{2}_{loc}$. Consider an i.i.d. sequence of driving PRMs $ \pi^{kj } , 1 \le k \le {{\cal K}}, j \geq 1 .$ Define for each $N\in{\mathbb{N}}, $ the $N$-dimensional Hawkes process $${\left( }Z^{N},X^{N},A^{N} {\right) }= \left( Z^{Nki}, X^{Nk}, A^{Nki} \right)_{k \le {{\cal K}}, i \le N_k } ,$$ given by , driven by $(\pi^{kj }),$ with weight functions ${\left( }N^{-1 }h^{N}_{kl}{\right) }$, spiking rate ${\left( }\psi^{k}{\right) }$ and initial processes ${\left( }R^{Nk}{\right) }.$ Applying with weight functions ${\left( }h_{kl}{\right) }$ and initial functions $\beta^{k} = r^{k},$ we obtain, for any $ 1 \le k \le {{\cal K}}$ and for all $ i \in {\mathbb{N}}, $ a solution $(Z^{ki}, X^k, A^{ki} ) $ to the equation $$Z^{ki}_{t} = \int_{0}^{t}\int_{0}^{\infty}{\mathbbm{1}}{\left\lbrace}z\leq \psi^{k} {\left( }\sum_{l=1}^{{{\cal K}}}p_{l}\int_{0}^{s} h_{kl}{\left( }s-u {\right) }d{\mathbb E} Z^{lj }_{u}+r^{k}_{s},A^{ki}_{s}{\right) }{\right\rbrace}\pi^{ki}{\left( }dz,ds{\right) },$$ $1 \le k \le {{\cal K}}, i \in {\mathbb{N}},$ driven by the same sequence of PRMs. \[theo:prop\] Consider the framework described above.\ The following convergences hold for all $ t \geq 0$ $$\begin{aligned} \label{conresult} {\mathbb E} {\left\vert}d{\left( }Z^{Nki}_{t}-Z^{ki}_{t} {\right) }{\right\vert}{\stackrel{}{\rightarrow}} 0, \text{ for } \; N {\rightarrow}\infty , \end{aligned}$$ for all $k\leq {{\cal K}},i\in {\mathbb{N}}$. In particular, for any finite set of indices ${\left( }k_{1}, i_{1},\dots ,k_{n},i_{n}{\right) },$ we have weak convergence $$\begin{aligned} {\left( }Z^{Nk_{1}i_{1}},\dots ,Z^{Nk_{n}i_{n}}{\right) }_{t\geq 0}{\stackrel{wk}{\rightarrow}}{\left( }Z^{k_{1}i_{1}},\dots ,Z^{k_{n}i_{n}}{\right) }_{t\geq 0} \end{aligned}$$ as $ N \to \infty $ (in $ D ( {\mathbb{R}}_+, {\mathbb{R}}_+^n ), $ endowed with the topology of locally-uniform convergence). To prove this theorem we shall need the following lemma. \[weightapproximation\] Let $( h_{kl})_{ 1 \le k, l \le {{\cal K}}} ,( \tilde{h}_{kl})_{1 \le k,l \le {{\cal K}}} $ be sets included in a family ${\mathcal E}$ of real-valued functions defined on $ {\mathbb{R}}_+$ which is uniformly integrable on $[0,T] .$ Define $( Z,X,A ),( \tilde{Z},\tilde{X},\tilde{A} )$ as the $N-$dimensional Age Dependent Hawkes process with weight functions $( N^{-1 }h_{kl})_{ 1 \le k, l \le {{\cal K}}} ,$ $( N^{-1 }\tilde{h}_{kl})_{1 \le k,l \le {{\cal K}}} $, rate functions ${\left( }\psi^{k}{\right) }_{ k \le {{\cal K}}} $, and with initial conditions $A_{0}$, $ R^{k}$. There exists $C>0$ depending on the family ${\mathcal E} $, on $T,L,{{\cal K}}$ and on $\sup_{k\leq {{\cal K}}}{\left\Vert}{\mathbb E} R^{k}{\right\Vert}_{T}$ (but not on $N$) such that $$\begin{aligned} \sum_{k=1 }^{\mathcal{K} }{\mathbb E} {\left\vert}d{\left( }Z^{k1}_{t}-\tilde{Z}^{k1}_{t}{\right) }{\right\vert}\leq C_{T}\sum_{k,l=1 }^{\mathcal{K} }\int_{0}^{t}{\left\vert}h_{kl }-\tilde{h}_{kl }{\right\vert}{\left( }s {\right) }ds , \end{aligned}$$ for all $t\leq T. $ The proofs of and may be found in the Appendix. The result shows that finitely many units will be asymptotically independent for $N{\rightarrow}\infty.$ The mean-field limit in the case of a hard refractory period ------------------------------------------------------------ In this section we consider the mean-field limit of Age Dependent Hawkes processes with one single population (${{\cal K}}= 1 $) and a weight function given by an Erlang kernel as in Example \[ex:erlang\], $$h(t) = b e^{ - \nu t } \frac{t^n }{n!} ,$$ for some fixed constants $b \in {\mathbb{R}}, \nu > 0 , $ $n \in {\mathbb{N}}.$ Throughout this section we suppose that $$\label{eq:hard} \psi (x, a) = f(x) {\mathbbm{1}}\{ a \geq \delta \}.$$ For the limit system, we have $$\label{eq:limitlambda} \phi_t = \int_0^t E ( \psi ( x^{(0)}_s , A_s) ) ds ,$$ and we write $$\bar \lambda_t = E ( \psi ( x^{(0)}_t , A_t) )$$ for the expected jump rate at time $t.$ In the above equation, $$\begin{aligned} \label{eq:cascade} d A_{t+} &= &dt - A_{t} \int_{{\mathbb{R}}_+} {\mathbbm{1}}{\{ z \le \psi ( x^0_t, A_{t})\}} \pi (dz, dt ) ,\nonumber \\ dx^{(0)}_t&=&x^{(1)}_tdt-\nu x^{(0)}_tdt ,\\ &\vdots& \nonumber \\ dx^{(n-1)}_t&=&x^{(n)}_tdt-\nu x^{(n-1)}_tdt , \nonumber\\ dx^{(n)}_t&=&-\nu x^{(n)}_tdt+b d \phi_t. \nonumber\end{aligned}$$ We start by studying the age process within the limit system. Write $ \tau_t = \sup \{ 0 \le s \le t : \Delta A_s \neq 0 \} $ for the last jump time of the process before time $t,$ where by convention, $ \sup \emptyset := 0 .$ Then obviously, $$A_{t+} = (t- \tau_t) {\mathbbm{1}}{\{ \tau_t > 0 \}} + (A_0 + t ) {\mathbbm{1}}{\{ \tau_t = 0 \}}.$$ Due to , we have the following $${\mathcal L} ( \tau_t) (dz) = E( e^{- \int_0^t f(x_s^{(0)} ) {\mathbbm{1}}\{ A_0 + s \geq \delta \} ds } ) \delta_0 (dz) + f(x_z^{(0)}) p_{z} e^{ - \int_{z+ \delta}^t f( x_s^{(0)}) ds } {\mathbbm{1}}{ \{0 < z < t \}} dz,$$ where $p_t = P ( A_t \geq \delta ) $ is given by $$\begin{aligned} p_t &=& E \left( {\mathbbm{1}}{\{ A_0 \geq \delta - t \}} e^{- \int_{(\delta - A_0) \vee 0} ^t f(x_s^{(0)} ) ds } \right) + \int_0^{t- \delta} f(x_s^{(0)}) p_s e^{- \int_{s + \delta }^t f( x_u^{(0)}) du } ds \\ &=& \int_{ (\delta - t )\vee 0}^\infty \pi_0 ( da) e^{- \int_{(\delta - a) \vee 0} ^t f(x_s^{(0)} ) ds } + \int_0^{t- \delta} f(x_s^{(0)}) p_s e^{- \int_{s + \delta }^t f( x_u^{(0)}) du } ds ,\end{aligned}$$ where $ A_0 \sim \pi_0 ( da) .$ In particular, the above representation shows that, even starting from a non-smooth initial trajectory, $p_t$ is eventually smooth. For any starting law $\pi_0 ( da) , $ $t \mapsto p_t$ is continuous on $ ] \delta , \infty ],$ and thus, taking into account , $ C^1 ( ] 2 \delta , \infty [, {\mathbb{R}}) ,$ solving $$\dot p_t = - f ( x_t^{(0)} ) p_t dt + f(x_{t- \delta }^{(0)}) p_{t- \delta } dt , \mbox{ for all } t > 2 \delta .$$ If the starting law is smooth, we can say more. If $ \pi_0 ( da) = \pi_0 (a) da , $ with $\pi_0 \in C ( {\mathbb{R}}, {\mathbb{R}}_+), $ then for all $ t < \delta , $ $$p_t = \int_{ \delta- t}^\infty \pi_0 (a) e^{- \int_{(\delta - a) \vee 0} ^t f(x_s^{(0)} ) ds } da$$ is continuous and thus, taking into account , $ C^1 ( [0, \delta [, {\mathbb{R}}) .$ In particular, on $ [0, \delta [, $ $t \mapsto p_t$ solves $$\dot p_t = \pi_0 (\delta - t ) - f ( x_t^{(0)} ) p_t dt .$$ By induction, this implies that $ t \mapsto p_t $ is continuous on $ {\mathbb{R}}_+ $ and $C^1 $ on $ ] \delta , \infty [, $ with $$\dot p_t = - f ( x_t^{(0)} ) p_t dt + f(x_{t- \delta }^{(0)}) p_{t- \delta } dt , \mbox{ for all } t > \delta .$$ Moreover, at $t = \delta, $ we have $$\dot p_{\delta - } = \pi_0 ( 0 ) - f ( x_\delta^{(0)} ) p_\delta \; \mbox { and } \; \dot p_{\delta + } = - f ( x_\delta^{(0)} ) p_\delta + f ( x_0^{(0)} ) p_0 .$$ Stationary solutions {#stationary-solutions .unnumbered} -------------------- We are looking for stationary solutions of . At equilibrium, we necessarily have that $$x^{(0)} \equiv x^*$$ such that $ A_t $ is a renewal process with dynamics $$\label{eq:renewal} d A_{t+} = dt - A_{t} \int_{{\mathbb{R}}_+} 1_{\{ z \le \psi ( x^* , A_{t})\}} \pi (dz, dt ) .$$ $A$ is recurrent in the sense of Harris, if $ \int_0^\infty \psi ( x^* , A_{t}) dt = \infty $ almost surely. This is granted by the following condition. For all $x, $ there exists $ r(x) \geq 0 $ such that $ \psi ( x, a) $ is lower bounded for all $ a \geq r( x) .$ The stationary distribution of has the density (see Proposition 21 of [@evafou]) $$g_{x^* } (a) = \kappa e^{ - \int_0^a \psi ( x^* , z) dz }$$ on ${\mathbb{R}}_+,$ where $ \kappa $ is such that $ \int_0^\infty g_{x^* } (a) da = 1.$ Recall that $$\bar \lambda_t = \frac{ d \phi_t}{dt}$$ denotes the (expected) jump rate of the limit system at time $t.$ Then at equilibrium, the total jump rate is constant and given by $ \bar \lambda_t =\bar \lambda.$ From we get that $$\bar \lambda = \kappa \int_0^\infty \psi ( x^* , a ) e^{ - \int_0^a \psi (x^* , z ) dz } da = \kappa ,$$ where we have used the change of variables $ y = \int_0^a \psi ( x^* , z) dz, dy = \psi ( x^* , a ) da .$ As a consequence, $$\bar \lambda = \kappa$$ implying that at equilibrium, the jump rate of the system is solution of $$\label{eq:fixedpoint} \frac{1}{\bar \lambda } = \int_0^\infty \exp \left( { -\int_0^a \psi (\frac{b}{\nu^{n+1} }\bar \lambda , z) dz}\right) da .$$ Here we have used that at equilibrium $$x^* = x^{(0 )} = \frac{1}{\nu } x^{(1)} = \ldots = \frac{b}{\nu^{n+1}} \bar \lambda ,$$ which follows from . Suppose that $ x \mapsto \psi ( x, a ) $ is strictly increasing for any fixed $a \geq 0 $ and that $ b < 0 .$ There exists a unique solution $ \lambda^* $ to . Recall that we suppose that $ \psi (x, a ) = f(x) {\mathbbm{1}}{ \{a \geq \delta \}},$ for some $\delta > 0 .$ We calculate the right hand side of and obtain the fixed point equation $$\label{eq:fp} \int_0^\infty \exp \left( { -\int_0^a \psi (\frac{b}{\nu^{n+1} }\bar \lambda , z) dz}\right) da = \delta + \frac{1}{ f( \frac{b}{\nu^{n+1} }\bar \lambda )}= \frac{1}{\bar \lambda } .$$ More generally, for any Hawkes process with mean field interactions, rate function $ \psi (x, a ) $ given by $ \psi ( x, a) = f(x) {\mathbbm{1}}{ \{a \geq \delta \}}$ and [**general weight function**]{} $ h \in {\mathcal L}^1 ( {\mathbb{R}}_+) , $ we obtain the fixed point equation $$\label{eq:fixedpoint2} \frac{1}{\bar \lambda } = \delta + \frac{1}{ f( \bar \lambda \int_0^\infty h (t) dt )}$$ for the limit intensity. This limit intensity depends on the length of the refractory period, we write $ \bar \lambda = \bar \lambda ( \delta ) $ to indicate this dependence. It is then natural to study the influence of the length of the refractory period $ \delta $ on the limit intensity. If $ f $ is increasing and $ \int_0^\infty h (t) dt < 0 , $ then clearly $$\delta \mapsto \bar \lambda ( \delta )$$ is decreasing: increasing the length of the refractory period “calms down the system”. If however the system is [**excitatory**]{}, that is, $ \int_0^\infty h(t) dt > 0 ,$ and if there exists a solution to the fixed point equation , then we would rather expect that $ \delta \mapsto \bar \lambda ( \delta ) $ is increasing. Suppose e.g. that $f$ is lower- and upper-bounded. Then the function $$\bar \lambda \mapsto \frac{1}{ f( \bar \lambda \int_0^\infty h (t) dt )}$$ is a strictly decreasing function mapping $ [0, \infty ] $ onto $ [ \frac{1}{ f (0) } , \frac{1}{f( \infty ) }].$ Therefore, there is exactly one fixed point solution of , and $ \delta \mapsto \bar \lambda ( \delta ) $ is increasing. Appendix ======== Here we prove Lemma \[odelm\], and Lemma 3.4. Then we collect some useful results about counting processes. Proofs {#proofs .unnumbered} ------ It suffices to show that a unique solution exists on ${\left[}0,T {\right]}$, for arbitrary $T\geq 0$. In the following proof, $C:=C_{T}$ will denote a dynamic constant depending on the parameters described in the lemma. It need not represent the same constant from line to line, nor from equation to equation. First we prove existence of a solution to with $\phi_{t}=\alpha_{t}$ using Picard-iteration. For $n\in {\mathbb{N}}$ define ${\left( }\phi^{n},x^{n},A^{n}{\right) }= {\left( }\phi^{nk},x^{nk},A^{nk} {\right) }_{k\leq {{\cal K}}}$ as follows. Initialize the system for $n=0$ by putting ${\left( }\phi^{0k},x^{0k},A^{0k} {\right) }\equiv {\left( }0,0,A_{0} {\right) }$. For general $n\in{\mathbb{N}}, n \geq 1 , $ the triple ${\left( }\phi^{n},x^{n},A^{n}{\right) }$ is defined as the solution to with $\alpha=\phi^{n-1}$ . Inductively it is seen that these processes are well-defined. Recall that $h=\sum_{k,l=1}^{{{\cal K}}} {\left\vert}h_{kl} {\right\vert}.$ Using we bound $x^{n}$ by $${\left\vert}x^{n}_{t}{\right\vert}\leq \sum_{l=1}^{{{\cal K}}} \int^{t}_{0} {\left\vert}h{\left( }t-s {\right) }{\right\vert}d \phi^{n-1,l}_{s} + | \beta_t| \leq C\int_{0}^{t} {\left\vert}h{\left( }s {\right) }{\right\vert}ds + C\int^{t}_{0} {\left\vert}h{\left( }t-s {\right) }{\right\vert}{\left\vert}x^{n-1}_{s}{\right\vert}ds + | \beta_t| .$$ It follows from in the Appendix that there exists a constant $C>0$ which bounds all ${\left\Vert}x^{n}{\right\Vert}_{T},n\in {\mathbb{N}}$. Using this upper bound on $x^{n}, $ we also bound the difference of two consecutive solutions. Define $$\begin{aligned} \delta^{n}_{t}=\sum_{k=1}^{{{\cal K}}}\int_{0}^{t} {\mathbb E} {\left\vert}\psi^{k}{\left( }x^{nk}_{s},A^{nk}_{s} {\right) }-\psi^{k}{\left( }x^{n-1,k}_{s},A_{s}^{n-1,k} {\right) }{\right\vert}ds . \end{aligned}$$ The Lipschitz property of $\psi$ and the bound on $x^{n}$ yield $$\delta^{n+1}_{t}\leq C\int_{0}^{t} \left( {\left\vert}x^{n+1}_{s}-x^{n}_{s}{\right\vert}+\sum_{k=1}^{{{\cal K}}}P{\left( }{\left\Vert}A^{n+1,k}-A^{nk}{\right\Vert}_{s}>0 {\right) }\right) ds.$$ For the probability term, we note that a necessity for the age processes to differ, is that one of their corresponding intensities catches a $\pi$-singularity which the other one does not catch. This leads to the inequality $$\begin{aligned} &\sum_{k=1}^{{{\cal K}}}P{\left( }{\left\Vert}A^{n+1, k}-A^{nk }{\right\Vert}_{t}>0 {\right) }\\\leq& \sum_{k=1}^{{{\cal K}}} P{\left( }\int_{0}^{t}\int_{0}^{\infty}{\left\vert}{\mathbbm{1}}{\left\lbrace}z\leq \psi^{k}{\left( }x^{n+1,k}_{s},A^{n+1,k}_{s} {\right) }{\right\rbrace}-{\mathbbm{1}}{\left\lbrace}z\leq \psi^{k}{\left( }x^{nk}_{s},A^{nk}_{s}{\right) }{\right\rbrace}{\right\vert}\pi^{k} {\left( }dz,ds {\right) }\geq 1 {\right) }\nonumber \\ \leq& \, \delta^{n+1}_{t},\end{aligned}$$ where the latter inequality follows by the Markov inequality. By Gronwall’s inequality we obtain $$\begin{aligned} \delta^{n+1}_{t}\leq C \int_{0}^{t} {\left\vert}x^{n+1}_{s}-x^{n}_{s}{\right\vert}\;ds.\label{eq1}\end{aligned}$$ Moreover, Lemma 22 of [@dfh] gives $$\begin{aligned} \int_{0}^{t}{\left\vert}x^{n+1}_{s}-x^{n}_{s}{\right\vert}ds \leq \sum_{l=1}^{{{\cal K}}}\int_{0}^{t}\int_{0}^{s} h ( s-u) {\left\vert}d{\left( }\phi^{nl}_{u}-\phi^{n-1,l}_{u}{\right) }{\right\vert}ds \leq \int_{0}^{t} h( t-s) \delta^{n}_{s} ds. \label{eq2}\end{aligned}$$ It therefore follows from in the Appendix that for all $1\le k\leq K,$ $$\sum_{n=1}^{\infty} \sup_{ t \le T } | \phi^{n+1,k}_{t}-\phi^{n,k}_{t}| \, \leq \, \sum_{n=1}^{\infty}\delta^{n}_{T}<\infty .$$ Thus, $\phi^{n} $ and therefore also $ x^{n}$ converge locally-uniformly to some $\phi,x$, respectively. Moreover, $$\begin{aligned} P{\left( }A^{n}_{s\leq T}\neq A^{n+1}_{s\leq T} \; i.o.{\right) }= P\bigcap_{m\in{\mathbb{N}}} \bigcup_{n\geq m} {\left( }{\left\Vert}A^{n}-A^{n+1} \; {\right\Vert}_{T}>0{\right) }\leq \lim_{m{\rightarrow}\infty}\sum_{n\geq m}^{\infty}\delta^{n+1}_{T}=0.\end{aligned}$$ It follows that almost surely, $A^{n}$ converges to some limit $A$ after finitely many iterations. We need to show that the limit triple ${\left( }\phi,x,A {\right) }$ satisfies with $ \phi_t = \alpha_t .$ Recall that $x\mapsto \psi^{k}{\left( }x,a{\right) }$ is continuous for fixed $a\in {\mathbb{R}}_{+}$. Since $A^n$ reaches its limit in finitely many iterations, and $\psi$ is continuous in $x$ for fixed $a, $ we obtain $\lim_{n{\rightarrow}\infty}\psi^{k}{\left( }x^{nk}_{s},A^{nk}_{s}{\right) }$ exists for all $s\leq T, $ almost surely. By dominated convergence and , it follows that ${\mathbb E} \psi^{k}{\left( }x^{nk}_{s},A^{nk}_{s}{\right) }$ converges as well. Therefore, once again by dominated convergence, $$\phi_t^k = \lim_{n \to \infty} \phi_t^{nk} = \lim_{n \to \infty} \int_0^t {\mathbb E} \psi^{k}{\left( }x^{nk}_{s},A^{nk}_{s}{\right) }ds =\int_0^t {\mathbb E} \psi^{k}{\left( }x^{k}_{s},A^{k}_{s}{\right) }ds,$$ that is, $ \phi$ satisfies . One shows similarly that $$\begin{aligned} x_t^k& =& \sum_{l=1}^{{{\cal K}}} \lim_{n \to \infty} \int_0^t h_{kl} (t-s) d \phi_s^{n l } + \beta_t^k \\ & =& \sum_{l=1}^{{{\cal K}}} \lim_{n \to \infty} \int_0^t h_{kl} (t-s) {\mathbb E} \psi^l ( x_s^{n l }, A_s^{n l } ) ds + \beta_t^k\\ & =& \sum_{l=1}^{{{\cal K}}} \int_0^t h_{kl} (t-s) {\mathbb E} \psi^l ( x_s^{ l }, A_s^{ l } ) ds + \beta_t^k\\ &=& \sum_{l=1}^{{{\cal K}}} \int_0^t h_{kl} (t-s) d \phi_s^{ l } + \beta_t^k,\end{aligned}$$ and $x$ satisfies as well. For the age process, notice that the càglàd process $$\begin{aligned} {\varepsilon}{\left( }t{\right) }=\sum_{k=1}^{{{\cal K}}}\int_{0}^{t-}\int_{0}^{\infty} {\mathbbm{1}}{\left\lbrace}z = \psi^{k}{\left( }x^{k}_{s},A^{k}_{s}{\right) }{\right\rbrace}\pi^{k}{\left( }dz,ds{\right) }\end{aligned}$$ has a compensator which is equal to zero for all $ t \geq 0, $ almost surely, by . Therefore, ${\varepsilon}_t = 0 $ for all $ t \geq 0 $ almost surely. This implies that with probability $1,$ ${\mathbbm{1}}{\left\lbrace}z\leq \psi^{k} {\left( }x^{nk}_{s},A_{s}^{k}{\right) }{\right\rbrace}$ converges $\pi^{k}-$a.e. to ${\mathbbm{1}}{\left\lbrace}z\leq \psi^{k} {\left( }x^{k}_{s},A_{s}^{k}{\right) }{\right\rbrace}$ for all $k\leq {{\cal K}}$. As a consequence, $$\begin{aligned} A^{k}_{t}-A^{k}_{0} &=t-\lim_{n{\rightarrow}\infty}\int_{0}^{t-} A^{nk}_{s}{\mathbbm{1}}{\left\lbrace}z\leq \psi^{k}{\left( }x^{nk}_{s},A^{nk}_{s}{\right) }{\right\rbrace}\pi^{k}{\left( }dz,ds{\right) }\\&=t-\lim_{n{\rightarrow}\infty}\int_{0}^{t-} A^{k}_{s}{\mathbbm{1}}{\left\lbrace}z\leq \psi^{k}{\left( }x^{nk}_{s},A^{k}_{s}{\right) }{\right\rbrace}\pi^{k}{\left( }dz,ds{\right) }\\&= t-\int_{0}^{t-}\int_{0}^{\infty} A^{k}_{s}{\mathbbm{1}}{\left\lbrace}z\leq \psi^{k}{\left( }x^{k}_{s},A^{k}_{s}{\right) }{\right\rbrace}\pi^{k}{\left( }dz,ds{\right) },\end{aligned}$$ where we have used dominated convergence. Since $x$ is locally bounded, it follows that $\phi$ is $C^{0}.$ To prove uniqueness, we assume that ${\left( }\tilde{\phi}, \tilde{x} , \tilde{A} {\right) }$ also solves with $$\tilde{x}^{k}_{t}=\sum_{l=1}^{{{\cal K}}}p_{l}\int_{0}^{t}h_{kl}{\left( }t-s {\right) }d\tilde{\phi} ^{l}_{s} + \beta_t^k .$$ Define $$\begin{aligned} \delta_{t} = \sum_{k=1}^{{{\cal K}}}\int_{0}^{t}{\mathbb E} {\left\vert}\psi^{k}{\left( }x^{k}_{s},A^{k}_{s} {\right) }-\psi^{k} {\left( }\tilde{x}^{k}_{s},\tilde{A} ^{k}_{s}{\right) }{\right\vert}ds.\end{aligned}$$ Considerations analogous to the ones given in the proof of existence, gives that $$\begin{aligned} {\left\vert}x_{t}-\tilde{x}_{t}{\right\vert}\leq \delta_{t}\leq C\int_{0}^{t} h{\left( }t-s {\right) }\delta_{s}ds .\end{aligned}$$ From Gronwall it follows that $\delta \equiv 0$, and therefore also that $x = \tilde{x} $ on ${\left[}0,T{\right]}.$ From it follows immediately $\phi = \tilde{\phi} $ and $A = \tilde{A} $ almost surely. Throughout this proof, $C$ is a dynamic constant with dependencies as declared in the theorem. Define the functions $h = \sum_{k,l } {\left\vert}h_{kl }{\right\vert}, \tilde{h} = \sum_{k,l } {\left\vert}\tilde{h}_{kl }{\right\vert}$. First we prove that the memory processes ${\mathbb E} |X_{t}|,{\mathbb E} |\tilde{X}_{t}| $ are bounded on $[0,T]$ by a suitable constant $C $. Note that $$\begin{aligned} {\mathbb E} {\left\vert}X_t {\right\vert}&\leq& \sum_{l=1}^{{{\cal K}}}{\left( }\int_{0}^{t} h{\left( }t-s{\right) }{\mathbb E} \psi^{l}{\left( }X^{l}_{s},A^{l}_{s}{\right) }ds+{\mathbb E} |R^{l}_{t}| {\right) }\\ &\leq& C\int_{0}^{t} h{\left( }t-s{\right) }{\mathbb E} {\left\vert}X_{s}{\right\vert}ds+C\int_{0}^{t}h{\left( }s{\right) }ds +{\mathbb E} |R_{t} | .\end{aligned}$$ Since $ {\mathcal E}$ is uniformly integrable, the direct sum ${\left\lbrace}\sum_{k,l=1 }^{\mathcal{K}}{\left\vert}f_{kl}{\right\vert},f_{kl }\in \mathcal{E} {\right\rbrace}$ is uniformly integrable as well. Thus, there exists $b>0$ satisfying $$\begin{aligned} \int_{0}^{T} \sum_{k,l = 1 }^{\mathcal{K}} {\left\vert}f_{kl} {\right\vert}{\left( }s{\right) }{\mathbbm{1}}{\left\lbrace}\sum_{k,l =1 }^{\mathcal{K}}{\left\vert}f_{kl}{\right\vert}{\left( }s{\right) }>b{\right\rbrace}ds<2^{-1}\end{aligned}$$ for all choices of ${\left( }f_{kl } {\right) }\subset {\mathcal E}$. It follows from that $ {\mathbb E} \| X\|_T \le C $ for a suitable $C$. The same argument shows that also $ {\mathbb E} \| \tilde{X}\|_T \le C .$ Define the total variation measure $\delta_t = \sum_{k=1}^{{{\cal K}}} {\mathbb E} {\left\vert}d{\left( }Z^{k1}_{t}-\tilde{Z}^{k1}_{t} {\right) }{\right\vert}$. We may write $$\begin{aligned} \delta_{t}&\leq &{\mathbb E}\sum_{k=1}^{{{\cal K}}}\int_{0}^{t} {\left\vert}\psi^{k}{\left( }\tilde{X}^{k}_{s},\tilde{A}^{k1}_{s}{\right) }-\psi^{k}{\left( }X^{k}_{s},A^{k1}_{s}{\right) }{\right\vert}ds\nonumber\\ &\leq &C\sum_{k=1}^{{{\cal K}}} \int_{0}^{t} {\mathbb E}{\left\vert}\tilde{X}^{k}_{s}-X^{k}_{s} {\right\vert}+P{\left( }{\left\Vert}\tilde{A}^{k1}-A^{k1}{\right\Vert}_{s}>0{\right) }ds\label{eqbom}. \end{aligned}$$ As in the proof of we apply Markov’s inequality to achieve $$\begin{aligned} \sum_{k=1}^{{{\cal K}}}P{\left( }{\left\Vert}\tilde{A}^{k1}-A^{k1}{\right\Vert}_{t}> 0{\right) }\leq \delta^{n}_{t}. \end{aligned}$$ We insert this inequality into to get $$\begin{aligned} \delta_{t}\leq C{\left( }\int_{0}^{t} {\mathbb E}{\left\vert}\tilde{X}_{s}-X_{s} {\right\vert}ds+\int_{0}^{t}\delta_{s} ds {\right) }.\label{normC2} \end{aligned}$$ We now wish to bound the difference of the memory processes. First, define $\gamma = \sum_{k,l =1}^{\mathcal{K}}{\left\vert}h_{kl }-\tilde{h}_{kl }{\right\vert}$, and note that for any fixed $k,l,j$ we have for any $s\geq 0 $ $$\begin{aligned} &&{\left\vert}\int_{0 }^{s- }h_{kl }{\left( }s-u {\right) }dZ^{lj }_{u} - \int_{0 }^{s- }\tilde{h}_{kl }{\left( }s-u {\right) }d\tilde{Z}^{lj }_{u}{\right\vert}\nonumber\\ &&\leq \int_{0 }^{s- }{\left\vert}h_{kl }-\tilde{h}_{kl }{\right\vert}{\left( }s-u {\right) }dZ^{lj }_{u} + \int_{0 }^{s- } {\left\vert}\tilde{h}_{kl }{\right\vert}{\left( }s-u {\right) }{\left\vert}d{\left( }Z^{lj }_{u}-\tilde{Z}^{lj }_{u} {\right) }{\right\vert}\nonumber \\&&\leq \int_{0 }^{s- }\gamma{\left( }s-u {\right) }d\sum_{l=1}^{\mathcal{K} } Z^{lj }_{u} + \int_{0 }^{s- } \tilde{h} {\left( }s-u {\right) }{\left\vert}d \sum_{l=1}^{\mathcal{K}}{\left( }Z^{lj }_{u}-\tilde{Z}^{lj }_{u} {\right) }{\right\vert}.\label{mixingintegral}\end{aligned}$$ We take expectation and apply Lemma 22 of [@dfh] to obtain $$\begin{gathered} \mathbb{E}\int_{0 }^{t}{\left\vert}\int_{0 }^{s- }h_{kl }{\left( }s-u {\right) }dZ^{lj }_{u} - \int_{0 }^{s- }\tilde{h}_{kl }{\left( }s-u {\right) }d\tilde{Z}^{lj }_{u}{\right\vert}ds\leq \\ \int_{0 }^{t }\gamma{\left( }t-s {\right) }L{\left( }1+\mathbb{E}{\left\Vert}X_{T }{\right\Vert}{\right) }ds + \int_{0 }^{t } \tilde{h} {\left( }t-s {\right) }\delta_{s } ds.\end{gathered}$$ Note that this expression does not depend on $k,l$ nor $j$. Thus we get $$\begin{aligned} &&\sum_{k=1}^{{{\cal K}}}\int_{0}^{t} {\mathbb E}{\left\vert}\tilde{X}^{k}_{s}-X^{k}_{s} {\right\vert}ds \nonumber \\ &&\leq\sum_{k=1}^{{{\cal K}}}\int_{0 }^{t }N^{-1 }\sum_{l=1}^{{{\cal K}}}\sum_{j = 1 }^{N_{l }}{\mathbb E}{\left\vert}\int_{0 }^{s- }h_{kl }{\left( }s-u {\right) }dZ^{lj }_{u} - \int_{0 }^{s-}\tilde{h}_{kl }{\left( }s-u {\right) }d\tilde{Z}^{lj }_{u}{\right\vert}ds \nonumber\\ &&\leq C{\left( }\int_{0 }^{t } \gamma{\left( }s {\right) }ds + \int_{0}^{t} {\left\vert}\tilde{h}{\left( }t-s {\right) }{\right\vert}\delta_{s} ds {\right) }.\label{h-inequality2} \end{aligned}$$ Inserting inequality into , we obtain $$\delta_{t}\leq C{\left( }\int_{0}^{t}\gamma{\left( }s {\right) }ds + \int_{0}^{t} {\left( }\tilde{h}{\left\vert}{\left( }t-s {\right) }{\right\vert}+1{\right) }\delta_{s} ds{\right) }.$$ The proof will be complete, after repeating the argument for bounded ${\mathbb E} {\left\vert}X{\right\vert},$ but with $\delta $ in place of ${\mathbb E} {\left\vert}X{\right\vert}$. Let $( \tilde{Z}^{N},\tilde{X}^{N},\tilde{A}^{N})$ be the $N-$dimensional Age Dependent Hawkes process induced by the same parameters as $(Z^N, X^N, A^N),$ except the weight functions $h_{kl}$ instead of $h^N_{kl}.$ Fix $ T > 0 $ and consider $t \in [0, T ].$ We have $$\begin{aligned} \sum_{k=1}^{{{\cal K}}}{\left\vert}d{\left( }Z^{Nk1}_{t}-Z^{k1}_{t} {\right) }{\right\vert}\leq \sum_{k=1}^{{{\cal K}}}{\mathbb E} {\left\vert}d{\left( }Z^{Nk1}_{t}-\tilde{Z}^{Nk1}_{t} {\right) }{\right\vert}+\sum_{k=1}^{{{\cal K}}}{\mathbb E} {\left\vert}d{\left( }\tilde{Z}^{Nk1}_{t}-Z^{k1}_{t} {\right) }{\right\vert}:= \tilde{\delta}_t^{Nk}+\delta_t^{Nk}. \end{aligned}$$ The first term converges by , and so it remains to prove convergence of $\delta^{N}_t.$ This part of the proof follows closely the proof given by Chevallier in [@chevallier], but we include it here for completeness. Let $C$ be a dynamic constant depending on $p_{k },L,T,{{\cal K}},{\left\Vert}r{\right\Vert}_{T}$ and ${\left( }h_{kl}{\right) }$. We use the symbol ${\varepsilon}{\left( }N{\right) }$ for any function depending on the same parameters as $C $, and $ N $ such that ${\varepsilon}{\left( }N {\right) }{\stackrel{N{\rightarrow}\infty}{\rightarrow}}0 $. Recall that ${\left\Vert}x{\right\Vert}_{T}$ is bounded by $C$ sufficiently large by . As in the proof of we obtain $$\begin{aligned} \delta^{N}_{t}\leq C{\left( }\int_{0}^{t} {\mathbb E}{\left\vert}\tilde{X}^{N}_{s}-x_{s} {\right\vert}ds+\int_{0}^{t}\delta^{N}_{s} ds {\right) }.\label{normC} \end{aligned}$$ This inequality prepares for an application of Gronwall’s inequality, but first we bound $\int_0^t {\mathbb E}{\left\vert}\tilde{X}_{s}-x_{s}{\right\vert}ds $ using $\delta^{N}_{t}$ as well. Indeed, set $\Lambda^{kj}_{t}:=\int_{0}^{t-} \psi{\left( }x^{k}_{s},A^{kj}_{s}{\right) }ds$, which is the compensator of $Z^{kj}$. We write $p_{k}=N_{k}/N+{\varepsilon}{\left( }N{\right) }$ and obtain $$x^{k}_{t}=N^{-1}\sum_{l=1}^{{{\cal K}}}\sum_{j=1}^{N_{l}}\int_{0}^{t-}h_{kl}{\left( }t-s{\right) }d\phi^{l}_{s}+r^{k}_{t}+{\varepsilon}{\left( }N{\right) }\sum_{l=1}^{{{\cal K}}}\int_{0}^{t} h_{kl}{\left( }t-s{\right) }d\phi^{l}_{s}.$$ Since $d \phi^{l}_{s} ={\mathbb E} \psi{\left( }x^{l}_{s},A_{s}^{lj}{\right) }ds$ and ${\mathbb E} \psi{\left( }x^{l}_{s},A_{s}^{lj}{\right) }$ are locally bounded, the entire right term may be replaced by an ${\varepsilon}$-function. For fixed $k\leq {{\cal K}},$ we apply the triangle inequality $$\begin{aligned} &&\int_{0}^{t}\left( {\mathbb E}{\left\vert}\tilde{X}^{Nk}_{s}-x^{k}_{s} {\right\vert}-{\mathbb E}{\left\vert}R^{Nk}_{s}-r^{k}_{s}{\right\vert}\right) ds\nonumber\leq {\varepsilon}{\left( }N{\right) }\\ &&+ \int_{0}^{t} N^{-1}\sum_{l=1}^{{{\cal K}}}{\mathbb E}{\left\vert}\sum_{j=1}^{N_{l}}\int_{0}^{s-}h_{kl}{\left( }s-u {\right) }d{\left( }\phi^{l}_{u}-\Lambda^{lj}_{u}{\right) }{\right\vert}ds \label{b1}\\ \;\;\;\;&& + \int_{0}^{t}N^{-1}\sum_{l=1}^{{{\cal K}}}{\mathbb E}{\left\vert}\sum_{j=1}^{N_{l}}\int_{0}^{s-}h_{kl}{\left( }s-u {\right) }d{\left( }\Lambda^{lj}_{u}-Z^{lj}_{u} {\right) }{\right\vert}ds \label{b2}\\ \;\; \;\;&& +\int_{0}^{t} N^{-1}\sum_{l=1}^{{{\cal K}}} {\mathbb E}{\left\vert}\sum_{j=1}^{N_{l}}\int_{0}^{s-}h_{kl}{\left( }s-u {\right) }d{\left( }Z^{lj}_{u}- \tilde{Z}^{Nlj}_{u} {\right) }{\right\vert}ds\\ :=&& {\varepsilon}{\left( }N{\right) }+B^{1k}_{t}+B^{2k}_{t}+B^{3k}_{t}.\nonumber \end{aligned}$$ We now proceed to bound $B^{i}:=\sum_{k=1}^{{{\cal K}}}B^{ik}$, $i\leq 3$. Define $h=\sum_{k,l=1}^{{{\cal K}}}{\left\vert}h_{kl}{\right\vert}$. Rewrite $\phi$ and $\Lambda$ in terms of their densities, and thereby obtain a bound for the inner-most sum in for $s\in {\left[}0,t {\right]},l\leq {{\cal K}}, $ which is given by $${\mathbb E}\int_{0}^{s} \sum_{j=1}^{N_{l}} h{\left( }s-u {\right) }|d ( \phi^{l}_{u}-\Lambda^{lj}_{u}) | \leq \int_{0}^{s} h{\left( }s-u {\right) }{\mathbb E} \sum_{j=1}^{N_{l}}{\left\vert}\psi^{l}{\left( }x^{l}_{u},A^{lj}_{u} {\right) }-{\mathbb E}\psi^{l}{\left( }x^{l}_{u},A^{lj}_{u} {\right) }{\right\vert}du.$$ Notice that the sum consists of i.i.d. terms, so we may apply Cauchy-Schwarz to bound it by $\sqrt{N_{l}\, \text{Var} ( \psi ( x^{l}_{u},A^{lj}_{u}) ) }$, which is bounded for $u\in {\left[}0,T {\right]}$ by $\sqrt{N_{l}} C{\left( }1+{\left\Vert}r^{l}{\right\Vert}_{T} {\right) }$ using . Insert this into to see that $$B_t^{1}=\sum_{k=1}^{{{\cal K}}}B^{1k}_{t} \leq{\varepsilon}{\left( }N{\right) }.$$ For $B_t^{2}$, recall that $(Z^{lj}-\Lambda^{lj})_j $ are i.i.d. for fixed $l.$ By Cauchy-Schwarz, we obtain a bound for the inner-most sum of $$N_{l}^{1/2}\sqrt{\text{Var}\int_{0}^{s}h_{kl}{\left( }s-u {\right) }d{\left( }Z^{l1}_{u}-\Lambda^{l1}_{u}{\right) }}\; . \label{B2}$$ To treat the process inside the root, fix $s\geq 0,l\leq {{\cal K}}$ and consider the process $$I:r\mapsto \int_{0}^{r\wedge s}h_{kl}{\left( }s-u {\right) }d{\left( }Z^{l1}_{u}-\Lambda^{l1}_{u}{\right) }.$$ Then $I$ is a martingale, and $$\text{Var }I_{s}={\mathbb E}[I]_{s}={\mathbb E}\int_{0}^{s} h_{kl}^2{\left( }s-u{\right) }d\Lambda^{l1}_{u}=\int_{0}^{s} h_{kl}^2{\left( }s-u{\right) }{\mathbb E} \psi^{l} {\left( }x^{l}_{u},A^{l1}_{u} {\right) }du.$$ Since $h_{kl}\in {\mathcal L}^{2}_{loc}$ it follows that $\text{Var} \;I_{s}$ is bounded on $s\in [0,T], $ and so $$B^{2}_{t}\leq {\varepsilon}{\left( }N{\right) }$$ for all $ t \le T. $ For $B^{3}$ the triangle inequality, and Lemma 22 [@dfh] gives $$\begin{aligned} B^{3}_{t}\leq C\int_{0}^t h{\left( }t-s {\right) }\delta_{s}^{N} ds . \end{aligned}$$ We plug the bounds for $B_{1},B_{2}$ and $B_{3}$ into to obtain $$\begin{aligned} \delta^{N}_{t}\leq C{\left( }\int_{0}^{t} h{\left( }t-s {\right) }\delta^{N}_{s}+ {\varepsilon}{\left( }N{\right) }+\sum_{k=1}^{{{\cal K}}}{\mathbb E} {\left\vert}R^{Nk}_{s}-r^{k}_{s}{\right\vert}ds {\right) }. \end{aligned}$$ Applying in the Appendix yields $$\begin{aligned} \delta^{N}_{t}\leq {\varepsilon}{\left( }N{\right) }+C\int_{0}^T {\mathbb E} {\left\vert}R_{s}^N-r_{s}{\right\vert}ds = {\varepsilon}{\left( }N{\right) }\end{aligned}$$ for all $t\leq T$, which implies the desired result. Results about counting processes {#results-about-counting-processes .unnumbered} -------------------------------- Measure theory on ${\left( }M_{E},{\mathcal M}_E {\right) }$ ------------------------------------------------------------- This section provides a brief overview of measure theory on the measurable space ${\left( }M_{E},{\mathcal M}_E {\right) }$ of bounded measures defined on a Polish space $E.$ We refer to A.2.6 in [@DALEY] for more details. Let $d$ be a distance so that ${\left( }E,d{\right) }$ is complete and separable. A measure $\nu$ on $E$ is said to be *boundedly finite* if $\text{diam}{\left( }A{\right) }<\infty $ implies that $ \nu ( A) <\infty$ for $A\in {\mathcal B}{\left( }E{\right) }$. Let $M_{E}$ be the space of all boundedly-finite measures on $(E, {\mathcal B} (E) )$. This space may now be equipped with the *weak-hat* metric $\hat{d}$, making $( M_{E},\hat{d})$ a complete and separable space itself. The Borel-algebra ${\mathcal M}_{E}$ is easily characterized by the projections $\Pi_{A}:M_{E}\ni \nu \mapsto \nu{\left( }A{\right) }$ in the sense that $$\begin{aligned} {\mathcal M}_{E} =\sigma{\left( }\Pi_{A}, \; A\in {\mathbb D} {\right) }, \label{generate}\end{aligned}$$ where ${\mathbb D}\subset {\mathcal B}{\left( }D{\right) }$ is a semi-ring of bounded sets. A random variable taking values in the space of measures is called a *random measure*. A particular interesting example is when $E={\mathbb{R}}\times {\mathbb{R}}_+ $ as considered in this paper. A Poisson random measure $\pi:\Omega\rightarrow M_{{\mathbb{R}}\times{\mathbb{R}}_{+}}$ on $ {\mathbb{R}}\times {\mathbb{R}}_+ $ with Lebesgue intensity measure is a random measure such that for any disjoint $A_{1},\dots ,A_{n}\in {\mathcal B} ( {\mathbb{R}}) \times {\mathcal B} ( {\mathbb{R}}_+) , $ - $\pi{\left( }A_{i}{\right) }\sim \text{Pois}{\left( }\int_{A_{i}}dsdz {\right) }$ - $\pi{\left( }A_{1}{\right) }{\rotatebox[origin=c]{90}{$\models$}}\dots {\rotatebox[origin=c]{90}{$\models$}}\pi{\left( }A_{n}{\right) }.$ Since the underlying space is on the form ${\mathbb{R}}\times {\mathbb{R}}_+, $ the first coordinate of $\pi$ may be thought of as the time coordinate; and concepts like stationarity and ergodicity transfer naturally to random measures. Define the shift operator as the automorphism on $M_{{\mathbb{R}}\times {\mathbb{R}}_+}$ given by $$\begin{aligned} {\left( }\theta^{r}\nu{\right) }{\left( }C{\right) }= \nu {\left( }{\left\lbrace}{\left( }t,x{\right) }\in {\mathbb{R}}\times {\mathbb{R}}_+ : {\left( }t- r,x {\right) }\in C {\right\rbrace}{\right) }.\end{aligned}$$ Then a random measure $\sigma$ on ${\mathbb{R}}\times {\mathbb{R}}_+ $ is said to be *stationary* if the distribution is invariant under shift $$\begin{aligned} {\mathcal L} ( \sigma) = {\mathcal L} {\left( }\theta^{r}\sigma{\right) }\end{aligned}$$ for all $ r \in {\mathbb{R}}.$ Stationarity is equivalent to have invariance of the finite dimensional distributions (fidi’s) (Proposition 6.2.III of [@DALEY]) $$\begin{aligned} P\bigcap_{i=1}^{n}{\left( }\sigma{\left( }A_{i}{\right) }\in B_{i}{\right) }=P\bigcap_{i=1}^{n}{\left( }\theta^{r}\sigma{\left( }A_{i}{\right) }\in B_{i}{\right) }\end{aligned}$$ for all $r\in {\mathbb{R}},n>0,A_{1},\dots , A_{n}\in {\mathcal B} ( {\mathbb{R}}\times {\mathbb{R}}_+) ,B_{1},\dots,B_{n}\in {\mathcal B} ( {\mathbb{R}}_+ ) .$ A stationary random measure $\sigma$ is *mixing* if $$\begin{aligned} \label{eq:mix} P{\left( }\sigma \in V, \theta^r \sigma\in W{\right) }{\stackrel{{\left\vert}r {\right\vert}{\rightarrow}\infty}{\rightarrow}} P{\left( }\sigma\in V{\right) }P{\left( }\sigma \in W{\right) }\end{aligned}$$ for all $ V, W \in {\mathcal M}_{{\mathbb{R}}\times {\mathbb{R}}_+}.$ We refer to Chapter 10.2-10.3 of [@DALEY] for a thorough introduction to ergodic theory for random measures. As is the case for processes, mixing implies that ${\mathcal L} (\sigma) $ is ergodic w.r.t. the shift operator $\theta^{r}$ for all $r>0$, meaning that all invariant events have probability $0/1$. Finally, we present a core measurability result, which can be applied to show measurability of all processes treated in this article. \[measurelemma\] Let $ D,E$ be complete and separable metric spaces, and let $H:D\times E{\rightarrow}{\mathbb{R}}_+ $ be measurable. The section integral $F: M_E \times D {\rightarrow}\overline{{\mathbb{R}}} $ $$\begin{aligned} F{\left( }\nu, d {\right) }=\int_E H{\left( }d ,s{\right) }d\nu{\left( }s{\right) }\end{aligned}$$ is ${\mathcal M}_{E} \times {\mathcal B} ({D} ) {\rightarrow}{{\mathcal B}} ( \overline {\mathbb{R}}) $ measurable. We start by defining $G: M_{ D\times E} {\rightarrow}\overline{{\mathbb{R}}_+}$ as $$\begin{aligned} G:\rho \mapsto \int H{\left( }x,y{\right) }\rho (dx, dy) .\end{aligned}$$ It is easily seen that $G$ is measurable. Consider the map $m : M_E \times D \to M_{D \times E } $ given by $ {\left( }\nu , d{\right) }\mapsto \delta_{d} \otimes \nu $. To prove measurability of $m$ it is sufficient to treat projections into bounded boxes $A\times B, A\in {\mathcal B} ({D}) ,B\in {\mathcal B} ({E}) $. Such projections are simply given as $\Pi_{A\times B} m : {\left( }\nu,d {\right) }\mapsto {\mathbbm{1}}{\left\lbrace}B{\right\rbrace}{\left( }d {\right) }\Pi_{A}{\left( }\nu {\right) }$ and are therefore measurable. We conclude that $$\begin{aligned} G\circ m {\left( }\nu,d {\right) }= \int\int H{\left( }u,s {\right) }\; d\delta_{d}{\left( }u{\right) }d\nu{\left( }s{\right) }=\int H{\left( }d,s{\right) }d\nu{\left( }s{\right) }=F{\left( }\nu,d{\right) },\end{aligned}$$ proving that $F$ is measurable. We shall need the following version of Gronwall’s lemma which has been proven in [@dfh]. Recall that for any function $g : {\mathbb{R}}_{+}{\rightarrow}{\mathbb{R}}_{}$ and any $ T > 0, $ we have introduced $\|g\|_T = \sup_{ t \le T } | g(t) |.$ \[convobound\] Let $h : {\mathbb{R}}_{+}{\rightarrow}{\mathbb{R}}_{+}$ be locally integrable and $g : {\mathbb{R}}_{+}{\rightarrow}{\mathbb{R}}_{+}$ be locally bounded. Let $T\geq 0.$\ 1. Let $u$ be a locally bounded nonnegative function satisfying $u_{t}\leq g_{t}+\int_{0}^{t}h{\left( }t-s{\right) }u_{s} ds$ for all $t\in [0,T]$. If $b> 0$ satisfies that $$\begin{aligned} \int_{0}^{T}h{\left( }s{\right) }{\mathbbm{1}}{\left\lbrace}h{\left( }s{\right) }\geq b{\right\rbrace}ds<\frac12 , \label{A-equality} \end{aligned}$$ then ${\left\Vert}u {\right\Vert}_{T}\leq 2e^{2bT}{\left\Vert}g{\right\Vert}_{T} =: C_T {\left\Vert}g{\right\Vert}_{T} .$\ 2. Let ${\left( }u^{n}{\right) }$ be a sequence of locally bounded nonnegative functions such that $u^{n+1}_{t}\leq g_{t}+ \int_{0}^{t}h{\left( }t-s{\right) }u^{n}_{s}ds$ for all $t\in [0,T]$. Then $\sup_{n}{\left\Vert}u^{n}{\right\Vert}_{T}\leq C_{T}{\left( }{\left\Vert}g {\right\Vert}_{T}+{\left\Vert}u^{0}{\right\Vert}_{T}{\right) }.$ Moreover, if the inequality is satisfied with $g\equiv 0, $ then $\sum_{n} u^{n}$ converges uniformly on $[0,T]$. Point Process Results --------------------- We collect some useful results on point processes known in the literature. \[compensator\] Let $H:{\left( }\Omega\times {\mathbb{R}}\times {\mathbb{R}}{\right) }{\rightarrow}{\mathbb{R}}$ be ${\mathcal P}\otimes {\mathcal B}\rightarrow {\mathcal B}$ measurable and assume that almost surely $$\begin{aligned} Z : t\mapsto \int_{0}^{t}\int_{0}^{\infty} H{\left( }s,z{\right) }\pi{\left( }dz,ds {\right) }\end{aligned}$$ does not explode; that is, for all $ t > 0, $ $$\int_{0}^{t}\int_{0}^{\infty} |H{\left( }s,z{\right) }|\pi{\left( }dz,ds {\right) }< \infty$$ almost surely. 1. If $ H $ is bounded, then the compensator $\Lambda$ of $Z$ is given by $$\begin{aligned} \Lambda : t\mapsto \int_{0}^{t}\int_{0}^{\infty} H{\left( }s,z{\right) }dz\; ds , \end{aligned}$$ i.e. $Z-\Lambda$ is a local ${\left( }{\mathcal F}_{t}{\right) }$-martingale. 2. If moreover $s\mapsto {\mathbb E} \int | H{\left( }s,z{\right) }| dz$ is locally integrable, then $Z-\Lambda$ is a martingale. 3. Fix $T\geq 0$ and assume that $\Lambda$ can be written as $$\begin{aligned} \Lambda_{t}=\int_{0}^{t} \lambda_{s} ds. \end{aligned}$$ Assume also that $\lambda{\left( }s{\right) }=F{\left( }Z{\left( }-\infty,s{\right) },s{\right) }+{\varepsilon}{\left( }s{\right) }$ where $F$ is ${\mathcal B} ( {\mathbb{R}}_+) \times {\mathcal B} ( {\mathbb{R}}) \mapsto {\mathcal B} ( {\mathbb{R}}) $ measurable and $t\mapsto {\varepsilon}{\left( }t{\right) }$ is ${\left( }{\mathcal F}_{t\wedge T}{\right) }$-predictable. It holds that $$\begin{aligned} P{\left( }Z{\left( }T,\infty{\right) }=0 \; \vert {\mathcal F}_{T}{\right) }\stackrel{a.s.}{= } \exp{\left( }-\int_{T}^{\infty} F{\left( }Z{\left( }-\infty,T{\right]},s{\right) }+{\varepsilon}{\left( }s{\right) }ds{\right) }. \end{aligned}$$ The first point follows from [@jacod], Theorem 1.8 of Chapter II, by using the localizing sequence $ T_n = \inf \{ t : \int_{0}^{t}\int_{0}^{\infty} |H{\left( }s,z{\right) }|\pi{\left( }dz,ds {\right) }\geq n \} , n \geq 1 ,$ since $$\int_0^{T_n}\int_{0}^{\infty} |H{\left( }s,z{\right) }|\pi{\left( }dz,ds {\right) }\le n + \| H \|_\infty .$$ For the second point, let $M_t := Z_t - \Lambda_t .$ It suffices to show that $ {\mathbb E}( \sup_{s \le t } | M_s|) < \infty $ which follows from $${\mathbb E}(\sup_{ s \le t } | M_{ s \wedge T_n } |) \le 2 {\mathbb E}\int_0^t \int_0^\infty | H ( s, z ) | ds dz < \infty$$ by monotone convergence. The third point is Lemma 1 in [@bm]. \[ergodiclemma\] Assume that $X,Y$ are ${\left( }{\mathcal F}_{t}{\right) }_{t\in {\mathbb{R}}_+}$-progressive and that for all $ s \geq 0$ $$\begin{aligned} P{\left( }X_{t} = Y_{t}\; \forall t>s\; \vert {\mathcal F}_{s} {\right) }\geq U_{s} -r {\left( }s{\right) }, \end{aligned}$$ where $U$ is ergodic, $P{\left( }U_{s}>0{\right) }>0$ and $r {\left( }t{\right) }{\stackrel{a.s.}{\rightarrow}}0$ for $ t \to \infty .$ Then almost surely, $X$ and $Y$ couple in finite time. Acknowledgements {#acknowledgements .unnumbered} ================ This research has been conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01). [99]{} Hawkes model for price and trades high-frequency dynamics. , 14 (2014), 1–20. Stability of nonlinear Hawkes processes. , 24(3) (1996) 1563-1588. Mean-field limit of generalized Hawkes processes. 127 (2017), 3870-3912. Microscopic approach of a time elapsed neural model. , 25(14) (2015) 2669–2719. An Introduction to the Theory of Point Processes. . Hawkes processes on large networks. 26 (2016), 216–261. Multi-class oscillating systems of interacting neurons. 127 (2017), 1840–1869. Stability, convergence to equilibrium and simulation of non-linear Hawkes Processes with memory kernels given by the sum of Erlang kernels. Submitted, arxiv.org/abs/1610.03300, 2017. A toy model of interacting neurons. 52 (2016), 1844–1876. , 13(2) (2017): e1005390. doi:10.1371/journal.pcbi.1005390. Interacting multi-class transmissions in large stochastic systems. , 6 (2009) 2334–2361. Measure theory. , 1974. Spectra of Some Self-Exciting and Mutually Exciting Point Processes. , 58 (1971) 83-90. A cluster process representation of a self-exciting process. , 11 (1974) 93-503. Limit theorems for stochastic processes. Second edition, Springer-Verlag, Berlin, 2003. Invariant measures of ultimately bounded stochastic processes , 49 (1973), 149–153. , 5 (2010), 2781–2822. Hawkes Point Processes for Social Media Analytics. In [*“Reshaping Society through Analytics, Collaboration, and Decision Support: Role of Business Intelligence and Social Media”*]{}, Springer International Publishing, 2015. [^1]: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen [^2]: Universit[é]{} de Cergy-Pontoise, AGM UMR-CNRS 8088, 2 avenue Adolphe Chauvin, F-95302 Cergy-Pontoise Cedex
--- author: - Sangsik Kim - Kyunghun Han - Cong Wang - 'Jose A. Jaramillo-Villegas' - Xiaoxiao Xue - Chengying Bao - Yi Xuan - 'Daniel E. Leaird' - 'Andrew M. Weiner' - Minghao Qi bibliography: - 'CRR\_arxiv\_SI.bib' title: 'Supplementary Information to “Frequency Comb Generation in 300 nm Thick SiN Concentric-Racetrack-Resonators: Overcoming the Material Dispersion Limit”' --- Type II comb with a concentric-racetrack-resonator ================================================== [![ [**Type II frequency comb with a concentric-racetrack-resonator.**]{} [**a,**]{} Layout image of the concentric-racetrack resonator. Geometric parameters are $h=300$ nm, $R_{\rm out}=100~\mu$m, $g=500$ nm, $w_{\rm out}=1400$ nm, $w_{\rm in}=2800$ nm, and $L_r=150~\mu$m. [**b,**]{} Type II comb spectrum and corresponding [**c,**]{} RF intensity noise. []{data-label="figS_type2"}](figS_type2 "fig:"){width="90.00000%"}]{} Figure \[figS\_type2\]a shows the layout image of the concentric-racetrack resonator, whose bending radius and widths are increased to $R_{\rm out}=100~\mu$m, $w_{\rm out}=1400$ nm, and $w_{\rm in}=2800$ nm, respectively. Other geometric parameters are $h=300$ nm, $g=500$ nm, and $L_{\rm st}=150~\mu$m. To increase the coupling efficiency between bus waveguide and resonators, we bent the bus waveguide with the same bending radius of $R=100~\mu$m. We were able to generate the frequency combs, and Figs \[figS\_type2\]b and \[figS\_type2\]c show the corresponding comb spectrum and RF noise, respectively. The generated comb exhibits non-native mode spacing (a type II comb), with high RF intensity noise attributed to beating between sub-combs [@ferdous2011spectral; @herr2012universal]. Line-by-line pulse shaping and autocorrelation measurements =========================================================== [![ [**Time domain autocorrelation (AC) measurements.**]{} [**a, b, c,**]{} Time domain AC measurements of the comb state III in Figs. 3e, 3f, and 3g, respectively. Phase correction via line-by-line pulse shaping is performed. Autocorrelations with (blue) and without (green) phase correction are displayed. The autocorrelations calculated based on the experimental power spectrum under the assumption of flat spectral phase are also shown (red). []{data-label="figS_AC"}](figS_AC "fig:"){width="100.00000%"}]{} To test the coherence of comb states III in Figs. 3e, 3f, and 3g from the main manuscript, we selected a subset of the comb lines of each comb, conducted line-by-line pulse shaping experiments, and measured the autocorrelation (AC) [@ferdous2011spectral]; figures \[figS\_AC\]a, \[figS\_AC\]b, and \[figS\_AC\]c are the corresponding AC results, respectively. Green and blue curves in each figure are the experimentally measured AC traces before and after the phase correction, respectively, and red is the calculated AC that is based on the measured power spectrum and assuming flat spectral phase. In Figs. \[figS\_AC\]a and \[figS\_AC\]c, notice the clear difference between before (green) and after (blue) the phase correction; the phase corrected results (blue) are close to the calculated ACs (red). These data indicate the coherence of comb states III in Figs. 3e and 3g, as we anticipated from the low RF noise [@ferdous2011spectral]. However, in Fig. \[figS\_AC\]b, a clear difference between the phase corrected (blue) and calculated (red) ACs remain, even with the phase correction via line-by-line shaping. In particular, the experimental AC trace shows very poor contrast ratio compared both to the calculated curve and to the AC traces in Figs. \[figS\_AC\]a and \[figS\_AC\]c. Poor contrast ratio in intensity autocorrelation is a hallmark of broadband intensity noise [@ferdous2011spectral; @weiner2011ultrafast]. Thus, the autocorrelation data indicate that the coherence of comb state III in Fig. 3f is poor, consistent with the high RF noise observed for this comb.
[Local stability and Lyapunov functionals\ for $n$-dimensional quasipolynomial conservative systems ]{} [Benito Hernández–Bermejo$^{\: 1}$ Víctor Fairén]{} [*Departamento de Física Matemática y Fluidos, Universidad Nacional de Educación a Distancia. Senda del Rey S/N, 28040 Madrid, Spain.*]{} [Abstract]{} We present a method for determining the local stability of equilibrium points of conservative generalizations of the Lotka-Volterra equations. These generalizations incorporate both an arbitrary number of species —including odd-dimensional systems— and nonlinearities of arbitrarily high order in the interspecific interaction terms. The method combines a reformulation of the equations in terms of a Poisson structure and the construction of their Lyapunov functionals via the energy-Casimir method. These Lyapunov functionals are a generalization of those traditionally known for Lotka-Volterra systems. Examples are given. [**Running title:**]{} Stability of nonlinear conservative systems. [**Keywords:**]{} Lyapunov functionals — Stability — Lotka-Volterra equations — Hamiltonian systems. $^1$ Corresponding author. E-mail: bhernand@apphys.uned.es Telephone: (+ 34 91) 398 72 19. Fax: (+ 34 91) 398 66 97. [1. Introduction]{} Consider the following Lotka-Volterra system \[\[lot1\],\[v1\]\] $$\label{lv0} \dot{x}_i = x_i \left( \lambda _i + \sum _{j=1}^n A_{ij} x_j \right) , \;\:\;\: i = 1, \ldots ,n$$ which is assumed to have a unique equilibrium point, ${\bf x}_0 \in \mbox{int}\{ I\!\!R_+^n\}$. One of the most relevant results about its stability is well summarized in a theorem originally enunciated by Kerner \[\[ker1\]\], and later generalized by many different authors \[\[goh1\],\[goh2\],\[har1\],\[hsu1\],\[kri1\],\[ltj1\],\[lyt1\],\[red1\],\[ta1\]-\[tat2\]\] (see also \[\[hs1\],\[log1\],\[tak1\]\] for detailed reviews of the subject). The result makes use of the well-known Lyapunov functional $$\label{hlv0} V({\bf x}) = \sum _{i=1}^{n} d_i \left( x_i-x_{0i} - x_{0i} \ln \frac{x_i}{x_{0i}} \right)$$ The time derivative of (\[hlv0\]) along the trajectories of (\[lv0\]) is $$\label{hp0} \dot{V}({\bf x}) = \frac{1}{2} ({\bf x} - {\bf x}_0)^T (D \cdot A + A^T \cdot D) ({\bf x} - {\bf x}_0)$$ where $D=$ diag$(d_1, \ldots ,d_n)$. Thus, it can be stated that if there exists a positive definite diagonal matrix $D$ such that $D \cdot A + A^T \cdot D$ is negative definite, ${\bf x}_0$ is Lyapunov asymptotically stable. Moreover if, instead, $D \cdot A + A^T \cdot D$ is negative semi-definite, then ${\bf x}_0$ is Lyapunov semi-stable and every solution of (\[lv0\]) in int$\{ I\!\!R_+^n \}$ tends to the maximal invariant set $M$ contained in the set (see \[\[las0\],\[las1\],\[lyt2\]\] and references therein) $$E = \left\{ {\bf x} \in \mbox{int} \{ I\!\!R_+^n \} \; / \; ({\bf x} - {\bf x}_0)^T (D \cdot A + A^T \cdot D) ({\bf x} - {\bf x}_0) = 0 \right\}$$ Every one of the two previous alternatives encompasses the already classical community models, respectively: The so-called Lotka-Volterra dissipative and conservative systems \[\[log1\]\]. In particular, Lotka-Volterra conservativeness implies that (\[hlv0\]) is a constant of motion, thus making conservative systems formally amenable to analysis by standard theoretical mechanics methods \[\[gas1\],\[ker3\],\[ker4\]\] and statistical mechanics considerations \[\[gmm1\],\[ker1\],\[ker2\]\]. On this respect, the Hamiltonization of classical Lotka-Volterra conservative systems \[\[ker3\],\[ker4\]\] proceeds by defining the canonical variables, $z_i$, as linear transforms of new dependent variables $y_i= \ln (x_i/x_{0i})$. If the Hamiltonian, $H$, is simply identified with the functional in (\[hlv0\]) appropriately rewritten in the canonical variables, $z_i$, the conservative Lotka-Volterra equations adopt the familiar symplectic form $$\label{slp0} \dot{\bf z} = S \cdot \nabla H({\bf z})$$ where $S$ is the classical symplectic matrix \[\[gold\]\]. Unfortunately, this constructive procedure cannot be carried out beyond the class of even-dimensional classical conservative Lotka-Volterra systems. This made the Hamiltonian description of rather limited use until Nutku \[\[nut1\]\] and Plank \[\[pla1\]\] suggested reconsidering it under the light of the more general Poisson structure representation (see \[\[olv1\]\] for an overview; see also references therein). Poisson systems constitute a natural extension of classical Hamiltonian dynamical systems, but have the advantage of embracing odd-dimensional flows as well. In the Poisson context, no prior transformation on the variables is necessary, and the conservative Lotka-Volterra equations can be put into Poisson form in terms of the original variables $$\label{p0} \dot{\bf x} = {\cal J} \cdot \nabla H({\bf x})$$ where the elements of the structure matrix ${\cal J}$ are defined as $J_{ij} = K_{ij}x_ix_j$, $K$ being a skew-symmetric matrix, and $H$ is the classical Volterra’s constant of motion (\[hlv0\]). In fact, form (\[p0\]) happens to be suitable for embracing a higher number of families of conservative systems than those of type (\[lv0\]), as stated in the following result (see \[\[byv5\]\]): [Theorem 1.]{} \[\[byv5\]\] Note that systems (\[glv\]) appear when we combine a quadratic structure matrix (first identified by Plank \[\[pla1\]\]) together with Hamiltonian (\[H\]), which is a generalization of Volterra’s constant of motion (\[hlv0\]). Important dynamical features of certain particular cases of such systems have recently deserved detailed attention in the literature \[\[pla2\]\]. In what follows, we shall denote systems described by Theorem 1 as quasipolynomial of Poisson form, or QPP in brief. QPP systems (\[glv\]) include the conservative Lotka-Volterra equations as a particular case when $m=n$, $B$ is the identity matrix, the dimension is even and $A$ is invertible. In such a case, Hamiltonian (\[H\]) also reduces to Volterra’s first integral, as it can be easily verified. The purpose of the present article is to investigate under which conditions the equilibrium points of the QPP systems are stable and compare the resulting generalization with what is known for conservative Lotka-Volterra models (\[lv0\]). In particular, we shall also carry out a generalization of the corresponding Lyapunov functionals (\[hlv0\]). In this way, we shall complete a treatment that simultaneously embraces arbitrary-dimensional systems and also arbitrary nonlinearities in the flow. The construction of suitable Lyapunov functionals for the QPP systems involved will be possible thanks to their Poisson structure, which allows the use of the energy-Casimir method (see \[\[hmrw1\]\] and references therein) in which the stability analysis of a given fixed point ${\bf x}_0$ proceeds by defining an [*ansatz*]{} for the Lyapunov functional, which takes the form: $$\label{ecf} H_C({\bf x}) = H + F(C_1, \ldots ,C_k)$$ where $F(z_1, \ldots ,z_k)$ is a $C^2$ real function to be determined and $\{ C_1, \ldots , C_k \}$ is a complete set of independent Casimir invariants. The method amounts to the search of one suitable $F$, by imposing two conditions on $H_C$: (i) $H_C$ must have a critical point at ${\bf x}_0$; and (ii) the second derivative of $H_C$ at ${\bf x}_0$ must be either positive or negative definite. Once one suitable $F$ has been found, stability of ${\bf x}_0$ follows automatically, and the method provides us with a Lyapunov functional for this point. The structure of the article is as follows: Section 2 is devoted to the establishment of the main stability theorem. Different consequences of the result are considered in the examples of Section 3. [2. Stability of QPP Systems]{} Let us start by recalling the following definition, valid for normed spaces (see \[\[hmrw1\]\]): [Definition 1.]{} \[\[hmrw1\]\] In what follows, stability shall denote local stability. We give now our main result: [Theorem 2.]{} [*Proof.*]{} The proof rests strongly on the quasimonomial formalism. The unfamiliar reader is referred to the basic references on the subject \[\[br1\]-\[byg1\],\[byv1\]-\[bvb1\],\[pym1\]\]. The strategy of the proof consists in reducing the problem to the Lotka-Volterra representative and analyzing there the stability of the fixed points. The resulting criteria and Lyapunov functionals are then mapped back into the original system. For the sake of clarity, we omit in what follows the proofs of the auxiliary lemmas, which can be found in the Appendix. [*Proof of the first statement of Theorem 2.*]{} We begin by examining the behaviour of stability properties under embeddings. Consider an arbitrary quasipolynomial system with $m>n$: $$\dot{x}_i = x_i \left(\lambda _{i} + \sum_{j=1}^{m}A_{ij}\prod_{k= 1}^{n}x_k^{B_{jk}} \right) , \;\:\;\: i = 1, \ldots ,n \label{glv2}$$ Let $\tilde{A}$, $\tilde{B}$ and $\tilde{\lambda}$ be the matrices of the expanded system which are defined in the following way: $$\label{exps} \tilde{A} = \left( \begin{array}{c} A_{n \times m} \\ O_{(m-n) \times m} \end{array} \right) \: , \;\;\: \tilde{B} = \left( \begin{array}{cc} B_{m \times n} \mid B'_{m \times (m-n)} \end{array} \right) \: , \;\;\: \tilde{\lambda} = \left( \begin{array}{c} \lambda _{n \times 1} \\ O_{(m-n) \times 1} \end{array} \right)$$ where we have explicitly indicated by means of indexes the sizes of the submatrices for the sake of clarity, $O$ denotes a null matrix, $B'$ is a matrix of arbitrary entries chosen in such a way that $\mid \tilde{B} \mid >0$, and $x_i =1$ for $i= n+1, \ldots, m$. [Lemma 1.]{} We can now examine the effect of quasimonomial transformations (QMT’s from now on) of the form: $$\label{qmt} x_{i} = \prod_{k=1}^{n} y _{k}^{\Gamma_{ik}} , \;\:\: i=1, \ldots ,n \:\: , \;\:\: \mid \Gamma \mid > 0$$ [Lemma 2.]{} In particular, Lemma 2 applies to the expanded QP system (\[exps\]). Let us choose a QMT such that $\Gamma$ in (\[qmt\]) is given by $\tilde{B}^{-1}$ in (\[exps\]). The result is a new QP system with characteristic matrices: $$\label{mlv} \tilde{A}' = \tilde{B} \cdot \tilde{A} \;\:\: , \;\:\: \tilde{B}' = I \;\:\: , \;\:\: \tilde{\lambda}' = \tilde{B} \cdot \tilde{\lambda}$$ and thus a Lotka-Volterra system ($\tilde{B}'$ is the identity matrix). The inverse transformation, leading from (\[mlv\]) to (\[exps\]) is also a QMT, thus validating Lemma 2 for (\[mlv\]). Alternatively, in the case $m=n$ no embedding is to be performed and Lemma 2 is applied directly to the original flow setting $\Gamma = B^{-1}$. In either case ($m>n$ or $m=n$) we have reduced the stability problem to that corresponding to the Lotka-Volterra representative: If we establish stability for the corresponding fixed point of the Lotka-Volterra system, the steady state of the original flow will automatically be stable. Note that these considerations hold irrespectively of the fact that now the Lotka-Volterra representative may have an infinity of fixed points, even if this is not the case for the original flow. Let us then consider an arbitrary $m$-dimensional QPP system of Lotka-Volterra form. Since the tildes and primes appearing in (\[mlv\]) will not be necessary in what follows, we drop them for the sake of clarity. We then have $A = K \cdot D$, $\lambda = K \cdot L$ and, according to \[\[byv5\]\], rank($A$) $=$ rank($K$) $\equiv r \leq m$. Steady states are given in parametric form by: $${\bf x}_0 (N) = - D^{-1} \cdot (L-N) \;\: , \; N \in \mbox{ker}(K)$$ We can now turn to the characterization of stability of steady-states by means of the energy-Casimir method. The ($m-r$) independent Casimir functions are of the form: $$C^{(N)} = \sum _{j=1}^m N_j \ln x_j \;\: , \; N \in \mbox{ker}(K)$$ and we can accordingly take the following convenient form for the energy-Casimir functional: $$\label{fecd} H_C \equiv \sum _{j=1}^m \left( D_{jj} x_j + (L_j + N_j) \ln x_j \right) \;\: ,$$ where $N \in \mbox{ker}(K)$ is to be determined. Let us concentrate on a particular steady state ${\bf x}_0^* = -D^{-1} \cdot (L-N_0)$. We can state: [Lemma 3.]{} Now notice that: $$L-N_0 = -D \cdot {\bf x}_0^*$$ Since we consider only steady states belonging to the positive orthant, Lemma 3 can be equivalently formulated in terms of positiveness or negativeness of matrix $D$. Since $D$ is invariant under QMTs and embeddings \[\[byv5\]\], the same result is valid for the original QPP system and the first part of Theorem 2 is demonstrated. The energy-Casimir functional (\[fecd\]) is mapped into a functional of the form (\[fecqmp\]) for the original QPP system \[\[byv5\]\]. We need to prove, however, that (\[fecqmp\]) is also an energy-Casimir functional. This is done in the following two lemmas: [Lemma 4.]{} And finally: [Lemma 5.]{} This completes the proof of Theorem 2. [*Remark 1.*]{} The stable character of the steady state is independent of important features of the system, such as the degree of nonlinearity or the number of fixed points present in the positive orthant. This implies that there are certain degrees of freedom available in the Hamiltonian which can be varied without destroying the stability of motion. This has relevant consequences that we shall illustrate in the next section. [*Remark 2.*]{} The criterion in Theorem 2 can be verified straightforwardly by simple inspection of the Hamiltonian. In particular, a precise knowledge of the coordinates of the fixed point(s) is not required. [*Remark 3.*]{} In the specific case of conservative Lotka-Volterra equations, we have from (\[kdl\]) that $B$ is the identity matrix and then $A = K \cdot D$. Therefore, if the hypothesis of Theorem 2 is verified then there exists a diagonal positive definite matrix $\bar{D}$, which is the absolute value of $D$, such that $\bar{D} \cdot A + A^T \cdot \bar{D} = 0$ due to the skew-symmetry of $K$. Accordingly, the classical stability criterion for conservative Lotka-Volterra systems is implied by Theorem 2 and now appears as a particular case. [3. Examples]{} [Example 1.]{} We first consider Volterra’s \[\[v1\]\] predator-prey equations: $$\label{lv2d} \begin{array}{ccl} \dot{x}_1 & = & x_1 ( a - bx_2) \vspace{2mm} \\ \dot{x}_2 & = & x_2 (-d + cx_1) \end{array}$$ Here $a$, $b$, $c$ and $d$ are positive constants. This system is QPP with: $$\label{mat2d} K = \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) \; , \:\; D = \left( \begin{array}{cc} -c & 0 \\ 0 & -b \end{array} \right) \; , \:\; L = \left( \begin{array}{c} d \\ a \end{array} \right)$$ The Hamiltonian is: $$\label{h1e1} H(x_1,x_2) = -cx_1 -bx_2 +d \ln x_1 +a \ln x_2$$ It is well known that there is a unique fixed point in the positive orthant, which is stable. We can immediately verify this from the point of view of Theorem 2, since $D$ in (\[mat2d\]) is negative definite. Therefore the steady state is stable. Moreover, (\[h1e1\]) is a Lyapunov functional for it, since flow (\[lv2d\]) is symplectic. [Example 2.]{} Taking the system of Example 1 as starting point, let us now consider the following generalization of the Hamiltonian: $$\label{h2e1} H(x_1,x_2) = -cx_1^{\alpha}x_2^{\beta} -bx_1^{\gamma}x_2^{\delta} + d \ln x_1 + a \ln x_2$$ Now the equations become $$\label{nlg1} \begin{array}{ccl} \dot{x}_1 & = & x_1 ( a - \beta c x_1^{\alpha}x_2^{\beta} - \delta b x_1^{\gamma}x_2^{\delta}) \vspace{2mm} \\ \dot{x}_2 & = & x_2 (-d + \alpha c x_1^{\alpha}x_2^{\beta} + \gamma b x_1^{\gamma}x_2^{\delta}) \vspace{2mm} \end{array}$$ Let us assume that $\alpha$, $\beta$, $\gamma$ and $\delta$ are all positive. Since in Volterra’s model $\alpha$ and $\delta$ are greater than $\beta$ and $\gamma$, we shall also extend this requirement here and consider: $$\mid B \mid = \alpha \delta - \beta \gamma > 0$$ Within these assumptions, which are not very restrictive, it is not difficult to prove that there exists a unique fixed point inside the positive orthant if and only if: $$\label{1.i} \frac{\delta}{\gamma} > \frac{a}{d} > \frac{\beta}{\alpha}$$ We have that matrix $D$ retains the same form than in (\[mat2d\]). Therefore, according to Theorem 2 the point is stable, Hamiltonian (\[h2e1\]) is also a Lyapunov functional of the generalized system (given that (\[nlg1\]) is a symplectic flow) and (\[1.i\]) remains as the only condition both for the positiveness of the fixed point and for its stability. It is clear that the generalized Hamiltonian (\[h2e1\]) must incorporate dynamical features not present in Volterra’s model. To see this, we first put (\[nlg1\]) into classical Hamiltonian form by means of transformation $y_i = \ln x_i$, for $i=1,2$ (see \[\[byv5\]\] for the general reduction algorithm of QPP systems into the Darboux canonical form). After that, we perform a phase-space translation with the new axes centered in the steady state: $y_i = y_i^0 + \varepsilon _i$, $i=1,2$. Finally, we consider the case of small oscillations around the steady state and neglect terms of order $\varepsilon ^3$. The resulting system has the following Hamiltonian: $$\label{h3e1} H(\varepsilon _1,\varepsilon _2) = \mu _1 \varepsilon _1^2 + \mu _2 \varepsilon _2^2 + 2 \mu \varepsilon _1 \varepsilon _2 \;\: ,$$ where $\mu_1$, $\mu_2$ and $\mu$ are negative constants. We shall first consider a particular case of (\[nlg1\]) $$\label{pc1} \begin{array}{ccl} \dot{x}_1 & = & x_1 ( a - (1+ \delta ^*) b x_2^{1+\delta ^*}) \vspace{2mm} \\ \dot{x}_2 & = & x_2 (-d + (1+ \alpha ^*) c x_1^{1+\alpha ^*}) \end{array}$$ where $\alpha ^*$ and $\delta ^*$ are greater than $-1$. It is a simple task to demonstrate that for (\[pc1\]) $\mu =0$ in (\[h3e1\]), and then the trajectories are ellipses aligned with the coordinate axes, similarly to what occurs in Volterra’s case. However, the frecuency of the oscillations is now generalized to: $$\omega = \sqrt{(1+ \alpha ^*)(1+ \delta ^*)ad}$$ If $\alpha ^*$ and $\delta ^*$ remain small, $\omega$ is of the order of Volterra’s frecuency $\omega _0 = \sqrt{ad}$. In the most general case, $\omega$ can take any positive value, and is not restricted to any particular range. There are some additional features not present in Volterra’s model which are due to the off-diagonal terms in matrix $B$. These are related to the phase shift between the oscillations of the predator and the prey. To see this, let us turn back to the general Hamiltonian (\[h3e1\]) for the case of small oscillations. It is well known that there exists a canonical transformation, which is a rotation of angle $\phi$ of the axes, such that in the new variables the Hamiltonian is $$H(\xi _1, \xi _2) = \lambda _1 \xi _1^2 + \lambda _2 \xi _2^2 \;\: ,$$ where $\lambda _1 $ and $\lambda _2$ are the eigenvalues of the ellipse. The solution for $(\xi _1, \xi _2)$ is straightforward. Then, if we transform back into the variables $(\varepsilon _1 , \varepsilon _2)$, a simple calculation shows that the phase shift between the predator and the prey is just: $$\Phi (\rho,\phi) = \frac{\pi}{2} + \arctan ( \rho \tan \phi ) - \arctan ( \rho ^{-1} \tan \phi )$$ where $\rho = \sqrt{\lambda _1 / \lambda _2}$. Thus, we now have phase shifts which may be different to $\pi /2$, which is the classical Volterra value ($\phi = 0$). Notice that, in the neighbourhood of $\phi = 0$ we have: $$\Phi (\rho,\phi) = \frac{\pi}{2} + \left( \rho - \frac{1}{ \rho } \right) \phi + o( \phi ^3)$$ Therefore, if the eigenvalues do not have exactly the same magnitude (which is a reasonable assumption) these models can reproduce, in particular, a whole range of phase shifts centered around $\pi /2$. This is consistent with observed time series in predator-prey systems (see, for example, \[\[hass1\], pp. 60, 92\] and \[\[mur1\], p. 67\]) in which the average phase shifts may differ from $\pi /2$. We can then conclude that generalization (\[h2e1\]) accounts for additional features observed in real systems, while retaining the advantages and the basic framework provided by a Hamiltonian formulation. [Example 3.]{} We shall start again with the Lotka-Volterra equations (\[lv2d\]). Let us now consider the addition to the Hamiltonian (\[h1e1\]) of two extra nonlinear terms: $$H(x_1,x_2) = -cx_1 -bx_2 + \sigma _1 x_1^{\alpha} + \sigma _2 x_2^{\beta} + d \ln x_1 +a \ln x_2$$ with both $\alpha$ and $\beta$ positive and different from 1. Notice that matrix $D$ is: $$D = \left( \begin{array}{cccc} -c & 0 & 0 & 0 \\ 0 & -b & 0 & 0 \\ 0 & 0 & \sigma _1 & 0 \\ 0 & 0 & 0 & \sigma _2 \end{array} \right)$$ The resulting generalized equations are: $$\label{e2eq1} \begin{array}{ccl} \dot{x}_1 & = & x_1 ( a - bx_2 + \beta \sigma_2 x_2^{\beta}) \vspace{2mm} \\ \dot{x}_2 & = & x_2 (-d + cx_1 - \alpha \sigma_1 x_1^{\alpha}) \end{array}$$ Before considering the existence of steady states, note from the form of $H$ and $D$ and from Theorem 2 that every fixed point of the positive orthant is stable if $\sigma _1 <0$ and $\sigma _2 <0$, independently of the values of $\alpha$ and $\beta$. Let us assume that this is the case. It is then simple to prove that there exists a unique point in the interior of the positive orthant which verifies the fixed point conditions: $$\begin{array}{ccl} cx_1 - \alpha \sigma_1 x_1^{\alpha} & = & d \\ bx_2 - \beta \sigma_2 x_2^{\beta} & = & a \end{array}$$ Therefore there is a unique steady state inside the positive orthant, it is stable and $H$ is a Lyapunov functional for it. The analytic determination of the coordinates of the point may be a nontrivial problem, since $\alpha$ and $\beta$ are real constants in general. However, it is now possible to establish stability even without knowing the exact position of the point, but only by demonstrating its existence. [Example 4.]{} We shall finally look upon the following system, characterized by Nutku \[\[nut1\]\]: $$\label{lv3d} \begin{array}{ccl} \dot{x}_1 &=& x_1( \rho + cx_2 + x_3) \vspace{2mm} \\ \dot{x}_2 &=& x_2( \mu + x_1 + ax_3) \vspace{2mm} \\ \dot{x}_3 &=& x_3( \nu + bx_1 + x_2) \vspace{2mm} \end{array}$$ As Nutku has pointed out, this is a Poisson system if $$\label{clv3d} abc=-1 \; , \;\;\; \nu = \mu b - \rho ab$$ In fact, if conditions (\[clv3d\]) hold the system is QPP with Hamiltonian: $$\label{hlv3d} H = ab x_1 + x_2 - a x_3 + \nu \ln x_2 - \mu \ln x_3$$ The associated QPP matrices are: $$\label{mlv3d} K = \left( \begin{array}{ccc} 0 & c & bc \\ -c & 0 & -1 \\ -bc & 1 & 0 \end{array} \right) \; , \;\;\; D = \left( \begin{array}{ccc} ab & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -a \end{array} \right) \; , \;\;\; L = \left( \begin{array}{c} 0 \\ \nu \\ - \mu \end{array} \right)$$ System (\[lv3d\]), being odd-dimensional, falls out of the scope of the traditional Hamiltonian domain. However, the previous results hold in this context as well. If we apply Theorem 2 to this case, from $D$ in (\[mlv3d\]) together with (\[clv3d\]) we immediately obtain that the fixed points of the positive orthant are stable if $a<0$, $b<0$ and $c<0$. Notice that system (\[lv3d\]) has an infinite number of fixed points, so stability is simultaneously demonstrated for all those belonging to int$\{I \!\! R^3_+\}$. Notice also that the flow is not symplectic, and we have one independent Casimir invariant: $$C = ab \ln x_1 -b \ln x_2 + \ln x_3 = \mbox{constant}$$ Thus, according to (\[fecqmp\]) the Lyapunov functional of every positive steady state will be of the form: $$H_C = H + \kappa C = ab x_1 + x_2 - a x_3 + \kappa ab \ln x_1 + ( \nu - \kappa b) \ln x_2 + ( \kappa - \mu ) \ln x_3 \; , \:\; \kappa \in I \!\! R$$ Obviously, the Lyapunov functional (i.e., the appropriate value of the parameter $\kappa$) will be different for every fixed point and can be determined without difficulty by following the constructive procedure given in the proof of Theorem 2. We do not elaborate further on this issue for the sake of conciseness. Finally, notice that the flow can be easily generalized to account for higher order nonlinearities while preserving stability, by means of the same techniques employed in Examples 2 and 3. Such techniques are completely general. [Appendix]{} [*Proof of Lemma 1.*]{} For (i), we have: $$\sum_{j=1}^m \tilde{A}_{ij} \prod _{k=1}^m (x_{0k})^{\tilde{B}_{jk}} + \tilde{\lambda}_i = 0 \;\: , \: \forall i = 1, \ldots ,m \Longrightarrow$$ $$\sum_{j=1}^m A_{ij} \prod _{k=1}^n (x_{0k})^{B_{jk}} + \lambda_i = 0 \;\: , \: \forall i$$ The converse follows after noting that the sense of these implications can be reversed. The proof of (ii) is a consequence of the fact that the removal or addition of variables of constant value 1 does not affect the stable character of the point. [**Q.E.D.**]{} [*Proof of Lemma 2.*]{} It is a consequence of the fact that QMTs (\[qmt\]) are orientation-preserving diffeomorphisms and therefore relate topologically orbital equivalent systems. [**Q.E.D.**]{} [*Proof of Lemma 3.*]{} The gradient of the energy-Casimir functional vanishes identically at ${\bf x}_0^*$ if we set $N = -N_0$ in $H_C$. For the second part of the criterion, we note that the Hessian of $H_C$ at ${\bf x}_0^*$ is diagonal due to the simple form of $H$ in the case of Lotka-Volterra equations, and takes the value $$\mbox{Hess}(H_C \mid _{{\bf x}_0^*}) = \mbox{diag} \left( \frac{(N_0-L)_1}{(x_{01}^*)^2}, \ldots , \frac{(N_0-L)_m}{(x_{0m}^*)^2} \right)$$ [**Q.E.D.**]{} [*Proof of Lemma 4.*]{} It is a simple consequence of the chain rule for $C^2$ functions of $m$ real arguments. 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--- abstract: 'The twistor space of representations on an open variety maps to a weight two space of local monodromy transformations around a divisor component at infinty. The space of $\sigma$-invariant sections of this slope-two bundle over the twistor line is a real $3$ dimensional space whose parameters correspond to the complex residue of the Higgs field, and the real parabolic weight of a harmonic bundle.' address: | CNRS, Laboratoire J. A. Dieudonné, UMR 6621\ Université de Nice-Sophia Antipolis\ 06108 Nice, Cedex 2, France author: - Carlos Simpson title: A weight two phenomenon for the moduli of rank one local systems on open varieties --- Introduction {#sec-introduction} ============ Let $X$ be a smooth projective variety and $D\subset X$ a reduced effective divisor with simple normal crossings. We would like to define a Deligne glueing for the Hitchin twistor space of the moduli of local systems over $X-D$. Making the construction presents new difficulties which are not present in the case of compact base, so we only treat the case of local systems of rank $1$. Every local system comes from a vector bundle on $X$ with connection logarithmic along $D$, however one can make local meromorphic gauge transformations near components of $D$, and this changes the structure of the bundle as well as the eigenvalues of the residue of the connection. The change in eigenvalues is by subtracting an integer. There is no reasonable algebraic quotient by such an operation: for our main example §\[sec-p1\], that would amount to taking the quotient of ${{\mathbb A}}^1$ by the translation action of ${{\mathbb Z}}$. Hence, we are tempted to look at the moduli space of logarithmic connections and accept the fact that the Riemann-Hilbert correspondence from there to the moduli space of local systems, is many-to-one. We first concentrate on looking at the simplest case, which is when $X:= {{\mathbb P}}^1$ and $D:= \{ 0 , \infty \} $ and the local systems have rank $1$. In this case, much as in Goldman and Xia [@GoldmanXia], one can explicitly write down everything, in particular we can write down a model. This will allow observation of the weight two phenomenon which is new in the noncompact case. The residue of a connection takes values in a space which represents the local monodromy around a puncture. As might be expected, this space has weight two, so when we do the Deligne glueing we get a bundle of the form ${{\mathcal O}}_{{{\mathbb P}}^1}(2)$. There is an antipodal involution $\sigma$ on this bundle, and the preferred sections corresponding to harmonic bundles are $\sigma$-invariant. The space of $\sigma$-invariant sections of ${{\mathcal O}}_{{{\mathbb P}}^1}(2)$ is ${{\mathbb R}}^3$, in particular it doesn’t map isomorphically to a fiber over one point of ${{\mathbb P}}^1$. Then kernel of the map to the fiber is the parabolic weight parameter. Remarkably, the parabolic structure appears “out of nowhere”, as a result of the holomorphic structure of the Deligne-Hitchin twistor space constructed only using the notion of logarithmic $\lambda$-connections. After §\[sec-p1\] treating in detail the case of ${{\mathbb P}}^1 - \{ 0 , \infty \} $, we look in §\[sec-tatetwistor\] more closely at the bundle ${{\mathcal O}}_{{{\mathbb P}}^1}(2)$ which occurs: it is the [*Tate twistor structure*]{}, and is also seen as a twist of the tangent bundle $T{{\mathbb P}}^1$. Then §\[sec-general\] concerns the case of rank one local systems when $X$ has arbitrary dimension. In §\[sec-strict\] we state a conjecture about strictness which should follow from a full mixed theory as we are suggesting here. Since we are considering rank one local systems, the tangent space is Deligne’s mixed Hodge structure on $H^1(X -D, {{\mathbb C}})$ (see Theorem \[mhs-ident\]). However, a number of authors, such as Pridham [@PridhamMHS] [@PridhamQl] and Brylinski-Foth [@BrylinskiFoth] [@Foth] have already constructed and studied a mixed Hodge structure on the deformation space of representations of rank $r>1$ over an open variety. These structures should amount to the local version of what we are looking for in the higher rank case, and motivate the present paper. They might also allow a direct proof of the infinitesimal version of the strictness conjecture \[conj-strictness\]. In the higher rank case, there are a number of problems blocking a direct generalization of what we do here. These are mostly related to non-regular monodromy operators. In a certain sense, the local structure of a connection with diagonalizable monodromy operators, is like the direct sum of rank $1$ pieces. However, the action of the gauge group contracts to a trivial action at $\lambda = 0$, so there is no easy way to cut out an open substack corresponding only to regular values. We leave this generalization as a problem for future study. This will necessitate using contributions from other works in the subject, such as Inaba-Iwasaki-Saito [@InabaIwasakiSaito] [@InabaIwasakiSaito2] and Gukov-Witten [@GukovWitten]. This paper corresponds to my talk in the conference “Interactions with Algebraic Geometry” in Florence (May 30th-June 2nd 2007), just a week after the Augsberg conference. Sections \[sec-tatetwistor\]–\[sec-strict\] were added later. We hope that the observation we make here can contribute to some understanding of this subject, which is related to a number of other works such as the notion of $tt^{\ast}$-geometry [@Hertling] [@Schafer], geometric Langlands theory [@GukovWitten], Deligne cohomology [@EsnaultViehweg] [@Gajer], harmonic bundles [@Biquard] [@Mochizuki] and twistor ${{\mathcal D}}$-modules [@Sabbah], Painlevé equations [@Boalch] [@InabaIwasakiSaito] [@InabaIwasakiSaito2], and the theory of rank one local systems on open varieties [@Budur] [@Dimca] [@DimcaMaisonobeSaito] [@DimcaPapadimaSuciu] [@Libgober]. Preliminary definitions {#sec-prelim} ======================= It is useful to follow Deligne’s way of not choosing a square root of $-1$. This serves as a guide to making constructions more canonically, which in turn serves to avoid encountering unnecessary choices later. We do this because one of the goals below is to understand in a natural way the Tate twistor structure $T(1)$. In particular, this has served as a useful guide for finding the explanation given in §\[sub-tate-integer\] for the sign change necessary in the logarithmic version $T(1,\log )$. We have tried, when possible, to explain the motivation for various other minus signs too. [*Caution:*]{} there may remain sign errors specially towards the end. Let ${{\mathbb C}}$ be an algebraic closure of ${{\mathbb R}}$, but without a chosen $\sqrt{-1}$. Nevertheless, occasional explanations using a choice of $i = \sqrt{1} \in {{\mathbb C}}$ are admitted so as not to leave things too abstruse. Complex manifolds {#sub-complex} ----------------- There is a notion of ${{\mathbb C}}$-linear complex manifold $M$. This means that at each point $m\in M$ there should be an action of ${{\mathbb C}}$ on the real tangent space $T_{{{\mathbb R}}}(M)$. Holomorphic functions are functions $M\rightarrow {{\mathbb C}}$ whose $1$-jets are compatible with this action. Usual Hodge theory still goes through without refering to a choice of $i\in {{\mathbb C}}$. We get the spaces $A^{p,q}(M)$ of forms on $M$, and the operators $\partial$ and $\overline{\partial}$. Let ${{\mathbb R}}^{\perp}$ denote the imaginary line in ${{\mathbb C}}$. This is what Deligne would call ${{\mathbb R}}(1)$ however we don’t divide by $2\pi$. If $h$ is a metric on $M$, there is a naturally associated two-form $\omega\in A^2(M, {{\mathbb R}}^{\perp})$. The Kähler class is $[\omega ]\in H^2(X,{{\mathbb R}}^{\perp}) = H^2(X,{{\mathbb R}}(1))$. Classically this is brought back to a real-valued $2$-form by multiplying by a choice of $\sqrt{-1}$, but we shouldn’t do that here. Then, the operators $L$ and $\Lambda$ are defined independently of $\sqrt{-1}$, but they take values in ${{\mathbb R}}^{\perp}$. The Kähler identities now hold without $\sqrt{-1}$ appearing; but it is left to the reader to establish a convention for the signs. Note that $M$ may not be canonically oriented. If $Q = \{ \pm \sqrt{-1}\}$ as below, then the orientation of $M$ is canonically defined in the $n$-th power $Q^n\subset {{\mathbb C}}$ where $n=dim_{{{\mathbb C}}}M$. In particular, the orientation in codimension $1$ is always ill-defined. If $D$ is a divisor, this means that $[D]\in H^2(M, {{\mathbb R}}^{\perp})$. This agrees with what happens with the Kähler metric. Similarly, if $L$ is a line bundle then $c_1(L)\in H^2(M, {{\mathbb R}}^{\perp})$. If $X$ is a quasiprojective variety over ${{\mathbb C}}$ then $X({{\mathbb C}})$ has a natural topology. Denote this topological space by $X^{\rm top}$. It is the topological space underlying a structure of complex analytic space. In the present paper, we don’t distinguish too much between algebraic and analytic varieties, so we use the same letter $X$ to denote the analytic space. Let $\overline{X}$ denote the conjugate variety, where the structural map is composed with the complex conjugation $Spec({{\mathbb C}})\rightarrow Spec({{\mathbb C}})$. In terms of coordinates, $\overline{X}$ is given by equations whose coefficients are the complex conjugates of the coefficients of the equations of $X$. There is a natural isomorphism $\varphi : X^{\rm top}\stackrel{\cong}{\rightarrow} \overline{X}^{\rm top}$, which in terms of equations is given by $x\mapsto \overline{x}$ conjugating the coordinates of each point. The imaginary scheme of a group {#sub-imaginary} ------------------------------- Let $Q\subset {{\mathbb C}}$ be the zero set of the polynomial $x^2+1$, in other words $Q = \{ \pm \sqrt{-1}\}$. Multiplication by $-1$ is equal to multiplicative inversion, which is equal to complex conjugation, and these all define an involution $$c_Q:Q\rightarrow Q.$$ Suppose $Y$ is a set provided with an involution $\tau _Y$. Then we define a new set denoted $Y^{\perp}$ starting from $Hom (Q,Y)$ with its two involutions $$f\mapsto \tau _Y\circ f, \;\;\; f\mapsto f\circ c_Q.$$ Let $Y^{\perp}$ be the equalizer of these two involutions, in other words $$Y^{\perp}:= \{ f\in Hom (Q,Y), \;\; \tau _Y\circ f = f\circ c_Q \} .$$ Thus, an element of $G^{\perp}$ is a function $\gamma : q \mapsto \gamma (q)$ such that $\gamma (-q)= \tau _Y(\gamma (q))$. The two equal involutions will be denoted $\tau _{Y^{\perp}}$. If we choose $i = \sqrt{-1}\in {{\mathbb C}}$, then $Y^{\perp}$ becomes identified with $Y$ via $\gamma \mapsto \gamma (i)$. For the opposite choice of $i$ this isomorphism gets composed with $\tau _Y$. If $G$ is a scheme defined over ${{\mathbb R}}$ then $G^{\perp}$ is also defined over ${{\mathbb R}}$. For example, if $G={{\mathbb R}}$ with involution $x\mapsto -x$ then $G^{\perp}$ is the imaginary line ${{\mathbb R}}^{\perp}$ defined above. Using the involution $x\mapsto -x$ we could also define ${{\mathbb C}}^{\perp}$. However there is a natural isomorphism ${{\mathbb C}}\cong {{\mathbb C}}^{\perp}$ sending $a$ to the function $\gamma : q \mapsto qa$. In view of this, and in order to lighten notation, we don’t distinguish between ${{\mathbb C}}$ and ${{\mathbb C}}^{\perp}$ even in places where that might be natural for example throughout §\[sec-exact\]. If $G$ is an algebraic group over ${{\mathbb C}}$, it has an involution $g\mapsto g^{-1}$, which doesn’t preserve the group structure unless $G$ is abelian. Using this involution yields a scheme denoted $G^{\perp}$. If $G$ is abelian then $G^{\perp}$ has a natural group structure. In general there is a natural action of $G$ on $G^{\perp}$ by conjugation: if $g\in G$ and $q\mapsto \gamma (q)$ is an element of $G^{\perp}$ then the element $q\mapsto g\gamma (q)g^{-1}$ is again an element of $G^{\perp}$. Since we will be looking mostly at rank one local systems, we are particularly interested in the case $G={{{\mathbb G}}_m}$. Then $${{{\mathbb G}}_m}^{\perp} = \{ (x,y)\in {{\mathbb C}}^2, \;\; x^2 + y^2 = 1 \} .$$ The equality is given as follows: to an element $\gamma : q \mapsto \gamma (q)$ of ${{{\mathbb G}}_m}^{\perp}$, associate the point $(x,y)$ given by $$x:= \frac{1}{2}\sum _{q\in Q}\gamma (q),$$ $$y:= \frac{1}{2}\sum _{q\in Q}q^{-1}\gamma (q).$$ Call $(x,y)$ the [*circular coordinates*]{} on ${{{\mathbb G}}_m}^{\perp}$. It is well-known that the exponential should really be considered as a map ${\rm exp}: {{\mathbb C}}(1) \rightarrow {{{\mathbb G}}_m}$. Alternatively, we can view the exponential as a map $${\rm exp}^{\perp} : {{\mathbb C}}\rightarrow {{{\mathbb G}}_m}^{\perp}.$$ given in circular coordinates by $${\rm exp}^{\perp}(\theta ) := (\cos (2\pi \theta ), \sin (2\pi \theta )).$$ It is useful to include $2\pi$ here because of the relationship with residues, see below. The kernel of ${\rm exp}^{\perp}$ is the usual ${{\mathbb Z}}\subset {{\mathbb C}}$. We call $\theta$ a [*circular logarithm*]{} of its image point. Logarithmic connections {#sub-logarithmic} ----------------------- Suppose $X$ is a smooth projective variety and $D\subset X$ is a normal crossings divisor. Let $U:= X-D$ and $j: U\hookrightarrow X$ be the inclusion. Recall that the sheaf of [*logarithmic forms*]{} on $(X,D)$ denoted $\Omega ^1_X(\log D)$ is the locally free sheaf, subsheaf of $j_{\ast}\Omega ^1_{U}$, which is generated in local coordinates by $d\log z_1,\ldots , d\log z_k , dz_{k+1},\ldots , dz_n$ whenever $(z_1,\ldots , z_n)$ is a system of local coordinates in which $D$ is given by $z_1\cdots z_k = 0$. A [*logarithmic connection*]{} $\nabla$ on a vector bundle $E$ over $X$, is a morphism of sheaves $$\nabla : E\rightarrow E\otimes _{{{\mathcal O}}_X}\Omega ^1_X(\log D)$$ such that $\nabla (af) = a\nabla (f) + da \cdot f$. More generally, for $\lambda \in {{\mathbb C}}$ a [*logarithmic $\lambda$-connection*]{} is a map $\nabla$ as above such that $\nabla (af) = a\nabla (f) + \lambda da \cdot f$. For $\lambda = 1$ this is a usual connection, and for any $\lambda \neq 0$ we get a usual connection $\lambda ^{-1}\nabla$. For $\lambda = 0$ it is a Higgs field. The [*Riemann-Hilbert correspondence*]{} takes a vector bundle with logarithmic connection $(E,\nabla )$ to its monodromy representation $\rho$. This is well-defined independent of the choice of $\sqrt{-1}\in {{\mathbb C}}$. In the compact case, it is an equivalence of categories between vector bundles with connection, and representations up to conjugacy. However, in our open case there are many possible choices of $(E,\nabla )$ which give the same representation $\rho$, because of the possibility of making [*meromorphic gauge transformations*]{} along the components of the divisor $D$, see §\[sub-meromorphicgauge\] below. For any $\lambda \neq 0$, the [*monodromy representation*]{} of a $\lambda$-connection is by definition that of the normalized connection $\lambda ^{-1}\nabla$. Local monodromy {#sub-localmonodromy} --------------- The reason for introducing the imaginary scheme $G^{\perp}$ was to discuss local monodromy. Keep the notation that $(X,D)$ is a smooth variety with a normal crossings divisor. For each component $D_i$ of $D$, choose a point $x_i$ near $D_i$. Choose a local coordinate system $(z_1,\ldots , z_n)$ for $X$ near a smooth point of $D_i$, such that $D_i$ is given by $z_1=0$ and $x_i$ is the point $(\epsilon , 0 , \ldots , 0)$. We get a map from $Q$ to $\pi _1(X,x_i)$ as follows: for $q\in Q$, consider the path $t\mapsto (\epsilon \cdot e^{2\pi q t}, 0, \ldots , 0)$. For $-q$ we get the inverse path, in other words we really have an element of $\pi _1(X,x_i) ^{\perp}$. Conjugating by a choice of path from $x$ to $x_i$, we get an element $$\gamma _{D_i} \in \pi _1(X,x)^{\perp}.$$ It is well-defined up to the conjugation action of $\pi _1(X,x)$. If $\rho : \pi _1(X,x)\rightarrow G$ is a representation, we obtain by functoriality of the construction $(\; ) ^{\perp}$ a map $$\rho ^{\perp}: \pi _1(X,x)^{\perp}\rightarrow G^{\perp},$$ so we get the [*local monodromy element*]{} $${\rm mon}(\rho , D_i):= \rho ^{\perp} (\gamma _{D_i}) \in G^{\perp},$$ which is well-defined up to the conjugation action of $G$. If $G$ is abelian, such as $G={{{\mathbb G}}_m}$, then the local monodromy element is well-defined. Meromorphic gauge group {#sub-meromorphicgauge} ----------------------- Since we will mostly be working with line bundles, we describe the meromorphic gauge group only in this case. It is much easier than in general. Decompose $D=D_1+\ldots + D_k$ into a union of smooth irreducible components. The gauge group is just $${{\mathcal G}}:= {{\mathbb Z}}^k ,$$ acting as follows. Suppose $(L,\nabla )$ is a line bundle with logarithmic $\lambda$-connection on $(X,D)$ and $g=(g_1,\ldots , g_k)$ is an element of ${{\mathcal G}}$. Then the new line bundle is defined by $$L^g:= L(g_1D_1 + \ldots + g_kD_k),$$ and $\nabla ^g$ is the unique logarithmic $\lambda$-connection on $L^g$ which coincides with $\nabla$ over the open set $U$ via the canonical isomorphism $L^g|_{U}\cong L|_{U}$. The gauge transformation affects the first Chern class: $$\label{c1formula} c_1(L^g) = c_1(L) + g_1[D_1] + \ldots + g_k [D_k] ,$$ and the residue: $$\label{residueformula} {\rm res}(\nabla ^g ; D_i ) = {\rm res}(\nabla ; D_i) - \lambda g_i.$$ For convenience, here is the proof of . If $u$ is a nonvanishing holomorphic ection of $L$ near a point of $D_i$ (but not near the other divisor components), then $u$ may also be considered as a meromorphic section of $L^g$, but it has a zero of order $g_i$ along $D_i$. Hence, $u':=z_i^{-g_i}u$ is a nonvanishing holomorphic section of $L^g$ near our point of $D_i$. Let $R_i:= {\rm res}(\nabla ; D_i)$, so $$\nabla (au) = \lambda d(a)u + R_i \frac{dz_i}{z_i}a u + \ldots .$$ Generically, $\nabla $ and $\nabla ^g$ are the same connection. However, a section of $L^g$ is written in terms of the unit section $u'$ as $au' = az_i^{-g_i}u$, so $$\nabla ^g (au') = \nabla (az_i^{-g_i}u) = \lambda d(a) u' - \lambda g_i \frac{dz_i}{z_i}a u' + R_i \frac{dz_i}{z_i}a u +\ldots .$$ The residue of $\nabla ^g$ is ${\rm res}(\nabla ^g ; D_i ) = R_i - \lambda g_i$ as claimed in . The restrictions to the open set are isomorphic: $$(L^g, \nabla ^g) |_{U} \cong (L,\nabla )|_{U},$$ hence the monodromy representations are the same in the case $\lambda \neq 0$. Conversely, again in the case $\lambda \neq 0$, given $(L,\nabla )$ and $(L',\nabla ')$ two logarithmic $\lambda$-connections with the same monodromy representations, there is a unique meromorphic gauge transformation $g\in{{\mathcal G}}$ such that $(L',\nabla ') \cong (L^g, \nabla ^g)$. Throughout the paper, make the convention that spaces and maps are in the complex analytic category. The reader will notice which parts of these analytic spaces have natural algebraic structures, for example the Betti spaces or the charts $M_{\rm Hod}(X,\log D)$. Often these algebraic charts will be divided by a group action or glued to other charts in an analytic way, so the result only has a structure of analytic space. The Deligne glueing in the compact case {#sec-deligneglue} ======================================= In this section we recall the Deligne glueing construction for the twistor space, in the case of a compact base variety $X$, that is $D=\emptyset$. The hyperkähler structure on the moduli space was constructed by Hitchin [@Hitchin], who also considered the Penrose twistor space associated to the quaternionic structure. Deligne in [@DeligneLett] proposed a construction of the twistor space using a deformation called the space of [*$\lambda$-connections*]{} closely related to the Hodge filtration, plus the Riemann-Hilbert correspondence relating connections on $X$ and the conjugate variety $\overline{X}$. Apparently Witten contributed something too because Deligne’s letter [@DeligneLett] starts off: > [“As I understand, Hitchin’s understanding of why one has a hyperkähler structure—as explained to me by Witten—works in your case. …”.]{} The twistor space structure is related to the notion of $tt^{\ast}$ geometry [@CecottiVafa] [@Hertling] [@Schafer]. The idea of a deformation relating de Rham and Dolbeault cohomology goes back further, to the theory of $\Gamma$-factors [@Deninger], Esnault’s notion of $\tau$-connection [@Esnault], Dolbeault homotopy theory [@NeisendorferTaylor], to the relation between cyclic and Hochschild cohomology [@Connes] [@Kaledin], and to singular perturbation theory [@Voros]. Moduli spaces {#sec-moduli} ------------- Fix a basepoint $x\in X$. Complex conjugation provides a map of topological spaces $\varphi : X^{\rm top}\rightarrow \overline{X}^{\rm top}$, which is antiholomorphic for the complex structures. In particular, we get $$\varphi _{\ast}: \pi _1(X,x)\stackrel{\cong}{\rightarrow} \pi _1(\overline{X},\overline{x}).$$ Recall the following moduli spaces or moduli stacks. Usually we don’t distinguish between moduli stacks or their universal categorical quotients which are moduli spaces. Also we are fixing the target group as $GL(r,{{\mathbb C}})$ which will be left out of the notation. In this section we let $r$ be arbitrary, although the next sections will specialize to $r=1$. Write $M_{\rm Hod}(X)\rightarrow {{\mathbb A}}^1$ for the moduli space or stack of semistable vector bundles of rank $r$ with $\lambda$-connection with vanishing Chern classes. The fibers over $0$ and $1$ are denoted respectively $M_{\rm Dol}(X)$ and $M_{\rm DR}(X)$. The group ${{{\mathbb G}}_m}$ acts, and over ${{{\mathbb G}}_m}\subset {{\mathbb A}}^1$ this action provides an isomorphism $$M_{\rm Hod}(X)\times _{{{\mathbb A}}^1}{{{\mathbb G}}_m}\cong {{{\mathbb G}}_m}\times M_{\rm DR}(X).$$ These natural constructions applied to the conjugate variety give conjugate varieties: $$M_{\rm Hod}(\overline{X}) \cong \overline{M_{\rm Hod}(X)}, \;\;\; M_{\rm DR}(\overline{X}) \cong \overline{M_{\rm DR}(X)}, \;\;\; M_{\rm Dol}(\overline{X}) \cong \overline{M_{\rm Dol}(X)}.$$ We also have the Betti moduli space [@LubotskyMagid] $$M_{\rm B}(X) = \frac{{\rm Hom}(\pi _1(X,x), GL(n,{{\mathbb C}}))}{GL(n,{{\mathbb C}})}$$ where the quotient is either a stack quotient or a universal categorical quotient depending on which framework we are using. The Riemann-Hilbert correspondence gives an isomorphism of analytic spaces or stacks $$M_{DR}(X)^{\rm an} \cong M_{B}(X)^{\rm an}.$$ It doesn’t depend on a choice of square root of $-1$. Glueing {#sec-glue} ------- The [*Deligne glueing*]{} is an isomorphism of complex analytic spaces $${\bf d} : M_{\rm Hod}(X)\times _{{{\mathbb A}}^1}{{{\mathbb G}}_m}\cong M_{\rm Hod}(\overline{X})\times _{{{\mathbb A}}^1}{{{\mathbb G}}_m}.$$ It is defined as follows. A point in the source is a triple $(\lambda , E, \nabla )$ where $\lambda \in {{{\mathbb G}}_m}\subset {{\mathbb A}}^1$, where $E$ is a vector bundle on $X$, and $\nabla $ is a $\lambda$-connection on $E$. This corresponds to the point $(\lambda , (E, \lambda ^{-1}\nabla ))$ in ${{{\mathbb G}}_m}\times M_{\rm DR}(X)$. Let $\rho (\lambda ^{-1}\nabla )$ denote the monodromy representation of $\pi _1(X,x)$ corresponding to the connection $\lambda ^{-1}\nabla$. Then $\rho (\lambda ^{-1}\nabla ) \circ \varphi _{\ast}^{-1}$ is a representation of $\pi _1(\overline{X},\overline{x})$. It corresponds to a vector bundle with connection $(F, \Phi )$ on $\overline{X}$. This vector bundle with connection may be characterized as follows: —we have a natural identification $F_{\overline{x}}\cong E_x$; and —the monodromy of $(F,\Phi )$ around a loop $\gamma $ in $\pi _1(\overline{X}, \overline{x})$ is equal, via this identification, to the monodromy of $(E,\lambda ^{-1}\nabla )$ around the loop $\varphi ^{-1}(\gamma )$ in $\pi _1(X,x)$. To continue with the definition of ${\bf d}$, choose the point $\mu = \lambda ^{-1} \in {{{\mathbb G}}_m}$, and look at the point $$(\mu , (F,\Phi ))\in {{{\mathbb G}}_m}\times M_{\rm DR}(\overline{X}).$$ It corresponds to a point $$(\mu , F, \mu \Phi )\in M_{\rm Hod}(\overline{X})\times _{{{\mathbb A}}^1}{{{\mathbb G}}_m}.$$ We set $${\bf d}(\lambda , E, \nabla ):= (\mu , F, \mu \Phi ).$$ Note that by definition ${\bf d}$ covers the map ${{{\mathbb G}}_m}\rightarrow {{{\mathbb G}}_m}$ given by $\lambda \mapsto \mu := \lambda ^{-1}$. This isomorphism can now be used to glue together the two analytic spaces $M_{\rm Hod}(X)^{\rm an}$ and $M_{\rm Hod}(\overline{X})^{\rm an}$ along their open sets which are the source and target of ${\bf d}$. The resulting space is denoted $M_{\rm DH}(X)$ for [*Deligne-Hitchin*]{}. It is Hitchin’s twistor space [@Hitchin], constructed as suggested by Deligne [@DeligneLett]. Interpreting ${{\mathbb P}}^1$ as obtained by glueing two copies of ${{\mathbb A}}^1$ along the map $\mu = \lambda ^{-1}$, we get a map $M_{\rm DH}(X)\rightarrow {{\mathbb P}}^1$. It is essential to make some remarks on the choices above. The space $M_{\rm Hod}(X)$ and its conjugate counterpart admit numerous natural automorphisms, for example multiplication by any element of ${{{\mathbb G}}_m}$, but also taking the dual of an object. In particular, it would have been possible to insert these operations in the middle of the definition of ${\bf d}$. They would extend to automorphisms of either of the two sides being glued, so the resulting space would be isomorphic. We feel that it is reasonable at each step of the way to use the simplest choice. This will nonetheless result in more complicated choices in the definition of preferred sections later. Note that we have not used any choice of $\sqrt{-1}\in {{\mathbb C}}$ in the construction, so $M_{\rm DH}(X)$ is independant of that. According to the construction, notice that we have two inclusions $$u: M_{\rm Hod}(X)\hookrightarrow M_{\rm DH}(X), \;\;\; v:M_{\rm Hod}(\overline{X})\hookrightarrow M_{\rm DH}(X).$$ We have $u(\lambda , E,\nabla ) = v (\mu , F , \mu \Phi )$ exactly when ${\bf d}(\lambda , E, \nabla ) = (\mu , F, \mu \Phi )$ as constructed above. We leave to the reader the problem of comparison of $M_{\rm DH}(X)$ and $M_{\rm DH}(\overline{X})$. The antipodal involution {#sec-antipodal} ------------------------ A crucial part of the structure is an antilinear involution $\sigma : M_{\rm DH}(X)\rightarrow M_{\rm DH}(X)$, covering the antipodal involution of ${{\mathbb P}}^1$. Note that the antipodal involution exchanges the two charts ${{\mathbb A}}^1$ of ${{\mathbb P}}^1$. Thus, in order to define $\sigma$ it suffices to define an antiholomorphic map $$\sigma _{{\rm Hod}, X} : M_{\rm Hod}(X)\rightarrow M_{\rm Hod}(\overline{X})$$ which is an antilinear isomorphism, and involutive: that is $\sigma _{{\rm Hod},\overline{X}}\circ \sigma _{{\rm Hod},X} = {\rm Id}$. Suppose we have a point $(\lambda , E, \nabla )$. Taking the complex conjugate of everything gives a point $(\overline{\lambda}, \overline{E},\overline{\nabla})\in M_{\rm Hod}(\overline{X})$. This gives an antiholomorphic map denoted $$C _{{\rm Hod}, X} : M_{\rm Hod}(X)\rightarrow M_{\rm Hod}(\overline{X}).$$ We need to show that it is compatible with the glueing ${\bf d}$ in the sense that $$\label{cdcompatible} C_{{\rm Hod},\overline{X}} \circ {\bf d} = {\bf d}^{-1} \circ C_{{\rm Hod},X}.$$ First of all $C_{{\rm Hod},X}$ and $C_{{\rm Hod},\overline{X}}$ intertwine the multiplication action of ${{{\mathbb G}}_m}$, with the complex conjugation ${{{\mathbb G}}_m}\cong \overline{\mathbb G}_m$. Also, $C$ and ${\bf d}$ both fix the de Rham fiber over $\lambda = 1$. Hence, to verify the compatibility , it suffices to verify it over $\lambda = 1$. Here $$C_{{\rm DR},X}: M_{\rm DR}(X) \rightarrow M_{\rm DR}(\overline{X}), \;\;\; (E,\nabla )\mapsto (\overline{E},\overline{\nabla})$$ and composing with the isomorphism ${\bf d}^{-1}$ which comes from $\pi _1(X,x)\cong \pi _1(\overline{X},\overline{x})$ we get an antilinear automorphism of $M_{\rm DR}(X)$. It is easy to see that, in terms of the isomorphism with $M_B(X)$, this antilinear automorphism is just the complex conjugation action on representations, $\rho \mapsto \overline{\rho}$ where $\overline{\rho} (\gamma ) := \overline{\rho (\gamma )}$. Similarly, $C_{{\rm Hod},\overline{X}} \circ {\bf d}$ is also seen to be the same automorphism $\rho \mapsto \overline{\rho}$. This proves the equality . With this compatibility, $C_{{\rm Hod},X}$ and $C_{{\rm Hod},\overline{X}}$ glue to give an antiholomorphic involution $$C: M_{\rm DH}(X)\rightarrow M_{\rm DH}(X).$$ covering the involution $\lambda \mapsto \overline{\lambda}^{\, -1}$ of ${{\mathbb P}}^1$. As described above, on the fiber over $\lambda = 1$ which is $M_{\rm DR}(X)\cong M_B(X)$, the involution is $C(\rho ) = \overline{\rho}$. The dual of a vector bundle with $\lambda$-connection $(E,\nabla )$ is again a vector bundle with $\lambda$-connection denoted $(E^{\ast}, \nabla ^{\ast})$. This operation is compatible with multiplication by ${{{\mathbb G}}_m}$, and with the operation of taking the dual of a local system via the Riemann-Hilbert correspondence. Therefore, it is compatible with the glueing ${\bf d}$ and gives an involution, holomorphic this time, denoted $$D: M_{\rm DH}(X)\rightarrow M_{\rm DH}(X)$$ which covers the identity of ${{\mathbb P}}^1$. Finally, multiplication by $-1\in {{{\mathbb G}}_m}$ gives an involution of $M_{\rm DH}(X)$ denoted by $N$, covering the involution $\lambda \mapsto -\lambda $ of ${{\mathbb P}}^1$. The involutions $C$, $D$ and $N$ commute. Their product is an involution $\sigma$ of $M_{\rm DH}(X)$ covering the antipodal involution $\lambda \mapsto -\overline{\lambda}^{\, -1}$ of ${{\mathbb P}}^1$. [*Proof:*]{} The complex conjugate of the dual of a vector bundle is naturally isomorphic to the dual of the complex conjugate. These also clearly commute with the operation of multiplying the connection by $-1$. Hence, the three involutions commute, which implies that the product $CDN$ is again an involution. It is antilinear because $C$ is antilinear whereas $D$ and $N$ are ${{\mathbb C}}$-linear. Since $C$, $D$ and $N$ cover respectively the involutions $\lambda \mapsto \overline{\lambda}^{\, -1}$, $\lambda \mapsto \lambda$ and $\lambda \mapsto -\lambda$, their product covers the product of these three, which is the antipodal involution. [ $\Box$]{} Preferred sections and the twistor property {#sec-preferred-twistor} ------------------------------------------- Deligne proposed to construct a family of “preferred sections” of the glued space $M_{\rm DH}(X)$, one for each harmonic bundle on $X$. \[prop-pref\] Suppose $(E,\partial , {\overline{\partial}}, \theta , \overline{\theta} )$ is a harmonic bundle on $X$. Then it leads to a section ${{\mathcal P}}: {{\mathbb P}}^1 \rightarrow M_{\rm DH}(X)$ which is $\sigma$-invariant and which sends $\lambda \in {{\mathbb A}}^1$ to the holomorphic bundle $(E, {\overline{\partial}}+ \lambda \overline{\theta})$ with $\lambda$-connection $\nabla = \lambda \partial + \theta$. [*Proof:*]{} We first define the value of the section at $\lambda \in {{\mathbb A}}^1$. On the ${{\mathcal C}}^{\infty}$ bundle $E$, consider the holomorphic structure $${\overline{\partial}}_{\lambda} := {\overline{\partial}}+ \lambda \overline{\theta}.$$ The holomorphic bundle $E^{\lambda}:= (E,{\overline{\partial}}_{\lambda})$ admits a $\lambda$-connection operator $\nabla _{\lambda} := \lambda \partial + \theta$. This gives a point $(E^{\lambda}, \nabla _{\lambda})$ in $M_{\rm Hod}(X)_{\lambda}$. One checks that over ${{{\mathbb G}}_m}\subset {{\mathbb A}}^1$, this section is invariant under the antipodal involution operator. Hence, taking image of the graph of our section already defined over ${{\mathbb A}}^1$, by $\sigma$, gives the section over the other chart ${{\mathbb A}}^1$ at infinity, and over ${{{\mathbb G}}_m}$ these glue together. One can also define directly the value of the section on the complex conjugate chart, see for example [@twistor pp 20-24]. [ $\Box$]{} In Hitchin’s original point of view [@HitchinH] [@Hitchin], the twistor space $M_{\rm DH}(X)$ came from the Penrose construction for the quaternionic structure on $M(X)$ whose different complex structures were those of $M_{\rm Dol}$ and $M_{\rm DR}$. The Penrose twistor space has a natural product structure of the form ${{\mathbb P}}^1 \times M(X)$. In Deligne’s reinterpretation [@DeligneLett] we can first construct the space $M_{\rm DH}(X)$ using the notion of $\lambda$-connection, complex conjugation and the Riemann-Hilbert correspondence as described above. The product structure is obtained by considering the family of preferred sections as described in the previous proposition. This leads back to the quaternionic structure by looking at the tangent space near a preferred section. The key to this beautiful procedure is the observation that the relative tangent space, or equivalently the normal bundle, along a prefered section is a semistable bundle of slope $1$ on ${{\mathbb P}}^1$, which is to say it is isomorphic to ${{\mathcal O}}_{{{\mathbb P}}^1}(1)^{\oplus a}$. This weight one property is equivalent to having a quaternionic structure, as was observed in [@HitchinH]. There is an equivalence of categories between quaternionic vector spaces, and vector bundles of slope $1$ over ${{\mathbb P}}^1$ with involution $\sigma$ covering the antipodal involution. If $V={{\mathcal O}}_{{{\mathbb P}}^1}(1)^d$ is a slope one bundle, the space of sections is $H^0({{\mathbb P}}^1,V) \cong {{\mathbb C}}^{2d}$. If $\sigma$ is an antipodal involution, the space of $\sigma$-invariant sections is a real form of the space of sections, thus $$H^0({{\mathbb P}}^1, V)^{\sigma}\cong {{\mathbb R}}^{2d},$$ and the twistor property says that the map from here to any of the fibers $V_{\lambda}$ is an isomorphism. This is what provides the single real vector space ${{\mathbb R}}^{2d}$ with a whole sphere of different complex structures. With this equivalence, saying that the various complex structures on $M(X)$ correspond to a quaternionic structure is equivalent to saying that the normal bundle to a preferred section has slope $1$. One can show directly the weight $1$ property given the construction of $M_{\rm DH}(X)$ and the preferred sections of Proposition \[prop-pref\], see [@hfnac]. It then follows that the deformation space of a preferred section in the world of $\sigma$-invariant sections of the fibration $M_{\rm DH}(X)\rightarrow {{\mathbb P}}^1$, maps isomorphically to the tangent space of any of the moduli space fibers (for example $M_{\rm DR}(X)$ over $\lambda = 1$ or $M_{\rm Dol}(X)$ over $\lambda = 0$). It implies that, locally, there is a unique $\sigma$-invariant section going through any point, and gives an alternative proof of Hitchin’s theorem that the moduli space has a quaternionic structure. For rank one bundles, this property can be globalized: \[uniglobal\] For bundles of rank $1$ on a compact $X$, the evaluation morphism at any point $p\in{{\mathbb P}}^1$ $$\Gamma ({{\mathbb P}}^1,M_{\rm DH}(X))^{\sigma} \rightarrow M_{\rm DH}(X)_p$$ is an isomorphism. In the rank one case, the moduli space is a Lie group so we can use its exponential exact sequence. The tangent at the identity preferred section is purely semistable of slope $1$. There is a lattice $A= H^1(X,{{\mathbb Z}})\cong {{\mathbb Z}}^a$ and a finite group $B= H^2(X,{{\mathbb Z}})$ such that we have an exact sequence $$0\rightarrow A \rightarrow A\otimes {{\mathcal O}}_{{{\mathbb P}}^1} (1)\rightarrow M_{\rm DH}(X) \rightarrow B \rightarrow 0.$$ We have $H^1({{\mathbb P}}^1,A)= H^1({{\mathbb P}}^1, A\otimes {{\mathcal O}}_{{{\mathbb P}}^1})=H^1({{\mathbb P}}^1,B)=0$, so taking sections gives an exact sequence. The subgroups of $\sigma$-invariants again form an exact sequence. The weight one property, equivalent to the quaternionic structure, says $$\Gamma ({{\mathbb P}}^1, A\otimes {{\mathcal O}}_{{{\mathbb P}}^1} (1))^{\sigma} \stackrel{\cong}{\rightarrow} A\otimes {{\mathcal O}}_{{{\mathbb P}}^1} (1)_p.$$ Comparing with the exact sequence of values at $p$ gives the result for $M_{\rm DH}(X)$. Deligne gave the construction of a quaternionic structure associated to a weight $1$ real Hodge structure in [@DeligneLett]. Given a vector space $V$ with two filtrations $F$ and $\overline{F}$, we can form a bundle $\xi (V,F,\overline{F})\rightarrow {{\mathbb P}}^1$ and this bundle has slope $1$ if and only if the two filtrations are $1$-opposed, i.e. they define a Hodge structure of weight $1$. In this sense, the twistor property is analogous to saying that a Hodge structure has weight $1$. These slightly different points of view are compatible for a preferred section which comes from a variation of Hodge structure, where the tangent space to the moduli space has a natural weight $1$ Hodge structure. The bundles of the form $\xi (V,F,\overline{F})$ are the slope $1$ bundles together with additional structure of an action of ${{{\mathbb G}}_m}$. A somewhat different collection of additional structure involving a connection is used in the notion of $tt^{\ast}$ geometry [@CecottiVafa] [@Hertling], which also has its physical roots in Hitchin’s twistor space. Schmid mentionned, during his courses on Hodge theory, the similarity between the equations governing the local structure of variations of Hodge structure, and the monopole or Nahm’s equations. It is interesting that these objects from physics are so closely related to variations of Hodge structure, the analytic incarnation of the idea of motives. This suggests a relationship between physics and motives which might be philosophically compelling. The twistor space for $X={{\mathbb P}}^1 - \{ 0 , \infty \} $ {#sec-p1} ============================================================= We would now like to mimic the Deligne-Hitchin construction for a quasiprojective curve. For simplicity of calculation, let us take the easiest case which is $X:= {{\mathbb P}}^1$ and $D:= \{ 0 , \infty \} $. Let $U:=X-D$ and fix $x= 1$ as basepoint in $X$ or $U$. Then $\pi _1(U, x)\cong {{\mathbb Z}}^{\perp}$. A choice of $q=\sqrt{-1}\in Q\subset {{\mathbb C}}$ yields a choice of generator $\gamma _0(q)\in \pi _1(U, x)$ going once around the origin, counterclockwise if $1$ is pictured to the right of the origin and $q$ is pictured above the origin. Changing the choice of $q$ changes the generator to its inverse, which is why we get ${{\mathbb Z}}^{\perp}$ rather than ${{\mathbb Z}}$. For the local monodromy transformations, this yields $$\pi _1(U,x) ^{\perp} \cong {{\mathbb Z}}$$ with a distinguished generator denoted $\gamma _0$. Let $z$ denote the standard coordinate on $X$. A logarithmic $\lambda$-connection on the trivial bundle $E:= {{\mathcal O}}_X$ is of the form $$\nabla = \lambda d + \alpha \frac{dz}{z}.$$ In particular, we can write $$M_{\rm Hod}(X,\log D) = {{\mathbb A}}^1\times {{\mathbb A}}^1 = \{ (\lambda , \alpha )\} .$$ The first coordinate is the parameter $\lambda$ and the second, the residue parameter $\alpha$. For $\lambda \neq 0$ a point $(\lambda , \alpha )$ corresponds to the $1$-connection $d+ \lambda ^{-1}\alpha (dz/z)$. Let $\rho : \pi _1(U,x)\rightarrow {{{\mathbb G}}_m}$ be its monodromy representation. The local monodromy transformation at the origin (see §\[sub-localmonodromy\] above) is $$\rho ^{\perp} (\gamma _0) = {\rm exp}^{\perp}(\alpha /\lambda ) = (\cos (2\pi \alpha /\lambda ), \sin (2\pi \alpha /\lambda ))\in {{{\mathbb G}}_m}^{\perp}.$$ In order to write the global monodromy, note that any loop $\gamma \in \pi _1(U,x)$ can be expressed as a unique function $$\gamma : [0,1]\rightarrow U$$ such that $|\gamma (t)|=1$ and $\gamma$ proceeds at a uniform speed i.e. $\left| \frac{d\gamma}{dt} \right|$ is constant. The function $\gamma$ is then real analytic and extends by analytic continuation to a unique map $\gamma : {{\mathbb C}}\rightarrow U$. In usual terms choosing $i=\sqrt{-1}\in {{\mathbb C}}$, the path $\gamma$ is written as $t\mapsto e^{2\pi i t}$ and this expression is valid for any $t\in {{\mathbb C}}$. The global monodromy of our differential equation $d+ \lambda ^{-1}\alpha (dz/z)$ can now be expressed by the formula $$\rho (\gamma ) = \gamma (\alpha /\lambda ).$$ Next, note that $\overline{X} = {{\mathbb P}}^1 - \{ 0,\infty \}$ too, and $\overline{x} = 1$ still, so we can write $(\overline{X},\overline{D},\overline{x}) = (X,D,x)$. In terms of this equality, $\varphi$ is just the geometric operation of complex conjugation on $U^{\rm top}$. Hence, for any $\gamma \in \pi _1(U,x)$, the complex conjugation map $\varphi$ takes $\gamma$ to the loop $$\varphi (\gamma ) = \overline{\gamma} = \gamma ^{-1}.$$ Computation of the Deligne glueing {#sub-computation} ---------------------------------- We also have $$M_{\rm Hod}(\overline{X},\overline{D}) = {{\mathbb A}}^1\times {{\mathbb A}}^1 = \{ (\lambda , \alpha )\}$$ via the identification between $X$ and $\overline{X}$. In order to compute the Deligne glueing map ${\bf d}$, suppose we start with a point $(\lambda , a)$ in $M_{\rm Hod}(X,\log D)$. This corresponds to a monodromy representation $\gamma \mapsto \gamma (\alpha /\lambda ) $ as explained above. The fact that $\varphi$ interchanges $\gamma$ and $\gamma ^{-1}$ means that, after re-identifying $\overline{X}$ with $X$, the image of this representation by $\varphi^{\ast}$ is $$\gamma \mapsto \gamma ^{-1}(\alpha /\lambda ).$$ There is a unique way to lift this to a logarithmic connection, if we want to send the point $\alpha=0$ to the point $\alpha=0$ and keep everything continuous: it is the connection $\Phi = d-\lambda ^{-1}\alpha (dz/z)$. Finally, in the prescription for the Deligne glueing we set $\mu := \lambda ^{-1}$ and transform this to a $\mu$-connection $\mu \Phi$. This yields the point $${\bf d}(\lambda , \alpha ) = (\mu , \beta ) = (\lambda ^{-1}, -\lambda ^{-2}\alpha ).$$ We can now glue the two charts to get the Deligne-Hodge twistor space: $$M_{\rm DH}(X,\log D) := M_{\rm Hod}(X,D) \sqcup ^{{\bf d}} M_{\rm Hod}(\overline{X},\log \overline{D}).$$ The weight two property {#sub-weighttwo} ----------------------- Notice that the glueing map is linear in $\alpha$, so in this case the result is a vector bundle over ${{\mathbb P}}^1$, in fact it is clearly the bundle ${{\mathcal O}}_{{{\mathbb P}}^1}(2)$ with glueing function $-\lambda ^{-2}$. The minus sign will have an effect on the antilinear involution below. Suppose $\lambda \mapsto P(\lambda )$ is a polynomial considered as a section of $M_{\rm Hod}(X,\log D)$. Then its graph is the set of points $(\lambda , P(\lambda ))$ and these correspond to points of the form $(\lambda ^{-1}, -\lambda ^{-2}P(\lambda ))$ in $M_{\rm Hod}(\overline{X},\overline{D})$ for $\lambda$ invertible. Taking the closure over $\mu = \lambda ^{-1}\rightarrow 0$ yields the set of points of the form $(\mu , -\mu ^2P(\mu ^{-1}))$. This is a holomorphic section in the $\mu$ chart if and only if $P$ is a polynomial of degree $\leq 2$. This is one way to see that $M_{\rm DH}(X,\log D) \cong {{\mathcal O}}_{{{\mathbb P}}^1}(2)$. The global sections are those which are, in the standard chart $M_{\rm Hod}(X,\log D)$, polynomials of degree $\leq 2$. The main point of the title of this paper is that, since this bundle has slope $2$, it corresponds in some sense to a Hodge structure of weight $2$. That contrasts with the normal bundle of a preferred section in the compact case (§\[sec-preferred-twistor\]), which has slope $1$. The weight two behavior is to be expected in the present situation, by analogy with the usual mixed Hodge theory of open varieties, where $H^1({{\mathbb P}}^1- \{ 0, \infty \} )$ has a pure Hodge structure of weight two. In the present case, the Deligne-Hitchin space $M_{\rm DH}(X,\log D)$ will again have an involution $\sigma$ to be calculated below. Given that it is a line bundle of slope $2$, the space of sections has dimension $3$: $$\Gamma ({{\mathbb P}}^1, M_{\rm DH}(X,\log D)) \cong \Gamma ({{\mathbb P}}^1, {{\mathcal O}}_{{{\mathbb P}}^1}(2))\cong {{\mathbb C}}^3,$$ and the $\sigma$-invariant sections are a real form $$\Gamma ({{\mathbb P}}^1, M_{\rm DH}(X,\log D))^{\sigma} \cong {{\mathbb R}}^3.$$ The map from here to any one of the fibers over $\lambda \in {{\mathbb P}}^1$ will be surjective but have a real one-dimensional kernel. It turns out that this real kernel corresponds to the real parameter involved in a parabolic structure, even though we have seen the existence of this additional real parameter without refering [*a priori*]{} to the notion of parabolic structure. The antipodal involution {#sub-antipodal} ------------------------ We now calculate explicitly the involution $\sigma$. Recall that it is a product of the three involutions $C$, $D$ and $N$. The duality involution is trivial on the underlying bundles because we are using the trivial bundle: $E^{\ast} = {{\mathcal O}}^{\ast} = {{\mathcal O}}= E$. The connection on $E\otimes E^{\ast}$ should be trivial so we see that for $\nabla = d + \alpha (dz/z)$ the dual connection is $\nabla ^{\ast} = d - \alpha (dz/z)$. Thus $$D(\lambda , \alpha ) = (\lambda , -\alpha ).$$ Similarly, by definition $$N(\lambda , \alpha ) = (-\lambda , -\alpha ).$$ Putting these together gives $DN(\lambda , \alpha ) = (-\lambda , \alpha )$. These are expressed within a single chart $M_{\rm Hod}(X,\log D)$. On the other hand, the involution $C$ goes from the chart $M_{\rm Hod}(X,\log D)$ to the chart $M_{\rm Hod}(\overline{X},\overline{D})$, and with respect to these charts it is given by $$C(\lambda , \alpha ) = (\overline{\lambda}, \overline{\alpha }).$$ For $\lambda$ invertible we would like to put this back in the original chart. Recall that a point of the form $(\mu , \beta )$ in the chart $M_{\rm Hod}(\overline{X},\overline{D})$ corresponds to $(\mu ^{-1}, -\mu ^{-2}\beta )$ in the chart $M_{\rm Hod}(X,\log D)$. Thus, within the same chart $M_{\rm Hod}(X,\log D)$ and for $\lambda$ invertible, the involution $C$ can be expressed as $$C(\lambda , \alpha ) = (\overline{\lambda}^{\, -1}, -\overline{\lambda}^{\, -2}\overline{\alpha }).$$ Putting these together gives our expression for $\sigma = CDN$ again within the original chart and for $\lambda$ invertible: $$\sigma (\lambda , \alpha ) = (-\overline{\lambda}^{\, -1}, -\overline{\lambda}^{\, -2}\overline{\alpha }).$$ We would now like to calculate which are the $\sigma$-invariant sections. Recall from §\[sub-computation\] that a global section of $M_{\rm DH}(X,\log D)$ is, in the first chart, a polynomial $P$ of order $\leq 2$. Thus we can write our section as $$P:\lambda \mapsto (\lambda , a_0 + a_1\lambda + a_2\lambda ^2).$$ Its graph is the set of image points. The transformed section $P^{\sigma}$ has graph which is the closure of the set of points of the form $$\sigma P(\lambda ) = \left( -\overline{\lambda}^{\, -1}, -\overline{\lambda}^{\, -2}\overline{(a_0 + a_1\lambda + a_2\lambda ^2)} \right) \\ = \left( -\overline{\lambda}^{\, -1}, -\overline{a_2} - \overline{a_1}\overline{\lambda}^{\, -1} -\overline{a_0}\overline{\lambda}^{\, -2} \right) .$$ Substituting in the above expression $-\overline{\lambda}^{\, -1}$ by $t$, the graph becomes the set of points of the form $$(t,-\overline{a_2} + \overline{a_1}t - \overline{a_0}t^2).$$ This is the graph of the polynomial $t\mapsto -\overline{a_2} + \overline{a_1}t - \overline{a_0}t^2$. Thus, writing our polynomials generically with a variable $u$ we can write $$\label{eq-sigma} \left( a_0 + a_1u + a_2 u^2 \right) ^{\sigma} = \left( -\overline{a_2} + \overline{a_1}u - \overline{a_0}u^2 \right) .$$ The $\sigma$-invariant sections are the polynomials with $a_2 = - \overline{a_0}$ and $a_1 = \overline{a_1}$. In other words, an invariant section corresponds to a pair $(a,\alpha )\in {{\mathbb R}}\times {{\mathbb C}}\cong {{\mathbb R}}^3$ with the formula $$\label{like-mochizuki} P(\lambda = \psi (a,\alpha )(\lambda ) := \alpha - a \lambda - \overline{\alpha} \lambda ^2.$$ The reader will recognize this as the formula from Mochizuki [@Mochizuki 2.1.7, p. 25]. Gauge transformations {#sub-gauge} --------------------- A logarithmic connection is not uniquely determined by its monodromy representation. This situation is complicated in higher rank, but is understood easily in our case from the fact that the monodromy associated to a $\lambda$-connection $(\lambda , \alpha )$ is ${\rm exp}^{\perp}( \alpha /\lambda )$, or with a choice of $i=\sqrt{-1}$ it is $e^{2\pi i\alpha /\lambda }$. If we replace $\alpha $ by $\alpha -k \lambda$ for $k\in {{\mathbb Z}}$ we get the same monodromy representation. This process may be viewed as making the meromorphic gauge transformation $v\mapsto z^{-k} v$ on the bundle $E= {{\mathcal O}}_X$, or equivalently changing the bundle $E$ to $E(kD_0 - kD_{\infty})$ where $D_0=\{ 0\}$ and $D_{\infty} = \{ \infty \}$. The zeros or poles of the gauge transformation at points of $D$ change the residues of the $\lambda$-connection by integer multiples of $\lambda$. The sign here and in the definition of $\psi (a,\alpha )$ comes from the formula . In terms of our space $M_{\rm Hod}(X,\log D)$ we have an action of ${{\mathbb Z}}$ obtained by letting $k\in {{\mathbb Z}}$ act as $(\lambda , \alpha )\mapsto (\lambda , \alpha - \lambda k)$. This action extends to the other chart, and gives an action of ${{\mathbb Z}}$ on $M_{\rm Hod}(X,\log D)$. Over ${{{\mathbb G}}_m}\subset {{\mathbb P}}^1$ the action is discrete and the quotient is ${{{\mathbb G}}_m}\times {{\mathbb C}}/ {{\mathbb Z}}= {{{\mathbb G}}_m}\times M_B(U)$. Note that the action degenerates to a trivial action on the fibers over $0$ and $\infty$, the quotients of these actions are trivial $B{{\mathbb Z}}$-gerbs over $M_{\rm Dol}(X,\log D)$ and $M_{\rm Dol}(\overline{X},\log \overline{D})$. The ${{\mathbb Z}}$ action respects the involution $\sigma$ so it gives an action on the space of $\sigma$-invariant sections. In terms of the previous formulae, this clearly acts on the degree $1$ term in the polynomials, or in terms of the coordinates $(a,\alpha )\in {{\mathbb R}}\times {{\mathbb C}}$ it acts with generator $(1,0)$. Thus, we can write $$\frac{\Gamma ({{\mathbb P}}^1, M_{\rm DH}(X,\log D))^{\sigma}}{{{\mathbb Z}}} \cong \frac{{{\mathbb R}}\times {{\mathbb C}}}{(1,0)\cdot {{\mathbb Z}}}.$$ Given a harmonic bundle on $U$ we get a $\sigma$-invariant preferred section, $a$ is the parabolic weight of the Higgs bundle and $\alpha$ is the residue of the Higgs field, see Theorem \[harmoniccompose\] below. We recover in this way the space of possible residues of parabolic $\lambda$-connections, with the residue of the $\lambda$-connection being given by the previous formula . Note that the action of ${{\mathbb Z}}$ is by meromorphic gauge transformations, so moving the parabolic index once around the circle induces an elementary transformation of the bundle. The Tate twistor structure {#sec-tatetwistor} ========================== Before getting to the general rank one case, we investigate the structures associated to the bundle ${{\mathcal O}}_{{{\mathbb P}}^1}(2)$ which occurs above. Recall that $T{{\mathbb P}}^1 \cong {{\mathcal O}}_{{{\mathbb P}}^1}(2)$. Furthermore, the sign $-\lambda ^2$ which occurs in the glueing function for residues of points in $M_{\rm DH}(X,\log D)$ is the same as in the glueing function for $T{{\mathbb P}}^1$. The Tate motive is a pure Hodge structure of type $(1,1)$ hence weight $2$. In view of this, we define the [*additive Tate twistor structure*]{} to be the bundle $$T(1):= T{{\mathbb P}}^1 ,$$ with its natural antilinear involution $$\sigma _{T(1)} := \sigma _{T{{\mathbb P}}^1}.$$ See also Mochizuki [@Mochizuki §3.10.2] and Sabbah [@Sabbah §2.1.3]. On the other hand, we define the [*logarithmic Tate twistor structure*]{} to be the same bundle $T(1,\log ):= T{{\mathbb P}}^1$, but here the antilinear involution should have a sign change with respect to the natural one on $T{{\mathbb P}}^1$, $$\sigma _{T(1,\log )} := - \sigma _{T{{\mathbb P}}^1}.$$ The reason for the difference between these two will be explained below. Integer subgroups {#sub-tate-integer} ----------------- The action of ${{{\mathbb G}}_m}$ on ${{\mathbb P}}^1$ preserving $0$ and $\infty$ gives an action of ${{{\mathbb G}}_m}$ on $T{{\mathbb P}}^1$. The derivative of this action is a section of the tangent bundle, defining the integer subgroup ${{\mathbb Z}}\times {{\mathbb P}}^1 \subset T{{\mathbb P}}^1$. This section is antipreserved by the standard involution $\sigma _{T{{\mathbb P}}^1}$. For the additive Tate twistor structure, we therefore use the imaginary version of this integer subgroup, the set of integer multiples of $\pm 2\pi \sqrt{-1}$ denoted $${{\mathbb Z}}(1)\cong {{\mathbb Z}}^{\perp} \subset \Gamma ({{\mathbb P}}^1,T(1))^{\sigma}.$$ For the logarithmic Tate twistor structure, we use the integer subgroup itself $${{\mathbb Z}}(1,\log ):= {{\mathbb Z}}\subset \Gamma ({{\mathbb P}}^1,T(1,\log ))^{\sigma}.$$ Take for generator of ${{\mathbb Z}}(1,\log )$ the vector field $-\lambda \frac{\partial}{\partial \lambda}$ in the standard chart ${{\mathbb A}}^1$. This minus sign comes from the formula , see also §\[sub-gauge\] above. If we go into the other chart then the generator changes sign once again. See the paragraph above equation for a reflection of this sign change in the action of the gauge group on the complex conjugate chart. The sign change for $\sigma$ on $T(1,\log )$ guarantees that the integer sections are preserved by $\sigma$. The action of ${{{\mathbb G}}_m}$ on the additive Tate twistor structure $T(1)$ corresponds to the usual Tate Hodge structure of type $(1,1)$ with its integral subgroup ${{\mathbb Z}}(1)$. Define the [*multiplicative Tate twistor structure*]{} to be $$\label{gm1} {{{\mathbb G}}_m}(1) := T(1,\log ) / {{\mathbb Z}}(1,\log ).$$ The fiber over $\lambda = 1$, the de Rham version, is naturally identified with ${{\mathbb C}}/ {{\mathbb Z}}$. The exponential map gives the isomorphism $${\rm exp}^{\perp} : {{\mathbb C}}/ {{\mathbb Z}}\stackrel{\cong}{\rightarrow} {{{\mathbb G}}_m}^{\perp}.$$ This explains why we needed to change the sign of $\sigma$ for the logarithmic Tate structure: the exponential map interchanges objects before and after $(\; )^{\perp}$. Thus, to get a Deligne-Tate type twist on the multiplicative group corresponding to the local monodromy operator of a connection, we need to undo this twist which occurs naturally in $T(1)=T{{\mathbb P}}^1$. Over $\lambda = 0$ and $\lambda = \infty$, the quotient defining ${{{\mathbb G}}_m}(1)$ is to be taken in the stack sense. Hence, $${{{\mathbb G}}_m}(1)_{\rm Dol} = {{\mathbb C}}\times B{{\mathbb Z}}.$$ The antipodal involution in the additive case {#sub-tangent-antipodal} --------------------------------------------- Start by computing the natural antipodal involution $\sigma _{T{{\mathbb P}}^1}$ of the tangent bundle or equivalently the additive Tate structure $T(1)$. For this subsection, the notation $\sigma$ represents $\sigma _{T{{\mathbb P}}^1}= \sigma _{T(1)}$. The vector field $\lambda \frac{\partial}{\partial \lambda}$ goes radially outward from $0$ towards $\infty$. Up to a scalar it is the unique vector field with zeros at $0$ and $\infty$, and this property is preserved by $\sigma$. Geometrically we see that the antipodal involution changes the sign of this radial vector field. Acting on this section considered as a section of $T{{\mathbb P}}^1$ we get $$\sigma ^{\ast}\lambda \frac{\partial}{\partial \lambda} = -\lambda \frac{\partial}{\partial \lambda} .$$ Similar geometric consideration shows that $\sigma$ interchanges the vector fields $\frac{\partial}{\partial \lambda}$ and $\lambda ^2\frac{\partial}{\partial \lambda}$, this time with no sign change. Thus $$\sigma ^{\ast}\frac{\partial}{\partial \lambda} = \lambda ^2\frac{\partial}{\partial \lambda}$$ and vice-versa. A point of the total space of the tangent bundle, in the first standard chart, has the form $ \left( \lambda , v\frac{\partial}{\partial \lambda} \right)$. We know that $\sigma$ acts on the first coordinate by sending $\lambda$ to $-\overline{\lambda} ^{\, -1}$. Thinking of the above sections as corresponding to their graphs which are sets of points, and noting that $\sigma$ is antilinear in the coordinate $v$, the formula for $\sigma$ on points of the total bundle is $$\sigma \left( \lambda , b\frac{\partial}{\partial \lambda} \right) = \left( -\overline{\lambda} ^{\, -1} , \overline{\lambda} ^{\, -2}\overline{v} \frac{\partial}{\partial \lambda} \right) .$$ The antipodal involution in the logarithmic or multiplicative case {#sub-tate-antipodal} ------------------------------------------------------------------ Recall that $\sigma _{T(1,\log )}= - \sigma _{T{{\mathbb P}}^1}$, with the minus sign acting only in the bundle fiber direction. Hence, for $\sigma = \sigma _{T(1,\log )}$ the formulae from the previous section become $$\sigma \left( \lambda , v\frac{\partial}{\partial \lambda} \right) = \left( -\overline{\lambda} ^{\, -1} , -\overline{\lambda} ^{\, -2}\overline{v} \frac{\partial}{\partial \lambda} \right) ,$$ and $$\sigma ^{\ast}(u+v\lambda + w\lambda ^2)\frac{\partial}{\partial \lambda} = (-\overline{w}+ \overline{v}\lambda -\overline{u}\lambda ^2) \frac{\partial}{\partial \lambda} .$$ This fits with the formula : a $\sigma$-invariant section has the form $$\psi (a,\alpha ) = \lambda \mapsto (\alpha - a \lambda - \overline{\alpha} \lambda ^2)\frac{\partial}{\partial \lambda}$$ with $\alpha \in {{\mathbb C}}$ and $a\in{{\mathbb R}}$. On the quotient ${{{\mathbb G}}_m}(1) = T(1,\log )/{{\mathbb Z}}(1,\log ) $ we get the involution $\sigma _{{{{\mathbb G}}_m}(1)}$. The space of invariant sections {#sub-invariant} ------------------------------- The space of $\sigma$-invariant sections of $T(1,\log )$ inherits some canonical structure. For any point $p\in {{\mathbb P}}^1$ we get a distinguished $\sigma$-invariant direction in $\Gamma ({{\mathbb P}}^1,T(1,\log ))$: the sections having simple zeros at $p$ and $\sigma (p)$. This space $\Gamma ({{\mathbb P}}^1, T(1,\log )(-p-\sigma (p)))^{\sigma}$ is naturally identified with the Lie algebra of the one parameter group of $\sigma$-antipreserving homotheties of ${{\mathbb P}}^1$ which fix $p$ and $\sigma (p)$. One must say “antipreserving” here because of the sign change in $\sigma _{T(1,\log )}$. The elements of this group are radial homotheties; the group is isomorphic to ${{{\mathbb G}}_m}({{\mathbb R}})$ and its Lie algebra is isomorphic to ${{\mathbb R}}$. In particular, there is a distinguished generator which is the vector field going inward from $\sigma (p)$ towards $p$ attaining speed $1$ at the equator between the two fixed points. Equivalently, we can require that the expansion factor at $p$ be equal to $-1$. This is normalized so that when $p=0$ it gives the generator of ${{\mathbb Z}}(1,\log )$. The expansion factor of a vector field with a zero, is a well-defined complex number: it is dual to the residue, and can be defined as the value of the vector field on the differential form $\frac{dz}{z}$. Let $\nu ^p\in \Gamma ({{\mathbb P}}^1, T(1,\log )(-p-\sigma (p)))^{\sigma}$ denote the generator normalized to have expansion factor $-1$ at $p$. We get a canonical isomorphism $$\Gamma ({{\mathbb P}}^1, T(1,\log )(-p-\sigma (p)))^{\sigma} \cong {{\mathbb R}}, \;\;\; \nu _p \mapsto 1 .$$ Evaluating at $p$ gives a map ${\rm ev}_p$ from the space of sections to the fiber $T(1,\log )_p$. \[exactatp\] These maps fit into a canonical exact sequence depending on $p\in{{\mathbb P}}^1$, $$0\rightarrow {{\mathbb R}}\stackrel{\nu_p}{\rightarrow} \Gamma ({{\mathbb P}}^1,T(1,\log ))^{\sigma} \stackrel{{\rm ev}_p}{\rightarrow} T(1,\log )_p \rightarrow 0.$$ It is exact in the middle because ${{\mathbb R}}\cdot \nu _p$ is exactly the space of sections vanishing at $p$. Exactness on the left and right follow by dimension count. The residue evaluation {#sub-reseval} ---------------------- The standard translation action of ${\mathbb G}_a$ on ${{\mathbb P}}^1$ fixing the point $\infty$ gives a trivialization $$T(1,\log )|_{{{\mathbb A}}^1}\cong {{\mathcal O}}.$$ For any point $p\in {{\mathbb A}}^1$, let ${\rm res}_p$ denote the composition of this trivialization at $p$, with the evaluation map ${\rm ev}_p$. Then the exact sequence \[exactatp\] can be written $$\label{eq-exactatp} 0\rightarrow {{\mathbb R}}\stackrel{\nu_p}{\rightarrow} \Gamma ({{\mathbb P}}^1,T(1,\log ))^{\sigma} \stackrel{{\rm res}_p}{\rightarrow} {{\mathbb C}}\rightarrow 0.$$ As calculated above, the $\sigma$-invariant sections are identified with the polynomials of the form $$\psi (a,\alpha )= \left( \alpha - a \lambda - \overline{\alpha} \lambda ^2\right) \frac{\partial}{\partial \lambda}$$ with $a\in {{\mathbb R}}$ and $\alpha \in {{\mathbb C}}$. The residue evaluation at $p$ is $$\label{for-reseval} {\rm res}_p(\psi (a,\alpha )) = \alpha - ap -\overline{\alpha}p^2.$$ Notice that ${\rm res}_p(\psi (a_p,\alpha _p))=0$, since $\nu _p$ is a section vanishing at $p$. The generator $\nu _p$ {#sub-onedim} ---------------------- Suppose given a point $p\in {{\mathbb A}}^1$. In coordinates, $\sigma (p) = -\overline{p}^{\, -1}$. The line of $\sigma$-invariant sections which vanish to first order at $p$ is $$\alpha - a p - \overline{\alpha} p^2 = 0.$$ Vanishing at $\sigma (p)$ is a consequence, because a $\sigma$-invariant section vanishing at $p$ also has to vanish at $\sigma (p)$. Recall that $a\in {{\mathbb R}}$. We get a real one-dimensional space of solutions generated for example by $(a,\alpha )$ with $$a = 1- |p|^2, \;\;\; \alpha = p.$$ Then $$\psi (a,\alpha ) (\lambda )= \left( p + (|p|^2-1)\lambda - \overline{p}\lambda ^2\right) \frac{\partial}{\partial \lambda}.$$ Let us calculate the expansion factor at $p$ of the vector field corresponding to our section $\psi (a_1,\alpha _1)$. For this, express the vector field in the form $$\psi (a,\alpha ) = \eta \cdot (\lambda -p)\frac{\partial}{\partial\lambda} + \ldots$$ where the $\ldots $ signify higher order terms at $p$. The constant $\eta$ is the expansion factor. We have $$\eta = \frac{d}{d\lambda} \left. \left( p + (|p|^2-1)\lambda - \overline{p}\lambda ^2 \right) \right| _{\lambda = p}$$ $$= (|p|^2 -1 - 2\overline{p}\lambda )|_{\lambda = p} = -(1+|p|^2)$$ We can normalize to get the canonical generator $\nu _p = \psi (a_p,\alpha _p)$ whose expansion factor is $-1$: $$\label{generator-one} a_p = \frac{1-|p|^2}{1+|p|^2}, \;\;\; \alpha _p = \frac{p}{1+|p|^2}.$$ For $p=1$ it is the generator $(1,0)$ of ${{\mathbb Z}}(1,\log )$. A natural inner product {#sub-innerprod} ----------------------- For all the above vectors , we have $a_p^2 + 4|\alpha _p|^2 = 1$. The equation $a^2 + 4|\alpha |^2 = 1$ defines an $S^2\subset {{\mathbb R}}\times {{\mathbb C}}$, and the function $p\mapsto (a_p,\alpha _p)$ provides an isomorphism between ${{\mathbb P}}^1$ and this $S^2$. This is the unit sphere for the inner product $$\label{scalarprod} (a,\alpha )\cdot (b,\beta):= ab + 2( \alpha \overline{\beta} + \overline{\alpha}\beta ).$$ \[so3-innerprod\] The group $SO(3)$ acts naturally as the group of $\sigma$-intertwining complex automorphisms of ${{\mathbb P}}^1$, so it acts on the space of sections $\Gamma ({{\mathbb P}}^1,T(1,\log ))^{\sigma}$. The above inner product is the $SO(3)$-invariant one, unique up to a scalar. Any $\sigma$-intertwining automorphism $f$ of ${{\mathbb P}}^1$ acts on $T(1,\log )$ because that bundle is naturally defined as the tangent bundle with a twisted $\sigma$. Hence it acts on the space of $\sigma$-invariant sections. The section $\nu _p$ is canonically defined depending on the point $p$ and the involution $\sigma$. Hence $f_{\ast}$ takes $\nu _p$ to $\nu _{f(p)}$. It follows that the action of $f$ preserves the sphere $S^2$ image of the map $\nu$. Therefore $f$ is in the orthogonal group $O(3)$ for this scalar product. It has determinant $1$ because of holomorphicity. We get $f\in SO(3)$, the special orthogonal group for the scalar product . The parabolic weight function {#sub-parabolicweight} ----------------------------- The space of $\sigma$-invariant sections of $T(1,\log )$ is an ${{\mathbb R}}^3$, and at any $p\in {{\mathbb P}}^1$ it is naturally an extension of $T(1,\log )_p$ by ${{\mathbb R}}$ (Lemma \[exactatp\]). The quotient $T(1,\log )_p$ represents the residue of a $\lambda$-connection. The extra real parameter corresponds to the real parabolic weight of a parabolic structure. However, we need to discuss the normalization of this identification splitting the exact sequence. We use the coordinates $a,\alpha$ for the set of invariant sections denoted $\psi (a,\alpha )$, giving an isomorphism between this space and ${{\mathbb R}}\times {{\mathbb C}}$. The point $(1,0)\in {{\mathbb R}}\times {{\mathbb C}}$ is the generator of the subgroup ${{\mathbb Z}}(1, \log )$. For each $p$ we have the point $\nu _p = \psi (a_p,\alpha _p)$ given by . The quotient of ${{\mathbb R}}\times {{\mathbb C}}$ by the line generated by $(a_p,\alpha _p)$, is the space of residues at $p$. We would like to define a [*parabolic weight function*]{} $\varpi _p : {{\mathbb R}}\times {{\mathbb C}}\rightarrow {{\mathbb R}}$, depending on the point $p$, such that $\varpi _p(1,0) = 1$ for compatibility with local gauge transformations; and $\varpi _p(a_p,\alpha _p) \neq 0$ so that the local residue map is an isomorphism between $\ker (\varpi _p)$ and ${{\mathbb C}}$. Using the inner product $(a,\alpha )\cdot (a',\alpha ') = aa' + 2( \alpha \overline{\alpha}' + \overline{\alpha}\alpha ')$, the simplest thing to do is to let $\varpi _p$ be given by the inner product with the average of the two vectors $(1,0)$ and $(a_p,\alpha _p)$, then normalize to get $\varpi _p(1,0)=1$. This is $$\begin{aligned} \label{for-paraweight} \varpi _p(a,\alpha )& := &\frac{(a, \alpha )\cdot (1,0) + (a,\alpha )\cdot (a_p,\alpha _p)}{(1,0)\cdot (1,0) + (1,0)\cdot (a_p,\alpha _p)} \nonumber \\ & = & \frac{(1+|p|^2)a + (1-|p|^2)a + 2 (\alpha \overline{p} + \overline{\alpha}p)}{(1+|p|^2) + (1-|p|^2)} \nonumber \\ & =& a + \alpha \overline{p} +\overline{\alpha}p .\end{aligned}$$ The parabolic weight and residue functions are the same as Mochizuki’s functions ${\mathfrak p}$ and ${\mathfrak e}$ of [@Mochizuki §2.1.7], however we have preferred to motivate their introduction independently above. \[prop-parabolicweight\] For any point $p\in {{\mathbb A}}^1$, the parabolic weight function and the residue give an isomorphism $$(\varpi _p, {\rm res}_p): \Gamma ({{\mathbb P}}^1,T(1,\log ))^{\sigma} \stackrel{\cong}{\rightarrow} {{\mathbb R}}\times {{\mathbb C}}.$$ Let ${{\mathbb Z}}= {{\mathbb Z}}(1,\log )$ act on ${{\mathbb R}}\times {{\mathbb C}}$ with generator $(1,-p)$ in keeping with . Then the above isomorphism descends to the quotient to give $$(\varpi _p, {\rm res}_p): \frac{\Gamma ({{\mathbb P}}^1,T(1,\log ))^{\sigma}}{{{\mathbb Z}}(1,\log )} \stackrel{\cong}{\rightarrow} \frac{{{\mathbb R}}\times {{\mathbb C}}}{(1,-p){{\mathbb Z}}}.$$ In terms of the coordinates $(a, \alpha )$ given by the construction $\psi$, we have $$(\varpi _p, {\rm res}_p)(\psi (a,\alpha )) = (a + \alpha \overline{p} +\overline{\alpha}p, \alpha - ap -\overline{\alpha}p^2).$$ We have chosen $\varpi _p$ so that $\varpi _p (\nu _p) = 1$. From the exact sequence of Lemma \[exactatp\], this implies that $\varpi _p$ and ${\rm res}_p$ are linearly independent so by dimension count we get the first isomorphism. For the second part, it suffices to recall that the subgroup ${{\mathbb Z}}(1,\log )\subset \Gamma ({{\mathbb P}}^1,T(1,\log ))^{\sigma}$ is given by generator $\psi (1,0)$, and to note that $$(\varpi _p,{\rm res}_p)(\psi (1,0)) = (1, -p).$$ The formula in terms of coordinates just recalls our calculations above. Some questions {#sub-questions} -------------- The residue of a logarithmic $\lambda$-connection at a singular point lies in the fiber $T(1,\log )_{\lambda}$. This was seen by direct calculation: using the standard frame for $T(1,\log )$ over ${{\mathbb A}}^1$ and the expression of the residue as a well-defined complex number, we obtain this identification over the chart ${{\mathbb A}}^1$. Then by calculation, it is compatible with the corresponding identification for $\overline{X}$ over the chart at infinity, using the Riemann-Hilbert correspondence. This passage by an explicit calculation is unsatisfactory but I haven’t seen any way of improving it, so we formulate a question: \[quest-twistorp1\] Is there some more natural geometric way of identifying the residue of a logarithmic $\lambda$-connection at a singular point, with a tangent vector to the $\lambda$-line? One possible approach would be to give a geometric description of the meaning of points in the twistor line ${{\mathbb P}}^1$. Similarly, we picked the definition of the parabolic weight function “out of the hat”. Taking the scalar product with the average of the two vectors, then normalizing, is certainly the easiest way to make sure that the function takes on nonzero values on the two vectors, furthermore the resulting formula for $\varpi _p$ is relatively simple. Nonetheless, it would be better to have a more motivated reason for this choice. \[quest-paraweight\] Is there a geometric interpretation of the meaning of the parabolic weight function, preferably going with the geometric interpretation we are looking for in Question \[quest-twistorp1\]? Another direction of questions is the relationship with $SO(3)$. The group of $\sigma$-invariant automorphisms of ${{\mathbb P}}^1$ is the group of metric automorphisms of $S^2$, in other words it is $SO(3)$ by Lemma \[so3-innerprod\]. The space of $\sigma$-invariant sections of $T(1,\log )$, which are the $\sigma$-antiinvariant sections of the tangent bundle, may be identified with the perpendicular of the Lie algebra ${\mathfrak s}{\mathfrak o}(3)^{\perp}$. By naturality, $T(1,\log )$ also has an action of $SO(3)$, corresponding to the adjoint action on the Lie algebra. \[quest-so3\] What is the significance of this action of $SO(3)$ and the identification of elements of the space of parabolic weights and residues, with vectors in the Lie algebra? It seems to be one of the subjects of Gukov and Witten’s paper [@GukovWitten]. The general rank one case {#sec-general} ========================= Consider now the following situation: $X$ is a smooth projective variety, and $D\subset X$ is a reduced strict normal crossings divisor written as $D=D_1 + \ldots + D_k$ where $D_i$ are its distinct smooth connected irreducible components. Let $U:= X-D$. The Hodge moduli space {#sec-hodge} ---------------------- Let $M_{\rm Hod}(X,\log D)$ denote the moduli space of triples $(\lambda , L , \nabla )$ where $\lambda \in {{\mathbb A}}^1$, $L$ is a line bundle on $X$ such that $$\label{c1cond} c_1(L)_{{{\mathbb Q}}} \in {{\mathbb Q}}\cdot [D_1] + \cdots + {{\mathbb Q}}\cdot [D_k] \subset H^2(X, {{\mathbb Q}}^{\perp}),$$ and $$\nabla : L\rightarrow L\otimes _{{{\mathcal O}}_X}\Omega ^1_X(\log D)$$ is an integrable logarithmic $\lambda$-connection on $L$. The first coordinate is a map $$\lambda : M_{\rm Hod}(X,\log D)\rightarrow {{\mathbb A}}^1.$$ Let ${\rm res}(\nabla ; D_i )\in {{\mathbb C}}$ denote the residue of $\nabla$ along $D_i$. Recall that the residue is a locally constant function and $D_i$ is connected so it is a complex scalar. For $\lambda \neq 0$, we have $$\label{c1compatible} \lambda c_1(L) = - \sum _i {\rm res}(\nabla ; D_i)\cdot [D_i] \;\; \mbox{in} \;\; H^2(X, {{\mathbb C}}^{\perp}),$$ as can be calibrated by comparing with the gauge transformation formulae and . So the condition about $c_1(L)$ in the definition of $M_{\rm Hod}(X,\log D)$ is automatically satisfied when $\lambda \neq 0$, however for $\lambda = 0$ this condition is nontrivial. Tensor product gives $M_{\rm Hod}(X,\log D)$ a structure of abelian group scheme relative to ${{\mathbb A}}^1$. We need to use the condition about Chern classes to prove that it is smooth over ${{\mathbb A}}^1$, otherwise there would be additional irreducible components lying over $\lambda = 0$. The morphism $M_{\rm Hod}(X,\log D)\rightarrow {{\mathbb A}}^1$ is smooth. [*Proof:*]{} Suppose $\phi : Y\rightarrow M_{\rm Hod}(X,\log D)$ is a morphism from an artinian scheme. Suppose $Y\subset Y'$ is an artinian extension provided with a morphism $\lambda ' : Y'\rightarrow {{\mathbb A}}^1$. We need to extend to $Y'\rightarrow M_{\rm Hod}(X,\log D)$ lifting $\lambda '$. The map $\phi$ corresponds to a line bundle with integrable $\lambda$-connection $(L,\nabla )$ on $X\times Y$. By smoothness of the Picard scheme of $X$, this extends to a line bundle $L'$ on $X\times Y'$. The condition about the Chern class of $L$ implies that there exists some logarithmic connection $\nabla _{1,y}$ on $L_y$ where $y\in Y$ denotes the closed point and $L_y$ is the restriction of $L$ to the fiber over $y$. By smoothness of $M_{DR}(X,\log D)$, which follows because of its group structure under tensor product, we can extend $\nabla _{1,y}$ to some integrable connection $\nabla '_1$ on $L'$. Then $\lambda ' \nabla '_1$ is an integrable $\lambda '$-connection on $L'$. Restricted to $X\times Y$, we can write $$\nabla = \lambda ' \nabla '_1 |_{X\times Y} + A$$ where $$A\in H^0(X\times Y, \Omega ^1_X(\log D)\otimes _{{{\mathcal O}}_X}{{\mathcal O}}_{X\times Y}) \cong H^0(X,\Omega ^1_X(\log D)) \otimes _{{{\mathbb C}}}{{\mathcal O}}_Y .$$ Now extend $A$ in any way to a section $$A' \in H^0(X\times Y', \Omega ^1_X(\log D)\otimes _{{{\mathcal O}}_X}{{\mathcal O}}_{X\times Y'}) \cong H^0(X,\Omega ^1_X(\log D)) \otimes _{{{\mathbb C}}}{{\mathcal O}}_{Y'}$$ and set $$\nabla ' := \lambda ' \nabla '_1 + A'.$$ This provides the required extension. [ $\Box$]{} Gauge group action {#sec-gauge} ------------------ Recall the action of the local meromorphic gauge group ${{\mathcal G}}:= {{\mathbb Z}}^k$ on $M_{\rm Hod}(X,\log D)$. A vector $g = (g _1, \ldots , g _k)$ sends $(\lambda , L , \nabla )$ to $(\lambda , L (g_1D_1 + \cdots + g _kD_k), \nabla ^{\alpha})$ where $\nabla ^{\alpha}$ is the logarithmic $\lambda$-connection on $L (\alpha _1D_1 + \cdots + \alpha _kD_k)$ which coincides with $\nabla$ over $U$, via the isomorphism $$L (g _1D_1 + \cdots + g _kD_k)|_U \cong L|_U.$$ We have $${\rm res}(\nabla ^{g} ; D_i) = {\rm res}(\nabla ; D_i) - \lambda g _i .$$ The vector of residues, viewed in the standard framing $\frac{\partial}{\partial\lambda}$ for $T(1,\log )$, provides a morphism $$R : M_{\rm Hod}(X,\log D)\rightarrow T(1,\log ) ^k$$ where the $i$-th coordinate of $R(\lambda , L , \nabla )$ is by definition ${\rm res}(\nabla ; D_i)\cdot \frac{\partial}{\partial\lambda}$. The morphism $R$ is compatible with the action of ${{\mathcal G}}= {{\mathbb Z}}^k = {{\mathbb Z}}(1,\log )^k$, where $g \in {{\mathbb Z}}^k$ acts on $T(1,\log )^k$ by $$(v_1,\ldots ,v_k)\mapsto (v_1-g_1\lambda \frac{\partial}{\partial\lambda}, \ldots , v_k-g_1\lambda \frac{\partial}{\partial\lambda}),$$ adding $g$ times our standard generator of ${{\mathbb Z}}(1,\log )$. Let $$M_{\rm Hod}(X,\log D)_{{{{\mathbb G}}_m}} := M_{\rm Hod}(X,\log D)\times _{{{\mathbb A}}^1}{{{\mathbb G}}_m}.$$ Then ${{\mathcal G}}$ acts properly discontinuously on $M_{\rm Hod}(X,\log D)_{{{{\mathbb G}}_m}}$ because this action lies over the proper discontinuous action on $T(1,\log )^k_{{{{\mathbb G}}_m}}$ via the map $R$. In the analytic category, we can form the quotient, and the Riemann-Hilbert correspondence gives an isomorphism $$M_{\rm Hod}(X,\log D)_{{{{\mathbb G}}_m}}^{\rm an}/{{\mathcal G}}\cong {{{\mathbb G}}_m}\times M_B(U)$$ where $M_B(U):= Hom (\pi _1(U), {{{\mathbb G}}_m})$. In view of the Riemann-Hilbert correspondence, we define $M_{\rm Hod}(U)$ to be the stack-theoretical quotient $$M_{\rm Hod}(U):= M_{\rm Hod}(X,\log D)/{{\mathcal G}},$$ and similarly for the fibers over $\lambda = 0,1$: $$M_{\rm Dol}(U):= M_{\rm Dol}(X,\log D)/{{\mathcal G}}, \;\;\; M_{\rm DR}(U):= M_{\rm DR}(X,\log D)/{{\mathcal G}}.$$ Note that ${{\mathcal G}}$ acts trivially on $M_{\rm Dol}(X,\log D)$ so the quotient $M_{\rm Dol}(U)$ is a stack with ${{\mathcal G}}$ in the general stabilizer group. If we started with a stack version of $M_{\rm Dol}(X,\log D)$ then the general stabilizer also contains the automorphism group ${{{\mathbb G}}_m}$ of a rank one Higgs bundle. In the rank one case, the stabilizer groups are all the same. So the full stabilizer group of any point of $M_{\rm Dol}(U)$ would be ${{{\mathbb G}}_m}\times {{\mathcal G}}$. Using these definitions, the RH correspondence again says $M_{\rm DR}(U)\cong M_B(U)$, and the Deligne glueing process applies as in §\[sec-glue\] to give an analytic stack $$M_{\rm DH}(U) \rightarrow {{\mathbb P}}^1$$ whose charts are $M_{\rm Hod}(U)$ and $M_{\rm Hod}(\overline{U})$. Note however that these charts don’t have algebraic structures. We would like to investigate how to lift to a Deligne-Hitchin glueing on the space of logarithmic connections, to get an analytic stack $M_{\rm DH}(X,\log D)$ which would have nicer geometric properties—its charts would be the Artin algebraic stacks. We would then have a quotient expression $$M_{\rm DH}(U) = M_{\rm DH}(X,\log D)/{{\mathcal G}}\rightarrow {{\mathbb P}}^1 .$$ One way of looking at this question would be to calculate the fundamental group of $M_{\rm DH}(U)\rightarrow {{\mathbb P}}^1 $ and see if it has a covering which resolves the stackiness over $0$ and $\infty$. Instead, we construct directly the covering. The Riemann-Hilbert correspondence and glueing {#sec-rhglue} ---------------------------------------------- Our goal in this subsection is to define $M_{DH}(X,\log D)$ by Deligne glueing of $M_{Hod}(X,\log D)$ with $M_{Hod}(\overline{X}, \log \overline{D})$. It will be useful to have a Betti version of $M_{DR}(X,\log D)$ to intervene in the glueing. Suppose $\rho \in M_B(U)$. Recall from §\[sub-localmonodromy\] that for each component $D_i$ of the divisor, we get a well-defined local monodromy element $\rho ^{\perp} (\gamma _{D_i})\in {{{\mathbb G}}_m}^{\perp}$. Consider the following diagram: $$M_B(U) \rightarrow ({{{\mathbb G}}_m}^{\perp}) ^k \stackrel{{\rm exp}^{\perp}}{\longleftarrow} {{\mathbb C}}^k,$$ where the $k$ copies are for the $k$ components of the divisor $D= D_1 + \ldots + D_k$, the first map sends $\rho$ to its vector of local monodromy transformations, and $${\rm exp}^{\perp}:(a_1,\ldots , a_k)\mapsto \left( (\cos (2\pi a_1), \sin (2\pi a_1)),\ldots , (\cos (2\pi a_k), \sin (2\pi a_k)) \right) .$$ Let $M_B(X, \log D)$ denote the fiber product. Thus, a point in $M_B(X,\log D)$ is an uple $(\rho ; a_1,\ldots , a_k )$ where $\rho$ is a representation of rank one over $U$ and $a_i\in {{\mathbb C}}$ are choices of circular logarithms of the monodromy operators $\rho ^{\perp}(\gamma _{D_i})\in {{{\mathbb G}}_m}^{\perp}$. Define an action of the gauge group ${{\mathcal G}}= {{\mathbb Z}}^k$ on $M_B(X,\log D)$ by $$g= (g_1,\ldots , g_k): (\rho ; a_1,\ldots , a_k)\mapsto (\rho ; a_1 - g_1, \ldots , a_k - g_k).$$ \[rhlift\] The Riemann-Hilbert correspondence lifts to $$M_{DR}(X,\log D) \cong M_B(X,\log D).$$ [*Proof:*]{} Given a line bundle with integrable connection $(L,\nabla )$, associate the point $$(\rho , {\rm res}(\nabla ;D_1),\ldots , {\rm res}(\nabla ; D_k)) \in M_B(U)\times _{({{\mathbb C}}^{\ast})^k} {{\mathbb C}}^k = M_B(X,\log D).$$ This is a morphism of analytical groups with the same dimension, so it suffices to prove that it is injective and surjective. Before doing that, we verify compatibility with the gauge group action. Given $g= (g_1,\ldots , g_k)\in{{\mathcal G}}$, notice that the monodromy representation of $(L^g,\nabla ^g)$ is the same as $\rho$. For the circular logarithms, the formula for residues $${\rm res}(\nabla ^g; D_i) = {\rm res}(\nabla ; D_i) -g_i$$ implies the required compatibility with the gauge group action. For injectivity, suppose $(L,\nabla )$ and $(L',\nabla ')$ are two line bundles with connection, with the same monodromy representation and the same residues. There is a unique isomrphism $\psi : L|_{U}\cong L'|_{U}$ compatible with the monodromy representation or equivalently with $\nabla$ and $\nabla '$ on $U$. Then, the poles or zeros of $\psi$ along a component $D_i$ are determined by the difference between the residues of $\nabla$ and $\nabla '$. The condition that the residues are the same means that $\psi$ has neither pole nor zero along each component $D_i$. Thus, $\psi$ is an isomorphism of bundles over $X$. For surjectivity, suppose $(\rho , a_1,\ldots , a_k)$ is a point in $M_B(X,\log D)$. Choose a line bundle with logarithmic connection $(L,\nabla )$ inducing the monodromy representation $\rho$ on $U$. Let $a'_1,\ldots , a'_k$ be the residues of $\nabla '$ along the $D_i$. We have $a_i = a'_i - g_i$ with $g_i\in {{\mathbb Z}}$. Now $(L^g, \nabla ^g)$ maps to $(\rho , a_1,\ldots , a_k )$. [ $\Box$]{} We have the conjugation isomorphism $\varphi : U \cong \overline{X}-\overline{D}$. If $\rho$ is a local system on $U$ then we obtain $\varphi _{\ast}(\rho )$ a local system on $\overline{X}-\overline{D}$, defined by $$\varphi _{\ast}(\rho ) (\eta ) := \rho (\varphi ^{-1}\eta ).$$ The divisor $\overline{D}$ breaks up into components $\overline{D}_1 + \ldots + \overline{D}_k$. Let $\gamma _{\overline{D}_i}\in \pi _1(\overline{U},\overline{x})^{\perp}$ denote the local monodromy operator around $\overline{D}_i$. Then we have $$\varphi ^{-1}(\gamma _{\overline{D}_i}) = \gamma _{D_i} ^{-1}$$ because $\varphi $ reverses orientation. Thus, $$\varphi _{\ast}(\rho )^{\perp}(\gamma _{\overline{D}_i}) = \rho ^{\perp}(\gamma _{D_i}) ^{-1}.$$ Given a logarithm $a_i$ of $\rho ^{\perp}(\gamma _{D_i})$, its negative $-a_i$ is a logarithm of $\varphi _{\ast}(\rho )^{\perp}(\gamma _{\overline{D}_i})$. Therefore define the isomorphism $$\varphi : M_B(X,\log D) \stackrel{\cong}{\rightarrow}M_B(\overline{X}, \log \overline{D})$$ by $$\varphi (\rho ; a_1,\ldots , a_k) := (\varphi _{\ast}(\rho ); -a_1, \ldots , -a_k ).$$ Using the Riemann-Hilbert correspondence of Lemma \[rhlift\] we can do the Deligne glueing exactly as before to get a moduli space $$M_{DH}(X,\log D)\rightarrow {{\mathbb P}}^1.$$ The gauge group of meromorphic gauge transformations along the divisors ${{\mathcal G}}= {{\mathbb Z}}^k$ acts on $M_{DH}(X,\log D)$ in the following way. It acts in the canonical way on the first chart $M_{Hod}(X,\log D)$. On the other hand, a point $(g_1,\ldots , g_k)$ acts by the canonical action of $(-g_1, \ldots , -g_k)$ on the chart $M_{Hod}(\overline{X},\log \overline{D})$, because of the sign change in the definition of $\varphi $. The global quotient is the Deligne glueing considered previously: $$\label{dhgauge} M_{DH}(U) = M_{\rm DH}(X,\log D) /{{\mathcal G}}\rightarrow {{\mathbb P}}^1.$$ Over ${{{\mathbb G}}_m}\subset {{\mathbb P}}^1$ this quotient is isomorphic to $M_B(U)\times {{{\mathbb G}}_m}$ so it has a reasonable structure; however near the fibers over $0$ and $\infty$ the quotient is analytically stacky, with ${{\mathcal G}}$ contributing to the stabilizer group. Exact sequences {#sec-exact} --------------- Since we are treating the case $r=1$, our moduli spaces are really just abelian cohomology groups, for example[^1] $$M_B(U) = H^1(U,{{{\mathbb G}}_m}).$$ This may also be interpreted as a Deligne cohomology group, see [@EsnaultViehweg] [@Gajer] for example; we leave to the reader to make the link between our Hodge filtration and the Hodge filtration on Deligne cohomology. The exponential exact sequence for $U=X-D$ is $$0\rightarrow H^1(U,{{\mathbb Z}}^{\perp})\rightarrow H^1(U,{{\mathbb C}}) \rightarrow M_B(U)\rightarrow H^2(U,{{\mathbb Z}}^{\perp})\rightarrow H^2(U,{{\mathbb C}}).$$ The exact sequence for the gauge group action is $$0 \rightarrow {{\mathcal G}}= {{\mathbb Z}}^k \rightarrow M_B(X,\log D) \rightarrow M_B(U)\rightarrow 1.$$ Let $W_1H^1(U)= H^1(X, {{\mathbb Z}})$ denote the weight $1$ piece of the weight filtration. We have an exact sequence $$0\rightarrow W_1H^1(U) = H^1(X, {{\mathbb Z}}^{\perp}) \rightarrow H^1(U,{{\mathbb Z}}^{\perp}) \rightarrow {{\mathcal G}}= {{\mathbb Z}}^k = H^2(X,U, {{\mathbb Z}}^{\perp}) \rightarrow$$ $$\rightarrow H^2(X,{{\mathbb Z}}^{\perp}) \rightarrow H^2(U,{{\mathbb Z}}^{\perp}) \rightarrow H^3(X,U,{{\mathbb Z}}^{\perp})\rightarrow \ldots$$ The exponential exact sequence lifts to an exact sequence for the logarithmic space $$0\rightarrow W_1H^1(U,{{\mathbb Z}}^{\perp})\rightarrow H^1(U,{{\mathbb C}}) \rightarrow M_B(X,\log D)\rightarrow H^2(X,{{\mathbb Z}}^{\perp} )\rightarrow H^2(X,{{\mathbb C}}).$$ The image of the connecting map in the first exponential exact sequence, is the torsion subgroup of $H^2(U)$. A duality calculation relates $H^3(X,U,{{\mathbb Z}}^{\perp} )$ to the $H^1(D_i, {{\mathbb Z}}^{\perp})$ so this is torsion-free. Hence, any torsion element in $H^2(U,{{\mathbb Z}}^{\perp})$ comes from $H^2(X,{{\mathbb Z}}^{\perp})$. This fits in with the fact that any element of $M_B(U)$ admits a Deligne canonical extension to a line bundle with logarithmic connection on $X$. These exact sequences all fit together into a diagram $$\begin{array}{rclclclc} & & & & & 0 &\!\rightarrow\! & H^1(X ,{{\mathbb Z}}^{\perp}) \\ & & & & & \downarrow & & \downarrow \\ & & & & & 0 &\!\rightarrow \! & H^1(U ,{{\mathbb Z}}^{\perp}) \\ & & & & & \downarrow & & \downarrow \\ & 0 &\! \rightarrow \! & 0 &\!\rightarrow\! & {{\mathcal G}}= {{\mathbb Z}}^k &\!\stackrel{=}{\rightarrow}\! & H^2(X, U,{{\mathbb Z}}^{\perp}) \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow \\ 0\!\rightarrow \! & H^1(X,{{\mathbb Z}}^{\perp}) &\! \rightarrow \! & H^1(U,{{\mathbb C}}) &\!\rightarrow\! & M_B(X,\log D) &\!\rightarrow \! & H^2(X,{{\mathbb Z}}^{\perp}) \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow \\ 0\!\rightarrow \! & H^1(U,{{\mathbb Z}}^{\perp}) &\! \rightarrow\!& H^1(U,{{\mathbb C}}) &\!\rightarrow\! & M_B(U) &\!\rightarrow \! & H^2(U,{{\mathbb Z}}^{\perp}) \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow \\ {{\mathcal G}}\stackrel{=}{\rightarrow}\! & H^2(X,U,{{\mathbb Z}}^{\perp})&\! \rightarrow \! & 0 &\!\rightarrow\! & 0 &\!\rightarrow \! & H^3(X,U,{{\mathbb Z}}^{\perp}) \, . \end{array}$$ There are also exact sequences for localization near the singular points. The main one is $$\label{basicexact} 0\rightarrow M_B(X) \rightarrow M_B(X, \log D) \stackrel{{\rm res}}{\rightarrow} {{\mathbb C}}^k \rightarrow H^2(X,{{{\mathbb G}}_m}) .$$ It fits with the exponential exact sequence to give a diagram $$\begin{array}{cccccccc} & 0 & & 0 & & & & \\ & \downarrow & & \downarrow & & & & \\ 0\rightarrow & H^1(X,{{\mathbb Z}}^{\perp}) &\rightarrow & H^1(X,{{\mathbb Z}}^{\perp}) &\rightarrow & 0 &\rightarrow & H^2(X,{{\mathbb Z}}^{\perp}) \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow \\ 0\rightarrow & H^1(X,{{\mathbb C}}) &\rightarrow & H^1(U,{{\mathbb C}}) &\rightarrow & {{\mathbb C}}^k &\rightarrow & H^2(X,{{\mathbb C}}) \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow \\ 0\rightarrow & M_B(X) &\rightarrow & M_B(X, \log D)&\rightarrow & {{\mathbb C}}^k &\rightarrow & H^2(X, {{{\mathbb G}}_m}) \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow \\ 0\rightarrow & H^2(X,{{\mathbb Z}}^{\perp}) &\rightarrow & H^2(X,{{\mathbb Z}}^{\perp}) &\rightarrow & 0 &\rightarrow & H^3(X,{{\mathbb Z}}^{\perp})\, . \\ \end{array}$$ Dividing by the gauge group gives the diagram $$\begin{array}{cccccccc} & 0 & & 0 & & 0 & & \\ & \downarrow & & \downarrow & & \downarrow & & \\ 0\rightarrow & H^1(X,{{\mathbb Z}}^{\perp}) &\rightarrow & H^1(U,{{\mathbb Z}}^{\perp}) &\rightarrow & {{\mathbb Z}}^k & \rightarrow & H^2(X,{{\mathbb Z}}^{\perp}) \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow \\ 0\rightarrow & H^1(X,{{\mathbb C}}) &\rightarrow & H^1(U,{{\mathbb C}}) &\rightarrow & {{\mathbb C}}^k &\rightarrow & H^2(X,{{\mathbb C}}) \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow \\ 0\rightarrow & M_B(X) &\rightarrow & M_B(U)&\rightarrow & ({{{\mathbb G}}_m}^{\perp} ) ^k&\rightarrow & H^2(X, {{{\mathbb G}}_m}) \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow \\ 0\rightarrow & H^2(X,{{\mathbb Z}}^{\perp}) &\rightarrow & H^2(U,{{\mathbb Z}}^{\perp})&\rightarrow & 0 & \rightarrow & H^3(X,{{\mathbb Z}}^{\perp}) \, . \\ \end{array}$$ The connected component of $M_B(U)$ containing the identity representation, is a quotient: $$M_B(U)^o = H^1(U,{{\mathbb C}})/ H^1(U,{{\mathbb Z}}^{\perp}).$$ This identification is valid in the analytic category. The logarithmic space is obtained by dividing out instead by $W_1H^1(X -D, {{\mathbb Z}}^{\perp}) = H^1(X, {{\mathbb Z}}^{\perp})$: $$M_B(X,\log D)^o = H^1(U,{{\mathbb C}})/ W_1H^1(U,{{\mathbb Z}}^{\perp}).$$ Let ${{\mathcal G}}^o := \ker \left( {{\mathbb Z}}^k \rightarrow H^2(X,{{\mathbb Z}}^{\perp}) \right) $. Then ${{\mathcal G}}^o$ acts on $M_B(X,\log D)^o$ with quotient $M_B(U)^o$. The above diagrams show that this is compatible with the quotient descriptions. Compatibility with Hodge filtration {#sec-compatibility} ----------------------------------- The above diagrams can be replaced with the corresponding diagrams of twistor spaces over ${{\mathbb P}}^1$. This raises the question of showing that the maps preserve the twistor structure, another way of saying that they should be compatible with the Hodge filtrations. Recall that the Hodge filtration and its complex conjugate for $H^1(U, {{\mathbb C}})$ lead to the twistor bundle $\xi (H^1(U, {{\mathbb C}}),F,\overline{F})$ over ${{\mathbb P}}^1$, see [@icm]. We can again take the quotient by the action of $H^1(U,{{\mathbb Z}})$ or $W_1H^1(U,{{\mathbb Z}})$. This gives an identification of the Deligne-Hitchin twistor space, at least for the connected component of the identity representation. \[mhs-ident\] Denote by a superscript $(\; )^o$ the connected component of the identity representation. We have identifications of analytic spaces over ${{\mathbb P}}^1$, $$M_{\rm DH}(X,\log D)^{o} \cong \xi (H^1(U, {{\mathbb C}}),F,\overline{F}) / W_1H^1(U,{{\mathbb Z}}^{\perp})$$ and $$M_{\rm DH}(X,\log D)^{o}/{{\mathcal G}}^o \cong \xi (H^1(U, {{\mathbb C}}),F,\overline{F}) / H^1(U,{{\mathbb Z}}^{\perp}).$$ Thus, the maps in the above big diagrams are compatible with the twistor structures. Use a cocycle description of $H^1(U,{{\mathbb C}})$. Suppose we are given an open analytic covering of $X$ by open sets $U_i$, and let $U_{ij}:= U_i\cap U_j$ etc. Recall Grothendieck’s theorem $$H^1(U,{{\mathbb C}}) = {\mathbb H}^1\left( {{\mathcal O}}_X \rightarrow \Omega ^1_X (\log D) \rightarrow \Omega ^2_X(\log D)\rightarrow \ldots \right) .$$ An element here is given by a pair $(\{ g_{ij} \} , \{ a_i\} )$ where $$g_{ij}\in {{\mathcal O}}_X(U_{ij}), \;\;\; a_i \in \Omega ^1_X(\log D) (U_i)$$ and these satisfy the cocycle condition $g_{ij}+ g_{jk} + g_{ki}=0$, the compatibility condition $d(g_{ij}) = a_i -a_j$, and $d(a_i) = 0$. The image of this pair in $M_{DR}(X,\log D)$ is $(L,\nabla )$ where $L$ is the line bundle whose transition functions are $e^{g_{ij}}$, and $\nabla := d + a_i$ over $U_i$, with $d$ being the constant connection with respect to the trivialization $L|_{U_i}\cong {{\mathcal O}}_{U_i}$. This image is compatible with the exponential map $H^1(U,{{\mathbb C}})\rightarrow M_B(U)$ via the Riemann-Hilbert correspondence. The Hodge filtration or Rees-bundle $\xi (H^1(U,{{\mathbb C}}), F)\rightarrow {{\mathbb A}}^1$ may also be described as the bundle of triples $(\lambda , \{ g_{ij}\} , \{ a_i\} )$ subject to the conditions, analogues of the notion of $\lambda$-connection: $$g_{ij}+ g_{jk} + g_{ki}=0, \;\;\; \lambda d(g_{ij}) = a_i -a_j, \;\;\; \lambda d(a_i) = 0.$$ Map this triple to $(\lambda , L, \nabla )$ where $L$ is again given by transition functions $e^{g_{ij}}$, and $\nabla := \lambda d + a_i$ over $U_i$. This gives a map $$\xi (H^1(U,{{\mathbb C}}), F) \rightarrow M_{\rm Hod}(X,\log D).$$ It is compatible with the action of ${{{\mathbb G}}_m}$, and is the same as the previous map in the fiber over $\lambda = 1$, so it is compatible with the exponential map on Betti cohomology. Note that the complex conjugate of the Hodge filtration on $H^1(U,{{\mathbb C}})$ is the same as the pullback by $\varphi : U^{\rm top}\cong \overline{U}^{\rm top}$, of the Hodge filtration on $H^1(\overline{U},{{\mathbb C}})$. Indeed, $\varphi$ is antiholomorphic so the pullback by $\varphi$ of a cohomology class containing at least a certain number of $dz_i$, is a cohomology class containing at least that many $d\overline{z}_i$. Using all of these things, our map glues together with the corresponding map for $\overline{U}= \overline{X}-\overline{D}$ to give a map of twistor spaces over ${{\mathbb P}}^1$, $$\xi (H^1(U,{{\mathbb C}}), F,\overline{F}) \rightarrow M_{\rm DH}(X,\log D).$$ This is the required compatibility. From the cocycle description, we get that the map is surjective to the connected component $M_{\rm DH}(X,\log D)^{o} $, even in the fibers over $\lambda = 0,\infty $. Using smoothness of both sides over ${{\mathbb P}}^1$ and a dimension count, we see that the kernel is discrete and flat over ${{\mathbb P}}^1$. In the general fiber it is $W_1H^1(U,{{\mathbb Z}})$. The closure of the graph of this subgroup is again a subgroup of the form $W_1H^1(U,{{\mathbb Z}})\times {{\mathbb P}}^1\subset \xi (H^1(U,{{\mathbb C}}), F,\overline{F})$. Hence the isomorphism $$M_{\rm DH}(X,\log D)^{o} \cong \xi (H^1(U, {{\mathbb C}}),F,\overline{F}) / W_1H^1(U,{{\mathbb Z}}).$$ The other one is obtained by dividing out by the gauge group ${{\mathcal G}}^o$. [*Problem:*]{} Find a similar description for the twistor spaces of other connected components of $M_B(U)$ corresponding to torsion elements in $H^2(U,{{\mathbb Z}})$. This should be doable using the fact that the points of finite order in $M_B(U)$ occur in every connected component, as can be seen from the analogue of the exponential exact sequence $$0\rightarrow H^1(U,{{\mathbb Z}}) \rightarrow H^1(X_D, {{\mathbb Q}}) \rightarrow H^1(U, \mu _{\infty}) \rightarrow H^2(U,{{\mathbb Z}}) \rightarrow H^2(U,{{\mathbb Q}}).$$ Preferred sections {#sec-preferred} ------------------ We now describe how a tame harmonic bundle of rank one on $U=X-D$ gives rise to a section of the fibration . In this discussion, we use the notion of parabolic structure and in particular Mochizuki’s notion of KMS-spectrum [@Mochizuki]. See also Budur [@Budur] for a discussion of the rank one case. In Theorem \[pref-id\], the space of harmonic bundles will be identified with the space of $\sigma$-invariant sections of $M_{\rm DH}(U)$. The latter doesn’t refer to the notion of parabolic structure, but the identification map between them does. Fix a Kähler metric $\omega$ on $X$, which restricts to a K"’ahler metric on $U$. Recall that a tame harmonic bundle over $U$ is a vector bundle $E$ together with operators $D'$ and $D''$ and a metric $h$, with respect to which these operators satisfy certain equations [@Hitchin] [@Corlette] [@hbls]. Our preferred section will not depend on changes of $h$ by multiplying by a positive constant. Decompose $$\label{decomp} D'' = {\overline{\partial}}+ \theta , \;\;\; D' = \partial + \overline{\theta} .$$ Use the notation ${{\mathcal E}}= (E,D',D'',h)$ for our harmonic bundle. Fix $\lambda \in {{\mathbb P}}^1$ and for now we suppose it is in the first standard chart ${{\mathbb A}}^1$ so we think of $\lambda \in {{\mathbb C}}$. Then we get a holomorphic structure ${\overline{\partial}}+ \lambda \overline{\theta}$ on the bundle $E$, and a $\lambda$-connection $\lambda \partial + \theta$. Denote the holomorphic bundle with this holomorphic structure by ${{\mathcal E}}^{\lambda}$ and the $\lambda$-connection by $\nabla ^{\lambda}$ By [@Mochizuki], for any vector $a= (a_1,\ldots , a_k)$ of real numbers, we get an extension of ${{\mathcal E}}^{\lambda}$ to a holomorphic bundle denoted $E^{\lambda}_a$ on $X$, and $\nabla ^{\lambda}$ extends to a logarithmic $\lambda$-connection on $E^{\lambda}_a$. If we pick $\lambda _0$ and any $i=1,\ldots , k$, then there is a set of critical values of $a_i$ called the [*KMS-spectrum*]{} [@Mochizuki]. For $a= (a_1,\ldots , a_k)$ with $a_i$ not in the KMS-spectrum at $\lambda _0$ and $D_i$, there is a neighborhood $\lambda _0\in L\subset {{\mathbb P}}^1$ such that for $\lambda \in L$, the bundles with logarithmic connection $({{\mathcal E}}^{\lambda}_a, \nabla ^{\lambda})$ vary holomorphically in $\lambda$. For each divisor component and fixed $\lambda _0$, the KMS-spectrum is a ${{\mathbb Z}}$-translation orbit in ${{\mathbb R}}$, that is it consists of everything of the form $a_i+u_i$ for $u_i\in {{\mathbb Z}}$. This is special to the rank one case, where there is only one KMS spectrum element in ${{\mathbb R}}/{{\mathbb Z}}$. The [*KMS-critical locus*]{} at $\lambda _0$ is the set of all $a$ such that some $a_i$ is in the KMS-spectrum for $\lambda _0$ and $D_i$. This locus is a union of translates of the $k$ coordinate hyperplanes. The translates included are all of the form $(a_1 + u_1,\ldots , a_k + u_k)$ where $a_i$ are some elements of the KMS-spectrum, and $u_i$ are any integers. The [*KMS-chambers*]{} are the connected components of the complement of the KMS-spectrum. Note that ${{\mathbb Z}}^k$ acts simply transitively on the set of KMS-chambers for any $\lambda_0$. Furthermore, the set of KMS-chambers varies continuously with $\lambda _0$: a point which is well in the middle of a chamber for $\lambda _1$, will remain in a uniquely determined chamber for $\lambda _1$ nearby, or to put it another way the KMS-critical locus varies continuously as a function of $\lambda$. In particular, if for any $\lambda _0$ we choose a particular KMS-chamber, then by following this around it determines a KMS-chamber for all other $\lambda \in {{\mathbb A}}^1$. For different values of $a$ in the same KMS-chamber, the bundles with logarithmic connection $({{\mathcal E}}^{\lambda}_a, \nabla ^{\lambda})$ are all canonically isomorphic. Hence, the choice of a KMS-chamber for a given $\lambda _0$ determines a choice of KMS-chamber for all $\lambda \in {{\mathbb A}}^1$; let $a(\lambda )$ denote a function taking values in this chamber for each $\lambda$. We thus get the collection of bundles with logarithmic connection depending on $\lambda$, $$\lambda \mapsto ({{\mathcal E}}^{\lambda}_{a(\lambda )}, \nabla ^{\lambda}).$$ This is our preferred section of $M_{DH}(X,\log D)$ over ${{\mathbb A}}^1$. If we choose a different chamber to begin with, then the section is modified by the corresponding element of the local meromorphic gauge group ${{\mathcal G}}= {{\mathbb Z}}^k$. The projection to the quotient gives a uniquely defined section of the fibration , at least over ${{\mathbb A}}^1$. This construction patches together with the corresponding construction on the other chart ${{\mathbb A}}^1$ at $\infty$. See [@Mochizuki Chapter 11]. The construction we have described here is an isomorphism between harmonic bundles and $\sigma$-invariant sections of the fibration $M_{\rm DH}(X,\log D)/{{\mathcal G}}\rightarrow {{\mathbb P}}^1$. \[pref-id\] Let $M_{\rm har}(U)$ denote the group of tame harmonic line bundles on $U$. The map described above goes from here to the space of $\sigma$-invariant sections of $M_{\rm DH}(X,\log D)$ modulo the gauge group action: $${{\mathcal P}}: M_{\rm har}(U) \rightarrow \Gamma ({{\mathbb P}}^1 , M_{\rm DH}(X,\log D)) ^{\sigma} /{{\mathcal G}}.$$ This map is an isomorphism. The map is given by the discussion above. The proof that it is an isomorphism, which requires techniques from the next subsections, will be given in Corollary \[pref-id-inj\] and §\[sub-proof\] below. This theorem, which is only in the rank one case, nevertheless suggests that in the correspondence between harmonic bundles and pure twistor ${{\mathcal D}}$-modules of [@Mochizuki2] and [@Sabbah] the parabolic weight should come out of the structure of twistor ${{\mathcal D}}$-module, without having to impose an additional parabolic structure on the ${{\mathcal D}}$-module side. It isn’t clear to me to what extent this statement may already be contained in [@Mochizuki2] and [@Sabbah]. Residues and parabolic structures {#sub-parabolic} --------------------------------- We now get to one of the main observations in this article: that the three dimensional space of $\sigma$-invariant sections of $T(1,\log )$ encodes the data of residues and parabolic weights for a harmonic bundle. Fix a divisor component $D_i$, and a point $p\in {{\mathbb A}}^1$. The fiber $T(1,\log )_p$ is identified with ${{\mathbb C}}$ by the frame $\frac{\partial}{\partial \lambda}$. Hence the residue map can be composed with this identification to give $${\rm res}_{D_i,p}:M_{\rm Hod}(X,\log D)_p \rightarrow {{\mathbb C}}\cong T(1,\log )_p.$$ It sends a logarithmic $p$-connection $(E,\nabla )$ to ${\rm res}(\nabla ; D_i) \frac{\partial}{\partial \lambda}(p)$. The glueing function for residues of logarithmic $\lambda$-connections is $-\lambda ^2$, the same as for $T(1,\log )= T{{\mathbb P}}^1$. Therefore, this map glues with the same map on the chart $M_{\rm Hod}(\overline{X},\log \overline{D})$ to give a bundle map over ${{\mathbb P}}^1$, $${\rm res}^{\rm DH}_{D_i}: M_{\rm DH}(X,\log D)\rightarrow T(1,\log ).$$ \[residuesigma\] The residue map ${\rm res}^{\rm DH}_{D_i}$ is compatible with the antipodal involutions on $M_{\rm DH}(X,\log D)$ and $T(1,\log )$, so it gives a map on $\sigma$-invariant sections $$\Gamma ({{\mathbb P}}^1, M_{\rm DH}(X,\log D)) ^{\sigma} \rightarrow \Gamma ({{\mathbb P}}^1, T(1,\log )) ^{\sigma}.$$ The calculation for $X$ near $D$ is the same as that of §\[sub-antipodal\]. Comparing with the calculation of §\[sub-tate-antipodal\], we see that the residue is compatible with $\sigma$ and it induces a map on $\sigma$-invariant sections. Next, consider the projection ${\rm pr}_i: {{\mathcal G}}\rightarrow {{\mathbb Z}}(1,\log )$ which sends $(g_1,\ldots , g_k)$ to $g_i$. \[residuegauge\] The residue map ${\rm res}^{\rm DH}_{D_i}$ is compatible with the action of the local meromorphic gauge group ${{\mathcal G}}$ via the projection ${\rm pr}_i$ composed with the morphism ${{\mathbb Z}}(1,\log )\rightarrow T(1,\log )$, so it gives a map on quotients $$\Gamma ({{\mathbb P}}^1, M_{\rm DH}(X,\log D)) ^{\sigma} /{{\mathcal G}}\rightarrow \Gamma ({{\mathbb P}}^1, T(1,\log )) ^{\sigma}/{{\mathbb Z}}(1,\log ).$$ At each point $p$, the action of the gauge group is compatible by equation with the map ${\rm pr}_i$ via the standard morphism ${{\mathbb Z}}(1,\log ) \rightarrow T(1,\log )_p$ which sends the generator to $-p\frac{\partial}{\partial\lambda}$. This gives the compatibility on global sections. For any $p\in{{\mathbb A}}^1\subset {{\mathbb P}}^1$, we can compose the map of Lemma \[residuesigma\] with the isomorphism $(\varpi _p,{\rm res}_p)$ of Proposition \[prop-parabolicweight\], comprising the parabolic weight function $\varpi _p$ and the residue or evaluation at $p$. This gives a map $$\label{globalsigma} (\varpi _p, {\rm res}_p)_{D_i} : \Gamma ({{\mathbb P}}^1, M_{\rm DH}(X,\log D)) ^{\sigma} \rightarrow {{\mathbb R}}\times {{\mathbb C}}.$$ Dividing by the action of the local meromorphic gauge group corresponds to dividing by the action of ${{\mathbb Z}}$ on ${{\mathbb R}}\times {{\mathbb C}}$ generated by $(\varpi _p, {\rm res}_p)(\psi (1,0))= (1,-p)$. We get a quotient map $$\label{globalsigmagauge} (\varpi _p, {\rm res}_p)^{{{\mathcal G}}}_{D_i} : \Gamma ({{\mathbb P}}^1, M_{\rm DH}(X,\log D)) ^{\sigma}/{{\mathcal G}}\rightarrow \frac{{{\mathbb R}}\times {{\mathbb C}}}{(1,-p)\cdot {{\mathbb Z}}} .$$ Compose with the preferred-sections map ${{\mathcal P}}$ of Theorem \[pref-id\]. Our main observation is that this encodes the parabolic weight and residue of a harmonic bundle. These were defined for the case of curves, at $\lambda = 0$ and $\lambda = 1$, in [@hbnc]. They were defined in higher dimensions and for all $\lambda$ in [@Mochizuki]. Given a parabolic bundle $F=\{ F_b\}$ filtered by bundles indexed in the increasing sense by $b\in {{\mathbb R}}^k$, suppose we have chosen $E$ as one of these bundles. Define the [*parabolic weight*]{} to be the element $b=(b_1,\ldots , b_k)$ with $b_i$ as small as possible so that $E=F_b$. Given a harmonic bundle ${{\mathcal E}}= (E,D',D'',h)\in M_{\rm Har}(U)$, we obtain for any $\lambda$ a parabolic logarithmic $\lambda$-connection ${{\mathcal E}}^{\lambda}$ by [@Mochizuki]. Its underlying parabolic bundle has a parabolic weight as defined at the start of this paragraph, and the parabolic $\lambda$-connection on ${{\mathcal E}}^{\lambda}$ has a residue along each $D_i$. The parabolic weight of ${{\mathcal E}}^{\lambda}$ is determined by the rate of growth of the harmonic metric: if $u$ is a unit section near a point of $D_i$, and if $D_i$ is cut out by the equation $z=0$, then $|u|_h\sim |z|^{-b_i}$ where $b_i$ is the parabolic weight along $D_i$. \[harmoniccompose\] Suppose $D_i$ is a divisor component and $p\in {{\mathbb A}}^1\subset {{\mathbb P}}^1$. Suppose ${{\mathcal E}}= (E,D',D'',h)\in M_{\rm Har}(U)$ is a rank one harmonic bundle on $U$. Then $$(\varpi _p,{\rm res}_p)^{{{\mathcal G}}}_{D_i} ({{\mathcal P}}({{\mathcal E}})) \in \frac{{{\mathbb R}}\times {{\mathbb C}}}{(1,-p)\cdot {{\mathbb Z}}}$$ is the parabolic weight and residue of the parabolic logarithmic $\lambda$-connection ${{\mathcal E}}^{\lambda}$ at $\lambda = p$. Fix an extension of the logarithmic Higgs bundle $({{\mathcal E}}^0,\nabla ^0)$ to a line bundle over $X$. It then has a harmonic metric $h$. Let $a'$ be the parabolic weight along $D_i$. Let $\alpha '$ be the residue of the Higgs field along $D_i$. Mochizuki defines functions ${\mathfrak p}(\lambda ,a,\alpha )$ and ${\mathfrak e}(\lambda ,a,\alpha )$ in [@Mochizuki §2.1.7], and in Corollary 7.71, [@Mochizuki §7.3.3] he points out that the rule obeyed by the KMS-spectrum of a harmonic bundle is given by the transformation $({\mathfrak p},{\mathfrak e})$. In the rank one case, the KMS-spectrum has only one element. Hence, the transformation rule [@Mochizuki Cor. 7.71] means that the parabolic weight and residue of ${{\mathcal E}}^{\lambda }$ are respectively $${\mathfrak p}(\lambda ,a',\alpha ')\;\;\; \mbox{and} \;\;\; {\mathfrak e}(\lambda ,a',\alpha ').$$ By inspection, the functions ${\mathfrak p},{\mathfrak e}$ of [@Mochizuki §2.1.7] are the same as the parabolic weight functions and residue functions occuring in Proposition \[prop-parabolicweight\]: $$\label{equmochi} (\varpi _p,{\rm res}_p)(\psi (a,\alpha )) = ({\mathfrak p}(p,a,\alpha ), {\mathfrak e}(p,a,\alpha )).$$ Recall that $\lambda \mapsto ({{\mathcal E}}^{\lambda} ,\nabla ^{\lambda})$ is exactly our preferred section ${{\mathcal P}}({{\mathcal E}})$ (lifted over the gauge group action). Let $(a,\alpha )$ denote the Higgs coordinates for the residue section, so $${\rm res}_{D_i}({{\mathcal P}}({{\mathcal E}}))=\psi (a,\alpha ) \in \Gamma ({{\mathbb P}}^1,T(1,\log ))^{\sigma} \cong {{\mathbb R}}\times {{\mathbb C}}.$$ The value of this section at the point $\lambda$, which is the residue of the logarithmic $\lambda$-connection $\nabla ^{\lambda}$, is given by the residue function ${\rm res}_{\lambda}(\psi (a,\alpha ))$ calculated in §\[sub-reseval\] above. We conclude that for all $\lambda \in {{\mathbb A}}^1$, $${\rm res}_{\lambda}(\psi (a,\alpha )) = {\mathfrak e}(\lambda ,a',\alpha ').$$ The identity between the functions ${\rm res}_{\lambda}(\psi (a,\alpha ))$ and ${\mathfrak e}(\lambda ,a,\alpha )$, writing them out per §\[sub-reseval\], means that $$\alpha - a\lambda -\overline{\alpha}\lambda ^2 = \alpha ' - a'\lambda -\overline{\alpha}'\lambda ^2$$ for all $\lambda \in {{\mathbb A}}^1$. It follows that $a=a'$ and $\alpha = \alpha '$. This proves the statement of the theorem at $p=\lambda = 0$. At a general value of $\lambda = p$, the parabolic weight and residue of the harmonic bundle are given as ${\mathfrak p}(\lambda ,a,\alpha )$ and ${\mathfrak e}(\lambda ,a,\alpha )$ respectively, by Corollary 7.71, [@Mochizuki §7.3.3]. The identity shows that these are the same as $(\varpi _p,{\rm res}_p)_{D_i} ({{\mathcal P}}({{\mathcal E}}))$. Modulo the action of the gauge group (which absorbs our initial choice of extension of the bundle), this gives the statement of the theorem. We now have enough to do half of the isomorphism in Theorem \[pref-id\]. \[pref-id-inj\] The preferred-sections morphism ${{\mathcal P}}$ in Theorem \[pref-id\] is injective. Suppose ${{\mathcal E}}$ and ${{\mathcal F}}$ are rank one harmonic bundles, such that ${{\mathcal P}}({{\mathcal E}}) \cong {{\mathcal P}}({{\mathcal F}})$. The local parabolic weight and residue data of the harmonic bundles coincide, because these functions factor through ${{\mathcal P}}$ by Theorem \[harmoniccompose\]. The line bundles with connection ($\lambda = 1$) associated to ${{\mathcal E}}$ and ${{\mathcal F}}$ correspond to filtered local systems of rank $1$ [@hbnc] [@Mochizuki] [@GukovWitten]. The filtration weight of the filtered local system is obtained from the parabolic weight and residues of the line bundles with connection, see the third column of the table in [@hbnc p. 720]. Therefore, the filtration weights of the filtered local systems associated to ${{\mathcal E}}$ and ${{\mathcal F}}$ are the same. The fact that ${{\mathcal P}}({{\mathcal E}}) \cong {{\mathcal P}}({{\mathcal F}})$ at $\lambda = 1$ restricted over $\lambda = 1$ means that the associated logarithmic connections are the same up to local meromorphic gauge transformation, hence the associated monodromy representations are the same. Now, in rank one a filtered local system is determined uniquely by its monodromy representation and its filtration weight. A filtered local system corresponds to a unique harmonic bundle. Therefore ${{\mathcal E}}\cong {{\mathcal F}}$. Comparison with [@hbnc] {#comp-hbnc} ----------------------- It is interesting to comment on the particular cases $p=0$ and $p=1$. The transformation for going from $p=0$ to $p=1$ gives back the transformation between the first two columns of the table on p. 720 of [@hbnc], which has remained mysterious to me up until now. The parabolic weights and residues for the different points $\lambda = p$, are different coordinate systems on the same three dimensional space $\Gamma ({{\mathbb P}}^1, T(1,\log )) ^{\sigma}$. Going between two different values of $p$ gives a change of coordinates. In [@hbnc] only the values $\lambda = 0$ (Higgs bundles) and $\lambda = 1$ (logarithmic connections) were considered. However there are some changes of notation: we have adopted Mochizuki’s coordinates [@Mochizuki §6.1.1] at the Higgs point $(a,\alpha )$ for the reader’s convenience. We have also adopted the standard convention that the parabolic structure is indexed by an increasing filtration. In [@hbnc], the parabolic structure was given by a decreasing filtration. So, here $a\in {{\mathbb R}}$ is the parabolic weight of the Higgs bundle in the increasing sense, which corresponds to $-\alpha$ in the notation of [@hbnc]. In [@hbnc] the sheaf $E_{\alpha}$ corresponded to sections whose growth was bounded by $|z|^{\alpha}$ whereas here $E_a$ corresponds to sections whose growth is bounded by $|z|^{-a}$. And here, $\alpha$ is the residue of the Higgs field, which was denoted by $b+ci$ on p. 720 of [@hbnc]. These coordinates coincide with our parabolic weight and residue at $p=0$. The parabolic weight and residue at $p=1$ are given by the formulae $$\varpi _1(a,\alpha ) = a+\alpha +\overline{\alpha}, \;\;\; {\rm res}_1 (a,\alpha ) = \alpha - \overline{\alpha}-a.$$ In terms of the notation $(\alpha , b,c)$ of [@hbnc]—where $\alpha$ has a different meaning from the rest of the present paper, and where $i\in {{\mathbb C}}$ is chosen—we get $$\varpi _1 = -\alpha + 2 b, \;\;\; {\rm res}_1 = \alpha + 2 i c.$$ These are the values in the second column of the table on page 720 of [@hbnc], taking into account that the “jump” there is $-\varpi _1$. The conclusion is that the three-dimensional space, and the coordinate transformation in the table of [@hbnc], come from the fact that the twistor bundle of residues is $T(1, \log )\cong {{\mathcal O}}_{{{\mathbb P}}^1}(2)$, in other words the local monodromy around singular divisors is in a weight-two twistor bundle. The weight filtration {#sec-weight} --------------------- Because we look at line bundles, the moduli spaces form groups under tensor product. For $M_{\rm DH}$ we get a group structure relative to ${{\mathbb P}}^1$. Define the [*weight filtration*]{}: $$W_1M_{\rm DH}(X,\log D) := M_{\rm DH}(X)^o, \;\;\; W_2M_{\rm DH}(X,\log D) := M_{\rm DH}(X,\log D).$$ In our arguments below, it has seemed most natural to use only the connected component of the identity representation in the weight $1$ piece. Define the second graded piece as the quotient using the group structure $$Gr^W_2(M_{\rm DH}(X,\log D)) := W_2/W_1 = \frac{M_{\rm DH}(X,\log D)}{M_{\rm DH}(X)} .$$ Using only the connected component $M_{\rm DH}(X)^o$ for $W_1$, leads to a nontrivial finite group as $Gr^W_2$ even in the compact case $D=\emptyset$. There is a version modulo the gauge group: $$Gr^W_2(M_{\rm DH}(U)) = \frac{M_{\rm DH}(U)}{M_{\rm DH}(X)^o} = \frac{Gr^W_2(M_{\rm DH}(X,\log D))}{{{\mathcal G}}} .$$ In the second equality the quotient is taken in a stacky sense over $\lambda = 0,\infty$. The inclusion $M_{\rm DH}(X)^o\subset M_{\rm DH}(U)$ is strict, injective on each fiber over $\lambda \in {{\mathbb P}}^1$. There is also an induced weight filtration on the connected component given by $$W_1M_{\rm DH}(X,\log D)^o := M_{\rm DH}(X)^o.$$ \[weight-compatible\] The exponential map giving an isomorphism in Theorem \[mhs-ident\] is strictly compatible with the weight filtrations for connected components on both sides, in other words it sends the usual weight filtration on $H^1(U,{{\mathbb C}})$ to the weight filtration on $M_{\rm DH}(X,\log D)^o$. The exact diagrams in §\[sec-exact\] extend to exact diagrams of bundles over ${{\mathbb P}}^1$, with the Tate twistor structure $T(1,\log )$ inserted in place of ${{\mathbb C}}$ at appropriate places. The exponential map is one of the middle vertical maps in the exact squares. The weight foliation is the kernel foliation of the map $R$, and the weight filtration on abelian cohomology is the kernel of the corresponding map $H^1(U,{{\mathbb C}})\rightarrow {{\mathbb C}}^k$. Exactness then implies that the exponential isomorphism is compatible with weight filtrations. The goal of this subsection is to identify $Gr^W_2M_{\rm DH}(X,\log D)$ and show that it has “weight $2$”. Write $NS(X)$ for the Neron-Severi group of divisors modulo algebraic equivalence, which is contained in $H^2(X,{{\mathbb Z}})$. Let $NS(X,D)\subset NS(X)$ be the subgroup generated by divisor components $D_i$ of $D$. Let $NS(X,D)^{\rm sat}$ be the saturation of this subgroup, in other words the subgroup of all elements $A\in NS(X)$ such that some multiple $mA$ is in $NS(X,D)$. This may be seen as the kernel in the sequence $$0\rightarrow NS(X,D)^{\rm sat} \rightarrow H^2(X,{{\mathbb Z}}) \rightarrow \frac{H^{1,1}(X,{{\mathbb C}})}{{{\mathbb C}}\cdot [D_1] + {{\mathbb C}}\cdot [D_k]}$$ and it includes the subgroup of torsion $NS(X)^{\rm tors}$. For example if $D=\emptyset$ then $NS(X,D)^{\rm sat} = NS(X)^{\rm tors}$. We close this paragraph by noting that $NS(X,D)^{\rm sat}$ is the preimage of $NS(U)^{\rm tors}$ under the restriction map $NS(X)\rightarrow NS(U)$, and $$\label{nsutors} \frac{NS(X,D)^{\rm sat}}{NS(X,D)} = NS(U)^{\rm tors}.$$ Putting together the residue maps at divisor components $D_1,\ldots , D_k$, we get a map $$R: M_{\rm DH}(X,\log D)\rightarrow T(1,\log ) ^k.$$ The condition on $c_1(L)$ for a point $(L,\nabla )\in M_{\rm DH}(X,\log D)$ is equivalent to saying that $c_1(L)\in NS(X,D)^{\rm sat}$. These give a map $$(c_1,R): M_{\rm DH}(X,\log D)\rightarrow NS(X,D)^{\rm sat}\times T(1,\log )^k.$$ Consider the map defined using the divisor components $$\Sigma :{{\mathbb C}}^k \rightarrow H^{1,1}(X,{{\mathbb C}}), \;\;\; (a_1,\ldots , a_k)\mapsto \sum a_i[D_i].$$ The cohomology $H^2(X,{{\mathbb C}})$ has a pure weight two Hodge structure, whose twistor bundle is semistable of slope $2$. The twistor bundle $\xi (H^2(X,{{\mathbb C}}), F,\overline{F})$ has a natural subbundle corresponding to $H^{1,1}$, and since that space is pure of Hodge type $(1,1)$ we have a natural isomorphism $$\xi (H^{1,1}(X,{{\mathbb C}}), F,\overline{F}) \cong H^{1,1}(X,{{\mathbb C}})\otimes T(1,\log ).$$ With respect to these constructions, the map $\Sigma$ extends over ${{\mathbb P}}^1$ to give a map $$\Sigma _{\rm DH}: T(1,\log )^k \rightarrow H^{1,1}(X,{{\mathbb C}})\otimes T(1,\log ).$$ There is a natural morphism of groups over ${{\mathbb P}}^1$, $$\Lambda _{\rm DH}: NS(X)\times {{\mathbb P}}^1\rightarrow H^{1,1}(X,{{\mathbb C}})\otimes T(1,\log )\cong \xi (H^{1,1}X,{{\mathbb C}}), F,\overline{F})$$ which sends an element of $NS(X)$ to a section which has a simple pole at $0$ and another simple pole at $\infty$. In terms of the usual trivialization of $T(1,\log )$ over ${{\mathbb A}}^1$, this corresponds to multiplying by $\lambda$ the usual map from $NS(X)$ to $H^{1,1}(X,{{\mathbb C}})$. Restrict it to the subgroup $NS(X,D)^{\rm sat}$. Adding to $\Sigma _{\rm DH}$ gives a morphism $$\Lambda _{\rm DH} + \Sigma _{\rm DH}: NS(X,D)^{\rm sat}\times T(1,\log )^k \rightarrow H^{1,1}(X,{{\mathbb C}})\otimes T(1,\log ).$$ The Chern classes and residues of logarithmic $\lambda$-connections on line bundles $(L,\nabla )$ satisfy the condition , which in terms of the present notation says that the composed map $(\Lambda _{\rm DH} + \Sigma _{\rm DH})\circ (c_1,R)$ is zero. The following exact sequence is an analogue of the basic exact sequence and following diagram in §\[sec-exact\]. \[weight-exact\] The weight filtrations for $M_{\rm DH}(X,\log D)$ and $M_{\rm DH}(U)$ with group structure given by tensor product, fit into a strict exact sequence of analytic groups over ${{\mathbb P}}^1$ $$\label{exseq} 1\rightarrow M_{\rm DH}(X)^o\rightarrow M_{\rm DH}(X,\log D) \stackrel{(c_1,R)}{\longrightarrow} NS(X,D)^{\rm sat}\times T(1,\log )^k \cdots$$ $$\cdots \stackrel{\Lambda _{\rm DH} + \Sigma _{\rm DH}}{\longrightarrow} H^{1,1}(X,{{\mathbb C}})\otimes T(1,\log ).$$ It suffices to prove this in the fiber over a fixed $\lambda$, and we may assume $\lambda \in {{\mathbb A}}^1$. Injectivity on the left is easy and was mentionned previously: given $(L,\nabla )$ and $(L',\nabla ')$ on $X$, an isomorphism of logarithmic $\lambda$-connections on $(X,D)$ between them is also an isomorphism of $\lambda$-connections on $X$. For exactness at $M_{\rm DH}(X,\log D)$, suppose $(L,\nabla )$ is a logarithmic $\lambda$-connection such that $c_1(L)=0$ and $R(\nabla )=0$, that is ${\rm res}_{D_i}(\nabla )= 0$ for each component $D_i$. Then $\nabla$ is a connection over $X$ and even if $\lambda = 0$, the condition $c_1=0$ insures inclusion in $M_{\rm DH}(X)$. The fact that $c_1=0$ in the Neron-Severi group means that $L$ is algebraically equivalent to $0$. From the structure of the Picard group this implies that $L$ is in the connected component of the trivial bundle and $M_{\rm DH}(X)_{\lambda}\rightarrow Pic(X)$ is smooth, so $(L,\nabla )$ is in the connected component $M_{\rm DH}(X)^o_{\lambda}$. We prove exactness at $T(1,\log )^k \times NS(X,D)^{\rm sat}$. Use the standard frame for $T(1,\log )$. A point in $\ker (\Lambda _{\rm DH} + \Sigma _{\rm DH})$ is $\zeta \in NS(X,D)^{\rm sat}$ together with a $k$-uple $(a_1,\ldots , a_k)\in {{\mathbb C}}^k$ such that $$\lambda \zeta + \sum a_i[D_i] = 0 \;\;\; \mbox{in}\;\; H^{1,1}(X,{{\mathbb C}}).$$ Choose a line bundle $L$ such that $c_1(L) = \zeta $. The elements of $NS(X,D)^{\rm sat}$ restrict to torsion elements on $U$ by . Line bundles whose Chern class are torsion, have flat regular singular connections, which in the rank $1$ case are automatically logarithmic. Thus we can choose an initial $\lambda$-connection $\nabla '$ on $L$ logarithmic with respect to $(X,D)$. Let $a'_i$ denote the residues of $\nabla '$ along $D_i$. Then $$\sum (a_i-a'_i)[D_i] = 0 \;\; \;\mbox{in}\;\; H^{1,1}(X,{{\mathbb C}}).$$ Hence there is a logarithmic one-form $\beta$ on $(X,D)$ having residues $a_i-a'_i$ along $D_i$. Now $\nabla = \nabla ' + \beta $ is a logarithmic $\lambda$-connection with $(c_1,R)(L,\nabla ) = (\zeta , (a_1,\ldots , a_k))$. \[grw2-ident\] The exact sequence identifies the graded piece of the weight filtration as $$Gr^W_2 M_{\rm DH}(X,\log D)=$$ $$\ker \left( NS(X,D)^{\rm sat} \times T(1, \log ) ^k \stackrel{\Lambda _{\rm DH} + \Sigma _{\rm DH} }{\longrightarrow} H^{1,1}(X,{{\mathbb C}})\otimes T(1,\log ) \right) .$$ We can now describe the weight two phenomenon in the title of the paper: \[weight2\] There is $b$ such that $$\ker \left( T(1,\log ) ^k \rightarrow H^{1,1}(X,{{\mathbb C}})\otimes T(1,\log ) \right) \cong T(1,\log )^b.$$ There is an exact sequence $$0 \rightarrow T(1,\log )^b \rightarrow Gr^W_2 M_{\rm DH}(X,\log D) \rightarrow NS(X,D)^{\rm sat} \rightarrow 0.$$ On the connected component of the identity representation $$T(1,\log )^b \cong Gr^W_2 M_{\rm DH}(X,\log D)^o .$$ Modulo the gauge group we have $$0 \rightarrow {{{\mathbb G}}_m}(1)^b \rightarrow Gr^W_2 M_{\rm DH}(U) \rightarrow NS(U)^{\rm tors} \rightarrow 0.$$ A map between pure twistor structures of weight $2$ has a kernel which is again a pure twistor structure of weight $2$. Hence, there is $b$ as in the first claim. The first exact sequence comes from Proposition \[weight-exact\] and the fact that every element of $NS(X,D)^{\rm sat}$ goes into $H^{1,1}(X,{{\mathbb C}})$ to something which comes from ${{\mathbb C}}^k$. For the second exact sequence, note that $$Gr^W_2 \left( M_{\rm DH}(X,\log D)^o\right) = \left( Gr^W_2 M_{\rm DH}(X,\log D)^o\right)$$ because $W_1M_{\rm DH}(X,\log D)^o = M_{\rm DH}(X,\log D)$. For the last exact sequence, divide out by the gauge group ${{\mathcal G}}= {{\mathbb Z}}^k$, which means dividing the first exact sequence by the exact sequence $$0\rightarrow {{\mathbb Z}}^b \rightarrow {{\mathbb Z}}^k \rightarrow NS(X,D)\rightarrow 0.$$ Equation identifies the quotient of $NS(X,D)^{\rm sat}$ by $NS(X,D)$ with $NS(U)^{\rm tors}$. The weight equivalence relation induces an equivalence relation on sections: two sections are equivalent if and only if their values are equivalent over each $\lambda \in {{\mathbb P}}^1$. For this discussion, we work modulo the gauge group with $M_{\rm DH}(U)$. \[sectionsquotient\] There is a finite abelian group $K$ and an exact sequence $$0\rightarrow \Gamma ({{\mathbb P}}^1, T(1,\log )^b)^{\sigma}\rightarrow Gr ^W_2 \Gamma ({{\mathbb P}}^1, M_{\rm DH}(U))^{\sigma} \rightarrow K \rightarrow 0 .$$ For any $p\in {{\mathbb A}}^1$ the parabolic weight and residue give an exact sequence $$0\rightarrow \left( \frac{{{\mathbb R}}\times{{\mathbb C}}}{(1,-p){{\mathbb Z}}} \right) ^b \rightarrow Gr ^W_2 \Gamma ({{\mathbb P}}^1, M_{\rm DH}(U))^{\sigma} \rightarrow K \rightarrow 0 .$$ The second exact sequence comes from the first via Proposition \[prop-parabolicweight\]. The map $$\label{interchange} Gr ^W_2 \Gamma ({{\mathbb P}}^1, M_{\rm DH}(U))^{\sigma}\rightarrow \Gamma ({{\mathbb P}}^1, Gr ^W_2M_{\rm DH}(U))^{\sigma}$$ is injective. Suppose we have a $\sigma$-invariant section of $Gr ^W_2M_{\rm DH}(U)$. The obstruction to lifting it to a section in $Gr ^W_2 \Gamma ({{\mathbb P}}^1, M_{\rm DH}(U))^{\sigma}$ lies in $H^1({{\mathbb P}}^1,M_{\rm DH}(X)^o)$. In view of the exact sequence used in Lemma \[uniglobal\], we have $H^1({{\mathbb P}}^1,M_{\rm DH}(X)^o)=H^2({{\mathbb P}}^1,A)$ which is discrete. Therefore, on the connected component of the space of sections, the map is surjective. There is a finite subgroup $K\subset NS(U)^{\rm tors}$ representing the components in the image of . In fact $A=H^1(X,{{\mathbb Z}})$ and there is an exact sequence of the form $$0\rightarrow \Gamma ({{\mathbb P}}^1, T(1,\log )^b)^{\sigma}\rightarrow Gr ^W_2 \Gamma ({{\mathbb P}}^1, M_{\rm DH}(U))^{\sigma} \rightarrow NS(U)^{\rm tors} \rightarrow H^2({{\mathbb P}}^1,H^1(X,{{\mathbb Z}}))$$ I don’t know whether there are any examples where the last connecting map is nonzero. Proof of Theorem \[pref-id\] {#sub-proof} ---------------------------- Injectivity is proven in Corollary \[pref-id-inj\]. Suppose we are given a $\sigma$-invariant section in the target of the map ${{\mathcal P}}$. Lift it over the quotient of the action of the gauge group ${{\mathcal G}}$, to get a section $$\epsilon \in \Gamma ({{\mathbb P}}^1,M_{\rm DH}(X,\log D))^{\sigma}.$$ We would like to construct a harmonic bundle mapping to $\epsilon$. For this, we will use the correspondence [@hbnc] [@Mochizuki] [@Mochizuki2] between harmonic bundles and parabolic logarthmic $\lambda$-connections for some fixed $\lambda \in {{\mathbb A}}^1$. It would be sufficient to use the Higgs case $\lambda = 0$ or the de Rham case $\lambda = 1$ but it is interesting to treat a general $\lambda$. The value $\epsilon (\lambda )$ corresponds to a logarithmic $\lambda$-connection $(E,\nabla )$, with residue ${\rm res}_{\lambda ,D_i}(\epsilon )$ along each $D_i$. On the other hand, consider the parabolic weight parameter $$b_i:=\varpi _{\lambda ,D_i}(\epsilon )\in {{\mathbb R}}.$$ Put a parabolic structure onto $(E,\nabla )$ using these weights. This gives a parabolic logarithmic $\lambda$-connection $(E(\sum b_iD_i),\nabla )$. We claim that $c_1( E(\sum b_iD_i))=0$. To see this, look at the exact sequence of Proposition \[weight-exact\]. Take spaces of $\sigma$-invariant sections, and use the identification of Proposition \[prop-parabolicweight\] at our fixed $\lambda$. In these terms, $\epsilon$ maps to an element of $$\ker \left( NS(X,D)^{\rm sat} \times ({{\mathbb R}}\times {{\mathbb C}}) ^k \rightarrow H^{1,1}(X,{{\mathbb R}})\otimes _{{{\mathbb R}}} ({{\mathbb R}}\times {{\mathbb C}}) \right) .$$ The coefficient in $NS(X,D)^{\rm sat}$ is $\zeta = c_1(E)$, whereas the coefficient in ${{\mathbb R}}^k$ is $(b_1,\ldots , b_k)$. The image in the first factor $H^{1,1}(X,{{\mathbb R}})\otimes _{{{\mathbb R}}} {{\mathbb R}}$ is $$\varpi _{\lambda} (\Lambda _{\rm DH}(\zeta ) + \Sigma _{\rm DH}(b_1,\ldots , b_k) ).$$ Notice that $\varpi _{\lambda}\circ \Lambda _{\rm DH}$ is equal to the usual map $NS(X,D)^{\rm sat}\rightarrow H^{1,1}(X,{{\mathbb R}})$. This is because of the normalization condition that $\varpi _{\lambda}(1,0)=1$ used in §\[sub-parabolicweight\]. Similarly, $$\varpi _{\lambda} \Sigma _{\rm DH}(b_1,\ldots , b_k) = b_1[D_1]+ \ldots + b_k[D_k]\;\; \in H^{1,1}(X,{{\mathbb R}}).$$ We conclude that $$c_1( E(\sum b_iD_i)) = c_1(E) + b_1[D_1]+\ldots + b_k[D_k] = 0$$ as claimed. Then the harmonic theory for parabolic logarithmic connections [@EellsSampson] [@Corlette] [@DonaldsonApp] [@hbnc] [@Mochizuki2] [@Budur] provides a rank $1$ tame harmonic bundle ${{\mathcal E}}$ over $U$, whose associated parabolic $\lambda$-connection is $(E(\sum b_iD_i),\nabla )$. By Theorem \[harmoniccompose\], the parabolic weight of ${{\mathcal E}}$ is the same as the parabolic weight of the harmonic bundle, that is $$\varpi _{1,D_i}({{\mathcal P}}({{\mathcal E}})) = b_i .$$ This coincides with the parabolic weight of $\epsilon$. Furthermore, by construction the values of ${{\mathcal P}}({{\mathcal E}})$ and $\epsilon$ at $p=1$ are the same, both equal to $(E,\nabla )$. We conclude that ${{\mathcal P}}({{\mathcal E}}) = \varepsilon$, which concludes the proof of Theorem \[pref-id\], by the following lemma. \[section-unique\] Suppose $p\in {{\mathbb P}}^1$, and suppose $\xi , \epsilon$ are two $\sigma$-invariant sections of $M_{\rm DH}(X,\log D)$. Suppose that for each $D_i$, the parabolic weights agree $\varpi _{p,D_i}(\xi ) = \varpi _{p,D_i}(\epsilon )$. Suppose furthermore that $\xi (p) = \epsilon (p)$. Then $\xi = \epsilon$. The weight filtration exact sequence gives an exact sequence on spaces of $\sigma$-invariant sections. Then, identify the space of sections of $T(1,\log )^k$ with $({{\mathbb R}}\times {{\mathbb C}})^k$ using $(\varpi _p, {\rm res}_p)$ as in Proposition \[prop-parabolicweight\]. If $\xi (p)=\epsilon (p)$ then their residues at $p$ agree. By hypothesis the parabolic weight coordinates agree. Therefore, $\xi$ and $\epsilon$ go into the same section of $T(1,\log )^k$. They go to the same section of the discrete group $NS(X,D)^{\rm sat}$, because the values at $p$ are the same by hypthesis. By Lemma \[weight2\], $\xi$ and $\epsilon$ go to the same section of $Gr^W_2 M_{\rm DH}(X,\log D)$. Therefore the difference $\xi \otimes \epsilon ^{-1}$ comes from a $\sigma$-invariant section of $M_{\rm DH}(X)$. As was noted in Lemma \[uniglobal\], the weight $1$ property of $M_{\rm DH}(X)$ says that the space of $\sigma$-invariant sections here maps isomorphically to any fiber. The condition $\xi (p) = \epsilon (p)$ thus implies that $\xi \otimes \epsilon ^{-1}$ is trivial. Strictness consequences {#sec-strict} ======================= One of the most useful things about weights in Hodge theory is that they lead to a notion of strictness. Here we formulate a conjecture which would be the corresponding strictness property coming from the weight two piece of the nonabelian $H^1$. Since it is just a conjecture, we consider representations of any rank. Suppose $(X,D)$ and $(Y, E)$ are smooth projective varieties with simple normal crossings divisors, such that $D$ has $k$ components and $E$ has $m$ components. Suppose ${{\mathcal F}}$ is some natural construction from local systems on $U:=X-D$ to local systems on $V:=Y-E$. This could include any combination of pullbacks, higher direct images, tensor products, duals, etc. For the present purposes, denote by $M_B(U)$ and $M_B(V)$ the full unions of spaces of representations of all ranks. There will be a stratification of $M_B(U)$ into locally closed subsets such that ${{\mathcal F}}$ is algebraic on each stratum. Assume that this stratification is maximal, that is $M_B(U)_{\alpha}$ is the full subset of representations $\rho$ on $U$ of a given rank, such that the image ${{\mathcal F}}(\rho )$ has a given rank on $V$. In the higher rank case, the eigenvalues of the local monodromy transformations may be considered all at once, with their multiplicities, as divisors on ${{{\mathbb G}}_m}^{\perp}$. The group of such divisors is denoted $Div({{{\mathbb G}}_m}^{\perp})$, and for a divisor $D$ decomposing into $k$ irreducible components, the full collection of residual data is a point in $Div({{{\mathbb G}}_m}^{\perp})^k$. \[conj-strictness\] Let $M_B(U)_{\alpha}$ be a stratum on which ${{\mathcal F}}={{\mathcal F}}_{\alpha}$ is defined as an algebraic map into $M_B(V)$. (1) There should be a diagram expressing the effect of the construction ${{\mathcal F}}_{\alpha}$ on residues: $$\begin{array}{ccc} M_B(U)_{\alpha} & \rightarrow & Div({{{\mathbb G}}_m}^{\perp})^k \\ \downarrow && \downarrow \\ M_B(V)_{\alpha} & \rightarrow & Div({{{\mathbb G}}_m}^{\perp})^m \, . \end{array}$$ (2) The following strictness property holds: suppose $\rho _1, \rho _2 \in M_B(U)_{\alpha}$ are two semisimple representations such that ${{\mathcal F}}_{\alpha} (\rho _1)$ and ${{\mathcal F}}_{\alpha}(\rho _2)$ have the same residues in $Div({{{\mathbb G}}_m}^{\perp})^m$. Then there exists a semisimple representation $\rho _3\in M_B(U)_{\alpha}$ such that $\rho _3$ has the same residues as $\rho _1$ in $Div({{{\mathbb G}}_m}^{\perp})^k$, but ${{\mathcal F}}(\rho _3) \cong {{\mathcal F}}(\rho _2)$. To phrase it differently, this conjecture says that any variation of the image representation ${{\mathcal F}}(\rho )$, within a locus of representations on $V$ all having the same residues, obtained by possibly varying the residues of $\rho$, can equally well be obtained while keeping the residues of $\rho$ fixed. It would be the analogue of the same statement in abelian Hodge theory for the diagram $$\begin{array}{ccc} H^1(U) & \rightarrow & Gr^W_2(H^1(U)) \\ \downarrow && \downarrow \\ H^1(V) & \rightarrow & Gr^W_2(H^1(V)) \end{array} .$$ In the abelian case, pretty much the only possibility for the construction ${{\mathcal F}}$ is pullback for a map $V \rightarrow U$. The strictness statement says that if $a_1,a_2$ are classes in $H^1(U)$ whose pullbacks to $V$ have the same residues along $D$, then there is a class $a_3$ with the same residues as $a_2$, whose pullback coincides with the pullback of $a_2$. Our observation of the weight two phenomenon in the case of rank one local systems should provide a proof of this conjecture for the rank one case. We don’t discuss that here: it would go beyond the scope of the paper. One can also expect an infinitesimal formulation of the strictness property, which might be easier to prove. It would be the same statement, in the case where $\rho _1$ and $\rho _2$ are infinitesimally close, and we would look for $\rho _3$ also infinitesimally close. This should be a consequence of having a mixed Hodge structure on the local deformation theory [@BrylinskiFoth] [@Foth] [@PridhamMHS] [@PridhamQl], plus a compatibility of the construction ${{\mathcal F}}$ with this mixed Hodge structure. Again, this goes out of the scope of the present discussion. One should also be able to formulate a similar conjecture for harmonic bundles with the parabolic residual data characterized by points in $Div(\frac{{{\mathbb R}}}{{{\mathbb Z}}} \times {{\mathbb C}})^k$. 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--- abstract: 'We introduce the notions of 2-covering maps and 2-fundamental groups of graphs, and investigate their basic properties. These concepts are closely related to Hom complexes and neighborhood complexes. Indeed, we prove that the fundamental group of a neighborhood complex is isomorphic to a subgroup of the 2-fundamental group whose index is 1 or 2. We prove that the 2-fundamental group and the fundamental group of a neighborhood complex for a connected graph whose chromatic number is 3 have group homomorphisms onto $\mathbb{Z}$.' author: - Takahiro Matsushita title: Fundamental groups of neighborhood complexes --- Introduction ============ Neighborhood complexes were defined by Lov$\acute{\rm a}$sz in [@lov] in the context of the graph coloring problem. He proved the connectivity of the neighborhood complex gives the lower bound of the chromatic number and determined the chromatic number of Kneser’s graphs. In this paper, we give another interpretation of the fundamental group of the neighborhood complex, and obtain a necessary condition for a graph that its chromatic number is 3. Usually, a relation between the chromatic number and the topology of the graph complex is obtained by using characteristic classes of principal bundles of finite groups (see [@Koz07], [@Kriz], and [@MZ04]), but our method is different. We introduce the notions of 2-covering maps and 2-fundamental groups of graphs. 2-covering maps are essentially different from usual covering maps of graphs. In fact, we prove that the connected 2-covering over a complete graph $K_n$ on $n$ vertices for $n\geq 4$ are only $K_n$ and $K_2 \times K_n$ (Corollary 5.2). We study that 2-fundamental groups are closely related to 2-covering maps, as is the case of the covering space theory in topology. For example, there is a natural correspondence between subgroups of the 2-fundamental group of a graph $G$ and the connected based 2-coverings over $G$ (Theorem 4.16). After establishing the basic theory of 2-covering maps and 2-fundamental groups, we prove that a 2-fundamental group of a connected graph whose chromatic number is 3 has a surjection to $\mathbb{Z}$ (Corollary 5.4). Finally, we prove that the subgroup of a 2-fundamental group, called the even part, whose index is 1 or 2 is isomorphic to the fundamental group of the neighborhood complex (Theorem 6.1). Then we prove that the 1-dimensional homology group of $\mathcal{N}(G)$ has $\mathbb{Z}$ as a direct summand (Corollary 6.2). The rest of this paper organized as follows. In Section 2 we review the all necessary definitions and facts related to graphs, simplicial complexes, neighborhood complexes, and Hom complexes. In Section 3 we provide the definition of 2-covering maps and study the basic properties of 2-covering maps. Here we prove that 2-covering maps induce covering maps of neighborhood complexes and Hom complexes. In Section 4 we provide the definition of 2-fundamental groups and study their basic properties and investigate the relation between 2-fundamental groups and based 2-covering maps. In Section 5, we compute 2-fundamental groups for basic graphs and establish van Kampen theorem for 2-fundamental groups. In Section 6, we prove the even part of the 2-fundamental group of $G$ is isomorphic to the fundamental group of the neighborhood complex of $G$. Definitions and Facts ===================== In this section, we review the basic definitions and facts used in this paper relating graphs following [@BK06], [@Koz07] and [@Koz08]. For the covering space theory in topology, we refer to [@H].\ \ **Graphs :** A *graph* is a pair $(V,E)$ where $V$ is a set and $E$ is a subset of $V\times V$ such that $(x,y)\in E$ implies $(y,x)\in E$. Therefore our graphs are undirected, simple and may have loops. For a graph $G=(V,E)$, $V$ is called the [*vertex set*]{} of $G$ and $E$ is called the [*edge set*]{} of $G$. We write $V(G)$ for the vertex set of $G$ and $E(G)$ for the edge set of $G$. For vertices $v,w$ of $G$, we write $v \sim w$ if $(v,w)\in E(G)$. Let $G$ and $H$ be graphs. A map $f :V(G) \rightarrow V(H)$ is called a [*graph homomorphism*]{} or a [*graph map*]{} from $G$ to $H$ if $f\times f (E(G)) \subset E(H)$. Let $G$ be a graph. A subset $A$ of $V(G)$ is called independent if $A\times A \cap E(G) =\emptyset$. We define the graph $K_n$ for $n \in \mathbb{N} =\{ 0,1,2, \cdots \}$ by $V(K_n)$=$\{ 0,1, \cdots , n-1\}$ and $E(K_n)=\{ (x,y)\in V(G) \times V(G) |x\neq y\}$, which is called the complete graph on $n$-vertices. For a graph $G$, a graph homomorphism from $G$ to $K_n$ is called an [*$n$-coloring*]{} of $G$. Set $$\chi(G) = {\rm min}\{ n \; | {\rm \; There \; is \; a \; graph \; homomorphism \; from \;} G \; {\rm to} \; K_n. \}.$$ In this paper, we set $\chi (G) =\infty$ if there is no $n$ such that there is a graph homomorphisms from $G$ to $K_n$. $\chi (G)$ is called the [*chromatic number*]{} of $G$. We say that $G$ is [*bipartite*]{} if $\chi(G) =2$. A graph $G$ is said to be [*connected*]{} if $V(G)\neq \emptyset$ and each pair $(v,w) \in V(G)\times V(G)$ there exists a finite sequence $(v_0, \cdots , v_n)$ of vertices of $G$ such that $v_0=v$, $v_n=w$, and $(v_{i-1},v_i)\in E(G)$ for every $i \in \{1, \cdots ,n \}$. Let $G$ be a graph. A graph $H$ is called a [*subgraph*]{} of $G$ if $V(H) \subset V(G)$ and $E(H) \subset E(G)$. Let $\{ H_{\alpha}\}_{\alpha \in A}$ be a family of subgraphs of $G$. The subgraph $(\bigcup_{\alpha \in A} V(H_{\alpha}), \bigcup_{\alpha \in A} E(H_{\alpha}))$ of $G$ is called the [*union*]{} of $\{ H_{\alpha}\}_{\alpha \in A}$, written by $\bigcup_{\alpha \in A}H_{\alpha}$, and the subgraph $(\bigcap _{\alpha \in A} V(H_{\alpha}), \bigcap _{\alpha \in A} E(H_{\alpha}))$ of $G$ is called the [*intersection*]{} of $\{ H_{\alpha}\}_{\alpha \in A}$, and written by $\bigcap_{\alpha \in A}H_{\alpha}$. Let $\{ G_{\alpha }\}_{\alpha \in A}$ be a family of graphs. We define the [*product*]{} $\prod_{\alpha \in A} G_{\alpha}$ of $\{ G_{\alpha}\}_{\alpha \in A}$ by setting $V(\prod_{\alpha \in A} G_{\alpha})=\prod_{\alpha \in A} V(G_{\alpha})$ and $E(\prod_{\alpha \in A} G_{\alpha})= \{ ((x_{\alpha})_{\alpha \in A} ,(y_{\alpha})_{\alpha \in A}) \; | \; (x_{\alpha},y_{\alpha}) \in E(G)$ for each $\alpha \in A .\}$. We define the [*coproduct*]{} $\coprod_{\alpha \in A} G_{\alpha}$ of $\{ G_{\alpha}\}_{\alpha \in A}$ by setting $V(\coprod_{\alpha \in A} G_{\alpha})=\coprod_{\alpha \in A} V(G_{\alpha})$ and $E(\coprod_{\alpha \in A} G_{\alpha})=\coprod_{\alpha \in A} E(G_{\alpha})$. A right action of a group $\Gamma$ on a graph $G$ is a right action $\alpha :V(G)\times \Gamma \rightarrow V(G)$ of $\Gamma$ on $V(G)$ as a set such that $\alpha (x,\gamma) \sim \alpha (y,\gamma )$ if $x\sim y$ for all $x,y \in V(G)$ and $\gamma \in \Gamma$. A graph $G$ with a right action of $\Gamma$ on $G$ is called a [*right $\Gamma$-graph*]{}. We define similarly a left $\Gamma$-graph. Let $G$ be a graph and $R$ an equivalence relation on $V(G)$. We define the graph $G/R$ by setting $$V(G/R)=V(G)/R$$ $$E(G/R)=\{ (\alpha ,\beta )\; | \; \alpha \times \beta \cap E(G) \neq \emptyset \}.$$ Then the quotient map $q:V(G)\rightarrow V(G/R)$ is a graph homomorphism. For a graph homomorphism $f:G\rightarrow H$ such that $f(v)=f(w)$ for all $(v,w) \in R$, we can easily see that there exists a unique graph homomorphism $\overline{f}:G/R\rightarrow H$ such that $\overline{f}\circ q=f$. Let $\Gamma$ be a group and $G$ a right $\Gamma$-graph . We define the equivalence relation $R_{\Gamma}$ on $V(G)$ by setting $$R_{\Gamma}=\{ (v,v\gamma) \; | \; v\in V(G) , \gamma \in \Gamma \}.$$ We write $G/\Gamma$ for $G/R_{\Gamma}$. We define similarly $\Gamma \setminus H$ for a left $\Gamma$-graph $H$. Let $G$ be a graph and $v\in V(G)$. We write $N(v)$ for the set $\{ w\in V(G) \; | \; (v,w) \in E(G)\}$. $N(v)$ is called the [*neighborhood*]{} of $v$ in $G$. We say that $v$ is [*isolated*]{} if $N(v)=\emptyset$. In general, for a subset $A$ of $V(G)$, we write $N(A)$ for the set $\{ w\in V(G) \; |$ There is $u\in A$ such that $(u,w)\in E(G). \}$. We write $N_2(v)$ for $N(N(v))$. A [*based graph*]{} is a pair $(G,v)$ where $G$ is a graph and $v$ is a vertex of $G$. Let $(G,v)$ and $(H,w)$ be based graphs. A graph homomorphism $f$ from $G$ to $H$ such that $f(v)=w$ is called a [*based graph homomorphism*]{} or a [*based graph map*]{} from $(G,v)$ to $(H,w)$.\ \ **Simplicial complex :** A pair $(V, \Delta)$ is called an [*abstract simplicial complex*]{} or a [*simplicial complex*]{} if $(V,\Delta)$ satisfies the following properties.\ (0) $V$ is a set and $\Delta$ is a subset of $2^V$.\ (1) Each $\sigma \in \Delta$ is a finite subset of $V$.\ (2) For each $v\in V$, $\{ v\} \in \Delta$.\ (3) Let $\tau ,\sigma \in 2^{V}$ with $\tau \subset \sigma$. If $\sigma \in \Delta$, then $\tau \in \Delta$. Let $(V, \Delta)$ be a simplicial complex. $V$ is called a [*vertex set*]{} of $(V,\Delta)$. We often abbreviate $(V,\Delta)$ to $\Delta$ for a simplicial complex $(V,\Delta)$. In this notation, the vertex set of $\Delta$ is written by $V(\Delta)$. Let $\Delta$ and $\Delta'$ be simplicial complexes. We say that $\Delta'$ is a [*subcomplex*]{} of $\Delta$ if $V(\Delta') \subset V(\Delta)$ and $\Delta' \subset \Delta$, and written by $\Delta' \subset \Delta$. Let $\Delta$ be a simplicial complex and $v\in V(\Delta)$. We define the [*star*]{} of $v$, written by ${\rm st}(v)$, as the subcomplex $\{ \sigma \in \Delta \; | \; \sigma \cup \{ v\} \in \Delta \}$ of $\Delta$. Let $\Delta_1$ and $\Delta_2$ be simplicial complexes. A map $f:V(\Delta_1) \rightarrow V(\Delta _2)$ is called a [*simplicial map*]{} if $f(\sigma) \in \Delta_2$ for each $\sigma \in \Delta_1$. A partially orderd set is called a [*poset*]{}. Let $P$ be a poset. A subset $P'$ of $P$ is called a [*chain*]{} of $P$ if the restriction of the partial order of $P$ to $P'$ is a total order of $P'$. We set $\Delta (P)=\{ P' \; | \; P'$ is a finite chain of $P$.$\}$. Then $\Delta (P)$ forms a simplicial complex and is called the [*order complex*]{} of $P$. Let $f:P\rightarrow Q$ be an order preserving map. Since $f$ preserves finite chains, we have a simplicial map $\Delta (f): \Delta (P)\rightarrow \Delta (Q)$. Let $V$ be a set. We write $\mathbb{R}^{(V)}$ for a free $\mathbb{R}$-module generated by $V$. We regard $\mathbb{R}^{(V)}$ as a topological space with the direct limit topology of finite dimensional vector subspaces of $\mathbb{R}^{(V)}$. For a finite subset $S\subset V$, the topological subspace $\{ \sum_{i=0}^{n} a_i v_i \; |\; v_i \in S, a_i \geq 0, \sum_{i=1}^{n} a_i =1 \}$ of $\mathbb{R}^{(V)}$ is written by $\Delta_S$. Let $\Delta$ be a simplicial complex. The topological subspace $$|\Delta |=\bigcup_{\sigma \in \Delta } \Delta_{\sigma}$$ of $\mathbb{R}^{(V(\Delta))}$ is called the [*geometrical realization*]{} of $\Delta$. For a poset $P$, the geometrical realization of $\Delta (P)$ is called the geometrical realization of $P$, and we write $|P|$ for $|\Delta(P)|$.\ \ **Hom complex :** Let $G$ and $H$ be graphs. A [*multihomomorphism*]{} from $G$ to $H$ is a map $\eta :V(G) \rightarrow 2^{V(H)}$ such that $\eta (v)\times \eta(w) \subset E(H)$ for all $(v,w) \in E(G)$. For multihomomorphisms $\eta ,\eta '$ from $G$ to $H$, we write $\eta \leq \eta'$ if $\eta(v) \subset \eta'(v)$ for each $v\in V(G)$. Then $\leq $ is a partial order of the set of all multihomomorphisms from $G$ to $H$ and we write ${\rm Hom}(G,H)$ for this poset. ${\rm Hom}(G,H)$ is called the [*Hom complex*]{} from $G$ to $H$. We remark that a minimal point of ${\rm Hom}(G,H)$ is identified with a graph homomorphism from $G$ to $H$. Let $G,H_1$ and $H_2$ be graphs and $f:H_1\rightarrow H_2$ a graph homomorphism. Then we have the poset map $f_* :{\rm Hom}(G,H_1 )\rightarrow {\rm Hom}(G,H_2)$ by setting $(f_* \eta )(x) = f(\eta (x))$ for $x\in V(G)$ and $\eta \in {\rm Hom}(G,H_1)$. And we have the poset map $f^* :{\rm Hom}(H_2,G)\rightarrow {\rm Hom}(H_1,G)$ by setting $f^* (\eta)(x)=\eta (f(x))$ for $x\in V(H_1)$ and $\eta \in {\rm Hom}(H_2,G)$. In some literature, Hom complex is defined as follows for finite graphs. Let $G$ and $H$ be finite graphs. We remark that there is a natural correspondence between the set of the cells of $|\Delta ^{V(H)}|$ and the set of subsets of $V(H)$. In this correpondence, a multihomomorphism from $G$ to $H$ determines a cell of $\prod_{v\in V(G)} |\Delta^{V(H)}|$. We write $X_{G,H}$ for the union of all cells of $\prod_{v\in V(G)} |\Delta^{V(H)}|$ determined by the multihomomorphisms from $G$ to $H$. We remark that the definition of $X_{G,H}$ needs the assumption that $G$ and $H$ are finite graphs. Some people say that $X_{G,H}$ is the Hom complex from $G$ to $H$. However, since $X_{G,H}$ is a regular CW-complex and the face poset of $X_{G,H}$ is identified with ${\rm Hom}(G,H)$ we defined, $X_{G,H}$ is naturally homeomorphic to $|{\rm Hom}(G,H)|$. But since we want to consider the infinite graphs, we defined Hom complex as above. For more details and interests about Hom complexes, see [@BK06], [@Koz07], [@Koz08], [@Sch09].\ \ **$\times$-homotopy theory :** Dochtermann established a homotopy theory of graphs in [@Doca] called $\times$-homotopy theory as follows. For $n \in \mathbb{N}=\{ 0,1,2, \cdots \}$, we define the graph $I_n$ by $V(I_n)=\{ 0,1, \cdots ,n \}$ and $E(I_n)=\{ (x,y) \; | \; |x-y| \leq 1\}$. Let $f,g : G\rightarrow H$ be graph homomorphisms. A $\times$-homotopy from $f$ to $g$ is a graph homomorphism $F: G\times I_n \rightarrow H$ for some $n \in \mathbb{N}$ such that $F(x,0)=f(x)$ and $F(x,n)=g(x)$ for each $x\in V(G)$. If there exists a $\times$-homotopy from $f$ to $g$, $f$ is said to be [*$\times$-homotopic*]{} to $g$ and written by $f \simeq _{\times} g$ or simply by $f\simeq g$. We can see that $f$ is $\times$-homotopic to $g$ if and only if $f$ and $g$ are in the same connected component of ${\rm Hom}(G,H)$. In fact, for graph homomorphisms $f,g$ from $G$ to $H$, the set map $F:V(G\times I_1) \rightarrow V(H)$ defined by $(x,0) \mapsto f(x)$ and $(x,1) \mapsto g(x)$ is a graph homomorphism if and only if the set map $V(G) \rightarrow 2^{V(H)} -\{ \emptyset \}$ defined by $x\mapsto \{ f(x) ,g(x)\}$ is a multihomomorphism. A graph homomorphism $f:G\rightarrow H$ is called a [*$\times$-homotopy equivalence*]{} if there exists a graph homomorphism $g:H\rightarrow G$ such that $gf \simeq _{\times} {\rm id}_G$ and $fg \simeq_{\times} {\rm id}_H$. An important example of $\times$-homotopy equivalences is a folding homomorphism defined as follows. Given a graph $G$ and $v \in V(G)$, we write $G\setminus v$ for the graph $V(G\setminus v) =V(G)\setminus \{ v\}$ and $E(G\setminus v)=E(G) \cap V(G\setminus v)\times V(G\setminus v)$. The graph $G \setminus v$ is called a [*fold*]{} of $G$ if there exists $w \in V(G)$ such that $N(v) \subset N(w)$. In this case $f_v:V(G) \rightarrow V(G\setminus v)$ defined by $x\mapsto x$ for $x\neq v$ and $v\mapsto w$ is called a [*folding map*]{}. It is easy to see that $f_v$ is a $\times$-homotopy equivalence and its $\times$-homotopy inverse is the inclusion $G \setminus v \rightarrow G$.\ \ [**Neighborhood complex :**]{} For a graph $G$, a neighborhood complex is the simplicial complex $$\mathcal{N}(G)=\{ A\subset V(G) \; | \; \sharp A <\infty \; {\rm and \; there \; exists \;}v\in V(G) \; {\rm such \; that \;} A\subset N(v). \}.$$ We remark that a vertex of $\mathcal{N}(G)$ is a non-isolated vertex of $G$. If $G$ is finite, $\mathcal{N} (G)$ is known to be homotopy equivalent to ${\rm Hom}(K_2,G)$ (see [@BK06]). Moreover, $\mathcal{N} (G)$ is known to be simple homotopy equivalent to ${\rm Hom}(K_2,G)$ (see [@Koz06]). For more details and interests about neighborhood complexes, see [@Cs], [@lov], [@Ziv]. 2-covering maps of graphs ========================= In this section, we introduce the notion of a 2-covering map and investigate its basic properties.\ [**Definition of 2-covering maps :**]{} Given a graph $G$ and a vertex $v$ of $G$. Recall that the set $\{ w\in V(G) \; | \; (v,w) \in E(G)\}$ is written by $N(v)$ and $\{ w\in V(G) \; |$ There is $u\in V(G)$ such that $(v,u),(u,w)\in E(G).\}$ is written by $N_2(v)$. A graph homomorphism $p:G\rightarrow H$ is called a [*2-covering map*]{} [if $p|_{N(v)}: N(v) \rightarrow N(pv)$ and $p|_{N_2(v)}:N_2(v) \rightarrow N_2(pv)$ are both bijective for each $v \in V(G)$. A 2-covering map $p:G\rightarrow H$ is said to be [*connected*]{} if $G$ is connected. ]{} We do not assume that a 2-covering map is surjective. Thus the inclusion $\emptyset \rightarrow G$ is a 2-covering map. [ (1) An identity map is a 2-covering map.\ (2) For a graph $G$, the second projection $K_2\times G \rightarrow G$ is a 2-covering map. We remark that $K_2\times G$ is 2-colorable for any $G$. Suppose $G$ is connected. Then $K_2 \times G$ is connected if and only if $\chi (G) \geq 3$. In Section 5, we prove that the only connected 2-coverings over $K_n$ for $n\geq 4$ are $K_n$ and $K_2 \times K_n$.\ (3) The cyclic graph $C_n$ for $n\geq 3$ is defined by $V(C_n)=\mathbb{Z} /n\mathbb{Z}$ and $E(C_n)=\{ (x,x+1) ,(x+1,x) \; | \; x\in \mathbb{Z}/n\mathbb{Z}\}$. If $n\neq 4$, then the graph homomorphism $p:C_{nk} \rightarrow C_n$ defined by $p(x\; {\rm mod.}nk)=p(x\; {\rm mod.}n)$ is a 2-covering map. Since $K_3=C_3$, there exist infinitely many connected 2-coverings over $K_3$. We remark that these graph homomorphisms do not preserve $N_3(x)$ bijectively for each $x\in C_{nk}$, where $N_3(x)=N(N_2(x))$, except for $C_6\cong K_2 \times K_3 \rightarrow K_3$.\ (4) The graph $L$ is defined by $V(L)=\mathbb{Z}$ and $E(L)=\{ (x,y)\; |\; |x-y|=1 \}$. If $n\geq 3$ and $n\neq 4$, the graph homomorphism $p$ from $L$ to $C_n$ defined by $p(x)=(x \; {\rm mod.}n)$ is a 2-covering map. ]{} Let $p$ be a graph homomorphism from a graph $G$ to a graph $H$. If $p|_{N(v)}:N(v)\rightarrow N(p(v))$ is surjective and $p|_{N_2(v)}: N_2(v)\rightarrow N_2(p(v))$ is injective for each $v\in V(G)$, then $p$ is a 2-covering map. Let $v\in V(G)$. We want to show that $p|_{N(v)}$ is injective and $p|_{N_2(v)}$ is surjective. Let $w_1, w_2 \in N(v)$ with $p(w_1) =p(w_2)$. Since $w_1, w_2 \in N_2(w_1)$, we have $w_1 =w_2$ from the injectivity of $p|_{N_2(w_1)}$. Therefore $p|_{N(v)}$ is injective. Let $x\in N_2(p(v))$. Then there exists $y\in N(p(v))$ such that $x\in N(y)$. We have $w\in N(v)$ with $p(w) = y$ from the surjectivity of $p|_{N(v)}$, and $u\in N(w)$ with $p(u) =x$ from the surjectivity of $p|_{N(w)}$. Since $u \in N_2(v)$, we have $p|_{N_2(v)}$ is surjective. Let $f:G\rightarrow H$, $g:H\rightarrow K$ be graph homomorphisms. Then the followings hold.\ (1) If $g$ and $f$ are 2-covering maps, then $gf$ is a 2-covering map.\ (2) If $g$ and $gf$ are 2-covering maps, then $f$ is a 2-covering map.\ (3) If $f$ is a surjective 2-covering map and $gf$ is a 2-covering map, then $g$ is a 2-covering map. Let $v \in V(G)$. We have the following commutative diagrams. $$\begin{CD} N(v) @>f|_{N(v)}>> N(f(v)) \\ @Vgf|_{N(v)}VV @VVg|_{N(f(v))}V \\ N(gf(v)) @= N(gf(v)) \end{CD} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \begin{CD} N_2(v) @>f|_{N_2(v)}>> N_2(f(v)) \\ @Vgf|_{N_2(v)}VV @VVg|_{N_2(f(v))}V \\ N_2(gf(v)) @= N_2(gf(v)) \end{CD}$$ If one of (1), (2), (3) holds, two of three arrows in each diagram are bijective. Hence so is third. Let $G$ and $H$ be graphs and $p:G\rightarrow H$ be a 2-covering map. Then the map $|\mathcal{N}(G)|\rightarrow |\mathcal{N}(H)|$ induced by $p$ is a covering map in a topological sense. Firstly, we remark that a vertex of a neighborhood complex is a non-isolated vertex. It is sufficient to prove that $p^{-1}({\rm st}(v))=\coprod_{v_i \in p^{-1}(v)} {\rm st}(v_i)$ for each $v\in V(H)$ with $N(v)\neq \emptyset$, where $p^{-1}({\rm st}(v))$ is the subcomplex of $\mathcal{N}(G)$ whose simplex is $\sigma \in \mathcal{N}(G)$ such that $p(\sigma) \in {\rm st}(v)$. Suppose $w\in V({\rm st}(v_i)) \cap V({\rm st}(v_j))$ for $v_i,v_j \in p^{-1}(v)$. Since $v_i ,v_j \in N_2(w)$ and $p(v_i)=p(v_j)$, we have $v_i =v_j$. Therefore ${\rm st}(v_i)$ for $v_i \in p^{-1}(v)$ are disjoint. Let $\sigma \in \mathcal{N}(G)$. Suppose $\emptyset \neq \sigma \in p^{-1}({\rm st}(v))$. Then there exists $v' \in V(G)$ such that $p(\sigma)\cup \{ v\} \subset N(v')$. Since $p$ is a 2-covering map, there exists $w'\in p^{-1}(v')$ such that $\sigma \subset N(w')$. Let $w \in N(w')$ with $p(w) =v$. Then we have $\sigma \in {\rm st}(w)$ with $w\in p^{-1}(v)$. Therefore we have $p^{-1}({\rm st}(v))\subset \coprod_{v_i \in p^{-1}(v)} {\rm st}(v_i)$. On the other hand, $\coprod_{v_i \in p^{-1}(v)} {\rm st}(v_i) \subset p^{-1}({\rm st}(v))$ is obvious. Let $T$ be a connected graph having no isolated points and $p:G\rightarrow H$ be a 2-covering map and $\eta_0 ,\eta_1 \in {\rm Hom}(T,G)$ with $p_*\eta_0 =p_*\eta_1$. If there exists $x \in V(T)$ such that $\eta_0(x)\cap \eta_1(x) \neq \emptyset$, then $\eta_0 =\eta_1$. Let $v \in \eta_0(x) \cap \eta_1(x)$. Since $T$ has no isolated points, we have $\eta_0(x) \subset N_2(v)$ and $\eta_1 (x) \subset N_2(v)$. Since $p(\eta_0(x))=p(\eta_1(x))$ and $p|_{N_2(v)}$ is injective, we have $\eta_0(x)=\eta_1(x)$. Since $T$ is connected and $\eta_0(x)=\eta_1(x)$, it is sufficient to show that, for each $(y,z)\in E(G)$, $\eta_0(y)=\eta_1(y)$ implies $\eta_0(z)=\eta_1(z)$. Thus let $(y,z)\in E(G)$ and suppose $\eta_0(y)=\eta_1(y)$ and let $w\in \eta_0(y)$. Since $\eta_0(z) \subset N(w)$ and $\eta_1(z) \subset N(w)$, $\eta_0(z)=\eta_1(z)$ from the injectivity of $p|_{N(w)}$. Let $T$ be a graph having no isolated points and $p: G\rightarrow H$ be a 2-covering map. Let $\eta \in {\rm Hom}(T,G)$ and put $\zeta =p_{*} \eta$.\ (1) For each $\zeta _0 \leq \zeta$ , there exists a unique $\eta_0 \in {\rm Hom}(T,G)$ such that $p_{*}\eta _0 =\zeta _0 $ and $\eta_0 \leq \eta$.\ (2) For each $\zeta _1 \geq \zeta$ , there exists a unique $\eta_1\in {\rm Hom}(T,G)$ such that $p_{*}\eta _1 =\zeta _1 $ and $\eta_1 \geq \eta$. Firstly, we prove (2). Choose $v_x \in \eta_0 (x)$ for each $x\in V(T)$. Define a map $\eta_1: V(T) \rightarrow 2^{V(G)}-\{ \emptyset \} $ by $\eta_1(x)=(p|_{N_2(v_x)})^{-1}(\zeta _1(x))$. Since $T$ has no isolated points, $\emptyset \neq \zeta_1(x) \subset N_2(p(v_x))$. Hence $\eta_1(x)$ is not empty. We must prove $\eta_1$ is a multihomomorphism. Let $(x,y)\in E(T)$, and $v\in \eta_1(x)$ and $w \in \eta_1(y)$. We want to show $(v,w)\in E(G)$. Since $p(v) \in \zeta_1(x) \subset N(p(v_y))$, there exists $v' \in N(v_y)$ such that $p(v)=p(v')$. Since $v,v' \in N_2(v_x)$, we have $v=v'$ from the injectivity of $p|_{N_2(v_x)}$. Therefore we have $v\sim v_y$. Similarly we have $v_x \sim w$. So we have $$v\sim v_y \sim v_x \sim w$$ and $p(v) \sim p(w)$. There exists $w' \in N(v)$ such that $p(w')=p(w)$. Since $w,w' \in N_2(v_y)$, we have $w=w'$. Therefore we have $v\sim w$ and $\eta _1$ is a multihomomorphism. We prove the uniqueness of $\eta _1$. Let $\eta' \in {\rm Hom}(T,G)$ such that $\eta \leq \eta'$ and $p_{*}(\eta ')=\zeta_1$. Let $x \in V(T)$. Since $\eta _1(x)$ and $\eta'(x)$ are subset of $N_2(v_x)$, and $p(\eta_1(x))=p(\eta'(x))$, we have $\eta_1(x)=\eta'(x)$. The proof of (1) is similar and much easier. Indeed, we construct $\eta_0$ by setting $\eta_0(x)=(p|_{N_2(v_x)})^{-1}(\zeta _0(x))$. $\eta_0$ is obviously a multihomomorphism since $\eta_0(x) \subset \eta(x)$. (Homotopy Lifting Property) Let $G,H$ and $T$ be graphs and $p:G\rightarrow H$ be a 2-covering map. Suppose $T$ has no isolated vertices. Given graph homomorphisms $F:T\times I_n \rightarrow G$ and $f:T\rightarrow H$ such that $F(x,0)=pf(x)$ for each $x\in V(T)$. Then there exists a unique graph homomorphism $\tilde{F}:T\times I_n \rightarrow G$ such that $\tilde{F}(x,0)=f(x)$ for each $x\in V(T)$ and $p\tilde{F}=F$. We can assume $n=1$. The map $\eta:V(T)\rightarrow 2^{V(H)} , x \mapsto \{ F(x,0),F(x,1)\}$ forms a multihomomorphism. From Proposition 3.7, we have a multihomomorphism $\tilde{\eta}:V(T)\rightarrow 2^{V(G)}$ such that $f\leq \tilde{\eta}$, and a graph homomorphism $g:T\rightarrow G$ such that $g\leq \tilde{\eta}$ and $pg(x)=F(x,1)$. We define a set map $\tilde{F}:V(T\times I_1) \rightarrow V(G)$ by $\tilde{F}(x,0)=f(x)$ and $\tilde{F}(x,1)=g(x)$. Then $F$ is a graph homomorphism since $f,g \leq \tilde{\eta}$. The uniqueness of $\tilde{F}$ is obvious since $\tilde{F}(x,1)$ is the unique element of $N_2(f(x))$ mapped to $F(x,1)$ by $p$. Let $T$ be a graph having no isolated points and $p: G\rightarrow H$ be a 2-covering map. Then the map $|p_*|:|{\rm Hom}(T,G)| \rightarrow |{\rm Hom}(T,H)|$ is a covering map in a topological sense. This is obtained from Proposition 3.7 and the following lemma. Let $P,Q$ be posets and $f:P\rightarrow Q$ be an order preserving map. If the following two conditions are satisfied, then $|f|: |P| \rightarrow |Q|$ is a covering map.\ (1) For $x \in P$ and $y \in Q$ with $y \leq p(x)$, there exists a unique $x' \in P$ such that $p(x')=y$ and $x' \leq y$.\ (2) For $x \in P$ and $y \in Q$ with $y \geq p(x)$, there exists a unique $x' \in P$ such that $p(x')=y$ and $x' \geq y$. Firstly, we remark that, for an arbitrary poset $P$ and $x,y\in P$, $y \in V({\rm st}(x))$ if and only if $y$ is comparable with $x$. Let $f: P\rightarrow Q$ be an order preserving map satisfying (1) and (2). It is sufficient to show $$p^{-1}({\rm st}(y))= \coprod_{y' \in p^{-1}(y)} {\rm st}(y')$$ for each $y\in Q$. Firstly, we prove that the union of the right of the above equation is disjoint. Let $y_1 ,y_2 \in p^{-1}(y)$ with $y_1 \neq y_2$. Suppose there exists $x\in V({\rm st}(y_1)) \cap V({\rm st}(y_2))$. Then one of the following conditions holds.\ (i) $x\leq y_1$ and $x\leq y_2$.\ (ii) $y_1 \leq x \leq y_2$.\ (iii) $y_1 \leq x$ and $y_2 \leq x$.\ (iv) $y_2 \leq x \leq y_1$. \(i) contradicts to (2) and (iii) contradicts to $(1)$. If (ii) holds, we have $y_1 \leq y_2$ and $p(y_1)=p(y_2)$. Then we have $y_1 = y_2$ from (1) (or (2)) and this contradicts to the assumption of $y_1,y_2$. The proof of the fact that the case (iv) contradicts is similar to the case (ii). So we have that ${\rm st}(y')$ $(y' \in p^{-1}(y))$ are disjoint. The relation $ p^{-1}({\rm st}(y)) \supset \coprod_{y' \in p^{-1}(y)} {\rm st}(y') $ is obvious. Let $\emptyset \neq \sigma \in p^{-1}({\rm st}(y))$ and $z'\in \sigma$. Suppose $y\leq p(z')$. Then there exists $y' \in p^{-1}(y)$ such that $y' \leq z'$. We want to show that $\sigma \in {\rm st}(y')$. Let $ x' \in \sigma$. If $x' \geq z'$, then $x' \geq y'$ and we have $x' \in V({\rm st}(y'))$. So we assume $x'\leq z'$. Suppose $p(x') \leq y$. Then there exists $x''\leq y'$ such that $p(x')=p(x'')$. But since $x' \leq z'$ and $x'' \leq z'$, we have $x'=x''$. The case $y \leq p(x')$ is similar. Hence we have $\sigma \in {\rm st}(y')$ if $y\leq p(z')$. The case $y\geq p(z')$ is similar, and we have $ p^{-1}({\rm st}(y)) \subset \coprod_{y' \in p^{-1}(y)} {\rm st}(y') $. \ [**Relation to group actions :**]{} Let $\Gamma$ be a group and $G$ a graph and $\alpha$ a right $\Gamma$-action on $G$. $\alpha$ is called a [*2-covering action*]{} if $N_{2}(v)\cap N_2(v\gamma)=\emptyset$ for every $v\in V(G)$ and $\gamma \in \Gamma \setminus \{ e_{\Gamma}\}$. We define similary that a left $\Gamma$-action on a graph is a 2-covering action. Let $G$ be a graph having no isolated vertices, $\Gamma$ a group, and $\alpha$ a right $\Gamma$-action on $G$. Consider the following three conditions.\ (1) $\alpha$ is a 2-covering action.\ (2) $\alpha$ is free and the quotient map $p:G\rightarrow G/\Gamma$ is a 2-covering map.\ (3) $\alpha$ is effective and the quotient map $p:G\rightarrow G/\Gamma$ is a 2-covering map. In any case, (1) and (2) are equivalent. If $G$ is connected, then the above conditions are equivalent. \(1) $\Rightarrow$ (2) : Since we assume that $G$ has no isolated vertices, we have $v\in N_2 (v)$ for every vertex $v$ of $G$. Therefore $N_2(v)\cap N_2(v\gamma )=\emptyset $ implies $v\neq v\gamma$ and we have the 2-covering action is free. Let $v\in V(G)$. It is sufficient to show that $p|_{N(v)}$ is surjective and $p|_{N_2(v)}$ is injective (see Lemma 3.3). Let $a \in N(p(v))$. Then there are $w\in a$ and $\gamma_0 ,\gamma_1 \in \Gamma$ such that $(v\gamma_0 , w\gamma_1) \in E(G)$. Therefore we have $w\gamma_1 \gamma_0 ^{-1} \in N(v)$ and $p(w \gamma_1 \gamma _0 ^{-1})=a$. Hence $p|_{N(v)}$ is surjective. Let $w_0, \; w_1 \in N_2(v)$ and suppose $p(w_0) =p(w_1)$. Then there exists $\gamma \in \Gamma$ such that $w_0 \gamma = w_1$. Since $v\in N_2(w_0) \cap N_2(w_1)$, we have that $\gamma$ is the identity of $\Gamma$ from the definition of 2-covering action. Hence $w_0 =w_1$, and we have $p|_{N_2(v)}$ is injective. \(2) $\Rightarrow$ (1) : Let $v \in V(G)$ and $\gamma \in \Gamma$. Suppose $N_2(v) \cap N_2(v \gamma ) \neq \emptyset$ and let $w\in N_2(v) \cap N_2(v \gamma )$. Since $v,v\gamma \in N_2(w)$ and $p(v)=p(v\gamma)$ and $p$ is a 2-covering, we have that $v=v\gamma$. Since the $\Gamma$-action $\alpha$ of $G$ is free, we have that $\gamma$ is the identity of $\Gamma$. Therefore $\alpha$ is a 2-covering action. \(2) $\Rightarrow$ (3) is obvious. We suppose that $G$ is connected and prove $(3) \Rightarrow (2)$ in this case. Let $v\in V(G)$ and $\gamma \in \Gamma$ and suppose $v=v\gamma$. Then the map $f_{\gamma} : G\rightarrow G,$ $v\mapsto v\gamma$ has a fixed point $v$. Since $G$ is connected, we have $f_{\gamma} = {\rm id}_G$ from Proposition 3.6. Since the action $\alpha$ is effective, we have $\gamma$ is the identity of $\Gamma$. Hence we have $\alpha$ is free. [[(1)]{} Let $n$ and $k$ be positive integers such that $k\geq 3$ and $k\neq 4$. Then, the action of $\mathbb{Z}/n\mathbb{Z}$ on $C_{nk}$ by $x\tau=x+k$, where $\tau = (1 \; {\rm mod.}n) \in\mathbb{Z}/n\mathbb{Z}$, is a 2-covering action, and its quotient graph is $C_k$.\ [(2)]{} Let $k$ be an integer with $k\geq 3$ and $k\neq 4$. Then the right action of $\mathbb{Z}$ on $L$ defined by $x\cdot n=x+kn$ is a 2-covering action, and its quotient graph is $C_k$.\ [(3)]{} For $r\geq 2$, the action of $\mathbb{Z}/2 \mathbb{Z}$ on $C_{2r}$ defined by $x\tau =1-2x$, where $\tau$ is the generator of $\mathbb{Z}/2\mathbb{Z}$, is a 2-covering action, and its quotient graph has two looped vertices.\ [(4)]{} For $r\geq 1$, the action of $\mathbb{Z}/2 \mathbb{Z}$ on $C_{2r+1}$ defined by $x\tau =1-2x$ is not a 2-covering action, since this action fixes $r \in V(C_{2r+1})$. ]{} [**Pullbacks of 2-coverings :**]{} Let $f:G\rightarrow H$ be a 2-covering map and $p:K\rightarrow H$ be a 2-covering over $H$. Then we define the graph $f^* K$ by $$V(f^*K) =\{ (x,y)\in V(G)\times V(K) \; | \; f(x)=p(y)\}$$ $$E(f^*K) =\{ ((x_1,y_1),(x_2,y_2)) \; | \; (x_1,x_2)\in E(G) \textit{ and } (y_1,y_2)\in E(K) \}.$$ The projection $q: f^*K \rightarrow G, (x,y) \mapsto x$ is a 2-covering map. Let $(x,y) \in f^*K$. Let $x_0\in N(x)$. Since $f(x)=p(y)$, there is $y'$ such that $p(y')=x'$. Therefore $(x',y')\in N(x,y)$ and $q(x',y')=x'$. Therefore $q|_{N(x,y)}$ is surjective. Let $(x_0,y_0),(x_1,y_1)\in N_2(x,y)$ with $x_0 =x_1$. Then since $p(y_0)=f(x_0)=f(x_1)=p(y_1)$ and $y_0,y_1 \in N_2(y)$, we have $y_0=y_1$. Hence $q|_{N_2(x,y)}$ is injective. Therefore $q$ is a 2-covering map. Let $G$ be a graph having no isolated vertices and $n$ a nonnegative integer. Let $i_k$ denote the graph homomorphism $G\rightarrow G\times I_n , x\mapsto (x,k)$ for $k=0,1,\cdots n$. Then for a 2-covering map $p:E\rightarrow G\times I_n$, $i_0^*E\cong i_n^*E$. We can assume that $n=1$. Let $(x,e) \in i_0^* E$. Since $(x,1) \in N_2(x,0)$, there exists $f(e) \in N_2(e)$ such that $p(f(e))=(x,0)$. We want to show that $f$ is a graph map. Let $(x',e') \in i_0^* E$ such that $(x,e) \sim (x',e')$. Then $f'(e) \sim e \sim e' \sim f(e)$ and $p(f(e))=(x,1) \sim (x',1)=p(f(e'))$. Hence there is $e'' \in N(f(e))$ such that $p(e'')=pf(e)$. Since $f(e),e'' \in N_2(e')$, we have $f(e)=e''$ and $f(e)\sim f(e')$. Let $G,H$ be graphs having no isolated points and $f,g:G\rightarrow H$ be graph homomorphisms with $f\simeq g$ and $p:E\rightarrow H$ be a 2-covering map. Then $f^*E \cong g^*E$ as a 2-covering over $G$. $F:G\times I_n \rightarrow H$ be a $\times$-homotopy from $f$ to $g$. Then $f^*E \cong i_0^* F^* E \cong i_n^* F^* E \cong g^*E .$ 2-fundamental groups ==================== In this section, we give the definition of a 2-fundamental group of a based graph and study its basic property. After that, we investigate the relation between based 2-covering maps and 2-fundamental groups.\ \ [**Definition of 2-fundamental groups :**]{} Let $n$ be a nonnegative integer. The graph $L_n$ is defined by $V(L_n)=\{ 0, 1, \cdots , n\}$ and $E(L_n)=\{ (x,y) \; | \; |x-y| =1\}$. A graph homomorphism from $L_n$ to a graph $G$ is called a [*path*]{} of $G$ with length $n$. Given a path $\varphi$ of $G$, the length of $\varphi$ is denoted by $l(\varphi )$ and $\varphi (0)$ is called the [*initial point*]{} of $\varphi$ and $\varphi (l(\varphi))$ is called the [*terminal point*]{} of $\varphi$. For vertices $v,w \in V(G)$, a path from $v$ to $w$ is a path whose initial point is $v$ and whose terminal point is $w$. We denote the set of all paths from $v$ to $w$ by $P(G; v, w)$. We consider the following two conditions for $(\varphi ,\psi) \in P(G;v,w)\times P(G;v,w)$.\ (i) $l(\varphi)+2=l(\psi)$ and there exists $x\in \{ 0,1, \cdots , l(\varphi)\}$ such that $\varphi(i)=\psi(i)$ for $i\leq x$ and $\varphi(i+2)=\psi (i)$ for $i\geq x$.\ (ii) $ l(\varphi) =l(\psi)$ and $\varphi \times \psi (E(L_{l(\varphi)})) \subset E(G)$. We write $\simeq$ for the equivalence relation on $P(G; v,w)$ generated by the above two conditions. Consider the next condition for $(\varphi ,\psi) \in P(G;v,w)\times P(G;v,w)$.\ (ii)$'$ $l(\varphi)=l(\psi)$ and there exists $x\in \{ 0,1 \cdots ,l(\varphi)\}$ such that $\varphi(i)=\psi (i)$ for all $i\neq x$. Then the equivalence relation $\simeq'$ generated by (i) and (ii)$'$ is equal to $\simeq$. In fact, if $(\varphi,\psi)$ satisfies the condition (ii)$'$ , then $(\varphi,\psi)$ satisfies the condition (ii). Thus $\varphi \simeq' \psi$ implies $\varphi \simeq \psi$. If $(\varphi, \psi)$ satisfies the condition (ii), let $\eta_k$ for $1\leq k \leq l(\varphi)$ denote by the graph homomorphism defined by $\eta_k(i)=\psi(i)$ for $i< k$ and $\eta_k(i)=\varphi(i)$ for $i\geq k$. Since $(\eta_k ,\eta_{k-1})$ satisfies the condition (ii)$'$ and $\eta_0 =\varphi$ and $\eta_{l(\varphi)}= \psi$, we have $\varphi \simeq' \psi$. Hence we have that $\simeq'$ is equal to $\simeq$. We write $\pi^2_1(G; v,w)$ for the quotient set $P(G; v,w) / \simeq$. For $\varphi , \; \psi \in P(G;v,w)$, we say that $\varphi$ is 2-homotopic to $\psi$ if $\varphi \simeq \psi$. For $\varphi \in P(G; v,w)$, the equivalence class of $\simeq$ represented by $\varphi$ with respect to $\simeq$ is denoted by $[\varphi]$, and is called the 2-homotopy class of $\varphi$. For $\varphi :L_n\rightarrow G$ and $\psi :L_m\rightarrow G$ such that $\varphi (n)=\psi (0)$, we define the composition of $\varphi$ and $\psi$ by the path $\psi \cdot \varphi :L_{n+m}\rightarrow G$ such that $(\psi \cdot \varphi)(i)=\varphi (i)$ $(i\leq n)$ and $(\psi \cdot \varphi (i))= \psi (i-n)$ $(i\geq n)$. Obviously this operation is associative, i.e. $(\varphi \cdot \psi) \cdot \eta=\varphi \cdot (\psi \cdot \eta)$ if these are composable. Let $G$ be a graph and $u,v,w \in V(G)$ and $\varphi ,\varphi' \in P(G; v,w), \psi ,\psi ' \in P(G; u,v)$. If $\varphi \simeq \varphi'$ and if $\psi \simeq \psi'$, then $\varphi \cdot \psi \simeq \varphi' \cdot \psi'$. We can assume $\varphi = \varphi'$ or $\psi =\psi'$. Suppose $\varphi = \varphi'$. We can assume $(\psi,\psi')$ satisfies the condition (i) in the definition of $\simeq$ or (ii)$'$ in Remark 4.1. But in this case $\varphi \cdot \psi \simeq \varphi \cdot \psi'$ is obvious. The case $\psi =\psi'$ is similar. From Lemma 4.2, the composition of paths induces the composition of 2-homotopy classes of paths $$\pi^2_1(G;v,w) \times \pi^2_1(G;u,w)\longrightarrow \pi^2_1(G;u,w), \;\; ([\varphi], [\psi]) \mapsto [\varphi \cdot \psi].$$ We write $\alpha \cdot \beta$ for the composition of 2-homotopy classes, for $\alpha \in \pi^2_1(G;v,w)$ and for $\beta \in \pi^2_1(G;u,v)$. Given a graph $G$ and $v\in V(G)$, then the path $L_0 \rightarrow G, \; 0\mapsto v$ is denoted by $*_v$. It is obvious that $*_v \cdot \varphi =\varphi$ and $\psi \cdot *_v =\psi$ if composable. Let $\varphi : L_n\rightarrow G$ be a path of a graph $G$. The path $L_n \rightarrow G$ defined by $i \mapsto \varphi(n-i)$ is denoted by $\overline{\varphi}$. From the condition (i), we have $\overline{\varphi} \cdot \varphi \simeq *_v$ and $\varphi \cdot \overline{\varphi} \simeq *_w$, where $v$ is the initial point of $\varphi$ and $w$ is the terminal point of $\varphi$. Let $G$ be a graph and $\varphi, \psi$ paths of $G$. If $\varphi \simeq \psi$, then we have $l(\varphi) = l(\psi)$ $({\rm mod}.2)$ from the definition of 2-homotopy. Thus we say that the 2-homotopy class $\alpha$ is [*even*]{} if the length of the representative of $\alpha$ is even, and we say that $\alpha$ is [*odd*]{} if $\alpha$ is not even. A [*loop*]{} of a based graph $(G,v)$ is a path of $G$ from $v$ to $v$. For a based graph $(G,v)$, $\pi^2_1(G;v,v)$ is denoted by $\pi^2_1(G,v)$. $\pi^2_1(G,v)$ is a group with the composition and is called the [*2-fundamental group*]{} of $(G,v)$. Obviously, the identity element of $\pi^2_1(G,v)$ is $[*_v]$, and the inverse element of $[\varphi] \in \pi_1(G,v)$ is $[\overline{\varphi}]$. In this paper, we only consider the 2-fundamental groups. Hence we write $\pi_1(G,v)$ for $\pi^2_1(G,v)$ and $\pi_1(G;v,w)$ for $\pi^2_1(G;v,w)$. All even elements of $\pi_1(G,v)$ forms a subgroup of $\pi_1(G,v)$ and we write $\pi_1(G,v)_{\rm ev}$ for this subgroup. $\pi_1(G,v)_{\rm ev}$ is called the [*even part of the 2-fundamental group*]{} of $(G,v)$. $\pi_1(G,v)_{\rm ev}$ is a normal subgroup of $\pi_1(G,v)$ whose index is $1$ or $2$. Indeed $\pi_1(G,v)_{\rm ev}$ is the kernel of the group homomorphism $$\pi_1(G,v)\longrightarrow \mathbb{Z}/2\mathbb{Z} \; ,\;\; [\varphi] \longmapsto (l(\varphi) \;\; {\rm mod}.2).$$ We remark that $\pi_1(G,v)_{\rm ev}=\pi_1(G,v)$ if and only if the connected component of $G$ containing $v$ is 2-colorable. In Section 6, we will prove that $\pi_1(G,v)_{\rm ev}$ is isomorphic to the fundamental group of $(|\mathcal{N}(G)|,v)$ if $v$ is not isolated. Let $f:G\rightarrow H$ be a graph homomorphism and let $\varphi ,\psi \in P(G;v,w)$ where $v, \; w$ are vertices of $G$. We can easily see that $f\circ \varphi \simeq f\circ \psi$ if $\varphi \simeq \psi$. If $\varphi $ and $\psi$ are composable, then $f\circ(\varphi \cdot \psi) =(f\circ \varphi) \cdot (f\circ \psi)$. Therefore a based graph map $f:(G,v) \rightarrow (H,w)$ induces a group homomorphism $\pi_1(G,v)\rightarrow \pi_1(H,w)$, and is denoted by $\pi_1(f)$ or $f_*$. Obviously $\pi_1({\rm id})={\rm id}$ and $\pi_1(g\circ f) =\pi_1 (g) \circ \pi_1(f)$ and these preserve even parts. Therefore $\pi_1$ and the even part of $\pi_1$ are functors from the category of based graphs $({\rm \bf Graphs})_*$ to the category of groups $({\rm \bf Groups})$. We investigate how $\pi_1(f)$ depends on the choice of a based graph homomorphism $f$. Then we need to consider the $\times$-homotopy for the based graph case. Dochtermann defined the based case of $\times$-homotopy in [@Docb], but he assumed that basepoint has a loop. For based graph homomorphisms $f,g:(G,v)\rightarrow (H,w)$, we say that $f$ is $\times$-homotopic to $g$ if there exists a nonnegative integer $n$ and a graph homomorphism $F:G\times I_n \rightarrow H$ such that $F(v,i)=w$ for all $0\leq i\leq n$ and $F(x,0)=f(x)$ and $F(x,n)=g(x)$ for all $x\in V(G)$. Let $(G,v)$ and $(H,w)$ be based graphs. A [*based multihomomorphism*]{} from $(G,v)$ to $(H,w)$ is $\eta \in {\rm Hom}(G,H)$ such that $\eta(v)=\{ w\}$. The poset of all based multihomomorphisms from $(G,v)$ to $(H,w)$ is denoted by ${\rm Hom}((G,v),(H,w))$. Then we can easily see that, for based graph homomorphisms $f,g:(G,v)\rightarrow (H,w)$, $f$ is $\times$-homotopic to $g$ in the based sense if and only if $f$ and $g$ are contained in the same connected component of ${\rm Hom}((G,v),(H,w))$. We remark that ${\rm Hom}((G,v),(H,w))$ does not have a canonical basepoint. Indeed ${\rm Hom}((G,v),(H,w))$ may be empty. Let $f,g:(G,v) \rightarrow (H,w)$ be based graph homomorphisms. If $f$ is $\times$-homotopic to $g$ in the based sense, then we have $\pi_1(f)=\pi_1(g)$. We can assume that there is $\eta \in {\rm Hom}((G,v),(H,w))$ such that $f\leq \eta$ and $g\leq \eta$. In this case $f\times g (E(G)) \subset E(H)$. Therefore for a loop $\varphi :L_n \rightarrow G$, $$(f\circ \varphi)\times (g\circ \varphi) (E(L_n))=(f\times g)\circ (\varphi \times \varphi)(E(L_n)) \subset E(H)$$ and we have $f \circ \varphi \simeq g\circ \varphi$. Let $G$ be a graph and $v,w \in V(G)$, $\alpha \in \pi_1(G; v,w)$. We define the group homomorphism ${\rm Ad}(\alpha) : \pi_1(G,v) \rightarrow \pi_1(G,w)$ by ${\rm Ad}(\alpha)(\beta)= \alpha \cdot \beta \cdot \overline{\alpha}$. We write ${\rm Ad}(\varphi) ={\rm Ad}([\varphi])$ for $\varphi \in P(G; v,w)$. We remark that ${\rm Ad}(\alpha)$ preserves even parts. From the definition of ${\rm Ad}$, we can easily see that ${\rm Ad}([*_v])={\rm id}_{\pi_1(G,v)}$ and ${\rm Ad}(\beta)\circ {\rm Ad}(\alpha)={\rm Ad}(\beta \cdot \alpha)$ for $\beta \in \pi_1(G;v,w)$ and $\alpha \in \pi_1(G; u,v)$. Therefore ${\rm Ad}(\overline{\alpha}) \circ {\rm Ad}(\alpha) ={\rm id}$ and ${\rm Ad}(\alpha)\circ {\rm Ad}(\overline{\alpha})={\rm id}$. So we have ${\rm Ad}(\alpha)$ is an isomorphism and ${\rm Ad}(\alpha)^{-1}={\rm Ad}(\overline{\alpha})$. Thus if $G$ is connected, the group $\pi_1(G,v)$ is, up to isomorphisms, independent of the choice of the basepoint $v$. In this case, the notation $\pi_1(G,v)$ is often abbreviated to $\pi_1(G)$. Let $(G,v)$ be a based graph and $\alpha \in \pi_1(G,v)$. In this case, ${\rm Ad}(\alpha)$ is an inner automorphism of $\pi_1(G,v)$ with respect to $\alpha$. Let $\alpha ,\beta \in \pi_1(G;v,w)$. In general, ${\rm Ad}(\alpha)$ is not equal to ${\rm Ad}(\beta)$. However, we have that $\beta \cdot \overline{\alpha} \in \pi_1(G,w)$ and ${\rm Ad}(\beta)={\rm Ad}(\beta \cdot \overline{\alpha}) \circ {\rm Ad}(\alpha)$. Since an inner automorphism fixes the abelianization, the abelian group homomorphisms $\pi_1(G,v)/[\pi_1(G,v),\pi_1(G,v)] \rightarrow \pi_1(G,w)/[\pi_1(G,w) ,\pi_1(G,w)]$ induced by ${\rm Ad(\alpha)}$ is independent of the choice of $\alpha$. We can define the functor $H_1$ from the category of graphs $({\rm \bf Graphs})$ to the category of abelian groups ([**Abels**]{}) as follows. Let $G$ be a graph. Let $G=\coprod G_{\alpha}$ be a decomposition of connected components of $G$. We choose one vertex $v_{\alpha}$ for each connected component $G_{\alpha}$ of $G$, and we define $H_1(G)=\bigoplus \pi_1(G,v_{\alpha}) / [\pi_1(G,v_{\alpha}), \pi_1(G,v_{\alpha})]$. Then the functor $H_1$ is independent of the choice of $v_{\alpha}$ up to natural isomorphisms. Let $G,H$ be graphs and $f,g$ graph homomorphisms from $G$ to $H$. Let $v\in V(G)$ be a non-isolated vertex. If $f \simeq_{\times} g$ then there exists a path $\gamma$ from $f(v)$ to $g(v)$ such that $$\pi_1(g)={\rm Ad}(\gamma) \circ \pi_1(f): \pi_1(G,v)\rightarrow \pi_1(H,g(v))$$ Therefore if $G$ has no isolated vertices and if $f\simeq _{\times} g$, then $H_1(f)=H_1(g)$. Let $F:G\times I_n \rightarrow G$ be a $\times$-homotopy from $f$ to $g$. Since $v$ is not isolated, there is $w\in N(v)$. Define a path $\gamma':L_{2n} \rightarrow G\times I_n$ by $\gamma'(2i)=(v,i)$ $(0\leq i \leq n)$ and $\gamma' (2i-1)=(w,i)$ $(1\leq i\leq n)$. Set $\gamma =F\circ \gamma'$. We prove $\pi_1(g)={\rm Ad}(\gamma ) \circ \pi_1(f)$. Let $k$ be an integer such that $0\leq k \leq n$. Let $\gamma'_k$ denote $\gamma'|_{L_{2k}}$ and $i_k$ denote the inclusion $G\rightarrow G\times I_n$, $x\mapsto (x,k)$. Let $\varphi :L_n\rightarrow G$ be a loop of $(G,v)$ and set $\varphi_k=\overline{\gamma'_k}\cdot \varphi'_k\cdot \gamma'_k$. Firstly, we prove $\varphi_n\simeq \varphi_0$. It is sufficient to show that $\varphi_k \simeq \varphi_{k-1}$ for $1\leq k\leq n$. Define a path $\psi_k:L_{4k+n} \rightarrow G\times I_n$ by $\psi_k(i) =\varphi_k(i)$ ($i\leq 2k-2$ or $i\geq 2k+n+2$) and $\psi_k(2k-1)=\psi_k(2k+n+1)=(w,k-1)$, $\psi_k(2k)=(v,k-1)=\psi_k(2k+n) $, $\psi_k(x)=(\varphi (x), k-1)$ $(2k\leq x \leq 2k+n)$. We can easily see $(\varphi_k \times \psi_k)(E(L_n)) \subset E(G \times I_n)$. From (i) in the definition of 2-homotopy of paths, $\psi_k \simeq \varphi_{k-1}$ and we prove $\varphi _n \simeq \varphi_0$. Therefore we have $${\rm Ad}(\overline{\gamma}) \circ g_* ([\varphi])=F_*([\overline{\gamma'} \cdot (i_n \circ \varphi ) \cdot \gamma' ])=[F\circ \varphi_n]=[F\circ \varphi_0]= f_* [\varphi].$$ Hence $g_* ={\rm Ad}(\gamma) \circ f_*$. Let $G,H$ be graphs and $f$ a graph homomorphism from $G$ to $H$, $v\in V(G)$ a non-isolated vertex. If $f$ is $\times$-homotopy equivalence, $f_*:\pi_1(G,v)\rightarrow \pi_1(H,f(v))$ is an isomorphism. Let $g$ be a $\times$-homotopy inverse of $f$. Since $g\circ f \simeq_{\times} {\rm id}$, we have $g_* \circ f_*$ is an isomorphism from previous proposition. Hence we have $f_*$ is injective. On the other hand, since $f\circ g \simeq {\rm id}$, we have $f_* \circ g_*$ is an isomorphism. Hence $f_*$ is surjective. [**Relation to 2-covering maps :**]{} A based graph homomorphism $p:(G,v)\rightarrow (H,w)$ is called a [*based 2-covering map*]{} or a [*based 2-covering over*]{} $(H,w)$ if $p:G\rightarrow H$ is 2-covering map. A based 2-covering $p:(G,v)\rightarrow (H,w)$ over $(H,w)$ is called [*connected*]{} if $G$ is connected. Let $p:(G,v)\rightarrow (H,w)$ be a based covering map.\ (1) Let $\varphi :(L_n , 0)\rightarrow (H,w)$. Then there exists a unique $\tilde{\varphi }:(L_n ,0) \rightarrow (G,v)$ with $\varphi=p \tilde{\varphi}$.\ (2) We write $\tilde {\varphi}$ for a graph homomorphism $(L_n ,0) \rightarrow (G,v)$ such that $\varphi = p\tilde{\varphi}$ for a based graph homomorphism $\varphi :(L_n ,0) \rightarrow (H,w)$. Let $u\in V(G)$ and $\varphi$, $\psi \in P(G; w,u)$. If $\varphi \simeq \psi$, then the terminal point of $\tilde{\varphi}$ is equal to the terminal point of $\tilde{\psi}$, and we have $\tilde{\varphi}\simeq \tilde{\psi}$. Since the case of $n=0$ is obvious, we assume $n\geq 1$. Therefore $L_n$ has no isolated points.\ (1) The uniqueness of $\tilde{\varphi}$ follows from Proposition 3.6. Let $x_0=v$. Let $x_1 \in N(x_0)$ such that $\varphi(1) \in N(\varphi(0))=N(w)$. By induction, we obtain a sequence $(x_0 , \cdots ,x_n)$ of vertices of $G$ such that $x_0 =v$ and $p(x_i) =\varphi (i)$. Set $\tilde{\varphi }(i)= x_i$ for $0\leq i \leq n$, then $\tilde{\varphi}$ is the lift of $\varphi$.\ (2) We can assume that $\varphi$ and $\psi$ satisfy the condition (i) or (ii) in the definition of 2-homotopy of paths. Suppose $\varphi$ and $\psi$ satisfy the condition (i) and let $n$ be the length of $\varphi$. Then $l(\psi) =n+2$ and there is $x\in \{ 0, \cdots , n\}$ such that $\varphi(i) = \psi (i)$ for $i\leq x$ and $\varphi(i) =\psi (i+2)$ for $i\geq x$. From the uniqueness of (1), we have $\tilde{\varphi}(i)= \tilde{\psi}(i)$ for $i\leq x$. Since $\tilde{\psi}(x)$ and $\tilde{\psi}(x+2)$ are elements of $N(\tilde{\psi}(x))$ and $p(\tilde{\psi}(x))=\psi(x)=\psi(x+2)=p(\tilde{\psi}(x+2))$, we have $\tilde{\psi}(x+2)=\tilde{\psi}(x)=\tilde{\varphi}(x)$. From the uniqueness of (1), we have $\tilde{\varphi}(i)=\tilde{\psi} (i+2)$ for $x\leq i$. Therefore $\tilde{\varphi}$ and $\tilde{\psi}$ satisfy the condition (i). Suppose $\varphi$ and $\psi$ satisfy the condition (ii). Then the map $F:V(L_n \times I_1) \rightarrow V(H)$ where $F(x,0)=\varphi(x)$ and $F(x,1)=\psi(x)$ is a graph map. From Corollary 3.8, we have a graph map $\tilde{F}: L_n \times I_1 \rightarrow G$ such that $\tilde{F}(x,0)=f(x)$ and $p\tilde{F}=F$. Since $\tilde{F}(x,1)\in N_2(\tilde{F}(x,0))$ and $p\tilde{F}(0,0)=p\tilde{F}(0,1)$ and $p\tilde{F}(n,0)=p\tilde{F}(n,0)$, we have $\tilde{F}(0,0)=\tilde{F}(0,1)$ and $\tilde{F}(n,0)=\tilde{F}(n,1)$. Hence we have $\tilde{\psi}(x)=\tilde{F}(x,1)$ and $\tilde{\psi}(n)=\tilde{\varphi}(n)$. Let $p:(G,v) \rightarrow (H,w)$ be a based 2-covering map. Then $\pi_1(p)$ is injective. Let $\varphi$ be a loop of $(H,w)$. Then $[\varphi] \in p_*(\pi_1(G,v))$ if and only if the lift $\tilde{\varphi}$ of $\varphi$ with respect to $p :(G,v)\rightarrow (H,w)$ whose initial point is $v$ is a loop of $(G,v)$. Let $\psi$ be a loop of $(G,v)$ with $p\psi \simeq *_w$. Since $*_v$ is the lift of $*_w$, we have $\psi \simeq *_v$ from the previous lemma. Therefore $\pi_1(f) :\pi_1(G,v) \rightarrow \pi_1(H,w)$ is injective. Let $\varphi$ be a loop of $(H,w)$. If $\tilde{\varphi}$ is a loop of $(G,v)$, then we obviously have $[\varphi] \in p_*\pi_1(G,v)$. Suppose $[\varphi] \in p_*\pi_1(G,v)$. Then there exists a loop $\psi$ of $(G,v)$ such that $p\psi \simeq \varphi$. From the previous lemma, the terminal point of $\tilde{\varphi}$ is equal to the terminal point of $\psi$. So $\tilde{\varphi}$ is a loop of $(G,v)$. Let $p:(G,v) \rightarrow (H,w)$ be a based 2-covering and $(T,x)$ a connected based graph and $f:(T,x) \rightarrow (H,w)$ a based graph map. Then there exists a graph map $\tilde{f}:(T,x) \rightarrow (G,v)$ such that $p\tilde{f}=f$ if and only if $f_*(\pi_1(T,x)) \subset p_*(\pi_1(G,v))$. Suppose there exists $\tilde{f}: (T,x) \rightarrow (G,v)$ such that $p\tilde{f}=f$. Then we have $f_*(\pi_1(T,x)) = p_* \tilde{f}_* (\pi_1(T,x)) \subset p_* \pi_1(G,v)$. Suppose $f_* \pi_1(T,x) \subset p_* \pi_1(G,v)$. We define the graph map $\tilde{f} :(T,x) \rightarrow (G,v)$ as follows. Let $y \in V(T)$, and $\varphi$ a path of $T$ from $x$ to $y$. Let $\tilde{\varphi}$ be the lift of $f\varphi$ whose initial point is $v$. We want to define $\tilde{f} (y)$ by the terminal point of $\tilde{\varphi}$. Let $\psi$ be another path of $T$ from $x$ to $y$, and $\tilde{\psi}$ the lift of $f\psi$ with respect to $p$ whose initial point is $v$. We want to show the terminal point of $\tilde{\psi}$ is equal to $\tilde{\psi}$. Since $f_* \pi_1(T,x) \subset p_* \pi_1(G,v)$, the lift of $f(\overline{\psi} \cdot \varphi)$ is a loop of $(G,v)$. We write $\gamma$ for the lift of $f(\overline{\psi} \cdot \varphi)$. Since $f(\psi \cdot \overline{\psi} \cdot \varphi) \simeq f(\varphi)$, the terminal point of the lift of $f(\psi \cdot \overline{\psi} \cdot \varphi)$ is equal to the terminal point of $\tilde{\varphi}$. Since the lift of $f(\psi \cdot \overline{\psi} \cdot \varphi)$ is $\tilde{\psi} \cdot \gamma$, we have that the terminal point of $\tilde{\psi}$ is equal to the terminal point of $\tilde{\varphi}$ from Lemma 4.7. Therefore $\tilde{f}$ is well-defined as a set map. We want to show that $\tilde{f}$ is a graph homomorphism. Let $(y,z)\in E(T)$, and $\varphi :L_n \rightarrow T$ be a path from $x$ to $y$. We define the path $\varphi':L_{n+1} \rightarrow T$ by $\varphi'|_{L_n}=\varphi$ and $\varphi'(n+1)=z$. Let $\tilde{\varphi}$ be the lift of $f\varphi$ whose initial point is $v$. Since $f(z) \in N(f(y))=N(p(\varphi(n)))$, there exists $z' \in N(\tilde{\varphi}(n))$ with $p(z')= f(z)=f(\varphi'(n+1))$. Then the lift $\tilde{\varphi'}:L_{n+1} \rightarrow G$ whose initial point is $v$ is obtained by $\tilde{\varphi'}|_{L_n} =\tilde{\varphi}$ and $\varphi'(n+1)=z'$. Therefore $\tilde{f}(z)=z' \in N(\tilde{f}(y))$ and we have $\tilde{f}$ is a graph homomorphism. From Proposition 3.6 the lift $\tilde{f}$ in the previous proposition is unique. Let $p:(G,v) \rightarrow (H,w)$ be a connected based 2-covering map. Then there exists a bijection $$\Phi: \pi_1(H,w)/ p_*\pi_1(G,v) \cong p^{-1}(w).$$ This bijection $\Phi$ is obtained as follows. Let $\varphi$ be a loop of $(H,w)$ and $\tilde{\varphi}$ a lift of $\varphi$. Then we set $\Phi([\varphi])$ be the terminal point of $\tilde{\varphi}$. Firstly, we want to show that the map $\Phi$ is well-defined. Let $\varphi ,\psi$ be loops of $(H,w)$ with $[\varphi] = [\psi]$ in $\pi_1(H,w)/ p_*\pi_1(G,v)$. Then there is a loop $\gamma$ of $(H,w)$ such that the lift $\tilde{\gamma}$ is a loop of $(G,v)$ and $\varphi \simeq \psi \cdot \gamma$. From Lemma 4.7, the terminal point of $\tilde{\varphi}$ is equal to the terminal point of the lift of $\psi \cdot \gamma$. But the lift of $\psi \cdot \gamma $ is $\tilde{\psi} \cdot \tilde{\gamma}$ since $\tilde{\gamma}$ is a loop. Therefore $\Phi$ is well-defined. Since $(G,v)$ is connected, $\Phi$ is surjective. Let $\varphi , \psi$ be loops of $(H,w)$ with $\Phi([\varphi])=\Phi([\psi])$. Then $\overline{\tilde{\varphi}} \cdot \tilde{\psi}$ is a loop of $(G,v)$ and $\psi \simeq \varphi \cdot \overline{\varphi} \cdot \psi\simeq \varphi \cdot p(\overline{\tilde{\varphi}}\cdot \tilde{\psi})$. Therefore $[\varphi]=[\psi]$ in $\pi_1(H,w)/ p_*\pi_1(G,v)$. Hence $\Phi$ is injective. Let $(G,v)$ be a based graph. A connected based 2-covering map $(\tilde{G},\tilde{v})\rightarrow (G,v)$ is called a [*universal 2-covering over*]{} $(G,v)$ if $\pi_1(\tilde{G},\tilde{v})$ is trivial. We deduce that the universal 2-covering over $(G,v)$ is unique up to isomorphism over $(G,v)$ from Proposition 4.9 and Remark 4.10. The following corollary is useful to compute 2-fundamental groups. Let $(G,v)$ be a connected based graph and $p:(\tilde{G},\tilde{v})\rightarrow (G,v)$ be a universal 2-covering over $(G,v)$. For each $x\in p^{-1} (v)$, let $\varphi_x$ be a path from $\tilde{v}$ to $x$. Then $[p\circ \varphi_x]\neq [p\circ \varphi_y]$ for $x\neq y$, and $\pi_1(G,v)=\{ [p\circ \varphi_x] \; | \; x\in p^{-1}(v) \}$. Let $(G,v)$ be a graph. Then there exists a universal 2-covering over $(G,v)$. Let $V(\tilde{G})=\coprod_{w\in V(G)} \pi_1(G; v,w)$, $E(\tilde{G})=\{(\alpha ,\beta ) \; |$ There is $\varphi \in \beta $ such that $\varphi|L_{l(\varphi)-1}\in \alpha$ $\}$, $p(\pi_1(G; v,w)) \subset \{ w\}$ and $\tilde{v} = [*_v]$. We show that $\tilde{G} = (V(\tilde{G}) , E(\tilde{G}))$ is a graph. Firstly, we claim that the following three conditions for $(\alpha , \beta )\in V(\tilde{G})\times V(\tilde{G})$ is equivalent.\ (1) $(\alpha ,\beta ) \in E(\tilde{G})$.\ (2) For each $\varphi \in \alpha$, the map $\varphi' : V(L_{l(\varphi)+1})\rightarrow V(G)$ defined by $\varphi' |_{V(L_{ l(\varphi)})}=\varphi$ and $\varphi' (l(\varphi) +1)=p(\beta)$ is a graph homomorphism and an element of $\beta$.\ (3) There exists $\varphi' \in \alpha$ such that the map $\varphi' : V(L_{l(\varphi)+1}\rightarrow V(G))$ defined by $\varphi' |_{V(L_{ l(\varphi)})}=\varphi$ and $\varphi' (l(\varphi) +1)=p(\beta)$ is a graph homomorphism and an element of $\beta$. In fact (2) $\Rightarrow $ (3) $\Leftrightarrow $ (1) is obvious, and (3) $\Rightarrow$ (2) is deduced from Lemma 4.2. Let $(\alpha ,\beta ) \in E(\tilde{G})$, the path $\varphi \in \alpha$ and $n=l(\varphi)$. Then $\varphi'$ defined above is an element of $\beta$. Let $\varphi'':L_{n +2} \rightarrow G$ denote the path defined by $\varphi''|_{L_{n+1}}=\varphi'$ and $\varphi''(n+2)= \varphi(n)=p(\alpha)$. We have $\varphi'' \in \alpha$ since $\varphi'' \simeq \varphi$. Therefore $(\beta ,\alpha) \in E(\tilde{G})$ and $\tilde{G}=(V(\tilde{G}),E(\tilde{G}))$ is a graph. We show $p$ is a 2-covering map of graphs. It is obvious that $p$ is a graph homomorphism. Let $\alpha \in V(\tilde{G})$ and $x=p(\alpha)$. Let $y \in N(x)$ and $\varphi \in \alpha$ a path with length $n$. Let $\varphi':L_{n+1} \rightarrow G$ be a graph map defined by $\varphi'|_{L_n}=\varphi$ and $\varphi'(n+1)=y$. Then we have $[\varphi']\in N(\alpha)$ from the definition of $E(\tilde{G})$ and $p([\varphi'])=y$. Therefore $p|_{N(\alpha)} :N(\alpha) \rightarrow N(x)$ is surjective. Let $\beta_1, \beta_2 \in N_2(\alpha)$ such that $p(\beta_1)=p(\beta_2)=z$. Then there are $\gamma_1$, $\gamma_2 \in N(\alpha)$ such that $\beta_i \in N(\gamma_i)$ for $i=1, \; 2$. Let $\varphi \in \alpha$ and $n$ be the length of $\varphi$. Let $\varphi'_i :L_{n+1}\rightarrow G$ for $i=1, \;2$ be a path defined by $\varphi'|_{L_n}=\varphi$ and $\varphi'(n+1)=p(\gamma_i)$. Let $\varphi_i'':L_{n+2}\rightarrow G$ for $i=1, \; 2$ be a path defined by $\varphi''_i|_{L_{n+1}}=\varphi'_i$ and $\varphi''(n+2)=z$. Then $\varphi'_i \in \gamma_i$ and $\varphi''_i \in \beta_i$. Since $\varphi_1''$ and $\varphi_2''$ satisfy the condition (ii) in the definition of homotopy, we have $\beta_1=\beta_2$ hence $p|_{N_2(\alpha)}$ is injective. From Lemma 3.3, we obtain that $p$ is a 2-covering map. We show that $\tilde{G}$ is connected. We remark that, for a based graph map $(L_n ,0) \rightarrow (G,v)$, the map $\tilde{\varphi} :V(L_n) \rightarrow V(\tilde{G})$ defined by $\tilde{\varphi}(i)=[\varphi|_{L_i}]$ is a lift of $\varphi$, and $\tilde{\varphi}$ is a path from $\tilde{v}$ to $[\varphi]$. Therefore $\tilde{G}$ is connected. Finally, we show that $\pi_1(\tilde{G},\tilde{v})$ is trivial. Since $p$ is a 2-covering map, it is sufficient to show that $p_*\pi_1(\tilde{G},\tilde{v})$ is trivial. Let $[\varphi] \in p_*\pi_1(\tilde{G},\tilde{v})$. Then the terminal point of the lift $\tilde{\varphi}$ of $\varphi$ with respect to $p$ is $\tilde{v} =[*_v]$ from Corollary 4.8. But since the terminal point of $\tilde{\varphi}$ is $[\varphi]$ from the construction of $\tilde{\varphi}$, we have $[\varphi]=[*_v]$. Therefore $p_*\pi_1(\tilde{G},\tilde{v})$ is trivial. Let $(G,v)$ be a graph and $(\tilde{G} ,\tilde{v})$ the universal 2-covering over $(G,v)$ constructed in the proof of Proposition 4.13. Then the action $$V(\tilde{G}) \times \pi_1(G,v) \rightarrow V(\tilde{G}), \; (\alpha ,\beta )\mapsto \alpha \cdot \beta$$ is a 2-covering action. Let $\alpha \in \pi_1(G,v)$ and $(\beta_1 ,\beta_2) \in E(\tilde{G})$. Then there is $\varphi \in \beta_2$ such that $\varphi|_{L_{n-1}} \in \beta_1$ where $n$ is the length of $\varphi$. Let $\psi \in \alpha$ and $m$ be the length of $\psi$. Then $\varphi \cdot \psi \in \beta_2 \cdot \alpha$ and $\varphi \cdot \psi |_{L_{m+n-1}} =\psi |_{L_{n-1}} \cdot \varphi \in \beta_1 \cdot \alpha$. Therefore $(\beta_1 \cdot \alpha, \beta_2 \cdot \alpha) \in E(\tilde{G})$ from the definition of $E(\tilde{G})$. Since $\tilde{v} \cdot \alpha =\alpha$ for each $\alpha \in \pi_1(G,v)$, this action is effective and the quotient $\tilde{G}\rightarrow \tilde{G}/\pi_1(G,v)\cong G$ is a 2-covering, this action is a 2-covering action from Proposition 3.12. Let $(G,v)$ be a based graph and $\Gamma$ a subgroup of $\pi_1(G,v)$. Then there exists a connected based 2-covering $p_{\Gamma} :(G_{\Gamma} ,v_{\Gamma}) \rightarrow (G,v)$ such that $p_{\Gamma *} \pi_1(G_{\Gamma},v_{\Gamma}) =\Gamma$. Let $p:(\tilde{G} ,\tilde{v})\rightarrow (G,v)$ be the universal 2-covering over $(G,v)$ constructed in the proof of Proposition 4.12. We define $G_{\Gamma}$ by $\tilde{G} / \Gamma$ and $v_{\Gamma}$ is the image of $\tilde{v}$ of the quotient map $ q :\tilde{G} \rightarrow G_{\Gamma}$. From the universality of a quotient map, we have a graph homomorphism $p_{\Gamma} :G_{\Gamma} \rightarrow G$ such that $p_{\Gamma} \circ q =p$. Since $q$ is a surjective 2-covering map, $p_{\Gamma}$ is a 2-covering map from Lemma 3.4. We want to show that $p_{\Gamma *} \pi_1(G_{\Gamma},v_{\Gamma})$ is $\Gamma$. Let $[\varphi] \in \Gamma$ and $n$ be the length of $\varphi$. Since the lift $\tilde{\varphi} :(L_n ,0) \rightarrow (\tilde{G} ,\tilde{v})$ of $\varphi$ is defined by $\tilde{\varphi}(i) =[\varphi|_{L_i}]$, we have $\tilde{\varphi}(n)=[\varphi] \in \Gamma$. Therefore $q\tilde{\varphi}$ is a loop of $(G_{\Gamma},v_{\Gamma})$. So we have $[\varphi] \in p_{\Gamma *}\pi_1(G_{\Gamma},v_{\Gamma})$. Suppose $[\varphi] \in p_*\pi_1(G_{\Gamma} ,v_{\Gamma})$. Then $q\tilde{\varphi}$ is the lift of $\varphi$ of $(G_{\Gamma},v_{\Gamma})$ where $\tilde{\varphi}$ is the lift of $\varphi$ with respect to $(\tilde{G},\tilde{v})\rightarrow (G,v)$. We have the terminal point of $\tilde{\varphi}$ is an element of $\Gamma$. So we have $[\varphi] \in \Gamma$. The next theorem summarizes this section. Let $(G,v)$ be a based graph. Let $\mathcal{C}$ denote the category whose objects are connected based 2-coverings over $(G,v)$ and morphisms are based graph homomorphisms over $(G,v)$. Let $\mathcal{D}$ denote the small category whose objects are subgroups of $\pi_1(G,v)$ and the morphisms are inclusions. Then the functor $F:\mathcal{C}\rightarrow \mathcal{D}$, $(p:(\tilde{G} ,\tilde{v})\rightarrow (G,v))\mapsto {\rm Im}(p_{*}:\pi_1(\tilde{G} , \tilde{v})\rightarrow \pi_1(G,v))$ is a categorical equivalence. Before giving the proof, we recall some terminologies of the category theory. Let $F$ be a functor from a catgory $\mathcal{C}$ to a category $\mathcal{D}$. $F$ is said to be [*essentially surjective*]{} if, for each object $X$ of $\mathcal{D}$, there exists an object $A$ of $\mathcal{C}$ such that $FA \cong X$. $F$ is said to be [*fully faithful*]{} if, for objects $A,B$ of $\mathcal{C}$, the map $\mathcal{C}(A,B)\rightarrow \mathcal{D}(FA,FB)$ is bijective. It is known that $F$ is an equivalence of categories if and only if $F$ is fully faithful and essentially surjective. (see [@Mac]) It is obvious that $F$ is essentially surjective from Proposition 4.15. Let $p_i:(G_i ,v_i) \rightarrow (G,v)$ for $i=1,2$ be connected based covering. From the Proposition 4.9, there exists a morphism of $\mathcal{C}$ from $p_1$ to $p_2$ if and only if $p_{1*} \pi_1(G_1,v_1)\subset p_{2*} \pi_1(G_2,v_2)$. And if $p_{1*} \pi_1(G_1,v_1)\subset p_{2*} \pi_1(G_2,v_2)$, then there is a unique the morphism of $\mathcal{C}$ from $p_1$ to $p_2$ (see Remark 4.10). This indicates that $F$ is fully faithful. Let $(G,v)$ be a connected based graph such that $\chi(G)\geq 3$. Then the connected based 2-covering corresponding to $\pi_1(G,v)_{\rm ev}$ is the second projection $p:(K_2\times G,(0,v))\rightarrow (G,v)$. Let $\varphi:L_n \rightarrow G$ be a loop of $(G,v)$. Since the lift of $\varphi$ with respect to $p$ is $\tilde{\varphi}:L_n \rightarrow K_2\times G$ defined by $\tilde{\varphi}(i)=(i \; {\rm mod.}2, \varphi(i))$. Therefore the terminal point of $\tilde{\varphi}$ is $(0,v)$ if and only if $\varphi$ is even. Hence $p_* \pi_1(K_2 \times G)=\pi_1(G)_{\rm ev}$. Let $f:(G,v)\rightarrow (H,w)$ be a based graph map and $p:(K,u)\rightarrow (H,w)$ be a based 2-covering map. In this case, we consider the basepoint or $f^*K$ is $(v,u)$. We remark that the connectedness of $K$ does not imply the connectedness of $f^*K$. We write $q$ for the projection $f^*K \rightarrow G$. Then $q_*(\pi_1(f^*K))=f_*^{-1}(p_*(\pi_1(K)))$. Let $\overline{f}$ denote the second projection $f^{*}G \rightarrow G, (x,y) \mapsto y$. Since $p\overline{f} =fq$, we have $q_* (\pi_2(f^*G))\subset f_*^{-1}(p_*\pi_1(K))$. Let $[\varphi] \in f_*^{-1}(p_*\pi_1(K))$ and $\psi$ denote the lift of $f\varphi$ with respect to $p$. Since $[f\varphi] \in p_*\pi_1(K)$, we have $\psi$ is a loop of $(K,u)$. Then $x\mapsto (\varphi(x),\psi(x))$ is a loop of $f^{*}K$ and hence $q_* (\pi_1(f^*G))=f_*^{-1}(p_*(\pi_1(G)))$. Computations ============ In this section, we compute 2-fundamental groups of some graphs including $C_n$ and $K_n$. Then we obtain a condition for a graph whose chromatic number is 3. For the application of it, we investigate involutions on bipartite graphs. Next we construct some CW-complex whose fundamental group is equal to the 2-fundamental group, and prove van Kamepen theorem for 2-fundamental groups. Computations of 2-fundamental groups of graphs ----------------------------------------------- Here we compute 2-fundamental groups of $C_n$ and $K_n$. First we remark that $\pi_1(K_2)$ is trivial by definitions, and hence $\pi_1(C_4)$ is trivial since $C_4$ is $\times$-homotopy equivalent to $K_2$. Next we consider the case of $K_n$ for $n\geq 4$. For $n\geq 4$, $\pi_1(K_n)_{\rm ev}$ is trivial. Hence $\pi_1(K_n) \cong \mathbb{Z}/2\mathbb{Z}$ for $n\geq 4$. Let $0 \in K_n$ be a base point of $K_n$. Let $\varphi:L_m \rightarrow G$ be a loop of $K_n$. Suppose $m \geq 4$. Let $a \in V(K_n)$ be a vertex such that $a$ is not $0=\varphi(0)$, $\varphi(2)$ or $\varphi(4)$. We define a loop $\psi : L_m \rightarrow K_n$ by $\psi(i) =\varphi (i)$ for $i\neq 1,3$ and $\psi(1) =\psi (3)=a$. We define a loop $\gamma : L_{m-2} \rightarrow K_n$ by $\gamma (1)= a$ and $\gamma (i)=\varphi (i+2)$ for $i \geq 2$. Then we have $\varphi \simeq \psi \simeq \gamma$. Therefore for a loop $\varphi : L_m \rightarrow K_n$ with $m\geq 4$ is homotopic to a loop with length $m-2$. Since a loop whose length is 2 is 2-homotopic to a trivial path, we have $\pi_1(K_n)_{\rm ev}$ is trivial. For $n\geq 4$, connected 2-coverings over $K_n$ are only $K_n$ and $K_2 \times K_n$. Next we consider the case of $C_n$ for $n\geq 5$ or $n=3$. $\pi_1(C_r) \cong \mathbb{Z}$ for $r=3$ or $r\geq 5$. Therefore $\pi_1(K_3) \cong \mathbb{Z}$. First we prove that $\pi_1(L,0)$ is trivial. Since $L$ is bipartite, it is sufficient to prove that $\pi_1(L,0)_{\rm ev}$ is trivial. Let $\varphi: L_{2n}\rightarrow L$ be a loop of $(L,0)$. Suppose $n\geq 1$. Then $M=\max \{ |\varphi(x)| \; | \; x\in\{ 0,1, \cdots ,n\} \}$ is positive. Let $x_0 \in \{ 0,1, \cdots ,n\}$ with $|\varphi(x_0)|=M$. Then $x_0 \neq 0,2n$ since $M \neq 0$, and we have $\varphi(x_0 -1)=\varphi(x_0+1)$. Therefore $\varphi$ is homotopic to a loop with length $2n-2$. Hence $\pi_1(G,v)_{\rm ev}$ is trivial. Therefore $(L,0) \rightarrow (C_r,0)$ is a universal 2-covering over $(C_r,0)$ for $r\geq 5$ or $r=3$. For a nonnegative integer $n$, we let $\varphi_{n}:L_{rn}\rightarrow C_r$ be a loop of $(C_r,0)$ defined by $\varphi_n(i)= i$ $({\rm mod.}r)$. Then the lift of $\varphi_n$ with respect to $(L,0) \rightarrow (C_r,0)$ is $\tilde{\varphi}_n :L_{rn} \rightarrow L$ defined by $\tilde{\varphi}_n(i)=i$. For a negative integer $n$, we let $\varphi_n$ be a loop of $(C_r ,0)$ defined by $\varphi_n(i)=-i$ $({\rm mod}.r)$, and the lift of $\varphi_n$ with respect to $(L, 0) \rightarrow (C_r,0)$ is a loop $\tilde{\varphi}_n:L_{|rn|}\rightarrow L$ where $\tilde{\varphi}(i)=-i$. From Corollary 4.12, $\pi_1 (C_r,0)= \{[\varphi_n] \; | \; n\in \mathbb{Z}\}$ and $[\varphi_ n]\neq [\varphi_m]$ for $n\neq m$. Since $[\varphi_n]\cdot [\varphi_m]=[\varphi_{n+m}]$ for $n,m\in \mathbb{Z}$, we have $\mathbb{Z}\cong \pi_1(C_r,v)$. Let $G$ be a graph. If $\chi (G) =3$, $H_1(G)$ has $\mathbb{Z}$ as a direct summand. We can assume that $G$ is connected. Suppose $\chi(G)=3$ and $H_1(G)$ does not have $\mathbb{Z}$ as a direct summand. Then there exists a graph homomorphism $f :G\rightarrow K_3$. Since the hypothesis means that there is no surjective group homomorphism from $\pi_1(G)$ to $\mathbb{Z}$, the induced map $\pi_1(f) : \pi_1(G) \rightarrow \pi_1(K_3) \cong \mathbb{Z}$ is trivial. So there exists a lift $G\rightarrow L$, and we have $\chi(G) \leq \chi (L)=2$. This contradicts $\chi (G)=3$. Involutions of connected bipartite graphs ----------------------------------------- For an application of Corollary 5.5, we consider involutions of bipartite graphs. For a graph $G$, a graph homomorphism $f:G\rightarrow G$ is called an [*involution*]{} of $G$ if $f^2={\rm id}$. Let $G$ be a connected bipartite graph and $f: G\rightarrow G$ a graph homomorphism. We can easily show that it is independent of the choice of $v\in V(G)$ and a path $\varphi$ from $v$ to $f(v)$ whether the length of $\varphi$ is even or odd. We say that a graph homomorphism $f$ is [*even*]{} if the length of a path from $v$ to $f(v)$ is even for $v \in V(G)$. We say that $f$ is [*odd*]{} if $f$ is not even. Let $G$ be a connected bipartite graph. Then there exist $A_0 ,A_1 \subset V(G)$ such that\ (i) $A_0$ and $A_1$ are independent in $G$.\ (ii) $A_0 \cap A_1 =\emptyset$ and $A_0 \cup A_1 =V(G)$. This unordered pair $\{ A_0,A_1 \}$ is unique. Let $f:G\rightarrow G$ be a graph homomorphism. Then we have $$f \; {\rm is \; even.} \; \Leftrightarrow \; f(A_0) \subset A_0. \; \Leftrightarrow \; f(A_1) \subset A_1.$$ and $$f \; {\rm is \; odd.} \; \Leftrightarrow \; f(A_0) \subset A_1. \; \Leftrightarrow \; f(A_1) \subset A_0.$$ Therefore, if $\tau :G\rightarrow G$ is an odd involution for a connected bipartite graph $G$, the action of $\mathbb{Z}/2\mathbb{Z}$ induced by $\tau$ is a 2-covering action. Indeed, for $v\in A_0$, $N_2(v) \subset A_0$ and $N_2(\tau v) \subset A_1$, hence $N_2(v)\cap N_2(\tau v)=\emptyset$. The case $v\in A_1$ is similar. Let $G$ be a connected bipartite graph. Suppose $H_1(G)$ does not have $\mathbb{Z}$ as a direct summand. Let $\tau$ be an involution of $G$. Then we have;\ (1) if $\tau$ is even, for any 3-coloring $f: G\rightarrow K_3$, there exists $v\in V(G)$ such that $f(v)=f(\tau v)$.\ (2) if $\tau$ is odd, for any 3-coloring $f: G\rightarrow K_3$, there exists $v\in V(G)$ such that $f(v) \neq f(\tau v)$. Let $v\in V(G)$. We consider $v$ is a basepoint of $G$.\ (1) : Let $\tau$ be an even involution of $G$. We define a graph $G_{\tau}$ by $V(G_{\tau})=V(G)$ and $E(G_{\tau})=E(G) \cup \{ (x,\tau x ) \; | \; x\in V(G)\}$. It is sufficient to prove that $\chi(G_{\tau}) \geq 4$. Since $\tau$ is even, there exists a path $\varphi : L_{2r} \rightarrow G$ such that $\varphi(0)=v$ and $\varphi(2r)=\tau v$. Then we have a loop $\varphi ':L_{2r+1} \rightarrow G_{\tau}$ defined by $x\mapsto \varphi(x)$ for $x\leq 2r$ and $\varphi(2r+1)=v$. Therefore $\chi (G)\geq 3$. Let $\varphi:L_n\rightarrow G_{\tau}$ be a loop of $(G_{\tau},v)$. Let $n(\varphi)$ denote $\sharp \{ x\in \{ 0,1,\cdots \ n-1\} \; | \; (\varphi (x),\varphi(x+1))\notin E(G)\}$. We show if $n(\varphi)\geq 2$, $\varphi$ is homotopic to $\psi$ such that $n(\psi)=n(\varphi)-2$. Let $x_{\varphi}$ be the minimum of $\{ x\in \{ 0,1,\cdots \ n-1\} \; | \; (\varphi (x),\varphi(x+1))\notin E(G)\}$. Since $n(\varphi) \geq 2$, $x_{\varphi} \neq n-1$. If $(\varphi (x_{\varphi}+1),\varphi(x_{\varphi}+2))\in E(G)$, $\varphi$ is homotopic to $\varphi'$ defined by $\varphi'(i)=\varphi(i)$ for $i\neq x+1$ and $\varphi'(x+1)=\tau \varphi(x+1)$. Then $x_{\varphi'}=x_{\varphi}+1$ and $n(\varphi)=n(\varphi')$. Therefore $\varphi$ is homotopic to a loop $\psi$ such that $(\psi(x_{\psi}+1),\psi(x_{\psi}+2))\notin E(G) $ and $n(\psi)=n(\phi)$. Since $\psi(x_{\psi}+2)=\tau \psi(x_{\psi}+1)=\tau ^2 \psi(x_{\psi})=\psi(x_{\psi})$, $\psi$ is homotopic to the loop $\eta$ defined by $\eta(i)=\psi(i)$ for $i\leq x_{\psi}$ and $\eta(i)=\psi(i+2)$ for $i\geq x_{\psi}$. Then we have $n(\eta)=n(\varphi)-2$. Therefore if $n(\varphi)$ is even, $\varphi$ is homotopic to a loop of $(G,v)$. Since $n(\varphi \cdot \varphi)=n(\varphi)+n(\varphi)$, $\alpha^2 \in i_* \pi_1(G,v)$ for every $\alpha \in \pi_1(G_{\tau},v)$. Suppose that there exists a surjective group homomorphism $\Phi:\pi_1(G_{\tau},v)\rightarrow \mathbb{Z}$. Let $\alpha \in \Phi^{-1}(1)$. Since $\alpha ^2 \in i_{*} \pi_1(G,v)$, the composition $$\begin{CD} \pi_1(G,v) @>i_{*}>> \pi_1(G_{\tau},v) @>\Phi>> \mathbb{Z} \end{CD}$$ is not trivial. This contradicts to the assumption of $\pi_1(G,v)$.\ (2) : We write $H$ for the quotient graph $G/(\mathbb{Z}/2\mathbb{Z})$, and $w$ be the image of $v$ of the quotient map $p:G\rightarrow H$. We want to show that $\chi(H)\geq 4$. Since $\tau$ is odd, there is a path $\varphi:L_{2r+1}\rightarrow G$ such that $\varphi(0)=v$ and $\varphi(2r+1)=\tau v$. Then $p\varphi$ is a loop of $(H,w)$ with odd length, and we have $\chi(H) \geq 3$. Since $\tau$ is odd, the quotient map $p$ is a connected based double 2-covering over $(H,w)$. Therefore $p_* \pi_1(G,v)$ is a subgroup of $\pi_1(H,w)$ whose index is 2 (see Lemma 4.11.). Hence there are no surjective group homomorphisms from $\pi_1(H,w)$ to $\mathbb{Z}$. Therefore we have $\chi(H)\neq 3$ from Corollary 5.4. [(1) We can see that the chromatic number of $G_{\tau}$ in the proof of (1) of previous theorem is 4 if $\tau$ has no fixed point. In fact, let 2-coloring $f:G\rightarrow K_2$ and $A\subset V(G)$ such that $A\cap \tau A =\emptyset$ and $A\cup \tau A =V(G)$. Then we can obtain a graph homomorphism $g:V(G_{\tau}) \rightarrow K_4$ defined by $g(x)=2f(x)$ for $x\in A$ and $g(x)=2f(x)+1$ for $x \in \tau A$.\ (2) The chromatic number of $H$ in the proof of (2) of previous theorem is not determined. In fact, If $G$ is $K_2\times K_n$ and $\tau$ is defined by $\tau (\epsilon , x) =(\epsilon +1 \; {\rm mod.}2, x)$, then $H$ is $K_n$. ]{} Let $r$ be a positive integer. Let $n_1,n_2,\cdots ,n_r$ be positive integers and $s$ a integer with $0\leq s\leq r-2$. We define the graph $G(n_1,\cdots,n_r;s)$ by $$V(G(n_1,\cdots ,n_r;s))=\{ (x_1,\cdots ,x_r) \; | \; x_i \in \{ 0,1,\cdots ,n_i\} \text{ and } \sharp\{ i \; | x_i = 0 \text{ or }n_i \}\geq s\}$$ $$E(G(n_1,\cdots,n_r;s))=\{ ((x_1,\cdots ,x_r),(y_1,\cdots,y_r)) \; | \; \sum_{i=1}^n |x_i-y_i|=1\}.$$ $G(n_1,\cdots ,n_r;s)$ is bipartite since there exists a graph map $f:G(n_1,\cdots ,n_r;s) \rightarrow K_2$ defined by $f(x_1,\cdots ,x_r)=x_1+\cdots +x_r \; ({\rm mod.}2)$. We will prove that $\pi_1(G(n_1,\cdots ,n_r;s))$ is trivial in Example 5.15. $G$ has an involution $\tau$ defined by $\tau (x_1,\cdots x_r)\mapsto (n_1-x_1,\cdots ,n_r-x_r)$. $\tau$ is even if $N=n_1+\cdots +n_r$ is even and is odd if $N$ is odd. Therefore, for each 3-coloring $f:G(n_1,\cdots ,n_r;s) \rightarrow K_3$, there is $x\in V(G(n_1,\cdots ,n_r;s))$ such that $f(\tau x)=f(x)$ ($f(\tau x)\neq f(x)$) if $N$ is even (odd, respectively). (1): A 3-coloring $f$ of $C_8$ such that there exists no $v\in C_8$ such that $f(\tau v)=f(v)$, where $\tau$ is an antipodal map of $C_8$. We remark $\tau$ is even.\ (2): A 3-coloring $g$ of $C_6$ such that $f\tau=f$ where $\tau$ is an antipodal map of $C_6$. We remark $\tau$ is odd.\ \ \ Van Kampen’s theorem for 2-fundamental groups ---------------------------------------------- First, for a given graph $G$, we construct a 2-dimensional CW complex $|G|$ whose fundamental group is isomorphic to $\pi_1(G)$. In the construction of $|G|$ we need the following definitions. Let $G$ be a graph. A graph map $C_4 \rightarrow G$ is called a [*square*]{} of $G$. A square $\sigma : C_4 \rightarrow G$ of $G$ is said to be [*degenerate*]{} if $\sigma (0 )=\sigma(2)$ or $\sigma(1)=\sigma(3)$. A square $\sigma$ of $G$ is said to be [*nondegenerate*]{} if $\sigma$ is not degenerate. Let $\sigma, \sigma'$ be squares of $G$. We say that $\sigma$ is [*equivalent*]{} to $\sigma'$ if there is $k \in \{ 0,1,2,3\}$ such that $\sigma(i)=\sigma' (i+k)$ or $\sigma(i)=\sigma'(k-i)$ for each $i\in V(C_4)$. The examples of nondegenerate squares\ \ \ We begin to construct $|G|$. First the set of 0-cells of $|G|$ is $V(G)$. We regard $E(G)$ as a $(\mathbb{Z}/2\mathbb{Z})$-set by $\tau (x,y)=(y,x)$ where $\tau $ is a generator of $\mathbb{Z}/2\mathbb{Z}$. For each $\{ (x,y),(y,x)\} \in E(G)/(\mathbb{Z}/2\mathbb{Z})$, we attach a 1-cell whose end points are $x$ and $y$. We fix an orientation of a loop of each looped vertex. 2-cells of $|G|$ are attached as follows. For each looped vertex, we attach a 2-cell twice along given orientation of the loop. For each equivalence class $\alpha$ of a nondegenerate square, we let some representative $\sigma_{\alpha} \in \alpha$ and attached a cell along the loop $$\sigma (0) \rightarrow \sigma(1) \rightarrow \sigma(2) \rightarrow \sigma(3).$$ This is the construction of $|G|$. $\pi_1(G,v)\cong \pi_1(|G|,v)$ for a based graph $(G,v)$. $H_1(G)\cong H_1(|G|; \mathbb{Z})$ for a graph $G$. For the proof of Theorem 5.12, we need the next lemma. We write $I$ for the topological space $[0,1]$. Let $X$ be a 1-dimensional CW complex, $v,w \in X^0$. Let $\alpha$ be a homotopy class of paths of $X$ from $v$ to $w$ which is not homotopic to a constant path. Then there exists a unique representative $\varphi_{\alpha}$ of $\alpha$ satisfying following properties.\ (1) There exists a positive integer $N_{\alpha}$ such that $\varphi_{\alpha}^{-1}(X^0)=\{ \frac{i}{N_{\alpha}} \; | \; i\in \{ 0, 1, \cdots ,N_{\alpha}\} \}.$\ (2) For each $i\in \{ 0, 1, \cdots ,N_{\alpha}-1\}$, a path $I\rightarrow X$, $t\mapsto \varphi_{\alpha}(\frac{i+t}{N_{\alpha}})$ or $t\mapsto \varphi_{\alpha}(\frac{i+1-t}{N_{\alpha}})$ is a characteristic map of 1-cell whose endpoints are $\varphi_{\alpha}(\frac{i}{N_{\alpha}})$ and $\varphi_{\alpha}(\frac{i+1}{N_{\alpha}})$.\ (3) There is no $i\in \{ 1,\cdots ,N_{\alpha}-1\}$ such that $\varphi_{\alpha}(\frac{i-t}{N_{\alpha}})=\frac{i+t}{N_{\alpha}}$ for all $t\in I$. Let $X$ be a 1-dimensional CW complex, $v,w \in X^0$. Let $\alpha$ be the homotopy class of paths of $X$ from $v$ to $w$ and $\varphi \in \alpha$. Then $\varphi$ is called a [*canonical representative*]{} of $\alpha$ if $\varphi$ is constant or $\varphi=\varphi_{\alpha}$ where $\varphi_{\alpha}$ is in Lemma 5.13. We begin to prove Lemma 5.13. We can assume $X$ is simply-connected. Let $\varphi \in \alpha$. Let $x_0 =0$, $v_0 =v$, $x_1=\inf (\varphi^{-1}(X^0 -\{ v_0\}))$ and $v_1=\varphi(x_1)$. We remark that since $\varphi^{-1}(X^0 -\{ v_0\})$ is closed in $I$, $v_0 \neq v_1 \in X^0$. If $\varphi^{-1} (X^0 -\{ v_1 \}) \cap [x_1,1] \neq \emptyset$, set $x_2 =\inf (\varphi^{-1} (X^0 -\{ v_1 \}) \cap [x_1,1])$ and $v_2 =\varphi(x_2)$. After finite this operations, we have $x_n$ such that $\varphi^{-1}(X^0 -\{ v_n \})\cap [x_n ,1]=\emptyset$. In fact, if not, there exists an infinite sequence $(x_i)_{i}$ of $I$ such that $x_i <x_{i+1}$, $\varphi(x_i)\in X^0$ and $\varphi(x_i)\neq \varphi(x_{i+1})$. Set $x_{\infty}=\sup \{ x_i \; | \; i\in \mathbb{N}\}$. Since $\varphi^{-1}(X^0)$ is closed in $I$, $\varphi(x_{\infty}) \in X^0$. Since $\varphi^{-1}(x_{\infty})$ is open in $\varphi^{-1}(X^0)$, there exists $n\in \mathbb{N}$ such that for all $m\geq n$, we have $\varphi(x_m)=\varphi(x_{\infty})$. But this contradicts to $\varphi(x_n)\neq \varphi(x_{n+1})$. Therefore we have a finite sequence $v=x_0 , \cdots ,x_n=w$. Then $\varphi$ is homotopic to $\psi: I \rightarrow X$ satisfying $\psi^{-1}(X^0)=\{ \frac{i}{n} \; | \; i\in \{0,1,\cdots ,n \} \}$ and $\psi(\frac{i}{n})=x_i$ and (2). If $\psi$ does not satisfy (3), there exists $i\in \{ 1, \cdots ,N-1 \}$ such that $\psi(i+t)=\psi(i-t)$ for $t\in I$. We define a path $\psi': I\rightarrow X$ by setting $\psi'(\frac{j+t}{N-2})=\psi(\frac{j+t}{N})$ for $t\in I$, $j< i$ and $\psi'(\frac{j+t}{N-2})=\psi(\frac{j+2+t}{N})$ for $t\in I$, $j\geq i$. Then $\psi'$ is obviously homotopic to $\psi$. After finite these operations, we have $\varphi_{\alpha}$ satisfying (1), (2), (3). We prove the uniqueness of $\varphi_{\alpha}$. Suppose $\varphi_{\alpha}$ is not unique. Then we can easily see that there exists a loop $\varphi:I \rightarrow X$ satisfying (1), (2), (3). If there exists $\varphi (\frac{i}{N})=\varphi (\frac{j}{N})$ for $i< j$, let $\varphi'$ be a loop defined by the composition $$\begin{CD} [0,1] @>>> [\frac{i}{N} ,\frac{j}{N}] @>\varphi>> X \end{CD}.$$ Therefore we can assume $\varphi(\frac{i}{N}) \neq \varphi(\frac{j}{N})$ for $i < j$. Let $Y$ be a subcomplex $X\setminus \varphi((0,\frac{1}{N}))$. Then the composition $$\begin{CD} S^1 \approx [0,1]/\{0,1\} @>\varphi>> X @>>> X/Y \approx S^1 \end{CD}$$ has degree 1 or $-1$. This is a contradiction since $\pi_1(X)$ is trivial. We begin to prove Theorem 5.12. Let $\varphi:L_n \rightarrow G$ be a loop of $(G,v)$. We write $\Phi (\varphi)$ for the loop of $(|G|,v)$ such that $\varphi(0)\rightarrow \varphi(1) \rightarrow \cdots \rightarrow \varphi(n)$. Then $\Phi$ induces a group homomorphism $\overline{\Phi}: \pi_1(G,v)\rightarrow \pi_1(|G|,v)$. We define the group homomorphism $\Psi :\pi_1(|G|^1,v) \rightarrow \pi_1(G,v)$ as follows. We set $\Psi (1)=1$. For $1 \neq \alpha \in \pi_1(|G|^1,v)$, let $\varphi_{\alpha}$ be the canonical representation of $\alpha$. Then we define the loop $\Psi(\alpha) :L_{N_{\alpha}} \rightarrow G$ of $(G,v)$ by setting $\Psi(\alpha)(i)=\varphi_{\alpha}(\frac{i}{N_{\alpha}})$. Then $\Psi$ induces $\overline{\Psi} : \pi_1(|G|,v)\rightarrow \pi_1(G,v)$ and $\overline{\Psi}$ is the inverse of $\overline{\Phi}$. Let $G= G(n_1,\cdots ,n_r; s)$ $(s\leq r-2)$ in Example 5.8. From Theorem 5.13, $\pi_1(G)$ is isomorphic to the fundamental group of a topological space $Y_{r,s}$ where $$Y_{r,s}=\{ (x_1,\cdots ,x_r)\in [0,1]^n \; | \; \sharp\{i \; | x_i=0\text{ or }1\} \geq s\}.$$ First we remark that $Y_{r,0}$ is simply connected for $r\geq 2$, and $Y_{r,s}$ for $s\leq r-1$ is connected. Let $(r,s)$ for $r\geq s$, $$Y^+_{r,s}=\{ (x_1,\cdots ,x_r)\in [0,1]^n \; | \; \sharp\{i \; | x_i=0\text{ or }1\} \geq s, x_r<1 \}$$ $$Y^+_{r,s}=\{ (x_1,\cdots ,x_r)\in [0,1]^n \; | \; \sharp\{i \; | x_i=0\text{ or }1\} \geq s, x_r>0 \}$$ Since $Y_{r-1,s-1}$ is the deformation retract of $Y^+_{r,s}$ and $Y^-_{r,s}$ and $Y^+_{r,s} \cap Y^-_{r,s}$ is connected since $Y_{r-1,s}$ is the deformation retract of $Y^+_{r,s} \cap Y^-_{r,s}$. Hence $Y_{r,s}$ for $s\leq r-2$ is simply connected from van Kampen’s theorem. Before giving the statement of van Kampen’s theorem for 2-fundamental groups, we give some definitions about squares. Let $G$ be a graph and $\sigma,\sigma_1,\sigma_2$ be squares of $G$. We write $\sigma_1 \cup \sigma_2 =\sigma$ or $\sigma_2\cup \sigma_1=\sigma$ if one of the following two conditions are satisfied.\ (1) $\sigma(i)=\sigma_1(i)$ for $i\neq 2$ and $\sigma(i)=\sigma_2(i)$ for $i\neq 0$ and $\sigma_1(2)=\sigma_2(0)$.\ (2) $\sigma(i)=\sigma_1(i)$ for $i\neq 3$ and $\sigma(i)=\sigma_2(i)$ for $i\neq 1$ and $\sigma_1(3)=\sigma_2(1)$. Let $G$ be a graph and $\sigma$ be a square of $G$. We define a decomposition sequence of $\sigma$ as follows :\ (1) $(\sigma)$ is a decomposition sequence.\ (2) If $(\sigma_1 ,\cdots ,\sigma_n)$ be a decomposition sequence of $\sigma$ and $\sigma_i = \tau \cup \tau'$, then $(\sigma_1 , \cdots , \sigma_{i-1},\tau ,\tau' ,\sigma_{i+1} ,\cdots ,\sigma_n)$ is a decomposition sequence of $\sigma$. We say that $\sigma$ is decomposed into $\sigma_1,\sigma_2,\cdots,\sigma_n$ if $(\sigma_1,\cdots ,\sigma_n)$ is a decomposition sequence of $\sigma$. Let $(G,v)$ be a based graph. Let $\{ G_{\alpha}\}_{\alpha \in A}$ be a family of subraphs of $G$ such that $v \in V(G_{\alpha})$ for each $\alpha \in A$. We write $i_{\alpha}$ for the inclusion $G_{\alpha} \rightarrow G$ and $j_{\alpha \beta}$ for the inclusion $G_{\alpha}\cap G_{\beta}\rightarrow G_{\alpha}$. We define the group homomorphism $\Phi :* _{\alpha \in A}\pi_1(G_{\alpha},v) \rightarrow \pi_1(G,v)$ by the free product of all $\pi_1 (i_{\alpha})$. Let $N$ be the normal subgroup generated by $\{ j_{\alpha \beta *}^{-1}(x) j_{\beta \alpha *}(x) \; | \; \alpha,\beta \in A, \; x\in \pi_1(G_{\alpha} \cap G_{\beta}, v) \}$. Since $i_{\alpha} j_{\alpha \beta}=i_{\beta}j_{\beta \alpha}$, $N$ is in the kernel of $\Phi$. Therefore $\Phi$ induces the group homomorphism $\tilde{\Phi} :*_{\alpha \in A} \pi_1(G_{\alpha} ,v) /N \rightarrow \pi_1(G,v)$. The statement of the following theorem is that $\tilde{\Phi}$ is isomorphism under some conditions. This is an analogy of van Kampen theorem. Let $(G,v)$ be a graph. Let $\{ G_{\alpha } \}_{\alpha \in A}$ be a family of subgraphs of $G$ satisfying following properties.\ (i) $\displaystyle \bigcup_{\alpha \in A} G_{\alpha } =G$ and $v\in V(G_{\alpha})$ for all $\alpha \in A$.\ (ii) $G_\alpha \cap G_\beta \cap G_\gamma$ is connected for all $\alpha,\beta,\gamma \in A$.\ (iii) Every nondegenerate square of $G$ is decomposed into squares contained in some $G_{\alpha}$. Then the homomorphism $\tilde{\Phi} :*_{\alpha \in A} \pi_1(G_{\alpha},v)/N \rightarrow \pi_1(G,v)$ defined above is an isomorphism. In this case, $\pi_1(|G|,v)\cong \pi_1(\bigcup_{\alpha} |G_{\alpha}|,v )$ and $|G_{\alpha} \cap G_{\beta}|=|G_{\alpha}|\cap |G_{\beta}|$. Therefore we can deduce this theorem from van Kampen theorem of topology. Let $\{ (G_{\alpha},v_{\alpha}) \}_{\alpha}$ be a family of based graphs such that $v_{\alpha}$ is not looped for all $\alpha$. In this case, a nondegenerate square of $\bigvee_{\alpha} G_{\alpha}$ is contained in some $G_{\alpha}$. Therefore we have $\pi_1(\bigvee G_{\alpha})\cong *_{\alpha} \pi_1(G_{\alpha})$. Let $G$ be a graph. We write $H_0(G)$ for the free abelian group generated by the set of all connected components of $G$. Let $G$ be a graph and $K_1,K_2$ subgraphs of $G$. If every nondegenerate square of $G$ is decomposed in squares of $K_1$ or $K_2$, there exists an exact sequence $$H_1(K_1\cap K_2) \rightarrow H_1 (K_1) \oplus H_1(K_2) \rightarrow H_1(K_1 \cup K_2) \rightarrow H_0(K_1\cap K_2) \rightarrow H_0 (K_1) \oplus H_0(K_2) \rightarrow H_0(K_1 \cup K_2)\rightarrow 0$$ In this case, $H_1 (|K_1\cup K_2| ;\mathbb{Z})= H_1(|K_1|\cup |K_2| ;\mathbb{Z})$. Therefore we can deduce this theorem from Mayer-Vietoris sequence of $H_{*} (-;\mathbb{Z})$. Fundamental groups of neighborhood complexes ============================================ In this section, we prove the following theorem. Let $(G,v)$ be a based graph where $v$ is not isolated. Then we have a group isomorphism $$\pi_1(G,v)_{\rm ev} \cong \pi_1(|\mathcal{N}(G)|,v).$$ and this isomorphism is natural with respect to based graph homomorphisms. Let $G$ be a graph. If $\chi(G)=3$, $H_1(\mathcal{N}(G) ; \mathbb{Z})$ has $\mathbb{Z}$ as a direct summand. This is deduced from Theorem 6.1 and Corollary 5.4. For a locally finite graph $G$, the map $|{\rm Hom}(K_2,G)| \rightarrow |\mathcal{N}(G)|$ induced by the poset map $\eta \mapsto \eta(0)$ is a homotopy equivalence, see [@BK06]. Therefore we have similar statements of Theorem 6.1 and Corollary 6.2 for ${\rm Hom}(K_2,G)$. But since ${\rm Hom}(K_2,G)$ has no canonical point for a based graph $(G,v)$, and the statement and the proof of Theorem 6.1 become more complicated. Let $(G,v)$ and $(H,w)$ be based graphs where $v$ and $w$ are not isolated. Then$$(p_{1*},p_{2*}) : \pi_1(G\times H,(v,w))_{\rm ev} \rightarrow \pi_1(G,v)_{\rm ev}\times \pi_1(H,w)_{\rm ev}, \; \alpha \mapsto (p_{1*}\alpha,p_{2*}\alpha)$$ is an isomorphism where $p_1:G\times H \rightarrow G$ and $p_2 :G\times H \rightarrow H$ are projections. Then this is deduced from the fact that $(|p_{1*}|,|p_{2*}|):|\mathcal{N}(G\times H)|\rightarrow |\mathcal{N}(G)|\times |\mathcal{N}(H)|$ is a homotopy equivalence. We think this is well-known, but we give the proof for self-contained. We write $F\mathcal{N}(G)$ for the face poset of $\mathcal{N}(G)$. Then we have the two poset maps $$p: F\mathcal{N}(G\times H)\rightarrow F\mathcal{N}(G)\times F\mathcal{N}(H), \sigma \mapsto (p_1(\sigma),p_2(\sigma ))$$ $$i:F\mathcal{N}(G)\times F\mathcal{N}(H)\rightarrow F\mathcal{N}(G\times H),(\sigma,\tau)\mapsto \sigma \times \tau.$$ It is easy to show that $p$ and $i$ are well-defined poset maps, $pi={\rm id}$ and $ip(\sigma)\geq \sigma$ for $\sigma \in F\mathcal{N}(G\times H)$. Hence $ip$ is an ascending closure map and $|p|\cong |(p_{1*},p_{2*})|$ is a homotopy equivalence. To prove Theorem 6.1, we first construct the group homomorphism $\overline{\Phi}: \pi_1(G,v)\rightarrow \pi_1(\mathcal{N}(G),v)$. Let $\varphi$ be a loop with length $2n$. We remark that $\varphi (2i-2),\varphi(2i)\in N(\varphi(2i-1))$, so we have $\{ \varphi (2i-2) ,\varphi(2i)\}$ is a simplex of $\mathcal{N}(G)$. For a vertex $a,b \in \mathcal{N}(G)$ such that $\{ a,b\}$ is a simplex of $\mathcal{N}(G)$, let $\gamma _{ba}: I \rightarrow |\mathcal{N}(G)|$ denote a path $\gamma_{ba}(t)=(1-t)a +tb$. Let $(a_0 ,\cdots ,a_m)$ be a finite sequence of vertices of $\mathcal{N}(G)$ such that $\{ a_{i-1},a_i\} \subset \mathcal{N}(G)$ for $1\leq i \leq m$. Then we write $$a_0 \rightarrow a_1 \rightarrow \cdots \rightarrow a_m$$ for a path $t\mapsto \gamma_{a_i a_{i-1}}(mt-i)$ for $\frac{i-1}{m}\leq t \leq \frac{i}{m}$. In this notation, the loop $\Phi(\varphi)$ is defined by $$\varphi (0)\rightarrow \varphi(2)\rightarrow \cdots \rightarrow \varphi(2n).$$ We prove $\Phi$ induces a group homomorphism $\overline{\Phi}:\pi_1(G,v)\rightarrow \pi_1(|\mathcal{N}(G)|,v)$. Let $\varphi ,\psi$ be loops of $(G,v)$ with even lengths such that $\varphi \simeq \psi$. We want to say $\Phi (\varphi)\simeq \Phi (\psi)$. It is sufficient to show that $(\varphi,\psi)$ satisfies the condition (i) or (ii)$'$ in the definition of 2-homotopy of paths. Suppose ($\varphi ,\psi$) satisfies (i). Namely $l(\varphi)=2n$ then $l(\psi)=2n+2$ and there exists $x\in \{ 1,2 \cdots ,2n-1\}$ such that $\varphi(i)=\psi(i)$ for $(i\leq x)$ and $\varphi(i)=\psi(i+2)$ for $i\geq x$. If $x$ is even, $\Phi (\varphi)\simeq \Phi(\psi)$ is obvious since $\Phi (\varphi)$ is the path $$v=\varphi(0) \rightarrow \varphi(2)\rightarrow \cdots \rightarrow \varphi(x-2)\rightarrow \varphi(x)\rightarrow \varphi(x+2)\rightarrow \cdots \rightarrow \varphi(2n)=v$$ and $\Phi(\psi)$ is the loop $$v=\varphi(0)\rightarrow \varphi(2)\rightarrow \cdots \rightarrow \varphi(x-2)\rightarrow \varphi(x)\rightarrow \varphi(x)\rightarrow \varphi(x+2)\rightarrow \cdots \rightarrow \varphi(2n)=v.$$ Suppose $x$ is odd. We remark that $\psi (x-1) (=\varphi(x-1))$, $\psi(x+1)$ and $\psi (x+3)(=\varphi(x+1))$ is in the neighborhood of $\psi(x)(=\varphi(x)=\varphi(x+2))$. Therefore $\Phi(\varphi)$ is homotopic to the loop $$\varphi(0)\rightarrow \varphi(2)\rightarrow \cdots \rightarrow \varphi(x-1)\rightarrow \psi (x+1) \rightarrow \varphi(x-1) \rightarrow \varphi(x+1) \rightarrow \cdots \rightarrow \varphi(2n)$$ and this is homotopic to the loop $$\varphi(0) \rightarrow \varphi(2)\rightarrow \varphi(x-1)\rightarrow \psi (x+1) \rightarrow \varphi(x+1)\rightarrow \cdots \rightarrow \varphi(2n).$$ Since the last one is equal to $\Phi(\psi)$, we have $\Phi(\varphi)\simeq \Phi(\psi)$. Suppose $(\varphi ,\psi )$ satisfies the condition (ii)$'$. Namely $l(\varphi)=l(\psi)=2n$ and there is $x\in \{ 1,2,\cdots ,2n-1\}$ such that $\varphi(i)=\psi (i)$ for $i\neq x$. If $x$ is odd, then $\Phi (\varphi)=\Phi(\psi)$. Suppose $x$ is even. We remark that $\{\varphi(x-2) ,\varphi(x),\psi(x)\} \subset N(\psi (x-1))$ and $\{ \varphi(x),\psi(x),\varphi(x+2)\} \subset N(\varphi(x+1))$. Therefore$$\varphi(0)\rightarrow \varphi(2) \rightarrow \cdots \rightarrow \varphi(x-2) \rightarrow \varphi(x)\rightarrow \varphi(x+2) \varphi \cdots \varphi(2n)$$ is homotopic to $$\varphi (0)\rightarrow \varphi(2) \rightarrow \cdots \rightarrow \varphi (x-2)\rightarrow \varphi(x)\rightarrow \psi(x)\rightarrow \varphi(x) \rightarrow \varphi(x+2) \rightarrow \cdots \rightarrow \varphi(2n)$$ and this is homotopic to $$\varphi(0) \rightarrow \varphi(2) \rightarrow \cdots \rightarrow \varphi(x-2)\rightarrow \psi (x) \rightarrow \varphi(x+2) \rightarrow \cdots \rightarrow \varphi(2n).$$ Therefore we have $\Phi(\varphi) \simeq \Phi(\psi)$. Hence $\Phi$ induces $\overline{\Phi}:\pi_1(G,v) \rightarrow \pi_1(|\mathcal{N}(G)|,v)$. From the definition of $\Phi$, $\overline{\Phi}$ is a group homomorphism and is natural with respect to based graph homomorphisms. Next we construct the inverse of $\overline{\Phi}$. We define the $\Psi:\pi_1(|\mathcal{N}(G)|^1,v)\rightarrow \pi_1(G,v)_{\rm ev}$ as follows. First we set $\Psi (1)=1$. Let $\alpha \in \pi_1(|\mathcal{N}(G)|^1,v)$ be a non-identity element and $\varphi_{\alpha}$ a canonical representative of $\alpha$. Then there exists $n \in \mathbb{N}$ such that $\varphi_{\alpha}^{-1}(|\mathcal{N}(G)|^0)=\{ \frac{i}{n} \; | \; 0\leq i \leq n\}$. Then we define $\Psi(\alpha)$ by the 2-homotopy class of $$\varphi(0)\rightarrow v_1 \rightarrow \varphi(\frac{1}{n}) \rightarrow v_2 \rightarrow \cdots \rightarrow v_n \rightarrow \varphi(1)$$ where $v_i$ is a vertex of $V(G)$ such that $\varphi(\frac{i-1}{n}),\varphi(\frac{i}{n})\in N(v_i)$. From the definition of 2-homotopy of paths, the homotopy class of the above path is independent of the choice of $v_i$. We can easily see that $\Psi$ is a group homomorphism. We want to show that $\Psi$ induces a group homomorphism $\overline{\Psi}: \pi_1(|\mathcal{N}(G)|,v) \rightarrow \pi_1(G,v)_{\rm ev}$. Then it is sufficient to show that, for each loop $\gamma$ of $(|\mathcal{N}(G)|^1,v)$ which is homotopic to an attaching map of a 2-cell of $|\mathcal{N}(G)|$, $\Psi ([\gamma])$ is nullhomotopic. Let $\{ y_0 ,y_1,y_2\}$ be a 2-simplex of $|\mathcal{N}(G)|$. Let $\gamma$ be the loop of $(|\mathcal{N}(G)|^1,v)$ written by $$v=x_0 \rightarrow x_1 \rightarrow \cdots \rightarrow x_n=y_0 \rightarrow y_1 \rightarrow y_2 \rightarrow y_0 =x_n \rightarrow \cdots \rightarrow x_1 \rightarrow x_0 .$$ Then $\Psi ([\gamma])$ is a homotopy class of a loop $\varphi :L_{4n+6} \rightarrow G$ such that $\varphi (2i)=x_i=\varphi(4n+6-2i)$ for $i=0 ,1, \cdots ,n$ and $\varphi(2n+2)=y_1$ and $\varphi(2n+4)=y_2$. Since $\{y_0 ,y_1 ,y_2\}$ is a simlex of $\mathcal{N}(G)$, there exists $w \in V(G)$ such that $\{y_0 ,y_1 ,y_2\} \subset N(w)$. Therefore we can assume $\varphi(2n+1)=\varphi(2n+3)=\varphi(2n+5)=w$. Thus $\varphi$ is homotopic to $\psi:L_{4n}\rightarrow G$ such that $\psi (2i)=x_i=\psi(4n-2i)$ for $i=0,1,\cdots n$. But $\psi$ is obviously nullhomotopic. Therefore $\Psi$ induces a group homomorphism $\overline{\Psi}:\pi_1(|\mathcal{N}(G)|,v)\rightarrow \pi_1(G,v)_{\rm ev}$. From the definition, $\overline{\Psi}$ is the inverse of $\overline{\Phi}$. This completes the proof of Theorem 6.1.\ [**Acknowledgement.**]{} The author would like to express his gratitude to Toshitake Kohno for helpful suggestions and stimulating discussions. [0]{} Eric Babson and Dmitry N. Kozlov, *Complexes of graph homomorphisms*, Israel J. Math, **152** (2006) 285-312, P${\rm \acute{e}}$ter Csorba, [*Homotopy types of box complexes.*]{} Combinatorica, [**27**]{} (6):669-682 Anton Dochtermann, *Hom complexes and homotopy theory in the category of graphs* European J. Combin., **30** (2):490-509, 2009. Anton Dochtermann, *Homotopy groups of Hom complexes of graphs*. J. Combin. Theory Ser. 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--- abstract: 'The explicit expressions for the electric, magnetic, axial and induced pseudoscalar form factors of the nucleons are derived in the [*ab initio*]{} quantized Skyrme model. The canonical quantization procedure ensures the existence of stable soliton solutions with good quantum numbers. The form factors are derived for representations of arbitrary dimension of the $SU(2)$ group. After fixing the two parameters of the model, $f_\pi$ and $e$, by the empirical mass and electric mean square radius of the proton, the calculated electric and magnetic form factors are fairly close to the empirical ones, whereas the the axial and induced pseudoscalar form factors fall off too slowly with momentum transfer.' address: - 'Institute of Theoretical Physics and Astronomy, Vilnius, 2600 Lithuania' - 'Department of Physics, University of Helsinki, 00014 Finland' - Helsinki Institute of Physics author: - 'A. Acus' - 'E. Norvaišas' - 'D.O. Riska' title: Nucleon form factors in the canonically quantized Skyrme model --- Nucleon form factors ,Skyrme model 12.39.Dc ,13.40.Gp Introduction {#intro} ============ Quantum chromodynamics (QCD) describes the nucleon as a composite system with many internal degrees of freedom. In the nonperturbative region, which encompasses hadron structure and intermediate range observables, the large $N_C$ limit of the theory, which partly allows treatment in closed form, has proven to be of phenomenological utility. This limit of QCD may be realized either in terms of the constituent quark model, or, as was first suggested, in the form of effective meson field theory, in which the baryons appear as topologically stable soliton solutions [@Jenkins]. The generic chiral topological soliton model with topologically stable solutions, which represent baryons is that of T.H.R. Skyrme [@Sky1]. The first comprehensive phenomenological application of the model to nucleon structure was the semiclassical calculation of the static properties of the nucleon in ref. [@Adk]. In terms of agreement with the static baryon observables the results fell short by typically $\sim$ 30 $\%$, which could be viewed as an expected flaw of the large $N_C$ limit. That approach to the model did however have the more principal imperfection in that its lack of stable semiclassical solutions with good quantum numbers. The quantization was therefore realised by means of rigid body rotation of the classical skyrmion solutions. A strictly canonical quantization of Skyrme model was derived later in ref.[@Fujii] and shown to yield stable quantized skyrmion solutions in refs.[@Nor1; @Acus97; @Acus98]. In addition the Skyrme model was generalized to representations of arbitrary dimension of the SU(2) group. It was shown that - in fact an obvious consistency check - the classical skyrmion solutions are independent of the dimension of the representation, but that in contrast the canonically quantized Skyrme model gives results for baryon observables, which are representation dependent. An interesting consequence of the canonical [*ab initio*]{} quantization of the Skyrme model is the natural appearance of a finite effective pion mass even for the chirally symmetric Lagrangian. While the finite pion mass is conventionally introduced by adding an explicitly chiral symmetry breaking pion mass term to the Lagrangian density of the model [@Nappi], the canonical quantization procedure by itself gives rise to a finite pion mass. This realizes Skyrme’s original conjecture that “This (chiral) symmetry is, however, destroyed by the boundary condition ($U(\infty )=1$), and we believe that the mass (of pion) may arise as a self consistent quantal effect [@Sky2]”. To derive the explicit expressions for electric, magnetic, axial and pseudoscalar form factors of the nucleon we employ the expressions for the Noether currents derived in ref.[@Acus98]. Because of the appearance of a finite “effective” pion mass the asymptotic behavior of the chiral angle $F(r)$ has the required exponential falloff, which ensures finite radii and physical forms for the energy (mass) density. The expressions for the current operators are valid for representations of arbitrary dimension of SU(2). Numerical results are given for the representations with $j = 1/2; 1; 3/2$ and also for the reducible representation $j = 1\oplus 1/2\oplus 1/2\oplus 0$. The different representations of the quantized Skyrme model may be interpreted as different phenomenological models. The best agreement with experimental data on the form factors obtain with the reducible SU(2) representation, which in fact is the SU(3) group octet $(1,1)$ restricted to the SU(2). This paper is divided into 5 sections. In Section \[sec2\] the canonically quantized skyrmion is reviewed. In Section \[sec3\] we derive the electroweak form factors of the nucleon. Section \[sec4\] contains the numerical results and Section \[sec5\] a concluding discussion. Canonically quantized skyrmion {#sec2} ============================== The chirally symmetric Lagrangian density that defines the Skyrme model may be written in the form [@Adk]: $${\mathcal{L}}[U({{\bf r}},t)]=-\frac{f_\pi ^2}{4}\mathrm{Tr}\{R_\mu R^\mu \}+{\frac1{32e^2}\mathrm{Tr}}\{[R_\mu ,R_\nu ]^2\}, \label{f1}$$ where $R_\mu$ is the “right handed” chiral current $R_\mu=(\partial _\mu U)U^{\dagger }$. The unitary field $U({{\bf r}},t)$ may, in a general reducible representation of the SU(2) group, be expressed as a direct sum of Wigner’s D matrices: $$U({{\bf r}}, t)=\sum_k \oplus D^{j_k}[\vec\alpha({{\bf r}},t)]. \label{f0}$$ Here the vector $\vec\alpha$ represents a triplet of Euler angles $\alpha_1({\bf r},t)$, $\alpha_2({\bf r},t)$, $\alpha_3({\bf r},t)$. Quantization of the skyrmion field $U$ is brought about by means of rotation by collective coordinates that separate the variables, which depend on time and spatial coordinates: $$U({\bf r},{\bf q}(t))= A\left( {\bf q}(t)\right) U_0({\bf r})A^{\dagger}\left( {\bf q}(t)\right). \label{f2}$$ Here the matrix $U_0$ is the generalization of the classical hedgehog ansatz to a general reducible representation [@Acus98]. The collective coordinates ${\bf q}(t)$ (the Euler angles) are dynamical variables that satisfy the commutation relations $[\dot q^a,\,q^b]\neq 0$. The energy of the canonically quantized skyrmion, which represents a baryon with spin-isospin $\ell $, which corresponds to the Lagrangian density (\[f1\]) in an arbitrary reducible representation has the form: $$E(j,\ell ,F)=M(F)+\Delta M_j(F)+\frac{\ell (\ell +1)}{2a(F)}, \label{f3}$$ where $M(F)$ represents the classical skyrmion mass: $$M(F)=2\pi \frac{f_\pi}{e}\int \d\tilde{r}\tilde{r}^2\biggl( F^{\prime 2}+\frac{\sin ^2F}{\tilde{r}^2}\Bigl( 2+2F^{\prime 2} +\frac{\sin ^2F}{\tilde{r}^2}\Bigr) \biggr). \label{f4}$$ The dimensionless variable $\tilde{r}=ef_\pi r$ has been employed here. Above $a(F)$ represents the moment of inertia of the skyrmion: $$a(F)=\frac{8\pi }3\frac{1}{e^3f_\pi }\int \d\tilde{r}\tilde{r}^2\sin ^2F\Bigl( 1 +F^{\prime 2}+\frac{\sin ^2F}{\tilde{r}^2}\Bigr), \label{f5}$$ and $\Delta M_j(F)$ is a (negative) mass term, which appears in the canonically quantized version of the model: $$\begin{aligned} \Delta M_j(F)&=&\frac{-2\pi } {15e^3f_\pi {a}^2(F)}\int \d\tilde{r}\tilde{r}^2\sin ^2F \Bigl(15+4d_2\sin ^2F(1-F^{\prime 2}) \label{f6} \\ &&\phantom{ \frac{-2\pi }{5e^3f_\pi {a}^2(F)}\int \d\tilde{r}\tilde{r}^2\sin ^2F \Bigl(15} +2d_3\frac{\sin ^2F}{\tilde r^2} +2d_1F^{\prime 2}\Bigr)\nonumber.\end{aligned}$$ The coefficients $d_{i}$ in these expressions are given as $$\begin{aligned} N&=&\frac23\sum_{k}j_{k}(j_{k}+1)(2j_{k}+1). \label{F24a}\\ d_{1} &=&\frac{1}{N}\sum_{k}j_{k}(j_{k}+1)(2j_{k}+1)\bigl( 8j_{k}(j_{k}+1)-1\bigr) , \label{F24b} \\ d_{2} &=&\frac{1}{N}\sum_{k}j_{k}(j_{k}+1)(2j_{k}+1)(2j_{k}-1)(2j_{k}+3), \label{F24c} \\ d_{3} &=&\frac{1}{N}\sum_{k}j_{k}(j_{k}+1)(2j_{k}+1)\bigl( 2j_{k}(j_{k}+1)+1\bigr) . \label{F24d}\end{aligned}$$ Minimization of the mass expression in Eq. (\[f4\]) for $M(F)$, gives the classical solution for the chiral angle $F(r)$, which behaves as $1/\tilde r^2$ at large distances. In the semiclassical case, the quantum mass correction $\Delta M_j(F)$ drops out, and variation of the expression (\[f3\]) yields no stable solution [@Braaten]. Such a semiclassical skyrmion was considered in ref. [@Adk] as a “rotating” rigid-body skyrmion with fixed $F(r)$. The canonical quantization procedure in terms the collective coordinates approach leads to the expanded energy expression  (\[f3\]), variation of which yields a (self-consistent) integro-differential equation with the boundary conditions $F(0)=\pi $ and $F(\infty )=0$. In contrast to the semiclassical case, the asymptotic behavior of $F(\tilde r)$ at large $\tilde r$ falls off exponentially as: $$F(\tilde r)=k\left( \frac{\tilde m_\pi }{\tilde r}+\frac 1{\tilde r^2}\right) \exp(-\tilde m_\pi \tilde r), \label{f12}$$ with $$\tilde m_\pi ^2=-\frac{1}{3e^2f_\pi^2 a(F)}\left( 8\Delta {M_j}(F)+\frac{2\ell (\ell +1)+3}{ a(F)}\right). \label{f11}$$ The integrals (\[f4\]), (\[f5\]), (\[f6\]) are convergent, and therefore ensure the stability of the quantum skyrmion only if $\tilde m_\pi ^2>0$. The positive quantity $m_\pi =ef_\pi \tilde m_\pi $ admits an obvious interpretation as an effective pion mass. The appearance of this effective pion mass conforms with Skyrme’s original conjecture concerning the origin of the pion mass. Form factors {#sec3} ============ The electroweak form factors of the semiclassically quantized SU(2) skyrmion were studied systematically in ref.[@Braaten1]. An extension of this work to the SU(3) was made in ref. [@Meissner]. Analogous studies of the electroweak form factors in the related SU(3) chiral Quark-Soliton Model has been made in ref.[@Praszal]. The explicit expressions for the Noether current density operators of the canonically quantized Skyrme model were derived in ref.[@Acus98]. The isoscalar part of the nucleon electromagnetic current operator is proportional to the topological baryon current operator and therefore depends on the Lagrangian density only through the chiral angle. The isovector component of the vector current of the nucleon current is proportional to vector Noether current of the Lagrangian density [@Acus98]. Linear combinations of the isoscalar and isovector charge densities yield the expressions for the proton and the neutron charge densities: $$\begin{aligned} \rho _{p}(r) &=&-\frac{1}{4\pi ^{2}r^{2}}F^{\prime }(r)\sin ^{2}F(r) \nonumber \\ &&\phantom{-}+\frac{1}{3a(F)}\sin ^{2}F(r)\biggl( f_{\pi }^{2}+\frac{1}{e^{2}}\Bigl( F^{\prime 2}(r)+\frac{\sin ^{2}F(r)}{r^{2}}\Bigr) \biggr), \label{fa1} \\ \rho _{n}(r) &=&-\frac{1}{4\pi ^{2}r^{2}}F^{\prime }(r)\sin ^{2}F(r) \nonumber \\ &&\phantom{-}-\frac{1}{3a(F)}\sin ^{2}F(r)\biggl( f_{\pi }^{2}+\frac{1}{e^{2}}\Bigl( F^{\prime 2}(r)+\frac{\sin ^{2}F(r)}{r^{2}}\Bigr) \biggr). \label{fa2}\end{aligned}$$ respectively. The Fourier transforms of these charge densities, which are spherically symmetric scalar functions, give the electric form factors of proton and the neutron in the Breit frame as: $$G_{E}^{p}(Q^2)= 4\pi \int \d r r^{2}\rho _{p}(r)j_{0}(qr), \label{fa3}$$ $$G_{E}^{n}(Q^2)= 4\pi \int \d r r^{2}\rho _{n}(r)j_{0}(qr). \label{fa4}$$ Here $j_{n}(qr)$ is the spherical Bessel function of n-th order and Q is the transfer to the nucleon ($Q^2=-{\bf q}^2$). The isoscalar and isovector magnetic form factors for the nucleon may be expressed as $$\begin{aligned} G_{M}^{S}(Q^2)&=&\frac{-2m}{\pi a(F)q}\int \d r rF^{\prime }(r) \sin ^{2}F(r)j_{1}(qr), \label{fa5} \\[3pt] G_{M}^{V}(Q^2)&=&\frac{16\pi m}{3q}\int \d r r \biggl( f_{\pi }^{2}+\frac{1}{e^{2}}\Bigl( F^{\prime 2}(r)+\frac{\sin ^{2}F(r)}{r^{2}}\nonumber \\ &&\phantom{\frac{16\pi m}{3q}\int \d r r \biggl( } -\frac{2d_{2}-15}{4\cdot5a^{2}(F)}\sin ^{2}F(r)\Bigr) \biggr)\sin ^{2}F(r) j_{1}(qr). \label{fa6}\end{aligned}$$ Recombination into proton and neutron form factors yields $$\begin{aligned} G_{M}^{p}(Q^2)&=&\half\Bigl( G_{M}^{S}(Q^2)+G_{M}^{V}(Q^2)\Bigr), \label{fa7} \\ G_{M}^{n}(Q^2)&=&\half\Bigl( G_{M}^{S}(Q^2)-G_{M}^{V}(Q^2)\Bigr). \label{fa8} \end{aligned}$$ The magnetic form factors at zero-momentum transfer give the magnetic moments of nucleons as $$G_{M}^{p}(0) =\mu _{p},\qquad G_{M}^{n}(0) =\mu _{n}, \label{fa9}$$ in units of nuclear magnetons. The standard definition of the matrix element of the axial vector current of the nucleon is $$\begin{aligned} \left\langle N^{\prime }({\bf p_2})\left| A_{\mu }^{i}(0)\right| N({\bf p_1})\right\rangle &=&\overline{u}({\bf p_2})\tau ^{i}\biggl( \gamma _{\mu}\gamma _{5}G_{A}(Q^2)\label{fa10} \\ &&\phantom{\overline{u}({\bf p_2})\tau ^{i}\Bigl(\gamma _{\mu}\gamma _{5}} +q_{\mu }\gamma _{5}\frac{G_{P}(Q^2)} {2m}\biggr) u({\bf p_1}),\nonumber\end{aligned}$$ where $G_A(Q^2)$ and $G_P(Q^2)$ are the axial vector and induced pseudoscalar form factors respectively and ${\bf q}={\bf p_2}-{\bf p_1}$ In the non-relativistic limit the axial current operator takes the form $$\begin{aligned} \left\langle N^{\prime }({\bf p_{2}})\left| A_{b}^{a}(0)\right| N({\bf p_{1}})\right\rangle &=& \left\langle N^{\prime }\left| \tau ^{a}\sigma _{b^{\prime }}\right| N\right\rangle \biggl((-1)^{b} \delta _{b,-b^{\prime }} G_{A}(Q^2)\label{fc4} \\ && \phantom{\left\langle N^{\prime }\left| \tau ^{a}\sigma _{b^{\prime }}\right| N\right\rangle\biggl((-1)} -\frac{q^{2}}{4m^2} \hat{q}_{b}\hat{q}_{b^{\prime }}G_{P}(Q^2)\biggr) \nonumber\end{aligned}$$ Here it is convenient to employ the circular coordinates system for spin and isospin. The momentum transfer ${\bf q}=q\hat {\bf q}$ is then: $$\hat{q}_{a}=\frac{2\sqrt{\pi }}{\sqrt{3}}Y_{1,a}(\vartheta ,\varphi ). \label{fc3}$$ The induced pseudoscalar form factor now takes the form: $$\begin{aligned} G_{P}(Q^2) &=&-\frac{3\sqrt{2\cdot5}m^2}{\sqrt{\pi }q^{2}} \int {\rm d}\vartheta {\rm d}\varphi \sin \vartheta \left\langle p\left|A_{0}^{1}(0)\right| n\right\rangle Y_{2,0}(\vartheta ,\varphi ) \nonumber \\ &=&-\frac{16\pi m^2}{3q^2}\int r^{2}\d r \bigg( f_{\pi }^{2}\Bigl( 2F^{\prime }-\frac{\sin 2F}{r}\Bigr)-\frac{1}{e^{2}}\Bigl( F^{\prime 2}\frac{\sin 2F}{r}\nonumber \\ &&\phantom{-\frac{16\pi m^2}{3q^2}\int r^{2}\d r \bigg(} -4F^{\prime }\frac{\sin^{2}F}{r^{2}} +\frac{\sin ^{2}F\sin 2F}{r^{3}} \label{fc9} \\ && \phantom{-\frac{16\pi m^2}{3q^2}\int r^{2}\d r \bigg(} +\frac{\sin ^{2}F\sin 2F}{4a^{2}(F)r}+F^{\prime }\frac{\sin ^{2}F}{4a^{2}(F)}\Bigr) \biggr) j_{2}(qr). \nonumber\end{aligned}$$ The axial form factor is $$\begin{aligned} G_{A}(Q^2) &=& \frac{1}{\sqrt{2\pi }}\int \d \vartheta \d \varphi \sin \vartheta \left\langle p\left| A_{0}^{1}(0)\right| n\right\rangle \Bigl( Y_{0,0}(\vartheta ,\varphi )-\frac{\sqrt{5}}{ 2}Y_{2,0}(\vartheta ,\varphi )\Bigr) \nonumber \\ &=&-\frac{8\pi }{9}\int r^{2}\d r \biggl( f_{\pi }^{2}\Bigl( F^{\prime }+\frac{\sin 2F}{r}\Bigr) +\frac{1}{e^{2}}\Bigl( F^{\prime2}\frac{\sin 2F}{r}+2F^{\prime }\frac{\sin ^{2}F}{r^{2}} \nonumber \\ && \phantom{-\frac{8\pi }{9}\int r^{2}\d r \biggl(} +\frac{\sin ^{2}F\sin 2F}{r^{3}} -\frac{5\sin ^{2}F\sin 2F}{4a^{2}(F)r}\label{fa11} \\ && \phantom{-\frac{8\pi }{9}\int r^{2}\d r \biggl(} -F^{\prime }\frac{\sin^{2}F}{8a^{2}(F)}\Bigr) \biggr) j_{0}(qr) +\frac{q^2}{12m^2}G_{P}(Q^2).\nonumber\end{aligned}$$ The expression (\[fa11\]) equals that for the axial form factor given in ref.[@Nyman], with exception for the quantum corrections $\sim 1/a^{2}(F)$ which appear in the canonical quantization procedure. The axial current operator contains terms of fourth order in the components of ${\bf r}$ [@Acus98], and consequently its Fourier transform involves terms of fourth order in $ {\bf q} $. To avoid a redefinition of the axial current (\[fa10\]), we reduce it to $Y_{4,a}(\vartheta ,\phi )$, and terms of second and zero order in the components of ${\bf q}$. The electromagnetic mean square radii of nucleons is determined by means of the expression: $$\left\langle r^2 \right\rangle = -\frac{6}{G(0)} \frac\d {\d q^2}G(-q^2) \label{fb11}$$ The effect of Lorentz boosts for these form factors may be taken into account by means of the rescalings [@Ji]: $$\begin{aligned} {}_{rel}G_{E}^{p,n}(Q^{2})&=& G_{E}^{p,n}\left(\frac{Q^{2}}{1+Q^{2}/4m^{2}}\right), \label{fa12} \\ {}_{rel}G_{M}^{p,n}(Q^{2})&=& \frac{1}{1+Q^{2}/4m^{2}}G_{M}^{p,n}\left(\frac{Q^{2}}{1+Q^{2}/4m^{2}}\right), \label{fa13} \\ {}_{rel}G_{A}(Q^{2})&=& \frac{1}{\sqrt{1+Q^{2}/4m^{2}}}G_{A}\left( \frac{Q^{2}}{1+Q^{2}/4m^{2}}\right), \label{fa14} \\ {}_{rel}G_{P}(Q^{2})&=& \frac{1}{\sqrt{(1+Q^{2}/4m^{2})^{3}}}G_{P}\left( \frac{Q^{2}}{1+Q^{2}/4m^{2}}\right). \label{fa15} \end{aligned}$$ These boost corrections are numerically significant for large values of momentum transfer. Numerical results {#sec4} ================= -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- $\mathbf{j}$ **Classical [@Nappi]** $\mathbf{1/2}$ $\scriptscriptstyle \mathbf{1\oplus \frac 12 \oplus \frac $\mathbf{1}$ $\mathbf{3/2}$ **Expt.** 12}$ ------------------------------------- ------------------------ ---------------- ----------------------------------------------------------- -------------- ---------------- ---------------- $m$ Input Input Input Input Input $939$ MeV $\left\langle r^2\right\rangle^p_E$ $\infty$ Input Input Input Input $0.735$ fm$^2$ $f_\pi$ 64.5 64.8 60.3 59.4 57.5 $93$ MeV $e$ 5.45 4.76 4.31 4.19 3.86 $\left\langle r^2\right\rangle^n_E$ $\infty$ -0.368 -0.269 -0.249 -0.210 -0.114 fm$^2$ $\left\langle r^2\right\rangle^p_M$ $\infty$ 0.618 0.594 0.587 0.575 0.719 fm$^2$ $\left\langle r^2\right\rangle^n_M$ $\infty$ 0.687 0.609 0.594 0.567 0.637 fm$^2$ $\mu_p$ 1.87 1.96 2.32 2.39 2.54 2.79 $\mu_n$ -1.31 -1.37 -1.73 -1.81 -1.99 -1.91 $g_A$ 0.61 0.73 0.84 0.87 0.95 1.26 $m_\pi$ 0 110 171 191 246 138 MeV -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- : The predicted static baryon observables for different representations with fixed empirical values for the proton radius $\left\langle r^2\right\rangle_E^p=0.735$ fm$^2$ and nucleon mass $939$ MeV. The experimental data on the nucleon mean square radii are from ref.[@Grash]. []{data-label="table"} The nucleon form factors have been calculated numerically in the representations of the SU(2) group with $j = 1/2, 1, 3/2$ and in the reducible representation $1 \oplus 1/2 \oplus 1/2 \oplus 0$. The two parameters of the Lagrangian density, $f_\pi$ and $e$, have been determined here so that the empirical mass of the proton (938 MeV) and its electric mean square radius (0.735 fm$^2$) are reproduced for each value of $j$ (Table \[table\]). The chiral angle $F(r)$ for each one of these representations has been determined by self consistent numerical variation of the energy expression (\[f3\]). This procedure yields four pairs of model parameters $f_\pi$ and $e$, all of which are close to the values in [@Acus98]. The value of the axial coupling constant $g_A$, which is far too small in the semiclassical version of the Skyrme model remains below 1 in all the representations considered. The reason for this systematic underestimate is the absence of a quark contribution to the helicity of the nucleon as explained by a sum rule argument in ref.[@Mari]. The “effective” pion mass $m_{\pi}$ describes the behavior at infinity of the chiral angle $F(R)$ and the asymptotic falloff $e^{-2m_{\pi}r}$ of nucleon mass density. ![ Proton electric form factor $G_E^p(Q^2)$ with relativistic corrections.[]{data-label="pe"}](Afig1.eps) The calculated electric form factor of the proton as obtained with the boost corrections (\[fa12\]) are plotted in Fig. \[pe\]. In this case the form factor that is calculated in the reducible representation comes closest to the dipole fit to the empirical data. The corresponding magnetic form factors of the proton, again including the boost correction (\[fa13\]), are plotted in Fig. \[pm\]. In this case all the calculated form factors have a realistic falloff with momentum transfer at low values of momentum transfer, although the absolute predictions for the magnetic moment of the proton fall short by some $\sim$ 10-20%. In the semiclassical case the magnetic form factor is not well defined [@Adk]. ![ Proton magnetic form factor $G_M^p(Q^2)$ with relativistic corrections.[]{data-label="pm"}](Afig2.eps) ![ Neutron electric form factor $G_E^n(Q^2)$ with relativistic corrections.[]{data-label="ne"}](Afig3.eps) In Fig. \[ne\] the calculated electric form factors of neutron are shown. The results again include the boost correction (\[fa12\]). The experimental data in this case have too wide uncertainty margins for model discrimination. The new experimental results obtained by polarized electron scattering [@Pass; @Ost] indicates this form factor to much larger than what earlier data have suggested, and thus closer to the present calculated values, even though these are still much larger than the empirical results at intermediate values of momentum transfer. ![ Neutron magnetic form factor $G_M^n(Q^2)$ with relativistic corrections.[]{data-label="nm"}](Afig4.eps) In Fig. \[nm\] we plot the magnetic form factors of neutron as obtained with the boost correction (\[fa13\]). In terms of agreement with the empirical form factor values only the results for the fundamental representation in which $j=1/2$ is found to be wanting. This form factor is also ill defined in the semiclassical case. ![ Nucleon axial form factor $G_A(Q^2)$ with relativistic corrections.[]{data-label="ga"}](Afig5.eps) In Fig. \[ga\] we plot the axial form factor of nucleon with the boost correction (\[fa14\]). The empirical values for the axial form factor have a dipole-like behavior. The Skyrme model form factors tend to underestimate the falloff rate with momentum considerably, although it is possible to find parameter values that bring the axial coupling constant close to the empirical value in the case of the quantum skyrmion. ![ Nucleon pseudoscalar form factor $G_P(Q^2)$ with relativistic corrections.[]{data-label="gpa"}](Afig6.eps) In Fig. \[gpa\] we plot the pseudoscalar form factor of nucleon with the boost correction. This correction represents only about a 1% correction at $Q^2$ = 0.2 (GeV/c)$^2$ For this form factor the calculated values fall below the uncertainty margin of the experimental values, with exception of case of the semiclassical result, which is too large at small values of momentum transfer and too small at large values. Mathematica (Wolfram Research inc.) has been extensively used both for symbolic and numerical calculations [@wolf]. Discussion {#sec5} ========== The nucleon form factors are well defined in the Skyrme model if the chiral angle asymptotically falls faster than by the semiclassical rate $1/r^2$. The desired exponential fall of has to be brought about by a finite pion mass term, which implies breaking of chiral symmetry. While the pion mass term may be introduced at the classical level through an explicit chiral symmetry breaking term in the Lagrangian density, we have previously shown that such breaking of chiral symmetry also arises, without additional mass parameters, in the canonical [*ab initio*]{} quantization of the Skyrme model [@Acus98]. As shown here, this ensures well defined nucleon form factors, which - at least in the case of the electromagnetic form factors - do have phenomenologically adequate momentum dependence. It has also been noted elsewhere and in another context, that quantum corrections may generate a finite pion mass [@Torn]. 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